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+[    {        "title": "1508.00629v2.A_Critical_Analysis_of_the_Feasibility_of_Pure_Strain_Actuated_Giant_Magnetostrictive_Nanoscale_Memories.pdf",        "content": " 1 A Critical Analysis  of the Feasibility of Pure Strain -Actuated Giant Magnetostrictive \nNano scale Memories  \nP.G. Gowtham1, G.E. Rowlands1, and R.A. Buhrman1 \n1Cornell University, Ithaca, New York, 14853, USA \n \nAbstract  \n  Concepts for memories  based on the  manipulation  of giant magnetostrictive \nnano magnets  by stress pulses  have garnered recent attention  due to their potential for ultra -low \nenergy operation in the high storage density limit . Here we discuss the feasibility of making such \nmemories in light of the fact that the Gilbert damping of such materials is typically quite  high. \nWe report the results of numerical simulations for  several classes of  toggle precessional and non -\ntoggle dissipative  magnetoelastic  switching modes. M aterial candidates  for each o f the several \nclasses are analyzed  and f orms for the anisotropy energy density and range s of material \nparameters appropriate for each material class are employed.   Our study indicates that the Gilbert \ndamping  as well as the anisotropy and demagnetization energies are all crucial for  determining \nthe feasibility of  magnetoelastic toggle -mode precessional switching schemes.  The role s of \nthermal stability  and thermal fluctuations for stress -pulse switching of gia nt magnetostrictive \nnanomagnets are also discussed in detail and are shown to be important in the viability, design, \nand footprint of magnetostrictive switching schemes.  \n \n \n \n  2 I. Introduction  \n \nIn recent years pure electric -field based control of magnetization has become a subject of \nvery active research. It has been demonstrated in a variety of systems ranging from multiferroic \nsingle phase materials, gated dilute ferromagnetic semiconductors 1–3, ultra -thin metallic \nferromagnet/oxide interfaces 4–10 and piezoelectric /magnetoelastic composites 11–15. Beyond the \ngoal of establishing an understanding of the physics involved in each of these systems, this work \nhas been strongly motivated by the fact that  electrical -field based manipulation of magnetization \ncould form the basis for a new generation of ultra -low power, non -volatile memories.  Electric -\nfield based magnetic devices are not necessarily limited by Ohmic losses during the write cycle \n(as can be the case in current based memories such as spin -torque magnetic random access \nmemory (ST -MRAM) ) but rather by the capacitive charging/decharging energies incurred per \nwrite cycle. As the capacitance of these devices scale with area the write energies have the \npotential to be as low as 1 aJ per write cycle or less.  \nOne general approach to the electrical control of magnetism utilizes a magnetostrictive \nmagnet/piezoelectric transducer hybrid as the active component of a nanoscale memory element. \nIn this appro ach a mechanical strain is generated by an electric field within the piezoelectric \nsubstrate or film and is then transferred to a thin, nanoscale magnetostrictive magnet that is \nformed on top of the piezoelectric. The physical interaction driving the write  cycle of these \ndevices is the magnetoelastic interaction that describes the coupling between strain in a magnetic \nbody and the magnetic anisotropy energy. The strain imposed upon the magnet creates an \ninternal effective magnetic field via the magnetoelast ic interaction that can exert a direct torque \non the magnetization. If successfully implemented this torque can switch the magnet from one \nstable configuration to another, but whether imposed stresses and strains can be used to switch a  3 magnetic element be tween two bi -stable states depend s on the strength of the magnetoelastic \ncoupling (or the magnetostriction). Typical values of the magnetostriction ( = 0.5 -60 ppm) in \nmost ferromagnets yield strain and stress scales that make the process of strain -induced \nswitching inefficient or impossible. However, considerable advances have been made in \nsynthesizing materials both in bulk and in thin film form that have magnetostrictions that are one \nto two orders of magnitude larger than standard transition metal ferrom agnets.  These giant \nmagnetostrictive materials allow the efficient conversion of strains into torque on the \nmagnetization.  However it is important to note that a large magnetostrictive (or magnetoelastic) \neffect tends to also translate into very high magnetic damping by virtue of the strong coupling \nbetween magnons and the phonon thermal bath, which has important implicati ons, both positive \nand negative, for piezoelectric based magnetic devices.   \nIn this paper we provide an analysis of the switching modes  of several different \nimplementations of piezoelectric/magnetostrictive devices.  We discuss how the high damping \nthat is generally associated with giant magnetoelasticity affects the feasibility of different \napproaches, and we also take other key material  properties into consideration, including the \nsaturation magnetization of the magnetostrictive element, and the form and magnitude of its \nmagnetic anisotropy. Th e scope of th is work excludes device concepts and physics \ncircumscribed by magneto -elastic mani pulation of domain walls in magnetic films, wires, and \nnanoparticle arrays 11,12,16. Instead we focus  here on analyzing various magnetoelastic reversal \nmodes, principally within the single domain approximation, but we do extend this work to \nmicromagnetic modeling in cases where it is not clear that the macrospin approximation prov ides \na fully successful description of the essential physics. We enumerate potential material \ns 4 candidates for each of the modes evaluated and discuss the various challenges inherent in \nconstructing reliable memory cells based on each of the reversal modes t hat we consider.  \nII. Toggle -Mode Precessional Switching  \n \nStress pulsing of a magnetoelastic element can be used to construct a toggle mode \nmemory. The toggling mechanism between two stable states relies on transient dynamics of the \nmagnetization that are initi ated by an abrupt change in the anisotropy energy that is of fixed and \nshort duration. This change in the anisotropy is created by the stress pulse and under the right \nconditions can generate precessional dynamics about a new effective field. This effectiv e field \ncan take the magnetization on a path such that when the pulse is turned off the magnetization \nwill relax to the other stable state. This type of switching mode is referred to as toggle switching \nbecause the same sign of the stress pulse will take t he magnetization from one state to the other \nirrespective of the initial state. We can divide the consideration of the toggle switching modes \ninto two cases; one that utilizes a high \nsM  in-plane magnetized element, and the other that \nemploys perpendicular magnetic anisotro py (PMA) materials with a lower  \nsM. We make this \ndistinction largely because of differences in the structure of the torques and stress fields required \nto induce a switch in these two class es of systems.  The switching of in -plane giant \nmagnetostrictive nanomagnets with sizeable out -of-plane demagnetization fields relies on the use \nof in-plane uniaxial stress -induced effective fields that overcome the in -plane anisotropy (~O( 102 \nOe)). The mo ment will experience a torque canting the moment out of plane and causing \nprecession about the large demagnetization field. Thus the precessional time scales for toggling \nbetween stable in -plane states will be largely determined by the d emagnetization fiel d (and thus \nsM\n). The dynamics of this mode bears striking resemblance to the dynamics in hard -axis field  5 pulse switching of nanomagnets 17. On the other hand, the dominant energy scale in PMA giant \nmagnetostrictive materials is the perpendicular anisotropy energy. This energy scale can vary \nsubstantially (anywhere from \nuK  ~ 105-107 ergs/cm3) depending on the materials utilized and the \ndetails of their growth. The anisotropy energy scale in these materials can be tuned into a region \nwhere stress -induced anisotropy energies can be comparable to it. A biaxial stress -induced \nanisotropy energy, i n this geometry, can induce switching by cancelling and/or overcoming the \nperpendicular anisotropy energy. As we shall see, this fact and the low\nsM  of these systems \nimply dynamical time scales that are substantially different from the case where in -plane \nmagnetized materials are employed.  \nA. In-Plane Magnetized Magnetostrictive Materials  \n \nWe first treat the macrospin switching dynamics of an in -plane magnetized \nmagnetostrictive nanomagnet with uniaxial anisotropy under a simple rectangular uniaxial stress \npulse. Giant magnetostriction in in -plane magnetized systems have been demonstrated for \nsputtered polycrystalline Tb 0.3Dy0.7Fe2 (Terfenol -D) 18, and more recently in quenched Co xFe1-x  \nthin film systems 19. We assume that the uniaxial anisotropy is defined completely by the shape \nanisotropy of the elliptical element and that any magneto -crystalline anisotropy in the film is \nconsiderably weaker. This is a reasonabl e assumption for the materials considered here in the \nlimit where the grain size is considerably smaller than the nanomagnet’s dimensions. The stress \nfield is applied by voltage pulsing an anisotropic piezoelectric film that is in contact with the \nnanomagn et. The proper choice of the film orientation of a piezoelectric material such as <110> \nlead magnesium niobate -lead titanate (PMN -PT) can ensure that an effective uniaxial in -plane \nstrain develops along a particular crystalline axis after poling the piezo in the z -direction. We  6 assume that the nanomagnet major axis lies along such a crystalline direction (the <110> -\ndirection of PMN -PT) so that the shape anisotropy is coincident with the strain axis  (see Figure 1 \nfor the relevant geometry) . For the analysis below we use material values appropriate to \nsputtered, nanocrystalline Tb0.3Dy0.7Fe2 18 (\nsM= 600 emu/cm3, \ns = 670 ppm is the saturation \nmagnetostriction). Nanocrystalline Tb0.3Dy0.7Fe2  films, with a mean crystalline grain diameter \ngraind\n < 10 nm, can  have an extremely high magnetostriction while being relatively magnetica lly \nsoft with coercive fields,  \ncH  ~ 50-100 Oe, results which can be achieved by thermal processing \nduring sputter growth at T  ~ 375 ºC 20. The nanomagnet dimensions were as sumed to be 80 nm \n(minor axis) × 135 nm (major axis) × 5 nm (thickness) yielding a shape anisotropy field \n4 ( )k y x sH N N M\n = 323 Oe and \n4 ( )demag z y sH N N M = 5.97 kOe. We use \ndemagnetization factors that are correct for an elliptical cylinder 21. \nThe value of the Gilbert damping parameter \n  for the magnetostrictive element is quite \nimportant in determining its dynamical behavior during in -plane stress -induced toggle switching. \nPrevious simulation results 22–24 used a value (\n0.1  for Terfenol -D) that, at least arguably, is \nconsid erably lower than is reasonable since that value was extracted from spin pumping in a Ni \n(2 nm) /Dy(5 nm) bilayer 25. However, that bilayer material is not a good surrogate for a rare -\nearth transition -metal alloy (especially for \n0L  rare earth ions). In the latter case the loss \ncontribution from direct magnon to short w avelength phonon conversion is important, as has \nbeen directly confirmed by studies of \n0L  rare earth ion doping into transition metals 26,27. For \nexample in -plane magnetized nanocrystalline 10% Tb -doped Py shows \n~ 0.8  when magnetron \nsputtered at 5 mtorr Ar pressure, even though the magnetostriction is small within this region of \nTb doping 27. We contend that a substantial increase in the magnetoelastic interaction in alloys  7 with higher Tb content is likely to make \n even larger. Magnetization rotation in a highly \nmagnetostrictive magnet will efficiently generate longer wavelength acoustic  phonons as well \nand heat loss will be generated when these phonons thermalize. Unfortunately, measurements of \nthe magnetic damping parameter in polycrystalline Tb0.3Dy0.7Fe2 do not appear to be available in \nthe literature. However, some results on the amo rphous Tb x[FeCo] 1-x system, achieved by using \nrecent ultra -fast demagnetization techniques, have extracted  \n~ 0.5 for compositions (x  ~ 0.3) \nthat have high magnetostriction 28. We can also estimate the scale for the Gilbert damping by \nusing a formalism that takes into account direct magnon to long wavelength phonon conversion  \nvia the magnetoelastic interaction  and subsequent phonon relaxation to the thermal phonon \nbath29. The damping can be estimated by the following formula:  \n \n2\n2236 1 1\n22s\nsT s L s\neff ex eff exMc M c M\nAA\n\n\n\n \n   \n            \n(1) \n \nUsing \nsM  = 600 emu/cm3, the exchange stiffness \nexA = 0.7x10-6 erg/cm, a mass density ρ \n= 8.5 g/cm3, Young’s modulus of 65 GPa 30, Poisson ratio \n0.3 , and an acoustic damping time \n\n= 0.18 ps 29 the result is an estimate of  \n~1 . Given the uncertainties in the various  parameter s \ndetermining the Gilbert damping , we examine the magnetization dynamics for values of  \n\nranging from 0.3 to 1.0.  \n We simulate the switching dynamics of the magnetic moment of a Terfenol -D \nnanomagnet at T=300 K using the Landau -Lifshitz -Gilbert form of the equation  describing the \nprecession of a magnetic moment  \nm:  8  \n( ) ( )eff eff eff Langevinddttdt dt       mmm H m H m   \n(2) \n \nwhere\neff  is the gyromagnetic ratio. As Tb0.3Dy0.7Fe2 is a rare earth – transition metal (RE-TM) \nferrimagnet (or more accurately a speromagnet), the gyromagnetic ratio cannot simply be \nassumed to be the free electron value. Instead we use the value\neff  = 1.78 107 Hz/Oe as \nextracted from a spin wave resonance study in the TbFe 2 system 31 which appears appropriate \nsince Dy and Tb are similar in magnetic moment/atom (10\nB  and 9\nB  respectively) and g factor \n( ~4/3 and ~3/2 respectively).  \nThe first term in Equation (2) represents the torque on the magnetization from any \napplied fields, the effective stress field, and any anisotropy and demagnetization fields that might \nbe present. The third term in the LLG represents the damping torque that acts to relax the \nmagnetization towards the direction of the effective field and hence damp out precessional \ndynamics. The second term is the Gaussian -distributed Langevin field that takes into account the \neffect thermal fluctuations on the magnetization dynamics. From the fluctuation -dissipation \ntheorem, \n2RMS B\nLangevin\neff skTHM V t\n  where \nt  is the simulation time -step 32. Thermal fluc tuations \nare also accounted for in our modeling by assuming that the equilibrium azimuthal and polar \nstarting angles (\n0  and \n0 /2  respectively) have a random mean fluctuation given by \nequipartition as  \n00 2\n2RMS BkT\nEV\n\n\n\n and \n0 24 ( )RMS B\nz y skT\nN N M V  . A \nbiasH   of 100 Oe was \n 9 used for our simulations which creates two stable energy minima at  \n0arcsin ~ 18bias\nkH\nH\n\n and \n1162\n symmetric about \n/2 . This non -zero starting angle ensures that \n00RMS  . \nThis field bias is essential as the initial torque from a stress pulse depends on the initial starting \nangle. This angular dependence generates much larger thermally -induced fluctu ations in the \ninitial torque than a hard -axis field pulse. The hard axis bias field also reduces the energy barrier \nbetween the two stable states. For Hbias = 100 Oe the energy barrier between the two states is Eb \n= 1.2 eV yielding a room temperature \n/bBE k T = 49. This ensures the long term thermal \nstability required for a magnetic memory.  \nTo incorporate the effect of a stress pulse in Equation (2) we employ a free energy form \nfor the effective field,  \n( ) /efftE  Hm that expresses the effect of a stress pulse along the x -\ndirection of our in -plane nanomagnet with a uniaxial shape anisotropy in the x -direction. The \nstress enters the energy as an effective in -plane anisotropy term that adds to the shape anisotropy \nof the  magnet (first term in Equation (3) below). The sign convention here is such that \n0\nimplies a tensile stress on the x -axis while \n0 implies a compressive strain. We also include \nthe possibility of a bias field applied along the hard axis in the final term in Equation (3).     \n \n22\n223( , , ) [2 ( ) ( )]2\n2 ( )x y z y x s s x\nz y s z bias s yE m m m N N M t m\nN N M m H M m \n  \n     \n(3) \n \n The geometry that we have assumed allows only for fast compressive -stress pulse based \ntoggle mode switching. The application of a DC compressive stress along the x -axis only reduces \nthe magnitude of the anisotropy and changes the position of the equilibriu m magnetic angles \n0 10 and \n10180\n  while keeping the potential wells associated with these states symmetric as \nwell. Adiabatically increasing the value of the compressive stress moves the angles toward \n/2\n until \n3()2sutK  but obviously can never induce a magnetic switch.  \n Thus the magnetoelastic memory in this geometry must make use of the transient \nbehavior of the magnetization under a stress pulse as opposed to re lying on quasistatic changes \nto the energy landscape. A compressive stress pulse where \n3()2sutK  creates a sudden \nchange in the effective field. The resultant effective field\n32ˆsu\neff y bias\nsKmHM  \nHy  \npoints in the y -direction and causes a  torque that brings the magnetization out of plane. At this \npoint the magnetization rotates rapidly about the very large perpendicular demagnetization field\nˆ 4demag s z Mm Hz\n and if the pulse is turned off at the right time will relax down to the \nopposite state at \n1  = 163. Such a switching trajectory for our simulated nanomagnet is shown in \nthe red curve in  Figure 2. This mode of switching is set by a minimum characteristic time scale\n1~ 7.54sw\nspsM\n, but the precession time will in general be longer than \nsw  for moderate \nstress pulse amplitudes,  \n( ) 2 / 3us tK , as the magnetization then cants out of plane enough to \nsee only a fraction of the maximum possible  \ndemagH . Larger stress pulse amplitudes result in \nshorter pulse duratio ns being required as the magnetization has a larger initial excursion out of \nplane. For pulse durations that are longer than required for a rotation (blue and green curves \nin Figure 2) \nm will exhibit damped elliptical precession about  \n/2 . If the stress is released \nduring the correct portion of any of these subsequent precessional cycles the magnetization \n180\n 11 should relax down to the \n1  state [blue curve in  Figure 2], but otherwise it will relax down to the \noriginal state [green curve in  Figure 2].  \nThe prospect of a practical device working reliably in the long pulse regime appears to be \nrather poor. The high damping of giant magnetostrictive magnets and the large field scale of the \ndemagnetization field yield very stringent pulse timing requirements  and fast damping times for \nequilibration to \n/2 . The natural time scale for magnetization damping in the in -plane \nmagnetized thin film case is  \n1\n2d\nsM  , which ranges from 50 ps down to 15 ps for\n0.3 1\n with \nsM = 600 emu/cm3. This high damping also results in the influence of thermal \nnoise on the magnetization dynamics being quite strong since  \nLangevinH  . Thus large stress \nlevels with extremely short pulse durations are required in order to rotate the magnetization \naround the \n/2 minimum within the damping time, and to keep the precession amplitude \nlarge enough that the magnetization  will deterministically relax to the reversed state. Our \nsimulation results for polycrystalline Tb0.3Dy0.7Fe2 show  that a high stress pulse amplitude of\n85 MPa\nwith a pulse duration ~ 65  ps is required if  \n0.5  (Figure 3a). However, the \npulse duration window for which the magnetization will deterministically switch is extremely \nsmall in this case (<5 ps). This is due to the fact that the precession amplitude about the \n/2  \nminimum at this damping gets small enough that thermal fluctuations allow only a very small \nwindow for which switching is reliable. For the lowest damping that we consider reasonable to \nassume,  \n0.3 , reliable switching is possible between \npulse ~ 30-60 ps at \n85 MPa . At a \nlarger damping \n0.75  we find that the switching is non -deterministic for all pulse widths as \nthe magnetization damps too quickly; instead very high stresses , \n200 MPa are required to \n1 12 generate deterministic switching of the magnetization with a pulse duration w indow \npulse ~ 25-\n45 ps ( Figure 3b).  \nGiven the high value of the expected damping we have also simulated the magnetization \ndynamics in the Landau Lifshitz (LL) form:  \n \n2(1 ) ( ( ) ( ))LL eff Langevinddttdt dt       mmm H H m   \n(4) \n \nThe LL form and the LLG form are equivalent in low damping limit (\n1 ) but they \npredict different dynamics at higher damping values.  Which of these norm -preserving forms for \nthe dynamics has the right damping form is still a subject of debate 33–37. As one increases α in \nthe LL form the precessional speed is kept the same while the damping is assumed to affect only \nthe rate of decay of the precession amplitude. The damping in the LLG dynamics, on the other \nhand, is a viscosity term and retards the pre cessional speed. The effect of this retardation can be \nseen in the LLG dynamics as the precessional cycles move to longer times as a function of \nincreasing damping. Our simulations show that the LL form (for fixed \n ) predicts highe r \nprecessional speeds than the LLG and hence an even shorter pulse duration window for which \nswitching is deterministic than the LLG, ~12 ps for LL as opposed to ~ 30 ps for LLG ( Figure \n3c). \nThe damping clearly plays a crucial role in the stress amplitude scale and pulse duration \nwindows for which deterministic switching is possible, regardless of the form used to describe \nthe dynamics. Even though the magnetostriction of Tb 0.3Dy0.7Fe2 is high and the stress required \nto entirely overcome the anisotropy energy is only 9.6 MPa, the fast damping time scale and \nincreased thermal noise (set by the large damping and the out -of-plane demagnetization) means  13 that the stress -amplitude that is required to achieve deterministic toggle switching is 10 -20 times \nlarger. In addition, the pulse duration for in -plane toggling must be extremely short, with typical \npulse durations of 10 -50 ps with tight time windows of 20 -30 ps within which the acoustic pulse \nmust be turned off. Given ferroelectric switching rise times on the order of ~50 ps extracted from \nexperiment38 and considering the acoustical resonant response of the entire piezoelectric / \nmagnetostrictive nanostructure and acoustic ringing and inertial terms in the lattice dynamics, \ngeneration of such large stresses with the strict pulse time requirem ents needed for switching in \nthis mode is likely unfeasible.  In addition, the stress scales  required to successfully toggle switch \nthe giant magnetostrictive nanomagnet in this geometry are nearly as high or even higher than \nthat for transition metal ferromagnets such as Ni (\n~ 38 ppms with \n0.045 ). For example, \nwith a 70 nm × 130 nm elliptical Ni nanomagnet with a thickness of 6 nm and a hard axis bias \nfield of 120 Oe we should obtain  switching at stress values \n = +95 MPa and \npulse  = 0.75 ns. \nTherefore  the use of giant magnetostrictive nanomagnets with high damping in this toggle mode \nscheme confers no clear advantage over the use of a more conventional transition metal \nferromagnet, and in neither case does this approach appear particularly viable for t echnological \nimplementation.  \nB. Magneto -Elastic Materials with PMA: Toggle Mode Switching  \n \nCertain amorphous sputtered RE/TM alloy films with perpendicular magnetic anisotropy \nsuch as a -TbFe 2 39–42 and a - Tb0.3Dy0.7Fe2 43 have  properties that may make these materials \nfeasible for use in  stress -pulse toggle switching. In certain composition ranges they exhibit large \nmagnetostriction (\ns  > 270 ppm for a -TbFe 2, and both  \ns  and the effective out of plane  14 anisotropy can be tuned over fairly wide ranges by varying the process gas pressure during \nsputter deposition, the target atom -substrate incidence angle, and the substrate temperature.  \nWe consider the energy of such an out -of-plane magnetostrictive material under the \ninfluence of a magnetic field  \nbiasH  applied in the \nˆx  direction and a pulsed biaxial stress:  \n \n223( , , ) [ 2 ( )]2u\nx y z s s biaxial z s bias xE m m m K M t m M H m        \n(5) \n \nSuch a biaxial stress could be applied to the magnet if it is part of a patterned [001] -poled PZT \nthin film/ferromagnet bilayer. A schematic of this device geometry is depicted in Figure 4.When\n0biasH\n, it is straightforward to see the stress pulse will not result in reliable switching since, \nwhen the tensile biaxial stress is large enough, the out of plane anisotropy becomes an easy -plane \nanisotropy and the equator presents a zero -torque condition on t he magnetization, resulting in a \n50%, or random, probability of reversal when the pulse is removed. However, reliable switching \nis possible for \n0biasH  since that results in a finite canting of \nm towards the x -axis. This \ncanting is required for the same reasons a hard -axis bias field was needed for the toggle \nswitching of an in -plane magnetized element as discussed previously.  A pulsed biaxial stress \nfield can then in principle lead to deterministic precessional toggle switching between  the +z and \n–z energy minima . This mode of pulsed switching is analogous to voltage pulse switching in the \nultra-thin CoFeB|MgO using the voltage -controlled magnetic anisotropy effect.5,8 Previous \nsimulation results have also di scussed  this class of  macrospin  magnetoelast ic switc hing in the \ncontext  of a Ni|Barium -Titatate  multilayer44 and a zero -field, biaxial stress -pulse induced  toggle \nswitching  scheme  taking advantage of micromagnetic  inhomogeneities has recently appeared in \nthe literature45. Here we discuss biaxial stress -pulse  switching for a broad class of  giant  15 magnetostrictive  PMA  magnets where we argue  that the monodomain limit strictly applies \nthroughout the switching process and extend past previous macrospin modeling by \nsystematically think ing about  how pulse -timing requirements  and critical write stress amplitudes \nare determined  by the damping, the PMA strength, and \nsM  for values  reasonable for these \nmaterials.  \nFor our simulation study of stress -pulse toggle switching of a PMA magnet, we \nconsidered a Tb 33Fe67 nanomagnet with an \nsM  = 300 emu/cm3, \neffK  = 4.0×105 ergs/cm3 and \ns  \n= 270 ppm.  To estimate the appropriate value for the damping parameter we noted that ultrafast \ndemagnetization measurements on Tb 18Fe82 have yielded  \n0.27 . This 18 -82 composition lies \nin a region where the magnetostriction is moderate (\ns ~50 ppm) 43 so we assumed that the \ndamping will be on the same order or higher for a -TbFe 2 due to its high magnetostriction. \nTherefore we ran simulations for the range of  \n= 0.3 -1. For the gyromagnetic ratio  we used\neff  \n= 1.78×107 s-1G-1 which  is appropriate for a -TbFe 2 31. We assumed an effective exchange \nconstant \n611 10effA erg cm   46 implying an exchange length  \nexeff no stress\neffAlK\n = 15.8 nm (in \nthe absences of an applied str ess) and \n22exeff pulse\nsAlM = 13.3 nm (assuming that the stress pulse \namplitude is just enough to cancel the out of plane anisotropy). A monodomain crossover \ncriterion of \ncd ~  ~ 56 nm  (with the pulse off) and \ncd ~\n22ex\nsA\nM ~ 47 nm  (with the pulse \non) can be calculated by considering the minimum length -scale associated with supporting \nthermal λ/2 confined spin wave modes 47. The important point here is that the low \nsM  of these \nsystems ensures that the exchange length is still fairly long even during the switching process, \n4ex\nuA\nK 16 which suggests that the macrospin approximation should be valid for describing the switching \ndynamics of this system for reasonably sized nanomagnets.  \nWe simulated a circular element with a diameter of 60 nm and a thickness of 10 nm, \nunder an x -axis bias field, \nbiasH = 500 Oe which creates an initial canting angle of 11 degrees \nfrom the  vertical  (z-axis).  This starting angle is sufficient to enable deterministic toggle \nprecessional switching between the +z and –z minima via biaxial stress pulsing.  The assumed \ndevice geometry, anisotropy energy density and bias field corresponded to an energy barrier \nbE   \n= 4.6  eV for thermally activated reversal, and hence a room temperature thermal stability factor  \n\n = 185.  \nWe show selected results of the macrospin simulations of stress -pulse toggle switching of \nthis modeled TbFe 2 PMA nanomagnet. Typical switching trajectories are shown in Figure 5a. The \nswitching transition can be divided into two stages (see Figure 5b): the precessional stage that \noccurs when the stress field is applied, during which the dynamics of the magnetization are \ndominated by precession about the effective field that arises from the sum of the bias field and \nthe easy -plane anisotropy field \n3 ( ) 2eff\ns\nz\nstKmM , and the dissipative stage that begins when the \npulse is turned off and where the large \neffK and the large \n  result in a comparatively quick \nrelaxation to the other energy minimum. Thus most of the switching process is spent in the \nprecessional phase and the entire switching process is not much longer than the actual stress \npulse duration.  For pulse amplitudes a t or not too far above the critical stress for reversal,\n2 / 3eff\ns K\n the two relevant timescales for the dynamics are set approximately by the \nprecessional period\n1/ 100 pssw bias H of the nanomagnet and the damping time 17 \n~ 2 /d bias H  . Both of these timescales are much longer than the timescales set by precession \nand damping about the demagnetization field in the in -plane magnetized toggle switching case. \nThe result is that even with quite high damping one can have reliable s witching over much \nbroader pulse width windows, 200 -450 ps . (Figure 6a,b). The relatively large pulse duration \nwindows within which reliable switching is possible (as compared to the in -plane toggle mode) \nhold for both the LL and LLG damping. However, the diffe rence between the two forms is \nevident in the PMA case ( Figure 6c). At fixed \n , the LLG damping predicts a larger pulse \nduration window than the LL damping. Also the effective viscosity implicit within the LLG \nequation ensures that the switching time scales are slower than in the LL case as can also be seen \nin Figure 6c. \nAn additional and important point concerns the factors that determine the critical \nswitching amplitude. In the in -plane toggle mode switching of the previous section, it was found \nthat the in-plane anisotropy field was not the dominant factor in determining the stress scale \nrequired to transduce a deterministic toggle switch. Instead, we found that the stress scale was \nalmost exclusively dependent on the need to generate a high enough preces sion \namplitude/precession speed during the switching trajectory so as to not be damped out to the \ntemporary equilibrium at \n/2  (at least within the damping range considered). This means \nthat the critical stress scale to transduce a deterministic switch is essentially determined by the \ndamping. We find that the situation is fundamentally different for the PMA based toggle \nmemories.  The critical amplitude \nc  is nearly independent of the damping from a range of \n0.3 0.75\n up until \n~1 where the damping is sufficiently high (i.e. damping times equaling \nand/or exceeding the p recessional time scale) that at \n85  MPa the magnetization traverses \ntoo close to the minimum at \n/2 ,\n0 . The main reason for this difference between the  18 PMA toggle based memories and the in-plane toggle based memory lies in the role that the \napplication of stress plays in the dynamics. First, in the in -plane case, the initial elliptical \namplitude and the initial out of plane excursion of the magnetization is set by the stress pulse \nmagnitu de. Therefore the stress has to be high to generate a large enough amplitude such that the \ndamping does not take the trajectory too close to the minimum at which point Langevin \nfluctuations become an appreciable part of the total effective field. This is n ot true in the PMA \ncase where the initial precession amplitude about the bias field is large and the effective stress \nscale for initiating this precession about the bias field is the full cancellation of the perpendicular \nanisotropy.  \nSince the minimum stre ss-pulse amplitude required to initiate a magnetic reversal in out -\nof-plane toggle switching scales with \neffK in the range of damping values considered, lowering \nthe PMA of the nanomagnet is a straightforward way to reduce the stress and write energy \nrequirements for this type of memory cell. Such reductions can be achieved by strain engineering \nthrough the choice of substrate, base electrode and transducer layers, by the choice of deposition \nparameters, and/or by post -growth annealing  protocols. For example growing a TbFe 2 film with a \nstrong tensile biaxial strain can substantially lower \neffK . If the P MA of such a nanomagnet can \nbe reliably r educed to \neffK = 2105 ergs/cm3 our simulations indicate that this would result in \nreliable pulse toggle switching at \n ~ -50 MPa (corresponding to a strain amplitude on the TbFe 2 \nfilm of less than 0.1%) with \npulse  ≈ 400 ps, for 0.3 ≤ \n  ≤  0.75 and \nbiasH ~ 250  Oe . Electrical \nactuation of this level of stress/strain in the sub -ns regime, while challenging, may  be possible to \nachieve.48 If we again assume \nsM  =300 emu/cm3, a diameter of 60 nm and a thickness of  10 nm, \nthis low PMA nanomagnet would still have a high thermal stability with  \n92 . The challenge, \n 19 of course, is to consistently and uniformly control the residual strain  in the magnetostrictive \nlayer. It is important to note that no  such tailoring (short of systematically lowering the damping) \ncan exist in the in -plane toggle mode case.  \nIII. Two -State Non -Toggle Switching  \n \nSo far we have discussed toggle mode switching where the same polarity strain pulse is \napplied to reverse the magnetization between two bi -stable states. In this case the strain pulse \nacts to create a temporary field around which the magnetization precesse s and the pulse is timed \nso that the energy landscape and magnetization relax the magnetization to the new state with the \ntermination of the pulse.  Non -toggle mode magneto -elastic switching differs fundamentally \nfrom the precessional dynamics of toggle -mode switching, being an example of dissipative \nmagnetization dynamics where a strain pulse of one sign destabilizes the original state (A) and \ncreates a global energy minimum for the other state (B). The energy landscape and the damping \ntorque completely de termine the trajectory of the magnetization and the magnetization \neffectively “rolls” down to its new global energy minimum. Reversing the sign of the strain pulse \ndestabilizes state B and makes state A the global energy minimum – thus ensuring a switch ba ck \nto state A. There are some major advantages to this class of switching for magneto -elastic \nmemories over toggle mode memories. Precise acoustic pulse timing is no longer an issue. The \nswitching time scales, for reasonable stress values, can range  from q uasi-static to nanoseconds. \nIn addition, the large damping typical of magnetoelastic materials does not present a challenge \nfor achieving robust switching trajectories in deterministic switching as it does in toggle -mode \nmemories. Below we will discuss det erministic switching for magneto -elastic materials that have \ntwo different types of magnetic anisotropy.   20 C. The Case of Cubic Anisotropy  \n \nWe first consider magneto -elastic materials with cubic anisotropy under the influence of a \nuniaxial stress field pulse. T here are many epitaxial Fe -based magnetostrictive materials that \nexhibit a dominant cubic anisotropy when magnetron -sputter grown on oriented C u underlayers \non Si or on MgO,  GaAs , or PMN -PT substrates. For example, Fe 81Ga19 grown on MgO [100] or \non GaAs ex hibit a cubic anisotropy 49–51. Given the low cost of these Fe -based materials \ncompared to rare -earth alloys, it is worth investigating whether such films can be used to \nconstruct a two state memory. Fe 81Ga19 on MgO exhibits easy axes along <100>. In ad dition, \nepitaxial Fe 81Ga19 films have been found to have a reasonably high magnetostriction λ100=180 \nppm making them suitable for stress induced switching. If we assume that the  cubic \nmagnetoelastic thin-film nanomagnet has circular cross section, that the  stress field is applied by \na transducer along the [100] direction , and that a bias field  is applied  at \n4  degrees, the  \nmagnetic free energy is :  \n \n2 2 2 2 2\n11\n2( , ) (1 ) 2 ( )\n3( ) ( )2 2x y x y z z z s z\ns bias\nx y s xE m m K m m K m m N N M m\nMHm m t m\n    \n  \n   \n(6) \n \nEquation (6) shows that, in the absence of a bias field, the anisotropy energy is 4 -fold \nsymmetric in the film -plane. It is rather easy to see that it is im possible to make a two -state non -\ntoggle switching with a simple cubic anisotropy energy and uniaxial stress field along [100].  \nFigure 7a shows the free energy landscape described by Equation (6) without stress applied. To \ncreate a two -state deterministic magnetostrictive device ,  \nbiasH  needs to be strong enough to \neradicate the energy minima at \n and \n3 / 2  which strictly requires that  \n1 0.5 /bias sH K M .  21 Finite temperature considerations can lower this minimum bias field requirement considerably. \nThis is due to the fact that the bias field can make the lifetime to escape the energy minima in th e \nthird quadrant and fourth qua drant small and the energy bar rier to return them from the energy \nminima in the first quadrant extremely large.  We arbitrarily set this requirement for the bias \nfield to correspond to a lifetime of 75 μs. The typical energy barriers to hop from back to the \nmetastable minima in the thi rd and fourth quadrant for device volumes we will consider are on \nthe order of several eV.  \nThe requirement for thermal stability of the two minima in the first quadrant , given a \ndiameter\nd  and a thickness \nfilmt  for the nanomagnet,  sets an upper  bound on  \nbiasH  as we require \n/ 40bbE k T \n at room temp erature between the two states (see Figure 7c). It is desirable that \nthis upper bound is high enough that there is some degree of tolerance to the value of the bias \nfield at device dimensions that are employed. This sets requirement s on the minimum volume of \nthe cylindical  nanomagnet that are dependent on\n1K . \nFor a circular element with \nd  = 100 nm, \nfilmt  = 12.5 nm and  \n1K= 1.5 105 ergs/cm3, two -\nstate non -toggle switching with the required thermal stability can only occur for \nbiasH  between \n50 - 56 Oe. This is too small a range of acceptable bias fields. However , by increasing \nfilmt  to 15 \nnm the bias field range grows to \nbiasH = 50 - 90 Oe wh ich is an acceptable range. For\n1K = \n2.0×105 erg/cm3 with \nd= 100 nm and \nfilmt = 12.5 nm , there is an appreciable region of bias field \n(~65-120 Oe) for which \n/barrier BE k T  > 42. For\n1K  = 2.5 105 ergs/cm3, the bias range goes from \n90 – 190 Oe for the same volume.  The main po int here is that, given the scale  for the cubic \nanisotropy in Fe 81Ga19, careful attention must be paid to the  actual  values of the anisotropy \n\n 22 constants, device lateral dimensions, film thickness, and the exchange bias strength in order  to \nensure device stability  in the sub -100 nm diameter  regime .  \n We now discuss the dynamics for a simulated case where  \nd = 100 nm,  \nfilmt= 12.5 nm,  \n1K\n= 2.0×105 ergs/cm3, \nbiasH  = 85 Oe, and \nsM  = 1300 emu/cm3.  Two stable minima exist at \n\n=10o and \n = 80o. Figure 7b shows the effect of the stress pulse on the energy landscape.  When \na compressive stress \nc  is applied, the potential minimum at \n =10o is rendered unstable \nand the magnetization follows the free energy gradient to \n = 80o (green curve). Since the stress \nfield is applied along [100] the magnetization first switches to a minima very close to but greater \nthan \n = 80o and when the stress is released it gently relaxes down to the zero stress minimum at \n\n= 80o. In order to switch from \n = 80o  to \n = 10o we need to reverse the sign of the applied \nstress field to tensile (red curve). A memory constructed on these principles is thus non -toggle.  \nThe magnetization -switching trajectory is simple and follows the dissipative dynamics \ndictated by the free energy landscape (see Figure 8a). We have assumed a damping of \n0.1  \nfor the Fe 81Ga19 system,  based on previous measurements52 and as confirmed by our own. Higher \ndamping only ends up speeding up the sw itching and ri ng-down process. Figure 8b shows the \nsimulated stress amplitude and pulse switching probability phas e diagram at room temperature.  \nUltimately, we must take the macrospin estimates for device parameters as  only a roug h \nguide. The macrospin dynamics approximate the true micromagnetics less and less well as the \ndevice diameter gets larger. The mai n reason for this is the large\nsM  of Fe 81Ga19 and the \ntendency of the magnetization to curl at the sample edges. Accordingly we have  performed T  = 0 \nºK micromagnetic simulations in  OOMMF.53 An exchange bias field \nbiasH  = 85 Oe was applied  23 at \n  = 45º and we assume \n1K  = 2.0×105 ergs/cm3,  \nsM  = 1300 emu/cm3, and \nexA = 1.9 × 10-6 \nerg/cm. Micromagnetics show that the macrospin picture quantitatively captures the switching \ndynamics, the angular positions of the stables states (\n0~ 10 and\n1~ 80 ) and the critical \nstress amplitude at (\n ~ 30 MPa) when the device diameter \nd < 75 nm. The switching is \nessentially a rigid in -plane rotation of the magnetization from  \n0 to \n1 . However, we cho se to \nshow the switching for  an element with \nd  = 100 nm because it allowed for thermal stability of \nthe devices in a region of thicknes s (\nfilmt  = 12-15 nm) where \nbiasH ~ 50-100 Oe at room \ntemperature could be reasonably expected. The initial average magnetization angle is larger (\n0~ 19\nand \n1~ 71 ) than would b e predicted by macrospin for a \nd = 100 nm element. \nThis is due to the magnetization c urling at the devices edges at\nd = 100 nm (see Figure 8c). \nDespite the fact that magnetization profile differs from the macrospin picture we find that there \nis no appreciable difference between the stress scales required for switching , or the basi c \nswitching mechanism.  \nThe stress amplitude scale for writing the simulated Fe 81Ga19 element at ~ 30 MPa is not \nexcessively high and  there are essentially no demands on the acoustic pulse width requirements.  \nThese memories can thus be written at pulse amplitudes of ~ 30 MPa with acoustical pulse \nwidths of ~ 10 ns. These numbers do not represent a major challenge from the acoustical \ntransduction point of view.  The drawback s to this scheme are  the necessity of growing high \nquality single crystal thin film s of Fe 81Ga19 on a piezoelectric substrate that can generate large \nenough strain to switch the magnet (e.g. PMN -PT) and difficulties  associated with  tailoring the \nmagnetocrystalline anisotropy \n1K  and ensuring thermal stability at low lateral device \ndimensions.   24 D. The Case of Uniaxial Anisotropy   \n \nLastly we discuss deterministic (non -toggle) switching of an in -plane giant \nmagnetostrictive magnet with uniaxial anisotropy. In -plane magnetized polycrystalline TbDyFe \npatterned into ellipti cal nanomagnets could serve as a potential candidate material in  such a \nmemory scheme. To implement deterministic switching in this geometry a bias field  \nbiasH  is \napplied along the hard axis of the nanomagnet. This generates two stable minima at \n0 and \n0 180\n symmetric about the hard axis. The axis of the stress pulse then needs to be non -\ncollinear with respect to the e asy axis in order to break the symmetry of the potential wells and \ndrive the transition to the selected equilibrium position. Figure 9 below shows a schematic of the \nsituation. When a stress pulse is applied in the direction that makes an angle\n with respect to the \neasy axis of the nanom agnet,  \noo0 90 , the free energy within the macrospin approximation \nbecomes:  \n \n2 2 2 2\n2( , , ) [2 ( ) 2 ( )\n3( ) (cos( ) sin( ) )2x y z y x s x z y s z bias s y\ns y x\nsE m m m N N M m N N M m H M m\nt m mM\n      \n     \n(7) \n \nFrom Equation (7) it can be seen that a sufficiently strong compressive stress pulse can switch \nthe magnetization between \n0  and \no\n0 180 , but only if \n0 is between\n and . To see why \nthis condition is necessary, we look at the magnetization dynamics in the high stress limit when \n0 0\n.  During such a strong pulse the magnetization will s ee a hard axis appear at\n  \nand hence will rotate towards the new easy axis at \n90 , but when the stress pulse is \no90 25 turned off the magnetization will equilibrate back to \n0 . This situation is represented by the \ngreen trajectory shown in  Figure 11a.   \nBut when \no\n090 , a sufficiently strong compressive stress pulse defines a new easy \naxis close to \no90  and when the pulse is turned off the magnetization will relax to\n0 180\n (blue trajectory in Figure 11a). Similarly the possibility of switching from \no180  \nto \nwith a tensile  strain depends on whether \no o o90 180 90     . Thus\no45 is the \noptimal situation as then the energy landscape becomes mirror symmetric about the hard axis and \nthe amplitude of the required switching stress (voltage) are equal. This scheme is quite similar to \nthe case of deterministic switching in biaxial anisotropy systems (with the coordinate system \nrotated by ). We note that a set of papers54–56 have previously proposed this particular case as \na candidate for non -toggle magnetoelectric memory  and have experimentally demonstrated \noperation of such a memory in the large feature -size (i.e. extended film ) limit .55  \nWe argue here that in-plane giant magnetostrictive magnets operated in the non -toggle \nmode could be a good candidate for  construct ing memories with  low write stress amplitude, and \nnanosecond -scale write time operation. However , as we will discuss , the prospects of this type of \nswitching mode  being suitable  for implementation in ultrahigh density memory appear to be  \nrather poor. The m ain reason for this lies in the hard axis bias field requirements for maintaining \nlow write error rates and the effect that such a hard axis bias field will have on  the long term \nthermal stability  of the element . At T = 0 ºK the requirement on  \nbiasH  is only that it be strong \nenough that  \n0 > 45º. However, this is no longer sufficient  at finite temperature where thermal \nfluctuations impl y a thermal, Gaussian distribution of the initial orientation of the magnetization \no45 26 direction \n0 about  \n0. If a significant componen t of this angular distribution falls below 45 \ndegrees there will be a high write error rate. Thus we must ensure that \nbiasH  is high enough that \nthe probability of \n  < 45º is extremely low. We have selected the re quirement that \n  < 45º  is a \n8\n event  where \n  is the standard deviation of \n about  \n0 and is  given by the relation\n. However,  \nbiasH  must be low enough to be technologically feasible,  but also \nmust not exceed a value that compromises the energy barrier between the two potential minima – \nthus rendering the nanomagnet thermally unstable . These minimum and maximum  requirement s \non \nbiasH  puts significant constraints on the minimum size of the nanomagnet that can be used in \nthis device approach. It also sets some rather tight requirements on the hard axis bias field, as we \nshall see.  \nWe first disc uss the effects of these requirements  in the case of a relatively large \nmagnetostrictive  device.  We assume the use  of a polycrystalline Tb 0.3Dy0.7Fe2 element having \nsM\n = 600 emu/cm3 and an elliptical cross section of 400×900 nm2 and a thickness \nfilmt  = 12.5 \nnm. This results in a  shape anisotropy field \nkH  ≈ 260 Oe. We find that for an applied hard axis \nbias field \nbiasH  ~ 200 Oe, a field strength that can be reasonably engineered on -chip, the \nequilibrium angle of the  element is \n0 ≈ 51º and its root mean square (RMS)  angular fluctuation  \namplitude is \nRMS ≈ 0.75º. Thus element ’s anisotropy field  and the assumed hard axis biasing \ncondition s just satisfy  the assumed requirement that  \n08RMS   > 45º (see Figure 10b). The \nmagnetic energy barrier to thermal energy ratio for the element at \nbiasH  = 200 Oe is \n/bBE k T\n02\n2BkT\nEV\n\n\n\n 27 ≈ 350, which  easily  satisf ies the long-term thermal  stability  requirement  (see Figure 10a), and \nwhich also provides some latitude for the use of a slightly higher\nbiasH  if desired to further reduce \nthe write error rate .  \nIt is straightforward to see from these numbers that if the area of the magnetostrictive \nelement is substantially reduced below 400 ×900 nm2 there must be a corresponding increase in \nkH\n  and hence in\nbiasH if the write error rate for the device is to remain acceptable.  Of course an \nincrease in the thickness of the element can partially reduce the increase in fluctuation amplitude \ndue to the decrease in the magnetic a rea, but the feasible range of thickness variation cannot \nmatch the effect of, for example, reducing the cross -sectional area by a factor of 10 to 100, with \nthe latter, arguably, being the minimum required for high density memory applications.    While \nperhaps a strong shape anisotropy and an increased \nfilmt  can yield the required \nkH  ≥ 1 kOe, the \nfact that in this deterministic mode of magnetostrictive switching we must also have \nbiasH  ~ \nkH  \nresults in a bias field requirement that is not technologically feasible.  We could of course allow \nthe write error rate to be much larger than indicated by an 8\nfluctuation probability, but this \nwould only relax the requirement on \nbiasH  marginally, which always must be such that \n0  > \n45o.Thus the deterministic  magneto strictive  device  is not a viable  candidate for ultra -high density \nmemory.  Instead this approach is only feasible for device s with lateral area  ≥ 105 nm2 . \nWhile the requir ement of a large footprint is a limitation of the deterministic  \nmagneto strictive  memory element , this device does have the significant advantage that the stress \nscale required to switch the memory is quite low. We have simulated T = 300 ºK macrospin \nswitching dynamics for a 400×900 nm2 ellipse with thickness \nfilmt  = 12.5 nm with \nbiasH = 200 Oe \nsuch that \n0 ~ 51º.  The Gilbert damping parameter was set to  \n0.5 and magnetostriction  \ns =  28 670 ppm. The magnetization switches by simple rotation from \n0  = 51º to \n1129\n that is \ndriven by the stress pulse induced change in the energy landscape (see Figure 11a). Phase \ndiagram results are provided in Figure 11b where the switching from  \n0 = 51º to \n1 = 129 º \nshows a 100% switching probability for stresses as low as \n  = - 5 MPa for pulse widths as short \nas 1 ns.  \nSince the dimensions of the ellipse are large enough that t he macrospin picture is  not strictly \nvalid, we have also conducted T = 0 K micromagnetic simulations of the stress -pulse induced \nreversal in this geometry. We find that the trajectories are essentially well described by a quasi -\ncoherent rotation with  non-uniformities in the magnetization being more pronounced at the \nellipse edges (see Figure 11c). The minimum stress pulse amplitude for swi tching is even lower \nthan that predicted by macrospin at \n  = - 3 MPa. This stress scale for switching is substantially \nlower than any of the switching mode schemes discussed before. Despite the fact that this \nscheme is not scalable down into the 100 -200 nm size regime, it can be  appropriate for larger \nfootprint memori es that can be written at very low write stress pulse amplitudes.  \nIV. CONCLUSION  \n   \nThe physical properties of giant magnetostrictive magnets (particularly of the rare -earth \nbased TbFe 2 and Tb 0.3Dy0.7Fe2 alloys) place severe restrictions on the viability of such materials \nfor use in fast, ultra -high density , low energy consumption  data storage. We have enumerated the \nvarious potential problems that might arise from the characteristically high damping of giant \nmagnetostrictive nanoma gnets in toggle -mode switch ing. We have also discussed  the rol e that \nthermal fluctuation s have on the various switching modes and the challenges involved in  29 maintaining long -time device thermal stability that arise mainly from the necessity of employing \nhard axis bias fields .  \nIt is clear that the task of constructing a reliable memory using pure stress induced \nreversal of g iant magnetostrictive magnets will be , when pos sible, a question of trade -offs and \ncareful engineering . PMA based giant magnetostrictive nanomagnets can be made extremely \nsmall (\nd  < 50 nm) while still maintaining thermal stability.  The small diameter and low cross -\nsectional area of these PMA giant magnetostrictive devices could , in principle,  lead to very low \ncapacitive write energies.   The counterpoint is that  the stress fields required to switch the device \nare not  necessarily  small and the acoustical pulse timing  requirements  are demanding.  However, \nit might  be possible t o tune the magnetostriction  \ns ,\nK , and \nsM  (either by adjustment of the \ngrowth conditions of the magnetostrictive magnet or by engineering the RE-TM multilayers \nappropriately) in order to significantly reduce the pulse amplitudes required f or switching (down \ninto the 20-50 MPa  range) and reduce th e required in -plane bias field – without compromising \nthermal stability of the bit . Such tuning must be carried out carefully. As we have discussed , the \nGilbert dampi ng \n, \ns ,\nK , and \nsM  can all affect the pure stress -driven switching process  and \ndevice thermal stability  in ways that are certainly interlinked and not necessarily complementary.   \nTwo state  non-toggle memories  such as we described in Section  III D  could have  extremely low \nstress write amplitudes and non-restrictive pulse requirements . However, the trade -off arises \nfrom thermal stability considerations and such a switching scheme is not scala ble down into the \n100-200 nm size regime . Despite this limitation there  may well be a place for  durable  memories \nwith very low write stress pulse amplitudes  and low write energies that operate reliably in the \nnanosecond regime .  30 ACKNOWLEDGEMENTS  \nWe thank  R.B. van Dover,  W.E. Bailey, C. Vittoria,  J.T. Heron, T. Gosavi, and S. Bhave \nfor fruitful discussions.  We also thank D.C. Ralph and T. Moriyama for comments and \nsuggestions on the manuscript. This work was supported by the Office of Naval Research and the \nArmy Research Office.   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  31 REFERENCES  \n \n1 H. Ohno, D. Chiba, F. Matsukura, T. Omiya, E. Abe, T. Dietl, Y. Ohno, and K. Ohtani, Nature \n408, 944 (2000).  \n2 D. Chiba, M. Sawicki, Y. Nishitani, Y. Nakatani, F. Matsukura, and H. Ohno, Nature 455, 515 \n(2008).  \n3 D. Chiba, M. 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Appl. \nPhys. 113, 17C719 (2013).  \n56 S. Giordano, Y. Dusch, N. Tiercelin, P. Pernod, and V. Preobrazhensky, J. Phys. D. Appl. \nPhys. 46, 325002 (2013).  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  35  \n \n \nFigure 1. Magnetoelastic elliptical memory element schematic with associated coordinate system for in -\nplane stress -pulse induced toggle switching.  Here  \nM is the magnetization  vector  with \n and \n being \npolar and azimuthal angles . For the in -plane t oggle switching case, the initial normalized magnetization  \n0 0 0ˆˆ cos sinm x y\n and is in the film plane with \n0arcsin[ / ]bias kHH   and \nˆbias bias H Hy . \n \n \n \nFigure 2. Toggle switching trajectory for an in -plane magnetized polycrystalline Tb 0.3Dy 0.7Fe2 element \nwith \nLLG  = 0.3, \n  = -120 MPa, and \npulse  = 50 ps (red) and 125 ps (blue) and 160 ps (green).  \n 36  \nFigure 3. a) Effect of the Gilbert damping on pulse switching probability statistics for\n  = -85 MPa. b) \nEffect of increasing stress pulse amplitude for high damping  \nLLG  = 0.75. Very high stress pulses ( >200 \nMPa) are required to allow precession to be fast enough to cause a switch before dynamics are damped \nout. c) Comparison of switching statistics for the LL and LLG dynamics at \n  = -200 MPa, \n  = 0.75. \nThe LL dynamics exhibits faster precession than the LLG for a given torque implying shorter windows of \nreliability and requirements for faster pulses.  \n \n \nFigure 4.  Schematic of TbFe 2 magnetic element under biaxial stress generated by a PZT layer.  \nHere the initial normalized magnetization  \n0 0 0ˆˆ cos sinm z x   is predominantly out of the \nfilm plane with a cant  \n0arcsin[ / ]bias kHH   in the x -direction provided by  \nˆbias bias H Hx . \n  \n 37  \n \nFigure 5. a) Switching trajectories for a TbFe 2 nanomagnet under a pulsed biaxial stress \n  = -85 MPa, \npulse\n = 400 ps ( green ) and \n  = -120 MPa and\npulse  = 300 ps (blue ) b) Switching trajectory time \ntrace for {m x,my,mz} for \n  = -85 MPa . The pulse is initiated at t  = 500 ps. The blue region \ndenotes when precession about \nbiasH  dominates (i.e. while the pulse is on) and the red when the \ndissipative dynamics rapidly damp the system down to the other equilibrium point.  \n \n 38 Figure 6. a) Dependence of the simulated pulse switching probability on \n  for \n  = -85 MPa . b) \nDependence of pulse switching probability on stress amplitude. Stress -induced switching is possible even \nfor \n  = 1.0. c) Comparison of pulse switching probability  for LL and LLG dynamics for \n = -85 MPa \nand \n  = 0.75. Here the difference between the LL and LLG dynamics has a significant effect on the \nwidth of the pulse window where reliable switching is predicted by the simulations (\nLL  = 200 ps and \nLLG\n=320 ps.)  \n \n \nFigure 7. a) Energy (normalized to \n1K ) landscape as a function of angle for various values of exchange \nbias energy. b)  \n= 80º (\n= 10 º) is the only stab le equilibrium for compressive ( tensi le) stress. \nDissipative dynamics and the free energy landscape then dictate the non -toggle switching dynamics. c) \nShows the energy barrier dependence  on the [110] bias field for a \nd   = 100 nm, \nfilmt = 12.5 nm circular \nelement with (curve 1) \n1K = 2.5x105 ergs/cm3, (curve 2) \n1K = 2.0×105 ergs/cm3, and ( curve 4) \n1K\n=1.5×105 ergs/cm3. Curve 3 shows the energy barrier dependence for  \n1K=1.5x105 ergs/cm3 and \nd  = 100 \nnm & \nfilmt  = 15 nm . \n \n \n \n \n 39  \nFigure 8. a) Magnetoelastic switching trajectory for Fe 81Ga19 with \n  = -45 MPa and \npulse = 3 ns. The \nmain part of the switching occurs within 200 ps. The magnetization relaxes to the equilibrium defined \nwhen the pulse is on and then relaxes to the final equilibrium when the pulse is turned off. b) Switchin g \nprobability phase diagram for Fe 81Ga19 with biaxial anisotropy at T  = 300 ºK. c)  T = 0 ºK OOMMF \nsimulations showing the equilibrium m icromagnetic configuration for \n1K  = 2×105 ergs/cm3 and \nsM  = \n1300 emu/cm3. Subsequent shots show the rotational switching mode for a 45 MPa uniaxial compressive \nstress along [100].  Color scale is blue -white -red indicating the local projection \n1xm  (blue), \n0xm\n(white), \n1xm (red).  \n  \n 40  \n \n \nFigure 9. Schematic of magnetostrictive device geometry that utilizes uniaxial anisotropy to achieve \ndeterministic switching. Polycrystalline Tb 0.3Dy 0.7Fe2 on PMN -PT with 1 axis oriented at angle \n  with \nrespect to the easy axis.  In this geometry, \nM  lies in the x -y plane (film -plane) with the normalized  \nˆˆ cos sinm x y\n.  \n  \n 41  \n \nFigure 10. a) In-plane shape anisotropy field (\nkH ) and hard axis bias field (\nbiasH ) for a 400×900 nm2 \nellipse as a function of film thickness required to ensure \n0  = 51º . Thermal stability parameter\n  plotted \nversus film thickness with\nkH , \nbiasH  such that  \n0 = 51º .  b) Eight  times the RMS angle fluctuation \nabout three different  average \n0 > 45º versus film thickness for a 400×900 nm2 ellipse at T = 300 ºK. \n \n \n 42 Figure 11. a) Magnetization trajectories for\n = 45º, \n= -5 MPa ,\npulse = 3 ns, with ~ 200 Oe \nyielding \n0  = 51º  ( red) and\n = 45º,\n = -20 MPa with \nbiasH  = 120 Oe yielding  \n0 = 28º ( green). b) T = \n300 ºK stress pulse (compressive) switching prob ability phase diagram for a 400×90 0 nm2 ellipse with \nfilmt\n = 12.5 nm , \n= 45º, \n0 = 51º c) Micromagneti c switching trajectory of a 400×90 0 nm2 ellipse under \na DC compressive stress of -3 MPa transduced along 45 degrees.  Color scale is blue -white -red indicating \nthe local projection \n1xm  (blue), \n0xm (white), \n1xm (red).  \n \n \n \nbiasH"    },    {        "title": "1610.04598v2.Nambu_mechanics_for_stochastic_magnetization_dynamics.pdf",        "content": "arXiv:1610.04598v2  [cond-mat.mes-hall]  19 Jan 2017Nambu mechanics for stochastic magnetization\ndynamics\nPascal Thibaudeaua,∗, Thomas Nusslea,b, Stam Nicolisb\naCEA DAM/Le Ripault, BP 16, F-37260, Monts, FRANCE\nbCNRS-Laboratoire de Math´ ematiques et Physique Th´ eoriqu e (UMR 7350), F´ ed´ eration de\nRecherche ”Denis Poisson” (FR2964), D´ epartement de Physi que, Universit´ e de Tours, Parc\nde Grandmont, F-37200, Tours, FRANCE\nAbstract\nThe Landau-Lifshitz-Gilbert (LLG) equation describes the dynamic s of a damped\nmagnetization vector that can be understood as a generalization o f Larmor spin\nprecession. The LLG equation cannot be deduced from the Hamilton ian frame-\nwork, by introducing a coupling to a usual bath, but requires the int roduction of\nadditional constraints. It is shown that these constraints can be formulated ele-\ngantly and consistently in the framework of dissipative Nambu mecha nics. This\nhas many consequences for both the variational principle and for t opological as-\npects of hidden symmetries that control conserved quantities. W e particularly\nstudy how the damping terms of dissipative Nambu mechanics affect t he con-\nsistent interaction of magnetic systems with stochastic reservoir s and derive a\nmaster equation for the magnetization. The proposals are suppor ted by numer-\nical studies using symplectic integrators that preserve the topolo gical structure\nof Nambu equations. These results are compared to computations performed\nby direct sampling of the stochastic equations and by using closure a ssumptions\nfor the moment equations, deduced from the master equation.\nKeywords: Magnetization dynamics, Fokker-Planck equation, magnetic\nordering\n∗Corresponding author\nEmail addresses: pascal.thibaudeau@cea.fr (Pascal Thibaudeau),\nthomas.nussle@cea.fr (Thomas Nussle), stam.nicolis@lmpt.univ-tours.fr (Stam Nicolis)\nPreprint submitted to Elsevier September 18, 20181. Introduction\nIn micromagnetism, the transverse Landau-Lifshitz-Gilbert (LLG ) equation\n(1 +α2)∂si\n∂t=ǫijkωj(s)sk+α(ωi(s)sjsj−ωj(s)sjsi) (1)\ndescribes the dynamics of a magnetization vector s≡M/MswithMsthe sat-\nuration magnetization. This equation can be seen as a generalization of Larmor\nspin precession, for a collection of elementary classical magnets ev olving in an\neffective pulsation ω=−1\n¯hδH\nδs=γBand within a magnetic medium, charac-\nterized by a damping constant αand a gyromagnetic ratio γ[1].His here\nidentified as a scalar functional of the magnetization vector and ca n be consis-\ntently generalized to include spatial derivatives of the magnetizatio n vector [2]\nas well. Spin-transfer torques, that are, nowadays, of particula r practical rele-\nvance [3, 4] can be, also, taken into account in this formalism. In the following,\nwe shall work in units where ¯ h= 1, to simplify notation.\nIt is well known that this equation cannot be derived from a Hamiltonia n\nvariational principle, with the damping effects described by coupling t he magne-\ntization to a bath, by deforming the Poisson bracket of Hamiltonian m echanics,\neven though the Landau–Lifshitz equation itself is Hamiltonian. The r eason is\nthat the damping cannot be described by a “scalar” potential, but b y a “vector”\npotential.\nThis has been made manifest [5] first by an analysis of the quantum ve rsion\nof the Landau-Lifshitz equation for damped spin motion including arb itrary\nspin length, magnetic anisotropy and many interacting quantum spin s. In par-\nticular, this analysis has revealed that the damped spin equation of m otion is\nan example of metriplectic dynamical system [6], an approach which t ries to\nunite symplectic, nondissipative and metric, dissipative dynamics into one com-\nmon mathematical framework. This dissipative system has been see n afterwards\nnothing but a natural combination of semimetric dynamics for the dis sipative\npart and Poisson dynamics for the conservative ones [7]. As a conse quence, this\nprovided a canonical description for any constrained dissipative sy stems through\n2an extension of the concept of Dirac brackets developed originally f or conserva-\ntive constrained Hamiltonian dynamics. Then, this has culminated rec ently by\nobserving the underlying geometrical nature of these brackets a s certain n-ary\ngeneralizations of Lie algebras, commonly encountered in conserva tive Hamilto-\nnian dynamics [8]. However, despite the evident progresses obtaine d, no clear\ndirection emerges for the case of dissipative n-ary generalizations, and even\nno variational principle have been formulated, to date, that incorp orates such\nproperties.\nWhat we shall show in this paper is that it is, however, possible to de-\nscribe the Landau–Lifshitz–Gilbert equation by using the variationa l principle\nof Nambu mechanics and to describe the damping effects as the resu lt of in-\ntroducing dissipation by suitably deforming the Nambu–instead of th e Poisson–\nbracket. In this way we shall find, as a bonus, that it is possible to de duce\nthe relation between longitudinal and transverse damping of the ma gnetization,\nwhen writing the appropriate master equation for the probability de nsity. To\nachieve this in a Hamiltonian formalism requires additional assumptions , whose\nprovenance can, thus, be understood as the result of the prope rties of Nambu\nmechanics. We focus here on the essential points; a fuller account will be pro-\nvided in future work.\nNeglecting damping effects, if one sets H1≡ −ω·sandH2≡s·s/2, eq.(1)\ncan be recast in the form\n∂si\n∂t={si,H1,H2}, (2)\nwhere for any functions A,B,Cofs,\n{A,B,C} ≡ǫijk∂A\n∂si∂B\n∂sj∂C\n∂sk(3)\nis the Nambu-Poisson (NP) bracket, or Nambu bracket, or Nambu t riple bracket,\na skew-symmetric object, obeying both the Leibniz rule and the Fun damental\nIdentity [9, 10]. One can see immediately that both H1andH2are constants of\nmotion, because of the anti-symmetric property of the bracket. This provides the\ngeneralization of Hamiltonian mechanics to phase spaces of arbitrar y dimension;\n3in particular it does not need to be even. This is a way of taking into acc ount\nconstraints and provides a natural framework for describing the magnetization\ndynamics, since the magnetization vector has, in general, three co mponents.\nThe constraints–and the symmetries–can be made manifest, by no ting that\nit is possible to express vectors and vector fields in, at least, two wa ys, that can\nbe understood as special cases of Hodge decomposition.\nFor the three–dimensional case that is of interest here, this mean s that a\nvector field V(s) can be expressed in the “Helmholtz representation” [11] in the\nfollowing way\nVi≡ǫijk∂Ak\n∂sj+∂Φ\n∂si(4)\nwhereAis a vector potential and Φ a scalar potential.\nOn the other hand, this same vector field V(s) can be decomposed according\nto the “Monge representation” [12]\nVi≡∂C1\n∂si+C2∂C3\n∂si(5)\nwhich defines the “Clebsch-Monge potentials”, Ci.\nIf one identifies as the Clebsch–Monge potentials, C2≡H1,C3≡H2and\nC1≡D,\nVi=∂D\n∂si+H1∂H2\n∂si, (6)\nand the vector field V(s)≡˙s, then one immediately finds that eq. (2) takes the\nform\n∂s\n∂t={s,H1,H2}+∇sD (7)\nthat identifies the contribution of the dissipation in this context, as the expected\ngeneralization from usual Hamiltonian mechanics. In the absence of the Gilbert\nterm, dissipation is absent.\nMore generally, the evolution equation for any function, F(s) can be written\nas [13]\n∂F\n∂t={F,H1,H2}+∂D\n∂si∂F\n∂si(8)\nfor a dissipation function D(s).\n4The equivalence between the Helmholtz and the Monge representat ion im-\nplies the existence of freedom of redefinition for the potentials, CiandDand\nAiand Φ. This freedom expresses the symmetry under symplectic tra nsforma-\ntions, that can be interpreted as diffeomorphism transformations , that leave the\nvolume invariant. These have consequences for the equations of m otion.\nFor instance, the dissipation described by the Gilbert term in the Lan dau–\nLifshitz–Gilbert equation (1)\n∂D\n∂si≡α(˜ωi(s)sjsj−˜ωj(s)sjsi) (9)\ncannot be derived from a scalar potential, since the RHS of this expr ession is not\ncurl–free, so the function Don the LHS is not single valued; but it does conserve\nthe norm of the magnetization, i.e. H2. Because of the Gilbert expression,\nbothωandηare rescaled such as ˜ω≡ω/(1 +α2) andη→η/(1 +α2).\nSo there are two questions: (a) Whether it can lead to stochastic e ffects, that\ncan be described in terms of deterministic chaos and/or (b) Whethe r its effects\ncan be described by a bath of “vector potential” excitations. The fi rst case\nwas described, in outline in ref. [14], where the role of an external to rque was\nshown to be instrumental; the second will be discussed in detail in the following\nsections. While, in both cases, a stochastic description, in terms of a probability\ndensity on the space of states is the main tool, it is much easier to pre sent for\nthe case of a bath, than for the case of deterministic chaos, which is much more\nsubtle.\nTherefore, we shall now couple our magnetic moment to a bath of flu ctuating\ndegrees of freedom, that will be described by a stochastic proces s.\n2. Nambu dynamics in a macroscopic bath\nTo this end, one couples linearly the deterministic system such as (8) , to\na stochastic process, i.e. a noise vector, random in time, labelled ηi(t), whose\nlaw of probability is given. This leads to a system of stochastic differen tial\nequations, that can be written in the Langevin form\n∂si\n∂t={si,H1,H2}+∂D\n∂si+eij(s)ηj(t) (10)\n5whereeij(s) can be interpreted as the vielbein on the manifold, defined by the\ndynamical variables, s. It should be noted that it is the vector nature of the\ndynamical variables that implies that the vielbein, must, also, carry in dices.\nWe may note that the additional noise term can be used to “renorma lize”\nthe precession frequency and, thus, mix, non-trivially, with the Gilb ert term.\nThis means that, in the presence of either, the other cannot be ex cluded.\nWhen this vielbein is the identity matrix, eij(s) =δij, the stochastic cou-\npling to the noise is additive, whereas it is multiplicative otherwise. In th at\ncase, if the norm of the spin vector has to remain constant in time, t hen the\ngradient of H2must be orthogonal to the gradient of Dandeij(s)si= 0∀j.\nHowever, it is important to realize that, while the Gilbert dissipation te rm\nis not a gradient, the noise term, described by the vielbein is not so co nstrained.\nFor additive noise, indeed, it is a gradient, while for the case of multiplic ative\nnoise studied by Brown and successors there can be an interesting interference\nbetween the two terms, that is worth studying in more detail, within N ambu\nmechanics, to understand, better, what are the coordinate art ifacts and what\nare the intrinsic features thereof.\nBecause {s(t)}, defined by the eq.(10), becomes a stochastic process, we\ncan define an instantaneous conditional probability distribution Pη(s,t), that\ndepends, on the noise configuration and, also, on the magnetizatio ns0at the\ninitial time and which satisfies a continuity equation in configuration sp ace\n∂Pη(s,t)\n∂t+∂( ˙siPη(s,t)))\n∂si= 0. (11)\nAn equation for /an}b∇acketle{tPη/an}b∇acket∇i}htcan be formed, which becomes an average over all the\npossible realizations of the noise, namely\n∂/an}b∇acketle{tPη/an}b∇acket∇i}ht\n∂t+∂/an}b∇acketle{t˙siPη/an}b∇acket∇i}ht\n∂si= 0, (12)\nonce the distribution law of {η(t)}is provided. It is important to stress here\nthat this implies that the backreaction of the spin degrees of freed om on the\nbath can be neglected–which is by no means obvious. One way to chec k this is\nby showing that no “runaway solutions” appear. This, however, do es not ex-\n6haust all possibilities, that can be found by working with the Langevin equation\ndirectly. For non–trivial vielbeine, however, this is quite involved, so it is useful\nto have an approximate solution in hand.\nTo be specific, we consider a noise, described by the Ornstein-Uhlen beck\nprocess [15] of intensity ∆ and autocorrelation time τ,\n/an}b∇acketle{tηi(t)/an}b∇acket∇i}ht= 0\n/an}b∇acketle{tηi(t)ηj(t′)/an}b∇acket∇i}ht=∆\nτδije−|t−t′|\nτ\nwhere the higher point correlation functions are deduced from Wick ’s theorem\nand which can be shown to become a white noise process, when τ→0. We\nassume that the solution to eq.(12) converges, in the sense of ave rage over-the-\nnoise, to an equilibrium distribution, that is normalizable and, whose co rrelation\nfunctions, also, exist. While this is, of course, not at all obvious to p rove, evi-\ndence can be found by numerical studies, using stochastic integra tion methods\nthat preserve the symplectic structure of the Landau–Lifshitz e quation, even\nunder perturbations (cf. [16] for earlier work).\n2.1. Additive noise\nWalton [17] was one of the first to consider the introduction of an ad ditive\nnoise into an LLG equation and remarked that it may lead to a Fokker- Planck\nequation, without entering into details. To see this more thoroughly and to\nillustrate our strategy, we consider the case of additive noise, i.e. w heneij=\nδijin our framework. By including eq.(10) in (12) and in the limit of white\nnoise, expressions like /an}b∇acketle{tηiPη/an}b∇acket∇i}htmust be defined and can be evaluated by either an\nexpansion of the Shapiro-Loginov formulae of differentiation [18] an d taking the\nlimit ofτ→0, or, directly, by applying the Furutsu-Novikov-Donsker theore m\n[19, 20, 21]. This leads to\n/an}b∇acketle{tηiPη/an}b∇acket∇i}ht=−˜∆∂/an}b∇acketle{tPη/an}b∇acket∇i}ht\n∂si. (13)\n7where ˜∆≡∆/(1 +α2). Using the dampened current vector Ji≡ {si,H1,H2}+\n∂D\n∂si, the (averaged) probability density /an}b∇acketle{tPη/an}b∇acket∇i}htsatisfies the following equation\n∂/an}b∇acketle{tPη/an}b∇acket∇i}ht\n∂t+∂\n∂si(Ji/an}b∇acketle{tPη/an}b∇acket∇i}ht)−˜˜∆∂2/an}b∇acketle{tPη/an}b∇acket∇i}ht\n∂si∂si= 0 (14)\nwhere˜˜∆≡∆/(1 +α2)2and which is of the Fokker-Planck form [22]. This last\npartial differential equation can be solved directly by several nume rical methods,\nincluding a finite-element computer code or can lead to ordinary differ ential\nequations for the moments of s.\nFor example, for the average of the magnetization, one obtains th e evolution\nequation\nd/an}b∇acketle{tsi/an}b∇acket∇i}ht\ndt=−/integraldisplay\ndssi∂/an}b∇acketle{tPη(s,t)/an}b∇acket∇i}ht\n∂t=/an}b∇acketle{tJi/an}b∇acket∇i}ht. (15)\nFor the case of Landau-Lifshitz-Gilbert in a uniform precession field B, we\nobtain the following equations, for the first and second moments,\nd\ndt/an}b∇acketle{tsi/an}b∇acket∇i}ht=ǫijk˜ωj/an}b∇acketle{tsk/an}b∇acket∇i}ht+α[˜ωi/an}b∇acketle{tsjsj/an}b∇acket∇i}ht−˜ωj/an}b∇acketle{tsjsi/an}b∇acket∇i}ht] (16)\nd\ndt/an}b∇acketle{tsisj/an}b∇acket∇i}ht= ˜ωl(ǫilk/an}b∇acketle{tsksj/an}b∇acket∇i}ht+ǫjlk/an}b∇acketle{tsksi/an}b∇acket∇i}ht) +α[˜ωi/an}b∇acketle{tslslsj/an}b∇acket∇i}ht\n+ ˜ωj/an}b∇acketle{tslslsi/an}b∇acket∇i}ht−2˜ωl/an}b∇acketle{tslsisj/an}b∇acket∇i}ht] + 2˜˜∆δij (17)\nwhere ˜ω≡γB/(1 +α2). In order to close consistently these equations, one can\ntruncate the hierarchy of moments; either on the second /an}b∇acketle{t/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0 or third\ncumulants /an}b∇acketle{t/an}b∇acketle{tsisjsk/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0, i.e.\n/an}b∇acketle{tsisj/an}b∇acket∇i}ht=/an}b∇acketle{tsi/an}b∇acket∇i}ht/an}b∇acketle{tsj/an}b∇acket∇i}ht, (18)\n/an}b∇acketle{tsisjsk/an}b∇acket∇i}ht=/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acketle{tsk/an}b∇acket∇i}ht+/an}b∇acketle{tsisk/an}b∇acket∇i}ht/an}b∇acketle{tsj/an}b∇acket∇i}ht+/an}b∇acketle{tsjsk/an}b∇acket∇i}ht/an}b∇acketle{tsi/an}b∇acket∇i}ht\n−2/an}b∇acketle{tsi/an}b∇acket∇i}ht/an}b∇acketle{tsj/an}b∇acket∇i}ht/an}b∇acketle{tsk/an}b∇acket∇i}ht. (19)\nBecause the closure of the hierarchy is related to an expansion in po wers of\n∆, for practical purposes, the validity of eqs.(16,17) is limited to low v alues\nof the coupling to the bath (that describes the fluctuations). For example, if\none sets /an}b∇acketle{t/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0, eq.(16) produces an average spin motion independent of\nvalue that ∆ may take. This is in contradiction with the numerical expe riments\n8performed by the stochastic integration and noise average of eq.( 10) quoted in\nreference [23] and by experiments. This means that it is mandatory to keep\nat least eqs.(16) and (17) together in the numerical evaluation of t he thermal\nbehavior of the dynamics of the average thermal magnetization /an}b∇acketle{ts/an}b∇acket∇i}ht. This was\npreviously observed [24, 25] and circumvented by alternate secon d-order closure\nrelationships, but is not supported by direct numerical experiment s.\nThis can be illustrated by the following figure (1). For this given set of\nFigure 1: Magnetization dynamics of a paramagnetic spin in a constant magnetic field,\nconnected to an additive noise. The upper graphs (a) plot som e of the first–order moments of\nthe averaged magnetization vector over 102realizations of the noise, when the lower graphs\n(b) plot the associated model closed to the third-order cumu lant (eqs.(16)-(17), see text).\nParameters of the simulations : {∆ = 0.13 rad.GHz; α= 0.1;ω= (0,0,18) rad.GHz; timestep\n∆t= 10−4ns}. Initial conditions: s(0) = (1,0,0),/an}bracketle{tsi(0)sj(0)/an}bracketri}ht= 0 but /an}bracketle{ts1(0)s1(0)/an}bracketri}ht= 1.\nparameters, the agreement between the stochastic average an d the effective\nmodel is fairly decent. As expected, for a single noise realization, th e norm\nof the spin vector in an additive stochastic noise cannot be conserv ed during\nthe dynamics, but, by the average-over-the-noise accumulation process, this is\n9observed for very low values of ∆ and very short times. However, t his agreement\nwith the effective equations is lost, when the temperature increase s, because of\nthe perturbative nature of the equations (16-17). Agreement c an, however, be\nrestored by imposing this constraint in the effective equations, for a given order\nin perturbation of ∆, by appropriate modifications of the hierarchic al closing\nrelationships /an}b∇acketle{t/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acket∇i}ht=Bij(∆) or/an}b∇acketle{t/an}b∇acketle{tsisjsk/an}b∇acket∇i}ht/an}b∇acket∇i}ht=Cijk(∆).\nIt is of some interest to study the effects of the choice of initial con ditions. In\nparticular, how the relaxation to equilibrium is affected by choosing a c omponent\nof the initial magnetization along the precession axis in the effective m odel, e.g.\ns(0) = (1/√\n2,0,−1/√\n2) and by taking all the initial correlations,\n/an}b∇acketle{tsi(0)sj(0)/an}b∇acket∇i}ht=\n1\n20−1\n2\n0 0 0\n−1\n201\n2\n(20)\nThe results are shown in figure (2).\nBoth in figures (1) and (2), it is observed that the average norm of the spin\nvector increases over time. This can be understood with the above arguments.\nIn general, according to eq.(10) and because Jis a transverse vector,\n(1 +α2)sidsi\ndt=eij(s)siηj(t). (21)\nThis equation describes how the LHS depends on the noise realization ; so the\naverage over the noise can be found by computing the averages of the RHS. The\nsimplest case is that of the additive vielbein, eij(s) =δij. Assuming that the\naverage-over-the noise procedure and the time derivative commu te, we have\nd\ndt/angbracketleftbig\ns2/angbracketrightbig\n=2/an}b∇acketle{tsiηi/an}b∇acket∇i}ht\n1 +α2. (22)\nFor any Gaussian stochastic process, the Furutsu-Novikov-Don sker theorem\nstates that\n/an}b∇acketle{tsi(t)ηi(t)/an}b∇acket∇i}ht=/integraldisplay+∞\n−∞dt′/an}b∇acketle{tηi(t)ηj(t′)/an}b∇acket∇i}ht/angbracketleftbiggδsi(t)\nδηj(t′)/angbracketrightbigg\n. (23)\nIn the most general situation, the functional derivativesδsi(t)\nδηj(t′)can be calculated\n[26], and eq.(23) admits simplifications in the white noise limit. In this limit,\n10-2-1012\n0 1 2 3 4 5\nt (ns)-2-1012\nsxsy\nsz(a)\n(b)\nFigure 2: Magnetization dynamics of a paramagnetic spin in a constant magnetic field, con-\nnected to an additive noise. The upper graphs (a) plot some of the first–order moments of\nthe averaged magnetization vector over 103realizations of the noise, when the lower graphs\n(b) plot the associated model closed to the third-order cumu lant (eqs.(16)-(17), see text).\nParameters of the simulations : {∆ = 0.0655 rad.GHz; α= 0.1;ω= (0,0,18) rad.GHz;\ntimestep ∆ t= 10−4ns,s(0) =/an}bracketle{ts(0)/an}bracketri}ht= (1/√\n2,0,−1/√\n2),/an}bracketle{tsisj/an}bracketri}ht(0) = 0 except for (11)=1/2,\n(13)=(31)=-1/2, (33)=1/2 }.\n11the integration is straightforward and we have\n/angbracketleftbig\ns2(t)/angbracketrightbig\n=s2(0) + 6˜˜∆t, (24)\nwhich is a conventional diffusion regime. It is also worth noticing that w hen\ncomputing the trace of (17), the only term which remains is indeed\nd\ndt/an}b∇acketle{tsisi/an}b∇acket∇i}ht= 6˜˜∆ (25)\nwhich allows our effective model to reproduce exactly the diffusion re gime. Fig-\nure (3) compares the time evolution of the average of the square n orm spin\nvector. Numerical stochastic integration of eq.(10) is tested by in creasing the\n0 1 2 3 4 5\nt (ns)11,522,53\n<|s|2>mean over 103 runs\nmean over 104 runs\ndiffusion regime\nFigure 3: Mean square norm of the spin in the additive white no ise case for the following\nconditions: integration step of 10−4ns; ∆ = 0 .0655 rad.GHz; s(0) = (0 ,1,0);α= 0.1;\nω= (0,0,18) rad.GHz compared to the expected diffusion regime (see te xt).\nsize of the noise sampling and reveals a convergence to the predicte d linear\ndiffusion regime.\n122.2. Multiplicative noise\nBrown [27] was one of the first to propose a non–trivial vielbein, tha t takes\nthe form eij(s) =ǫijksk/(1 +α2) for the LLG equation. We notice, first of\nall, that it is present, even if α= 0, i.e. in the absence of the Gilbert term.\nAlso, that, since the determinant of this matrix [ e] is zero, this vielbein is not\ninvertible. Because of its natural transverse character, this vie lbein preserves the\nnorm of the spin for any realization of the noise, once a dissipation fu nctionD\nis chosen, that has this property. In the white-noise limit, the aver age over-the-\nnoise continuity equation (12) cannot be transformed strictly to a Fokker-Planck\nform. This time\n/an}b∇acketle{tηiPη/an}b∇acket∇i}ht=−˜∆∂\n∂sj(eji/an}b∇acketle{tPη/an}b∇acket∇i}ht), (26)\nwhich is a generalization of the additive situation shown in eq.(13). The conti-\nnuity equation thus becomes\n∂/an}b∇acketle{tPη/an}b∇acket∇i}ht\n∂t+∂\n∂si(Ji/an}b∇acketle{tPη/an}b∇acket∇i}ht)−˜˜∆∂\n∂si/parenleftbigg\neij∂\n∂sk(ekj/an}b∇acketle{tPη/an}b∇acket∇i}ht)/parenrightbigg\n= 0. (27)\nWhat deserves closer attention is, whether, in fact, this equation is invariant\nunder diffeomeorphisms of the manifold [28] defined by the vielbein, o r whether\nit breaks it to a subgroup thereof. This will be presented in future w ork. In the\ncontext of magnetic thermal fluctuations, this continuity equatio n was encoun-\ntered several times in the literature [22, 29], but obtaining it from fir st principles\nis more cumbersome than our latter derivation, a remark already qu oted [18].\nMoreover, our derivation presents the advantage of being easily g eneralizable\nto non-Markovian noise distributions [23, 30, 31], by simply keeping th e partial\nderivative equation on the noise with the continuity equation, and so lving them\ntogether.\nConsequently, the evolution equation for the average magnetizat ion is now\nsupplemented by a term provided by a non constant vielbein and one h as\nd/an}b∇acketle{tsi/an}b∇acket∇i}ht\ndt=/an}b∇acketle{tJi/an}b∇acket∇i}ht+˜˜∆/angbracketleftbigg∂eil\n∂skekl/angbracketrightbigg\n. (28)\nWith the vielbein proposed by Brown and assuming a constant extern al field,\n13one gets\nd/an}b∇acketle{tsi/an}b∇acket∇i}ht\ndt=ǫijk˜ωj/an}b∇acketle{tsk/an}b∇acket∇i}ht+α(˜ωi/an}b∇acketle{tsjsj/an}b∇acket∇i}ht−˜ωj/an}b∇acketle{tsjsi/an}b∇acket∇i}ht)\n−2∆\n(1 +α2)2/an}b∇acketle{tsi/an}b∇acket∇i}ht. (29)\nThis equation highlights both a transverse part, coming from the av erage over\nthe probability current Jand a longitudinal part, coming from the average\nover the extra vielbein term. By imposing, further, the second-or der cumulant\napproximation /an}b∇acketle{t/an}b∇acketle{tsisj/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0, i.e. “small” fluctuations to keep the distribution of\nsgaussian, a single equation can be obtained, in which a longitudinal rela xation\ntimeτL≡(1 +α2)2/2∆ may be identified.\nThis is illustrated by the content of figure (4). In that case, the ap proxima-\nFigure 4: Magnetization dynamics of a paramagnetic spin in a constant magnetic field, con-\nnected to a multiplicative noise. The upper graphs (a) plot s ome of the first–order moments\nof the averaged magnetization vector over 102realizations of the noise, when the lower graphs\n(b) plot the associated model closed to the third-order cumu lant (eq.(29), see text). Param-\neters of the simulations : {∆ = 0.65 rad.GHz; α= 0.1;ω= (0,0,18) rad.GHz; timestep\n∆t= 10−4ns}. Initial conditions: s(0) = (1,0,0),/an}bracketle{tsi(0)sj(0)/an}bracketri}ht= 0 but /an}bracketle{tsx(0)sx(0)/an}bracketri}ht= 1.\n14tion/an}b∇acketle{t/an}b∇acketle{tsisjsj/an}b∇acket∇i}ht/an}b∇acket∇i}ht= 0 has been retained in order to keep two sets of equations, three\nfor the average magnetization components and nine on the averag e second-order\nmoments, that have been solved simultaneously using an eight-orde r Runge-\nKutta algorithm with variable time-steps. This is the same numerical im ple-\nmentation that has been followed for the studies of the additive nois e, solving\neqs.(16) and (17) simultaneously. We have observed numerically tha t, as ex-\npected, the average second-order moments are symmetrical by an exchange of\ntheir component indices, both for the multiplicative and the additive n oise. In-\nterestingly, by keeping identical the number of random events tak en to evaluate\nthe average of the stochastic magnetization dynamics between th e additive and\nmultiplicative noise, we observe a greater variance in the multiplicative case.\nAs we have done in the additive noise case, we will also investigate briefl y the\nbehavior of this equation under different initial conditions, and in par ticular with\na non vanishing component along the z-axis. This is illustrated by the c ontent\nof figure (5). It is observed that for both figures (4) and (5), th e average spin\nconverges to the same final equilibrium state, which depends ultimat ely on the\nvalue of the noise amplitude, as shown by equation (27).\n3. Discussion\nMagnetic systems describe vector degrees of freedom, whose Ha miltonian\ndynamics implies constraints. These constraints can be naturally ta ken into\naccount within Nambu mechanics, that generalizes Hamiltonian mecha nics to\nphase spaces of odd number of dimensions. In this framework, diss ipation can\nbe described by gradients that are not single–valued and thus do no t define\nscalar baths, but vector baths, that, when coupled to external torques, can lead\nto chaotic dynamics. The vector baths can, also, describe non-tr ivial geometries\nand, in that case, as we have shown by direct numerical study, the stochastic\ndescription leads to a coupling between longitudinal and transverse relaxation.\nThis can be, intuitively, understood within Nambu mechanics, in the fo llowing\nway:\n15-1-0.500.51\n0 1 2 3 4 5\nt (ns)-1-0.500.51\nsxsy\nsz(a)\n(b)\nFigure 5: Magnetization dynamics of a paramagnetic spin in a constant magnetic field, con-\nnected to a multiplicative noise. The upper graphs (a) plot s ome of the first–order moments\nof the averaged magnetization vector over 104realizations of the noise, when the lower graphs\n(b) plot the associated model closed to the third-order cumu lant (eq.(29), see text). Param-\neters of the simulations : {∆ = 0.65 rad.GHz; α= 0.1;ω= (0,0,18) rad.GHz; timestep\n∆t= 10−4ns}. Initial conditions: s(0) =/parenleftbig\n1/√\n2,0,1/√\n2/parenrightbig\n,/an}bracketle{tsi(0)sj(0)/an}bracketri}ht= 0 except for\n/an}bracketle{ts1(0)s1(0)/an}bracketri}ht=/an}bracketle{ts1(0)s3(0)/an}bracketri}ht=/an}bracketle{ts3(0)s3(0)/an}bracketri}ht= 1/2.\n16The dynamics consists in rendering one of the Hamiltonians, H1≡ω·s,\nstochastic, since ωbecomes a stochastic process, as it is sensitive to the noise\nterms–whether these are described by Gilbert dissipation or couplin g to an\nexternal bath. Through the Nambu equations, this dependence is “transferred”\ntoH2≡ ||s||2/2. This is one way of realizing the insights the Nambu approach\nprovides.\nIn practice, we may summarize our numerical results as follows:\nWhen the amplitude of the noise is small, in the context of Langevin-\ndynamics formalism for linear systems and for the numerical modeling ofsmall\nthermal fluctuations in micromagnetic systems, as for a linearized s tochastic\nLLG equation, the rigorous method of Lyberatos, Berkov and Cha ntrell might\nbe thought to apply [32] and be expected to be equivalent to the app roach\npresented here. Because this method expresses the approach t o equilibrium of\nevery moment, separately, however, it is restricted to the limit of s mall fluctua-\ntions around an equilibrium state and, as expected, cannot captur e the transient\nregime of average magnetization dynamics, even for low temperatu re. This is a\nuseful check.\nWe have also investigated the behaviour of this system under differe nt sets of\ninitial conditions as it is well-known and has been thoroughly studied in [ 1] that\nin the multiplicative noise case (where the norm is constant) this syst em can\nshow strong sensitivity to initial conditions and it is possible, using ste reographic\ncoordinates to represent the dynamics of this system in 2D. In our additive noise\ncase however, as the norm of the spin is not conserved, it is not eas y to get long\nrun behavior of our system and in particular equilibrium solutions. Mor eover as\nwe no longer have only two independent components of spin, it is not p ossible\nto obtain a 2D representation of our system and makes it more comp licated to\nstudy maps displaying limit cycles, attractors and so on. Thus under standing\nthe dynamics under different initial conditions would require somethin g more\nand, as it is beyond the scope of this work, will be done elsewhere.\nTherefore, we have focused on studying the effects of the prese nce of an\ninitial longitudinal component and of additional, diagonal, correlation s. No\n17differences have been observed so far.\nAnother issue, that deserves further study, is how the probabilit y density\nof the initial conditions is affected by the stochastic evolution. In th e present\nstudy we have taken the initial probability density to be a δ−function; so it will\nbe of interest to study the evolution of other initial distributions in d etail, in\nparticular, whether the averaging procedures commute–or not. In general, we\nexpect that they won’t. This will be reported in future work.\nFinally, our study can be readily generalized since any vielbein can be ex -\npressed in terms of a diagonal, symmetrical and anti-symmetrical m atrices,\nwhose elements are functions of the dynamical variable s. Because ˙sis a pseu-\ndovector (and we do not consider that this additional property is a cquired by the\nnoise vector), this suggests that the anti-symmetric part of the vielbein should\nbe the “dominant” one. Interestingly, by numerical investigations , it appears\nthat there are no effects, that might depend on the choice of the n oise connection\nfor the stochastic vortex dynamics in two-dimensional easy-plane ferromagnets\n[33], even if it is known that for Hamiltonian dynamics, multiplicative and a d-\nditive noises usually modify the dynamics quite differently, a point that also\ndeserves further study.\nReferences\n[1] Giorgio Bertotti, Isaak D. Mayergoyz, and Claudio Serpico. Nonlinear\nMagnetization Dynamics in Nanosystems . 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Physical Review B ,\n59(17):11349–11357, May 1999.\n22"    },    {        "title": "1708.02008v2.Chiral_damping__chiral_gyromagnetism_and_current_induced_torques_in_textured_one_dimensional_Rashba_ferromagnets.pdf",        "content": "arXiv:1708.02008v2  [cond-mat.mes-hall]  31 Aug 2017Chiral damping, chiral gyromagnetism and current-induced torques in textured\none-dimensional Rashba ferromagnets\nFrank Freimuth,∗Stefan Bl¨ ugel, and Yuriy Mokrousov\nPeter Gr¨ unberg Institut and Institute for Advanced Simula tion,\nForschungszentrum J¨ ulich and JARA, 52425 J¨ ulich, German y\n(Dated: May 17, 2018)\nWe investigate Gilbert damping, spectroscopic gyromagnet ic ratio and current-induced torques\nin the one-dimensional Rashba model with an additional nonc ollinear magnetic exchange field. We\nfind that the Gilbert damping differs between left-handed and right-handed N´ eel-type magnetic\ndomain walls due to the combination of spatial inversion asy mmetry and spin-orbit interaction\n(SOI), consistent with recent experimental observations o f chiral damping. Additionally, we find\nthat also the spectroscopic gfactor differs between left-handed and right-handed N´ eel- type domain\nwalls, which we call chiral gyromagnetism. We also investig ate the gyromagnetic ratio in the Rashba\nmodel with collinear magnetization, where we find that scatt ering corrections to the gfactor vanish\nfor zero SOI, become important for finite spin-orbit couplin g, and tend to stabilize the gyromagnetic\nratio close to its nonrelativistic value.\nI. INTRODUCTION\nIn magnetic bilayer systems with structural inversion\nasymmetry the energies of left-handed and right-handed\nN´ eel-type domain walls differ due to the Dzyaloshinskii-\nMoriya interaction (DMI) [1–4]. DMI is a chiral interac-\ntion, i.e., it distinguishes between left-handed and right-\nhanded spin-spirals. Not only the energy is sensitive to\nthe chirality of spin-spirals. Recently, it has been re-\nported that the orbital magnetic moments differ as well\nbetween left-handed and right-handed cycloidal spin spi-\nrals in magnetic bilayers [5, 6]. Moreover, the experi-\nmental observation of asymmetry in the velocity of do-\nmain walls driven by magnetic fields suggests that also\nthe Gilbert damping is sensitive to chirality [7, 8].\nIn this work we show that additionally the spectro-\nscopic gyromagnetic ratio γis sensitive to the chirality\nof spin-spirals. The spectroscopic gyromagnetic ratio γ\ncan be defined by the equation\ndm\ndt=γT, (1)\nwhereTis the torque that acts on the magnetic moment\nmand dm/dtis the resulting rate of change. γenters\nthe Landau-Lifshitz-Gilbert equation (LLG):\ndˆM\ndt=γˆM×Heff+αGˆM×dˆM\ndt,(2)\nwhereˆMis a normalized vector that points in the direc-\ntionofthemagnetizationandthetensor αGdescribesthe\nGilbert damping. The chiralityofthe gyromagneticratio\nprovides another mechanism for asymmetries in domain-\nwall motion between left-handed and right-handed do-\nmain walls.\nNot only the damping and the gyromagnetic ratio\nexhibit chiral corrections in inversion asymmetric sys-\ntems but also the current-induced torques. Amongthese torques that act on domain-walls are the adia-\nbatic and nonadiabatic spin-transfer torques [9–12] and\nthe spin-orbit torque [13–16]. Based on phenomenologi-\ncal grounds additional types of torques have been sug-\ngested [17]. Since this large number of contributions\nare difficult to disentangle experimentally, current-driven\ndomain-wall motion in inversion asymmetric systems is\nnot yet fully understood.\nThe two-dimensionalRashbamodel with an additional\nexchange splitting has been used to study spintronics\neffects associated with the interfaces in magnetic bi-\nlayer systems [18–22]. Recently, interest in the role of\nDMI in one-dimensional magnetic chains has been trig-\ngered [23, 24]. For example, the magnetic moments in\nbi-atomic Fe chains on the Ir surface order in a 120◦\nspin-spiral state due to DMI [25]. Apart from DMI, also\nother chiral effects, such as chiral damping and chiral\ngyromagnetism, are expected to be important in one-\ndimensional magnetic chains on heavy metal substrates.\nThe one-dimensional Rashba model [26, 27] with an ad-\nditional exchange splitting can be used to simulate spin-\norbit driven effects in one-dimensional magnetic wires on\nsubstrates [28–30]. While the generalized Bloch theo-\nrem[31]usuallycannotbeusedtotreatspin-spiralswhen\nSOI is included in the calculation, the one-dimensional\nRashba model has the advantage that it can be solved\nwith the help of the generalized Bloch theorem, or with a\ngauge-field approach [32], when the spin-spiral is of N´ eel-\ntype. WhenthegeneralizedBlochtheoremcannotbeem-\nployed one needs to resort to a supercell approach [33],\nuse open boundary conditions [34, 35], or apply pertur-\nbation theory [6, 9, 36–39] in order to study spintronics\neffects in noncollinear magnets with SOI. In the case of\nthe one-dimensional Rashba model the DMI and the ex-\nchangeparameterswerecalculatedbothdirectlybasedon\nagauge-fieldapproachandfromperturbationtheory[38].\nThe results from the two approaches were found to be in\nperfect agreement. Thus, the one-dimensional Rashba2\nmodel provides also an excellent opportunity to verify\nexpressions obtained from perturbation theory by com-\nparisonto the resultsfromthe generalizedBlochtheorem\nor from the gauge-field approach.\nIn this work we study chiral gyromagnetism and chi-\nral damping in the one-dimensional Rashba model with\nan additional noncollinear magnetic exchange field. The\none-dimensional Rashba model is very well suited to\nstudy these SOI-driven chiral spintronics effects, because\nit can be solved in a very transparent way without the\nneed for a supercell approach, open boundary conditions\nor perturbation theory. We describe scattering effects by\nthe Gaussian scalar disorder model. To investigate the\nrole of disorder for the gyromagnetic ratio in general, we\nstudyγalso in the two-dimensional Rashba model with\ncollinear magnetization. Additionally, we compute the\ncurrent-induced torques in the one-dimensional Rashba\nmodel.\nThis paper is structured as follows: In section IIA we\nintroduce the one-dimensional Rashba model. In sec-\ntion IIB we discuss the formalism for the calculation\nof the Gilbert damping and of the gyromagnetic ratio.\nIn section IIC we present the formalism used to calcu-\nlate the current-induced torques. In sections IIIA, IIIB,\nand IIIC we discuss the gyromagnetic ratio, the Gilbert\ndamping, and the current-induced torques in the one-\ndimensionalRashbamodel, respectively. Thispaperends\nwith a summary in section IV.\nII. FORMALISM\nA. One-dimensional Rashba model\nThe two-dimensional Rashba model is given by the\nHamiltonian [19]\nH=−/planckover2pi12\n2me∂2\n∂x2−/planckover2pi12\n2me∂2\n∂y2+\n+iαRσy∂\n∂x−iαRσx∂\n∂y+∆V\n2σ·ˆM(r),(3)\nwhere the first line describes the kinetic energy, the first\ntwotermsin thesecondline describethe RashbaSOI and\nthe last term in the second line describes the exchange\nsplitting. ˆM(r) is the magnetization direction, which\nmay depend on the position r= (x,y), andσis the\nvector of Pauli spin matrices. By removing the terms\nwith the y-derivatives from Eq. (3), i.e., −/planckover2pi12\n2me∂2\n∂y2and\n−iαRσx∂\n∂y, one obtains a one-dimensional variant of the\nRashba model with the Hamiltonian [38]\nH=−/planckover2pi12\n2me∂2\n∂x2+iαRσy∂\n∂x+∆V\n2σ·ˆM(x).(4)\nEq. (4) is invariant under the simultaneous rotation\nofσand of the magnetization ˆMaround the yaxis.Therefore, if ˆM(x) describes a flat cycloidal spin-spiral\npropagating into the xdirection, as given by\nˆM(x) =\nsin(qx)\n0\ncos(qx)\n, (5)\nwe can use the unitary transformation\nU(x) =/parenleftBigg\ncos(qx\n2)−sin(qx\n2)\nsin(qx\n2) cos(qx\n2)/parenrightBigg\n(6)\nin order to transform Eq. (4) into a position-independent\neffective Hamiltonian [38]:\nH=1\n2m/parenleftbig\npx+eAeff\nx/parenrightbig2−m(αR)2\n2/planckover2pi12+∆V\n2σz,(7)\nwherepx=−i/planckover2pi1∂/∂xis thexcomponent of the momen-\ntum operator and\nAeff\nx=−m\ne/planckover2pi1/parenleftbigg\nαR+/planckover2pi12\n2mq/parenrightbigg\nσy (8)\nis thex-component of the effective magnetic vector po-\ntential. Eq. (8) shows that the noncollinearity described\nbyqacts like an effective SOI in the special case of the\none-dimensional Rashba model. This suggests to intro-\nduce the concept of effective SOI strength\nαR\neff=αR+/planckover2pi12\n2mq. (9)\nBased on this concept of the effective SOI strength\none can obtain the q-dependence of the one-dimensional\nRashba model from its αR-dependence at q= 0. That a\nnoncollinear magnetic texture provides a nonrelativistic\neffective SOI has been found also in the context of the\nintrinsic contribution to the nonadiabatic torque in the\nabsence of relativistic SOI, which can be interpreted as\na spin-orbit torque arising from this effective SOI [40].\nWhile the Hamiltonian in Eq. (4) depends on position\nxthrough the position-dependence of the magnetization\nˆM(x) in Eq. (5), the effective Hamiltonian in Eq. (7) is\nnot dependent on xand therefore easy to diagonalize.\nB. Gilbert damping and gyromagnetic ratio\nIn collinear magnets damping and gyromagnetic ratio\ncan be extracted from the tensor [16]\nΛij=−1\nVlim\nω→0ImGR\nTi,Tj(/planckover2pi1ω)\n/planckover2pi1ω, (10)\nwhereVis the volume of the unit cell and\nGR\nTi,Tj(/planckover2pi1ω) =−i∞/integraldisplay\n0dteiωt/angbracketleft[Ti(t),Tj(0)]−/angbracketright(11)3\nis the retarded torque-torque correlation function. Tiis\nthei-th component of the torque operator [16]. The dc-\nlimitω→0 in Eq. (10) is only justified when the fre-\nquency of the magnetization dynamics, e.g., the ferro-\nmagnetic resonance frequency, is smaller than the relax-\nationrateoftheelectronicstates. In thin magneticlayers\nand monoatomicchains on substratesthis is typically the\ncase due to the strong interfacial disorder. However, in\nvery pure crystalline samples at low temperatures the\nrelaxation rate may be smaller than the ferromagnetic\nresonance frequency and one needs to assume ω >0 in\nEq. (10) [41, 42]. The tensor Λdepends on the mag-\nnetization direction ˆMand we decompose it into the\ntensorS, which is even under magnetization reversal\n(S(ˆM) =S(−ˆM)), and the tensor A, which is odd un-\nder magnetization reversal ( A(ˆM) =−A(−ˆM)), such\nthatΛ=S+A, where\nSij(ˆM) =1\n2/bracketleftBig\nΛij(ˆM)+Λij(−ˆM)/bracketrightBig\n(12)\nand\nAij(ˆM) =1\n2/bracketleftBig\nΛij(ˆM)−Λij(−ˆM)/bracketrightBig\n.(13)\nOne can show that Sis symmetric, i.e., Sij(ˆM) =\nSji(ˆM), while Ais antisymmetric, i.e., Aij(ˆM) =\n−Aji(ˆM).\nThe Gilbert damping may be extracted from the sym-\nmetric component Sas follows [16]:\nαG\nij=|γ|Sij\nMµ0, (14)\nwhereMis the magnetization. The gyromagnetic ratio\nγis obtained from Λ according to the equation [16]\n1\nγ=1\n2µ0M/summationdisplay\nijkǫijkΛijˆMk=1\n2µ0M/summationdisplay\nijkǫijkAijˆMk.\n(15)\nIt is convenient to discuss the gyromagnetic ratio in\nterms of the dimensionless g-factor, which is related to\nγthrough γ=gµ0µB//planckover2pi1. Consequently, the g-factor is\ngiven by\n1\ng=µB\n2/planckover2pi1M/summationdisplay\nijkǫijkΛijˆMk=µB\n2/planckover2pi1M/summationdisplay\nijkǫijkAijˆMk.(16)\nDue to the presence of the Levi-Civita tensor ǫijkin\nEq. (15) and in Eq. (16) the gyromagnetic ratio and the\ng-factoraredetermined solelyby the antisymmetriccom-\nponentAofΛ.\nVarious different conventions are used in the literature\nconcerning the sign of the g-factor [43]. Here, we define\nthe sign of the g-factor such that γ >0 forg >0 and\nγ <0 forg <0. According to Eq. (1) the rate of change\nofthemagneticmomentisthereforeparalleltothetorqueforpositive gandantiparalleltothetorquefornegative g.\nWhile we are interested in this work in the spectroscopic\ng-factor, and hence in the relation between the rate of\nchange of the magnetic moment and the torque, Ref. [43]\ndiscusses the relation between the magnetic moment m\nandtheangularmomentum Lthatgeneratesit, i.e., m=\nγstaticL. Since differentiation with respect to time and\nuse ofT= dL/dtleads to Eq. (1) our definition of the\nsigns ofgandγagrees essentially with the one suggested\nin Ref. [43], which proposes to use a positive gwhen the\nmagnetic moment is parallel to the angular momentum\ngeneratingitandanegative gwhenthemagneticmoment\nis antiparallel to the angular momentum generating it.\nCombining Eq. (14) and Eq. (15) we can express the\nGilbert damping in terms of AandSas follows:\nαG\nxx=Sxx\n|Axy|. (17)\nIntheindependentparticleapproximationEq.(10)can\nbe written as Λij= ΛI(a)\nij+ΛI(b)\nij+ΛII\nij, where\nΛI(a)\nij=1\nh/integraldisplayddk\n(2π)dTr/angbracketleftbig\nTiGR\nk(EF)TjGA\nk(EF)/angbracketrightbig\nΛI(b)\nij=−1\nh/integraldisplayddk\n(2π)dReTr/angbracketleftbig\nTiGR\nk(EF)TjGR\nk(EF)/angbracketrightbig\nΛII\nij=1\nh/integraldisplayddk\n(2π)d/integraldisplayEF\n−∞dEReTr/angbracketleftbigg\nTiGR\nk(E)TjdGR\nk(E)\ndE\n− TidGR\nk(E)\ndETjGR\nk(E)/angbracketrightbigg\n.(18)\nHere,dis the dimension ( d= 1 ord= 2 ord= 3),GR\nk(E)\nis the retarded Green’s function and GA\nk(E) = [GR\nk(E)]†.\nEFis the Fermi energy. ΛI(b)\nijis symmetric under the\ninterchange of the indices iandjwhile ΛII\nijis antisym-\nmetric. The term ΛI(a)\nijcontains both symmetric and\nantisymmetric components. Since the Gilbert damping\ntensor is symmetric, both ΛI(b)\nijand ΛI(a)\nijcontribute to\nit. Since the gyromagnetic tensor is antisymmetric, both\nΛII\nijand ΛI(a)\nijcontribute to it.\nIn order to account for disorder we use the Gaus-\nsian scalardisordermodel, wherethe scatteringpotential\nV(r) satisfies /angbracketleftV(r)/angbracketright= 0 and /angbracketleftV(r)V(r′)/angbracketright=Uδ(r−r′).\nThis model is frequently used to calculate transport\nproperties in disordered multiband model systems [44],\nbut it has also been combined with ab-initio electronic\nstructure calculations to study the anomalous Hall ef-\nfect [45, 46] and the anomalous Nernst effect [47] in tran-\nsition metals and their alloys.\nIn the clean limit, i.e., in the limit U→0, the an-\ntisymmetric contribution to Eq. (18) can be written as4\nAij=Aint\nij+Ascatt\nij, where the intrinsic part is given by\nAint\nij=/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nn,m[fkn−fkm]ImTi\nknmTj\nkmn\n(Ekn−Ekm)2\n= 2/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nn/summationdisplay\nll′fknIm/bracketleftbigg∂/angbracketleftukn|\n∂ˆMl∂|ukn/angbracketright\n∂ˆMl′/bracketrightbigg\n×\n×/summationdisplay\nmm′ǫilmǫjl′m′ˆMmˆMm′.\n(19)\nThe second line in Eq. (19) expresses Aint\nijin terms of\nthe Berry curvature in magnetization space [48]. The\nscattering contribution is given by\nAscatt\nij=/planckover2pi1/summationdisplay\nnm/integraldisplayddk\n(2π)dδ(EF−Ekn)Im/braceleftBigg\n−/bracketleftbigg\nMi\nknmγkmn\nγknnTj\nknn−Mj\nknmγkmn\nγknnTi\nknn/bracketrightbigg\n+/bracketleftBig\nMi\nkmn˜Tj\nknm−Mj\nkmn˜Ti\nknm/bracketrightBig\n−/bracketleftbigg\nMi\nknmγkmn\nγknn˜Tj\nknn−Mj\nknmγkmn\nγknn˜Ti\nknn/bracketrightbigg\n+/bracketleftBigg\n˜Ti\nknnγknm\nγknn˜Tj\nkmn\nEkn−Ekm−˜Tj\nknnγknm\nγknn˜Ti\nkmn\nEkn−Ekm/bracketrightBigg\n+1\n2/bracketleftbigg\n˜Ti\nknm1\nEkn−Ekm˜Tj\nkmn−˜Tj\nknm1\nEkn−Ekm˜Ti\nkmn/bracketrightbigg\n+/bracketleftBig\nTj\nknnγknm\nγknn1\nEkn−Ekm˜Ti\nkmn\n−Ti\nknnγknm\nγknn1\nEkn−Ekm˜Tj\nkmn/bracketrightBig/bracerightBigg\n.\n(20)\nHere,Ti\nknm=/angbracketleftukn|Ti|ukm/angbracketrightare the matrix elements of\nthe torque operator. ˜Ti\nknmdenotes the vertex corrections\nof the torque, which solve the equation\n˜Ti\nknm=/summationdisplay\np/integraldisplaydnk′\n(2π)n−1δ(EF−Ek′p)\n2γk′pp×\n×/angbracketleftukn|uk′p/angbracketright/bracketleftBig\n˜Ti\nk′pp+Ti\nk′pp/bracketrightBig\n/angbracketleftuk′p|ukm/angbracketright.(21)\nThe matrix γknmis given by\nγknm=−π/summationdisplay\np/integraldisplayddk′\n(2π)dδ(EF−Ek′p)/angbracketleftukn|uk′p/angbracketright/angbracketleftuk′p|ukm/angbracketright\n(22)\nand the Berry connection in magnetization space is de-\nfined as\niMj\nknm=iTj\nknm\nEkm−Ekn. (23)\nThe scattering contribution Eq. (20) formally resembles\nthe side-jump contribution to the AHE [44] as obtainedfrom the scalar disorder model: It can be obtained by\nreplacing the velocity operators in Ref. [44] by torque\noperators. We find thatin collinearmagnetswithoutSOI\nthis scattering contribution vanishes. The gyromagnetic\nratio is then given purely by the intrinsic contribution\nEq. (19). This is an interesting difference to the AHE:\nWithout SOI all contributions to the AHE are zero in\ncollinear magnets, while both the intrinsic and the side-\njump contributions are generally nonzero in the presence\nof SOI.\nIn the absence of SOI Eq. (19) can be expressed in\nterms of the magnetization [48]:\nAint\nij=−/planckover2pi1\n2µB/summationdisplay\nkǫijkMk. (24)\nInserting Eq. (24) into Eq. (16) yields g=−2, i.e., the\nexpected nonrelativistic value of the g-factor.\nTheg-factor in the presence of SOI is usually assumed\nto be given by [49]\ng=−2Mspin+Morb\nMspin=−2M\nMspin,(25)\nwhereMorbis the orbital magnetization, Mspinis the\nspin magnetization and M=Morb+Mspinis the total\nmagnetization. The g-factor obtained from Eq. (25) is\nusually in good agreementwith experimental results [50].\nWhen SOI is absent, the orbital magnetization is zero,\nMorb= 0, and consequently Eq. (25) yields g=−2 in\nthat case. Eq. (16) can be rewritten as\n1\ng=Mspin\nMµB\n2/planckover2pi1Mspin/summationdisplay\nijkǫijkAijˆMk=Mspin\nM1\ng1,(26)\nwith\n1\ng1=µB\n2/planckover2pi1Mspin/summationdisplay\nijkǫijkAijˆMk. (27)\nFrom the comparison of Eq. (26) with Eq. (25) it follows\nthat Eq. (25) holds exactly if g1=−2 is satisfied. How-\never, Eq. (27) usually yields g1=−2 only in collinear\nmagnets when SOI is absent, otherwise g1/negationslash=−2. In the\none-dimensionalRashbamodel the orbitalmagnetization\nis zero,Morb= 0, and consequently\n1\ng=µB\n2/planckover2pi1Mspin/summationdisplay\nijkǫijkAijˆMk. (28)\nThe symmetric contribution can be written as Sij=\nSint\nij+SRR−vert\nij+SRA−vert\nij, where\nSint\nij=1\nh/integraldisplayddk\n(2π)dTr/braceleftbig\nTiGR\nk(EF)Tj/bracketleftbig\nGA\nk(EF)−GR\nk(EF)/bracketrightbig/bracerightbig\n(29)5\nand\nSRR−vert\nij=−1\nh/integraldisplayddk\n(2π)dTr/braceleftBig\n˜TRR\niGR\nk(EF)TjGR\nk(EF)/bracerightBig\n(30)\nand\nSAR−vert\nij=1\nh/integraldisplayddk\n(2π)dTr/braceleftBig\n˜TAR\niGR\nk(EF)TjGA\nk(EF)/bracerightBig\n,\n(31)\nwhereGR\nk(EF) =/planckover2pi1[EF−Hk−ΣR\nk(EF)]−1is the retarded\nGreen’s function, GA\nk(EF) =/bracketleftbig\nGR\nk(EF)/bracketrightbig†is the advanced\nGreen’s function and\nΣR(EF) =U\n/planckover2pi1/integraldisplayddk\n(2π)dGR\nk(EF) (32)\nis the retarded self-energy. The vertex corrections are\ndetermined by the equations\n˜TAR=T+U\n/planckover2pi12/integraldisplayddk\n(2π)dGA\nk(EF)˜TAR\nkGR\nk(EF) (33)\nand\n˜TRR=T+U\n/planckover2pi12/integraldisplayddk\n(2π)dGR\nk(EF)˜TRR\nkGR\nk(EF).(34)\nIn contrast to the antisymmetric tensor A, which be-\ncomes independent of the scattering strength Ufor suf-\nficiently small U, i.e., in the clean limit, the symmetric\ntensorSdepends strongly on Uin metallic systems in\nthe clean limit. Sint\nijandSscatt\nijdepend therefore on U\nthrough the self-energy and through the vertex correc-\ntions.\nIn the case of the one-dimensional Rashba model, the\nequations Eq. (19) and Eq. (20) for the antisymmet-\nric tensor Aand the equations Eq. (29), Eq. (30) and\nEq. (31) for the symmetric tensor Scan be used both\nfor the collinear magnetic state as well as for the spin-\nspiral of Eq. (5). To obtain the g-factor for the collinear\nmagnetic state, we plug the eigenstates and eigenvalues\nof Eq. (4) (with ˆM=ˆez) into Eq. (19) and into Eq. (20).\nIn the case of the spin-spiral of Eq. (5) we use instead the\neigenstates and eigenvalues of Eq. (7). Similarly, to ob-\ntain the Gilbert damping in the collinear magnetic state,\nwe evaluate Eq. (29), Eq. (30) and Eq. (31) based on\nthe Hamiltonian in Eq. (4) and for the spin-spiral we use\ninstead the effective Hamiltonian in Eq. (7).\nC. Current-induced torques\nThe current-induced torque on the magnetization can\nbe expressed in terms of the torkance tensor tijas [15]\nTi=/summationdisplay\njtijEj, (35)whereEjis thej-th component of the applied elec-\ntric field and Tiis thei-th component of the torque\nper volume [51]. tijis the sum of three terms, tij=\ntI(a)\nij+tI(b)\nij+tII\nij, where [15]\ntI(a)\nij=e\nh/integraldisplayddk\n(2π)dTr/angbracketleftbig\nTiGR\nk(EF)vjGA\nk(EF)/angbracketrightbig\ntI(b)\nij=−e\nh/integraldisplayddk\n(2π)dReTr/angbracketleftbig\nTiGR\nk(EF)vjGR\nk(EF)/angbracketrightbig\ntII\nij=e\nh/integraldisplayddk\n(2π)d/integraldisplayEF\n−∞dEReTr/angbracketleftbigg\nTiGR\nk(E)vjdGR\nk(E)\ndE\n− TidGR\nk(E)\ndEvjGR\nk(E)/angbracketrightbigg\n.(36)\nWe decompose the torkance into two parts that are,\nrespectively, even and odd with respect to magnetiza-\ntion reversal, i.e., te\nij(ˆM) = [tij(ˆM) +tij(−ˆM)]/2 and\nto\nij(ˆM) = [tij(ˆM)−tij(−ˆM)]/2.\nIn the clean limit, i.e., for U→0, the even torkance\ncan be written as te\nij=te,int\nij+te,scatt\nij, where [15]\nte,int\nij= 2e/planckover2pi1/integraldisplayddk\n(2π)d/summationdisplay\nn/negationslash=mfknImTi\nknmvj\nkmn\n(Ekn−Ekm)2(37)\nis the intrinsic contribution and\nte,scatt\nij=e/planckover2pi1/summationdisplay\nnm/integraldisplayddk\n(2π)dδ(EF−Ekn)Im/braceleftBigg\n/bracketleftBig\n−Mi\nknmγkmn\nγknnvj\nknn+Aj\nknmγkmn\nγknnTi\nknn/bracketrightBig\n+/bracketleftBig\nMi\nkmn˜vj\nknm−Aj\nkmn˜Ti\nknm/bracketrightBig\n−/bracketleftBig\nMi\nknmγkmn\nγknn˜vj\nknn−Aj\nknmγkmn\nγknn˜Ti\nknn/bracketrightBig\n+/bracketleftBig\n˜vj\nkmnγknm\nγknn˜Ti\nnn\nEkn−Ekm−˜Ti\nkmnγknm\nγknn˜vj\nknn\nEkn−Ekm/bracketrightBig\n+1\n2/bracketleftBig\n˜vj\nknm1\nEkn−Ekm˜Ti\nkmn−˜Ti\nknm1\nEkn−Ekm˜vj\nkmn/bracketrightBig\n+/bracketleftBig\nvj\nknnγknm\nγknn1\nEkn−Ekm˜Ti\nkmn\n−Ti\nknnγknm\nγknn1\nEkn−Ekm˜vj\nkmn/bracketrightBig/bracerightBigg\n.\n(38)\nis the scattering contribution. Here,\niAj\nknm=ivj\nknm\nEkm−Ekn=i\n/planckover2pi1/angbracketleftukn|∂\n∂kj|ukm/angbracketright(39)\nis the Berry connection in kspace and the vertex correc-\ntions of the velocity operator solve the equation\n˜vi\nknm=/summationdisplay\np/integraldisplaydnk′\n(2π)n−1δ(EF−Ek′p)\n2γk′pp×\n×/angbracketleftukn|uk′p/angbracketright/bracketleftbig\n˜vi\nk′pp+vi\nk′pp/bracketrightbig\n/angbracketleftuk′p|ukm/angbracketright.(40)6\nThe odd contribution can be written as to\nij=to,int\nij+\ntRR−vert\nij+tAR−vert\nij, where\nto,int\nij=e\nh/integraldisplayddk\n(2π)dTr/braceleftbig\nTiGR\nk(EF)vj/bracketleftbig\nGA\nk(EF)−GR\nk(EF)/bracketrightbig/bracerightbig\n(41)\nand\ntRR−vert\nij=−e\nh/integraldisplayddk\n(2π)dTr/braceleftBig\n˜TRR\niGR\nk(EF)vjGR\nk(EF)/bracerightBig\n(42)\nand\ntAR−vert\nij=e\nh/integraldisplayddk\n(2π)dTr/braceleftBig\n˜TAR\niGR\nk(EF)vjGA\nk(EF)/bracerightBig\n.(43)\nThe vertex corrections ˜TAR\niand˜TRR\niof the torque op-\nerator are given in Eq. (33) and in Eq. (34), respectively.\nWhile the even torkance, Eq. (37) and Eq. (38), be-\ncomes independent of the scattering strength Uin the\nclean limit, i.e., for U→0, the odd torkance to\nijdepends\nstrongly on Uin metallic systems in the clean limit [15].\nIn the case of the one-dimensional Rashba model, the\nequations Eq. (37) and Eq. (38) for the even torkance\nte\nijand the equations Eq. (41), Eq. (42) and Eq. (43) for\nthe odd torkance to\nijcan be used both for the collinear\nmagnetic state as well as for the spin-spiral of Eq. (5).\nTo obtain the even torkance for the collinear magnetic\nstate, we plug the eigenstates and eigenvalues of Eq. (4)\n(withˆM=ˆez) into Eq. (37) and into Eq. (38). In the\ncase of the spin-spiral of Eq. (5) we use instead the eigen-\nstates and eigenvalues of Eq. (7). Similarly, to obtain the\nodd torkance in the collinear magnetic state, we evaluate\nEq. (41), Eq. (42) and Eq. (43) based on the Hamilto-\nnian in Eq. (4) and for the spin-spiral we use instead the\neffective Hamiltonian in Eq. (7).\nIII. RESULTS\nA. Gyromagnetic ratio\nWe first discuss the g-factor in the collinear case, i.e.,\nwhenˆM(r) =ˆez. Inthis casetheenergybandsaregiven\nby\nE=/planckover2pi12k2\nx\n2m±/radicalbigg\n1\n4(∆V)2+(αRkx)2.(44)\nWhen ∆ V/negationslash= 0 orαR/negationslash= 0 the energy Ecan become\nnegative. The band structure of the one-dimensional\nRashba model is shown in Fig. 1 for the model param-\netersαR=2eV˚A and ∆ V= 0.5eV. For this choice of\nparameters the energy minima are not located at kx= 0\nbut instead at\nkmin\nx=±/radicalBig\n(αR)4m2−1\n4/planckover2pi14(∆V)2\n/planckover2pi12αR,(45)-0.4 -0.2 0 0.2 0.4\nk-Point kx [Å-1]00.511.5Band energy [eV]\nFIG. 1: Band structure of theone-dimensional Rashbamodel.\nand the corresponding minimum of the energy is given\nby\nEmin=−m(αR)4+1\n4/planckover2pi14\nm(∆V)2\n2/planckover2pi12(αR)2. (46)\nThe inverse g-factor is shown as a function of the SOI\nstrength αRin Fig. 2 for the exchange splitting ∆ V=\n1eV and Fermi energy EF= 1.36eV. At αR= 0 the\nscattering contribution is zero, i.e., the g-factor is de-\ntermined completely by the intrinsic Berry curvature ex-\npression, Eq. (24). Thus, at αR= 0 it assumes the value\n1/g=−0.5, which is the expected nonrelativistic value\n(see the discussion below Eq. (24)). With increasing SOI\nstrength αRthe intrinsic contribution to 1 /gis more and\nmore suppressed. However, the scattering contribution\ncompensates this decrease such that the total 1 /gis close\nto its nonrelativistic value of −0.5. The neglect of the\nscattering corrections at large values of αRwould lead in\nthis case to a strong underestimation of the magnitude\nof 1/g, i.e., a strong overestimation of the magnitude of\ng.\nHowever, at smaller values of the Fermi energy, the\ngfactor can deviate substantially from its nonrelativis-\ntic value of −2. To show this we plot in Fig. 3 the in-\nverseg-factor as a function of the Fermi energy when\nthe exchange splitting and the SOI strength are set to\n∆V= 1eV and αR=2eV˚A, respectively. As discussed in\nEq. (44) the minimal Fermi energyis negativ in this case.\nThe intrinsic contribution to 1 /gdeclines with increas-\ning Fermi energy. At large values of the Fermi energy\nthis decline is compensated by the increase of the vertex\ncorrections and the total value of 1 /gis close to −0.5.\nPrevious theoretical works on the g-factor have not\nconsidered the scattering contribution [52]. It is there-\nfore important to find out whether the compensation\nof the decrease of the intrinsic contribution by the in-7\n00.511.52\nSOI strength αR [eVÅ]-0.5-0.4-0.3-0.2-0.101/gscattering\nintrinsic\ntotal\nFIG. 2: Inverse g-factor vs. SOI strength αRin the one-\ndimensional Rashba model.\n0 1 2 3 4 5 6\nFermi energy [eV]-0.6-0.4-0.201/gscattering\nintrinsic\ntotal\nFIG. 3: Inverse g-factor vs. Fermi energy in the one-\ndimensional Rashba model.\ncrease of the extrinsic contribution as discussed in Fig. 2\nand Fig. 3 is peculiar to the one-dimensional Rashba\nmodel or whether it can be found in more general cases.\nFor this reason we evaluate g1for the two-dimensional\nRashba model. In Fig. 4 we show the inverse g1-factor\nin the two-dimensional Rashba model as a function of\nSOI strength αRfor the exchange splitting ∆ V= 1eV\nand the Fermi energy EF= 1.36eV. Indeed for αR<\n0.5eV˚A the scattering corrections tend to stabilize g1at\nits non-relativistic value. However, in contrast to the\none-dimensional case (Fig. 2), where gdoes not deviate\nmuch from its nonrelativistic value up to αR= 2eV˚A,\ng1starts to be affected by SOI at smaller values of αR\nin the two-dimensional case. According to Eq. (26) the\nfullgfactor is given by g=g1(1+Morb/Mspin). There-\nfore, when the scattering corrections stabilize g1at its00.511.52\nSOI strength αR [eVÅ]-0.5-0.4-0.3-0.2-0.101/g1\nscattering\nintrinsic\ntotal\nFIG. 4: Inverse g1-factor vs. SOI strength αRin the two-\ndimensional Rashba model.\nnonrelativistic value the Eq. (25) is satisfied. In the two-\ndimensional Rashba model Morb= 0 when both bands\nare occupied. For the Fermi energy EF= 1.36eV both\nbands are occupied and therefore g=g1for the range of\nparameters used in Fig. 4.\nThe inverse g1of the two-dimensional Rashba model\nis shown in Fig. 5 as a function of Fermi energy for the\nparameters ∆ V= 1eV and αR= 2eV˚A. The scattering\ncorrection is as large as the intrinsic contribution when\nEF>1eV. While the scattering correction is therefore\nimportant, it is not sufficiently large to bring g1close to\nits nonrelativistic value in the energy range shown in the\nfigure, which is a major difference to the one-dimensional\ncase illustrated in Fig. 3. According to Eq. (26) the g\nfactor is related to g1byg=g1M/Mspin. Therefore, we\nshow in Fig. 6 the ratio M/Mspinas a function of Fermi\nenergy. AthighFermienergy(whenbothbandsareoccu-\npied) the orbital magnetization is zeroand M/Mspin= 1.\nAt low Fermi energy the sign of the orbital magnetiza-\ntionis oppositeto the signofthe spin magnetizationsuch\nthat the magnitude of Mis smaller than the magnitude\nofMspinresulting in the ratio M/Mspin<1.\nNext, we discuss the g-factor of the one-dimensional\nRashba model in the noncollinear case. In Fig. 7 we\nplot the inverse g-factor and its decomposition into the\nintrinsic and scattering contributions as a function of\nthe spin-spiral wave vector q, where exchange splitting,\nSOI strength and Fermi energy are set to ∆ V= 1eV,\nαR= 2eV˚A andEF= 1.36eV, respectively. Since\nthe curves are not symmetric around q= 0, the g-\nfactor at wave number qdiffers from the one at −q, i.e.,\nthegyromagnetism in the Rashba model is chiral . At\nq=−2meαR//planckover2pi12theg-factorassumesthevalueof g=−2\nand the scattering corrections are zero. Moreover, the\ncurves are symmetric around q=−2meαR//planckover2pi12. These8\n0 2 4 6\nFermi energy [eV]-0.5-0.4-0.3-0.2-0.101/g1\nscattering\nintrinsic\ntotal\nFIG. 5: Inverse g1-factor 1 /g1vs. Fermi energy in the two-\ndimensional Rashba model.\n-2 0 2 4 6\nFermi energy [eV]00.511.52M/Mspin\nFIG. 6: Ratio of total magnetization and spin magnetization ,\nM/Mspin, vs. Fermi energy in the two-dimensional Rashba\nmodel.\nobservationscan be explained by the concept of the effec-\ntive SOI introduced in Eq. (9): At q=−2meαR//planckover2pi12the\neffective SOI is zero and consequently the noncollinear\nmagnet behaves like a collinear magnet without SOI at\nthis value of q. As we have discussed above in Fig. 2, the\ng-factor of collinear magnets is g=−2 when SOI is ab-\nsent, which explains why it is also g=−2 in noncollinear\nmagnets with q=−2meαR//planckover2pi12. If only the intrinsic con-\ntribution is considered and the scattering corrections are\nneglected, 1 /gvaries much stronger around the point of\nzero effective SOI q=−2meαR//planckover2pi12, i.e., the scattering\ncorrections stabilize gat its nonrelativistic value close to\nthe point of zero effective SOI.-2 -1 0 1\nWave vector q [Å-1]-0.8-0.6-0.4-0.201/g\nscattering\nintrinsic\ntotal\nFIG. 7: Inverse g-factor 1 /gvs. wave number qin the one-\ndimensional Rashba model.\n0 1 2 3 4\nScattering strength U [(eV)2Å]-0.4-0.200.20.4Gilbert Damping αG\nxx\nRR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 8: Gilbert damping αG\nxxvs. scattering strength Uin the\none-dimensional Rashba model without SOI. In this case the\nvertex corrections and the intrinsic contribution sum up to\nzero.\nB. Damping\nWe first discuss the Gilbert damping in the collinear\ncase, i.e., we set ˆM(r) =ˆezin Eq. (4). The xxcom-\nponent of the Gilbert damping is shown in Fig. 8 as\na function of scattering strength Ufor the following\nmodel parameters: exchange splitting ∆ V=1eV, Fermi\nenergyEF= 2.72eV and SOI strength αR= 0. All\nthree contributions are individually non-zero, but the\ncontribution from the RR-vertex correction (Eq. (30)) is\nmuchsmallerthanthe onefromthe AR-vertexcorrection\n(Eq. (31)) and much smaller than the intrinsic contribu-\ntion (Eq. (29)). However, in this case the total damping\nis zero, because a non-zero damping in periodic crystals\nwith collinear magnetization is only possible when SOI\nis present [53].9\n1 2 3 4\nScattering strength U [(eV)2Å]050100150200250300Gilbert Damping αG\nxx\nRR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 9: Gilbert damping αG\nxxvs. scattering strength Uin the\none-dimensional Rashba model with SOI.\nIn Fig. 9 we show the xxcomponent of the Gilbert\ndamping αG\nxxas a function of scattering strength Ufor\nthe model parameters ∆ V= 1eV, EF= 2.72eV and\nαR= 2eV˚A. ThedominantcontributionistheAR-vertex\ncorrection. The damping as obtained based on Eq. (10)\ndiverges like 1 /Uin the limit U→0, i.e., proportional\nto the relaxation time τ[53]. However, once the relax-\nation time τis larger than the inverse frequency of the\nmagnetization dynamics the dc-limit ω→0 in Eq. (10)\nis not appropriate and ω >0 needs to be used. It has\nbeenshownthattheGilbertdampingisnotinfinite inthe\nballistic limit τ→ ∞whenω >0 [41, 42]. In the one-\ndimensional Rashba model the effective magnetic field\nexerted by SOI on the electron spins points in ydirec-\ntion. Since a magnetic field along ydirection cannot lead\ntoatorquein ydirectionthe yycomponentoftheGilbert\ndamping αG\nyyis zero (not shown in the Figure).\nNext, we discuss the Gilbert damping in the non-\ncollinear case. In Fig. 10 we plot the xxcomponent\nof the Gilbert damping as a function of spin spiral\nwave number qfor the model parameters ∆ V= 1eV,\nEF= 1.36eV,αR= 2eV˚A, and the scattering strength\nU= 0.98(eV)2˚A. The curves are symmetric around\nq=−2meαR//planckover2pi12, because the damping is determined by\nthe effective SOI defined in Eq. (9). At q=−2meαR//planckover2pi12\nthe effective SOI is zero and therefore the total damp-\ning is zero as well. The damping at wave number qdif-\nfers from the one at wave number −q, i.e.,the damp-\ning is chiral in the Rashba model . Around the point\nq=−2meαR//planckover2pi12the damping is described by aquadratic\nparabola at first. In the regions -2 ˚A−1< q <-1.2˚A−1\nand 0.2˚A−1< q <1˚A−1this trend is interrupted by a W-\nshape behaviour. In the quadratic parabola region the\nlowest energy band crosses the Fermi energy twice. As\nshown in Fig. 1 the lowest band has a local maximum at-2-1.5-1-0.500.51\nWave vector q [Å-1]05101520Gilbert damping αxxG RR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 10: Gilbert damping αG\nxxvs. spin spiral wave number q\nin the one-dimensional Rashba model.\nq= 0. In the W-shape region this local maximum shifts\nupwards, approaches the Fermi level and finally passes it\nsuch that the lowest energy band crosses the Fermi level\nfour times. This transition in the band structure leads to\noscillations in the density of states, which results in the\nW-shape behaviour of the Gilbert damping.\nSince the damping is determined by the effective SOI,\nwe can use Fig. 10 to draw conclusions about the damp-\ning in the noncollinear case with αR= 0: We only need\nto shift all curves in Fig. 10 to the right such that they\nare symmetric around q= 0 and shift the Fermi energy.\nThus, for αR= 0 the Gilbert damping does not vanish\nifq/negationslash= 0. Since for αR= 0 angular momentum transfer\nfrom the electronic system to the lattice is not possible,\nthe damping is purely nonlocal in this case, i.e., angular\nmomentum is interchanged between electrons at differ-\nent positions. This means that for a volume in which\nthe magnetization of the spin-spiral in Eq. (5) performs\nexactly one revolution between one end of the volume\nand the other end the total angular momentum change\nassociated with the damping is zero, because the angu-\nlar momentum is simply redistributed within this volume\nand there is no net change of the angular momentum.\nA substantial contribution of nonlocal damping has also\nbeen predicted for domain walls in permalloy [35].\nIn Fig. 11 we plot the yycomponent of the Gilbert\ndamping as a function of spin spiral wave number qfor\nthe model parameters ∆ V= 1eV,EF= 1.36eV,αR=\n2eV˚A, and the scattering strength U= 0.98(eV)2˚A. The\ntotaldampingiszerointhiscase. Thiscanbeunderstood\nfrom the symmetry properties of the one-dimensional\nRashba Hamiltonian, Eq. (4): Since this Hamiltonian is\ninvariant when both σandˆMare rotated around the\nyaxis, the damping coefficient αG\nyydoes not depend on\nthe position within the cycloidal spin spiral of Eq. (5).10\n-3 -2 -1 0 1 2\nWave vector q [Å-1]-0.4-0.200.20.4Gilbert Damping αG\nyyRR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 11: Gilbert damping αG\nyyvs. spin spiral wave number q\nin the one-dimensional Rashba model.\nTherefore, nonlocal damping is not possible in this case\nandαG\nyyhas to be zero when αR= 0. It remains to be\nshown that αG\nyy= 0 also for αR/negationslash= 0. However, this fol-\nlows directly from the observation that the damping is\ndetermined by the effective SOI, Eq. (9), meaning that\nany case with q/negationslash= 0 and αR/negationslash= 0 can always be mapped\nonto a case with q/negationslash= 0 and αR= 0. As an alternative\nargumentation we can also invoke the finding discussed\nabovethat αG\nyy= 0 in the collinearcase. Since the damp-\ning is determined by the effective SOI, it follows that\nαG\nyy= 0 also in the noncollinear case.\nC. Current-induced torques\nWe first discuss the yxcomponent of the torkance. In\nFig. 12 we show the torkance tyxas a function of the\nFermi energy EFfor the model parameters ∆ V= 1eV\nandαR= 2eV˚A when the magnetization is collinear and\npoints in zdirection. We specify the torkance in units of\nthe positive elementary charge e, which is a convenient\nchoice for the one-dimensional Rashba model. When\nthe torkance is multiplied with the electric field, we ob-\ntain the torque per length (see Eq. (35) and Ref. [51]).\nSince the effective magnetic field from SOI points in\nydirection, it cannot give rise to a torque in ydirec-\ntion and consequently the total tyxis zero. Interest-\ningly, the intrinsic and scattering contributions are indi-\nvidually nonzero. The intrinsic contribution is nonzero,\nbecause the electric field accelerates the electrons such\nthat/planckover2pi1˙kx=−eEx. Therefore, the effective magnetic\nfieldBSOI\ny=αRkx/µBchanges as well, i.e., ˙BSOI\ny=\nαR˙kx/µB=−αRExe/(/planckover2pi1µB). Consequently, the electron\nspin is no longer aligned with the total effective magnetic\nfield (the effective magnetic field resulting from both SOI-1 0 1 2 3 4 5 6\nFermi energy [eV]-0.2-0.100.10.2Torkance tyx [e]scattering\nintrinsic\ntotal\nFIG. 12: Torkance tyxvs. Fermi energy EFin the one-\ndimensional Rashba model.\nand from the exchange splitting ∆ V), when an electric\nfield is applied. While the total effective magnetic field\nlies in the yzplane, the electron spin acquires an xcom-\nponent, because it precesses around the total effective\nmagnetic field, with which it is not aligned due to the\napplied electric field [54]. The xcomponent of the spin\ndensity results in a torque in ydirection, which is the\nreason why the intrinsic contribution to tyxis nonzero.\nThe scattering contribution to tyxcancels the intrinsic\ncontribution such that the total tyxis zero and angular\nmomentum conservation is satisfied.\nUsing the concept of effective SOI, Eq. (9), we con-\nclude that tyxis also zero for the noncollinear spin-spiral\ndescribed by Eq. (5). Thus, both the ycomponent of the\nspin-orbit torque and the nonadiabatic torque are zero\nfor the one-dimensional Rashba model.\nTo show that tyx= 0 is a peculiarity of the one-\ndimensional Rashba model, we plot in Fig. 13 the\ntorkance tyxin the two-dimensional Rashba model. The\nintrinsic and scattering contributions depend linearly on\nαRfor small values of αR, but the slopes are opposite\nsuch that the total tyxis zero for sufficiently small αR.\nHowever, for largervalues of αRthe intrinsic and scatter-\ning contributions do not cancel each other and therefore\nthe total tyxbecomes nonzero, in contrast to the one-\ndimensional Rashba model, where tyx= 0 even for large\nαR. Several previous works determined the part of tyx\nthat is proportionalto αRin the two-dimensionalRashba\nmodel and found it to be zero [21, 22] for scalar disor-\nder, which is consistent with our finding that the linear\nslopes of the intrinsic and scattering contributions to tyx\nare opposite for small αR.\nNext, we discuss the xxcomponent of the torkance\nin the collinear case ( ˆM=ˆez). In Fig. 14 we plot\nthe torkance txxvs. scattering strength Uin the one-11\n00.511.52\nSOI strength αR [eVÅ]-0.00500.0050.01Torkance tyx [e/Å]\nscattering\nintrinsic\ntotal\nFIG. 13: Nonadiabatic torkance tyxvs. SOI parameter αRin\nthe two-dimensional Rashba model.\ndimensional Rashba model for the parameters ∆ V=\n1eV,EF= 2.72eV and αR= 2eV˚A. The dominant con-\ntribution is the AR-type vertex correction (see Eq. (43)).\ntxxdiverges like 1 /Uin the limit U→0 as expected for\nthe odd torque in metallic systems [15].\nIn Fig. 15 and Fig. 16 we plot txxas a function of\nspin-spiral wave number qfor the model parameters\n∆V= 1eV,EF= 2.72eV and U= 0.18(eV)2˚A. In Fig. 15\nthe case with αR= 2eV˚A is shown, while Fig. 16 illus-\ntrates the case with αR= 0. In the case αR= 0 the\ntorkance txxdescribes the spin-transfer torque (STT). In\nthe case αR/negationslash= 0 the torkance txxis the sum of contribu-\ntions from STT and spin-orbit torque (SOT). The curves\nwithαR= 0 andαR/negationslash= 0 are essentially related by a shift\nof ∆q=−2meαR//planckover2pi12, which can be understood based on\nthe concept of the effective SOI, Eq. (9). Thus, in the\nspecial case of the one-dimensional Rashba model STT\nand SOT are strongly related.\nIV. SUMMARY\nWe study chiral damping, chiral gyromagnetism and\ncurrent-induced torques in the one-dimensional Rashba\nmodel with an additional N´ eel-type noncollinear mag-\nnetic exchange field. In order to describe scattering ef-\nfects we use a Gaussian scalar disorder model. Scat-\ntering contributions are generally important in the one-\ndimensional Rashba model with the exception of the gy-\nromagnetic ratio in the collinear case with zero SOI,\nwhere the scattering correctionsvanish in the clean limit.\nIn the one-dimensional Rashba model SOI and non-\ncollinearity can be combined into an effective SOI. Us-\ning the concept of effective SOI, results for the mag-\nnetically collinear one-dimensional Rashba model can be\nused to predict the behaviour in the noncollinear case.1 2 3 4\nScattering strength U [(eV)2Å]-6-4-20Torkance txx [e]\nRR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 14: Torkance txxvs. scattering strength Uin the one-\ndimensional Rashba model.\n-2 -1 0 1\nWave vector q [Å-1]-4-2024Torkance txx [e]RR-Vertex\nAR-Vertex\nintrinsic\ntotal\nFIG. 15: Torkance txxvs. wave vector qin the one-\ndimensional Rashba model with SOI.\nIn the noncollinear Rashba model the Gilbert damp-\ning is nonlocal and does not vanish for zero SOI. The\nscattering corrections tend to stabilize the gyromagnetic\nratio in the one-dimensional Rashba model at its non-\nrelativistic value. 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Vehstedt, et al., Nature nanotechnology\n9, 211 (2014)."    },    {        "title": "2005.12756v1.A_transmission_problem_for_the_Timoshenko_system_with_one_local_Kelvin_Voigt_damping_and_non_smooth_coefficient_at_the_interface.pdf",        "content": "arXiv:2005.12756v1  [math.AP]  24 May 2020A TRANSMISSION PROBLEM FOR THE TIMOSHENKO SYSTEM WITH ONE LO CAL\nKELVIN-VOIGT DAMPING AND NON-SMOOTH COEFFICIENT AT THE INT ERFACE\nMOUHAMMAD GHADER AND ALI WEHBE\nLEBANESE UNIVERSITY, FACULTY OF SCIENCES 1, KHAWARIZMI LAB ORATORY OF MATHEMATICS AND\nAPPLICATIONS-KALMA, HADATH-BEIRUT.\nEMAILS: MHAMMADGHADER@HOTMAIL.COM AND ALI.WEHBE@UL.EDU .LB.\nAbstract. In this paper, we study the indirect stability of Timoshenko system with local or global Kelvin–Voigt damping, under\nfully Dirichlet or mixed boundary conditions. Unlike [ 43] and [ 39], in this paper, we consider the Timoshenko system with only\none locally or globally distributed Kelvin-Voigt damping D(see System ( 1.1)). Indeed, we prove that the energy of the system\ndecays polynomially of type t−1and that this decay rate is in some sense optimal. The method i s based on the frequency domain\napproach combining with multiplier method.\nMSC Classification. 35B35; 35B40; 93D20.\nKeywords. Timoshenko beam; Kelvin-Voigt damping; Semigroup; Stabil ity.\n1.Introduction\nIn this paper, we study the indirect stability of a one-dimen sional Timoshenko system with only one local or\nglobal Kelvin-Voigt damping. This system consists of two co upled hyperbolic equations:\n(1.1)ρ1utt−k1(ux+y)x= 0, (x,t)∈(0,L)×R+,\nρ2ytt−(k2yx+Dyxt)x+k1(ux+y) = 0,(x,t)∈(0,L)×R+.\nSystem ( 1.1) is subject to the following initial conditions:\n(1.2)u(x,0) =u0(x), ut(x,0) =u1(x), x∈(0,L),\ny(x,0) =y0(x), yt(x,0) =y1(x), x∈(0,L),\nin addition to the following boundary conditions:\n(1.3) u(0,t) =y(0,t) =u(L,t) =y(L,t) = 0, t∈R+,\nor\n(1.4) u(0,t) =yx(0,t) =u(L,t) =yx(L,t) = 0, t∈R+.\nHere the coefficients ρ1, ρ2, k1, andk2are strictly positive constant numbers. The function D∈L∞(0,L),\nsuch thatD(x)≥0,∀x∈[0,L]. We assume that there exist D0>0,α, β∈R,0≤α<β≤L,such that\n(H) D∈C([α,β])andD(x)≥D0>0∀x∈(α,β).\nThe hypothesis (H) means that the control Dcan be locally near the boundary (see Figures 1aand1b), or\nlocally internal (see Figure 2a), or globally (see Figure 2b). Indeed, in the case when Dis local damping (i.e.,\nα/ne}ationslash= 0orβ/ne}ationslash=L), we see that Dis not necessary continuous over (0,L)(see Figures 1a,1b, and 2a).\nThe Timoshenko system is usually considered in describing t he transverse vibration of a beam and ignoring\ndamping effects of any nature. Indeed, we have the following m odel, see in [ 40],\n/braceleftigg\nρϕtt= (K(ϕx−ψ))x\nIρψtt= (EIψx)x−K(ϕx−ψ),\nwhereϕis the transverse displacement of the beam and ψis the rotation angle of the filament of the beam.\nThe coefficients ρ, Iρ, E, I, andKare respectively the density (the mass per unit length), the polar moment\n1STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nx 0αβ=LD(x)\nD(x)\nFigure 1ax 0 =αβLD(x)\nD(x)\nFigure 1b\nx 0αβLD(x)D(x)\nD(x)\nFigure 2ax 0 =αL=βD(x)\nFigure 2b\nof inertia of a cross section, Young’s modulus of elasticity , the moment of inertia of a cross section and the\nshear modulus respectively.\nThe stabilization of the Timoshenko system with different ki nds of damping has been studied in number of\npublications. For the internal stabilization, Raposo and a l. in [ 34] showed that the Timoshenko system with\ntwo internal distributed dissipation is exponentially sta ble. Messaoudi and Mustafa in [ 27] extended the results\nto nonlinear feedback laws. Soufyane and Wehbe in [ 37] showed that Timoshenko system with one internal\ndistributed dissipation law is exponentially stable if and only if the wave propagation speeds are equal (i.e.,\nk1\nρ1=ρ2\nk2), otherwise, only the strong stability holds. Indeed, Rive ra and Racke in [ 30] they improved the\nresults of [ 37], where an exponential decay of the solution of the system ha s been established, allowing the\ncoefficient of the feedback to be with an indefinite sign. Wehbe and Youssef in [ 41] proved that the Timoshenko\nsystem with one locally distributed viscous feedback is exp onentially stable if and only if the wave propagation\nspeeds are equal (i.e.,k1\nρ1=ρ2\nk2), otherwise, only the polynomial stability holds. Tebou in [38] showed that\nthe Timoshenko beam with same feedback control in both equat ions is exponentially stable. The stability\nof the Timoshenko system with thermoelastic dissipation ha s been studied in [ 36], [12], [13], and [ 15]. The\nstability of Timoshenko system with memory type has been stu died in [ 3], [36], [14], [28], and [ 1]. For the\nboundary stabilization of the Timoshenko beam. Kim and Rena rdy in [ 19] showed that the Timoshenko beam\nunder two boundary controls is exponentially stable. Ammar -Khodja and al. in [ 4] studied the decay rate\nof the energy of the nonuniform Timoshenko beam with two boun dary controls acting in the rotation-angle\nequation. In fact, under the equal speed wave propagation co ndition, they established exponential decay results\nup to an unknown finite dimensional space of initial data. In a ddition, they showed that the equal speed wave\npropagation condition is necessary for the exponential sta bility. However, in the case of non-equal speed, no\ndecay rate has been discussed. This result has been recently improved by Wehbe and al. in [ 7]; i.e., the authors\nin [7], proved nonuniform stability and an optimal polynomial en ergy decay rate of the Timoshenko system\nwith only one dissipation law on the boundary. For the stabil ization of the Timoshenko beam with nonlinear\nterm, we mention [ 29], [2], [5], [27], [10], and [ 15].\nKelvin-Voigt material is a viscoelastic structure having p roperties of both elasticity and viscosity. There\nare a number of publications concerning the stabilization o f wave equation with global or local Kelvin-Voigt\ndamping. For the global case, the authors in [ 16,21], proved the analyticity and the exponential stability of t he\nsemigroup. When the Kelvin-Voigt damping is localized on an interval of the string, the regularity and stability\n2STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nof the solution depend on the properties of the damping coeffic ient. Notably, the system is more effectively\ncontrolled by the local Kelvin-Voigt damping when the coeffic ient changes more smoothly near the interface\n(see [22,35,42,25,23]).\nLast but not least, in addition to the previously cited paper s, the stability of the Timoshenko system with\nKelvin-Voigt damping has been studied in few papers. Zhao an d al. in [ 43] they considered the Timoshenko\nsystem with local distributed Kelvin–Voigt damping:\n(1.5)ρ1utt−[k1(ux+y)x+D1(uxt−yt)]x= 0, (x,t)∈(0,L)×R+,\nρ2ytt−(k2yx+D2yxt)x+k1(ux+y)x+D1(uxt−yt) = 0,(x,t)∈(0,L)×R+.\nThey proved that the energy of the System ( 1.5) subject to Dirichlet-Neumann boundary conditions has an\nexponential decay rate when coefficient functions D1, D2∈C1,1([0,L])and satisfy D1≤cD2(c >0).Tian\nand Zhang in [ 39] considered the Timoshenko System ( 1.5) under fully Dirichlet boundary conditions with\nlocally or globally distributed Kelvin-Voigt damping when coefficient functions D1, D2∈C([0,L]). First,\nwhen the Kelvin-Voigt damping is globally distributed, the y showed that the Timoshenko System ( 1.5) under\nfully Dirichlet boundary conditions is analytic. Next, for their system with local Kelvin-Voigt damping, they\nanalyzed the exponential and polynomial stability accordi ng to the properties of coefficient functions D1, D2.\nUnlike [ 43] and [ 39], in this paper, we consider the Timoshenko system with only one locally or globally\ndistributed Kelvin-Voigt damping D(see System ( 1.1)). Indeed, in this paper, under hypothesis (H), we show\nthat the energy of the Timoshenko System ( 1.1) subject to initial state ( 1.2) to either the boundary conditions\n(1.3) or (1.4) has a polynomial decay rate of type t−1and that this decay rate is in some sense optimal.\nThis paper is organized as follows: In Section 2, first, we show that the Timoshenko System ( 1.1) subject to\ninitial state ( 1.2) to either the boundary conditions ( 1.3) or (1.4) can reformulate into an evolution equation\nand we deduce the well-posedness property of the problem by t he semigroup approach. Second, using a criteria\nof Arendt-Batty [ 6], we show that our system is strongly stable. In Section 3, we show that the Timoshenko\nSystem ( 1.1)-(1.2) with the boundary conditions ( 1.4) is not uniformly exponentially stable. In Section 4, we\nprove the polynomial energy decay rate of type t−1for the System ( 1.1)-(1.2) to either the boundary conditions\n(1.3) or (1.4). Moreover, we prove that this decay rate is in some sense opt imal.\n2.Well-Posedness and Strong Stability\n2.1.Well-posedness of the problem. In this part, under condition (H), using a semigroup approac h, we\nestablish well-posedness result for the Timoshenko System (1.1)-(1.2) to either the boundary conditions ( 1.3) or\n(1.4). The energy of solutions of the System ( 1.1) subject to initial state ( 1.2) to either the boundary conditions\n(1.3) or (1.4) is defined by:\nE(t) =1\n2/integraldisplayL\n0/parenleftig\nρ1|ut|2+ρ2|yt|2+k1|ux+y|2+k2|yx|2/parenrightig\ndx.\nLet(u,y)be a regular solution for the System ( 1.1). Multiplying the first and the second equation of ( 1.1) by\nutandyt,respectively, then using the boundary conditions ( 1.3) or (1.4), we get\nE′(t) =−/integraldisplayL\n0D(x)|yxt|2dx≤0.\nThus System ( 1.1) subject to initial state ( 1.2) to either the boundary conditions ( 1.3) or (1.4) is dissipative in\nthe sense that its energy is non increasing with respect to th e timet. Let us define the energy spaces H1and\nH2by:\nH1=H1\n0(0,L)×L2(0,L)×H1\n0(0,L)×L2(0,L)\nand\nH2=H1\n0(0,L)×L2(0,L)×H1\n∗(0,L)×L2(0,L),\nsuch that\nH1\n∗(0,L) =/braceleftigg\nf∈H1(0,L)|/integraldisplayL\n0fdx= 0/bracerightigg\n.\n3STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nIt is easy to check that the space H1\n∗is Hilbert spaces over Cequipped with the norm\n/bardblu/bardbl2\nH1\n∗(0,L)=/bardblux/bardbl2,\nwhere/bardbl· /bardbldenotes the usual norm of L2(0,L). Both energy spaces H1andH2are equipped with the inner\nproduct defined by:\n/an}bracketle{tU,U1/an}bracketri}htHj=ρ1/integraldisplayL\n0vv1dx+ρ2/integraldisplayL\n0zz1dx+k1/integraldisplayL\n0(ux+y)((u1)x+y1)dx+k2/integraldisplayL\n0yx(y1)xdx\nfor allU= (u,v,y,z)andU1= (u1,v1,y1,z1)inHj,j= 1,2. We use /bardblU/bardblHjto denote the corresponding\nnorms. We now define the following unbounded linear operator sAjinHjby\nD(A1) =/braceleftbig\nU= (u,v,y,z)∈ H1|v, z∈H1\n0(0,L), u∈H2(0,L),(k2yx+Dzx)x∈L2(0,L)/bracerightbig\n,\nD(A2) =/braceleftbigg\nU= (u,v,y,z)∈ H2|v∈H1\n0(0,L), z∈H1\n∗(0,L), u∈H2(0,L),\n(k2yx+Dzx)x∈L2(0,L), yx(0) =yx(L) = 0/bracerightbigg\nand forj= 1,2,\nAjU=/parenleftbigg\nv,k1\nρ1(ux+y)x,z,1\nρ2(k2yx+Dzx)x−k1\nρ2(ux+y)/parenrightbigg\n,∀U= (u,v,y,z)∈D(Aj).\nIfU= (u,ut,y,yt)is the state of System ( 1.1)-(1.2) to either the boundary conditions ( 1.3) or (1.4), then the\nTimoshenko system is transformed into a first order evolutio n equation on the Hilbert space Hj:\n(2.1)/braceleftigg\nUt(x,t) =AjU(x,t),\nU(x,0) =U0(x),\nwhere\nU0(x) = (u0(x),u1(x),y0(x),y1(x)).\nProposition 2.1. Under hypothesis (H), for j= 1,2,the unbounded linear operator Ajis m-dissipative in\nthe energy space Hj.\nProof. Letj= 1,2, forU= (u,v,y,z)∈D(Aj), one has\nℜ/an}bracketle{tAjU,U/an}bracketri}htHj=−/integraldisplayL\n0D(x)|zx|2dx≤0,\nwhich implies that Ajis dissipative under hypothesis (H). Here ℜis used to denote the real part of a complex\nnumber. We next prove the maximality of Aj. ForF= (f1,f2,f3,f4)∈ Hj, we prove the existence of\nU= (u,v,y,z)∈D(Aj), unique solution of the equation\n−AjU=F.\nEquivalently, one must consider the system given by\n−v=f1, (2.2)\n−k1(ux+y)x=ρ1f2, (2.3)\n−z=f3, (2.4)\n−(k2yx+Dzx)x+k1(ux+y) =ρ2f4, (2.5)\nwith the boundary conditions\n(2.6) u(0) =u(L) =v(0) =v(L) = 0 and/braceleftigg\ny(0) =y(L) =z(0) =z(L) = 0,forj= 1,\nyx(0) =yx(L) = 0, forj= 2.\nLet(ϕ,ψ)∈ Vj(0,L), whereV1(0,L) =H1\n0(0,L)×H1\n0(0,L)andV2(0,L) =H1\n0(0,L)×H1\n∗(0,L). Multiplying\nEquations ( 2.3) and ( 2.5) byϕandψrespectively, integrating in (0,L), taking the sum, then using Equation\n(2.4) and the boundary condition ( 2.6), we get\n(2.7)/integraldisplayL\n0/parenleftig\nk1(ux+y)(ϕx+ψ)+k2yxψx/parenrightig\ndx=/integraldisplayL\n0/parenleftbig\nρ1f1¯ϕ+ρ2f4¯ψ+D(f3)xψx/parenrightbig\ndx,∀(ϕ,ψ)∈ Vj(0,L).\n4STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nThe left hand side of ( 2.7) is a bilinear continuous coercive form on Vj(0,L)×Vj(0,L), and the right hand side\nof (2.7) is a linear continuous form on Vj(0,L). Then, using Lax-Milligram theorem (see in [ 32]), we deduce\nthat there exists (u,y)∈ Vj(0,L)unique solution of the variational Problem ( 2.7). Thus, using ( 2.2), (2.4),\nand classical regularity arguments, we conclude that −AjU=Fadmits a unique solution U∈D(Aj)and\nconsequently 0∈ρ(Aj), whereρ(Aj)denotes the resolvent set of Aj. Then, Ajis closed and consequently\nρ(Aj)is open set of C(see Theorem 6.7 in [ 18]). Hence, we easily get λ∈ρ(Aj)for sufficiently small λ>0.\nThis, together with the dissipativeness of Aj, imply that D(Aj)is dense in Hjand that Ajis m-dissipative in\nHj(see Theorems 4.5, 4.6 in [ 32]). Thus, the proof is complete. /square\nThanks to Lumer-Phillips theorem (see [ 26,32]), we deduce that Ajgenerates a C0-semigroup of contraction\netAjinHjand therefore Problem ( 2.1) is well-posed. Then, we have the following result.\nTheorem 2.2. Under hypothesis (H), for j= 1,2,for anyU0∈ Hj, the Problem ( 2.1) admits a unique weak\nsolutionU(x,t) =etAjU0(x), such that\nU∈C(R+;Hj).\nMoreover, if U0∈D(Aj),then\nU∈C(R+;D(Aj))∩C1(R+;Hj).\n/square\nBefore starting the main results of this work, we introduce h ere the notions of stability that we encounter in\nthis work.\nDefinition 2.3. LetA:D(A)⊂H→Hgenerate a C 0−semigroup of contractions/parenleftbig\netA/parenrightbig\nt≥0onH. The\nC0-semigroup/parenleftbig\netA/parenrightbig\nt≥0is said to be\n1. strongly stable if\nlim\nt→+∞/bardbletAx0/bardblH= 0,∀x0∈H;\n2. exponentially (or uniformly) stable if there exist two po sitive constants Mandǫsuch that\n/bardbletAx0/bardblH≤Me−ǫt/bardblx0/bardblH,∀t>0,∀x0∈H;\n3. polynomially stable if there exists two positive constan tsCandαsuch that\n/bardbletAx0/bardblH≤Ct−α/bardblAx0/bardblH,∀t>0,∀x0∈D(A).\nIn that case, one says that solutions of ( 2.1) decay at a rate t−α. TheC0-semigroup/parenleftbig\netA/parenrightbig\nt≥0is said\nto be polynomially stable with optimal decay rate t−α(withα >0) if it is polynomially stable with\ndecay ratet−αand, for any ε>0small enough, there exists solutions of ( 2.1) which do not decay at a\nratet−(α+ε).\n/square\nWe now look for necessary conditions to show the strong stabi lity of theC0-semigroup/parenleftbig\netA/parenrightbig\nt≥0. We will rely\non the following result obtained by Arendt and Batty in [ 6].\nTheorem 2.4 (Arendt and Batty in [ 6]).LetA:D(A)⊂H→Hgenerate a C 0−semigroup of contractions/parenleftbig\netA/parenrightbig\nt≥0onH. If\n1.Ahas no pure imaginary eigenvalues,\n2.σ(A)∩iRis countable,\nwhereσ(A)denotes the spectrum of A, then theC0-semigroup/parenleftbig\netA/parenrightbig\nt≥0is strongly stable. /square\nOur subsequent findings on polynomial stability will rely on the following result from [ 9,24,8], which gives\nnecessary and sufficient conditions for a semigroup to be poly nomially stable. For this aim, we recall the\nfollowing standard result (see [ 9,24,8] for part (i) and [ 17,33] for part (ii)).\nTheorem 2.5. LetA:D(A)⊂H→Hgenerate a C 0−semigroup of contractions/parenleftbig\netA/parenrightbig\nt≥0onH. Assume\nthatiλ∈ρ(A),∀λ∈R. Then, the C0-semigroup/parenleftbig\netA/parenrightbig\nt≥0is\n5STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\n(i) Polynomially stable of order1\nℓ(ℓ>0)if and only if\nlim sup\nλ∈R,|λ|→∞|λ|−ℓ/vextenddouble/vextenddouble/vextenddouble(iλI−A)−1/vextenddouble/vextenddouble/vextenddouble\nL(H)<+∞.\n(ii) Exponentially stable if and only if\nlim sup\nλ∈R,|λ|→∞/vextenddouble/vextenddouble/vextenddouble(iλI−A)−1/vextenddouble/vextenddouble/vextenddouble\nL(H)<+∞.\n/square\n2.2.Strong stability. In this part, we use general criteria of Arendt-Batty in [ 6] (see Theorem 2.4) to show\nthe strong stability of the C0-semigroup etAjassociated to the Timoshenko System ( 2.1). Our main result is\nthe following theorem.\nTheorem 2.6. Assume that (H) is true. Then, for j= 1,2,theC0−semigroupetAjis strongly stable in Hj;\ni.e., for allU0∈ Hj, the solution of ( 2.1) satisfies\nlim\nt→+∞/vextenddouble/vextenddoubleetAjU0/vextenddouble/vextenddouble\nHj= 0.\nThe argument for Theorem 2.6relies on the subsequent lemmas.\nLemma 2.7. Under hypothesis (H), for j= 1,2,one has\nker(iλI−Aj) ={0},∀λ∈R.\nProof. Forj= 1,2, from Proposition 2.1, we deduce that 0∈ρ(Aj). We still need to show the result for\nλ∈R∗. Suppose that there exists a real number λ/ne}ationslash= 0andU= (u,v,y,z)∈D(Aj)such that\nAjU=iλU.\nEquivalently, we have\n(2.8)\n\nv=iλu,\nk1(ux+y)x=iρ1λv,\nz=iλy,\n(k2yx+Dzx)x−k1(ux+y) =iρ2λz.\nFirst, a straightforward computation gives\n0 =ℜ/an}bracketle{tiλU,U/an}bracketri}htHj=ℜ/an}bracketle{tAjU,U/an}bracketri}htHj=−/integraldisplayL\n0D(x)|zx|2dx,\nusing hypothesis (H), we deduce that\n(2.9) Dzx= 0 over(0,L)andzx= 0 over(α,β).\nInserting ( 2.9) in (2.8), we get\nu=yx= 0,over(α,β), (2.10)\nk1uxx+ρ1λ2u+k1yx= 0,over(0,L), (2.11)\n−k1ux+k2yxx+/parenleftbig\nρ2λ2−k1/parenrightbig\ny= 0,over(0,L), (2.12)\nwith the following boundary conditions\n(2.13) u(0) =u(L) =y(0) =y(L) = 0,ifj= 1 oru(0) =u(L) =yx(0) =yx(L) = 0,ifj= 2.\nIn fact, System ( 2.11)-(2.13) admits a unique solution (u,y)∈C2((0,L)). From ( 2.10) and by the uniqueness\nof solutions, we get\n(2.14) u=yx= 0,over(0,L).\n1. Ifj= 1, from ( 2.14) and the fact that y(0) = 0,we get\nu=y= 0,over(0,L),\nhence,U= 0. In this case the proof is complete.\n6STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\n2. Ifj= 2, from ( 2.14) and the fact that y∈H1\n∗(0,L)(i.e.,/integraltextL\n0ydx= 0),we get\nu=y= 0,over(0,L),\ntherefore,U= 0, also in this case the proof is complete.\n/square\nLemma 2.8. Under hypothesis (H), for j= 1,2,for allλ∈R, theniλI−Ajis surjective.\nProof. LetF= (f1,f2,f3,f4)∈ Hj, we look for U= (u,v,y,z)∈D(Aj)solution of\n(iλU−Aj)U=F.\nEquivalently, we have\nv=iλu−f1, (2.15)\nz=iλy−f3, (2.16)\nλ2u+k1\nρ1(ux+y)x=F1, (2.17)\nλ2y+ρ2−1[(k2+iλD)yx]x−k1\nρ2(ux+y) =F2, (2.18)\nwith the boundary conditions\n(2.19) u(0) =u(L) =v(0) =v(L) = 0 and/braceleftigg\ny(0) =y(L) =z(0) =z(L) = 0,forj= 1,\nyx(0) =yx(L) = 0, forj= 2.\nSuch that /braceleftigg\nF1=−f2−iλf1∈L2(0,L),\nF2=−f4−iλf3+ρ2−1(D(f3)x)x∈H−1(0,L).\nWe define the operator Ljby\nLjU=/parenleftbigg\n−k1\nρ1(ux+y)x,−ρ−1\n2[(k2+iλD)yx]x+k1\nρ2(ux+y)/parenrightbigg\n,∀ U= (u,y)∈ Vj(0,L),\nwhere\nV1(0,L) =H1\n0(0,L)×H1\n0(0,L)andV2(0,L) =H1\n0(0,L)×H1\n∗(0,L).\nUsing Lax-Milgram theorem, it is easy to show that Ljis an isomorphism from Vj(0,L)onto(H−1(0,L))2.\nLetU= (u,y)andF= (−F1,−F2), then we transform System ( 2.17)-(2.18) into the following form\n(2.20) U −λ2L−1\njU=L−1F.\nUsing the compactness embeddings from L2(0,L)intoH−1(0,L)and fromH1\n0(0,L)intoL2(0,L), and from\nH1\nL(0,L)intoL2(0,L), we deduce that the operator L−1\njis compact from L2(0,L)×L2(0,L)intoL2(0,L)×\nL2(0,L). Consequently, by Fredholm alternative, proving the exist ence ofUsolution of ( 2.20) reduces to\nprovingker/parenleftbig\nI−λ2L−1\nj/parenrightbig\n= 0. Indeed, if (ϕ,ψ)∈ker(I−λ2L−1\nj), then we have λ2(ϕ,ψ)− Lj(ϕ,ψ) = 0. It\nfollows that\n(2.21)\n\nλ2ϕ+k1\nρ1(ϕx+ψ)x= 0,\nλ2ψ+ρ2−1[(k2+iλD)ψx]x−k1\nρ2(ϕx+ψ) = 0,\nwith the following boundary conditions\n(2.22) ϕ(0) =ϕ(L) =ψ(0) =ψ(L) = 0,ifj= 1 orϕ(0) =ϕ(L) =ψx(0) =ψx(L) = 0,ifj= 2.\nIt is now easy to see that if (ϕ,ψ)is a solution of System ( 2.21)-(2.22), then the vector Vdefined by\nV= (ϕ,iλϕ,ψ,iλψ )\nbelongs toD(Aj)andiλV−AjV= 0.Therefore,V∈ker(iλI−Aj). Using Lemma 2.7, we getV= 0, and so\nker(I−λ2L−1\nj) ={0}.\n7STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nThanks to Fredholm alternative, the Equation ( 2.20) admits a unique solution (u,v)∈ Vj(0,L). Thus, using\n(2.15), (2.17) and a classical regularity arguments, we conclude that (iλ−Aj)U=Fadmits a unique solution\nU∈D(Aj). Thus, the proof is complete. /square\nWe are now in a position to conclude the proof of Theorem 2.6.\nProof of Theorem 2.6.Using Lemma 2.7, we directly deduce that Ajha non pure imaginary eigenvalues.\nAccording to Lemmas 2.7,2.8and with the help of the closed graph theorem of Banach, we ded uce that\nσ(Aj)∩iR={∅}. Thus, we get the conclusion by applying Theorem 2.4of Arendt and Batty. /square\n3.Lack of exponential stability of A2\nIn this section, our goal is to show that the Timoshenko Syste m (1.1)-(1.2) with Dirichlet-Neumann boundary\nconditions ( 1.4) is not exponentially stable.\n3.1.Lack of exponential stability of A2with global Kelvin–Voigt damping. In this part, assume that\n(3.1) D(x) =D0>0,∀x∈(0,L),\nwhereD0∈R+\n∗. We prove the following theorem.\nTheorem 3.1. Under hypothesis ( 3.1), forǫ>0(small enough ), we cannot expect the energy decay rate t−2\n2−ǫ\nfor all initial data U0∈D(A2)and for allt>0.\nProof. Following to Borichev [ 9] (see Theorem 2.4part (i)), it suffices to show the existence of sequences\n(λn)n⊂Rwithλn→+∞,(Un)n⊂D(A2), and(Fn)n⊂ H2such that (iλnI−A2)Un=Fnis bounded in\nH2andλ−2+ǫ\nn/bardblUn/bardbl →+∞. Set\nFn=/parenleftig\n0,sin/parenleftignπx\nL/parenrightig\n,0,0/parenrightig\n, Un=/parenleftig\nAnsin/parenleftignπx\nL/parenrightig\n,iλnAnsin/parenleftignπx\nL/parenrightig\n,Bncos/parenleftignπx\nL/parenrightig\n,iλnBncos/parenleftignπx\nL/parenrightig/parenrightig\nand\n(3.2) λn=nπ\nL/radicaligg\nk1\nρ1, An=−inπD0\nk1L/radicalbiggρ1\nk1+k2\nk1/parenleftbiggρ2\nk2−ρ1\nk1/parenrightbigg\n−ρ1L2\nk1π2n2, Bn=ρ1L\nk1nπ.\nClearly that Un∈D(A2),andFnis bounded in H2. Let us show that\n(iλnI−A2)Un=Fn.\nDetailing (iλnI−A2)Un, we get\n(iλnI−A2)Un=/parenleftig\n0,C1,nsin/parenleftignπx\nL/parenrightig\n,0,C2,ncos/parenleftignπx\nL/parenrightig/parenrightig\n,\nwhere\n(3.3)C1,n=/parenleftbiggk1\nρ1/parenleftignπ\nL/parenrightig2\n−λ2\nn/parenrightbigg\nAn+k1nπ\nρ1LBn, C2,n=nπk1\nρ2LAn+/parenleftbigg\n−λ2\nn+k1\nρ2+k2+iλnD0\nρ2/parenleftignπ\nL/parenrightig2/parenrightbigg\nBn.\nInserting ( 3.2) in (3.3), we get\nC1,n= 1 andC2,n= 0,\nhence we obtain\n(iλnI−A2)Un=/parenleftig\n0,sin/parenleftignπx\nL/parenrightig\n,0,0/parenrightig\n=Fn.\nNow, we have\n/bardblUn/bardbl2\nH2≥ρ1/integraldisplayL\n0/vextendsingle/vextendsingle/vextendsingleiλnAnsin/parenleftignπx\nL/parenrightig/vextendsingle/vextendsingle/vextendsingle2\ndx=ρ1Lλ2\nn\n2|An|2∼λ4\nn.\nTherefore, for ǫ>0(small enough ), we have\nλ−2+ǫ\nn/bardblUn/bardblH2∼λǫ\nn→+∞.\nFinally, following to Borichev [ 9] (see Theorem 2.4part (i)) we cannot expect the energy decay rate t−2\n2−ǫ./square\nNote that Theorem 3.1also implies that our system is non-uniformly stable.\n8STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\n3.2.Lack of exponential stability of A2with local Kelvin–Voigt damping. In this part, under the\nequal speed wave propagation condition (i.e.,ρ1\nk1=ρ2\nk2), we use the classical method developed by Littman\nand Markus in [ 20] (see also [ 11]), to show that the Timoshenko System ( 1.1)-(1.2) with local Kelvin–Voigt\ndamping, and with Dirichlet-Neumann boundary conditions ( 1.4) is not exponentially stable. For this aim,\nassume that\n(3.4)ρ1\nk1=ρ2\nk2andD(x) =/braceleftigg\n0,0<x≤α,\nD0α<x≤L,\nwhereD0∈R+\n∗andα∈(0,L). For simplicity and without loss of generality, in this part , we takeρ1\nk1= 1,\nD0=k2,L= 1, andα=1\n2, then hypothesis ( 3.4) becomes\n(3.5)ρ1\nk1=ρ2\nk2= 1andD(x) =/braceleftigg\n0,0<x≤1\n2,\nk21\n2<x≤1.\nOur main result in this part is following theorem.\nTheorem 3.2. Under hypothesis ( 3.5). The semigroup generated by the operator A2is not exponentially\nstable in the energy space H2.\nFor the proof of Theorem 3.2, we recall the following definitions: the growth bound ω0(A2)and the the spectral\nbounds(A2)ofA2are defined respectively as\nω(A2) = lim\nt→∞log/vextenddouble/vextenddoubleetA2/vextenddouble/vextenddouble\nL(H2)\ntands(A2) = sup{ℜ(λ) :λ∈σ(A2)}.\nFrom the Hille-Yoside theorem (see also Theorem 2.1.6 and Le mma 2.1.11 in [ 11]), one has that\ns(A2)≤ω0(A2).\nBy the previous results, one clearly has that s(A2)≤0and the theorem would follow if equality holds in the\nprevious inequality. It therefore amounts to show the exist ence of a sequence of eigenvalues of A2whose real\nparts tend to zero.\nSinceA2is dissipative, we fix α0>0small enough and we study the asymptotic behavior of the eige nvaluesλ\nofA2in the strip\nS={λ∈C:−α0≤ ℜ(λ)≤0}.\nFirst, we determine the characteristic equation satisfied b y the eigenvalues of A2. For this aim, let λ∈C∗be\nan eigenvalue of A2and letU= (u,λu,y,λy,ω )∈D(A2)be an associated eigenvector. Then the eigenvalue\nproblem is given by\nλ2u−uxx−yx= 0, x∈(0,1), (3.6)\nc2ux+/parenleftbig\nλ2+c2/parenrightbig\ny−/parenleftbigg\n1+D\nk2λ/parenrightbigg\nyxx= 0, x∈(0,1), (3.7)\nwith the boundary conditions\n(3.8) u(0) =yx(0) =u(1) =yx(1) = 0,\nwherec=/radicalig\nk1k−1\n2. We define\n/braceleftigg\nu−(x) :=u(x), y−(x) :=y(x), x∈(0,1/2),\nu+(x) :=u(x), y+(x) :=y(x), x∈[1/2,1),\nthen System ( 3.6)-(3.8) becomes\nλ2u−−u−\nxx−y−\nx= 0, x∈(0,1/2), (3.9)\nc2u−\nx+/parenleftbig\nλ2+c2/parenrightbig\ny−−y−\nxx= 0, x∈(0,1/2), (3.10)\nλ2u+−u+\nxx−y+\nx= 0, x∈[1/2,1), (3.11)\nc2u+\nx+/parenleftbig\nλ2+c2/parenrightbig\ny+−(1+λ)y+\nxx= 0, x∈[1/2,1), (3.12)\n9STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nwith the boundary conditions\nu−(0) =y−\nx(0) = 0, (3.13)\nu+(1) =y+\nx(1) = 0, (3.14)\nand the continuity conditions\nu−(1/2) =u+(1/2), (3.15)\nu−\nx(1/2) =u+\nx(1/2), (3.16)\ny−(1/2) =y+(1/2), (3.17)\ny−\nx(1/2) = (1+λ)y+\nx(1/2). (3.18)\nIn order to proceed, we set the following notation. Here and b elow, in the case where zis a non zero non-real\nnumber, we define (and denote) by√zthe square root of z; i.e., the unique complex number with positive real\npart whose square is equal to z. Our aim is to study the asymptotic behavior of the large eige nvaluesλofA2\ninS. By taking λlarge enough, the general solution of System ( 3.9)-(3.10) with boundary condition ( 3.13) is\ngiven by\n\n\nu−(x) =α1sinh(r1x)+α2sinh(r2x),\ny−(x) =α1λ2−r2\n1\nr1cosh(r1x)+α2λ2−r2\n2\nr2cosh(r2x),\nand the general solution of Equation ( 3.11)-(3.12) with boundary condition ( 3.14) is given by\n\n\nu+(x) =β1sinh(s1(1−x))+β2sinh(s2(1−x)),\ny+(x) =−β1λ2−s2\n1\ns1cosh(s1(1−x))−β2λ2−s2\n2\ns2cosh(s2(1−x)),\nwhereα1, α2, β1, β2∈C,\n(3.19) r1=λ/radicalbigg\n1+ic\nλ, r 2=λ/radicalbigg\n1−ic\nλ\nand\n(3.20) s1=/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbtλ+λ2\n2/parenleftbigg\n1+/radicalig\n1−4c2\nλ3−4c2\nλ4/parenrightbigg\n1+1\nλ, s 2=/radicaltp/radicalvertex/radicalvertex/radicalvertex/radicalbtλ+λ2\n2/parenleftbigg\n1−/radicalig\n1−4c2\nλ3−4c2\nλ4/parenrightbigg\n1+1\nλ.\nThe boundary conditions in ( 3.15)-(3.18) can be expressed by\nM\nα1\nα2\nβ1\nβ2\n= 0,\nwhere\nM=\nsinh/parenleftbigr1\n2/parenrightbig\nsinh/parenleftbigr2\n2/parenrightbig\n−sinh/parenleftbigs1\n2/parenrightbig\n−sinh/parenleftbigs2\n2/parenrightbig\nr1\nicλ2cosh/parenleftbigr1\n2/parenrightbigr2\nicλ2cosh/parenleftbigr2\n2/parenrightbigs1\nicλ2cosh/parenleftbigs1\n2/parenrightbigs2\nicλ2cosh/parenleftbigs2\n2/parenrightbig\nr2\n1sinh/parenleftbigr1\n2/parenrightbig\nr2\n2sinh/parenleftbigr2\n2/parenrightbig /parenleftbig\nλ3−(λ+1)s2\n1/parenrightbig\nsinh/parenleftbigs1\n2/parenrightbig /parenleftbig\nλ3−(λ+1)s2\n2/parenrightbig\nsinh/parenleftbigs2\n2/parenrightbig\nr−1\n1cosh/parenleftbigr1\n2/parenrightbig\nr−1\n2cosh/parenleftbigr2\n2/parenrightbig\ns−1\n1cosh/parenleftbigs1\n2/parenrightbig\ns−1\n2cosh/parenleftbigs2\n2/parenrightbig\n.\nDenoting the determinant of a matrix Mbydet(M), consequently, System ( 3.9)-(3.18) admits a non trivial\nsolution if and only if det(M) = 0. Using Gaussian elimination, det(M) = 0 is equivalent to det/parenleftig\n˜M/parenrightig\n= 0,\n10STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nwhere˜Mis given by\n˜M=\nsinh/parenleftbigr1\n2/parenrightbig\nsinh/parenleftbigr2\n2/parenrightbig\n−sinh/parenleftbigs1\n2/parenrightbig\n−1−e−s2\nr1\nicλ2cosh/parenleftbigr1\n2/parenrightbigr2\nicλ2cosh/parenleftbigr2\n2/parenrightbigs1\nicλ2cosh/parenleftbigs1\n2/parenrightbigs2\nicλ2(1+e−s2)\nr2\n1sinh/parenleftbigr1\n2/parenrightbig\nr2\n2sinh/parenleftbigr2\n2/parenrightbig /parenleftbig\nλ3−(λ+1)s2\n1/parenrightbig\nsinh/parenleftbigs1\n2/parenrightbig /parenleftbig\nλ3−(λ+1)s2\n2/parenrightbig\n(1−e−s2)\nr−1\n1cosh/parenleftbigr1\n2/parenrightbig\nr−1\n2cosh/parenleftbigr2\n2/parenrightbig\ns−1\n1cosh/parenleftbigs1\n2/parenrightbig\ns−1\n2(1+e−s2)\n.\nOne gets that\n(3.21)det/parenleftig\n˜M/parenrightig\n=g1cosh/parenleftigr1\n2/parenrightig\ncosh/parenleftigr2\n2/parenrightig\nsinh/parenleftigs1\n2/parenrightig\n+g2sinh/parenleftigr1\n2/parenrightig\ncosh/parenleftigr2\n2/parenrightig\ncosh/parenleftigs1\n2/parenrightig\n+g3cosh/parenleftigr1\n2/parenrightig\nsinh/parenleftigr2\n2/parenrightig\ncosh/parenleftigs1\n2/parenrightig\n+g4sinh/parenleftigr1\n2/parenrightig\nsinh/parenleftigr2\n2/parenrightig\ncosh/parenleftigs1\n2/parenrightig\n+g5cosh/parenleftigr1\n2/parenrightig\nsinh/parenleftigr2\n2/parenrightig\nsinh/parenleftigs1\n2/parenrightig\n+g6sinh/parenleftigr1\n2/parenrightig\ncosh/parenleftigr2\n2/parenrightig\nsinh/parenleftigs1\n2/parenrightig\n/parenleftbigg\n−g1cosh/parenleftigr1\n2/parenrightig\ncosh/parenleftigr2\n2/parenrightig\nsinh/parenleftigs1\n2/parenrightig\n−g2sinh/parenleftigr1\n2/parenrightig\ncosh/parenleftigr2\n2/parenrightig\ncosh/parenleftigs1\n2/parenrightig\n−g3cosh/parenleftigr1\n2/parenrightig\nsinh/parenleftigr2\n2/parenrightig\ncosh/parenleftigs1\n2/parenrightig\n+g4sinh/parenleftigr1\n2/parenrightig\nsinh/parenleftigr2\n2/parenrightig\ncosh/parenleftigs1\n2/parenrightig\n+g5cosh/parenleftigr1\n2/parenrightig\nsinh/parenleftigr2\n2/parenrightig\nsinh/parenleftigs1\n2/parenrightig\n+g6sinh/parenleftigr1\n2/parenrightig\ncosh/parenleftigr2\n2/parenrightig\nsinh/parenleftigs1\n2/parenrightig/parenrightbigg\ne−s2,\nwhere\n(3.22)\n\ng1=(λ+1)/parenleftbig\nr2\n1−r2\n2/parenrightbig/parenleftbig\ns2\n1−s2\n2/parenrightbig\nicr1r2λ2, g2=/parenleftbig\nr2\n2−s2\n1/parenrightbig/parenleftbig\n(λ+1)s2\n2−λ3−r2\n1/parenrightbig\nics1r2λ2,\ng3=−/parenleftbig\nr2\n1−s2\n1/parenrightbig/parenleftbig\n(λ+1)s2\n2−λ3−r2\n2/parenrightbig\nicr1s1λ2, g4=/parenleftbig\nr2\n1−r2\n2/parenrightbig/parenleftbig\ns2\n1−s2\n2/parenrightbig\nics1s2λ2,\ng5=/parenleftbig\nr2\n1−s2\n2/parenrightbig/parenleftbig\n(λ+1)s2\n1−λ3−r2\n2/parenrightbig\nics2r1λ2, g6=−/parenleftbig\nr2\n2−s2\n2/parenrightbig/parenleftbig\n(λ+1)s2\n1−λ3−r2\n1/parenrightbig\nicr2s2λ2.\nProposition 3.3. Under hypothesis ( 3.5), there exist n0∈Nsufficiently large and two sequences (λ1,n)|n|≥n0\nand(λ2,n)|n|≥n0of simple roots of det(˜M)(that are also simple eigenvalues of A2) satisfying the following\nasymptotic behavior:\nCase 1. If there exist no integers κ∈Nsuch thatc= 2κπ(i.e.,sin/parenleftbigc\n4/parenrightbig\n/ne}ationslash= 0andcos/parenleftbigc\n4/parenrightbig\n/ne}ationslash= 0), then\nλ1,n= 2inπ−2(1−isign(n))sin/parenleftbigc\n4/parenrightbig2\n/parenleftbig\n3+cos/parenleftbigc\n2/parenrightbig/parenrightbig/radicalbig\nπ|n|+O/parenleftbig\nn−1/parenrightbig\n, (3.23)\nλ2,n= 2inπ+πi+iarccos/parenleftig\ncos/parenleftigc\n4/parenrightig/parenrightig\n−(1−isign(n))cos/parenleftbigc\n4/parenrightbig2\n/parenleftig\n1+cos/parenleftbigc\n4/parenrightbig2/parenrightig/radicalbig\nπ|n|+O/parenleftbig\nn−1/parenrightbig\n. (3.24)\nCase 2. If there exists κ0∈Nsuch thatc= 2(2κ0+1)π, (i.e.,cos/parenleftbigc\n4/parenrightbig\n= 0), then\nλ1,n= 2inπ−1−isign(n)/radicalbig\nπ|n|+O/parenleftbig\nn−1/parenrightbig\n, (3.25)\nλ2,n= 2inπ+3πi\n2+ic2\n32πn−(8+i(3π−2))c2\n128π2n2+O/parenleftig\n|n|−5/2/parenrightig\n. (3.26)\nCase 3. If there exists κ1∈Nsuch thatc= 4κ1π, (i.e.,sin/parenleftbigc\n4/parenrightbig\n= 0), then\nλ1,n= 2inπ+ic2\n32πn−c2\n16π2n2+O/parenleftig\n|n|−5/2/parenrightig\n, (3.27)\nλ2,n= 2inπ+πi+ic2\n32πn−(4+iπ)c2\n64π2n2+O/parenleftig\n|n|−5/2/parenrightig\n. (3.28)\nHeresignis used to denote the sign function or signum function.\n11STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nThe argument for Proposition 3.3relies on the subsequent lemmas.\nLemma 3.4. Under hypothesis ( 3.5), letλbe a large eigenvalue of A2, thenλis large root of the following\nasymptotic equation:\n(3.29) F(λ) :=f0(λ)+f1(λ)\nλ1/2+f2(λ)\n8λ+f3(λ)\n8λ3/2+f4(λ)\n128λ2+f5(λ)\n128λ5/2+O/parenleftbig\nλ−3/parenrightbig\n= 0,\nwhere\n(3.30)\n\nf0(λ) = sinh/parenleftbigg3λ\n2/parenrightbigg\n+sinh/parenleftbiggλ\n2/parenrightbigg\ncos/parenleftigc\n2/parenrightig\n,\nf1(λ) = cosh/parenleftbigg3λ\n2/parenrightbigg\n−cosh/parenleftbiggλ\n2/parenrightbigg\ncos/parenleftigc\n2/parenrightig\n,\nf2(λ) =c2cosh/parenleftbigg3λ\n2/parenrightbigg\n−4ccosh/parenleftbiggλ\n2/parenrightbigg\nsin/parenleftigc\n2/parenrightig\n,\nf3(λ) =c2sinh/parenleftbigg3λ\n2/parenrightbigg\n−4cosh/parenleftbigg3λ\n2/parenrightbigg\n+12csinh/parenleftbiggλ\n2/parenrightbigg\nsin/parenleftigc\n2/parenrightig\n+4cosh/parenleftbiggλ\n2/parenrightbigg\ncos/parenleftigc\n2/parenrightig\n,\nf4(λ) =c2/parenleftbig\nc2−56/parenrightbig\nsinh/parenleftbigg3λ\n2/parenrightbigg\n−32c2cosh/parenleftbigg3λ\n2/parenrightbigg\n+8c2/parenleftig\ncsin/parenleftigc\n2/parenrightig\n−8cos/parenleftigc\n2/parenrightig\n+1/parenrightig\nsinh/parenleftbiggλ\n2/parenrightbigg\n−32c/parenleftig\n8sin/parenleftigc\n2/parenrightig\n+ccos/parenleftigc\n2/parenrightig/parenrightig\ncos/parenleftigc\n2/parenrightig\n,\nf5(λ) =−40c2sinh/parenleftbigg3λ\n2/parenrightbigg\n+/parenleftbig\nc4−88c2+48/parenrightbig\ncosh/parenleftbigg3λ\n2/parenrightbigg\n+32c/parenleftig\n5sin/parenleftigc\n2/parenrightig\n+ccos/parenleftigc\n2/parenrightig/parenrightig\nsinh/parenleftbiggλ\n2/parenrightbigg\n−/parenleftig\n8c3sin/parenleftigc\n2/parenrightig\n−16(4c2−3)cos/parenleftigc\n2/parenrightig\n−24c2/parenrightig\ncos/parenleftigc\n2/parenrightig\n.\nProof. Letλbe a large eigenvalue of A2, thenλis root ofdet/parenleftig\n˜M/parenrightig\n. In this lemma, we furnish an asymptotic\ndevelopment of the function det/parenleftig\n˜M/parenrightig\nfor largeλ. First, using the asymptotic expansion in ( 3.19) and ( 3.20),\nwe get\n(3.31)\n\nr1=λ+ic\n2+c2\n8λ−ic3\n16λ2+O/parenleftbig\nλ−3/parenrightbig\n, r2=λ−ic\n2+c2\n8λ+ic3\n16λ2+O/parenleftbig\nλ−3/parenrightbig\n,\ns1=λ−c2\n2λ2+O/parenleftbig\nλ−5/parenrightbig\n, s2=λ1/2−1\n2λ1/2+4c2+3\n8λ3/2+O/parenleftig\nλ−5/2/parenrightig\n.\nInserting ( 3.31) in (3.22), we get\n(3.32)\n\ng1= 2−c2\nλ2+O/parenleftbig\nλ−3/parenrightbig\n, g2= 1+ic\n2λ−(3c−16i)c\n8λ2+O/parenleftbig\nλ−3/parenrightbig\n,\ng3= 1−ic\n2λ−(3c+16i)c\n8λ2+O/parenleftbig\nλ−3/parenrightbig\n, g4= 2λ1/2−1\nλ3/2−4c2−3\n4λ5/2+O/parenleftig\nλ−7/2/parenrightig\n,\ng5=λ1/2−1−3ic\n2λ3/2−7c2−3−10ic\n8λ5/2+O/parenleftig\nλ−7/2/parenrightig\n,\ng6=λ1/2−1+3ic\n2λ3/2−7c2−3+10ic\n8λ5/2+O/parenleftig\nλ−7/2/parenrightig\n.\n12STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nInserting ( 3.32) in (3.21), then using the fact that real λis bounded in S, we get\n(3.33)det/parenleftig\n˜M/parenrightig\n= sinh(L1)+sinh(L2)cosh(L3)+cosh(L1)−cosh(L2)cosh(L3)\nλ1/2\n+iccosh(L2)sinh(L3)\n2λ−cosh(L1)−cosh(L2)cosh(L3)+3icsinh(L2)sinh(L3)\n2λ3/2\n−7c2sinh(L1)+8c2sinh(L2)cosh(L3)−32iccosh(L2)sinh(L3)−c2sinh(L4)\n16λ3/2\n−(11c2−6)cosh(L1)−(8c2−6) cosh(L2)cosh(L3)+20icsinh(L2)sinh(L3)−3c2cosh(L4)\n16λ5/2\n+/parenleftig\nsinh(L1)+sinh(L2)cosh(L3)+O/parenleftig\nλ−1/2/parenrightig/parenrightig\ne−s2+O/parenleftbig\nλ−3/parenrightbig\n,\nwhere\nL1=r1+r2+s1\n2, L2=s1\n2, L3=r1−r2\n2, L4=r1+r2−s1\n2.\nNext, from ( 3.31) and using the fact that real λis bounded S, we get\n(3.34)\n\nsinh(L1) = sinh/parenleftbigg3λ\n2/parenrightbigg\n+c2cosh/parenleftbig3λ\n2/parenrightbig\n8λ+c2/parenleftbig\nc2sinh/parenleftbig3λ\n2/parenrightbig\n−32cosh/parenleftbig3λ\n2/parenrightbig/parenrightbig\n128λ2+O/parenleftbig\nλ−3/parenrightbig\n,\ncosh(L1) = cosh/parenleftbigg3λ\n2/parenrightbigg\n+c2sinh/parenleftbig3λ\n2/parenrightbig\n8λ+c2/parenleftbig\nc2cosh/parenleftbig3λ\n2/parenrightbig\n−32sinh/parenleftbig3λ\n2/parenrightbig/parenrightbig\n128λ2+O/parenleftbig\nλ−3/parenrightbig\n,\nsinh(L2) = sinh/parenleftbiggλ\n2/parenrightbigg\n−c2cosh/parenleftbigλ\n2/parenrightbig\n4λ2+O/parenleftbig\nλ−4/parenrightbig\n,\ncosh(L2) = cosh/parenleftbiggλ\n2/parenrightbigg\n−c2sinh/parenleftbigλ\n2/parenrightbig\n4λ2+O/parenleftbig\nλ−4/parenrightbig\n,\nsinh(L3) =isin/parenleftigc\n2/parenrightig\n−ic3cos/parenleftbigc\n2/parenrightbig\n16λ2+O/parenleftbig\nλ−3/parenrightbig\n,\ncosh(L3) = cos/parenleftigc\n2/parenrightig\n+c3cos/parenleftbigc\n2/parenrightbig\n16λ2+O/parenleftbig\nλ−3/parenrightbig\n,\nsinh(L4) = sinh/parenleftbiggλ\n2/parenrightbigg\n+O/parenleftbig\nλ−1/parenrightbig\n,cosh(L4) = cosh/parenleftbiggλ\n2/parenrightbigg\n+O/parenleftbig\nλ−1/parenrightbig\n.\nOn the other hand, from ( 3.31) and ( 3.34), we obtain\n(3.35)/parenleftig\nsinh(L1)+sinh(L2)cosh(L3)+O/parenleftig\nλ−1/2/parenrightig/parenrightig\ne−s2=−/parenleftbigg\nsinh/parenleftbigg3λ\n2/parenrightbigg\n+sinh/parenleftbiggλ\n2/parenrightbigg\ncos/parenleftigc\n2/parenrightig/parenrightbigg\ne−√\nλ.\nSince real part of√\nλis positive, then\nlim\n|λ|→∞e−√\nλ\nλ3= 0,\nhence\n(3.36) e−√\nλ=o/parenleftbig\nλ−3/parenrightbig\n.\nTherefore, from ( 3.35) and ( 3.36), we get\n(3.37)/parenleftig\nsinh(L1)+sinh(L2)cosh(L3)+O/parenleftig\nλ−1/2/parenrightig/parenrightig\ne−s2=o/parenleftbig\nλ−3/parenrightbig\n.\nFinally, inserting ( 3.34) and ( 3.37) in (3.33), we getλis large root of F, whereFdefined in ( 3.29). /square\nLemma 3.5. Under hypothesis ( 3.5), there exist n0∈Nsufficiently large and two sequences (λ1,n)|n|≥n0and\n(λ2,n)|n|≥n0of simple roots of F(that are also simple eigenvalues of A2) satisfying the following asymptotic\nbehavior:\n(3.38) λ1,n= 2iπn+iπ+ǫ1,n,such that lim\n|n|→+∞ǫ1,n= 0\n13STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nand\n(3.39) λ2,n= 2nπi+iπ+iarccos/parenleftig\ncos2/parenleftigc\n4/parenrightig/parenrightig\n+ǫ2,n,such that lim\n|n|→+∞ǫ2,n= 0.\nProof. First, we look at the roots of f0. From ( 3.30), we deduce that f0can be written as\n(3.40) f0(λ) = 2sinh/parenleftbiggλ\n2/parenrightbigg/parenleftig\ncosh(λ)+cos2/parenleftigc\n4/parenrightig/parenrightig\n.\nThe roots of f0are given by\n\n\nµ1,n= 2nπi, n ∈Z,\nµ2,n= 2nπi+iπ+iarccos/parenleftig\ncos2/parenleftigc\n4/parenrightig/parenrightig\n, n∈Z.\nNow with the help of Rouché’s theorem, we will show that the ro ots ofFare close to f0.\nLet us start with the first family µ1,n. LetBn=B(2nπi,rn)be the ball of centrum 2nπiand radiusrn=1\n|n|1\n4\nandλ∈∂Bn; i.e.,λ= 2nπi+rneiθ, θ∈[0,2π). Then\n(3.41) sinh/parenleftbiggλ\n2/parenrightbigg\n= (−1)nsinh/parenleftbiggrneiθ\n2/parenrightbigg\n=(−1)nrneiθ\n2+O(r2\nn),cosh(λ) = cosh/parenleftbig\nrneiθ/parenrightbig\n= 1+O(r2\nn).\nInserting ( 3.41) in (3.40), we get\nf0(λ) = (−1)nrneiθ/parenleftig\n1+cos2/parenleftigc\n4/parenrightig/parenrightig\n+O(r3\nn).\nIt follows that there exists a positive constant Csuch that\n∀λ∈∂Bn,|f0(λ)| ≥Crn=C\n|n|1\n4.\nOn the other hand, from ( 3.29), we deduce that\n|F(λ)−f0(λ)|=O/parenleftbigg1√\nλ/parenrightbigg\n=O/parenleftigg\n1/radicalbig\n|n|/parenrightigg\n.\nIt follows that, for |n|large enough\n∀λ∈∂Bn,|F(λ)−f0(λ)|<|f0(λ)|.\nHence, with the help of Rouché’s theorem, there exists n0∈N∗large enough, such that ∀ |n| ≥n0(n∈Z∗),\nthe first branch of roots of F, denoted by λ1,nare close to µ1,n, hence we get ( 3.38). The same procedure yields\n(3.39). Thus, the proof is complete. /square\nRemark 3.6. From Lemma 3.5, we deduce that the real part of the eigenvalues of A2tends to zero, and this\nis enough to get Theorem 3.2. But we look forward to knowing the real part of λ1,nandλ2,n. Since in the\nnext section, we will use the real part of λ1,nandλ2,nfor the optimality of polynomial stability. /square\nWe are now in a position to conclude the proof of Proposition 3.3.\nProof of Proposition 3.3.The proof is divided into two steps.\nStep 1. Calculation of ǫ1,n. From ( 3.38), we have\n(3.42)\n\ncosh/parenleftbigg3λ1,n\n2/parenrightbigg\n= (−1)ncosh/parenleftbigg3ǫ1,n\n2/parenrightbigg\n,sinh/parenleftbigg3λ1,n\n2/parenrightbigg\n= (−1)nsinh/parenleftbigg3ǫ1,n\n2/parenrightbigg\n,\ncosh/parenleftbiggλ1,n\n2/parenrightbigg\n= (−1)ncosh/parenleftigǫ1,n\n2/parenrightig\n,sinh/parenleftbiggλ1,n\n2/parenrightbigg\n= (−1)nsinh/parenleftigǫ1,n\n2/parenrightig\n,\n14STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nand\n(3.43)\n\n1\nλ1,n=−i\n2πn+O/parenleftbig\nǫ1,nn−2/parenrightbig\n+O/parenleftbig\nn−3/parenrightbig\n,1\nλ2\n1,n=−1\n4π2n2+O/parenleftbig\nn−3/parenrightbig\n,\n1/radicalbig\nλ1,n=1−isign(n)\n2/radicalbig\nπ|n|+O/parenleftig\nǫ1,n|n|−3/2/parenrightig\n+O/parenleftig\n|n|−5/2/parenrightig\n,\n1/radicalig\nλ3\n1,n=−1−isign(n)\n4/radicalbig\nπ3|n|3+O/parenleftig\n|n|−5/2/parenrightig\n,1/radicalig\nλ5\n1,n=O/parenleftig\n|n|−5/2/parenrightig\n.\nOn the other hand, since lim|n|→+∞ǫ1,n= 0, we have the asymptotic expansion\n(3.44)\n\ncosh/parenleftbigg3ǫ1,n\n2/parenrightbigg\n= 1+9ǫ2\n1,n\n8+O(ǫ4\n1,n),sinh/parenleftbigg3ǫ1,n\n2/parenrightbigg\n=3ǫ1,n\n2+O(ǫ3\n1,n),\ncosh/parenleftigǫ1,n\n2/parenrightig\n= 1+ǫ2\n1,n\n8+O(ǫ4\n1,n),sinh/parenleftigǫ1,n\n2/parenrightig\n=ǫ1,n\n2+O(ǫ3\n1,n).\nInserting ( 3.44) in (3.42), we get\n(3.45)\n\ncosh/parenleftbigg3λ1,n\n2/parenrightbigg\n= (−1)n+9(−1)nǫ1,n\n8+O(ǫ4\n1,n),sinh/parenleftbigg3λ1,n\n2/parenrightbigg\n=3(−1)nǫ1,n\n2+O(ǫ3\n1,n),\ncosh/parenleftbiggλ1,n\n2/parenrightbigg\n= (−1)n+(−1)nǫ1,n\n8+O(ǫ4\n1,n),sinh/parenleftbiggλ1,n\n2/parenrightbigg\n=(−1)nǫ1,n\n2+O(ǫ3\n1,n).\nInserting ( 3.43) and ( 3.45) in (3.29), we get\n(3.46)ǫ1,n\n2/parenleftig\n3+cos/parenleftigc\n2/parenrightig/parenrightig\n+(1−isign(n))/parenleftbig\n1−cos/parenleftbigc\n2/parenrightbig/parenrightbig\n2/radicalbig\nπ|n|+ic/parenleftbig\n4sin/parenleftbigc\n2/parenrightbig\n−c/parenrightbig\n16πn\n+(1+isign(n))/parenleftbig\n1−cos/parenleftbigc\n2/parenrightbig/parenrightbig\n8/radicalbig\nπ3|n|3+8csin/parenleftbigc\n2/parenrightbig\n+/parenleftbig\n1+cos/parenleftbigc\n2/parenrightbig/parenrightbig\nc2\n16π2n2\n+O/parenleftig\n|n|−5/2/parenrightig\n+O/parenleftig\nǫ1,n|n|−3/2/parenrightig\n+O/parenleftig\nǫ2\n1,n|n|−1/2/parenrightig\n+O/parenleftbig\nǫ3\n1,n/parenrightbig\n= 0.\nWe distinguish two cases:\nCase 1. Ifsin/parenleftbigc\n4/parenrightbig\n/ne}ationslash= 0,then\n1−cos/parenleftigc\n2/parenrightig\n= 2sin2/parenleftigc\n4/parenrightig\n/ne}ationslash= 0,\ntherefore, from ( 3.46), we get\nǫ1,n\n2/parenleftig\n3+cos/parenleftigc\n2/parenrightig/parenrightig\n+sin2/parenleftbigc\n4/parenrightbig\n(1−isign(n))/radicalbig\n|n|π+O/parenleftbig\nǫ3\n1,n/parenrightbig\n+O/parenleftig\n|n|−1/2ǫ2\n1,n/parenrightig\n+O/parenleftbig\nn−1/parenrightbig\n= 0,\nhence, we get\n(3.47) ǫ1,n=−2sin2/parenleftbigc\n4/parenrightbig\n(1−isign(n))/parenleftbig\n3+cos/parenleftbigc\n2/parenrightbig/parenrightbig/radicalbig\n|n|π+O/parenleftbig\nn−1/parenrightbig\n.\nInserting ( 3.47) in (3.38), we get ( 3.23) and ( 3.25).\nCase 2. Ifsin/parenleftbigc\n4/parenrightbig\n= 0,then\n1−cos/parenleftigc\n2/parenrightig\n= 2sin2/parenleftigc\n4/parenrightig\n= 0,sin/parenleftigc\n2/parenrightig\n= 2sin/parenleftigc\n4/parenrightig\ncos/parenleftigc\n4/parenrightig\n= 0,\ntherefore, from ( 3.46), we get\n(3.48) 2ǫ1,n−ic2\n16πn+c2\n8π2n2+O/parenleftig\n|n|−5/2/parenrightig\n+O/parenleftig\nǫ1,n|n|−3/2/parenrightig\n+O/parenleftig\nǫ2\n1,n|n|−1/2/parenrightig\n+O/parenleftbig\nǫ3\n1,n/parenrightbig\n= 0.\nSolving Equation ( 3.48), we get\n(3.49) ǫ1,n=ic2\n32πn−c2\n16π2n2+O/parenleftig\n|n|−5/2/parenrightig\n.\n15STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nInserting ( 3.49) in (3.38), we get ( 3.27).\nStep 2. Calculation of ǫ2,n. We distinguish three cases:\nCase 1. Ifsin/parenleftbigc\n4/parenrightbig\n/ne}ationslash= 0andcos/parenleftbigc\n4/parenrightbig\n/ne}ationslash= 0, then0<cos2/parenleftbigc\n4/parenrightbig\n<1.Therefore\nζ:= arccos/parenleftig\ncos2/parenleftigc\n4/parenrightig/parenrightig\n∈/parenleftig\n0,π\n2/parenrightig\n.\nFrom ( 3.39), we have\n(3.50)1/radicalbig\nλ2,n=1−isign(n)\n2/radicalbig\nπ|n|+O/parenleftig\n|n|−3/2/parenrightig\nand1\nλ2,n=O(n−1).\nInserting ( 3.39) and ( 3.50) in (3.29), we get\n(3.51)2sinh/parenleftbiggλ2,n\n2/parenrightbigg/parenleftig\ncosh(λ2,n)+cos2/parenleftigc\n4/parenrightig/parenrightig\n+cosh/parenleftig\nλ2,n\n2/parenrightig/parenleftbig\ncosh(λ2,n)−cos2/parenleftbigc\n4/parenrightbig/parenrightbig\n(1−isign(n))\n/radicalbig\nπ|n|+O(n−1) = 0.\nFrom ( 3.39), we obtain\n(3.52)\n\ncosh(λ2,n) =−cos2/parenleftigc\n4/parenrightig\ncosh(ǫ2,n)−isin(ζ)sinh(ǫ2,n),\ncosh/parenleftbiggλ2,n\n2/parenrightbigg\n= (−1)n/parenleftbigg\n−sin/parenleftbiggζ\n2/parenrightbigg\ncosh/parenleftigǫ2,n\n2/parenrightig\n+icos/parenleftbiggζ\n2/parenrightbigg\nsinh/parenleftigǫ2,n\n2/parenrightig/parenrightbigg\n,\nsinh/parenleftbiggλ2,n\n2/parenrightbigg\n= (−1)n/parenleftbigg\n−sin/parenleftbiggζ\n2/parenrightbigg\nsinh/parenleftigǫ2,n\n2/parenrightig\n+icos/parenleftbiggζ\n2/parenrightbigg\ncosh/parenleftigǫ2,n\n2/parenrightig/parenrightbigg\n.\nSinceζ= arccos/parenleftbig\ncos2/parenleftbigc\n4/parenrightbig/parenrightbig\n∈/parenleftbig\n0,π\n2/parenrightbig\n, we have\n(3.53) sin(ζ) =/vextendsingle/vextendsingle/vextendsinglesin/parenleftigc\n4/parenrightig/vextendsingle/vextendsingle/vextendsingle/radicalbigg\n1+cos2/parenleftigc\n4/parenrightig\n,cos/parenleftbiggζ\n2/parenrightbigg\n=/radicalig\n1+cos2/parenleftbigc\n4/parenrightbig\n√\n2,sin/parenleftbiggζ\n2/parenrightbigg\n=/vextendsingle/vextendsinglesin/parenleftbigc\n4/parenrightbig/vextendsingle/vextendsingle\n√\n2.\nOn the other hand, since lim|n|→+∞ǫ2,n= 0, we have the asymptotic expansion\n(3.54)\n\ncosh(ǫ2,n) = 1+O(ǫ2\n2,n),sinh(ǫ2,n) =ǫ2,n+O(ǫ3\n2,n),\ncosh/parenleftigǫ2,n\n2/parenrightig\n= 1+O(ǫ2\n2,n),sinh/parenleftigǫ2,n\n2/parenrightig\n=ǫ2,n\n2+O(ǫ3\n2,n).\nInserting ( 3.53) and ( 3.54) in (3.52), we get\n(3.55)\n\ncosh(λ2,n) =−cos2/parenleftigc\n4/parenrightig\n−iǫ2,n/vextendsingle/vextendsingle/vextendsinglesin/parenleftigc\n4/parenrightig/vextendsingle/vextendsingle/vextendsingle/radicalbigg\n1+cos2/parenleftigc\n4/parenrightig\n+O(ǫ2\n2,n),\ncosh/parenleftbiggλ2,n\n2/parenrightbigg\n=(−1)n\n√\n2\niǫ2,n/radicalig\n1+cos2/parenleftbigc\n4/parenrightbig\n2−/vextendsingle/vextendsingle/vextendsinglesin/parenleftigc\n4/parenrightig/vextendsingle/vextendsingle/vextendsingle\n+O(ǫ2\n2,n),\nsinh/parenleftbiggλ2,n\n2/parenrightbigg\n=−(−1)n\n2√\n2/parenleftbigg/vextendsingle/vextendsingle/vextendsinglesin/parenleftigc\n4/parenrightig/vextendsingle/vextendsingle/vextendsingleǫ2,n−2i/radicalbigg\n1+cos2/parenleftigc\n4/parenrightig/parenrightbigg\n+O(ǫ2\n2,n).\nInserting ( 3.55) in (3.51), we get\n√\n2(−1)n/vextendsingle/vextendsingle/vextendsinglesin/parenleftigc\n4/parenrightig/vextendsingle/vextendsingle/vextendsingle/parenleftig\n1+cos2/parenleftigc\n4/parenrightig/parenrightig/parenleftigg\nǫ2,n+cos2/parenleftbigc\n4/parenrightbig\n(1−isign(n))/parenleftbig\n1+cos2/parenleftbigc\n4/parenrightbig/parenrightbig/radicalbig\nπ|n|/parenrightigg\n+O(n−1)+O(ǫ2\n2,n)+O/parenleftig\n|n|−1/2ǫ2,n/parenrightig\n= 0,\n16STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nsince in this case sin/parenleftbigc\n4/parenrightbig\n/ne}ationslash= 0, then we get\n(3.56) ǫ2,n=−cos2/parenleftbigc\n4/parenrightbig\n(1−isign(n))/parenleftbig\n1+cos2/parenleftbigc\n4/parenrightbig/parenrightbig/radicalbig\nπ|n|+O(n−1).\nInserting ( 3.56) in (3.46), we get ( 3.24).\nCase 2. Ifcos/parenleftbigc\n4/parenrightbig\n= 0, then\n(3.57) cos/parenleftigc\n2/parenrightig\n=−1,sin/parenleftigc\n2/parenrightig\n= 0.\nIn this case λ2,nbecomes\n(3.58) λ2,n= 2inπ+3πi\n2+ǫ2,n.\nTherefore, we have\n(3.59)\n\ncosh/parenleftbigg3λ2,n\n2/parenrightbigg\n=(−1)n\n√\n2/parenleftbigg\ncosh/parenleftbigg3ǫ2,n\n2/parenrightbigg\n+isinh/parenleftbigg3ǫ2,n\n2/parenrightbigg/parenrightbigg\n,\nsinh/parenleftbigg3λ2,n\n2/parenrightbigg\n=(−1)n\n√\n2/parenleftbigg\nicosh/parenleftbigg3ǫ2,n\n2/parenrightbigg\n+sinh/parenleftbigg3ǫ2,n\n2/parenrightbigg/parenrightbigg\n,\ncosh/parenleftbiggλ2,n\n2/parenrightbigg\n=(−1)n\n√\n2/parenleftig\n−cosh/parenleftigǫ2,n\n2/parenrightig\n+isinh/parenleftigǫ2,n\n2/parenrightig/parenrightig\n,\nsinh/parenleftbiggλ2,n\n2/parenrightbigg\n=(−1)n\n√\n2/parenleftig\nicosh/parenleftigǫ2,n\n2/parenrightig\n−sinh/parenleftigǫ2,n\n2/parenrightig/parenrightig\n.\nOn the other hand, since lim|n|→+∞ǫ2,n= 0, we have the asymptotic expansion\n(3.60)\n\ncosh/parenleftbigg3ǫ2,n\n2/parenrightbigg\n= 1+9ǫ2\n2,n\n8+O(ǫ4\n2,n),sinh/parenleftbigg3ǫ2,n\n2/parenrightbigg\n=3ǫ2,n\n2+O(ǫ3\n2,n),\ncosh/parenleftigǫ2,n\n2/parenrightig\n= 1+ǫ2\n2,n\n8+O(ǫ4\n2,n),sinh/parenleftigǫ2,n\n2/parenrightig\n=ǫ2,n\n2+O(ǫ3\n2,n).\nInserting ( 3.60) in (3.59), we get\n(3.61)\n\ncosh/parenleftbigg3λ2,n\n2/parenrightbigg\n=(−1)n\n√\n2/parenleftigg\n1+3iǫ2,n\n2+9ǫ2\n2,n\n8+O(ǫ3\n2,n)/parenrightigg\n,\nsinh/parenleftbigg3λ2,n\n2/parenrightbigg\n=(−1)n\n√\n2/parenleftigg\ni+3ǫ2,n\n2+9iǫ2\n2,n\n8+O(ǫ3\n2,n)/parenrightigg\n,\ncosh/parenleftbiggλ2,n\n2/parenrightbigg\n=(−1)n\n√\n2/parenleftigg\n−1+iǫ2,n\n2−ǫ2\n2,n\n8+O(ǫ3\n2,n)/parenrightigg\n,\nsinh/parenleftbiggλ2,n\n2/parenrightbigg\n=(−1)n\n√\n2/parenleftigg\ni−ǫ2,n\n2+iǫ2\n2,n\n8+O(ǫ3\n2,n)/parenrightigg\n.\nMoreover, from ( 3.58), we get\n(3.62)\n\n1\nλ2,n=−i\n2πn+3iπ\n8π2n2+O/parenleftbig\nǫ2,nn−2/parenrightbig\n+O/parenleftbig\nn−3/parenrightbig\n,1\nλ2\n2,n=−1\n4π2n2+O/parenleftbig\nn−3/parenrightbig\n,\n1/radicalbig\nλ2,n=1−isign(n)\n2/radicalbig\nπ|n|+3(−sign(n)+i)\n16/radicalbig\nπ|n|3+O/parenleftig\nǫ2,n|n|−3/2/parenrightig\n+O/parenleftig\n|n|−5/2/parenrightig\n,\n1/radicalig\nλ3\n2,n=−1−isign(n)\n4/radicalbig\nπ3|n|3+O/parenleftig\n|n|−5/2/parenrightig\n,1/radicalig\nλ5\n2,n=O/parenleftig\n|n|−5/2/parenrightig\n.\n17STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nInserting ( 3.57), (3.61), and ( 3.62) in (3.29), we get\n(3.63)iǫ2\n2,n\n2+/parenleftigg\n1+sign(n)+i\n2/radicalbig\nπ|n|+3c2\n64πn/parenrightigg\nǫ2,n−ic2\n32πn+(sign(n)−i)c2\n64/radicalbig\nπ3|n|3+/parenleftbig\n64−i/parenleftbig\nc2−24π+16/parenrightbig/parenrightbig\nc2\n1024π2n2\n+O/parenleftig\n|n|−5/2/parenrightig\n+O/parenleftig\nǫ2,n|n|−3/2/parenrightig\n+O/parenleftig\nǫ2\n2,n|n|−1/2/parenrightig\n+O/parenleftbig\nǫ3\n2,n/parenrightbig\n= 0.\nFrom ( 3.63), we get\nǫ2,n−ic2\n32πn+O/parenleftig\nǫ2,n|n|−1/2/parenrightig\n+O/parenleftbig\nǫ2\n2,n/parenrightbig\n= 0,\nhence\n(3.64) ǫ2,n=ic2\n32πn+ξn\nn,such that lim\n|n|→+∞ξn= 0.\nInserting ( 3.64) in (3.63), we get\nξn\nn+(8+i(3π−2))c2\n128π2n2+O/parenleftig\nξn|n|−3/2/parenrightig\n+O/parenleftig\n|n|−5/2/parenrightig\n= 0,\ntherefore\n(3.65) ξn=−(8+i(3π−2))c2\n128π2n+O(n−3/2).\nInserting ( 3.64) in (3.65), we get\n(3.66) ǫ2,n=ic2\n32πn−(8+i(3π−2))c2\n128π2n2+O(n−5/2).\nFinally, inserting ( 3.66) in (3.58), we get ( 3.26).\nCase 3. Ifsin/parenleftbigc\n4/parenrightbig\n= 0, then\n(3.67) cos/parenleftigc\n2/parenrightig\n= 1,sin/parenleftigc\n2/parenrightig\n= 0.\nIn this case λ2,nbecomes\n(3.68) λ2,n= 2inπ+iπ+ǫ2,n.\nSimilar to case 2, from ( 3.68) and using the fact that lim|n|→+∞ǫ2,n= 0, we have the asymptotic expansion\n(3.69)\n\ncosh/parenleftbigg3λ2,n\n2/parenrightbigg\n=−3i(−1)nǫ2,n\n2+O/parenleftbig\nǫ3\n2,n/parenrightbig\n,sinh/parenleftbigg3λ2,n\n2/parenrightbigg\n=−i(−1)n/parenleftigg\n1+9ǫ2\n2,n\n8/parenrightigg\n+O(ǫ4\n2,n),\ncosh/parenleftbigg3λ2,n\n2/parenrightbigg\n=i(−1)nǫ2,n\n2+O/parenleftbig\nǫ3\n2,n/parenrightbig\n,sinh/parenleftbigg3λ2,n\n2/parenrightbigg\n=i(−1)n/parenleftigg\n1+ǫ2\n2,n\n8/parenrightigg\n+O(ǫ4\n2,n).\nMoreover, from ( 3.68), we get\n(3.70)\n\n1\nλ2,n=−i\n2πn+iπ\n4π2n2+O/parenleftbig\nǫ2,nn−2/parenrightbig\n+O/parenleftbig\nn−3/parenrightbig\n,1\nλ2\n2,n=−1\n4π2n2+O/parenleftbig\nn−3/parenrightbig\n,\n1/radicalbig\nλ2,n=1−isign(n)\n2/radicalbig\nπ|n|+(1+isign(n))ǫ2,n+(−sign(n)+i)π\n8/radicalbig\nπ|n|3\n+3(1−isign(n))\n64/radicalbig\nπ|n|5+O/parenleftig\nǫ2,n|n|−5/2/parenrightig\n+O/parenleftig\n|n|−7/2/parenrightig\n,\n1/radicalig\nλ3\n2,n=−1−isign(n)\n4/radicalbig\nπ3|n|3+3(sign(n)+i)\n16/radicalbig\nπ3|n|5+O/parenleftig\nǫ2,n|n|−5/2/parenrightig\n+O/parenleftig\n|n|−7/2/parenrightig\n,\n1/radicalig\nλ5\n2,n=−1+isign(n)\n8/radicalbig\nπ5|n|5+O/parenleftig\n|n|−7/2/parenrightig\n,1\nλ3\n2,n=O/parenleftbig\nn−3/parenrightbig\n.\n18STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nInserting ( 3.67), (3.69), and ( 3.70) in (3.29), we get\n(3.71)−iǫ2\n2,n+/parenleftigg\n−sign(n)+i/radicalbig\nπ|n|−3c2\n32πn+sign(n)−i+(1+isign(n))π\n4/radicalbig\nπ3|n|3/parenrightigg\nǫ2,n−(sign(n)−i)c2\n32/radicalbig\nπ3|n|3\n+ic4\n512π2n2−3(3(sign(n)+i)−(1−isign(n))π)c2\n128/radicalbig\nπ5|n|5\n+O/parenleftbig\nn−3/parenrightbig\n+O/parenleftbig\nǫ2,nn−2/parenrightbig\n+O/parenleftbig\nǫ2\n2,nn−1/parenrightbig\n+O/parenleftbig\nǫ3\n2,n/parenrightbig\n= 0.\nSimilar to case 2, solving Equation ( 3.71), we get\n(3.72) ǫ2,n=ic2\n32πn−(4+iπ)c2\n64π2n2+O/parenleftig\n|n|−5/2/parenrightig\n.\nFinally, inserting ( 3.72) in (3.68), we get ( 3.28). Thus, the proof is complete. /square\nProof of Theorem 3.2.From Proposition 3.3the operator A2has two branches of eigenvalues with eigenvalues\nadmitting real parts tending to zero. Hence, the energy corr esponding to the first and second branch of\neigenvalues has no exponential decaying. Therefore the tot al energy of the Timoshenko System ( 1.1)-(1.2)\nwith local Kelvin–Voigt damping, and with Dirichlet-Neuma nn boundary conditions ( 1.4), has no exponential\ndecaying in the equal speed case. /square\n4.Polynomial stability\nIn this section, we use the frequency domain approach method to show the polynomial stability of/parenleftbig\netAj/parenrightbig\nt≥0\nassociated with the Timoshenko System ( 2.1). We prove the following theorem.\nTheorem 4.1. Under hypothesis (H), for j= 1,2,there exists C >0such that for every U0∈D(Aj), we\nhave\n(4.1) E(t)≤C\nt/bardblU0/bardbl2\nD(Aj), t>0.\nSinceiR⊆ρ(Aj),then for the proof of Theorem 4.1, according to Theorem 2.5, we need to prove that\n(H3) sup\nλ∈R/vextenddouble/vextenddouble/vextenddouble(iλI−Aj)−1/vextenddouble/vextenddouble/vextenddouble\nL(Hj)=O/parenleftbig\nλ2/parenrightbig\n.\nWe will argue by contradiction. Therefore suppose there exi sts{(λn,Un= (un,vn,yn,zn))}n≥1⊂R×D(Aj),\nwithλn>1and\n(4.2) λn→+∞,/bardblUn/bardblHj= 1,\nsuch that\n(4.3) λ2\nn(iλnUn−AjUn) = (f1,n,f2,n,f3,n,f4,n)→0inHj.\nEquivalently, we have\niλnun−vn=λ−2\nnf1,n→0inH1\n0(0,L), (4.4)\niλnvn−k1\nρ1((un)x+yn)x=λ−2\nnf2,n→0inL2(0,L), (4.5)\niλnyn−zn=λ−2\nnf3,n→0inWj(0,L), (4.6)\niλnzn−k2\nρ2/parenleftbigg\n(yn)x+D\nk2(zn)x/parenrightbigg\nx+k1\nρ2((un)x+yn) =λ−2\nnf4,n→0inL2(0,L), (4.7)\nwhere\nWj(0,L) =/braceleftigg\nH1\n0(0,L),ifj= 1,\nH1\n∗(0,L),ifj= 2.\n19STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nIn the following, we will check the condition (H3) by finding a contradiction with ( 4.2) such as /bardblUn/bardblHj=o(1).\nFor clarity, we divide the proof into several lemmas. From no w on, for simplicity, we drop the index n. Since\nUis uniformly bounded in H,we get from ( 4.4) and ( 4.6) respectively that\n(4.8)/integraldisplayL\n0|u|2dx=O/parenleftbig\nλ−2/parenrightbig\nand/integraldisplayL\n0|y|2dx=O/parenleftbig\nλ−2/parenrightbig\n,\nLemma 4.2. Under hypothesis (H), for j= 1,2,we have\n/integraldisplayL\n0D(x)|zx|2dx=o/parenleftbig\nλ−2/parenrightbig\n,/integraldisplayβ\nα|zx|2dx=o/parenleftbig\nλ−2/parenrightbig\n, (4.9)\n/integraldisplayβ\nα|yx|2dx=o/parenleftbig\nλ−4/parenrightbig\n. (4.10)\nProof. First, taking the inner product of ( 4.3) withUinHj, then using the fact that Uis uniformly bounded\ninHj, we get\n/integraldisplayL\n0D(x)|zx|2dx=−λ−2ℜ/parenleftig/angbracketleftbig\nλ2AjU,U/angbracketrightbig\nHj/parenrightig\n=λ−2ℜ/parenleftig/angbracketleftbig\nλ2(iλU−AjU),U/angbracketrightbig\nHj/parenrightig\n=o/parenleftbig\nλ−2/parenrightbig\n,\nhence, we get the first asymptotic estimate of ( 4.9). Next, using hypothesis (H) and the first asymptotic\nestimate of ( 4.9), we get the second asymptotic estimate of ( 4.9). Finally, from ( 4.3), (4.6), and ( 4.9), we get\nthe asymptotic estimate of ( 4.10). /square\nLetg∈C1([α,β])such that\ng(β) =−g(α) = 1,max\nx∈[α,β]|g(x)|=cgandmax\nx∈[α,β]|g′(x)|=cg′,\nwherecgandcg′are strictly positive constant numbers.\nRemark 4.3. It is easy to see the existence of g(x). For example, we can take g(x) = cos/parenleftig\n(β−x)π\nβ−α/parenrightig\nto get\ng(β) =−g(α) = 1,g∈C1([α,β]),|g(x)| ≤1and|g′(x)| ≤π\nβ−α. Also, we can take\ng(x) =x2−/parenleftig\nβ+α−2 (β−α)−1/parenrightig\nx+αβ−(β+α)(β−α)−1.\n/square\nLemma 4.4. Under hypothesis (H), for j= 1,2,we have\n|z(β)|2+|z(α)|2≤/parenleftigg\nρ2λ1\n2\n2k2+2cg′/parenrightigg/integraldisplayβ\nα|z|2dx+o/parenleftig\nλ−5\n2/parenrightig\n, (4.11)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D(x)\nk2zx/parenrightbigg\n(α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D(x)\nk2zx/parenrightbigg\n(β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n≤ρ2λ3\n2\n2k2/integraldisplayβ\nα|z|2dx+o/parenleftbig\nλ−1/parenrightbig\n. (4.12)\nProof. The proof is divided into two steps.\nStep 1. In this step, we prove the asymptotic behavior estimate of ( 4.11). For this aim, first, from ( 4.6), we\nhave\n(4.13) zx=iλyx−λ−2(f3)xinL2(α,β).\nMultiplying ( 4.13) by2gzand integrating over (α,β),then taking the real part, we get\n/integraldisplayβ\nαg(x)(|z|2)xdx=ℜ/braceleftigg\n2iλ/integraldisplayβ\nαg(x)yxzdx/bracerightigg\n−ℜ/braceleftigg\n2λ−2/integraldisplayβ\nαg(x)(f4)xzdx/bracerightigg\n,\nusing by parts integration in the left hand side of above equa tion, we get\n/bracketleftbig\ng(x)|z|2/bracketrightbigβ\nα=/integraldisplayβ\nαg′(x)|z|2dx+ℜ/braceleftigg\n2iλ/integraldisplayβ\nαg(x)yxzdx/bracerightigg\n−ℜ/braceleftigg\n2λ−2/integraldisplayβ\nαg(x)(f3)xzdx/bracerightigg\n,\n20STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nconsequently,\n(4.14) |z(β)|2+|z(α)|2≤cg′/integraldisplayβ\nα|z|2dx+2λcg/integraldisplayβ\nα|yx||z|dx+2λ−2cg/integraldisplayβ\nα|(f3)x||z|dx.\nOn the other hand, we have\n2λcg|yx||z| ≤ρ2λ1\n2|z|2\n2k2+2k2λ3\n2c2\ng\nρ2|yx|2and2λ−2|(f3)x||z| ≤cg′|z|2+c2\ngλ−4\ncg′|(f3)x|2.\nInserting the above equation in ( 4.14), then using ( 4.10) and the fact that (f3)x→0inL2(α,β), we get\n|z(β)|2+|z(α)|2≤/parenleftigg\nρ2λ1\n2\n2k2+2cg′/parenrightigg/integraldisplayβ\nα|z|2dx+o/parenleftig\nλ−5\n2/parenrightig\n,\nhence, we get ( 4.11).\nStep 2. In this step, we prove the following asymptotic behavior est imate of ( 4.12). For this aim, first,\nmultiplying ( 4.7) by−2ρ2\nk2g/parenleftig\nyx+D(x)\nk2zx/parenrightig\nand integrating over (α,β),then taking the real part, we get\n/integraldisplayβ\nαg(x)/parenleftigg/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D(x)\nk2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightigg\nxdx=2ρ2λ\nk2ℜ/braceleftigg\ni/integraldisplayβ\nαg(x)z/parenleftbigg\nyx+D(x)\nk2zx/parenrightbigg\ndx/bracerightigg\n+2k1\nk2ℜ/braceleftigg/integraldisplayβ\nαg(x)(ux+y)/parenleftbigg\nyx+D(x)\nk2zx/parenrightbigg\ndx/bracerightigg\n−2ρ2λ−2\nk2ℜ/braceleftigg/integraldisplayβ\nαg(x)f4/parenleftbigg\nyx+D(x)\nk2zx/parenrightbigg\ndx/bracerightigg\n,\nusing by parts integration in the left hand side of above equa tion, we get\n/bracketleftigg\ng(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D(x)\nk2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/bracketrightiggβ\nα=/integraldisplayβ\nαg′(x)/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D(x)\nk2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndx+2ρ2λ\nk2ℜ/braceleftigg\ni/integraldisplayβ\nαg(x)z/parenleftbigg\nyx+D(x)\nk2zx/parenrightbigg\ndx/bracerightigg\n+2k1\nk2ℜ/braceleftigg/integraldisplayβ\nαg(x)(ux+y)/parenleftbigg\nyx+D(x)\nk2zx/parenrightbigg\ndx/bracerightigg\n−2ρ2λ−2\nk2ℜ/braceleftigg/integraldisplayβ\nαg(x)f4/parenleftbigg\nyx+D(x)\nk2zx/parenrightbigg\ndx/bracerightigg\n,\nconsequently,\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D(x)\nk2zx/parenrightbigg\n(α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D(x)\nk2zx/parenrightbigg\n(β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n≤2ρ2cgλ\nk2/integraldisplayβ\nα|z|/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D(x)\nk2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingledx\ncg′/integraldisplayβ\nα/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D(x)\nk2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndx+2k1cg\nk2/integraldisplayβ\nα|ux+y|/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D(x)\nk2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingledx+2ρ2cgλ−2\nk2/integraldisplayβ\nα|f4|/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D(x)\nk2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingledx.\nNow, using Cauchy Schwarz inequality, Equations ( 4.9)-(4.10), the fact that f5→0inL2(α,β)and the fact\nthatux+yis uniformly bounded in L2(α,β)in the right hand side of above equation, we get\n(4.15)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D(x)\nk2zx/parenrightbigg\n(α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D(x)\nk2zx/parenrightbigg\n(β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n≤2ρ2cgλ\nk2/integraldisplayβ\nα|z|/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D(x)\nk2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingledx+o/parenleftbig\nλ−1/parenrightbig\n.\nOn the other hand, we have\n2ρ2cgλ\nk2|z|/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D(x)\nk2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤ρ2λ3\n2\n2k2|z|2+2ρ2λ1\n2c2\ng\nk2/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D(x)\nk2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n.\nInserting the above equation in ( 4.15), then using Equations ( 4.9)-(4.10), we get\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D(x)\nk2zx/parenrightbigg\n(α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D(x)\nk2zx/parenrightbigg\n(β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n≤ρ2λ3\n2\n2k2/integraldisplayβ\nα|z|2dx+o/parenleftbig\nλ−1/parenrightbig\n,\nhence, we get ( 4.12). Thus, the proof is complete. /square\n21STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nLemma 4.5. Under hypothesis (H), for j= 1,2,we have\n|ux(α)+y(α)|2=O(1),|ux(β)+y(β)|2=O(1). (4.16)\n|u(α)|2=O/parenleftbig\nλ−2/parenrightbig\n,|u(β)|2=O/parenleftbig\nλ−2/parenrightbig\n, (4.17)\n|v(α)|2=O(1),|v(β)|2=O(1). (4.18)\nProof. Multiplying Equation ( 4.5) by−2ρ1\nk1g(ux+y)and integrating over (α,β),then taking the real part\nand using the fact that ux+yis uniformly bounded in L2(α,β),f2→0inL2(α,β), we get\n(4.19)/integraldisplayβ\nαg(x)/parenleftig\n|ux+y|2/parenrightig\nxdx−2ρ1λ\nk1ℜ/braceleftigg\ni/integraldisplayβ\nαg(x)uxvdx/bracerightigg\n=2ρ1λ\nk1ℜ/braceleftigg\ni/integraldisplayβ\nαg(x)yvdx/bracerightigg\n+o/parenleftbig\nλ−2/parenrightbig\n.\nNow, we divided the proof into two steps.\nStep 1. In this step, we prove the asymptotic behavior estimates of ( 4.16)-(4.17). First, from ( 4.4), we have\n−iλv=λ2u+iλ−1f1.\nInserting the above equation in the second term in left of ( 4.19), then using the fact that uxis uniformly\nbounded in L2(α,β)andf1→0inL2(α,β), we get\n/integraldisplayβ\nαg(x)/parenleftig\n|ux+y|2/parenrightig\nxdx+ρ1λ2\nk1/integraldisplayβ\nαg(x)/parenleftig\n|u|2/parenrightig\nxdx=−2ρ1λ2\nk1ℜ/braceleftigg/integraldisplayβ\nαg(x)uydx/bracerightigg\n+o/parenleftbig\nλ−1/parenrightbig\n.\nUsing by parts integration and the fact that g(β) =−g(α) = 1 in the above equation, we get\n|ux(β)+y(β)|2+ρ1λ2\nk1|u(β)|2+|ux(α)+y(α)|2+ρ1λ2\nk1|u(α)|2=/integraldisplayβ\nαg′(x)|ux+y|2dx\n+ρ1λ2\nk1/integraldisplayβ\nαg′(x)|u|2dx−2ρ1λ2\nk1ℜ/braceleftigg/integraldisplayβ\nαg(x)uydx/bracerightigg\n+o/parenleftbig\nλ−1/parenrightbig\n,\nconsequently,\n|ux(β)+y(β)|2+ρ1λ2\nk1|u(β)|2+|ux(α)+y(α)|2+ρ1λ2\nk1|u(α)|2≤cg′/integraldisplayβ\nα|ux+y|2dx\n+ρ1cg′λ2\nk1/integraldisplayβ\nα|u|2dx+2ρ1cgλ2\nk1/integraldisplayβ\nα|u||y|dx+o/parenleftbig\nλ−1/parenrightbig\n.\nNext, since λu, λy andux+yare uniformly bounded, then from the above equation, we get ( 4.16)-(4.17).\nStep 2. In this step, we prove the asymptotic behavior estimates of ( 4.18). First, from ( 4.4), we have\n−iλux=vx−λ−2(f1)x.\nInserting the above equation in the second term in left of ( 4.19), then using the fact that vis uniformly bounded\ninL2(α,β)and(f1)x→0inL2(α,β), we get\n/integraldisplayβ\nαg(x)/parenleftig\n|ux+y|2/parenrightig\nxdx+ρ1\nk1/integraldisplayβ\nαg(x)/parenleftig\n|v|2/parenrightig\nxdx=−2ρ1λ2\nk1ℜ/braceleftigg/integraldisplayβ\nαg(x)uydx/bracerightigg\n+o/parenleftbig\nλ−1/parenrightbig\n.\nSimilar to step 1, by using by parts integration and the fact t hatg(β) =−g(α) = 1 in the above equation,\nthen using the fact that v, λu, λy andux+yare uniformly bounded in L2(α,β), we get ( 4.18). Thus, the\nproof is complete. /square\nLemma 4.6. Under hypothesis (H), for j= 1,2,forλlarge enough, we have\n/integraldisplayβ\nα|z|2dx=o/parenleftig\nλ−5\n2/parenrightig\n,/integraldisplayβ\nα|y|2dx=o/parenleftig\nλ−9\n2/parenrightig\n, (4.20)\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D(x)\nk2zx/parenrightbigg\n(α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n=o/parenleftbig\nλ−1/parenrightbig\n,/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D(x)\nk2zx/parenrightbigg\n(β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n=o/parenleftbig\nλ−1/parenrightbig\n. (4.21)\n22STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nProof. The proof is divided into two steps.\nStep 1. In this step, we prove the following asymptotic behavior est imate\n(4.22)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleik1\nρ2λ/integraldisplayβ\nα(ux+y)zdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/parenleftbigg1\n4+k2cg′\nρ2λ1\n2+k1\nρ2λ2/parenrightbigg/integraldisplayβ\nα|z|2dx+o/parenleftbig\nλ−3/parenrightbig\n.\nFor this aim, first, we have\n(4.23)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleik1\nρ2λ/integraldisplayβ\nα(ux+y)zdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleik1\nρ2λ/integraldisplayβ\nαyzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleik1\nρ2λ/integraldisplayβ\nαuxzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nNow, from ( 4.6) and using the fact that f3→0inL2(α,β)andzis uniformly bounded in L2(α,β), we get\n(4.24)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleik1\nρ2λ/integraldisplayβ\nαyzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤k1\nρ2λ2/integraldisplayβ\nα|z|2dx+o/parenleftbig\nλ−4/parenrightbig\n.\nNext, by using by parts integration, we get/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleik1\nρ2λ/integraldisplayβ\nαuxzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle−ik1\nρ2λ/integraldisplayβ\nαuzxdx+ik1\nρ2λu(β)z(β)−ik1\nρ2λu(α)z(α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle,\nconsequently,\n(4.25)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleik1\nρ2λ/integraldisplayβ\nαuxzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤k1\nρ2λ/integraldisplayβ\nα|u||zx|dx+k1\nρ2λ(|u(β)||z(β)|+|u(α)||z(α)|),\nOn the other hand, we have\nk1\nρ2λ(|u(β)||z(β)|+|u(α)||z(α)|)≤k2\n1\n2k2ρ2λ3\n2/parenleftig\n|u(α)|2+|u(β)|2/parenrightig\n+k2\n2ρ2λ1\n2/parenleftig\n|z(α)|2+|z(β)|2/parenrightig\n.\nInserting ( 4.11) and ( 4.17) in the above equation, we get\nk1\nρ2λ(|u(β)||z(β)|+|u(α)||z(α)|)≤/parenleftbigg1\n4+k2cg′\nρ2λ1\n2/parenrightbigg/integraldisplayβ\nα|z|2dx+o/parenleftbig\nλ−3/parenrightbig\n.\nInserting the above equation in ( 4.25), then using ( 4.9) and the fact that λuis bounded in L2(α,β), we get\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleik1\nρ2λ/integraldisplayβ\nαuxzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/parenleftbigg1\n4+k2cg′\nρ2λ1\n2/parenrightbigg/integraldisplayβ\nα|z|2dx+o/parenleftbig\nλ−3/parenrightbig\n.\nFinally, inserting the above equation and Equation ( 4.24) in (4.23), we get ( 4.22).\nStep 2. In this step, we prove the asymptotic behavior estimates of ( 4.20)-(4.21). For this aim, first, multiplying\n(4.7) by−iλ−1ρ−1\n2zand integrating over (α,β),then taking the real part, we get\n/integraldisplayβ\nα|z|2dx=−k2\nρ2λℜ/braceleftigg\ni/integraldisplayβ\nα/parenleftbigg\nyx+D\nk2zx/parenrightbigg\nxzdx/bracerightigg\n+k1\nρ2λℜ/braceleftigg\ni/integraldisplayβ\nα(ux+y)zdx/bracerightigg\n−λ−3ℜ/braceleftigg\ni/integraldisplayβ\nαf4zdx/bracerightigg\n,\nconsequently,\n(4.26)/integraldisplayβ\nα|z|2dx≤k2\nρ2λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayβ\nα/parenleftbigg\nyx+D\nk2zx/parenrightbigg\nxzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleik1\nρ2λ/integraldisplayβ\nα(ux+y)zdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+λ−3/integraldisplayβ\nα|f4||z|dx.\nFrom the fact that zis uniformly bounded in L2(α,β)andf5→0inL2(α,β), we get\n(4.27) λ−3/integraldisplayβ\nα|f4||z|dx=o/parenleftbig\nλ−3/parenrightbig\n.\nInserting ( 4.22) and ( 4.27) in (4.26), we get\n(4.28)/integraldisplayβ\nα|z|2dx≤k2\nρ2λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayβ\nα/parenleftbigg\nyx+D\nk2zx/parenrightbigg\nxzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle+/parenleftbigg1\n4+k2cg′\nρ2λ1\n2+k1\nρ2λ2/parenrightbigg/integraldisplayβ\nα|z|2dx+o/parenleftbig\nλ−3/parenrightbig\n.\n23STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nNow, using by parts integration and ( 4.9)-(4.10), we get\n(4.29)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayβ\nα/parenleftbigg\nyx+D\nk2zx/parenrightbigg\nxzdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftbigg/parenleftbigg\nyx+D\nk2zx/parenrightbigg\nz/bracketrightbiggβ\nα−/integraldisplayβ\nα/parenleftbigg\nyx+D\nk2zx/parenrightbigg\nzxdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D\nk2zx/parenrightbigg\n(β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle|z(β)|+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D\nk2zx/parenrightbigg\n(α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle|z(α)|+/integraldisplayβ\nα/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D\nk2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle|zx|dx\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D\nk2zx/parenrightbigg\n(β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle|z(β)|+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D\nk2zx/parenrightbigg\n(α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle|z(α)|+o/parenleftbig\nλ−2/parenrightbig\n.\nInserting ( 4.29) in (4.28), we get\n(4.30)/parenleftbigg3\n4−k2cg′\nρ2λ1\n2−k1\nρ2λ2/parenrightbigg/integraldisplayβ\nα|z|2dx\n≤k2\nρ2λ/parenleftbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D\nk2zx/parenrightbigg\n(β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle|z(β)|+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D\nk2zx/parenrightbigg\n(α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle|z(α)|/parenrightbigg\n+o/parenleftbig\nλ−3/parenrightbig\n.\nNow, forζ=βorζ=α, we have\nk2\nρ2λ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D\nk2zx/parenrightbigg\n(ζ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle|z(ζ)| ≤k2λ−1\n2\n2ρ2|z(ζ)|2+k2λ−3\n2\n2ρ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D\nk2zx/parenrightbigg\n(ζ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n.\nInserting the above equation in ( 4.30), we get\n/parenleftbigg3\n4−k2cg′\nρ2λ1\n2−k1\nρ2λ2/parenrightbigg/integraldisplayβ\nα|z|2dx≤k2λ−3\n2\n2ρ2/parenleftigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D\nk2zx/parenrightbigg\n(α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D\nk2zx/parenrightbigg\n(β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightigg\n+k2λ−1\n2\n2ρ2/parenleftbig\n|z(α)|2+|z(β)|2/parenrightbig\n+o/parenleftbig\nλ−3/parenrightbig\n.\nInserting Equations ( 4.11) and ( 4.12) in the above inequality, we obtain\n/parenleftbigg3\n4−k2cg′\nρ2λ1\n2−k1\nρ2λ2/parenrightbigg/integraldisplayβ\nα|z|2dx≤/parenleftbigg1\n2+k2cg′\nρ2λ1\n2/parenrightbigg/integraldisplayβ\nα|z|2dx+o/parenleftig\nλ−5\n2/parenrightig\n,\nconsequently,/parenleftbigg1\n4−2k2cg′\nρ2λ1\n2−k1\nρ2λ2/parenrightbigg/integraldisplayβ\nα|z|2dx≤o/parenleftig\nλ−5\n2/parenrightig\n,\nsinceλ→+∞, forλlarge enough, we get\n0</parenleftbigg1\n4−2k2cg′\nρ2λ1\n2−k1\nρ2λ2/parenrightbigg/integraldisplayβ\nα|z|2dx≤o/parenleftig\nλ−5\n2/parenrightig\n,\nhence, we get the first asymptotic estimate of ( 4.20). Then, inserting the first asymptotic estimate of ( 4.20) in\n(4.6), we get the second asymptotic estimate of ( 4.20). Finally, inserting ( 4.20) in (4.12), we get ( 4.21). Thus,\nthe proof is complete. /square\nLemma 4.7. Under hypothesis (H), for j= 1,2,forλlarge enough, we have\n(4.31)/integraldisplayβ\nα|ux|2dx=o(1)and/integraldisplayβ\nα|v|2dx=o(1).\nProof. The proof is divided into two steps.\nStep 1. In this step, we prove the first asymptotic behavior estimate of (4.31). First, multiplying Equation\n(4.7) byρ2\nk1(ux+y)and integrating over (α,β), we get\n/integraldisplayβ\nα|ux+y|2dx−k2\nk1/integraldisplayβ\nα/parenleftbigg\nyx+D\nk2zx/parenrightbigg\nx(ux+y)dx=−iρ2λ\nk1/integraldisplayβ\nαz(ux+y)dx+ρ2\nk1λ2/integraldisplayβ\nαf4(ux+y)dx,\n24STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nusing by parts integration in the second term in the left hand side of above equation, we get\n(4.32)/integraldisplayβ\nα|ux+y|2dx+k2\nk1/integraldisplayβ\nα/parenleftbigg\nyx+D\nk2zx/parenrightbigg\n(ux+y)xdx=k2\nk1/bracketleftbigg/parenleftbigg\nyx+D\nk2zx/parenrightbigg\n(ux+y)/bracketrightbiggβ\nα\n−iρ2λ\nk1/integraldisplayβ\nαz(ux+y)dx+ρ2\nk1λ2/integraldisplayβ\nαf4(ux+y)dx.\nNext, multiplying Equation ( 4.5) byρ1k2\nk2\n1/parenleftig\nyx+D\nk2zx/parenrightig\nand integrating over (α,β), then using the fact that\nf2→0inL2(0,L)and Equations ( 4.9)-(4.10), we get\n−k2\nk1/integraldisplayβ\nα/parenleftbigg\nyx+D\nk2zx/parenrightbigg\n(ux+y)xdx=−iρ1k2λ\nk2\n1/integraldisplayβ\nαv/parenleftbigg\nyx+D\nk2zx/parenrightbigg\ndx+ρ1k2\nk2\n1λ2/integraldisplayβ\nαf2/parenleftbigg\nyx+D\nk2zx/parenrightbigg\ndx,\nconsequently,\n(4.33)−k2\nk1/integraldisplayβ\nα/parenleftbigg\nyx+D\nk2zx/parenrightbigg\n(ux+y)xdx=iρ1k2λ\nk2\n1/integraldisplayβ\nαv/parenleftbigg\nyx+D\nk2zx/parenrightbigg\ndx+ρ1k2\nk2\n1λ2/integraldisplayβ\nαf2/parenleftbigg\nyx+D\nk2zx/parenrightbigg\ndx.\nAdding ( 4.32) and ( 4.33), we obtain\n/integraldisplayβ\nα|ux+y|2dx=−iρ2λ\nk1/integraldisplayβ\nαz(ux+y)dx+k2\nk1/bracketleftbigg/parenleftbigg\nyx+D\nk2zx/parenrightbigg\n(ux+y)/bracketrightbiggβ\nα\n+iρ1k2λ\nk2\n1/integraldisplayβ\nαv/parenleftbigg\nyx+D\nk2zx/parenrightbigg\ndx+ρ2\nk1λ2/integraldisplayβ\nαf4(ux+y)dx+ρ1k2\nk2\n1λ2/integraldisplayβ\nαf2/parenleftbigg\nyx+D\nk2zx/parenrightbigg\ndx,\ntherefore\n(4.34)/integraldisplayβ\nα|ux+y|2dx≤ρ2λ\nk1/integraldisplayβ\nα|z||ux+y|dx+k2\nk1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D\nk2zx/parenrightbigg\n(β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle|ux(β)+y(β)|\n+k2\nk1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D\nk2zx/parenrightbigg\n(α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle|ux(α)+y(α)|+ρ1k2λ\nk2\n1/integraldisplayβ\nα|v|/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D\nk2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingledx\n+ρ2\nk1λ2/integraldisplayβ\nα|f4||ux+y|dx+ρ1k2\nk2\n1λ2/integraldisplayβ\nα|f2|/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D\nk2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingledx.\nFrom ( 4.3), (4.9), (4.10), (4.16), (4.20), (4.21) and the fact that v, ux+yare uniformly bounded in L2(α,β),\nwe obtain\n\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D\nk2zx/parenrightbigg\n(β)/vextendsingle/vextendsingle/vextendsingle/vextendsingle|ux(β)+y(β)|=o/parenleftig\nλ−1\n2/parenrightig\n,/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg\nyx+D\nk2zx/parenrightbigg\n(α)/vextendsingle/vextendsingle/vextendsingle/vextendsingle|ux(α)+y(α)|=o/parenleftig\nλ−1\n2/parenrightig\n,\nλ/integraldisplayβ\nα|z||ux+y|dx=o/parenleftig\nλ−1\n4/parenrightig\n, λ/integraldisplayβ\nα|v|/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D\nk2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingledx=o(1),\nλ−2/integraldisplayβ\nα|f4||ux+y|dx=o/parenleftbig\nλ−2/parenrightbig\n, λ−2/integraldisplayβ\nα|f2|/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D\nk2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingledx=o/parenleftbig\nλ−3/parenrightbig\n.\nInserting the above equation in ( 4.34), we get\n/integraldisplayβ\nα|ux+y|2dx=o(1).\nFrom the above equation and ( 4.20), we get the first asymptotic estimate of ( 4.31).\nStep 2. In this step, we prove the second asymptotic behavior estima te of ( 4.31). Multiplying ( 4.5) by−iλ−1v\nand integrating over (α,β),then taking the real part, we get\n/integraldisplayβ\nα|v|2dx=−k1\nρ1λℜ/braceleftigg\ni/integraldisplayβ\nα(ux+y)xvdx/bracerightigg\n−λ−3ℜ/braceleftigg\ni/integraldisplayβ\nαf2vdx/bracerightigg\n,\nusing by parts integration in the second term in the right han d side of above equation, we get\n/integraldisplayβ\nα|v|2dx=k1\nρ1λℜ/braceleftigg\ni/integraldisplayβ\nα(ux+y)vxdx/bracerightigg\n−k1\nρ1λℜ/braceleftig\ni[(ux+y)v]β\nα/bracerightig\n−λ−3ℜ/braceleftigg\ni/integraldisplayβ\nαf2vdx/bracerightigg\n.\n25STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nconsequently,\n(4.35)/integraldisplayβ\nα|v|2dx≤k1\nρ1λ/integraldisplayβ\nα|ux+y||vx|dx+k1\nρ1λ(|ux(β)+y(β)||v(β)|+|ux(α)+y(α)|v(α)|)\n+λ−3/integraldisplayβ\nα|f2||v|dx.\nFinally, from ( 4.16), (4.18), (4.20), the first asymptotic behavior estimate of ( 4.31), the fact that λ−1vx, vare\nuniformly bounded in L2(α,β)and the fact that f2→0inL2(α,β), we get the second asymptotic behavior\nestimate of ( 4.20). Thus, the proof is complete. /square\nFrom what precedes, under hypothesis (H), for j= 1,2,from Lemmas 4.5,4.6and4.7, we deduce that\n(4.36) /bardblU/bardblHj=o(1),over(α,β).\nLemma 4.8. Under hypothesis (H), for j= 1,2,we have\n/bardblU/bardblHj=o(1),over(0,L).\nProof. Letφ∈H1\n0(0,L)be a given function. We proceed the proof in two steps.\nStep 1. Multiplying Equation ( 4.5) by2ρ1φuxand integrating over (α,β),then using the fact that uxis\nbounded in L2(0,L),f2→0inL2(0,L), and use Dirichlet boundary conditions to get\n(4.37) ℜ/braceleftigg\n2iρ1λ/integraldisplayL\n0φvuxdx/bracerightigg\n+k1/integraldisplayL\n0φ′|ux|2dx−ℜ/braceleftigg\n2k1/integraldisplayL\n0φuxyxdx/bracerightigg\n=o(λ−2).\nFrom ( 4.4), we have\niλux=−vx−λ−2(f1)x.\nInserting the above equation in ( 4.37), then using the fact that (f1)x→0inL2(0,L)and the fact that vis\nbounded in L2(0,L), we get\n(4.38) ρ1/integraldisplayL\n0φ′|v|2dx+k1/integraldisplayL\n0φ′|ux|2dx−ℜ/braceleftigg\n2k1/integraldisplayL\n0φuxyxdx/bracerightigg\n=o(λ−2).\nSimilarly, multiplying Equation ( 4.7) by2ρ2φ/parenleftig\nyx+D\nk1zx/parenrightig\nand integrating over (α,β),then using by parts\nintegration and Dirichlet boundary conditions to get\n(4.39)ℜ/braceleftigg\n2iρ2λ/integraldisplayL\n0φzyxdx/bracerightigg\n+k2/integraldisplayL\n0φ′/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D\nk2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndx+ℜ/braceleftigg\n2k1/integraldisplayL\n0φyxuxdx/bracerightigg\n=−λ−1ℜ/braceleftigg\n2k1/integraldisplayL\n0φλyyxdx/bracerightigg\n−ℜ/braceleftigg\n2iρ2\nk1λ/integraldisplayL\n0D(x)φzzxdx/bracerightigg\n−ℜ/braceleftigg\n2/integraldisplayL\n0D(x)φzxuxdx/bracerightigg\n−ℜ/braceleftigg\n2/integraldisplayL\n0D(x)φzxydx/bracerightigg\n+ℜ/braceleftigg\n2ρ2λ−2/integraldisplayL\n0φf3yxdx/bracerightigg\n+ℜ/braceleftigg\n2ρ2\nk1λ−2/integraldisplayL\n0D(x)φf3zxdx/bracerightigg\n.\nFor all bounded h∈L2(0,L), using Cauchy-Schwarz inequality, the first estimation of ( 4.9), and the fact that\nD∈L∞(0,L), to obtain\n(4.40) ℜ/braceleftigg/integraldisplayL\n0D(x)hzxdx/bracerightigg\n≤/parenleftigg\nsup\nx∈(0,L)D1/2(x)/parenrightigg/parenleftigg/integraldisplayL\n0D(x)|zx|2dx/parenrightigg1/2/parenleftigg/integraldisplayL\n0|h|2dx/parenrightigg1/2\n=o(λ−1).\nFrom ( 4.39) and using ( 4.40), the fact that z, λy, y xare bounded in L2(0,L), the fact that f3→0inL2(0,L),\nwe get\n(4.41) ℜ/braceleftigg\n2iρ2λ/integraldisplayL\n0φzyxdx/bracerightigg\n+k2/integraldisplayL\n0φ′/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D\nk2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndx+ℜ/braceleftigg\n2k1/integraldisplayL\n0φyxuxdx/bracerightigg\n=o(1).\nOn the other hand, from ( 4.6), we have\niλyx=−zx−λ−2(f3)x.\n26STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nInserting the above equation in ( 4.41), then using the fact that (f3)x→0inL2(0,L)and the fact that zis\nbounded in L2(0,L), we get\n(4.42) ρ2/integraldisplayL\n0φ′|z|2dx+k2/integraldisplayL\n0φ′/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D\nk2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndx+ℜ/braceleftigg\n2k1/integraldisplayL\n0φyxuxdx/bracerightigg\n=o(1).\nAdding ( 4.38) and ( 4.42), we get\n(4.43)/integraldisplayL\n0φ′/parenleftigg\nρ1|v|2+ρ2|z|2+k1|ux|2+k2/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D\nk2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightigg\ndx=o(1).\nStep 2. Letǫ>0such thatα+ǫ<β and define the cut-off function ς1inC1([0,L])by\n0≤ς1≤1, ς1= 1on[0,α]andς1= 0on[α+ǫ,L].\nTakeφ=xς1in (4.43), then use the fact that /bardblU/bardblHj=o(1)on(α,β)(i.e., ( 4.36)), the fact that α<α+ǫ<β,\nand (4.9)-(4.10), we get\n(4.44)/integraldisplayα\n0/parenleftigg\nρ1|v|2+ρ2|z|2+k1|ux|2+k2/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D\nk2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightigg\ndx=o(1).\nMoreover, using Cauchy-Schwarz inequality, the first estim ation of ( 4.9), the fact that D∈L∞(0,L), and\n(4.44), we get\n(4.45)/integraldisplayα\n0|yx|2dx≤2/integraldisplayα\n0/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D\nk2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndx+2\nk2\n2/integraldisplayα\n0D(x)2|zx|2dx,\n≤2/integraldisplayα\n0/vextendsingle/vextendsingle/vextendsingle/vextendsingleyx+D\nk2zx/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndx+2/parenleftig\nsupx∈(0,α)D(x)/parenrightig\nk2\n2/integraldisplayα\n0D(x)|zx|2dx,\n=o(1).\nUsing ( 4.44) and ( 4.45), we get\n/bardblU/bardblHj=o(1)on(0,α).\nSimilarly, by symmetry, we can prove that /bardblU/bardblHj=o(1)on(β,L)and therefore\n/bardblU/bardblHj=o(1)on(0,L).\nThus, the proof is complete. /square\nProof of Theorem 4.1.Under hypothesis (H), for j= 1,2,from Lemma 4.8, we have /bardblU/bardblHj=o(1),over\n(0,L), which contradicts ( 4.2). This implies that\nsup\nλ∈R/vextenddouble/vextenddouble/vextenddouble(iλId−Aj)−1/vextenddouble/vextenddouble/vextenddouble\nL(Hj)=O/parenleftbig\nλ2/parenrightbig\n.\nThe result follows from Theorem 2.5part (i). /square\nIt is very important to ask the question about the optimality of (4.1). For the optimality of ( 4.1), we first\nrecall Theorem 3.4.1 stated in [ 31].\nTheorem 4.9. LetA:D(A)⊂H→Hgenerate a C 0−semigroup of contractions/parenleftbig\netA/parenrightbig\nt≥0onH. Assume\nthatiR∈ρ(A). Let(λk,n)1≤k≤k0, n≥1denote the k-th branch of eigenvalues of Aand(ek,n)1≤k≤k0, n≥1the\nsystem of normalized associated eigenvectors. Assume that for each 1≤k≤k0there exist a positive sequence\nµk,n→ ∞ asn→ ∞ and two positive constant αk>0,βk>0such that\n(4.46) ℜ(λk,n)∼ −βk\nµαk\nk,nandℑ(λk,n)∼µk,nasn→ ∞.\nHereℑis used to denote the imaginary part of a complex number. Furt hermore, assume that for u0∈D(A),\nthere exists constant M >0independent of u0such that\n(4.47)/vextenddouble/vextenddoubleetAu0/vextenddouble/vextenddouble2\nH≤M\nt2\nℓk/bardblu0/bardbl2\nD(A), ℓk= max\n1≤k≤k0αk,∀t>0.\n27STABILIZATION OF THE TIMOSHENKO SYSTEM WITH KELVIN-VOIGT D AMPING\nThen the decay rate ( 4.47) is optimal in the sense that for any ǫ>0we cannot expect the energy decay rate\nt−2\nℓk−ǫ. /square\nIn the next corollary, we show that the optimality of ( 4.1) in some cases.\nCorollary 4.10. For everyU0∈D(A2), we have the following two cases:\n1. If condition ( 3.1) holds, then the energy decay rate in ( 4.1) is optimal.\n2. If condition ( 3.4) holds and if there exists κ1∈Nsuch thatc=/radicalig\nk1\nk2= 2κ1π, then the energy decay\nrate in ( 4.1) is optimal.\nProof. We distinguish two cases:\n1. If condition ( 3.1) holds, then from Theorem 3.1, forǫ>0(small enough ), we cannot expect the energy\ndecay ratet−2\n2−ǫfor all initial data U0∈D(A2)and for allt>0.Hence the energy decay rate in ( 4.1)\nis optimal.\n2. If condition ( 3.4) holds, first following Theorem 4.1, for all initial data U0∈D(A2)and for all t>0,\nwe get ( 4.47) withℓk= 2. Furthermore, from Proposition 3.3(case 2 and case 3), we remark that:\nCase 1. If there exists κ0∈Nsuch thatc= 2(2κ0+1)π, we have\n\n\nℜ(λ1,n)∼ −1\nπ1/2|n|1/2,ℑ(λ1,n)∼2nπ,\nℜ(λ2,n)∼ −c2\n16π2n2,ℑ(λ2,n)∼/parenleftbigg\n2n+3\n2/parenrightbigg\nπ,\nthen ( 4.46) holds with α1=1\n2andα2= 2. Therefore, ℓk= 2 = max( α1,α2).Then, applying Theorem\n4.9, we get that the energy decay rate in ( 4.1) is optimal.\nCase 2. If there exists κ1∈Nsuch thatc= 4κ1π, we have\n\nℜ(λ1,n)∼ −c2\n16π2n2,ℑ(λ1,n)∼2nπ,\nℜ(λ2,n)∼ −c2\n16π2n2,ℑ(λ2,n)∼(2n+1)π,\nthen ( 4.46) holds with α1= 2andα2= 2. 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We consider the inverse hyperbolic problem of recovering all spatial\ndependent coefficients, which are the wave speed, the damping coe fficient, po-\ntential coefficient and gradient coefficient, in a second-order hype rbolic equation\ndefined on an open bounded domain with smooth enough boundary. W e show\nthat by appropriately selecting finite pairs of initial conditions we can uniquely\nand Lipschitz stably recover all those coefficients from the corres ponding bound-\nary measurements of their solutions. The proofs are based on sha rp Carleman\nestimate, continuous observability inequality and regularity theory for general\nsecond-order hyperbolic equations.\nKeywords : Inverse hyperbolic problem, finite sets of measurements, Carlem an\nestimates, uniqueness and stability\n2010 Mathematics Subject Classifications : 35R30; 35L10\n1.Introduction and Main Results\nLet Ω⊂Rn,n≥2, be an open bounded domain with smooth enough (e.g., C2)\nboundary Γ = ∂Ω =Γ0∪Γ1, where Γ 0∩Γ1=∅. We refer Γ 1as the observed\npart of the boundary where the measurements are taken, and Γ 0as the unobserved\npart of the boundary. We consider the following general second-o rder hyperbolic\nequation for w=w(x,t) defined on Q= Ω×[−T,T], along with initial conditions\n{w0,w1}and Dirichlet boundary condition hon Σ = Γ ×[−T,T] that are given in\nappropriate function spaces:\n(1)\n\nwtt−c2(x)∆w+q1(x)wt+q0(x)w+q(x)·∇w= 0 inQ\nw(x,0) =w0(x);wt(x,0) =w1(x) in Ω\nw(x,t) =h(x,t) in Σ .\nHere the wave speed c(x) satisfies\nc∈C={c∈C1(Ω) :c−1\n0≤c(x)≤c0,for some c0>0}\nDate: October 11, 2022.\n12 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS\nq1∈L∞(Ω),q0∈L∞(Ω), and q∈(L∞(Ω))nare the damping, potential, and\ngradient coefficients, respectively.\nWe then consider the following inverse problem for the system (1): R ecover all\ntogether the wave speed c(x), the damping coefficient q1(x), the potential coefficient\nq0(x), and the gradient coefficient q(x) from measurements of Neumann boundary\ntraces of the solution w=w(w0,w1,h,c,q 1,q0,q) over the observed part Γ 1of\nthe boundary and over the time interval [ −T,T]. Of course here T >0 should\nbe sufficiently large due to the finite propagation speed of the syste m (1). In\naddition, to make the observed part Γ 1of the boundary more precise, in this paper\nwe assume the following standard geometrical assumptions on the d omain Ω and\nthe unobserved part of the boundary Γ 0:\n(A.1) There exists a strictly convex function d:Ω→Rin the metric g=\nc−2(x)dx2, and of class C3(Ω), such that the following two properties hold true\n(through translation and rescaling if necessary):\n(i) The normal derivative of don the unobserved part Γ 0of the boundary is\nnon-positive. Namely,\n∂d\n∂ν=/a\\}⌊∇a⌋ketle{tDd(x),ν(x)/a\\}⌊∇a⌋ket∇i}ht ≤0,∀x∈Γ0,\nwhereDd=∇gdis the gradient vector field on Ω with respect to g.\n(ii)\nD2d(X,X) =/a\\}⌊∇a⌋ketle{tDX(Dd),X/a\\}⌊∇a⌋ket∇i}htg≥2|X|2\ng,∀X∈Mx,min\nx∈Ωd(x) =m0>0\nwhereD2dis the Hessian of d(a second-order tensor) and Mxis the tangent space\natx∈Ω.\n(A.2)d(x) has no critical point on Ω. In other words,\ninf\nx∈Ω|Dd|>0,so that we may take inf\nx∈Ω|Dd|2\nd>4.\nRemark 1.1. The geometrical assumptions above permit the construction of a v ec-\ntor field that enables a pseudo-convex function necessary for allo wing a Carleman\nestimate containing no lower-order terms for the general second -order equation (1)\n(see Section 2). These assumptions are first formulated in [16] und er the framework\nof a Euclidean metric, with [22] employing them under the more genera l Riemann-\nian framework. For examples and detailed illustrations of large gener al classes of\ndomains {Ω,Γ1,Γ0}satisfying the aforementioned assumptions we refer to [22, Ap-\npendix B]. One canonical example is to take d(x) =|x−x0|2, withx0being a point\noutsideΩ, if the wave speed csatisfies/vextendsingle/vextendsingle/vextendsingle∇c(x)·(x−x0)\n2c(x)/vextendsingle/vextendsingle/vextendsingle≤rc<1 for some rc∈(0,1).\nThe classical inverse hyperbolic problems usually involve recovering a single un-\nknown coefficient, typically the damping coefficient orthe potential c oefficient, fromRECOVER ALL COEFFICIENTS 3\nasingleboundary measurement of the solution. To some extent, those se tup are ex-\npectedsince theunknown coefficient, whether itisthedampingorth epotentialone,\ndepends on nindependent variables and the corresponding boundary measurem ent\nalso depends on nfree variables. In fact, under proper conditions it is even possible\nto recover both potential and damping coefficients in one shot by ju st one single\nboundary measurement [19]. In the case of a gradient coefficient, t he unknown\nfunction is vector-valued and containing ndifferent real-valued functions. Hence a\nsingle measurement does not seem to be sufficient to recover all of t hem, which is\nprobably why such problem is much less studied in the literature. Neve rtheless, it is\npossible to recover the coefficient by properly making nsets of boundary measure-\nments [8]. Last, in the case of recovering the variable unknown wave speed, since\nthe unknown function is at the principle order level, one typically need s to rewrite\nthe hyperbolic equation as a Riemannian wave equation so that the pr inciple part\nbecomes constant coefficients on an appropriate Riemannian manifo ld [3, 20].\nIn this paper, we seek to recover all together the aformentioned coefficients in the\nsecond-order hyperbolic equation (1). To the best of our knowled ge, this is the first\npaper that addresses the uniqueness and stability of recovering a ll these coefficients\nat once through finitely many boundary measurements. Note that all together\nthese coefficients contain a total of n+3 unknown functions, so naturally one may\nexpect to be able to recover them by making n+3 sets of boundary measurements.\nThis is entirely possible to do following the ideas in this paper (see Remar k (1)\nin Section 4). Nevertheless, in the following we will show that by appro priately\nchoosing ⌊n+4\n2⌋1pairs of initial conditions {w0,w1}and a boundary condition h, we\ncan uniquely and Lipschitz stably recover the coefficients c,q1, q0,andqall at once\nfrom the corresponding Neumann boundary measurements of the ir solutions. The\nprecise results are stated in Theorem 1.1 and Theorem 1.2 below.\nAs mentioned above recovering a single coefficient from a single bound ary mea-\nsurement isastandardformulationininversehyperbolicproblemsan dsuchproblem\nhasbeenstudiedextensively intheliterature. Hereweonlymentiont hemonographs\nand lecture notes [4, 6, 7, 9, 10, 17, 21] and refer to the substan tial lists of refer-\nences therein. The standard approach for this type of inverse hy perbolic problems\ntypically involves using Carleman-type estimates for the second-or der hyperbolic\nequations. To certain extent, such methods can all be seen as var iations or improve-\nments of the so called Bukhgeim–Klibanov (BK) method which was origin ated in\nthe seminal paper [5]. Our approach to solve the present inverse pr oblem also relies\non a sharp Carleman estimate for general second-order hyperbo lic equations and\nin particular a post Carleman estimate route that was introduced by Isakov in [7,\nTheorem 8.2.2]. Another standard feature of the BK method is the n eed of certain\npositivity assumptions on the initial conditions. Of course the precis e assumption\ndepends on what coefficient(s) one is trying to recover.\n1Here⌊·⌋denotes the usual floor function.4 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS\nOn the other hand, let us also mention that there is another standa rd formula-\ntion of inverse hyperbolic problems that usually does not require pos itivity on the\ninitial conditions. In this formulation, one tries to recover informat ion of second-\norder hyperbolic equations from all possible boundary measuremen ts, which are\noften modeled by the Dirichlet to Neumann or Neumann to Dirichlet ope rator. In\nparticular, in this case it is possible to recover all coefficients in syste m (1) up to\nnatural gauge transformations [12], following the powerful Bound ary Control (BC)\nmethod developed by Belishev [1]. For more inverse hyperbolic problem s with in-\nfinitely many measurements and the BC method, we refer to the rev iew paper [2]\nand the monograph [11].\nLet us now state the main theorems in this paper.\nTheorem 1.1. Under the geometrical assumptions (A.1) and (A.2) and let\n(2) T > T 0= 2/radicalbigg\nmax\nx∈Ωd(x).\nSuppose the initial and boundary conditions are in the follo wing function spaces\n(3) {w0,w1} ∈Hγ+1(Ω)×Hγ(Ω), h∈Hγ+1(Σ),whereγ >n\n2+4\nalong with all compatibility conditions (trace coincidenc e) which make sense. In ad-\ndition, dependingon the dimension nof the space, we assume the following positivity\ncondition: There exists r0>0such that\nCase I: If nis odd, i.e., n= 2m+1for some m∈N, then we choose m+2pairs\nof initial conditions {w(i)\n0,w(i)\n1},i= 1,...,m+ 2, and a boundary condition hso\nthat they satisfy (3) and\n(4) |detW(x)| ≥r0, a.e. x∈Ω\nwhereW(x)is the(n+3)×(n+3)matrix defined by\n(5)\nW(x) =\nw(1)\n0(x)w(1)\n1(x)∂x1w(1)\n0(x)···∂xnw(1)\n0(x) ∆w(1)\n0(x)\nw(1)\n1(x)w(1)\ntt(x)∂x1w(1)\n1(x)···∂xnw(1)\n1(x) ∆w(1)\n1(x)\n..................\nw(m+2)\n0(x)w(m+2)\n1(x)∂x1w(m+2)\n0(x)···∂xnw(m+2)\n0(x) ∆w(m+2)\n0(x)\nw(m+2)\n1(x)w(m+2)\ntt(x)∂x1w(m+2)\n1(x)···∂xnw(m+2)\n1(x) ∆w(m+2)\n1(x)\n\nCase II: If nis even, i.e., n= 2mfor some m∈N, then we choose m+2pairs of\ninitial conditions {w(i)\n0,w(i)\n1},i= 1,...,m+2, and a boundary condition hso thatRECOVER ALL COEFFICIENTS 5\nthey satisfy (3) and\n(6) |det/tildewiderW(x)| ≥r0, a.e. x∈Ω\nwhere/tildewiderW(x)is the(n+3)×(n+3)matrix defined by\n(7)\n/tildewiderW(x) =\nw(1)\n0(x)w(1)\n1(x)∂x1w(1)\n0(x)···∂xnw(1)\n0(x) ∆w(1)\n0(x)\nw(1)\n1(x)w(1)\ntt(x)∂x1w(1)\n1(x)···∂xnw(1)\n1(x) ∆w(1)\n1(x)\n..................\nw(m+1)\n0(x)w(m+1)\n1(x)∂x1w(m+1)\n0(x)···∂xnw(m+1)\n0(x) ∆w(m+1)\n0(x)\nw(m+1)\n1(x)w(m+1)\ntt(x)∂x1w(m+1)\n1(x)···∂xnw(m+1)\n1(x) ∆w(m+1)\n1(x)\nw(m+2)\n0(x)w(m+2)\n1(x)∂x1w(m+2)\n0(x)···∂xnw(m+2)\n0(x) ∆w(m+2)\n0(x)\n\nLetw(i)(c,q1,q0,q)andw(i)(˜c,p1,p0,p)be the corresponding solutions of equation\n(1) with different coefficients {c,q1,q0,q}and{˜c,p1,p0,p}, as well as the initial and\nboundary conditions {w(i)\n0,w(i)\n1,h},i= 1,···,m+2. If we have the same Neumann\nboundary traces over the observed part Γ1of the boundary and over the time interval\n[−T,T], i.e., for i= 1,···,m+2,\n(8)∂w(i)(c,q1,q0,q)\n∂ν(x,t) =∂w(i)(˜c,p1,p0,p)\n∂ν(x,t),(x,t)∈Γ1×[−T,T],\nthen we must have that all the coefficients coincide, namely,\n(9)c(x) = ˜c(x), q1(x) =p1(x), q0(x) =p0(x),q(x) =p(x)a.e. x∈Ω.\nAfter proving the above uniqueness theorem, we may also get the f ollowing Lips-\nchitz stability result for recovering all coefficients {c,q1,q0,q}from the correspond-\ning finite sets of boundary measurements.\nTheorem 1.2. Under the assumptions in Theorem 1.1, again let w(i)(c,q1,q0,q)\nandw(i)(˜c,p1,p0,p)denote the corresponding solutions of equation (1) with coe ffi-\ncients{c,q1,q0,q}and{˜c,p1,p0,p}, as well as the initial and boundary conditions\n{w(i)\n0,w(i)\n1,h},i= 1,···,m+2(eithernis odd or even). Then there exists C >0\ndepends on Ω,T,Γ1,c,q1,q0,q,w(i)\n0,w(i)\n1,hsuch that\n/⌊a∇d⌊lc2−˜c2/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lq1−p1/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lq0−p0/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lq−p/⌊a∇d⌊l2\nL2(Ω)\n≤Cm+2/summationdisplay\ni=1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂w(i)\ntt(c,q1,q0,q)\n∂ν−∂w(i)\ntt(˜c,p1,p0,p)\n∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(Σ1), (10)6 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS\nfor all such coefficients c,˜c,q1,q0,p1,p0∈H1\n0(Ω),q,p∈(H1\n0(Ω))n, where/⌊a∇d⌊l·/⌊a∇d⌊lL2(Ω)\nis defined as\n/⌊a∇d⌊lr/⌊a∇d⌊lL2(Ω)=/parenleftBigg/integraldisplay\nΩn/summationdisplay\ni=1|ri(x)|2dx/parenrightBigg1\n2\n,forr(x) = (r1(x),···,rn(x)).\nInverse source problem . The first step to solve the inverse problem above is to\nconvert it into a corresponding inverse source problem. Indeed, if we let\nf2(x) =c2(x)−˜c2(x), f1(x) =p1(x)−q1(x),\nf0(x) =p0(x)−q0(x),f(x) =p(x)−q(x);\nu(x,t) =w(c,q1,q0,q)−w(˜c,p1,p0,p), R(x,t) =w(˜c,p1,p0,p)(x,t),(11)\nthenu=u(x,t) is readily seen to satisfy the following homogeneous mixed problem\n(12)\n\nutt−c2(x)∆u+q1(x)ut+q0(x)u+q(x)·∇u=S(x,t) inQ\nu(x,0) =ut(x,0) = 0 in Ω\nu(x,t) = 0 in Σ,\nwhere\n(13)S(x,t) =f0(x)R(x,t)+f1(x)Rt(x,t)+f(x)·∇R(x,t)+f2(x)∆R(x,t).\nHere we assume that c∈C,q0,q1∈L∞(Ω) and q∈(L∞(Ω))nare given fixed\nandR=R(x,t) is a given function that can be suitably chosen. On the other\nhand, the source coefficients f0,f1,f2∈L2(Ω) and f∈(L2(Ω))nare assumed to\nbe unknown. The inverse source problem is to determine f0,f1,f2andffrom the\nNeumann boundary measurements of uover the observed part Γ 1of the boundary\nand over a sufficiently long time interval [ −T,T]. More specifically, corresponding\nwith Theorems 1.1 and 1.2, we will prove the following uniqueness and st ability\nresults.\nTheorem 1.3. Under geometrical assumptions (A.1) and (A.2) and let Tsatisfy\n(2). Depending on the dimension n, we assume the following regularity and posi-\ntivity conditions:\nCase I: If nis odd, i.e., n= 2m+ 1for some m∈N, then we choose m+ 2\nfunctions R(1),···,R(m+2)such that they satisfy\n(14) R(i),R(i)\nt,R(i)\ntt,R(i)\nttt∈W2,∞(Q), i= 1,···,m+2\nand there exists r0>0such that\n(15) |detU(x)| ≥r0, a.e. x∈ΩRECOVER ALL COEFFICIENTS 7\nwhereU(x)is the(n+3)×(n+3)matrix defined by\n(16)\nU(x) =\nR(1)(x,0)R(1)\nt(x,0)∂x1R(1)(x,0)···∂xnR(1)(x,0) ∆R(1)(x,0)\nR(1)\nt(x,0)R(1)\ntt(x,0)∂x1R(1)\nt(x,0)···∂xnR(1)\nt(x,0) ∆R(1)\nt(x,0)\n..................\nR(m+2)(x,0)R(m+2)\nt(x,0)∂x1R(m+2)(x,0)···∂xnR(m+2)(x,0) ∆R(m+2)(x,0)\nR(m+2)\nt(x,0)R(m+2)\ntt(x,0)∂x1R(m+2)\nt(x,0)···∂xnR(m+2)\nt(x,0) ∆R(m+2)\nt(x,0)\n\nCase II: If nis even, i.e., n= 2mfor some m∈N, then we choose m+ 2\nfunctions R(1),···,R(m+2)such that they satisfy (14) and there exists r0>0such\nthat\n(17) |det/tildewideU(x)| ≥r0, a.e. x∈Ω\nwhere/tildewideU(x)is the(n+3)×(n+3)matrix defined by\n(18)\n/tildewideU(x) =\nR(1)(x,0)R(1)\nt(x,0)∂x1R(1)(x,0)···∂xnR(1)(x,0) ∆R(1)(x,0)\nR(1)\nt(x,0)R(1)\ntt(x,0)∂x1R(1)\nt(x,0)···∂xnR(1)\nt(x,0) ∆R(1)\nt(x,0)\n..................\nR(m+1)(x,0)R(m+1)\nt(x,0)∂x1R(m+1)(x,0)···∂xnR(m+1)(x,0) ∆R(m+1)(x,0)\nR(m+1)\nt(x,0)R(m+1)\ntt(x,0)∂x1R(m+1)\nt(x,0)···∂xnR(m+1)\nt(x,0) ∆R(m+1)\nt(x,0)\nR(m+2)(x,0)R(m+2)\nt(x,0)∂x1R(m+2)(x,0)···∂xnR(m+2)(x,0) ∆R(m+2)(x,0)\n\nLetu(i)(f0,f1,f2,f)be the solutions of equation (12) with the functions R(i),i=\n1,···,m+2. If\n(19)∂u(i)(f0,f1,f2,f)\n∂ν(x,t) = 0,(x,t)∈Γ1×[−T,T], i= 1,···,m+2,\nthen we must have\n(20) f0(x) =f1(x) =f2(x) =f(x) = 0,a.e.x∈Ω.\nTheorem 1.4. Under the assumptions in Theorem 1.3, again let u(i)(f0,f1,f2,f)\ndenote the solutions of equation (12) with the functions R(i),i= 1,···,m+2(either\nnis odd or even). Then there exists C >0depends on Ω,T,Γ1,c,q1,q0,q,w(i)\n0,\nw(i)\n1,hsuch that\n(21)/⌊a∇d⌊lf0/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lf1/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lf2/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lf/⌊a∇d⌊l2\nL2(Ω)≤Cm+2/summationdisplay\ni=1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂u(i)\ntt(f0,f1,f2,f)\n∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(Σ1)8 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS\nfor allf0,f1,f2∈H1\n0(Ω)andf∈(H1\n0(Ω))n.\nThe rest of the paper is organized as follows. In the next section we recall\nsome necessary tools to solve the inverse problem. This includes the sharp Carle-\nman estimate, continuous observability inequality and regularity the ory for general\nsecond-order hyperbolic equations with Dirichlet boundary conditio n. In Section\n3 we provide the proofs of Theorems 1.1, 1.2, 1.3 and 1.4, and in the las t section\nwe give some examples where the positivity conditions (4), (6), (15) and (17) are\nsatisfied and some concluding remarks.\n2.Carleman Estimate, Continuous Observability Inequality a nd\nRegularity Theory for Second-Order Hyperbolic Equations\nIn this section we recall some key ingredients of the proofs used in t he next sec-\ntion. This includes Carleman estimate, continuous observability inequ ality, as well\nas regularity theory for general second-order hyperbolic equat ions with Dirichlet\nboundary condition. For simplicity here we only state the main results and refer to\n[22] and [13] for greater details.\nTo begin with, consider a Riemannian metric g(·,·) =/a\\}⌊∇a⌋ketle{t·,·/a\\}⌊∇a⌋ket∇i}htand squared norm\n|X|2=g(X,X),on a smooth finite dimensional manifold M. On the Riemannian\nmanifold ( M,g) we define Ω as an open bounded, connected set of Mwith smooth\nboundary Γ = Γ0∪Γ1, where Γ 0∩Γ1=∅. Letνdenote the unit outward normal\nfield along the boundary Γ. Furthermore, we denote by ∆ gthe Laplace–Beltrami\noperator on the manifold Mand byDthe Levi–Civita connection on M.\nConsider the following second-order hyperbolic equation with energ y level terms\ndefined on Q= Ω×[−T,T] for some T >0:\n(22)wtt(x,t)−∆gw(x,t)+F(w) =G(x,t),(x,t)∈Q= Ω×[−T,T]\nwhere the forcing term G∈L2(Q) and the energy level differential term F(w) is\ngiven by\nF(w) =/a\\}⌊∇a⌋ketle{tP(x,t),Dw/a\\}⌊∇a⌋ket∇i}ht+P1(x,t)wt+P0(x,t)w.\nHereP0,P1arefunctionsonΩ ×[−T,T],P(x,t)isavectorfieldon Mfort∈[−T,T],\nand they satisfy the following estimate: there exists a constant CT>0 such that\n|F(w)| ≤CT[w2+w2\nt+|Dw|2],∀(x,t)∈Q.\nPseudo-convex function. Having chosen, on the strength of geometrical as-\nsumption (A.1), a strictly convex function d(x), we can define the function ϕ(x,t) :\nΩ×R→Rof classC3by setting\nϕ(x,t) =d(x)−αt2, x∈Ω, t∈[−T,T],\nwhereT > T 0as in (2). Moreover, α∈(0,1) is selected as follows: Let T > T 0be\ngiven, then there exists δ >0 such that\nT2>4max\nx∈Ωd(x)+4δ.RECOVER ALL COEFFICIENTS 9\nFor thisδ >0, there exists a constant α∈(0,1), such that\nαT2>4max\nx∈Ωd(x)+4δ.\nIt is easy to check such function ϕ(x,t) carries the following properties:\n(a) For the constant δ >0 fixed above, we have\nϕ(x,−T) =ϕ(x,T)≤max\nx∈Ωd(x)−αT2≤ −δuniformly in x∈Ω;\nand\nϕ(x,t)≤ϕ(x,0),for anyt∈[−T,T] and any x∈Ω.\n(b) There are t0andt1, with−T < t0<0< t1< T, say, chosen symmetrically\nabout 0, such that\nmin\nx∈Ω,t∈[t0,t1]ϕ(x,t)≥σ,where 0< σ < m 0= min\nx∈Ωd(x).\nMoreover, let Q(σ) be the subset of Q= Ω×[−T,T] defined by\n(23) Q(σ) ={(x,t) :ϕ(x,t)≥σ >0,x∈Ω,−T≤t≤T},\nThen we have\n(24) Ω ×[t0,t1]⊂Q(σ)⊂Ω×[−T,T].\nCarleman estimate for general second-order hyperbolic equ ations. We\nnow return to the equation (22), and consider solutions w(x,t) in the class\n(25)/braceleftBigg\nw∈H1,1(Q) =L2(−T,T;H1(Ω))∩H1(−T,T;L2(Ω));\nwt,∂w\n∂ν∈L2(−T,T;L2(Γ)).\nThen for these solutions with geometrical assumptions (A.1) and (A .2) on Ω, the\nfollowing one-parameter family of estimates hold true, with β >0 being a suitable\nconstant ( βis positive by virtue of (A.2)), for all τ >0 sufficiently large and ǫ >0\nsmall:\n(26)\nBT(w)+2/integraldisplay\nQe2τϕ|G|2dQ+C1,Te2τσ/integraldisplay\nQw2dQ+cTτ3e−2τδ[Ew(−T)+Ew(T)]\n≥C1,τ/integraldisplay\nQe2τϕ[w2\nt+|Dw|2]dQ+C2,τ/integraldisplay\nQ(σ)e2τϕw2dxdt\nwhere\n(27) C1,τ=τǫ(1−α)−2CT, C2,τ= 2τ3β+O(τ2)−2CT.10 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS\nHereδ >0,σ >0 are the constants as in above, CT,cTandC1,Tare positive\nconstants depending on T, as well as d(but not on τ). The energy function Ew(t)\nis defined as\nEw(t) =/integraldisplay\nΩ[w2(x,t)+w2\nt(x,t)+|Dw(x,t)|2]dΩ.\nIn addition, BT(w) stands for boundary terms and can be explicitly calculated as\nBT(w) = 2τ/integraldisplay\nΣe2τϕ/parenleftbig\nw2\nt−|Dw|2/parenrightbig\n/a\\}⌊∇a⌋ketle{tDd,ν/a\\}⌊∇a⌋ket∇i}htdΣ\n+ 4τ/integraldisplay\nΣe2τϕ/a\\}⌊∇a⌋ketle{tDd,Dw/a\\}⌊∇a⌋ket∇i}ht/a\\}⌊∇a⌋ketle{tDw,ν/a\\}⌊∇a⌋ket∇i}htdΣ+8ατ/integraldisplay\nΣe2τϕtwt/a\\}⌊∇a⌋ketle{tDw,ν/a\\}⌊∇a⌋ket∇i}htdΣ\n+ 4τ2/integraldisplay\nΣe2τϕ/bracketleftbigg\n|Dd|2−4α2t2+∆d−α−1\n2τ/bracketrightbigg\nw/a\\}⌊∇a⌋ketle{tDw,ν/a\\}⌊∇a⌋ket∇i}htdΣ\n+ 2τ/integraldisplay\nΣe2τϕ/bracketleftbig\n2τ2/parenleftbig\n|Dd|2−4α2t2/parenrightbig\n+τ(3α+1)/bracketrightbig\nw2/a\\}⌊∇a⌋ketle{tDd,ν/a\\}⌊∇a⌋ket∇i}htdΣ.\nClearly if we have w|Γ×[−T,T]= 0 and∂w\n∂ν=/a\\}⌊∇a⌋ketle{tDw,ν/a\\}⌊∇a⌋ket∇i}ht= 0 on Γ 1×[−T,T], then in\nview of the geometrical assumption (A.1) we may compute\n(28) BT(w) = 2τ/integraldisplayT\n−T/integraldisplay\nΓ0e2τϕ|Dw|2/a\\}⌊∇a⌋ketle{tDd,ν/a\\}⌊∇a⌋ket∇i}htdΓ0dt≤0.\nContinuous observability inequality . As a corollary of the Carleman estimate,\nwe also have the following continuous observability inequality\n(29) CTEw(0)≤/integraldisplayT\n−T/integraldisplay\nΓ1/parenleftbigg∂w\n∂ν/parenrightbigg2\ndΓdt+/⌊a∇d⌊lG/⌊a∇d⌊l2\nL2(Q)\nfor the equation (22) with homogeneous Dirichlet boundary conditio nw|Σ= 0.\nHereT > T 0as in (2) and Ω satisfies the geometrical assumptions (A.1) and (A.2) .\nRemark 2.1. The continuous observability inequality (29) may be interpreted as\nfollows: If the second-order hyperbolic equation equation (22) ha s homogeneous\nDirichlet boundary condition and nonhomogeneous forcing term G∈L2(Q), and\nNeumann boundary trace∂w\n∂ν∈L2(Σ1), then necessarily the initial conditions\n{w(·,0),wt(·,0)}must lie in the natural energy space H1\n0(Ω)×L2(Ω). This fact\nwill be used in the proofs in Section 3.\nRegularity theory for general second-order hyperbolic equ ations with\nDirichlet boundary condition . Consider the second-order hyperbolic equation\n(22) with initial conditions w(x,0) =w0(x),wt(x,0) =w1(x) and Dirichlet bound-\nary condition w|Σ=h(x,t). Then the following interior and boundary regularityRECOVER ALL COEFFICIENTS 11\nresults for the solution whold true: For γ≥0 (not necessarily an integer), if the\ngiven data satisfy the following regularity assumptions\n/braceleftBigg\nG∈L1(0,T;Hγ(Ω)), ∂(γ)\ntG∈L1(0,T;L2(Ω)),\nw0∈Hγ+1(Ω), w1∈Hγ(Ω), h∈Hγ+1(Σ)\nwith all compatibility conditions (trace coincidence) which make sense . Then, we\nhave the following regularity for the solution w:\n(30)w∈C([0,T];Hγ+1(Ω)), ∂(γ+1)\ntw∈C([0,T];L2(Ω));∂w\n∂ν∈Hγ(Σ).\n3.Main Proofs\nIn this section we give the main proofs of the uniqueness and stability results\nestablished in the first section. We focus on proving Theorems 1.3 an d 1.4 for the\ninverse source problem since Theorems 1.1 and 1.2 of the original inve rse problem\nwill then follow from the relation (11) between the two problems and t he regularity\ntheory result recalled in Section 2. Henceforth for convenience we useCto denote a\ngeneric positive constant which may depend on Ω, T,c,q1,q0,q,r0,w(i),u(i),R(i),\ni= 1,···,m+2, but not on the free large parameter τappearing in the Carleman\nestimate.\nProof of Theorem 1.3 . First we consider the case when nis odd, i.e., n= 2m+1,\nfor some m∈N. Then corresponding with the choice of R(i),i= 1,···,m+2, we\nhavem+2 equations of the form (12) with solutions u(i)=u(i)(x,t) that satisfy\n(31)\n\nu(i)\ntt−c2(x)∆u(i)+q1(x)u(i)\nt+q0(x)u(i)+q(x)·∇u(i)=S(i)(x,t) inQ\nu(i)(x,0) =u(i)\nt(x,0) = 0 in Ω\nu(i)|Γ×[−T,T]= 0,∂u(i)\n∂ν|Γ1×[−T,T]= 0 in Σ ,Σ1,\nwhereS(i)(x,t) is defined in (13) with Rbeing replaced by R(i).\nNote since c∈C,q1,q0∈L∞(Ω) andq∈(L∞(Ω))n, the equation in (31) can be\nwritten as a Riemannian wave equation with respect to the metric g=c−2(x)dx2,\nmodulo lower-order terms2\nu(i)\ntt−∆gu(i)+“lower-order terms” = S(i)(x,t).\nMoreover, by the regularity assumption (14), we have that S(i)∈L2(Q) and by\nCauchy–Schwarz inequality\n|S(i)(x,t)|2≤C/parenleftbig\n|f0(x)|2+|f1(x)|2+|f(x)|2+|f2(x)|2/parenrightbig\n.\n2More precisely, we have ∆ gu=c2∆u+cn∇(c2−n)·∇u12 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS\nThus we can apply the Carleman estimate (26) for solution u(i)in the class (25)\nand get the following inequality for sufficiently large τ:\nτ/integraldisplay\nQe2τϕ[(u(i)\nt)2+|Du(i)|2]dQ+τ3/integraldisplay\nQ(σ)e2τϕ(u(i))2dxdt\n≤C/integraldisplay\nQe2τϕ/parenleftbig\n|f0(x)|2+|f1(x)|2+|f(x)|2+|f2(x)|2/parenrightbig\ndQ+Ce2τσ.(32)\nNote here we have dropped the unnecessary terms in the Carleman estimate (26)\nas well as the boundary terms BT(u(i)) since the homogeneous boundary data\nu(i)|Γ×[−T,T]=∂u(i)\n∂ν|Γ1×[−T,T]= 0 imply BT(u(i))≤0, as suggested in (28).\nDifferentiate the u(i)-system (31) in time t, we get the following u(i)\nt-problem\n(33)\n\n(u(i)\nt)tt−c2(x)∆u(i)\nt+q1(x)(u(i)\nt)t+q0(x)u(i)\nt+q(x)·∇u(i)\nt=S(i)\nt(x,t) inQ\n(u(i)\nt)(x,0) = 0,(u(i)\nt)t(x,0) =S(i)(x,0) in Ω\nu(i)\nt|Γ×[−T,T]= 0,∂u(i)\nt\n∂ν|Γ1×[−T,T]= 0 in Σ ,Σ1.\nNote again by (14) we have S(i)\nt∈L2(Q) and by Cauchy–Schwarz inequality\n|S(i)\nt(x,t)|2≤C/parenleftbig\n|f0(x)|2+|f1(x)|2+|f(x)|2+|f2(x)|2/parenrightbig\n.\nIn addition, BT(u(i)\nt)≤0 sinceu(i)\nt|Γ×[−T,T]=∂u(i)\nt\n∂ν|Γ1×[−T,T]= 0. Thus similar to\n(32) we can apply Carleman estimate (26) for solutions u(i)\ntin the class (25) and\nget the following inequality for sufficiently large τ:\nτ/integraldisplay\nQe2τϕ[(u(i)\ntt)2+|Du(i)\nt|2]dQ+τ3/integraldisplay\nQ(σ)e2τϕ(u(i)\nt)2dxdt\n≤C/integraldisplay\nQe2τϕ/parenleftbig\n|f0(x)|2+|f1(x)|2+|f(x)|2+|f2(x)|2/parenrightbig\ndQ+Ce2τσ.(34)\nContinue with this process, we differentiate (33) in ttwo more times, and get the\ncorresponding u(i)\nttandu(i)\nttt-systems\n(35)\n\n(u(i)\ntt)tt−c2(x)∆u(i)\ntt+q1(x)(u(i)\ntt)t+q0(x)u(i)\ntt+q(x)·∇u(i)\ntt=S(i)\ntt(x,t)\nu(i)\ntt(x,0) =S(i)(x,0),(u(i)\ntt)t(x,0) =S(i)\nt(x,0)−q1(x)S(i)(x,0)\nu(i)\ntt|Γ×[−T,T]= 0,∂u(i)\ntt\n∂ν|Γ1×[−T,T]= 0RECOVER ALL COEFFICIENTS 13\nand\n(36)\n\n(u(i)\nttt)tt−c2(x)∆u(i)\nttt+q1(x)(u(i)\nttt)t+q0(x)u(i)\nttt+q(x)·∇u(i)\nttt=S(i)\nttt(x,t)\n(u(i)\nttt)(x,0) =S(i)\nt(x,0)−q1(x)S(i)(x,0)\n(u(i)\nttt)t(x,0) =S(i)\ntt(x,0)+c2∆S(i)(x,0)−q1S(i)\nt(x,0)−q0S(i)(x,0)−q·∇S(i)(x,0)\nu(i)\nttt|Γ×[−T,T]= 0,∂u(i)\nttt\n∂ν|Γ1×[−T,T]= 0.\nAgain by (14), Cauchy–Schwarz inequality and the homogeneous Dir ichlet and\nNeumann boundary data, we can apply Carleman estimate (26) to th e correspond-\ningu(i)\ntt,u(i)\nttt-systems above and get the following inequalities that are similar to (3 2)\nand (34), for τsufficiently large\nτ/integraldisplay\nQe2τϕ[(u(i)\nttt)2+|Du(i)\ntt|2]dQ+τ3/integraldisplay\nQ(σ)e2τϕ(u(i)\ntt)2dxdt\n≤C/integraldisplay\nQe2τϕ/parenleftbig\n|f0(x)|2+|f1(x)|2+|f(x)|2+|f2(x)|2/parenrightbig\ndQ+Ce2τσ.(37)\nτ/integraldisplay\nQe2τϕ[(u(i)\ntttt)2+|Du(i)\nttt|2]dQ+τ3/integraldisplay\nQ(σ)e2τϕ(u(i)\nttt)2dxdt\n≤C/integraldisplay\nQe2τϕ/parenleftbig\n|f0(x)|2+|f1(x)|2+|f(x)|2+|f2(x)|2/parenrightbig\ndQ+Ce2τσ.(38)\nAdd the four inequalities (32), (34), (37), (38) together, we get\nτ/integraldisplay\nQe2τϕ[(u(i)\ntttt)2+(u(i)\nttt)2+(u(i)\ntt)2+(u(i)\nt)2+|Du(i)\nttt|2+|Du(i)\ntt|2+|Du(i)\nt|2+|Du(i)|2]dQ\n+τ3/integraldisplay\nQ(σ)e2τϕ[(u(i)\nttt)2+(u(i)\ntt)2+(u(i)\nt)2+(u(i))2]dxdt\n≤C/integraldisplay\nQe2τϕ/parenleftbig\n|f0(x)|2+|f1(x)|2+|f(x)|2+|f2(x)|2/parenrightbig\ndQ+Ce2τσ.(39)\nWe now analyze the integral term on the right-hand side of (39). Fir st note that\nby estimating the u(i)-equation in (31) and u(i)\nt-equation in (33) at time t= 0, we14 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS\ncan get\n(40)\n\nu(i)\ntt(x,0) =S(i)(x,0)\nu(i)\nttt(x,0) =S(i)\nt(x,0)−q1(x)S(i)(x,0).\nNote the above equations hold for any i, 1≤i≤m+2, so putting all of them\ntogether we get a ( n+3)×(n+3) linear system\n(41)/bracketleftBig\nu(1)\ntt(x,0),u(1)\nttt(x,0),···,u(m+2)\ntt(x,0),u(m+2)\nttt(x,0)/bracketrightBigT\n=Uq1(x)[f0(x),f1(x),f(x),f2(x)]T\nwhere the coefficient matrix Uq1(x) is defined as\n(42)\nUq1(x) =\nR(1)(x,0)R(1)\nt(x,0)∂x1R(1)(x,0)···∂xnR(1)(x,0) ∆R(1)(x,0)\n˜a(1)(x)˜b(1)(x) ˜ m(1)\n1(x)··· ˜m(1)\nn(x) ˜ℓ(1)(x)\n..................\nR(m+2)(x,0)R(m+2)\nt(x,0)∂x1R(m+2)(x,0)···∂xnR(m+2)(x,0) ∆R(m+2)(x,0)\n˜a(m+2)(x)˜b(m+2)(x) ˜m(m+2)\n1(x)···˜m(m+2)\nn(x)˜ℓ(m+2)(x)\n\nwith\n(43) ˜a(i)(x) =R(i)\nt(x,0)−q1(x)R(i)(x,0),˜b(i)(x) =R(i)\ntt(x,0)−q1(x)R(i)\nt(x,0),\n˜m(i)\nk(x) =∂xkR(i)\nt(x,0)−q1(x)∂xkR(i)(x),˜ℓ(i)(x,0) = ∆R(i)\nt(x,0)−q1(x)∆R(i)(x,0).\nNotice that from doing elementary row operations, specifically, add ingq1multiplied\nby an odd row to the subsequent even row, the matrix Uq1(x) andU(x) as defined\nin (16) have the same determinant. Thus the positivity assumption ( 15) implies\nthat we may invert Uq1(x) in (42) to obtain\n|f0(x)|2+|f1(x)|2+|f2(x)|2+|f(x)|2≤Cm+2/summationdisplay\ni=1/parenleftBig\n|u(i)\ntt(x,0)|2+|u(i)\nttt(x,0)|2/parenrightBig\n=C/parenleftbig\n|utt(x,0)|2+|uttt(x,0)|2/parenrightbig(44)\nwherewedenote u(x,t) = (u(1)(x,t),u(2)(x,t),···,u(m+2)(x,t)). Thusbyproperties\nof the pseudo-convex function ϕand the Cauchy–Schwarz inequality, we can get theRECOVER ALL COEFFICIENTS 15\nfollowing estimate\n/integraldisplay\nQe2τϕ(x,t)/parenleftbig\n|f0(x)|2+|f1(x)|2+|f(x)|2+|f2(x)|2/parenrightbig\ndQ (45)\n≤C/integraldisplay\nΩ/integraldisplayT\n−Te2τϕ(x,0)/parenleftbig\n|utt(x,0)|2+|uttt(x,0)|2/parenrightbig\ndtdΩ\n≤C/parenleftbigg/integraldisplay\nΩ/integraldisplay0\n−Td\nds[e2τϕ(x,s)/parenleftbig\n|utt(x,s)|2+|uttt(x,s)|2/parenrightbig\n]dsdΩ\n+/integraldisplay\nΩe2τϕ(x,−T)/parenleftbig\n|utt(x,−T)|2+|uttt(x,−T)|2/parenrightbig\ndΩ/parenrightbigg\n≤C/parenleftbigg\nτ/integraldisplay\nΩ/integraldisplay0\n−Te2τϕ(x,s)/parenleftbig\n|utt(x,s)|2+|uttt(x,s)|2/parenrightbig\n]dsdΩ\n+ 2/integraldisplay\nΩ/integraldisplay0\n−Te2τϕ(x,s)(utt·uttt+uttt·utttt)]dsdΩ\n+/integraldisplay\nΩe2τϕ(x,−T)/parenleftbig\n|utt(x,−T)|2+|uttt(x,−T)|2/parenrightbig\ndΩ/parenrightbigg\n≤C/parenleftbigg\nτ/integraldisplay\nQe2τϕ|utt|2dQ+τ/integraldisplay\nQe2τϕ|uttt|2dQ+/integraldisplay\nQe2τϕ|utttt|2dQ/parenrightbigg\n.\nTaking (45) into (39), and note (39) holds for all i= 1,···m+2, thus summing\noveriin (39) and dropping the non-negative gradient terms on the left-h and side,\nwe get that for τsufficiently large\nτ/integraldisplay\nQe2τϕ/parenleftbig\n|utttt|2+|uttt|2+|utt|2+|ut|2/parenrightbig\ndQ (46)\n+τ3/integraldisplay\nQ(σ)e2τϕ/parenleftbig\n|uttt|2+|utt|2+|ut|2+|u|2/parenrightbig\ndxdt\n≤Cτ/integraldisplay\nQe2τϕ(|utt|2+|uttt|2)dQ+C/integraldisplay\nQe2τϕ|utttt|2dQ+Ce2τσ.\nSincee2τϕ< e2τσonQ\\Q(σ) from the definition of Q(σ) (23), we have the\nfollowing\n/integraldisplay\nQe2τϕ/parenleftbig\n|utt|2+|uttt|2/parenrightbig\ndQ\n=/integraldisplay\nQ(σ)e2τϕ/parenleftbig\n|utt|2+|uttt|2/parenrightbig\ndxdt+/integraldisplay\nQ\\Q(σ)e2τϕ/parenleftbig\n|utt|2+|uttt|2/parenrightbig\ndxdt\n≤/integraldisplay\nQ(σ)e2τϕ/parenleftbig\n|utt|2+|uttt|2/parenrightbig\ndxdt+e2τσ/integraldisplay\nQ\\Q(σ)/parenleftbig\n|utt|2+|uttt|2/parenrightbig\ndxdt16 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS\nand therefore (46) becomes\nτ/integraldisplay\nQe2τϕ/parenleftbig\n|utttt|2+|uttt|2+|utt|2+|ut|2/parenrightbig\ndQ (47)\n+τ3/integraldisplay\nQ(σ)e2τϕ/parenleftbig\n|uttt|2+|utt|2+|ut|2+|u|2/parenrightbig\ndxdt\n≤Cτ/integraldisplay\nQ(σ)e2τϕ/parenleftbig\n|utt|2+|uttt|2/parenrightbig\ndxdt+C/integraldisplay\nQe2τϕ|utttt|2dQ+Ce2τσ.\nNote that in (47) the first and second terms on the right-hand side can be ab-\nsorbed by the corresponding terms on the left-hand side when τis taken large\nenough. Hence we may get the following estimate for sufficiently large τ:\nτ3/integraldisplay\nQ(σ)e2τϕ/parenleftbig\n|uttt|2+|utt|2+|ut|2+|u|2/parenrightbig\ndxdt≤Cτe2τσ.\nUse again the fact that ϕ(x,t)≥σonQ(σ) we hence get\nτ2/integraldisplay\nQ(σ)|uttt|2+|utt|2+|ut|2+|u|2dxdt≤C.\nSinceτ >0 in a free large parameter and the constants Cdo not depend on τ,\nthe above inequality implies we must have u=0a.e. onQ(σ). Note from (24)\nthe subspace Q(σ) satisfies the property Ω ×[t0,t1]⊂Q(σ)⊂Qwitht0<0< t1,\ntherefore by evaluating the uandut-systems of equations at t= 0, we get the\n(n+3)×(n+3) linear system (see (41))\nUq1(x)[f0(x),f1(x),f(x),f2(x)]T=0, a.e. x∈Ω.\nAs the coefficient matrix Uq1(x) is invertible from assumption (15), we must have\nthe desired conclusion\nf0(x) =f1(x) =f2(x) =f(x) = 0, a.e. x∈Ω.\nFor the case when nis even, i.e., n= 2m,m∈N, we can basically repeat\nthe above proof with obvious adjustments. The only difference her e is that since\nn= 2mis even, the linear system (41) contains an odd number ( n+3) of equations.\nTherefore we only need m+1 pairs of equations from (40) plus one more equationRECOVER ALL COEFFICIENTS 17\nfromu(m+2)\ntt(x,0). Doing this yields the matrix /tildewideUq1(x), where\n(48)\n/tildewideUq1(x) =\nR(1)(x,0)R(1)\nt(x,0)∂x1R(1)(x,0)···∂xnR(1)(x,0) ∆R(1)(x,0)\n˜a(1)(x)˜b(1)(x) ˜ m(1)\n1(x)··· ˜m(1)\nn(x) ˜ℓ(1)(x)\n..................\nR(m+1)(x,0)R(m+1)\nt(x,0)∂x1R(m+1)(x,0)···∂xnR(m+1)(x,0) ∆R(m+1)(x,0)\n˜a(m+1)(x)˜b(m+1)(x) ˜m(m+1)\n1(x)···˜m(m+1)\nn(x)˜ℓ(m+1)(x)\nR(m+2)(x,0)R(m+2)\nt(x,0)∂x1R(m+2)(x,0)···∂xnR(m+2)(x,0) ∆R(m+2)(x,0)\n\nwith ˜a(i),˜b(i), ˜m(i)\nkand˜ℓ(i)defined as in (43). Again since elementary row operations\ndo notchange thedeterminant, /tildewideUq1(x) will have thesamedeterminant asthematrix\n/tildewideU(x) in the assumption (17). This completes the proof of Theorem 1.3.\nProof of Theorem 1.4 . After achieving the uniqueness for the inverse source\nproblem, we now prove the corresponding stability estimate (21). T he proof below\nworks essentially for both of the cases whether nis odd or even, the only difference\nis in the choices of the functions R(i),i= 1,···,m+2, as indicated in the Theorem\n1.3. First we go back to the inequality (44), integrate over Ω gives\n(49)\n/⌊a∇d⌊lf0/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lf1/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lf2/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lf/⌊a∇d⌊l2\nL2(Ω)≤Cm+2/summationdisplay\ni=1/parenleftBig\n/⌊a∇d⌊lu(i)\ntt(·,0)/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lu(i)\nttt(·,0)/⌊a∇d⌊l2\nL2(Ω)/parenrightBig\n.\nFor each i, 1≤i≤m+2, we return to the u(i)\ntt-system:\n(50)\n\n(u(i)\ntt)tt−c2(x)∆u(i)\ntt+q1(x)(u(i)\ntt)t+q0(x)u(i)\ntt+q(x)·∇u(i)\ntt=S(i)\ntt(x,t)\nu(i)\ntt(x,0) =S(i)(x,0),(u(i)\ntt)t(x,0) =S(i)\nt(x,0)−q1(x)S(i)(x,0)\nu(i)\ntt|Γ×[−T,T]= 0\nwithS(i)(x,t) =f0(x)R(i)+f1(x)R(i)\nt+f(x)·∇R(i)+f2(x)∆R(i). Here we assume\n(51)c∈C, q0,q1,q2∈L∞(Ω),q∈(L∞(Ω))n,f0,f1,f2∈H1\n0(Ω),f∈/parenleftbig\nH1\n0(Ω)/parenrightbign\nandR(i)satisfies (14) and (15) (or (17) if nis even). By linearity, we split u(i)\ntt\ninto two systems, u(i)\ntt=y(i)+z(i), wherey(i)=y(i)(x,t) satisfies the homogeneous18 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS\nforcing term and nonhomogeneous initial conditions\n(52)\n\ny(i)\ntt−c2(x)∆y(i)+q1(x)y(i)\nt+q0(x)y(i)+q(x)·∇y(i)= 0 in Q\ny(i)(x,0) =u(i)\ntt(x,0) =S(i)(x,0) in Ω\ny(i)\nt(x,0) = (u(i)\ntt)t(x,0) =S(i)\nt(x,0)−q1(x)S(i)(x,0) in Ω\ny(i)|Γ×[−T,T]= 0 in Σ\nandz(i)=z(i)(x,t) has the nonhomogeneous forcing term and homogeneous initial\nconditions\n(53)\n\nz(i)\ntt−c2(x)∆z(i)+q1(x)z(i)\nt+q0(x)z(i)+q(x)·∇z(i)=S(i)\ntt(x,t) inQ\nz(i)(x,0) =z(i)\nt(x,0) = 0 in Ω\nz(i)|Γ×[−T,T]= 0 in Σ.\nFor they(i)-system, note by assumptions (51) and (14) we have\nS(i)(·,0)∈H1\n0(Ω) and S(i)\nt(·,0)−q1(·)S(i)(·,0)∈L2(Ω).\nThuswemayapplythecontinuousobservabilityinequality(29)(with g=c−2(x)dx2)\nto get\n/⌊a∇d⌊ly(i)(·,0)/⌊a∇d⌊l2\nH1\n0(Ω)+/⌊a∇d⌊ly(i)\nt(·,0)/⌊a∇d⌊l2\nL2(Ω)=/⌊a∇d⌊lu(i)\ntt(·,0)/⌊a∇d⌊l2\nH1\n0(Ω)+/⌊a∇d⌊lu(i)\nttt(·,0)/⌊a∇d⌊l2\nL2(Ω)≤C/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂y(i)\n∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(Σ1).\nSum the above inequality over i, use (49) and the decomposition u(i)\ntt=y(i)+z(i),\nas well as Poincar´ e’s inequality, we have\n/⌊a∇d⌊lf0/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lf1/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lf2/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lf/⌊a∇d⌊l2\nL2(Ω) (54)\n≤Cm+2/summationdisplay\ni=1/parenleftBig\n/⌊a∇d⌊lu(i)\nttt(·,0)/⌊a∇d⌊l2\nH1\n0(Ω)+/⌊a∇d⌊lu(i)\nttt(·,0)/⌊a∇d⌊l2\nL2(Ω)/parenrightBig\n≤Cm+2/summationdisplay\ni=1/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂y(i)\n∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(Σ1)\n=Cm+2/summationdisplay\ni=1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂u(i)\ntt\n∂ν−∂z(i)\n∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(Σ1)\n≤Cm+2/summationdisplay\ni=1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂u(i)\ntt\n∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(Σ1)+Cm+2/summationdisplay\ni=1/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂z(i)\n∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(Σ1).RECOVER ALL COEFFICIENTS 19\nNote this is the desired stability estimate (21) polluted by the z(i)terms. Next we\nshow those terms can be absorbed through a compactness–uniqu eness argument,\nwhere the uniqueness relies on Theorem 1.3. To start, note for the z(i)-system (53),\nwe have the following proposition.\nProposition 3.1. For each i= 1,···,m+2, the operator define by\nKi:L2(Ω)×L2(Ω)×L2(Ω)×/parenleftbig\nL2(Ω)/parenrightbign→L2(Σ1) (55)\n(f0,f1,f2,f)/mapsto→∂z(i)\n∂ν|Σ1,\nis a compact operator.\nProof.Note assumptions (51) and (14) imply S(i)\ntt∈H1(Q), thus by the regularity\nresult (30) we have\nS(i)\ntt∈H1(Q)⇒∂z(i)\n∂ν∈H1(Σ1) continuously .\nThis then implies the map ( f0,f1,f2,f)/mapsto→Ki(f0,f1,f2,f)∈H1(Σ1) is continuous\nand hence ( f0,f1,f2,f)/mapsto→Ki(f0,f1,f2,f)∈L2(Σ1) is compact. /square\nCombine K(i),i= 1,···,m+ 2, being compact, together with the uniqueness\nresult in Theorem 1.3, we may drop the z(i)terms in (54) to get the desired stability\nestimate (21). To carry this out, suppose by contradiction the st ability estimate\n(21) does not hold, then there exist sequences {fk\n0},{fk\n1},{fk\n2}and{fk}, with\nfk\n0,fk\n1,fk\n2∈H1\n0(Ω) andfk∈(H1\n0(Ω))n,∀k∈N, such that\n(56)/vextenddouble/vextenddoublefk\n0/vextenddouble/vextenddouble2\nL2(Ω)+/vextenddouble/vextenddoublefk\n1/vextenddouble/vextenddouble2\nL2(Ω)+/vextenddouble/vextenddoublefk\n2/vextenddouble/vextenddouble2\nL2(Ω)+/vextenddouble/vextenddoublefk/vextenddouble/vextenddouble2\nL2(Ω)= 1\nand\n(57) lim\nk→∞m+2/summationdisplay\ni=1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂u(i)\ntt(fk\n0,fk\n1,fk\n2,fk)\n∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2(Σ1)= 0\nwhereu(i)(fk\n0,fk\n1,fk\n2,fk) solves the system (31) with f0=fk\n0,f1=fk\n1,f2=fk\n2and\nf=fk. From (56), there exist subsequences, still denoted as {fk\n0},{fk\n1},{fk\n2}and\n{fk}, such that\n(58)\nfik⇀ f∗\niandfk⇀f∗weakly for some f∗\ni∈L2(Ω) andf∗∈/parenleftbig\nL2(Ω)/parenrightbign,i= 0,1,2.\nMoreover, in view of the compactness of Ki,i= 1,···,m+ 2, we also have the\nstrong convergence\n(59) lim\nk,l→∞/vextenddouble/vextenddoubleKi(fk\n0,fk\n1,fk\n2,fk)−Ki(fl\n0,fl\n1,fl\n2,fl)/vextenddouble/vextenddouble\nL2(Σ1)= 0,∀i= 1,···,m+2.20 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS\nOn the other hand, since the map ( f0,f1,f2,f)/mapsto→u(i)(f0,f1,f2,f) is linear, we\nhave from (54) that\n/vextenddouble/vextenddoublefk\n0−fl\n0/vextenddouble/vextenddouble2\nL2(Ω)+/vextenddouble/vextenddoublefk\n1−fl\n1/vextenddouble/vextenddouble2\nL2(Ω)+/vextenddouble/vextenddoublefk\n2−fl\n2/vextenddouble/vextenddouble2\nL2(Ω)+/vextenddouble/vextenddoublefk−fl/vextenddouble/vextenddouble2\nL2(Ω)\n≤Cm+2/summationdisplay\ni=1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂u(i)\ntt(fk\n0,fk\n1,fk\n2,fk)\n∂ν−∂u(i)\ntt(fl\n0,fl\n1,fl\n2,fl)\n∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(Σ1)\n+Cm+2/summationdisplay\ni=1/vextenddouble/vextenddoubleKi(fk\n0,fk\n1,fk\n2,fk)−Ki(fl\n0,fl\n1,fl\n2,fl)/vextenddouble/vextenddouble2\nL2(Σ1)\n≤Cm+2/summationdisplay\ni=1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂u(i)\ntt(fk\n0,fk\n1,fk\n2,fk)\n∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(Σ1)+Cm+2/summationdisplay\ni=1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂u(i)\ntt(fl\n0,fl\n1,fl\n2,fl)\n∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(Σ1)\n+Cm+2/summationdisplay\ni=1/vextenddouble/vextenddoubleKi(fk\n0,fk\n1,fk\n2,fk)−Ki(fl\n0,fl\n1,fl\n2,fl)/vextenddouble/vextenddouble2\nL2(Σ1)\nand therefore by (57) and (59) we get\nlim\nk,l→∞/vextenddouble/vextenddoublefk\ni−fl\ni/vextenddouble/vextenddouble\nL2(Ω)= lim\nk,l→∞/vextenddouble/vextenddoublefk−fl/vextenddouble/vextenddouble\nL2(Ω)= 0, i= 0,1,2.\nNamely, {fk\n0},{fk\n1},{fk\n2}are Cauchy sequences in L2(Ω) and {fk}is a Cauchy\nsequence in ( L2(Ω))n. By uniqueness of limit and in view of (58), we must have\n{fk\ni}converges to f∗\nistrongly, i= 0,1,2, and{fk}converges to f∗strongly. Hence\nwe have from (56)\n(60) /⌊a∇d⌊lf∗\n0/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lf∗\n1/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lf∗\n2/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lf∗/⌊a∇d⌊l2\nL2(Ω)= 1.\nNow again for the u(i)\ntt-system (50), by the regularity theory (30) we have that\nthe map ( f0,f1,f2,f)/mapsto→∂u(i)\ntt(f0,f1,f2,f)\n∂ν∈L2(Σ) is continuous and hence\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂u(i)\ntt(f0,f1,f2,f)\n∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(Σ)≤C/parenleftBig\n/⌊a∇d⌊lf0/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lf1/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lf2/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lf/⌊a∇d⌊l2\nL2(Ω)/parenrightBig\n.\nSince the map ( f0,f1,f2,f)/mapsto→u(i)\ntt(f0,f1,f2,f)|Σis linear, we thus have\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂u(i)\ntt(fk\n0,fk\n1,fk\n2,fk)\n∂ν−∂u(i)\ntt(f∗\n0,f∗\n1,f∗\n2,f∗)\n∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(Σ1)(61)\n≤C/parenleftBig\n/⌊a∇d⌊lfk\n0−f∗\n0/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lfk\n1−f∗\n1/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lfk\n2−f∗\n2/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lfk−f∗/⌊a∇d⌊l2\nL2(Ω)/parenrightBig\n.RECOVER ALL COEFFICIENTS 21\nThis then implies, by virtue of fk\ni→f∗\ni,i= 0,1,2 andfk→f∗strongly, that\nlim\nk→∞/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂u(i)\ntt(fk\n0,fk\n1,fk\n2,fk)\n∂ν−∂u(i)\ntt(f∗\n0,f∗\n1,f∗\n2,f∗)\n∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2(Σ1)= 0\nand hence∂u(i)\ntt(f∗\n0,f∗\n1,f∗\n2,f∗)\n∂ν= 0 inL2(Σ1) in view of (57). In other words,\n∂u(i)\nt(f∗\n0,f∗\n1,f∗\n2,f∗)\n∂νisaconstantin t∈[−T,T]. Weclaimthat∂u(i)\nt(f∗\n0,f∗\n1,f∗\n2,f∗)\n∂ν=\n0 on Σ 1. To see this, we consider the u(i)\nt(fk\n0,fk\n1,fk\n2,fk)-system\n(62)\n\n(u(i)\nt)tt−c2(x)∆u(i)\nt+q1(x)(u(i)\nt)t+q0(x)u(i)\nt+q(x)·∇u(i)\nt= (S(i)\nk)t(x,t) inQ\n(u(i)\nt)(x,0) = 0,(u(i)\nt)t(x,0) =S(i)\nk(x,0) in Ω\nu(i)\nt|Γ×[−T,T]= 0 in Σ\nwhere for i= 1,···,m+2,R(i)=R(i)(x,t) and\nS(i)\nk(x,t) =fk\n0(x)R(i)+fk\n1(x)R(i)\nt+fk(x)·∇R(i)+fk\n2(x)∆R(i).\nThe standard regularity theory (30) and trace theory implies\n/vextenddouble/vextenddouble/vextenddoubleu(i)\nt(fk\n0,fk\n1,fk\n2,fk)−u(i)\nt(f∗\n0,f∗\n1,f∗\n2,f∗)/vextenddouble/vextenddouble/vextenddouble2\nC([0,T];H1\n0(Ω))\n≤C/parenleftBig\n/⌊a∇d⌊lfk\n0−f∗\n0/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lfk\n1−f∗\n1/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lfk\n2−f∗\n2/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lfk−f∗/⌊a∇d⌊l2\nL2(Ω)/parenrightBig\nand\n/vextenddouble/vextenddouble/vextenddoubleu(i)\nt(fk\n0,fk\n1,fk\n2,fk)−u(i)\nt(f∗\n0,f∗\n1,f∗\n2,f∗)/vextenddouble/vextenddouble/vextenddouble2\nC([0,T];H1\n2(Σ)\n≤C/parenleftBig\n/⌊a∇d⌊lfk\n0−f∗\n0/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lfk\n1−f∗\n1/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lfk\n2−f∗\n2/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lfk−f∗/⌊a∇d⌊l2\nL2(Ω)/parenrightBig\n.\nNoteu(i)\nt(fk\n0,fk\n1,fk\n2,fk)(x,0) = 0 as well as the strong convergence of fk\ni→f∗\ni,\ni= 0,1,2 andfk→f∗. Thus letting k→ ∞we getu(i)\nt(f∗\n0,f∗\n1,f∗\n2,f∗)(x,0) = 0 in\nΩ andu(i)\nt(f∗\n0,f∗\n1,f∗\n2,f∗)|Σ= 0. Hence∂u(i)\nt(f∗\n0,f∗\n1,f∗\n2,f∗)\n∂ν(x,0) = 0 on Σ. Since we\nknow∂u(i)\nt(f∗\n0,f∗\n1,f∗\n2,f∗)\n∂νis a constant in t, we must have∂u(i)\nt(f∗\n0,f∗\n1,f∗\n2,f∗)\n∂ν= 0\non Σ1, as desired.22 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS\nThe above then implies∂u(i)(f∗\n0,f∗\n1,f∗\n2,f∗)\n∂νis also a constant in t. By repeating\nthe same argument, this time using the regularity theory for the u(i)(fk\n0,fk\n1,fk\n2,fk)-\nsystem and taking limit k→ ∞, we finally get∂u(i)(f∗\n0,f∗\n1,f∗\n2,f∗)\n∂ν= 0 on Σ 1.\nHence we have that u(i)(f∗\n0,f∗\n1,f∗\n2,f∗) satisfies the following\n(63)\n\nu(i)\ntt−c2(x)∆u(i)+q1(x)u(i)\nt+q0(x)u(i)+q(x)·∇u(i)=S(i)\n∗(x,t) inQ\nu(i)(x,0) =u(i)\nt(x,0) = 0 in Ω\nu(i)|Γ×[−T,T]= 0,∂u(i)\n∂ν|Γ1×[−T,T]= 0 in Σ ,Σ1\nwith\nS(i)\n∗(x,t) =f∗\n0(x)R(i)+f∗\n1(x)R(i)\nt+f∗(x)·∇R(i)+f∗\n2(x)∆R(i),i= 1,···,m+2.\nBy the uniqueness result we proved in Theorem 1.3, this must imply f∗\n0=f∗\n1=\nf∗\n2=f∗= 0, which contradicts with (60). Hence we must be able to drop the z(i)\nterms in (54). This completes the proof of Theorem 1.4.\nProof of Theorems 1.1 and 1.2 . Finally, we provide the proofs of uniqueness\nand stability of the original inverse problem. These results are pret ty much direct\nconsequences of Theorems 1.3 and 1.4 given the relationship (11) be tween the orig-\ninal inverse problem and the inverse source problem. More precisely , we have the\npositivity conditions (4) and (6) imply (15) and (17). In addition, by t he regularity\ntheory (30) the assumption (3) on the initial and boundary conditio ns{w(i)\n0,w(i)\n1,h}\nimplies the solutions w(i),i= 1,···,m+2, satisfy\n{w(i),w(i)\nt,w(i)\ntt,w(i)\nttt} ∈C/parenleftbig\n[−T,T];Hγ+1(Ω)×Hγ(Ω)×Hγ−1(Ω)×Hγ−2(Ω)/parenrightbig\n.\nAsγ >n\n2+ 4, we have the following embedding Hγ−2(Ω)֒→W2,∞(Ω) and hence\nthe regularity assumption (3) implies the corresponding regularity a ssumption (14)\nfor the inverse source problem. This completes the proof of all the theorems.\n4.Some Examples and Concluding Remarks\nIn this last section we first provide some concrete examples such th at the key\npositivity conditions (4), (6), (15), (17) are satisfied, and then g ive some general\nremarks.RECOVER ALL COEFFICIENTS 23\nExample 1 . Consider the following functions R(i)(x,t),x= (x1,···,xn)∈Ω,\nt∈[−T,T],i= 1,···,m+2, defined by\nR(1)(x,t) =t, R(i)(x,t) =x2i−3+tx2i−2,2≤i≤m+1,\nR(m+2)(x,t) =/braceleftBigg\nx2m+1+1\n2tx2\n1ifn= 2m+1 is odd\n1\n2x2\n1 ifn= 2mis even.\nThen we may easily see that the matrices U(x) and/tildewideU(x) are lower triangular\nmatrices with all 1s at the diagonal after swapping the first two colu mns. Thus\nthe determinants of the matrices U(x) and/tildewideU(x) are both −1 and hence conditions\n(15), (17) are satisfied. Correspondingly, we may choose the m+2 pairs of initial\nconditions {w(i)\n0(x),w(i)\n1(x)}as\nw(1)\n0(x) = 0, w(1)\n1(x) = 1,\nw(i)\n0(x) =x2i−3, w(i)\n1(x) =x2i−2,2≤i≤m+1,\nw(m+2)\n0(x) =/braceleftBigg\nx2m+1ifn= 2m+1 is odd\n1\n2x2\n1ifn= 2mis even\nw(m+2)\n1(x) =/braceleftBigg1\n2x2\n1ifn= 2m+1 is odd\n0 ifn= 2mis even.\nThen the matrices W(x) and/tildewiderW(x) are also lower triangular matrices with all 1s at\nthe diagonal after swapping the first two columns and hence condit ions are (4), (6)\nare satisfied.\nExample 2 . Considering the following functions R(i)(x,t),x= (x1,···,xn)∈Ω,\nt∈[−T,T],i= 1,···,m+2, defined by\nR(1)(x,t) = sin t, R(i)(x,t) = costex2i−3+sintex2i−2,2≤i≤m+1,\nR(m+2)(x,t) =/braceleftBigg\ncostex2m+1+sinte−x1ifn= 2m+1 is odd\ncoste−x1 ifn= 2mis even.24 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS\nThen the matrix U(x) becomes\nU(x) =\n0 1 0 0 ···0 0 0\n1 0 0 0 ···0 0 0\nex1ex2ex10 0 ···0ex1\nex2−ex10ex20···0ex2\n........................\nexn−1−exn−20···0exn−10exn−1\nexne−x10···0 0 exnexn\ne−x1−exn−e−x10···0 0 e−x1\n(64)\nNotice that U(x) is not a lower triangular matrix. However, we can easily transform\ntheitinto alower triangularmatrixbyswapping thefirst twocolumns a ndsubtract-\ning the 3rd, 4th, ..., (n+2)th column from the last column. As a conseq uence we get\ndetU(x) =−2/producttextn\ni=2exi. Inasimilar fashionwecanalsogetdet ˜U(x) =−2/producttextn\ni=2exi.\nAs Ω is a bounded domain, we hence have the conditions (15) and (17) are satisfied.\nCorrespondingly we may choose the m+2 pairs of initial conditions {w(i)\n0,w(i)\n1}\nas\nw(1)\n0(x) = 0, w(1)\n1(x) = 1,\nw(i)\n0(x) =ex2i−3, w(i)\n1(x) =ex2i−2,2≤i≤m+1,\nw(m+2)\n0(x) =/braceleftBigg\nexnifn= 2m+1 is odd\ne−x1ifn= 2mis even\nw(m+2)\n1(x) =/braceleftBigg\ne−x1ifn= 2m+1 is odd\n0 if n= 2mis even.\nThen the determinants of both W(x) and/tildewiderW(x) are also −2/producttextn\ni=2exi, calculated in\nthe same manner as in the case of U(x) and/tildewideU(x). Hence the conditions (4) and\n(6) are satisfied.\nExample 3 . In general if we have f(j)∈C2(Ω) with f(j)(x) =f(j)(x1,···,xj),\n1≤j≤nandg,h∈C2[−T,T] that satisfies\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂f(j)\n∂xj/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥rj>0,1≤j≤n,/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂2f(1)\n∂x2\n1/vextendsingle/vextendsingle/vextendsingle/vextendsingle≥˜r1>0\ng(0) =h′(0) = 1, g′(0) =h(0) = 0.RECOVER ALL COEFFICIENTS 25\nfor some positive rj, 1≤j≤n, and ˜r1. Then we may consider the functions\nR(i)(x,t),x= (x1,···,xn)∈Ω,t∈[−T,T],i= 1,···,m+ 2, of the following\nform:\nR(1)(x,t) =h(t), R(i)(x,t) =f(2i−3)(x)g(t)+f(2i−2)(x)h(t),2≤i≤m+1,\nR(m+2)(x,t) =/braceleftBigg\nf(n)(x)g(t)+f(1)(ax)h(t) ifn= 2m+1 is odd\nf(1)(ax)g(t) if n= 2mis even.\nwherea <0 so that ax∈Ω.\nCorrespondingly we may choose the m+2 pairs of initial conditions {w(i)\n0,w(i)\n1}\nas\nw(1)\n0(x) = 0, w(1)\n1(x) = 1,\nw(i)\n0(x) =f(2i−3)(x), w(i)\n1(x) =f(2i−2)(x),2≤i≤m+1,\nw(m+2)\n0(x) =/braceleftBigg\nf(n)(x) ifn= 2m+1 is odd\nf(1)(ax) ifn= 2mis even\nw(m+2)\n1(x) =/braceleftBigg\nf(1)(ax) ifn= 2m+1 is odd\n0 if n= 2mis even.\nIn this case, after swapping the first and second column, the last c olumn with the\npreceding ( n+2)th, (n+1)th,···, and finally the 3rd column, as well as swapping\nthe last row with the preceding ( n+2)th, (n+1)th,···, and finally the 3rd row. We\nmay get the determinants of the matrices U(x),/tildewideU(x),W(x) and/tildewiderW(x) are equal to\n/parenleftbig\na∂x1f(1)(ax)∂2\nx1f(1)(x)−a2∂2\nx1f(1)(ax)∂x1f(1)(x)/parenrightbign/productdisplay\nj=2∂xjf(j)(x).\nSincef(1)∈C2(Ω),|∂x1f(1)| ≥r1>0 and|∂2\nx1f(1)| ≥˜r1>0,∂x1f(1)and∂2\nx1f(1)\ndo not change sign. Hence we have\n|/parenleftbig\na∂x1f(1)(ax)∂2\nx1f(1)(x)−a2∂2\nx1f(1)(ax)∂x1f(1)(x)/parenrightbign/productdisplay\nj=2∂xjf(j)(x)| ≥(a2+|a|)˜r1n/productdisplay\nj=1rj.\nHence the positivity conditions (4), (6), (15) and (17) are satisfie d.\nFinally, we end the paper with some comments and remarks.\n(1) We have shown in this paper that in order to recover all the coeffi cients,\nwe need to appropriately choose ⌊n+4\n2⌋pairs of initial conditions {w0,w1}and a\nboundary condition h, and then use their corresponding boundary measurements.\nAs mentioned earlier, since in total there are n+3 unknown functions, it is natural26 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS\nto expect to recover them from n+3boundary measurements. Indeed, following the\napproach of this paper, we can also achieve the recovery by appro priately choosing\nn+ 3 initial positions w0with an initial velocity w1and a boundary condition h,\nand then use their corresponding boundary measurements. In pa rticular, in this\ncase the positivity condition becomes\ndet\nw(1)\n0(x)w1(x)∂x1w(1)\n0(x)···∂xnw(1)\n0(x) ∆w(1)\n0(x)\nw(2)\n0(x)w1(x)∂x1w(2)\n0(x)···∂xnw(2)\n0(x) ∆w(2)\n0(x)\n..................\nw(n+3)\n0(x)w1(x)∂x1w(n+3)\n0(x)···∂xnw(n+3)\n0(x) ∆w(n+3)\n0(x)\n≥r0>0.\nNote although in this case we need more measurements, an advanta ge is that we\nonly need to differentiate the u-equation with respect to ttwice, rather than three\ntimes. We may also get a better stability estimate of the form\n/⌊a∇d⌊lc2−˜c2/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lq1−p1/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lq0−p0/⌊a∇d⌊l2\nL2(Ω)+/⌊a∇d⌊lq−p/⌊a∇d⌊l2\nL2(Ω)\n≤Cn+3/summationdisplay\ni=1/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∂w(i)\nt(c,q1,q0,q)\n∂ν−∂w(i)\nt(˜c,p1,p0,p)\n∂ν/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(Σ1)\nin the sense that we only need to differentiate the measurements in t ime once.\n(2) In our problem formulation we use the time interval [ −T,T] and regard the\nmiddlet= 0 as initial time. This is not essential since a simple change of variable\nt→t−Ttransforms t= 0 tot=−T. However, this present choice allows\nthe recovery of all coefficients with fewer choices of initial condition s and hence\nfewer boundary measurements. This is because we may use both eq uations in (40),\ncomparetojustoneifweassumethetimeintervalas[0 ,T]andthenextendsolutions\nto [−T,0].\n(3) It is also possible to set up the inverse problem by assuming Neuma nn bound-\nary condition∂w\n∂νon Σ = Γ ×[−T,T] and making measurements of Dirichlet bound-\nary traces of the solution wover Σ 1= Γ1×[−T,T], such as in [18]. This, however,\nwould require more demanding geometrical assumption on the unobs erved portion\nof the boundaryΓ 0. Forexample, we may need to assume∂d\n∂ν=/a\\}⌊∇a⌋ketle{tDd,ν/a\\}⌊∇a⌋ket∇i}ht= 0 onΓ 0in\nthe geometrical assumption to account for the Neumann boundar y condition [22].\nIn addition, the more delicate regularity theory of second-order h yperbolic equation\nwith nonhomogeneous Neumann boundary condition will also need to b e invoked\n[14], [15]. Nevertheless, the main ideas of solving the inverse problem r emain the\nsame.\nAcknowledgementsRECOVER ALL COEFFICIENTS 27\nThe first author would like to thank Professor Yang Yang for many v ery useful\ndiscussions.\nReferences\n[1] M. Belishev, An approach to multidimensional inverse problems for the wave equation, Dokl.\nAkad. Nauk SSSR 297(1987), no.3, 524–527.\n[2] M. Belishev, Boundary control in reconstruction of manifolds an d metrics (BC method),\nInverse Problems, 13(1997), R1–R45.\n[3] M. Bellassoued, Uniqueness and stability in determining the speed o f propagation of second-\norder hyperbolic equation with variable coefficients, Appl. Anal., 83(2004), 983–1014.\n[4] M. Bellassoued and M. Yamamoto, “Carleman Estimates and Applica tions to Inverse Prob-\nlems for Hyperbolic Systems,” Springer Monographs in Mathematics S eries, 2017.\n[5] A. Bukhgeim and M. Klibanov, Global uniqueness of a class of multidimensional inverse\nproblems , Dokl. Akad. Nauk SSSR, 260(1981), 269–272.\n[6] V. Isakov, “Inverse Source Problems,” American Mathematical Society, 2000.\n[7] V.Isakov,“InverseProblemsforPartialDifferentialEquation s,”2ndedition, Springer-Verlag,\nNew York, 2006.\n[8] D. Jellali, An inverse problem for the acoustic wave equation with finite sets of boundary data ,\nJournal of Inverse and Ill-posed Problems, 14(2006), 665–684.\n[9] M. Klibanov, “Carleman Estimates for Coefficient Inverse Problem s and Numerical Applica-\ntions,” VSP, Utrecht, 2004.\n[10] M. Klibanov and J. Li, “Inverse Problems and Carleman Estimates : Global Uniqueness,\nGlobal Convergence and Experimental Data,” Inverse and Ill-pose d Problems Series 63, De\nGruyter, 2021.\n[11] A. Katchalov, Y. Kurylev and M. Lassas, “Inverse Boundary S pectral Problems,” Chapman\n& Hall/CRC, 2001.\n[12] Y. Kurylev and M. Lassas, Hyperbolic inverse boundary-value problem and time-conti nuation\nof the non-stationary Dirichlet-to-Neumann map , Proc. Roy. Soc. Edinburgh Sect. A, 132\n(2002), no. 4, 931–949.\n[13] I. Lasiecka, J.L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second\norder hyperbolic operators , J. Math. Pures Appl., 65(1986), 149–192.\n[14] I. Lasiecka and R. Triggiani, Sharp regularity theory for second order hyperbolic equati ons\nof Neumann type. Part I. L2Nonhomogeneous data , Ann. Mat. Pura. Appl. (IV), CLVII\n(1990), 285–367.\n[15] I. Lasiecka and R. Triggiani, Regularity theory of hyperbolic equations with non-homoge neous\nNeumann boundary conditions. II. General boundary data , Journal of Differential Equations,\n94(1991), 112–164.\n[16] I. Lasiecka, R. Triggianiand X. Zhang, Nonconservative wave equations with unobserved Neu-\nmann B.C.: Global uniqueness and observability in one shot , Differential geometric methods\nin the control of partial differential equations, 227–325, Amer. M ath. Soc., Providence, RI,\n2000.\n[17] M.M. Lavrentev, V.G. Romanov and S.P. Shishataskii, “Ill-Posed P roblems of Mathematical\nPhysics and Analysis,” Vol. 64, Amer. Math. Soc., Providence, RI, 1986.\n[18] S.Liu andR.Triggiani, Global uniqueness and stability in determining the damping coefficient\nof an inverse hyperbolic problem with non-homogeneous Neum ann B.C. through an additional\nDirichlet boundary trace , SIAM J. Math. Anal., 43(2011), 1631–1666.28 SHITAO LIU, ANTONIO PIERROTTET AND SCOTT SCRUGGS\n[19] S. Liu and R. Triggiani, Global uniqueness and stability in determining the damping and\npotential coefficients of an inverse hyperbolic problem , Nonlinear Anal. Real World Appl., 91\n(2011), 1562–1590.\n[20] S. Liu, Recovery of the sound speed and initial displacement for the wave equation by means of\na single Dirichlet boundary measurement , Evolution Equations and Control Theory, 2(2013),\n355–364.\n[21] S. Liu and R. Triggiani, Boundary control and boundary inverse theory for non-homog eneous\nsecond order hyperbolic equations: a common Carleman estim ates approach , HCDTE Lecture\nNotes, AIMS on Applied Mathematics Vol 6 (2012), 227–343.\n[22] R. Triggiani and P. F. Yao, Carleman estimates with no lower-order terms for general Ri e-\nmannian wave equations. Global uniqueness and observabili ty in one shot , Appl. Math. Op-\ntim.,46(2002), 331–375.\nSchool of Mathematical and Statistical Sciences, Clemson U niversity, Clemson,\nSC 29634\nEmail address :liul@clemson.edu\nEmail address :srscrug@g.clemson.edu\nEmail address :ampierr@g.clemson.edu"    },    {        "title": "2109.03684v2.Room_Temperature_Intrinsic_and_Extrinsic_Damping_in_Polycrystalline_Fe_Thin_Films.pdf",        "content": "Room-Temperature Intrinsic and Extrinsic Damping in\nPolycrystalline Fe Thin Films\nShuang Wu,1David A. Smith,1Prabandha Nakarmi,2Anish Rai,2Michael Clavel,3Mantu\nK. Hudait,3Jing Zhao,4F. Marc Michel,4Claudia Mewes,2Tim Mewes,2and Satoru Emori1\n1Department of Physics, Virginia Polytechnic Institute\nand State University, Blacksburg, VA 24061, USA\n2Department of Physics and Astronomy,\nThe University of Alabama, Tuscaloosa, AL 35487 USA\n3Department of Electrical and Computer Engineering,\nVirginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA\n4Department of Geosciences, Virginia Polytechnic Institute\nand State University, Blacksburg, VA 24061, USA\nAbstract\nWe examine room-temperature magnetic relaxation in polycrystalline Fe \flms. Out-of-plane fer-\nromagnetic resonance (FMR) measurements reveal Gilbert damping parameters of \u00190.0024 for Fe\n\flms with thicknesses of 4-25 nm, regardless of their microstructural properties. The remarkable\ninvariance with \flm microstructure strongly suggests that intrinsic Gilbert damping in polycrys-\ntalline metals at room temperature is a local property of nanoscale crystal grains, with limited\nimpact from grain boundaries and \flm roughness. By contrast, the in-plane FMR linewidths of\nthe Fe \flms exhibit distinct nonlinear frequency dependences, indicating the presence of strong\nextrinsic damping. To \ft our in-plane FMR data, we have used a grain-to-grain two-magnon scat-\ntering model with two types of correlation functions aimed at describing the spatial distribution of\ninhomogeneities in the \flm. However, neither of the two correlation functions is able to reproduce\nthe experimental data quantitatively with physically reasonable parameters. Our \fndings advance\nthe fundamental understanding of intrinsic Gilbert damping in structurally disordered \flms, while\ndemonstrating the need for a deeper examination of how microstructural disorder governs extrinsic\ndamping.\n1arXiv:2109.03684v2  [cond-mat.mtrl-sci]  24 Feb 2022I. INTRODUCTION\nIn all magnetic materials, magnetization has the tendency to relax toward an e\u000bective\nmagnetic \feld. How fast the magnetization relaxes governs the performance of a variety\nof magnetic devices. For example, magnetization relaxation hinders e\u000ecient precessional\ndynamics and should be minimized in devices such as precessional magnetic random access\nmemories, spin-torque oscillators, and magnonic circuits1{4. From the technological perspec-\ntive, it is important to understand the mechanisms behind magnetic relaxation in thin-\flm\nmaterials that comprise various nanomagnetic device applications. Among these materials,\nbcc Fe is a prototypical elemental ferromagnet with attractive properties, including high sat-\nuration magnetization, soft magnetism5, and large tunnel magnetoresistance6,7. Our present\nstudy is therefore motivated by the need to uncover magnetic relaxation mechanisms in Fe\nthin \flms { particularly polycrystalline \flms that can be easily grown on arbitrary substrates\nfor diverse applications.\nTo gain insights into the contributions to magnetic relaxation, a common approach is to\nexamine the frequency dependence of the ferromagnetic resonance (FMR) linewidth. The\nmost often studied contribution is viscous Gilbert damping8{13, which yields a linear increase\nin FMR linewidth with increasing precessional frequency. In ferromagnetic metals, Gilbert\ndamping arises predominately from \\intrinsic\" mechanisms14{16governed by the electronic\nband structure17. Indeed, a recent experimental study by Khodadadi et al.18has shown\nthat intrinsic, band-structure-based Gilbert damping dominates magnetic relaxation in high-\nquality crystalline thin \flms of Fe, epitaxially grown on lattice-matched substrates. However,\nit is yet unclear how intrinsic damping is impacted by the microstructure of polycrystalline\nFe \flms.\nMicrostructural disorder in polycrystalline Fe \flms can also introduce extrinsic magnetic\nrelaxation. A well-known extrinsic relaxation mechanism is two-magnon scattering, where\nthe uniform precession mode with zero wave vector scatters into a degenerate magnon mode\nwith a \fnite wave vector19{22. Two-magnon scattering generally leads to a nonlinear fre-\nquency dependence of the FMR linewidth, governed by the nature of magnon scattering\ncenters at the surfaces23,24or in the bulk of the \flm25{28. While some prior experiments\npoint to the prominent roles of extrinsic magnetic relaxation in polycrystalline ferromag-\nnetic \flms29{31, systematic studies of extrinsic relaxation (e.g., two-magnon scattering) on\n2polycrystalline Fe thin \flms are still lacking.\nHere, we investigate both the intrinsic and extrinsic contributions to magnetic relax-\nation at room temperature in polycrystalline Fe \flms. We have measured the frequency\ndependence of the FMR linewidth with (1) the \flm magnetized out-of-plane (OOP), where\ntwo-magnon scattering is suppressed25such that intrinsic Gilbert damping is quanti\fed re-\nliably, and (2) the \flm magnetized in-plane (IP), where two-magnon scattering is generally\nexpected to coexist with intrinsic Gilbert damping.\nFrom OOP FMR results, we \fnd that the intrinsic Gilbert damping of polycrystalline Fe\n\flms at room temperature is independent of their structural properties and almost identical\nto that of epitaxial \flms. Such insensitivity to microstructure is in contrast to disorder-\nsensitive Gilbert damping recently shown in epitaxial Fe at cryogenic temperature18. Our\npresent work implies that Gilbert damping at a su\u000eciently high temperature becomes a\nlocal property of the metal, primarily governed by the structure within nanoscale crystal\ngrains rather than grain boundaries or interfacial disorder. This implication refutes the\nintuitive expectation that intrinsic Gilbert damping should depend on structural disorder in\npolycrystalline \flms.\nIn IP FMR results, the frequency dependence of the FMR linewidth exhibits strong\nnonlinear trends that vary signi\fcantly with \flm microstructure. To analyze the nonlin-\near trends, we have employed the grain-to-grain two-magnon scattering model developed\nby McMichael and Krivosik25with two types of correlation functions for capturing inho-\nmogeneities in the \flm. However, neither of the correlation functions yields quantitative\nagreement with the experimental results or physically consistent, reasonable parameters.\nThis \fnding implies that a physical, quantitative understanding of extrinsic magnetic re-\nlaxation requires further corrections of the existing two-magnon scattering model, along\nwith much more detailed characterization of the nanoscale inhomogeneities of the magnetic\n\flm. Our study stimulates opportunities for a deeper examination of fundamental magnetic\nrelaxation mechanisms in structurally disordered ferromagnetic metal \flms.\nII. FILM DEPOSITION AND STRUCTURAL PROPERTIES\nPolycrystalline Fe thin \flms were deposited using DC magnetron sputtering at room\ntemperature on Si substrates with a native oxide layer of SiO 2. The base pressure of the\n3chamber was below 1 \u000210\u00007Torr and all \flms were deposited with 3 mTorr Ar pressure. Two\nsample series with di\u000berent seed layers were prepared in our study: subs./Ti(3 nm)/Cu(3\nnm)/Fe(2-25 nm)/Ti(3 nm) and subs./Ti(3 nm)/Ag(3 nm)/Fe(2-25 nm)/Ti(3 nm). In this\npaper we refer to these two sample series as Cu/Fe and Ag/Fe, respectively. The layer\nthicknesses are based on deposition rates derived from x-ray re\rectivity (XRR) of thick\ncalibration \flms. The Ti layer grown directly on the substrate ensures good adhesion of\nthe \flm, whereas the Cu and Ag layers yield distinct microstructural properties for Fe\nas described below. We note that Cu is often used as a seed layer for growing textured\npolycrystalline ferromagnetic metal \flms32,33. Our initial motivation for selecting Ag as an\nalternative seed layer was that it might promote qualitatively di\u000berent Fe \flm growth34,\nowing to a better match in bulk lattice parameter 𝑎between Fe ( 𝑎\u0019286\u0017A) and Ag\n(𝑎p\n2\u0019288\u0017A) compared to Fe and Cu ( 𝑎p\n2\u0019255\u0017A).\nWe performed x-ray di\u000braction (XRD) measurements to compare the structural properties\nof the Cu/Fe and Ag/Fe \flms. Figure 1(a,b) shows symmetric 𝜃-2𝜃XRD scan curves\nfor several \flms from both the Cu/Fe and Ag/Fe sample series. For all Cu/Fe \flms, the\n(110) body-center-cubic (bcc) peak can be observed around 2 𝜃=44°\u000045°(Fig. 1(a)). This\nobservation con\frms that the Fe \flms grown on Cu are polycrystalline and textured, where\nthe crystal grains predominantly possess (110)-oriented planes that are parallel to the sample\nsurface. For Ag/Fe (Fig. 1(b)), the (110) bcc peak is absent or extremely weak, from\nwhich one might surmise that the Fe \flms grown on Ag are amorphous or only possess\nweak crystallographic texture. However, we \fnd that the Ag/Fe \flms are, in fact, also\npolycrystalline with evidence of (110) texturing. In the following, we elaborate on our XRD\nresults, \frst for Cu/Fe and then Ag/Fe.\nWe observe evidence for a peculiar, non-monotonic trend in the microstructural properties\nof the Cu/Fe \flms. Speci\fcally, the height of the 𝜃-2𝜃di\u000braction peak (Fig. 1(a)) increases\nwith Fe \flm thickness up to \u001910 nm but then decreases at higher Fe \flm thicknesses. While\nwe do not have a complete explanation for this peculiar nonmonotonic trend with \flm\nthickness, a closer inspection of the XRD results (Fig. 1) provides useful insights. First, the\nFe \flm di\u000braction peak shifts toward a higher 2 𝜃value with increasing \flm thickness. This\nsigni\fes that thinner Fe \flms on Cu are strained (with the Fe crystal lattice tetragonally\ndistorted), whereas thicker Fe \flms undergo structural relaxation such that the out-of-plane\nlattice parameter converges toward the bulk value of \u00192.86 \u0017A, as summarized in Fig. 1(e).\n4354 04 55 05 5Ag/Fe2 nm6 nmIntensity [arb. unit]  \nCu/Febulk bcc Fe (110)1\n0 nm15 nm25 nm8\n nm(\na)\n4045502\nθ [deg]10 nm2\n5 nm \n2\nθ [deg]10 nm15 nm6\n nm2\n nm8 nm(\nb)\n16182022242628Ag/Fe2 nm6 nmIntensity [arb. unit]  \nCu/Febulk bcc Fe (110)1\n0 nm15 nm25 nm8\n nm(\nc)2\n5 nm \nθ\n [deg]10 nm15 nm6\n nm2\n nm8 nm(\nd)\n2.842.862.882.902.922.940\n5 10152025051015Bulk value 2.86 Cu/Fe \nAg/FeOut-of-planel\nattice parameter [Å](\ne)Crystallite size [nm]T\nhickness [nm](f)FIG. 1. (Color online) 𝜃-2𝜃X-ray di\u000braction scan curves for (a) Cu/Fe (blue lines) and (b) Ag/Fe\n(red lines) sample series. The inset in (b) is the grazing-incidence XRD scan curve for 10 nm thick\nAg/Fe \flm. Rocking curves for (c) Cu/Fe (blue lines) and (d) Ag/Fe (red lines) sample series.\n(e) Out-of-plane lattice parameter estimated via Bragg's law using the 2 𝜃value at the maximum\nof the tallest \flm di\u000braction peak. (f) Crystallite size estimated via the Scherrer equation using\nthe full-width-at-half-maximum of the tallest \flm di\u000braction peak. In (e) and (f), the data for the\nAg/Fe \flm series at a few thickness values are missing because of the absence of the bcc (110) peak\nin𝜃-2𝜃XRD scans.\nSecond, as the Fe \flm thickness approaches \u001910 nm, additional di\u000braction peaks appear to\nthe left of the tall primary peak. We speculate that these additional peaks may originate\nfrom Fe crystals that remain relatively strained (i.e., with an out-of-plane lattice parameter\nlarger than the bulk value), while the primary peak arises from more relaxed Fe crystals\n(i.e., with a lattice parameter closer to the bulk value). The coexistence of such di\u000berent\nFe crystals appears to be consistent with the rocking curve measurements (Fig. 1(c)), which\nexhibit a large broad background peak in addition to a small sharp peak for Cu/Fe \flms\nwith thicknesses near \u001910 nm. As we describe in Sec. IV, these \u001910 nm thick Cu/Fe samples\nalso show distinct behaviors in extrinsic damping (highly nonlinear frequency dependence of\n5the FMR linewidth) and static magnetization reversal (enhanced coercivity), which appear\nto be correlated with the peculiar microstructural properties evidenced by our XRD results.\nOn the other hand, it is worth noting that the estimated crystal grain size (Fig. 1(f)) {\nderived from the width of the 𝜃-2𝜃di\u000braction peak { does not exhibit any anomaly near the\n\flm thickness of\u001910 nm, but rather increases monotonically with \flm thickness.\nUnlike the Cu/Fe \flms discussed above, the Ag/Fe \flms do not show a strong (110) bcc\npeak in the 𝜃-2𝜃XRD results. However, the lack of pronounced peaks in the symmetric 𝜃-2𝜃\nscans does not necessarily signify that Ag/Fe is amorphous. This is because symmetric 𝜃-2𝜃\nXRD is sensitive to crystal planes that are nearly parallel to the sample surface, such that the\ndi\u000braction peaks capture only the crystal planes with out-of-plane orientation with a rather\nsmall range of misalignment (within \u00181°, dictated by incident X-ray beam divergence). In\nfact, from asymmetric grazing-incidence XRD scans that are sensitive to other planes, we\nare able to observe a clear bcc Fe (110) di\u000braction peak even for Ag/Fe samples that lack\nan obvious di\u000braction peak in 𝜃-2𝜃scans (see e.g. inset of Fig. 1(b)). Furthermore, rocking\ncurve scans (conducted with 2 𝜃\fxed to the expected position of the (110) Fe \flm di\u000braction\npeak) provide orientation information over an angular range much wider than \u00181°. As shown\nin Fig. 1(d), a clear rocking curve peak is observed for each Ag/Fe sample, suggesting that\nFe \flms grown on Ag are polycrystalline and (110)-textured { albeit with the (110) crystal\nplanes more misaligned from the sample surface compared to the Cu/Fe samples. The out-\nof-plane lattice parameters of Ag/Fe \flms (with discernible 𝜃-2𝜃di\u000braction \flm peaks) show\nthe trend of relaxation towards the bulk value with increasing Fe thickness, similar to the\nCu/Fe series. Yet, the lattice parameters for Ag/Fe at small thicknesses are systematically\ncloser to the bulk value, possibly because Fe is less strained (i.e., better lattice matched)\non Ag than on Cu. We also \fnd that the estimation of the crystal grain size for Ag/Fe {\nalthough made di\u000ecult by the smallness of the di\u000braction peak { yields a trend comparable\nto Cu/Fe, as shown in Fig. 1(f).\nWe also observe a notable di\u000berence between Cu/Fe and Ag/Fe in the properties of \flm\ninterfaces, as revealed by XRR scans in Fig. 2. The oscillation period depends inversely\non the \flm thickness. The faster decay of the oscillatory re\rectivity signal at high angles\nfor the Ag/Fe \flms suggests that the Ag/Fe \flms may have rougher interfaces compared to\nthe Cu/Fe \flms. Another interpretation of the XRR results is that the Ag/Fe interface is\nmore di\u000buse than the Cu/Fe interface { i.e., due to interfacial intermixing of Ag and Fe. By\n60.000.050.100.150.200.250.3010 nm2\n5 nmReflectivity [a.u.] \n(\na) Cu/Fe \nAg/Fe \nq\nz [Å-1](b)FIG. 2. (Color online) X-ray re\rectivity scans of 10 nm and 25 nm thick \flms from (a) Cu/Fe\n(blue circles) and (b) Ag/Fe (red squares) sample series. Black solid curves are \fts to the data.\n\ftting the XRR results35, we estimate an average roughness (or the thickness of the di\u000buse\ninterfacial layer) of .1 nm for the Fe layer in Cu/Fe, while it is much greater at \u00192-3 nm\nfor Ag/Fe36.\nOur structural characterization described above thus reveals key attributes of the Cu/Fe\nand Ag/Fe sample series. Both \flm series are polycrystalline, exhibit (110) texture, and\nhave grain sizes of order \flm thickness. Nevertheless, there are also crucial di\u000berences\nbetween Cu/Fe and Ag/Fe. The Cu/Fe series overall exhibits stronger 𝜃-2𝜃di\u000braction\npeaks than the Ag/Fe series, suggesting that the (110) bcc crystal planes of Fe grown on\nCu are aligned within a tighter angular range than those grown on Ag. Moreover, Fe grown\non Cu has relatively smooth or sharp interfaces compared to Fe grown on Ag. Although\nidentifying the origin of such structural di\u000berences is beyond the scope of this work, Cu/Fe\n7and Ag/Fe constitute two qualitatively distinct series of polycrystalline Fe \flms for exploring\nthe in\ruence of microstructure on magnetic relaxation.\nIII. INTRINSIC GILBERT DAMPING PROBED BY OUT-OF-PLANE FMR\nHaving established the di\u000berence in structural properties between Cu/Fe and Ag/Fe, we\ncharacterize room-temperature intrinsic damping for these samples with OOP FMR mea-\nsurements. The OOP geometry suppresses two-magnon scattering25such that the Gilbert\ndamping parameter can be quanti\fed in a straightforward manner. We use a W-band\nshorted waveguide in a superconducting magnet, which permits FMR measurements at high\n\felds ( &4 T) that completely magnetize the Fe \flms out of plane. The details of the mea-\nsurement method are found in Refs.18,37. Figure 3(a) shows the frequency dependence of\nhalf-width-at-half-maximum (HWHM) linewidth Δ𝐻OOP for selected thicknesses from both\nsample series. The linewidth data of 25 nm thick epitaxial Fe \flm from a previous study18\nis plotted in Fig. 3 (a) as well. The intrinsic damping parameter can be extracted from the\nlinewidth plot using\nΔ𝐻OOP=Δ𝐻0¸2𝜋\n𝛾𝛼OOP𝑓 (1)\nwhereΔ𝐻0is the inhomogeneous broadening38,𝛾=𝑔𝜇𝐵\nℏis the gyromagnetic ratio ( 𝛾2𝜋\u0019\n2.9 MHz/Oe [Ref.39], obtained from the frequency dependence of resonance \feld37), and\n𝛼OOP is the measured viscous damping parameter. In general, 𝛼OOP can include not only\nintrinsic Gilbert damping, parameterized by 𝛼int, but also eddy-current, radiative damping,\nand spin pumping contributions40, which all yield a linear frequency dependence of the\nlinewidth. Damping due to eddy current is estimated to make up less than 10% of the total\nmeasured damping parameter37and is ignored here. Since we used a shorted waveguide in\nour setup, the radiative damping does not apply here. Spin pumping is also negligible for\nmost of the samples here because the materials in the seed and capping layers (i.e., Ti, Cu,\nand Ag) possess weak spin-orbit coupling and are hence poor spin sinks31,41,42. We therefore\nproceed by assuming that the measured OOP damping parameter 𝛼OOP is equivalent to the\nintrinsic Gilbert damping parameter.\nThe extracted damping parameter is plotted as a function of Fe \flm thickness in Fig.\n3(b). The room-temperature damping parameters of all Fe \flms with thicknesses of 4-25\n80204060801001200306090120150180 \n25nm epitaxial Fe \n10nm Cu/Fe \n25nm Cu/Fe \n10nm Ag/Fe \n25nm Ag/FeΔHOOP [Oe]f\n [GHz](a)\n05101520250.0000.0010.0020.0030.004 epitaxial Fe \nCu/Fe \nAg/FeαOOPT\nhickness [nm](b)FIG. 3. (Color online) (a) OOP FMR half-width-at-half-maximum linewidth Δ𝐻OOPas a function\nof resonance frequency 𝑓. Lines correspond to \fts to the data. (b) Gilbert damping parameter\n𝛼𝑚𝑎𝑡ℎ𝑟𝑚𝑂𝑂𝑃 extracted from OOP FMR as a function of \flm thickness. The red shaded area\nhighlights the damping value range that contains data points of all \flms thicker than 4 nm. The\ndata for the epitaxial Fe sample (25 nm thick Fe grown on MgAl 2O4) are adapted from Ref.18.\nnm fall in the range of 0.0024 \u00060.0004, which is shaded in red in Fig. 3(b). This damping\nparameter range is quantitatively in line with the value reported for epitaxial Fe (black\nsymbol in Fig. 3(b))18. For 2 nm thick samples, the damping parameter is larger likely\ndue to an additional interfacial contribution43{45{ e.g., spin relaxation through interfacial\nRashba spin-orbit coupling46that becomes evident only for ultrathin Fe. The results in\nFig. 3(b) therefore indicate that the structural properties of the &4 nm thick polycrystalline\nbcc Fe \flms have little in\ruence on their intrinsic damping.\nIt is remarkable that these polycrystalline Cu/Fe and Ag/Fe \flms { with di\u000berent thick-\n9nesses and microstructural properties (as revealed in Sec. II) { exhibit essentially the same\nroom-temperature intrinsic Gilbert damping parameter as single-crystalline bcc Fe. This\n\fnding is qualitatively distinct from a prior report18on intrinsic Gilbert damping in single-\ncrystalline Fe \flms at cryogenic temperature, which is sensitive to microstructural disorder.\nIn the following, we discuss the possible di\u000berences in the mechanisms of intrinsic damping\nbetween these temperature regimes.\nIntrinsic Gilbert damping in ferromagnetic metals is predominantly governed by transi-\ntions of spin-polarized electrons between electronic states, within a given electronic band\n(intraband scattering) or in di\u000berent electronic bands (interband scattering) near the Fermi\nlevel15. For Fe, previous studies15,18,47indicate that intraband scattering tends to dominate\nat low temperature where the electronic scattering rate is low (e.g., \u00181013s\u00001); by contrast,\ninterband scattering likely dominates at room temperature where the electronic scattering\nrate is higher (e.g., \u00181014s\u00001). According to our results (Fig. 3(b)), intrinsic damping at\nroom temperature is evidently una\u000bected by the variation in the structural properties of the\nFe \flms. Hence, the observed intrinsic damping is mostly governed by the electronic band\nstructure within the Fe grains , such that disorder in grain boundaries or \flm interfaces has\nminimal impact.\nThe question remains as to why interband scattering at room temperature leads to Gilbert\ndamping that is insensitive to microstructural disorder, in contrast to intraband scattering\nat low temperature yielding damping that is quite sensitive to microstructure18. This dis-\ntinction may be governed by what predominantly drives electronic scattering { speci\fcally,\ndefects (e.g., grain boundaries, rough or di\u000buse interfaces) at low temperature, as opposed\nto phonons at high temperature. That is, the dominance of phonon-driven scattering at\nroom temperature may e\u000bectively diminish the roles of microstructural defects in Gilbert\ndamping. Future experimental studies of temperature-dependent damping in polycrystalline\nFe \flms may provide deeper insights. Regardless of the underlying mechanisms, the robust\nconsistency of 𝛼OOP (Fig. 3(b)) could be an indication that the intrinsic Gilbert damp-\ning parameter at a su\u000eciently high temperature is a local property of the ferromagnetic\nmetal, possibly averaged over the ferromagnetic exchange length of just a few nm48that is\ncomparable or smaller than the grain size. In this scenario, the impact on damping from\ngrain boundaries would be limited in comparison to the contributions to damping within\nthe grains.\n10Moreover, the misalignment of Fe grains evidently does not have much in\ruence on the\nintrinsic damping. This is reasonable considering that intrinsic Gilbert damping is predicted\nto be nearly isotropic in Fe at su\u000eciently high electronic scattering rates49{ e.g.,\u00181014s\u00001\nat room temperature where interband scattering is expected to be dominant15,18,47. It is\nalso worth emphasizing that 𝛼OOP remains unchanged for Fe \flms of various thicknesses\nwith di\u000berent magnitudes of strain (tetragonal distortion, as evidenced by the variation in\nthe out-of-plane lattice parameter in Fig. 1(e)). Strain in Fe grains is not expected to impact\nthe intrinsic damping, as Ref.18suggests that strain in bcc Fe does not signi\fcantly alter\nthe band structure near the Fermi level. Thus, polycrystalline Fe \flms exhibit essentially\nthe same magnitude of room-temperature intrinsic Gilbert damping as epitaxial Fe, as long\nas the grains retain the bcc crystal structure.\nThe observed invariance of intrinsic damping here is quite di\u000berent from the recent study\nof polycrystalline Co 25Fe75alloy \flms31, reporting a decrease in intrinsic damping with in-\ncreasing structural disorder. This inverse correlation between intrinsic damping and disorder\nin Ref.31is attributed to the dominance of intraband scattering, which is inversely propor-\ntional to the electronic scattering rate. It remains an open challenge to understand why the\nroom-temperature intrinsic Gilbert damping of some ferromagnetic metals might be more\nsensitive to structural disorder than others.\nIV. EXTRINSIC MAGNETIC RELAXATION PROBED BY IN-PLANE FMR\nAlthough we have shown via OOP FMR in Sec. III that intrinsic Gilbert damping is\nessentially independent of the structural properties of the Fe \flms, it might be expected\nthat microstructure has a pronounced impact on extrinsic magnetic relaxation driven by\ntwo-magnon scattering, which is generally present in IP FMR. IP magnetized \flms are more\ncommon in device applications than OOP magnetized \flms, since the shape anisotropy of\nthin \flms tends to keep the magnetization in the \flm plane. What governs the performance\nof such magnetic devices (e.g., quality factor50,51) may not be the intrinsic Gilbert damping\nparameter but the total FMR linewidth. Thus, for many magnetic device applications, it is\nessential to understand the contributions to the IP FMR linewidth.\nIP FMR measurements have been performed using a coplanar-waveguide-based spectrom-\neter, as detailed in Refs.18,37. Examples of the frequency dependence of IP FMR linewidth\n110501001502002500\n10203040506070050100150200250Cu/FeA\ng/Fe 2 nm \n6 nm \n8 nm \n10 nm \n15 nm \n25 nmΔHIP [Oe] \n12(\na) \nf\n [GHz](b)FIG. 4. (Color online) IP FMR half-width-at-half-maximum linewidth Δ𝐻IPas a function of\nresonance frequency 𝑓for (a) Cu/Fe and (b) Ag/Fe. The vertical dashed line at 12 GHz highlights\nthe hump in linewidth vs frequency seen for many of the samples.\nare shown in Fig. 4. In contrast to the linear frequency dependence that arises from in-\ntrinsic Gilbert damping in Fig. 3(a), a nonlinear hump is observed for most of the \flms\nin the vicinity of \u001912 GHz. In some \flms, e.g., 10 nm thick Cu/Fe \flm, the hump is so\nlarge that its peak even exceeds the linewidth at the highest measured frequency. Similar\nnonlinear IP FMR linewidth behavior has been observed in Fe alloy \flms52and epitaxial\nHeusler \flms53in previous studies, where two-magnon scattering has been identi\fed as a\nsigni\fcant contributor to the FMR linewidth. Therefore, in the following, we attribute the\nnonlinear behavior to two-magnon scattering.\nTo gain insight into the origin of two-magnon scattering, we plot the linewidth at 12\n122550751001251500\n5101520250255075100125150 Cu/Fe \nAg/Fe Cu/Fe \nAg/FeΔHIP @ 12 GHz [Oe](a)HC [Oe]T\nhickness [nm](b)FIG. 5. (Color online) (a) IP FMR half-width-at-half-maximum linewidth at 12 GHz { approxi-\nmately where the maximum (\\hump\") in linewidth vs frequency is seen (see Fig. 4) { as a function\nof \flm thickness for both Cu/Fe and Ag/Fe. (b) Coercivity 𝐻𝑐as a function of \flm thickness for\nboth Cu/Fe and Ag/Fe. The red shaded area highlights thickness region where the Cu/Fe sample\nseries show a peak behavior in both plots.\nGHz { approximately where the hump is seen in Fig. 4 { against the Fe \flm thickness in\nFig. 5(a). We do not observe a monotonic decay in the linewidth with increasing thickness\nthat would result from two-magnon scattering of interfacial origin54. Rather, we observe\na non-monotonic thickness dependence in Fig. 5(a), which indicates that the observed\ntwo-magnon scattering originates within the bulk of the \flms. We note that Ag/Fe with\ngreater interfacial disorder (see Sec. II) exhibits weaker two-magnon scattering than Cu/Fe,\nparticularly in the lower thickness regime ( .10 nm). This observation further corroborates\n13that the two-magnon scattering here is not governed by the interfacial roughness of Fe\n\flms. The contrast between Cu/Fe and Ag/Fe also might appear counterintuitive, since\ntwo-magnon scattering is induced by defects and hence might be expected to be stronger\nfor more \\defective\" \flms (i.e., Ag/Fe in this case). The counterintuitive nature of the\ntwo-magnon scattering here points to more subtle mechanisms at work.\nTo search for a possible correlation between static magnetic properties and two-magnon\nscattering, we have performed vibrating sample magnetometry (VSM) measurements with a\nMicrosense EZ9 VSM. Coercivity extracted from VSM measurements is plotted as a function\nof \flm thickness in Fig. 5(b), which shows a remarkably close correspondence with linewidth\nvs thickness (Fig. 5(a)). In particular, a pronounced peak in coercivity is observed for Cu/Fe\naround 10 nm, corresponding to the same thickness regime where the 12 GHz FMR linewidth\nfor Cu/Fe is maximized. Moreover, the 10 nm Cu/Fe sample (see Sec. II) exhibits a tall,\nnarrow bcc (110) di\u000braction peak, which suggests that its peculiar microstructure plays a\npossible role in the large two-magnon scattering and coercivity (e.g., via stronger domain\nwall pinning).\nWhile the trends shown in Fig. 5 provide some qualitative insights, we now attempt to\nquantitatively analyze the frequency dependence of FMR linewidth for the Cu/Fe and Ag/Fe\n\flms. We assume that the Gilbert damping parameter for IP FMR is equal to that for OOP\nFMR, i.e.,𝛼IP=𝛼OOP. This assumption is physically reasonable, considering that Gilbert\ndamping is theoretically expected to be isotropic in Fe \flms near room temperature49. While\na recent study has reported anisotropic Gilbert damping that scales quadratically with\nmagnetostriction55, this e\u000bect is likely negligible in elemental Fe whose magnetostriction is\nseveral times smaller56,57than that of the Fe 07Ga03alloy in Ref.55.\nThus, from the measured IP linewidth Δ𝐻IP, the extrinsic two-magnon scattering\nlinewidthΔ𝐻TMS can be obtained by\nΔ𝐻TMS=Δ𝐻IP\u00002𝜋\n𝛾𝛼IP (2)\nwhere2𝜋\n𝛾𝛼IPis the Gilbert damping contribution. Figure 6 shows the obtained Δ𝐻TMSand \ft\nattempts using the \\grain-to-grain\" two-magnon scattering model developed by McMicheal\nand Krivosik25. This model captures the inhomogeneity of the e\u000bective internal magnetic\n\feld in a \flm consisting of many magnetic grains. The magnetic inhomogeneity can arise\nfrom the distribution of magnetocrystalline anisotropy \feld directions associated with the\n14randomly oriented crystal grains52. In this model the two-magnon scattering linewidth\nΔ𝐻TMS is a function of the Gilbert damping parameter 𝛼IP, the e\u000bective anisotropy \feld\n𝐻𝑎of the randomly oriented grain, and the correlation length 𝜉within which the e\u000bective\ninternal magnetic \feld is correlated. Further details for computing Δ𝐻TMS are provided in\nthe Appendix and Refs.25,52,53. As we have speci\fed above, 𝛼IPis set to the value derived\nfrom OOP FMR results (i.e., 𝛼OOP in Fig. 3(b)). This leaves 𝜉and𝐻𝑎as the only free\nparameters in the \ftting process.\nThe modeling results are dependent on the choice of the correlation function 𝐶¹Rº, which\ncaptures how the e\u000bective internal magnetic \feld is correlated as a function of lateral distance\nRin the \flm plane. We \frst show results obtained with a simple exponentially decaying\ncorrelation function, as done in prior studies of two-magnon scattering25,52,53, i.e.,\n𝐶¹Rº=exp\u0012\n\u0000jRj\n𝜉\u0013\n (3)\nEquation 3 has the same form as the simplest correlation function used to model rough\ntopographical surfaces (when they are assumed to be \\self-a\u000ene\")58. Fit results with Eq. (3)\nare shown in dashed blue curves in Fig. 6. For most samples, the \ftted curve does not\nreproduce the experimental data quantitatively. Moreover, the \ftted values of 𝜉and𝐻𝑎\noften reach physically unrealistic values, e.g., with 𝐻𝑎¡104Oe and𝜉 1 nm (see Table I).\nThese results suggest that the model does not properly capture the underlying physics of\ntwo-magnon scattering in our samples.\nA possible cause for the failure to \ft the data is that the simple correlation function\n(Eq. 3) is inadequate. We therefore consider an alternative correlation function by again\ninvoking an analogy between the spatially varying height of a rough surface58and the spa-\ntially varying e\u000bective internal magnetic \feld in a \flm. Speci\fcally, we apply a correlation\nfunction (i.e., a special case of Eq. (4.3) in Ref.58where short-range roughness 𝛼=1) for\nthe so-called \\mounded surface,\" which incorporates the average distance 𝜆between peaks\nin topographical height (or, analogously, e\u000bective internal magnetic \feld):\n𝐶¹Rº=p\n2jRj\n𝜉𝐾1 p\n2jRj\n𝜉!\n𝐽0\u00122𝜋jRj\n𝜆\u0013\n (4)\nwhere𝐽0and𝐾1are the Bessel function of the \frst kind of order zero and the modi\fed Bessel\nfunction of the second kind of order one, respectively. This oscillatory decaying function is\nchosen because its Fourier transform (see Appendix) does not contain any transcendental\n15020406080100120 \nExperimental \nSelf-affine \nMoundedΔHTMS [Oe]Cu/FeA g/Fe6\n nm8\n nm1\n0 nm1\n5 nm2\n5 nm(a)( f)0\n50100150ΔHTMS [Oe](\nb)( g)0\n50100150ΔHTMS [Oe](\nc)( h)0\n255075100125ΔHTMS [Oe](\nd)( i)0\n2 04 06 0050100150200ΔHTMS [Oe]f\n [GHz](e)0\n2 04 06 0f\n [GHz](j)FIG. 6. (Color online) Extrinsic two-magnon scattering linewidth Δ𝐻TMSvs frequency 𝑓and \ftted\ncurves for 6, 8, 10, 15, and 25 nm Cu/Fe and Ag/Fe \flms. Black squares represent experimental\nFMR linewidth data. Dashed blue and solid red curves represent the \ftted curves using correlation\nfunctions proposed for modeling self-a\u000ene and mounded surfaces, respectively. In (d), (e), (h), (i),\ndashed blue curves overlap with solid red curves.\n16functions, which simpli\fes the numerical calculation. We also stress that while Eq. (4) in\nthe original context (Ref.58) was used to model topographical roughness, we are applying\nEq. (4) in an attempt to model the spatial \ructuations (\\roughness\") of the e\u000bective internal\nmagnetic \feld { rather than the roughness of the \flm topography.\nThe \ftted curves using the model with Eq. (4) are shown in solid red curves in Fig. 6. Fit\nresults for some samples show visible improvement, although this is perhaps not surprising\nwith the introduction of 𝜆as an additional free parameter. Nevertheless, the \ftted values\nof𝐻𝑎or𝜆still diverge to unrealistic values of ¡104Oe or¡104nm in some cases (see\nTable I), which means that the new correlation function (Eq. (4)) does not fully re\rect\nthe meaningful underlying physics of our samples either. More detailed characterization of\nthe microstructure and inhomogeneities, e.g., via synchrotron x-ray and neutron scattering,\ncould help determine the appropriate correlation function. It is also worth pointing out that\nfor some samples (e.g. 15 nm Cu/Fe and Ag/Fe \flms), essentially identical \ft curves are\nobtained regardless of the correlation function. This is because when 𝜆\u001d𝜉, the Fourier\ntransform of Eq. (4) has a very similar form as the Fourier transform of Eq. (3), as shown in\nthe Appendix. In such cases, the choice of the correlation function has almost no in\ruence\non the behavior of the two-magnon scattering model in the \ftting process.\nV. SUMMARY\nWe have examined room-temperature intrinsic and extrinsic damping in two series of\npolycrystalline Fe thin \flms with distinct structural properties. Out-of-plane FMR mea-\nsurements con\frm constant intrinsic Gilbert damping of \u00190.0024, essentially independent\nof \flm thickness and structural properties. This \fnding implies that intrinsic damping in\nFe at room temperature is predominantly governed by the crystalline and electronic band\nstructures within the grains, rather than scattering at grain boundaries or \flm surfaces. The\nresults from in-plane FMR, where extrinsic damping (i.e., two-magnon scattering) plays a\nsigni\fcant role, are far more nuanced. The conventional grain-to-grain two-magnon scatter-\ning model fails to reproduce the in-plane FMR linewidth data with physically reasonable\nparameters { pointing to the need to modify the model, along with more detailed character-\nization of the \flm microstructure. Our experimental \fndings advance the understanding of\nintrinsic Gilbert damping in polycrystalline Fe, while motivating further studies to uncover\n17TABLE I. Summary of IP FMR linewidth \ft results. Note the divergence to physically unreasonable\nvalues in many of the results. Standard error is calculated using equation√︁\nSSRDOF\u0002diag¹COVº,\nwhere SSR stands for the sum of squared residuals, DOF stands for degrees of freedom, and COV\nstands for the covariance matrix.\nSelf-a\u000ene Mounded\nSample\nSeriesThickness\n(nm)𝜉\n(nm)𝐻𝑎\n(Oe)𝜉\n(nm)𝐻𝑎\n(Oe)𝜆\n(nm)\nCu/Fe6 70\u000610 170\u000610 80\u000690 24\u00063 >1\u0002104\n8 200\u0006100 150\u000620 700\u00061000 25\u00062 900\u0006100\n10 140\u000640 200\u000620 160\u000650 33\u00061 800\u0006200\n15 9\u00062 800\u0006100 10\u000620 100\u000680 >1\u0002104\n25 0\u00065 >1\u000210460\u000630 >1\u000210410.41\u00060.01\nAg/Fe6 0\u000640 >1\u0002104150\u000640 >1\u000210411.7\u00060.7\n8 0\u000630 >1\u0002104170\u000650 >1\u000210412\u00064\n10 6\u00061 1500\u0006300 8\u000640 200\u0006500 >1\u0002104\n15 2\u00062 4000\u00063000 3\u00069 500\u0006900 >6\u0002103\n25 0\u00066 >1\u0002104140\u000650 >1\u000210415\u00066\nthe mechanisms of extrinsic damping in structurally disordered thin \flms.\nACKNOWLEDGMENTS\nS.W. acknowledges support by the ICTAS Junior Faculty Program. D.A.S. and S.E.\nacknowledge support by the National Science Foundation, Grant No. DMR-2003914. P.\nN. would like to acknowledge support through NASA Grant NASA CAN80NSSC18M0023.\nA. R. would like to acknowledge support through the Defense Advanced Research Project\nAgency (DARPA) program on Topological Excitations in Electronics (TEE) under Grant\nNo. D18AP00011. This work was supported by NanoEarth, a member of National Nan-\notechnology Coordinated Infrastructure (NNCI), supported by NSF (ECCS 1542100).\n18Appendix A: Details of the Two-Magnon Scattering Model\nIn the model developed by McMichael and Krivosik, the two-magnon scattering contri-\nbutionΔ𝐻TMS to the FMR linewidth is given by25,52,53\nΔ𝐻TMS=𝛾2𝐻2\n𝑎\n2𝜋𝑃𝐴¹𝜔º∫\nΛ0𝑘𝐶𝑘¹𝜉º𝛿𝛼¹𝜔\u0000𝜔𝑘ºd2𝑘 (A1)\nwhere𝜉is correlation length, 𝐻𝑎is the e\u000bective anisotropy \feld of the randomly oriented\ngrain.𝑃𝐴¹𝜔º=𝜕𝜔\n𝜕𝐻\f\f\n𝐻=𝐻FMR=√︃\n1¸¹4𝜋𝑀𝑠\n2𝜔𝛾º2accounts for the conversion between the fre-\nquency and \feld swept linewidth. Λ0𝑘represents the averaging of the anisotropy axis \ruc-\ntuations over the sample. It also takes into account the ellipticity of the precession for both\nthe uniform FMR mode and the spin wave mode52. The detailed expression of Λ0𝑘can\nbe found in the Appendix of Ref.52. The coe\u000ecients in the expression of Λ0𝑘depend on\nthe type of anisotropy of the system. Here, we used \frst-order cubic anisotropy for bcc Fe.\n𝛿𝛼¹𝜔\u0000𝜔𝑘ºselects all the degenerate modes, where 𝜔represents the FMR mode frequency\nand𝜔𝑘represents the spin wave mode frequency. The detailed expression of 𝜔𝑘can be found\nin Ref.25. In the ideal case where Gilbert damping is 0, 𝛿𝛼is the Dirac delta function. For a\n\fnite damping, 𝛿𝛼¹𝜔0\u0000𝜔𝑘ºis replaced by a Lorentzian function1\n𝜋¹𝛼IP𝜔𝑘𝛾º𝜕𝜔𝜕𝐻\n¹𝜔𝑘\u0000𝜔º2¸»¹𝛼IP𝜔𝑘𝛾º𝜕𝜔𝜕𝐻¼2,\nwhich is centered at 𝜔and has the width of ¹2𝛼IP𝜔𝑘𝛾º𝜕𝜔𝜕𝐻.\nFinally,𝐶𝑘¹𝜉º(or𝐶𝑘¹𝜉𝜆º) is the Fourier transform of the grain-to-grain internal \feld\ncorrelation function, Eq. (3) (or Eq. (4)). For the description of magnetic inhomogeneity\nanalogous to the simple self-a\u000ene topographical surface58, the Fourier transform of the\ncorrelation function, Eq. (3), is\n𝐶𝑘¹𝜉º=2𝜋𝜉2\n»1¸¹𝑘𝜉º2¼3\n2 (A2)\nas also used in Refs.25,52,53. For the description analogous to the mounded surface, the\nFourier transform of the correlation function, Eq. (4), is58\n𝐶𝑘¹𝜉𝜆º=8𝜋3𝜉2\u0010\n1¸2𝜋2𝜉2\n𝜆2¸𝜉2\n2𝑘2\u0011\n\u0014\u0010\n1¸2𝜋2𝜉2\n𝜆2¸𝜉2\n2𝑘2\u00112\n\u0000\u0010\n2𝜋𝜉2\n𝜆𝑘\u00112\u001532 (A3)\nWhen𝜆\u001d𝜉, Eq. (A3) becomes\n𝐶𝑘¹𝜉º\u00198𝜋3𝜉2\n\u0010\n1¸𝜉2\n2𝑘2\u00112 (A4)\n19100102104106108101010-2410-2210-2010-1810-1610-1410-1210-10 \nSelf-affine  \nMounded λ = 10 nm \nMounded λ = 100 nm \nMounded λ = 1000 nmCk [m2]k\n [m-1]ξ = 100 nmFIG. 7. Fourier transform of correlation function for mounded surfaces as a function of wavenumber\n𝑘for three di\u000berent 𝜆values. Fourier transform of correlation function for self-a\u000ene surfaces as a\nfunction of 𝑘is also included for comparison purpose. 𝜉is set as 100 nm for all curves.\nwhich has a similar form as Eq. (A2). This similarity can also be demonstrated graphically.\nFigure 7 plots a self-a\u000ene 𝐶𝑘curve (Eq. (A2)) at 𝜉=100 nm and three mounded 𝐶𝑘curves\n(Eq. (A3)) at 𝜆=10, 100, 1000 nm. 𝜉in mounded 𝐶𝑘curves is set as 100 nm as well. It\nis clearly shown in Fig. 7 that when 𝜆=1000 nm, the peak appearing in 𝜆=10 and 100\nnm mounded 𝐶𝑘curves disappears and the curve shape of mounded 𝐶𝑘resembles that of\nself-a\u000ene𝐶𝑘.\nThe hump feature in Fig. 4 is governed by both 𝛿𝛼and𝐶𝑘(see Eq. A1). 𝛿𝛼has the shape\nof1in reciprocal space ( 𝑘space), as shown in our videos in the Supplemental Material as\nwell as Fig. 5(b) of Ref.53and Fig 2 (b) of Ref.25. The size of the contour of the degenerated\nspin wave modes in 𝑘space increases as the microwave frequency 𝑓increases, which means\nthe number of available degenerate spin wave modes increases as 𝑓increases. As shown\nin Fig. 7, self-a\u000ene 𝐶𝑘is nearly constant with the wavenumber 𝑘until𝑘reaches\u00181𝜉.\nThis suggests that the system becomes e\u000bectively more uniform (i.e. weaker inhomogeneous\nperturbation) when the length scale falls below the characteristic correlation length 𝜉(i.e.,\n𝑘 ¡1𝜉). Because inhomogeneities serve as the scattering centers of two-magnon scattering\n20process, degenerate spin wave modes with 𝑘 ¡1𝜉are less likely to be scattered into.\nNow we consider the 𝑓dependence of the two-magnon scattering rate. When 𝑓is small,\nthe two-magnon scattering rate increases as 𝑓increases because more degenerate spin wave\nmodes become available as 𝑓increases. When 𝑓further increases, the wavenumber 𝑘of\nsome degenerate spin wave modes exceeds 1 𝜉. This will decrease the overall two-magnon\nscattering rate because the degenerate spin wave modes with 𝑘 ¡1𝜉are less likely to be\nscattered into, as discussed above. Furthermore, the portion of degenerate spin wave modes\nwith𝑘 ¡ 1𝜉increases as 𝑓continues to increase. When the impact of decreasing two-\nmagnon scattering rate for degenerate spin wave modes with high 𝑘surpasses the impact\nof increasing available degenerate spin wave modes, the overall two-magnon scattering rate\nwill start to decrease as 𝑓increases. Consequently, the nonlinear trend { i.e., a \\hump\" {\nin FMR linewidth Δ𝐻TMS vs𝑓appears in Fig. 4.\nHowever, the scenario discussed above can only happen when 𝜉is large enough, because\nthe wavenumber 𝑘of degenerate spin wave modes saturates (i.e., reaches a limit) as 𝑓\napproaches in\fnity. If the limit value of 𝑘is smaller than 1𝜉, the two-magnon scattering\nrate will increase monotonically as 𝑓increases. In that case the hump feature will not\nappear. See our videos in the Supplemental Material that display the 𝑓dependence of Λ0𝑘,\n𝛿𝛼¹𝜔\u0000𝜔𝑘º,����𝑘¹𝜉º\n2𝜋𝜉2,Λ0𝑘𝐶𝑘¹𝜉º𝛿𝛼¹𝜔\u0000𝜔𝑘º\n2𝜋𝜉2 , andΔ𝐻TMS for various𝜉values.\nPrevious discussions of the hump feature are all based on the self-a\u000ene correlation func-\ntion (Eq. 3). The main di\u000berence between the mounded correlation function (Eq. 4) and the\nself-a\u000ene correlation function (Eq. 3) is that the mounded correlation function has a peak\nwhen𝜆is not much larger than 𝜉as shown in Fig. 7. This means when the wavenumber\n𝑘of degenerate spin wave modes enters (leaves) the peak region, two-magnon scattering\nrate will increase (decrease) much faster compared to the self-a\u000ene correlation function. In\nother words, the mounded correlation function can generate a narrower hump compared to\nthe self-a\u000ene correlation function in the two-magnon linewidth Δ𝐻TMS vs𝑓plot, which is\nshown in Fig. 6 (b, c).\n1Z. Diao, Z. Li, S. Wang, Y. Ding, A. Panchula, E. Chen, L.-C. 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Lett. 88, 252510 (2006).\n24"    },    {        "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": "1901.01941v1.Giant_anisotropy_of_Gilbert_damping_in_epitaxial_CoFe_films.pdf",        "content": "Giant anisotropy of Gilbert damping in epitaxial CoFe \flms\nYi Li,1, 2Fanlong Zeng,3Steven S.-L. Zhang,2Hyeondeok Shin,4Hilal Saglam,2, 5Vedat Karakas,2, 6Ozhan\nOzatay,2, 6John E. Pearson,2Olle G. Heinonen,2Yizheng Wu,3, 7,\u0003Axel Ho\u000bmann,2,yand Wei Zhang1, 2,z\n1Department of Physics, Oakland University, Rochester, MI 48309, USA\n2Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA\n3State Key Laboratory of Surface Physics, Department of Physics, Fudan University, Shanghai 200433, China\n4Computational Sciences Division, Argonne National Laboratory, Argonne, IL 60439, USA\n5Department of Physics, Illinois Institute of Technology, Chicago IL 60616, USA\n6Department of Physics, Bogazici University, Bebek 34342, Istanbul, Turkey\n7Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China\n(Dated: January 8, 2019)\nTailoring Gilbert damping of metallic ferromagnetic thin \flms is one of the central interests in\nspintronics applications. Here we report a giant Gilbert damping anisotropy in epitaxial Co 50Fe50\nthin \flm with a maximum-minimum damping ratio of 400 %, determined by broadband spin-torque\nas well as inductive ferromagnetic resonance. We conclude that the origin of this damping anisotropy\nis the variation of the spin orbit coupling for di\u000berent magnetization orientations in the cubic lattice,\nwhich is further corroborate from the magnitude of the anisotropic magnetoresistance in Co 50Fe50.\nIn magnetization dynamics the energy relaxation rate\nis quanti\fed by the phenomenological Gilbert damping\nin the Landau-Lifshits-Gilbert equation [1], which is a\nkey parameter for emerging spintronics applications [2{\n6]. Being able to design and control the Gilbert damp-\ning on demand is crucial for versatile spintronic device\nengineering and optimization. For example, lower damp-\ning enables more energy-e\u000ecient excitations, while larger\ndamping allows faster relaxation to equilibrium and more\nfavorable latency. Nevertheless, despite abundant ap-\nproaches including interfacial damping enhancement [7{\n9], size e\u000bect [10, 11] and materials engineering [12{14],\nthere hasn't been much progress on how to manipulate\ndamping within the same magnetic device. The only\nwell-studied damping manipulation is by spin torque [15{\n18], which can even fully compensate the intrinsic damp-\ning [19, 20]. However the requirement of large current\ndensity narrows its applied potential.\nAn alternative approach is to explore the intrinsic\nGilbert damping anisotropy associated with the crys-\ntalline symmetry, where the damping can be continu-\nously tuned via rotating the magnetization orientation.\nAlthough there are many theoretical predictions [21{25],\nmost early studies of damping anisotropy are disguised\nby two-magnon scattering and linewidth broadening due\nto \feld-magnetization misalignment [26{29]. In addition,\nthose reported e\u000bects are usually too weak to be consid-\nered in practical applications [30, 31].\nIn this work, we show that a metallic ferromagnet can\nexhibit a giant Gilbert damping variation by a factor\nof four along with low minimum damping. We inves-\ntigated epitaxial cobalt-iron alloys, which have demon-\nstrated new potentials in spintronics due to their ultralow\ndampings [32, 33]. Using spin-torque-driven and induc-\ntive ferromagnetic resonance (FMR), we obtain a four-\nfold (cubic) damping anisotropy of 400% in Co 50Fe50thin\n\flms between their easy and hard axes. For each angle,the full-range frequency dependence of FMR linewidths\ncan be well reproduced by a single damping parame-\nter\u000b. Furthermore, from \frst-principle calculations and\ntemperature-dependent measurements, we argue that\nthis giant damping anisotropy in Co 50Fe50is due to the\nvariation of the spin-orbit coupling (SOC) in the cu-\nbic lattice, which di\u000bers from the anisotropic density of\nstate found in ultrathin Fe \flm [30]. We support our\nconclusion by comparing the Gilbert damping with the\nanisotropic magnetoresistance (AMR) signals. Our re-\nsults reveal the key mechanism to engineer the Gilbert\ndamping and may open a new pathway to develop novel\nfunctionality in spintronic devices.\nCo50Fe50(CoFe) \flms were deposited on MgO(100)\nsubstrates by molecular beam epitaxy at room temper-\nature, under a base pressure of 2 \u000210\u000010Torr [34]. For\nspin-torque FMR measurements, i) CoFe(10 nm)/Pt(6\nnm) and ii) CoFe(10 nm) samples were prepared. They\nwere fabricated into 10 \u0016m\u000240\u0016m bars by photolithog-\nraphy and ion milling. Coplanar waveguides with 100-\nnm thick Au were subsequently fabricated [18, 35]. For\neach layer structure, 14 devices with di\u000berent orienta-\ntions were fabricated, as shown in Fig. 1(a). The geom-\netry de\fnes the orientation of the microwave current, \u0012I,\nand the orientation of the biasing \feld, \u0012H, with respect\nto the MgO [100] axis (CoFe [1 10]).\u0012Iranges from 0\u000e\nto 180\u000ewith a step of 15\u000e(D1 to D14, with D7 and D8\npointing to the same direction). For each device we \fx\n\u0012H=\u0012I+ 45\u000efor maximal recti\fcation signals. In addi-\ntion, we also prepared iii) CoFe(20 nm) 40 \u0016m\u0002200\u0016m\nbars along di\u000berent orientations with transmission copla-\nnar waveguides fabricated on top for vector network an-\nalyzer (VNA) measurements. See the Supplemental Ma-\nterials for details [36].\nFig. 1(b) shows the angular-dependent spin-torque\nFMR lineshapes of CoFe(10 nm)/Pt devices from dif-\nferent samples (D1 to D4, hard axis to easy axis) atarXiv:1901.01941v1  [cond-mat.mtrl-sci]  7 Jan 20192\nFIG. 1. (a) Upper: crystalline structure, axes of bcc Co 50Fe50\n\flm on MgO(100) substrate and de\fnition of \u0012Hand\u0012I.\nLower: device orientation with respect to the CoFe crystal\naxis. (b) Spin-torque FMR lineshapes of i) CoFe(10 nm)/Pt\ndevices D1 to D4 measured. (c) Resonances of D1 and D4\nfrom (b) for \u00160Hres<0. (d) Resonances of iii) CoFe(20\nnm) for\u0012H= 45\u000eand 90\u000emeasured by VNA FMR. In (b-d)\n!=2\u0019= 20 GHz and o\u000bset applies.\n!=2\u0019= 20 GHz. A strong magnetocrystalline anisotropy\nas well as a variation of resonance signals are observed.\nMoreover, the linewidth increases signi\fcantly from easy\naxis to hard axis, which is shown in Fig. 1(c). We have\nalso conducted rotating-\feld measurements on a sec-\nond CoFe(10 nm)/Pt device from a di\u000berent deposition\nand the observations can be reproduced. This linewidth\nanisotropy is even more pronounced for the CoFe(20 nm)\ndevices without Pt, measured by VNA FMR (Fig. 1d).\nFor the CoFe(10 nm) devices, due to the absence of the\nPt spin injector the spin-torque FMR signals are much\nweaker than CoFe/Pt and completely vanish when the\nmicrowave current is along the easy axes.\nFigs. 2(a-b) show the angular and frequency de-\npendence of the resonance \feld Hres. In Fig. 2(a), the\nHresfor all four sample series match with each other,\nwhich demonstrates that the magnetocrystalline proper-\nties of CoFe(10 nm) samples are reproducible. A slightly\nsmallerHresfor CoFe(20 nm) is caused by a greater e\u000bec-\ntive magnetization when the thickness increases. A clear\nfourfold symmetry is observed, which is indicative of the\ncubic lattice due to the body-center-cubic (bcc) texture\nof Co 50Fe50on MgO. We note that the directions of the\nhard axes has switched from [100] and [010] in iron-rich\nalloys [33] to [110] and [1 10] in Co 50Fe50, which is con-\nω/2πμ0Hres  (T) μ0Hres  (T) [110]\n[110][100][010](a) (b) CoFe(10 nm)/Pt \nω/2π=2045o90 o135o\n135o180o 225oCoFe(10 nm)/Pt \nCoFe(10 nm) CoFe(20 nm) θH:\n[100]\n[110][010]FIG. 2. (a) Resonance \feld \u00160Hresas a function of \u0012Hat\n!=2\u0019= 20 GHz for di\u000berent samples. Diamonds denote the\nrotating-\feld measurement from the second CoFe(10 nm)/Pt\ndevice. The black curve denotes the theoretical prediction.\n(b)\u00160Hresas a function of frequency for the CoFe(10 nm)/Pt\ndevices. Solid curves denote the \fts to the Kittel equation.\nsistent with previous reports [37, 38].\nThe magnetocrystalline anisotropy can be quanti-\n\fed from the frequency dependence of \u00160Hres. Fig.\n2(b) shows the results of CoFe(10 nm)/Pt when HB\nis aligned to the easy and hard axes. A small uniax-\nial anisotropy is found between [1 10] (0\u000eand 180\u000e) and\n[110] (90\u000e) axes. By \ftting the data to the Kittel equa-\ntion!2=\r2=\u00162\n0(Hres\u0000Hk)(Hres\u0000Hk+Ms), where\n\r= 2\u0019(geff=2)\u000128 GHz/T, we obtain geff= 2:16,\n\u00160Ms= 2:47 T,\u00160H[100]\nk= 40 mT,\u00160H[010]\nk= 65 mT\nand\u00160H[110]\nk=\u00160H[110]\nk=\u000043 mT. Taking the disper-\nsion functions from cubic magnetocrystalline anisotropy\n[39, 40], we obtain an in-plane cubic anisotropy \feld\n\u00160H4jj= 48 mT and a uniaxial anisotropy \feld \u00160H2jj=\n12 mT. Fig. 2(a) shows the theoretical predictions from\nH4jjandH2jjin black curve, which aligns well with all\n10-nm CoFe samples.\nWith good magnetocrystalline properties, we now turn\nto the energy relaxation rate. Fig. 3(a) shows the full-\nwidth-half-maximum linewidths \u00160\u0001H1=2of the spin-\ntorque FMR signals at !=2\u0019= 20 GHz. Again, a fourfold\nsymmetry is observed for CoFe(10 nm)/Pt and CoFe(10\nnm), with the minimal (maximal) linewidth measured\nwhen the \feld lies along the easy (hard) axes. For\nCoFe(10 nm) devices, we did not measure any spin-torque\nFMR signal for HBalong the hard axes ( \u0012H= 45\u000e, 135\u000e\nand 225\u000e). This is due to the absence of the Pt spin\ninjector as well as the near-zero AMR ratio when the rf\ncurrent \rows along the easy axes, which will be discussed\nlater. For all other measurements, the linewidths of CoFe\ndevices are smaller than for CoFe/Pt by the same con-\nstant, independent of orientation (upper diagram of Fig.\n3a). This constant linewidth di\u000berence is due to the spin\npumping contribution to damping from the additional Pt\nlayer [41, 42]. Thus we can deduce the intrinsic damp-\ning anisotropy from CoFe(10 nm)/Pt devices, with the3\nω/2π 105, 195 deg 75, 165 deg 120, 210 deg 135, 225 deg(HA) 45, 135 deg (HA) \n60, 150 deg \n90, 180 deg(EA) θHCoFe(10 nm)/Pt \n(b) = -\n=-\n[100] [110] [110] [010](a) ω/2π=20 \nω/2π θH\n0, 90 deg 15, 75 deg 22.5, 67.5 deg 30, 60 deg 42.5, 50 deg \n40, 52.5 deg \n37.5, 55 deg CoFe(20 nm) (VNA) \n(c)CoFe(10 nm)/Pt CoFe(10 nm) \n90 deg (EA) \nfor CoFe \nFIG. 3. (a) \u00160\u0001H1=2as a function of \u0012Hat!=2\u0019= 20 GHz\nfor the CoFe(10 nm) series in Fig. 2(a). Top: Addtional\nlinewidth due to spin pumping of Pt. The green region de-\nnotes the additional linewidth as 4 :5\u00060:7 mT. (b-c) \u00160\u0001H1=2\nas a function of frequency for (b) CoFe(10 nm)/Pt and (c)\nCoFe(20 nm) samples. Solid lines and curves are the \fts to\nthe data.\ndamping shifted from CoFe(10 nm) devices by a constant\nand is much easier to measure.\nIn Fig. 3(b-c) we show the frequency dependence of\n\u00160\u0001H1=2of CoFe(10 nm)/Pt devices from spin-torque\nFMR and CoFe(20 nm) devices from VNA FMR. For\nboth the easy and hard axes, linear relations are ob-\ntained, and the Gilbert damping \u000bcan be extracted\nfrom\u00160\u0001H1=2=\u00160\u0001H0+ 2\u000b!=\r with the \fts shown\nas solid lines. Here \u00160\u0001H0is the inhomogeneous broad-\nening due to the disorders in lattice structures. In Fig.\n3(b) we also show the linewidths of the CoFe(10 nm)\ndevice along the easy axis ( \u0012H= 90\u000e), which has a\nsigni\fcant lower linewidth slope than the easy axis of\nCoFe(10 nm)/Pt. Their di\u000berences yield a spin pump-\ning damping contribution of \u0001 \u000bsp= 0:0024. By using\n\u0001\u000bsp=\r\u0016hg\"#=(4\u0019MstM), we obtain a spin mixing con-\nductance of g\"#(CoFe/Pt) = 25 nm\u00002, which is compa-\nrable to similar interfaces such as NiFe/Pt [43, 44]. For\n\u0012Hbetween the easy and hard axes, the low-frequency\nlinewidth broadenings are caused by the deviation of\nmagnetization from the biasing \feld direction, whereas\nat high frequencies the \feld is su\u000ecient to saturate the\nmagnetization. In order to \fnd the damping anisotropy,\nwe \ft the linewidths to the angular model developed bySuhl [45, 46], using a single \ft parameter of \u000band the\nextractedH2jjandH4jjfrom Fig. 2. The solid \ftting\ncurves in Fig. 3(b) nicely reproduce the experimental\npoints.\nThe obtained damping anisotropy for all the samples\nare summarized in Fig. 4, which is the main result of\nthe paper. For CoFe(10 nm)/Pt samples, \u000bvaries from\n0.0056 along the easy axis to 0.0146 along the hard axis.\nBy subtracting the spin pumping \u0001 \u000bspfrom both values,\nwe derive a damping anisotropy of 380%. For CoFe(20\nnm) samples measured by VNA FMR, \u000bvaries from\n0.0054 to 0.0240, which yields an anisotropy of 440% and\nreproduces the large anisotropy from spin-torque FMR.\nThis giant damping anisotropy implies, technologically,\nnearly four times smaller critical current to switch the\nmagnetization in a spin-torque magnetic random access\nmemory, or to excite auto-oscillation in a spin-torque os-\ncillator, by simply changing the magnetization orienta-\ntion from the hard axis to the easy axis within the same\ndevice. In addition, we emphasize that our reported\ndamping anisotropy is not subject to a dominant two-\nmagnon scattering contribution, which would be mani-\nfested as a nonlinear linewidth softening at high frequen-\ncies [28, 31]. For this purpose we have extended the fre-\nquency of spin-torque FMR on CoFe(10 nm)/Pt up to 39\nGHz, see the Supplemental Materials for details [36]. We\nchoose CoFe(10 nm)/Pt samples because they provide\nthe best signals at high frequencies and the additional Pt\nlayer signi\fcantly helps to excite the dynamics. Linear\nfrequency dependence of linewidth persists throughout\nthe frequency range and \u0001 H0is unchanged for the two\naxes, with which we can exclude extrinsic e\u000bects to the\nlinewidths. We also note that our result is substantially\ndi\u000berent from the recent report on damping anisotropy\nin Fe/GaAs [30], which is due to the interfacial SOC and\ndisappears quickly as Fe becomes thicker. In compari-\nson, the Gilbert damping anisotropy in Co 50Fe50is the\nintrinsic property of the material, is bonded to its bulk\ncrystalline structure, and thus holds for di\u000berent thick-\nnesses in our experiments.\nIn order to investigate the dominant mechanism for\nsuch a large Gilbert damping anisotropy, we perform\ntemperature-dependent measurements of \u000band the re-\nsistivity\u001a. Fig. 5(a) plots \u000bas a function of 1 =\u001afor\nthe CoFe(10 nm)/Pt and CoFe(20 nm) samples and for\nHBalong the easy and hard axes. The dominant lin-\near dependence reveals a major role of conductivitylike\ndamping behavior. This is described by the breathing\nFermi surface model for transition-metal ferromagnets,\nin which\u000bcan be expressed as [23, 24, 47{49]:\n\u000b\u0018N(EF)j\u0000\u0000j2\u001c (1)\nwhereN(EF) is the density of state at the Fermi level, \u001c\nis the electron relaxation time and \u0000\u0000=h[\u001b\u0000;Hso]iE=EF\nis the matrix for spin-\rip scatterings induced by the SOC\nHamiltonian Hsonear the Fermi surface [48, 49]. Here \u001c4\n(b) CoFe(10 nm) CoFe(20 nm) CoFe(20 nm) CoFe(10 nm)/Pt - ∆α sp \nCoFe(10 nm ) 400 %\n100 %\nFIG. 4. Renormalized damping and its anisotropy for\nCoFe(10 nm) and CoFe(20 nm), measured from spin-torque\nFMR and VNA FMR, respectively. For CoFe(20 nm)/Pt sam-\nples, \u0001\u000bsphas been subtracted from the measured damping.\nis proportional to the conductivity (1 =\u001a) from the Drude\nmodel, with which Eq. (1) gives rise to the behaviors\nshown in Fig. 5(a).\nFor the origin of damping anisotropy, we \frst check\nthe role of N(EF) by ab-initio calculations for di\u000berent\nordered cubic supercells, which is shown in the Supple-\nmental Materials [36]. However, a negligible anisotropy\ninN(EF) is found for di\u000berent magnetization orienta-\ntions. This is consistent with the calculated anisotropy\nin Ref. [30], where less than 0.4% change of N(EF) was\nobtained in ultrathin Fe \flms. The role of \u001ccan also be\nexcluded from the fact that the resistivity di\u000berence be-\ntween the easy and hard axes is less than 2% [36]. Thus\nwe deduce that the giant damping anisotropy of 400% is\ndue to the change of j\u0000\u0000j2, or the SOC, at di\u000berent crys-\ntalline directions. In particular, unlike the single element\nFe, disordered bcc Fe-Co alloy can possess atomic short-\nrange order, which gives rise to local tetragonal crystal\ndistortions due to the di\u000berent lattice constants of Fe and\nCo [2{4]. Such local tetragonal distortions will preserve\nglobal cubic symmetry but can have large e\u000bects on the\nSOC. We emphasize that our CoFe samples, which did\nnot experience annealing, preserve the random disorder.\nOur \frst principle calculations also con\frm the role of lo-\ncal tetragonal distortions and its enhancement on SOC,\nsee the Supplemental Materials for details [36].\nThe anisotropy of the SOC in Co 50Fe50can be re\rected\nby its AMR variation along di\u000berent crystalline orienta-\ntions. The AMR ratio can be de\fned as:\nAMR(\u0012I) =\u001ak(\u0012I)\n\u001a?(\u0012I)\u00001 (2)\nwhere\u001ak(\u0012I) and\u001a?(\u0012I) are measured for the biasing\n\feld parallel and perpendicular to the current direction,\nrespectively. The main contribution of AMR is the asym-\nmetrics-delectron scatterings where the s-orbitals are\nmixed with magnetization-containing d-orbitals due toSOC [53, 54]. Since both the damping and AMR origi-\nnate from SOC and, more precisely, are proportional to\nthe second order of SOC, a large damping anisotropy is\nexpected to be accompanied by a large AMR anisotropy\nand vice versa. Furthermore, due to the fourfold sym-\nmetry, the AMR should be invariant when the current\ndirection is rotated by 90 degrees, as the AMR is a func-\ntion of\u0012Ias de\fned in Eq. (1). Thus the damping and\nAMR should exhibit similar angular dependence on \u0012H\nand\u0012I, respectively.\nIn Fig. 5(b) we compare renormalized \u000b(\u0012H) with\nCoFe(20 nm) CoFe(10 nm)/Pt : (a)\n300 K 8 K F(θI)/F max (b) \n,10 nm \n20 nm 10 nm \n20 nm \nFIG. 5. (a) \u000b(T) as a function of 1 =\u001a(T).T= 8 K, 30 K, 70\nK, 150 K and 300 K for CoFe(10 nm)/Pt and T= 8 K and\n300 K for CoFe(20 nm). Dashed and dotted lines are guides\nto eyes. (b) Renormalized \u000b(\u0012H) and AMR( \u0012I) andF(\u0012I) for\nCoFe(10 nm)/Pt and CoFe(20 nm). Circles, crosses and +\ndenote\u000b, AMR and F, respectively.\nAMR(\u0012I) for 10-nm and 20-nm CoFe samples, where the\nAMR values are measured from Hall bars with di\u000berent\n\u0012I. The AMR ratio is maximized along h100iaxes and\nminimized alongh110iaxes, with a large anisotropy by a\nfactor of 10. This anisotropy is also shown by the inte-\ngrated spin-torque FMR intensity for CoFe(10 nm)/Pt,\nde\fned asF(\u0012I) = \u0001H1=2Vmax\ndc [17, 18] and plotted in\nFig. 5(b). The large AMR anisotropy and its symme-\ntry clearly coincide with the damping anisotropy mea-\nsured in the same samples, which con\frms our hypoth-\nesis of strong SOC anisotropy in CoFe. Thus we con-\nclude that the damping anisotropy is dominated by the\nvariation of SOC term in Eq. (1). In parallel, we also\ncompare\u000b(\u0012H) and AMR( \u0012I) for epitaxial Fe(10 nm)\nsamples grown on GaAs substrates [36]. Experimentally\nwe \fnd the anisotropy less is than 30% for both damping\nand AMR, which helps to explain the presence of weak\ndamping anisotropy in epitaxial Fe [30].5\nWe compare our results with prior theoretical works on\ndamping anisotropy [23, 24]. First, despite their propor-\ntional relationship in Fig. 5(a), the giant anisotropy in\n\u000bis not re\rected in 1 =\u001a. This is because the s-dscatter-\ning, which dominates in the anisotropic AMR, only con-\ntributes a small portion to the total resistivity. Second,\nneither the anisotropy of damping nor AMR are sensitive\nto temperature. This is likely because the thermal excita-\ntions at room temperature ( \u00180:025 eV) are much smaller\nthan the spin-orbit coupling ( \u00180:1 eV [47]). Third, the\ndamping tensor has been expressed as a function of M\nanddM=dt[24]. However in a fourfold-symmetry lat-\ntice and considering the large precession ellipticity, these\ntwo vectors are mostly perpendicular to each other, point\ntowards equivalent crystalline directions, and contribute\nequivalently to the symmetry of damping anisotropy.\nIn summary, we have experimentally demonstrated\nvery large Gilbert damping anisotropy up to 400% in\nepitaxial Co 50Fe50thin \flms which is due to their bulk,\ncubic crystalline anisotropy. We show that the damping\nanisotropy can be explained by the change of spin-orbit\ncoupling within the breathing Fermi surface model, which\ncan be probed by the corresponding AMR change. Our\nresults provide new insights to the damping mechanism\nin metallic ferromagnets, which are important for opti-\nmizing dynamic properties of future magnetic devices.\nWe are grateful for fruitful discussions with Bret Hein-\nrich. W.Z. acknowledges supports from the U.S. Na-\ntional Science Foundation under Grants DMR-1808892,\nMichigan Space Grant Consortium and DOE Visit-\ning Faculty Program. Work at Argonne, including\ntransport measurements and theoretical modeling, was\nsupported by the U.S. Department of Energy, Of-\n\fce of Science, Materials Science and Engineering Di-\nvision. Work at Fudan, including thin \flm growth\nand fabrication, was supported by the Nat'l Key Ba-\nsic Research Program (2015CB921401), Nat'l Key Re-\nsearch and Development Program (2016YFA0300703),\nNSFC (11734006,11474066,11434003), and the Program\nof Shanghai Academic Research Leader (17XD1400400)\nof China. O.O. and V.K. acknowledge supports\nfrom Bogazici University Research Fund (17B03D3),\nTUBITAK 2214/A and U.S. Department of State Ful-\nbright Visiting Scholar Program.\n\u0003wuyizheng@fudan.edu.cn\nyho\u000bmann@anl.gov\nzweizhang@oakland.edu\n[1] T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443\n(2004).\n[2] S. 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B 10, 4626 (1974).7\nSupplemental Materials:Giant anisotropy of Gilbert damping in epi-\ntaxial CoFe \flms\nbyYi Li, Fanlong Zeng, Steven S.-L. Zhang, Hyeondeok Shin, Hilal Saglam, Vedat Karakas, Ozhan Ozatay, John E.\nPearson, Olle G. Heinonen, Yizheng Wu, Axel Ho\u000bmann and Wei Zhang\nCrystallographic quality of Co 50Fe50\flms\nFIG. S-1. Crystallographic characterization results of CoFe \flms. (a) RHEED pattern of the CoFe(10 nm) \flm. (b) XRD\nof the CoFe(10 nm) and (20 nm) \flms. (c) X-ray re\rectometry measured for the CoFe(20 nm) \flm. (d) AFM scans of the\nCoFe(20 nm) \flm. (e) Rocking curves of the CoFe(20 nm) \flm for [100] and [110] rotating axes.\nFig. S-1 shows the crystallographic characterization for the epitaxial CoFe samples. Re\rection high-energy electron\ndi\u000braction (RHEED) shows very clear and sharp patterns which shows high quality of the epitaxal \flms. X-ray\ndi\u000braction (XRD) yields clear CoFe(002) peaks at 2 \u0012= 66:5\u000e. X-ray re\rectometry scan of the CoFe (20 nm) \flm\nshows a good periodic pattern and the \ft gives a total thickness of 19.84 nm. Atomic-force microscopy (AFM) scans\nfor 10\u0016m\u000210\u0016m and 100 nm\u0002100 nm scales show smooth surface with a roughness of 0.1 nm. Lastly XRD rocking\ncurves for [100] and [110] rotating axes show a consistent linewidth of 1.45\u000e, which indicates isotropic mosaicity of\nthe CoFe \flms.\nAs a result of the crystallographic characterizations, we believe our MBE-grown CoFe samples are epitaxial, have\nsmooth surfaces and exhibit excellent crystalline quality. Moreover, we can exclude the source of inhomogeneity\nfrom misorientation of crystallities (mosaicity) due to isotropic rocking curves. This means the inhomogeneous FMR\nlinewidth broadening is isotropic, as is consistent with the experiments.\nDevice geometries for Spin-torque FMR and VNA FMR measurements.\nFig. S-2 shows the device geometry for Spin-torque FMR and VNA FMR measurements. For spin-torque FMR,\nwe have prepared CoFe(10 nm)/Pt, CoFe(10 nm) and Fe(10 nm) devices. A second CoFe(10 nm)/Pt sample is also\nprepared for rotating-\feld measurements. For VNA FMR, we have prepared CoFe(20 nm) samples. All the CoFe\n\flms are grown on MgO(100) substrates; the Fe \flm is grown on a GaAs(100) substrate. Au (100 nm) coplanar\nwaveguides are subsequently fabricated on top of all devices. For VNA FMR samples, an additional SiO 2(100 nm) is8\nFIG. S-2. (a) Spin-torque FMR devices of CoFe(10 nm)/Pt samples. (b) Illustration of the Spin-torque FMR circuit. (c) Front\nand (d) back view of the VNA FMR devices for CoFe(20 nm) samples.\ndeposited between CoFe and Au for electric isolation. The CoFe(20 nm) bars is only visible from the back view in\nFig. S-2(d).\nSpin-torque FMR lineshapes\nFigure S-3 shows the full lineshapes of (a) CoFe(10 nm)/Pt(6 nm), (b) CoFe(10 nm) and (c) Fe(10 nm) devices\nmeasured at !=2\u0019= 20 GHz. The Fe \flms were deposited on GaAs substrates by MBE growth. (a) and (b) are used\nto extract the resonance \felds and linewidths in Figs. 2(a) and 3(a) of the main text. (c) is used to examine the\ncorrelation between damping anisotropy and AMR anisotropy.\nSpin-torque FMR linewidths as a function of frequency for CoFe(10 nm) devices.\nFigure S-4(a) shows the spin-torque FMR linewidths for CoFe(10 nm) devices. Because there is no spin torque\ninjection from Pt layer, the FMR signals are much weaker than CoFe(10 nm)/Pt and the extracted linewidths are\nmore noisy. The excitation of the dynamics is due to the magnon charge pumping e\u000bect [1] or inhomogeneities of the\nOersted \felds. No signal is measured for the rf current \rowing along the easy axis (magnetic \feld along the hard\naxis, see Fig. S-3b), because of the negligible AMR ratio.\nFigure S-4(b) shows the angular dependence of the extracted Gilbert damping for CoFe(10 nm)/Pt and CoFe(10\nnm). The former is extracted from Fig. 3(b) of the main text. The latter is extracted from Fig. S-4(a). The blue\ndata points for CoFe(10 nm)/Pt are obtained from the resonances at negative biasing \felds. Those data are used in\nFig. 4 of the main text.9\nFIG. S-3. Spin-torque FMR lineshapes of (a) CoFe(10 nm)/Pt, (b) CoFe(10 nm) and (c) Fe(10 nm) devices measured at\n!=2\u0019= 20 GHz. \u0012H\u0000\u0012Iis \fxed to 45\u000e.\nFIG. S-4. (a) \u00160\u0001H1=2as a function of frequency for CoFe(10 nm) devices. Solid lines and curves are the \fts to the experiments.\n\u0012H\u0000\u0012Iis \fxed to 45\u000e. (b)\u000bas a function of \u0012Hfor CoFe(10 nm)/Pt and CoFe(10 nm) devices.\nSpin-torque FMR for CoFe(10 nm)/Pt up to 39 GHz.\nFig. S-5 shows the spin-torque FMR lineshapes and linewidths up to 39 GHz for CoFe(10 nm)/Pt devices along\nthe easy and hard axes ( \u0012H= 90\u000eand45\u000e). At!=2\u0019= 32:1 GHz (Fig. S-5a), the spin-torque FMR amplitude is\n0.1\u0016V for the easy axis and 0.02 \u0016V for the hard axis. 10 seconds of time constant is used to obtained the signals.\nThroughout the frequency range, linewidths demonstrate good linear dependence on frequency as shown Fig. S-5(b).\nFor the hard axis the signal has reached the noise bottom limit at 32.1 GHz. For the easy axis the noise bottom limit\nis reached at 39 GHz. The two linear \fts yield \u000b= 0:0063 and\u00160\u0001H0= 1:8 mT for the easy axis and \u000b= 0:00153\nand\u00160\u0001H0= 1:5 mT for the hard axis. The two damping parameters are close to the values obtained below 20 GHz\nin the main text. Also the inhomogeneous linewidth \u00160\u0001H0nicely match between easy and hard axes.10\nFIG. S-5. High-frequency ST-FMR measurement of i) CoFe(10 nm)/Pt for the biasing \feld along the easy axis ( \u0012H= 90\u000e) and\nhard axis (\u0012H= 45\u000e). Left: lineshapes of ST-FMR at !=2\u0019= 32:1 GHz. Right: linewidth as a function of frequency. Lines\nare linear \fts to the data by setting both \u000band \u0001H0as free parameters.\nLow-temperature FMR linewidths and dampings for CoFe(10 nm)/Pt and CoFe(20 nm).\nFIG. S-6. (a-b) \u00160\u0001H1=2as a function of frequency for CoFe(10 nm)/Pt devices at di\u000berent temperatures. (c) Extracted\ndamping at di\u000berent temperatures, same as in Fig. 4 of the main text.\nFigure S-6 shows the frequency dependence of linewidths for extracting temperature-dependent Gilbert damping\nin Fig. 5(a) of the main text.\nFor CoFe(10 nm)/Pt samples, we plot both \u000band resistivity \u001ameasured at di\u000berent temperatures in Fig. S-6(c).\nThe measurements of \u001awere conducted with a biasing magnetic \feld of 1 Tesla parallel to the current direction, so\nthat the AMR in\ruence is excluded. Also the resistivity variation between the easy and hard axes is very small, about\n1%, which is much smaller than the damping anisotropy.\nWe have also conducted the low-temperature VNA FMR of the new CoFe(20 nm) samples at 8 K, in addition to\nthe room-temperature measurements. The linewidths data are shown in Fig. S-6(d) for both easy and hard axes.\nThe extracted damping are: \u000b= 0:0054 (EA, 300 K), 0.0061 (EA, 8 K), 0.0240 (HA, 300 K) and 0.0329 (HA, 8 K).\nThose values are used in Fig. 4(b) and Fig. 5(a) of the main text.\nFor CoFe(10 nm) the damping anisotropy decreases from 380 % at 300 K to 273 % at 30 K by taking out the spin\npumping damping enhancement (an unexpected reduction of alpha happens at 8 K for the hard axis). For CoFe(2011\nnm) the damping anisotropy increases from 440 % at 300 K to 540 % at 8 K. Thus a clear variation trend of damping\nanisotropy in CoFe \flms remains to be explored.\nFirst-principle calculation of N(EF)anisotropy for Co 50Fe50\nFIG. S-7. Density of states as a function of energy. EFis the Fermi level.\nFirst-principle calculations were done using QUANTUM ESPRESSO for a cubic lattice of Co 50Fe50of CsCl, Zintl\nand random alloy structures. Supercells consisting of 4 \u00024\u00024 unit cells were considered with a total of 128 atoms (64\ncobalt and 64 iron atoms). The calculations were done using plane-wave basis set with a 180 Ry kinetic energy cut-o\u000b\nand 1440 Ry density cut-o\u000b. For both Co and Fe atoms, fully relativistic PAW pseudopotentials were used. Figure\nS-7 shows the density of states (DOS) of the CsCl form for di\u000berent magnetization orientations \u0012in thexy-plane.\nClearly, DOS exhibits no anisotropy ( <0:1% variation at E=EF). No anisotropy was found in the Zintl form, either.\nThus, we conclude that the Gilbert damping anisotropy in Co 50Fe50cannot be caused by a variation of N(EF) with\nrespect to magnetization direction in ideal ordered structures.\nSOC induced by atomic short-range order (ASRO)\nIn our experiment, because the Co 50Fe50\flms were grown by MBE at low temperatures, they do not form the\nordered bcc B2 structure but instead exhibit compositional disorder. Transition metal alloys such as CoPt, NiFe, and\nCoFe tend to exhibit ASRO [2{4]. The ASRO in CoFe is likely to give rise to local tetragonal distortions because of the\ndi\u000berent lattice constants of bcc Fe and (metastable) bcc Co at 2.856 \u0017A and 2.82 \u0017A, respectively. Such local tetragonal\ndistortions will preserve global cubic (or four-fold in-plane) symmetry, but can have large e\u000bects on the SOC, with\nconcomitant e\u000bect on spin-orbit induced magnetization damping. For example, \frst-principle calculations using the\ncoherent-potential approximation for the substitutionally disordered system shows that a tetragonal distortion of 10%\nin the ratio of the tetragonal axes aandcgives rise to an magnetocrystalline anisotropy energy (MAE) density [2, 3]\nof about 1 MJ/m3. These results are consistent with our observed MAE in Co 50Fe50.\nTo con\frm this mechanism, we performed DFT-LDA calculations on 50:50 CoFe supercells consisting of a total\nof 16 atoms for CsCl, zintl, and random alloy structures; in the random alloy supercell, Co or Fe atoms randomly\noccupied the atomic positions in the supercell. Note that all CoFe geometries are fully relaxed, including supercell\nlattice vectors.\n1. Structural relaxation including spin-orbit coupling (SOC) shows local tetragonal distortions for random alloy\nsupercell. Among the three di\u000berent CoFe phases, tetragonal c/a ratio for the supercell in optimized geometry\nis largest (1.003) in the random alloy supercell with SOC, which means local tetragonal distortions are more12\nFIG. S-8. Density of states (DOS) for (a) CsCl, (b) Zintl, and (c) alloy form of CoFe with SOC (black solid) and without SOC\n(red solid).\nTABLE I. Relaxed atomic positions (including SOC) of the alloy structure. In the ideal CsCl or Zintl structures, the atomic\npositions are all multiples of 0.25 in units of the lattice vector components.\nAtom x-position y-position z-position\nCo 0.003783083 0.000000000 0.000000000\nFe -0.001339230 0.000000000 0.500000000\nFe -0.002327721 0.500000000 0.000000000\nFe 0.002079922 0.500000000 0.500000000\nFe 0.502327721 0.000000000 0.000000000\nFe 0.497920078 0.000000000 0.500000000\nCo 0.496216917 0.500000000 0.000000000\nFe 0.501339230 0.500000000 0.500000000\nCo 0.250000000 0.250000000 0.254117992\nFe 0.250000000 0.250000000 0.752628048\nFe 0.250000000 0.750000000 0.247371952\nCo 0.250000000 0.750000000 0.745882008\nCo 0.750000000 0.250000000 0.250415490\nCo 0.750000000 0.250000000 0.746688258\nCo 0.750000000 0.750000000 0.253311742\nCo 0.750000000 0.750000000 0.749584510\ndominant in random alloy compared to CsCl and Zintl structures. [c/a values : CsCl (0.999), Zintl (0.999),\nAlloy (1.003)]. In addition, the alloy system exhibited local distortions of Co and Fe position relative to their\nideal positions. In contrast, in CsCl and Zintl structures the Co and Fe atoms exhibited almost imperceptible\ndistortions. Table shows the relaxed atomic positions in the alloys structure in units of the lattice vectors. In\nthe ideal (unrelaxed) system, the positions are all at multiples of 0.25; the relaxed CsCl and Zintl structures no\ndeviations from these positions larger than 1 part in 106\n2. SOC changes the density of states (DOS) at the Fermi energy, notably for the random alloy but notfor the CsCl\nand Zintl structures. Figure S-8 shows DOS for (a) CsCl, (b) Zintl, and (c) random alloy structure with SOC\n(black lines) and without it red lines). We can see signi\fcant DOS di\u000berence for the random alloy supercell\nwith SOC where tetragonal distortions occurred, while almost no changes are observed in the CsCl and Zintl\nstructures.\n3. The local distortions in the alloy structure furthermore gave rise to an energy anisotropy with respect to the\nmagnetization direction. The energy (including SOC) of the relaxed alloy structure for di\u000berent directions of\nthe magnetization is shown in Fig. S-9. While the supercell was rather small, because of the computational\nexpense in relaxing the structure with SOC, so that no self-averaging can be inferred, the \fgure demonstrates\nan induced magnetic anisotropy that arises from the SOC and local distortions. No magnetic anisotropy was\ndiscernible in the CsCl and Zintl structures.\nAs a result from the DFT calculation, we attribute the large SOC e\u000bect in damping anisotropy of Co 50Fe50to local\ntetragonal distortions in disordered Co and Fe alloys. These distortions give rise to SOC-induced changes of DOS at\nthe Fermi level, as well as magnetic anisotropy energy with respect to the crystallographic axes.13\nFIG. S-9. Change in total energy (per supercell) of the alloy structure as function of the magnetization direction.\n\u0003wuyizheng@fudan.edu.cn\nyho\u000bmann@anl.gov\nzweizhang@oakland.edu\n[1] C. Ciccarelli, K. M. D. Hals, A. Irvine, V. Novak, Y. Tserkovnyak, H. Kurebayashi, A. Brataas and A. Ferguson, Nature\nNano. 10, 50 (2015)\n[2] S. Razee, J. Staunton, B. Ginatempo, E. Bruno, and F. Pinski, Phys. Rev. B 64, 014411 (2001).\n[3] Y. Kota and A. Sakuma, Appl. Phys. Express 5, 113002 (2012).\n[4] I. Turek, J. Kudrnovsk\u0013 y, and K. Carva, Phys. Rev. B 86, 174430 (2012)."    },    {        "title": "1909.08004v1.Microwave_induced_tunable_subharmonic_steps_in_superconductor_ferromagnet_superconductor_Josephson_junction.pdf",        "content": "arXiv:1909.08004v1  [cond-mat.supr-con]  17 Sep 2019Microwave induced tunable subharmonic steps in\nsuperconductor-ferromagnet-superconductor Josephson j unction\nM. Nashaat,1,2,∗Yu. M. Shukrinov,2,3,†A. Irie,4A.Y. Ellithi,1and Th. M. El Sherbini1\n1Department of Physics, Cairo University, Cairo, 12613, Egy pt\n2BLTP, JINR, Dubna, Moscow Region, 141980, Russian Federati on\n3Dubna State University, Dubna, 141982, Russian Federation\n4Department of Electrical and Electronic Systems Engineeri ng, Utsunomiya University, Utsunomiya, Japan.\nWe investigate the coupling between ferromagnet and superc onducting phase dynamics in\nsuperconductor-ferromagnet-superconductor Josephson j unction. The current-voltage character-\nistics of the junction demonstrate a pattern of subharmonic current steps which forms a devil’s\nstaircase structure. We show that a width of the steps become s maximal at ferromagnetic reso-\nnance. Moreover, we demonstrate that the structure of the st eps and their widths can be tuned\nby changing the frequency of the external magnetic field, rat io of Josephson to magnetic energy,\nGilbert damping and the junction size.\nThis paper is submitted to LTP Journal.\nI. INTRODUCTION\nJosephson junction with ferromagnet layer (F) is\nwidely considered to be the place where spintronics and\nsuperconductivity fields interact1. In these junctions\nthe supercurrent induces magnetization dynamics due\nto the coupling between the Josephson and magnetic\nsubsystems. The possibility of achieving electric con-\ntrol over the magnetic properties of the magnet via\nJosephson current and its counterpart, i.e., achieving\nmagnetic control over Josephson current, recently at-\ntracted a lot of attention1–7. The current-phase rela-\ntion in the superconductor-ferromagnet-superconductor\njunction (SFS) junctions is very sensitive to the mutual\norientation of the magnetizations in the F-layer8,9. In\nRef.[10] the authors demonstrate a unique magnetization\ndynamics with a series of specific phase trajectories. The\norigin of these trajectories is related to a direct coupling\nbetween the magnetic moment and the Josephson oscil-\nlations in these junctions.\nExternal electromagnetic field can also provide a cou-\npling between spin wave and Josephson phase in SFS\njunctions11–17. Spin waves are elementary spin excita-\ntions which considered to be as both spatial and time\ndependent variations in the magnetization18,19. The fer-\nromagnetic resonance(FMR) correspondsto the uniform\nprecession of the magnetization around an external ap-\nplied magnetic field18. This mode can be resonantly ex-\ncited by ac magnetic field that couples directly to the\nmagnetization dynamics as described by the Landau-\nLifshitz-Gilbert (LLG) equation18,19.\nIn Ref.[18] the authors show that spin wave resonance\nat frequency ωrin SFS implies a dissipation that is mani-\nfested as adepressionin the IV-characteristicofthe junc-\ntion when /planckover2pi1ωr= 2eV, where/planckover2pi1is the Planck’s constant,\ne is the electron charge and Vis the voltage across the\njunction. The ac Josephson current produces an oscil-\nlating magnetic field and when the Josephson frequencymatches the spin wave frequency, this resonantly excites\nthe magnetization dynamics M(t)18. Due to the non-\nlinearity of the Josephson effect, there is a rectification\nof current across the junction, resulting in a dip in the\naverage dc component of the suppercurrent18.\nIn Ref.[13] the authors neglect the effective field due\nto Josephson energy in LLG equation and the results re-\nveal that even steps appear in the IV-characteristic of\nSFS junction under external magnetic field. The ori-\ngin of these steps is due to the interaction of Cooper\npairs with even number of magnons. Inside the ferro-\nmagnet, if the Cooper pairs scattered by odd number of\nmagnons, no Josephson current flows due to the forma-\ntion of spin triplet state13. However, if the Cooper pairs\ninteract with even number of magnons, the Josephson\ncoupling between the s-wave superconductor is achieved\nand the spin singlet state is formed, resulting in flows of\nJosephsoncurrent13. In Ref.[20]weshowthat takinginto\naccount the effective field due to Josepshon energy and\nat FMR, additional subharmonic current steps appear in\nthe IV-characteristic for overdamped SFS junction with\nspin wave excitations (magnons). It is found that the po-\nsition of the current steps in the IV-characteristics form\ndevil’s staircase structure which follows continued frac-\ntion formula20. The positions of those fractional steps\nare given by\nV=\nN±1\nn±1\nm±1\np±..\nΩ, (1)\nwhere Ω = ω/ωc,ωis the frequency of the external ra-\ndiation, ωcis the is the characteristic frequency of the\nJosephson junction and N,n,m,pare positive integers.\nIn this paper, we present a detailed analysis for the\nIV-characteristics of SFS junction under external mag-\nnetic field, and show how we can control the position\nof the subharmonic steps and alter their widths. The\ncoupling between spin wave and Josephson phase in SFS\njunction is achieved through the Josephson energy and\ngauge invariant phase difference between the S-layers. In\nthe framework of our approach, the dynamics of the SFS2\njunction isfully describedbytheresistivelyshuntedjunc-\ntion (RSJ) model and LLG equation. These equations\nare solved numerically by the 4thorder Runge-Kutta\nmethod. The appearance and position of the observed\ncurrent steps depend directly on the magnetic field and\njunction parameters.\nII. MODEL AND METHODS\nF\nss\nHacxyz\nH0I\nI\nFIG. 1. SFS Josephson junction. The bias current is applied\nin x-direction, an external magnetic field with amplitude Hac\nand frequency ωis applied in xy-plane and an uniaxial con-\nstant magnetic field H0is applied in z-direction.\nIn Fig 1 we consider a current biased SFS junction\nwhere the two superconductors are separated by ferro-\nmagnet layer with thickness d. The area of the junction\nisLyLz. An uniaxial constant magnetic field H0is ap-\nplied in z-direction, while the magnetic field is applied in\nxy-plane Hac= (Haccosωt,Hacsinωt,0)withamplitude\nHacand frequency ω. The magnetic field is induced in\nthe F-layer through B(t) = 4πM(t), and the magnetic\nfluxes in z- and y-direction are Φ z(t) = 4πdLyMz(t),\nΦy(t) = 4πdLzMy(t), respectively. The gauge-invariant\nphase difference in the junction is given by21:\n∇y,zθ(y,z,t) =−2πd\nΦ0B(t)×n, (2)\nwhereθis the phase difference between superconducting\nelectrodes, and Φ 0=h/2eis the magnetic flux quantum\nandnis a unit vector normal to yz-plane. The gauge-\ninvariantphasedifference in terms ofmagnetizationcom-\nponents reads as\nθ(y,z,t) =θ(t)−8π2dMz(t)\nΦ0y+8π2dMy(t)\nΦ0z,(3)\nwhere Φ 0=h/(2e) is the magnetic flux quantum.\nAccordingtoRSJ model, the currentthroughthe junc-\ntion is given by13:\nI\nI0c= sinθ(y,z,t)+Φ0\n2πI0cRdθ(y,z,t)\ndt,(4)\nwhereI0\ncis the critical current, and R is the resistance\nin the Josephson junction. After taking into account thegaugeinvarianceincludingthemagnetizationoftheferro-\nmagnetandintegratingoverthejunction areatheelectric\ncurrent reads13:\nI\nI0c=Φ2\nosin(θ(t))sin/parenleftBig\n4π2dMz(t)Ly\nΦo/parenrightBig\nsin/parenleftBig\n4π2dMy(t)Lz\nΦo/parenrightBig\n16π4d2LzLyMz(t)My(t)\n+Φ0\n2πRI0cdθ(y,z,t)\ndt. (5)\nThe applied magnetic field in the xy-plane causes pre-\ncessionalmotionofthemagnetizationinthe F-layer. The\ndynamics of magnetization Min the F-layer is described\nby LLG equation\n(1+α2)dM\ndt=−γM×Heff−γ α\n|M|[M×(M×Heff)](6)\nThe total energy of junction in the proposed model is\ngivenby E=Es+EM+EacwhereEsistheenergystored\nin Josephson junction, EMis the energy of uniaxial dc\nmagnetic field (Zeeman energy) and Eacis the energy of\nac magnetic field:\nEs=−Φ0\n2πθ(y,z,t)I+EJ[1−cos(y,z,t)],\nEM=−VFH0Mz(t),\nEac=−VFMx(t)Haccos(ωt)−VFMy(t)Hacsin(ωt)(7)\nHere,EJ= Φ0I0\nc/2πis the the Josephson energy, H0=\nω0/γ,ω0is the FMR frequency, and VFis the volume of\nthe ferromagnet. We neglect the anisotropy energy due\nto demagnetizing effect for simplicity. The effective field\nin LLG equation is calculated by\nHeff=−1\nVF∇ME (8)\nThus, the effective field Hmdue to microwave radiation\nHacand uniaxial magnetic field H0is given by\nHm=Haccos(ωt)ˆex+Hacsin(ωt)ˆey+H0ˆez.(9)\nwhile the effective field ( Hs) due to superconducting part\nis found from\nHs=−EJ\nVFsin(θ(y,z,t))∇Mθ(y,z,t).(10)\nOne should take the integration of LLG on coordinates,\nhowever, the superconducting part is the only part which\ndepends on the coordinate so, we can integrate the ef-\nfective field due to the Josephson energy and insert the\nresult into LLG equation. Then, the y- and z-component\nare given by\nHsy=EJcos(θ(t))sin(πΦz(t)/Φ0)\nVFπMy(t)Φz(t)/bracketleftbigg\nΦ0cos(πΦy(t)/Φ0)\n−Φ2\n0sin(πΦy(t)/Φ0)\nπΦy(t)/bracketrightbigg\nˆey, (11)\nHsz=EJcos(θ(t))sin(πΦy(t)/Φ0)\nVFπMz(t)Φy(t)/bracketleftbigg\nΦ0cos(πΦz(t)/Φ0)\n−Φ2\n0sin(πΦz(t)/Φ0)\nπΦz(t)/bracketrightbigg\nˆez. (12)3\nAs a result, the total effective field is Heff=Hm+\nHs. In the dimensionless form we use t→tωc,ωc=\n2πI0\ncR/Φ0is the characteristic frequency, m=M/M0,\nM0=∝ba∇dblM∝ba∇dbl,heff=Heff/H0,ǫJ=EJ/VFM0H0,hac=\nHac/H0, Ω =ω/ωc, Ω0=ω0/ωc,φsy=4π2LydM0/Φo,\nφsz=4π2lzdM0/Φo. Finally, the voltage V(t) =dθ/dtis\nnormalized to /planckover2pi1ωc/(2e). The LLG and the effective field\nequations take the form\ndm\ndt=−Ω0\n(1+α2)/parenleftbigg\nm×heff+α[m×(m×heff)]/parenrightbigg\n(13)\nwith\nheff=haccos(Ωt)ˆex+(hacsin(Ωt)+ΓijǫJcosθ)ˆey\n+ (1+Γ jiǫJcosθ)ˆez, (14)\nΓij=sin(φsimj)\nmi(φsimj)/bracketleftbigg\ncos(φsjmi)−sin(φsjmi)\n(φsjmi)/bracketrightbigg\n,(15)\nwherei=y,j=z. The RSJ in the dimensionless form is\ngiven by\nI/I0\nc=sin(φsymz)sin(φszmy)\n(φsymz)(φszmy)sinθ+d��\ndt.(16)\nThe magnetization and phase dynamics of the SFS\njunction can be described by solving Eq.(16) together\nwith Eq.(13). To solve this system of equations, we em-\nploy the fourth-order Runge-Kutta scheme. At each cur-\nrent step, we find the temporal dependence of the volt-\nageV(t), phase θ(t), andmi(i=x,y,z) in the (0 ,Tmax)\ninterval. Then the time-average voltage Vis given by\nV=1\nTf−Ti/integraltext\nV(t)dt, whereTiandTfdetermine the in-\nterval for the temporal averaging. The current value is\nincreased or decreased by a small amount of δI (the bias\ncurrent step) to calculate the voltage at the next point\nof the IV-characteristics. The phase, voltage and mag-\nnetization components achieved at the previous current\nstep are used as the initial conditions for the next cur-\nrent step. The one-loop IV-characteristic is obtained by\nsweeping the bias current from I= 0 toI= 3 and back\ndown to I= 0. The initial conditions for the magnetiza-\ntion components are assumed to be mx= 0,my= 0.01\nandmz=/radicalBig\n1−m2x−m2y, while for the voltage and\nphase we have Vini= 0,θini=0. The numerical param-\neters (if not mentioned) are taken as α= 0.1,hac= 1,\nφsy=φsz= 4,ǫJ= 0.2 and Ω 0= 0.5.\nIII. RESULTS AND DISCUSSIONS\nItiswell-knownthatJosephsonoscillationscanbesyn-\nchronized by external microwave radiation which leads\nto Shapiro steps in the IV-characteristic22. The position\nof the Shapiro step is determined by relation V=n\nmΩ,\nwheren,mare integers. The steps at m= 1 are calledharmonics, otherwise we deal with synchronized subhar-\nmonic (fractional) steps. We show below the appearance\nof subharmonics in our case.\nFirst we present the simulated IV-characteristics at\ndifferent frequencies of the magnetic field. The IV-\ncharacteristics at three different values of Ω are shown\nin Fig 2(a).\nFIG. 2. (a) IV-characteristic at three different values of Ω.\nFor clarity, the IV-characteristics for Ω = 0 .5 and Ω = 0 .7\nhave been shifted to the right, by ∆ I= 0.5 and ∆ I= 1,\nrespectively with respect to Ω = 0 .2; (b) An enlarged part\nof the IV-characteristic with Ω = 0 .7. To get step voltage\nmultiply the corresponding fraction with Ω = 0 .7.\nAs we see, the second harmonic has the largest step\nwidth at the ferromagnetic resonance frequency Ω = Ω 0,\ni.e., the FMR is manifested itself by the step’s width.\nThere are also many subharmonic current steps in the\nIV-characteristic. We have analyzed the steps position\nbetween V= 0 and V= 0.7 for Ω = 0 .7 and found dif-\nferent level continued fractions, which follow the formula\ngiven by Eq.(1) and demonstrated in Fig.2(b). We see4\nthe reflection of the second level continued fractions 1 /n\nand 1−1/nwithN= 1. In addition to this, steps with\nthird level continued fractions 1 /(n−1/m) withN= 1\nis manifested. In the inset we demonstrate part of the\nfourth level continued fraction 1 −1/(n+ 1/(m+1/p))\nwithn= 2 and m= 2.\nIn case of external electromagnetic field which leads to\nthe additional electric current Iac=AsinΩt, the width\nof the Shapiro step is proportional to ∝Jn(A/Ω), where\nJnis the Bessel function of first kind. The preliminary\nresults (not presented here) show that the width of the\nShapiro-like steps under external magnetic field has a\nmore complex frequency dependence20. This question\nwill be discussed in detail somewhere else.\nThe coupling between Josephson phase and magneti-\nzation manifests itself in the appearance of the Shapiro\nsteps in the IV-characteristics at fractional and odd mul-\ntiplies of Ω20. In Fig.3 we demonstrate the effect of the\nratio of the Josephson to magnetic energy ǫJon appear-\nance of the steps and their width for Ω = 0 .5 where the\nenlarged parts of the IV-characteristics at three differ-\nent values of ǫJare shown. As it is demonstrated in\nthe figures, at ǫJ= 0.05 only two subharmonic steps\nappear between V= 1 and V= 1.5 (see hollow ar-\nrows). An enhanced staircase structure appears by in-\ncreasing the value of ǫJ, which can be see at ǫJ= 0.3\nand 0.5. Moreover, an intense subharmonic steps appear\nbetween V= 1.75 andV= 2 forǫJ= 0.5. The posi-\ntions for these steps reflect third level continued fraction\n(N−1)+1/(n+1/m)withN=4 andn=1 [see Fig.3(b)].\nLet us now demonstrate the effect of Gilbert damping\non the devil’s staircase structure. The Gilbert damping\nαis introduced into LLG equation23?to describe the\nrelaxation of magnetization dynamics. To reflect effect\nof Gilbert damping, we show an enlarged part of the IV-\ncharacteristic at three different values of αin Fig.4.\nThewidthofcurrentstepat V= 2Ωisalmostthesame\nat different values of α(e.g., see upward inset V= 2Ω).\nThe subharmonic current step width for V= (n/m)Ω (n\nis odd,mis integer) is decreasing with increasing α. In\naddition a horizontal shift for the current steps occurs.\nWe see the intense current steps in the IV-characteristic\nfor small value of α= 0.03 (see black solid arrows). With\nincrease in Gilbert damping (see α= 0.1, 0.16 and 0 .3)\nthe higher level subharmonic steps disappear. It is well-\nknownthatatlargevalueof αtheFMRlinewidthbecome\nmore broadening and the resonance frequency is shifted\nfrom Ω 0. Accordingly, the subharmonic steps disappear\nat large value of α. Furthermore, using the formula pre-\nsented in Ref.[20] the width at Ω = Ω 0for the fractional\nand odd current steps is proportional to (4 α2+α4)−q/2\n×(12+3α2)−k/2, whereqandkare integers.\nFinally, we demonstrate the effect of the junction size\non the devil’s staircase in the IV-characteristic under ex-\nternalmagneticfield. Thejunction sizechangesthe value\nofφsyandφsz. In Fig.5(a) we demonstrate the effect of\nthe junction thickness by changing φsz(φsyis qualita-\nFIG. 3. (a) An enlarged part of the IV-characteristic at\ndifferent values of ǫJin the interval between V= 1 and V=\n1.5; (b)Thesameintheintervalbetween V= 1.75andV= 2.\nFor clarity, the IV-characteristics for ǫJ= 0.3, and 0 .5 have\nbeen shifted to right, by ∆ I= 0.07, and 0 .14, respectively\nwith respect to the case with ǫJ= 0.05.\ntively the same).\nWe observe an enhanced subharmonic structure with\nincrease of junction size or the thickness of the ferro-\nmagnet. In Ref.[13] the authors demonstrated that the\ncritical current and the width of the step at V= 2Ω as a\nfunction of Lz/Lyfollow Bessel function of first kind. In\nFig.5(b), we can see the parts of continued fraction se-\nquences for subharmonic steps between V= 1 andV= 2\natφsz=φsy= 6. Current steps between V= 1 and\nV= 1.5 reflect the two second level continued fractions\n(N−1)+ 1/nandN−1/nwithN= 3 in both cases,\nwhile for the steps between V= 1.5 andV= 2 follow\nthe second level continued fraction ( N−1) + 1/nwith\nN= 4.\nFinally, wediscussthepossibilityofexperimentallyob-5\nFIG. 4. An enlarged part of IV-characteristic for four differ -\nent values of Gilbert damping for Ω = 0 .5. The inset shows an\nenlargedpartofcurrentstepwithconstantvoltage at V= 2Ω.\nserving the effects presented in this paper. For junction\nsized= 5nm, Ly=Lz= 80nm, critical current I0\nc≈\n200µA, saturation magnetization M0≈5×105A/m,\nH0≈40mT and gyromagnetic ratio γ= 3πMHz/T,\nwe find the value of φsy(z)=4π2Ly(z)dM0/Φ0= 4.8 and\nǫJ= 0.1. With the same junction parameters one can\ncontrol the appearance of the subharmonic steps by tun-\ning the strength of the constant magnetic field H0. Esti-\nmations showthat for H0= 10mT, the value of ǫJ= 0.4,\nand the fractional subharmonic steps are enhanced. In\ngeneral, the subharmonic steps are sensitive to junction\nparameters, Gilbert damping and the frequency of the\nexternal magnetic field.\nIV. CONCLUSIONS\nIn this work, we have studied the IV-characteristics\nof superconductor-ferromagnet-superconductor Joseph-\nson junction under external magnetic field. We used a\nmodified RSJ model which hosts magnetization dynam-\nics in F-layer. Due to the external magnetic field, the\ncouplingbetweenmagneticmomentandJosephsonphase\nis achieved through the effective field taking into account\nthe Josephson energy and gauge invariant phase differ-\nence between the superconducting electrodes. We have\nsolvedasystemofequationswhichdescribethe dynamics\nof the Josephson phase by the RSJ equation and magne-\ntization dynamics by Landau-Lifshitz-Gilbert equation.\nThe IV-characteristic demonstrates subharmonic current\nsteps. The pattern of the subharmonic steps can be con-\ntrolled by tuning the frequency of the ac magnetic field.\nWe show that by increasing the ratio of the Josephson to\nmagneticenergyanenhancedstaircasestructureappears.\nFinally, we demonstrate that Gilbert damping and junc-\nFIG. 5. (a) IV-characteristic at three different values of\nφsz= 0.7,3,6 andφsy=φsz. (b) An enlarged part of the IV-\ncharacteristic at φsz=φsy=6. The hollow arrows represent\nthe starting point of the sequences. To get step voltage we\nmultiply the corresponding fraction by Ω = 0 .5.\ntion parameters can change the subharmonic step struc-\nture. The observed features might find an application in\nsuperconducting spintronics.\nV. ACKNOWLEDGMENT\nWe thank Dr. D. V. Kamanin and Egypt JINR col-\nlaboration for support this work. The reported study\nwas partially funded by the RFBR research Projects No.\n18-02-00318 and No. 18-52-45011-IND. Numerical cal-\nculations have been made in the framework of the RSF\nProject No. 18-71-10095.6\nREFERENCES\n∗majed@sci.cu.edu.eg\n†shukrinv@theor.jinr.ru\n1J. Linder and K. Halterman, Phys. Rev. B 90, 104502\n(2014).\n2Yu. M. Shukrinov, A. Mazanik, I. Rahmonov, A. Botha,\nand A. Buzdin, EPL122, 37001 (2018).\n3Yu. M. Shukrinov, I. Rahmonov, K. Sengupta, and A.\nBuzdin, Appl. Phys. Lett. 110, 182407 (2017).\n4A. Buzdin, Phys. Rev. Lett. 101, 107005 (2008).\n5A. I. Buzdin, Rev. Mod. Phys. 77, 935 (2005).\n6F. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod.\nPhys.77, 1321 (2005).\n7A. A. Golubov, M. Y. Kupriyanov, and E. IlIchev, Rev.\nMod. Phys. 76, 411 (2004).\n8M. A. Silaev, I. V. Tokatly, and F. S. Bergeret, Phys. Rev.\nB95, 184508 (2017).\n9I. Bobkova, A. Bobkov, and M. Silaev, Phys. Rev. B 96,\n094506 (2017).\n10Y. M. Shukrinov, I. Rahmonov, and K. Sengupta, Phys.\nRev. B99, 224513 (2019).\n11M. Weides, M. Kemmler, H. Kohlstedt, R. Waser, D.\nKoelle, R. Kleiner, and E. 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Rev.\nB 97, 224514 (2018).\n21K. K. Likharev, Dynamics of Josephson junctions and cir-\ncuits, Gordon and Breach science publishers -Switzerland\n(1986).\n22S. Shapiro, Phys. Rev. Lett. 11, 80 (1963).\n23T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443-\n3449 (2004).\n24M. C. Hickey and J. S. Moodera, Phys. Rev. Lett. 102,\n137601(2009)."    },    {        "title": "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. 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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": "0807.5009v1.Scattering_Theory_of_Gilbert_Damping.pdf",        "content": "arXiv:0807.5009v1  [cond-mat.mes-hall]  31 Jul 2008Scattering Theory of Gilbert Damping\nArne Brataas,1,∗Yaroslav Tserkovnyak,2and Gerrit E. W. Bauer3\n1Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway\n2Department of Physics and Astronomy, University of Califor nia, Los Angeles, California 90095, USA\n3Kavli Institute of NanoScience, Delft University of Techno logy, Lorentzweg 1, 2628 CJ Delft, The Netherlands\nThe magnetization dynamics of a single domain ferromagnet i n contact with a thermal bath\nis studied by scattering theory. We recover the Landau-Lift shitz-Gilbert equation and express the\neffective fields and Gilbert damping tensor in terms of the sca ttering matrix. Dissipation of magnetic\nenergy equals energy current pumped out of the system by the t ime-dependent magnetization, with\nseparable spin-relaxation induced bulk and spin-pumping g enerated interface contributions. In\nlinear response, our scattering theory for the Gilbert damp ing tensor is equivalent with the Kubo\nformalism.\nMagnetization relaxation is a collective many-body\nphenomenon that remains intriguing despite decades of\ntheoretical and experimental investigations. It is im-\nportant in topics of current interest since it determines\nthe magnetization dynamics and noise in magnetic mem-\nory devices and state-of-the-art magnetoelectronic ex-\nperiments on current-induced magnetization dynamics\n[1]. Magnetization relaxation is often described in terms\nof a damping torque in the phenomenological Landau-\nLifshitz-Gilbert (LLG) equation\n1\nγdM\ndτ=−M×Heff+M×/bracketleftBigg˜G(M)\nγ2M2sdM\ndτ/bracketrightBigg\n, (1)\nwhereMis the magnetization vector, γ=gµB//planckover2pi1is the\ngyromagnetic ratio in terms of the gfactor and the Bohr\nmagnetonµB, andMs=|M|is the saturation magneti-\nzation. Usually, the Gilbert damping ˜G(M) is assumed\nto be a scalar and isotropic parameter, but in general it\nis a symmetric 3 ×3 tensor. The LLG equation has been\nderived microscopically [2] and successfully describes the\nmeasured response of ferromagnetic bulk materials and\nthin films in terms of a few material-specific parameters\nthatareaccessibletoferromagnetic-resonance(FMR) ex-\nperiments [3]. We focus in the following on small fer-\nromagnets in which the spatial degrees of freedom are\nfrozen out (macrospin model). Gilbert damping pre-\ndicts a striclylinear dependence ofFMR linewidts on fre-\nquency. This distinguishes it from inhomogenous broad-\nening associated with dephasing of the global precession,\nwhich typically induces a weaker frequency dependence\nas well as a zero-frequency contribution.\nThe effective magnetic field Heff=−∂F/∂Mis the\nderivative of the free energy Fof the magnetic system\nin an external magnetic field Hext, including the classi-\ncal magnetic dipolar field Hd. When the ferromagnet is\npart of an open system as in Fig. 1, −∂F/∂Mcan be\nexpressed in terms of a scattering S-matrix, quite anal-\nogous to the interlayer exchange coupling between ferro-\nmagnetic layers [4]. The scattering matrix is defined in\nthe space of the transport channels that connect a scat-\ntering region (the sample) to thermodynamic (left andleft\nreservoirF N Nright\nreservoir\nFIG. 1: Schematic picture of a ferromagnet (F) in contact\nwith a thermal bath via metallic normal metal leads (N).\nright) reservoirs by electric contacts that are modeled by\nideal leads. Scattering matrices also contain information\nto describe giant magnetoresistance, spin pumping and\nspin battery, and current-induced magnetization dynam-\nics in layered normal-metal (N) |ferromagnet (F) systems\n[4, 5, 6].\nIn the following we demonstrate that scattering the-\nory can be also used to compute the Gilbert damping\ntensor˜G(M).The energy loss rate of the scattering re-\ngion can be described in terms of the time-dependent\nS-matrix. Here, we generalize the theory of adiabatic\nquantum pumping to describe dissipation in a metallic\nferromagnet. Our idea is to evaluate the energy pump-\ningoutoftheferromagnetandtorelatethistotheenergy\nloss of the LLG equation. We find that the Gilbert phe-\nnomenology is valid beyond the linear response regime of\nsmall magnetization amplitudes. The only approxima-\ntion that is necessary to derive Eq. (1) including ˜G(M)\nis the (adiabatic) assumption that the frequency ωof the\nmagnetization dynamics is slow compared to the relevant\ninternal energy scales set by the exchange splitting ∆.\nThe LLG phenomenology works so well because /planckover2pi1ω≪∆\nsafely holds for most ferromagnets.\nGilbert damping in transition-metal ferromagnets is\ngenerally believed to stem from spin-orbit interaction in\ncombinationwith impurityscatteringthattransfersmag-\nnetic energy to itinerant quasiparticles [3]. The subse-\nquent drainage of the energy out of the electronic sys-\ntem,e.g.by inelastic scattering via phonons, is believed\nto be a fast process that does not limit the overall damp-\ning. Our key assumption is adiabaticiy, meaning that\nthe precession frequency goes to zero before letting the\nsample size become large. The magnetization dynam-\nics then heats up the entire magnetic system by a tiny2\namount that escapes via the contacts. The leakage heat\ncurrent then equals the total dissipation rate. For suf-\nficiently large samples, bulk heat production is insensi-\ntive to the contact details and can be identified as an\nadditive contribution to the total heat current that es-\ncapes via the contacts. The chemical potential is set\nby the reservoirs, which means that (in the absence of\nan intentional bias) the sample is then always very close\nto equilibrium. The S-matrix expanded to linear order\nin the magnetization dynamics and the Kubo linear re-\nsponse formalisms should give identical results, which we\nwill explicitly demonstrate. The role of the infinitesi-\nmal inelastic scattering that guarantees causality in the\nKubo approach is in the scattering approach taken over\nby the coupling to the reservoirs. Since the electron-\nphonon relaxation is not expected to directly impede the\noverall rate of magnetic energy dissipation, we do not\nneed to explicitly include it in our treatment. The en-\nergy flow supported by the leads, thus, appears in our\nmodel to be carried entirely by electrons irrespective of\nwhethertheenergyisactuallycarriedbyphonons, incase\nthe electrons relax by inelastic scattering before reaching\nthe leads. So we are able to compute the magnetization\ndamping, but not, e.g., how the sample heats up by it .\nAccording to Eq. (1), the time derivative of the energy\nreads\n˙E=Heff·dM/dτ= (1/γ2)˙ m/bracketleftBig\n˜G(m)˙ m/bracketrightBig\n,(2)\nin terms of the magnetization direction unit vector m=\nM/Msand˙ m=dm/dτ. We now develop the scatter-\ning theory for a ferromagnet connected to two reservoirs\nby normal metal leads as shown in Fig. 1. The total\nenergy pumping into both leads I(pump)\nEat low tempera-\ntures reads [11, 12]\nI(pump)\nE= (/planckover2pi1/4π)Tr˙S˙S†, (3)\nwhere˙S=dS/dτandSis the S-matrix at the Fermi\nenergy:\nS(m) =/parenleftbiggr t′\nt r′/parenrightbigg\n. (4)\nrandt(r′andt′) are the reflection and transmissionma-\ntrices spanned by the transport channels and spin states\nfor an incoming wave from the left (right). The gener-\nalization to finite temperatures is possible but requires\nknowledge of the energy dependence of the S-matrix\naround the Fermi energy [12]. The S-matrix changes\nparametrically with the time-dependent variation of the\nmagnetization S(τ) =S(m(τ)). We obtain the Gilbert\ndamping tensor in terms of the S-matrix by equating the\nenergy pumping by the magnetic system (3) with the en-\nergy loss expression (2), ˙E=I(pump)\nE. Consequently\nGij(m) =γ2/planckover2pi1\n4πRe/braceleftbigg\nTr/bracketleftbigg∂S\n∂mi∂S†\n∂mj/bracketrightbigg/bracerightbigg\n,(5)which is our main result.\nThe remainder of our paper serves three purposes. We\nshow that (i) the S-matrix formalism expanded to linear\nresponseis equivalentto Kubolinearresponseformalism,\ndemonstrate that (ii) energy pumping reduces to inter-\nface spin pumping in the absence ofspin relaxationin the\nscattering region, and (iii) use a simple 2-band toy model\nwith spin-flip scattering to explicitly show that we can\nidentify both the disorder and interface (spin-pumping)\nmagnetization damping as additive contributions to the\nGilbert damping.\nAnalogous to the Fisher-Lee relation between Kubo\nconductivity and the Landauer formula [15] we will now\nprove that the Gilbert damping in terms of S-matrix (5)\nis consistent with the conventionalderivation of the mag-\nnetization damping by the linear response formalism. To\nthis end we chose a generic mean-field Hamiltonian that\ndepends on the magnetization direction m:ˆH=ˆH(m)\ndescribes the system in Fig. 1. ˆHcan describe realistic\nband structures as computed by density-functional the-\nory including exchange-correlation effects and spin-orbit\ncouplingaswell normaland spin-orbitinduced scattering\noff impurities. The energy dissipation is ˙E=/angb∇acketleftdˆH/dτ/angb∇acket∇ight,\nwhere/angb∇acketleft.../angb∇acket∇ightdenotes the expectation value for the non-\nequilibriumstate. Inlinearresponse,weexpandthemag-\nnetization direction m(t) around the equilibrium magne-\ntization direction m0,\nm(τ)=m0+u(τ). (6)\nThe Hamiltonian can be linearized as ˆH=ˆHst+\nui(τ)∂iˆH, where ˆHst≡ˆH(m0) is the static Hamilto-\nnian and∂iˆH≡∂uiˆH(m0), where summation over re-\npeated indices i=x,y,zis implied. To lowest order\n˙E= ˙ui(τ)/angb∇acketleft∂iˆH/angb∇acket∇ight, where\n/angb∇acketleft∂iˆH/angb∇acket∇ight=/angb∇acketleft∂iˆH/angb∇acket∇ight0+/integraldisplay∞\n−∞dτ′χij(τ−τ′)uj(τ′).(7)\n/angb∇acketleft.../angb∇acket∇ight0denotes equilibrium expectation value and the re-\ntarded correlation function is\nχij(τ−τ′) =−i\n/planckover2pi1θ(τ−τ′)/angbracketleftBig\n[∂iˆH(τ),∂jˆH(τ′)]/angbracketrightBig\n0(8)\nin the interaction picture for the time evolution. In order\nto arrive at the adiabatic (Gilbert) damping the magne-\ntization dynamics has to be sufficiently slow such that\nuj(τ)≈uj(t) + (τ−t) ˙uj(t). Since m2= 1 and hence\n˙ m·m= 0 [7]\n˙E=i∂ωχij(ω→0)˙ui˙uj, (9)\nwhereχij(ω) =/integraltext∞\n−∞dτχij(τ)exp(iωτ). Next, we use\nthe scattering states as the basis for expressing the\ncorrelation function (8). The Hamiltonian consists of\na free-electron part and a scattering potential: ˆH=\nˆH0+ˆV(m). We denote the unperturbed eigenstates of3\nthe free-electron Hamiltonian ˆH0=−/planckover2pi12∇2/2mat en-\nergyǫby|ϕs,q(ǫ)/angb∇acket∇ight, wheres=l,rdenotes propagation\ndirection and qtransverse quantum number. The po-\ntentialˆV(m) scatters the particles between these free-\nelectron states. The outgoing (+) and incoming wave\n(-) eigenstates |ψ(±)\ns,q(ǫ)/angb∇acket∇ightof the static Hamiltonian ˆHst\nfulfill the completeness conditions /angb∇acketleftψ(±)\ns,q(ǫ)|ψ(±)\ns′,q′(ǫ′)/angb∇acket∇ight=\nδs,s′δq,q′δ(ǫ−ǫ′) [10]. These wave functions can be ex-\npressed as |ψ(±)\ns(ǫ)/angb∇acket∇ight= [1 +ˆG(±)\nstˆVst]|ϕs(ǫ)/angb∇acket∇ight, where the\nstatic retarded (+) and advanced (-) Green functions are\nˆG(±)\nst(ǫ) = (ǫ±iη−ˆHst)−1andηis a positive infinites-\nimal. By expanding χij(ω) in the basis of the outgo-\ning wave functions |ψ(+)\ns/angb∇acket∇ight, the low-temperature linear re-\nsponse leads to the followingenergydissipation (9) in the\nadiabatic limit\n˙E=−π/planckover2pi1˙ui˙uj/angbracketleftBig\nψ(+)\ns,q|∂iˆH|ψ(+)\ns′,q′/angbracketrightBig/angbracketleftBig\nψ(+)\ns′,q′|∂jˆH|ψ(+)\ns,q/angbracketrightBig\n,\n(10)\nwith wave functions evaluated at the Fermi energy ǫF.\nIn order to compare the linear response result, Eq.\n(10), withthat ofthe scatteringtheory, Eq. (5), weintro-\nduce the T-matrix ˆTasˆS(ǫ;m) = 1−2πiˆT(ǫ;m), where\nˆT=ˆV[1 +ˆG(+)ˆT] in terms of the full Green function\nˆG(+)(ǫ,m) = [ǫ+iη−ˆH(m)]−1. Although the adiabatic\nenergy pumping (5) is valid for any magnitude of slow\nmagnetization dynamics, in order to make connection to\nthe linear-response formalism we should consider small\nmagnetization changes to the equilibrium values as de-\nscribed by Eq. (6). We then find\n∂τˆT=/bracketleftBig\n1+ˆVstˆG(+)\nst/bracketrightBig\n˙ui∂iˆH/bracketleftBig\n1+ˆG(+)\nstˆVst/bracketrightBig\n.(11)\ninto Eq. (5) and using the completeness of the scattering\nstates, we recover Eq. (10).\nOur S-matrix approach generalizes the theory of (non-\nlocal) spin pumping and enhanced Gilbert damping in\nthin ferromagnets [5]: by conservation of the total an-\ngular momentum the spin current pumped into the\nsurrounding conductors implies an additional damping\ntorque that enhances the bulk Gilbert damping. Spin\npumping is an N |F interfacial effect that becomes impor-\ntant in thin ferromagnetic films [14]. In the absence of\nspin relaxation in the scattering region, the S-matrix can\nbe decomposed as S(m) =S↑(1+ˆσ·m)/2+S↓(1−ˆσ·\nm)/2, where ˆσis a vector of Pauli matrices. In this case,\nTr(∂τS)(∂τS)†=Ar˙ m2, whereAr= Tr[1−ReS↑S†\n↓]\nand the trace is over the orbital degrees of freedom only.\nWe recover the diagonal and isotropic Gilbert damping\ntensor:Gij=δijGderived earlier [5], where\nG=γMsα=(gµB)2\n4π/planckover2pi1Ar. (12)\nFinally, we illustrate by a model calculation that\nwe can obtain magnetization damping by both spin-\nrelaxationandinterfacespin-pumpingfromtheS-matrix.We consider a thin film ferromagnet in the two-band\nStoner model embedded in a free-electron metal\nˆH=−/planckover2pi12\n2m∇2+δ(x)ˆV(ρ), (13)\nwhere the in-plane coordinate of the ferromagnet is ρ\nand the normal coordinate is x.The spin-dependent po-\ntentialˆV(ρ) consists of the mean-field exchange interac-\ntion oriented along the magnetization direction mand\nmagnetic disorder in the form of magnetic impurities Si\nˆV(ρ) =νˆσ·m+/summationdisplay\niζiˆσ·Siδ(ρ−ρi),(14)\nwhich are randomly oriented and distributed in the film\natx= 0. Impurities in combination with spin-orbit cou-\npling will give similar contributions as magnetic impuri-\nties to Gilbert damping. Our derivation of the S-matrix\nclosely follows Ref. [8]. The 2-component spinor wave\nfunction can be written as Ψ( x,ρ) =/summationtext\nk/bardblck/bardbl(x)Φk/bardbl(ρ),\nwhere the transverse wave function is Φ k/bardbl(ρ) = exp(ik/bardbl·\nρ)/√\nAfor the cross-sectional area A. The effective one-\ndimensional equation for the longitudinal part of the\nwave function is then\n/bracketleftbiggd2\ndx2+k2\n⊥/bracketrightbigg\nck/bardbl(x) =/summationdisplay\nk′\n/bardbl˜Γk/bardbl,k′\n/bardblck/bardbl(0)δ(x),(15)\nwhere the matrix elements are defined by ˜Γk/bardbl,k′\n/bardbl=\n(2m//planckover2pi12)/integraltext\ndρΦ∗\nk/bardbl(ρ)ˆV(ρ)Φk′\n/bardbl(ρ)and the longitudinal\nwave vector k⊥is defined by k2\n⊥= 2mǫF//planckover2pi12−k2\n/bardbl. For\nan incoming electron from the left, the longitudinal wave\nfunction is\nck/bardbls=χs√k⊥/braceleftBigg\neik⊥xδk/bardbls,k′\n/bardbls′+e−ik⊥xrk/bardbls,k′\n/bardbls′,x<0\neik⊥xtk/bardbls,k′\n/bardbls′,x>0,\n(16)\nwheres=↑,↓andχ↑= (1,0)†andχ↓= (0,1)†. Inver-\nsion symmetry dictates that t′=tandr=r′. Continu-\nity of the wave function requires 1+ r=t. The energy\npumping (3) then simplifies to I(pump)\nE=/planckover2pi1Tr/parenleftbig˙t˙t†/parenrightbig\n/π.\nFlux continuity gives t= (1 +iˆΓ)−1, whereˆΓk/bardbls,k′\n/bardbls′=\nχ†\nsˆΓk/bardbls,k′\n/bardbls′χs′(4k⊥k⊥)−1/2.\nIn the absence of spin-flip scattering, the transmis-\nsion coefficient is diagonal in the transverse momentum:\nt(0)\nk/bardbl= [1−iη⊥σ·m]/(1+η2\n⊥), whereη⊥=mν/(/planckover2pi12k⊥).\nThe nonlocal (spin-pumping) Gilbert damping is then\nisotropic,Gij(m) =δijG′,\nG′=2ν2/planckover2pi1\nπ/summationdisplay\nk/bardblη2\n⊥\n(1+η2\n⊥)2. (17)\nIt can be shown that G′is a function of the ratio be-\ntween the exchange splitting versus the Fermi wave vec-\ntor,ηF=mν/(/planckover2pi12kF).G′vanishes in the limits ηF≪14\n(nonmagnetic systems) and ηF≫1 (strong ferromag-\nnet).\nWe include weak spin-flip scattering by expanding the\ntransmission coefficient tto second order in the spin-\norbit interaction, t≈/bracketleftbigg\n1+t0iˆΓsf−/parenleftBig\nt0iˆΓsf/parenrightBig2/bracketrightbigg\nt0, which\ninserted into Eq. (5) leads to an in general anisotropic\nGilbert damping. Ensemble averaging over all ran-\ndom spin configurations and positions after considerable\nbut straightforward algebra leads to the isotropic result\nGij(m) =δijG\nG=G(int)+G′(18)\nwhereG′is defined in Eq. (17). The “bulk” contribution\nto the damping is caused by the spin-relaxation due to\nthe magnetic disorder\nG(int)=NsS2ζ2ξ, (19)\nwhereNsis the number of magnetic impurities, Sis the\nimpurity spin, ζis the average strength of the magnetic\nimpurity scattering, and ξ=ξ(ηF) is a complicated ex-\npression that vanishes when ηFis either very small or\nvery large. Eq. (18) proves that Eq. (5) incorporates the\n“bulk” contribution to the Gilbert damping, which grows\nwith the number of spin-flip scatterers, in addition to in-\nterface damping. We could have derived G(int)[Eq. (19)]\nas well by the Kubo formula for the Gilbert damping.\nThe Gilbert damping has been computed before based\non the Kubo formalism based on first-principles elec-\ntronic band structures [9]. However, the ab initio appeal\nis somewhat reduced by additional approximations such\nas the relaxation time approximation and the neglect of\ndisorder vertex corrections. An advantage of the scatter-\ningtheoryofGilbertdampingisitssuitabilityformodern\nab initio techniques of spin transport that do not suffer\nfrom these drawbacks [16]. When extended to include\nspin-orbit coupling and magnetic disorder the Gilbert\ndamping can be obtained without additional costs ac-\ncording to Eq. (5). Bulk and interface contributions can\nbe readily separated by inspection of the sample thick-\nness dependence of the Gilbert damping.\nPhononsareimportantforthe understandingofdamp-\ning at elevated temperatures, which we do not explic-\nitly discuss. They can be included by a temperature-\ndependent relaxation time [9] or, in our case, structural\ndisorder. A microscopic treatment of phonon excitations\nrequires extension of the formalism to inelastic scatter-\ning, which is beyond the scope of the present paper.\nIn conclusion, we hope that our alternative formal-\nism of Gilbert damping will stimulate ab initio electronic\nstructure calculations as a function of material and dis-\norder. By comparison with FMR studies on thin ferro-\nmagnetic films this should lead to a better understanding\nof dissipation in magnetic systems.This work was supported in part by the Re-\nsearch Council of Norway, Grants Nos. 158518/143 and\n158547/431, and EC Contract IST-033749 “DynaMax.”\n∗Electronic address: Arne.Brataas@ntnu.no\n[1] For a review, see M. D. Stiles and J. Miltat, Top. Appl.\nPhys.101, 225 (2006) , and references therein.\n[2] B. Heinrich, D. Fraitov´ a, and V. Kambersk´ y, Phys. Sta-\ntus Solidi 23, 501 (1967); V. Kambersky, Can. J. Phys.\n48, 2906 (1970); V. Korenman, and R. E. Prange, Phys.\nRev. B6, 2769 (1972); V. S. 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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": "1307.7427v1.Theoretical_Study_of_Spin_Torque_Oscillator_with_Perpendicularly_Magnetized_Free_Layer.pdf",        "content": "arXiv:1307.7427v1  [cond-mat.mes-hall]  29 Jul 20131\nTheoretical Study of Spin-Torque Oscillator with\nPerpendicularly Magnetized Free Layer\nTomohiro Taniguchi, Hiroko Arai, Hitoshi Kubota, and Hiros hi Imamura∗\nSpintronics Research Center, AIST, Tsukuba, Ibaraki 305-8 568, Japan\nAbstract—The magnetization dynamics of spin torque oscilla-\ntor (STO) consisting of a perpendicularly magnetized free l ayer\nand an in-plane magnetized pinned layer was studied by solvi ng\nthe Landau-Lifshitz-Gilbert equation. We derived the anal ytical\nformula of the relation between the current and the oscillat ion\nfrequency of the STO by analyzing the energy balance between\nthe work done by the spin torque and the energy dissipation du e\nto the damping. We also found that the field-like torque break s\nthe energy balance, and change the oscillation frequency.\nIndex Terms —spintronics, spin torque oscillator, perpendicu-\nlarly magnetized free layer, the LLG equation\nI. INTRODUCTION\nSPIN torque oscillator (STO) has attracted much attention\ndue to its potential uses for a microwave generator and a\nrecording head of a high density hard disk drive. The self-\noscillation of the STO was first discovered in an in-plane\nmagnetized giant-magnetoresistive (GMR) system [1]. Afte r\nthat, the self-oscillation of the STO has been observed not\nonly in GMR systems [2]-[6] but also in magnetic tunnel\njunctions (MTJs) [7]-[11]. The different types of STO have\nbeen proposedrecently; for example,a point-contactgeome try\nwith a confinedmagneticdomainwall [12]-[14]whichenables\nusto controlthe frequencyfroma few GHz to a hundredGHz.\nRecently, Kubota et al.experimentally developed the MgO-\nbased MTJ consisting of a perpendicularly magnetized free\nlayer and an in-plane magnetized pinned layer [15],[16]. Th ey\nalso studied the self-oscillation of this type of MTJ, and\nobserved a large power ( ∼0.5µW) with a narrow linewidth\n(∼50MHz) [17].These results are great advancesin realizing\nthe STO device.However,the relationbetweenthe currentan d\nthe oscillation frequency still remains unclear. Since a pr ecise\ncontrol of the oscillation frequency of the STO by the curren t\nis necessary for the application, it is important to clarify the\nrelation between the current and the oscillation frequency .\nIn this paper, we derived the theoretical formula of the\nrelation between the current and the oscillation frequency\nof the STO consisting of the perpendicularly magnetized\nfree layer and the in-plane magnetized pinned layer. The\nderivation is based on the analysis of the energy balance\nbetween the work done by the spin torque and the energy\ndissipation due to the damping. We found that the oscillatio n\nfrequencymonotonicallydecreases with increasing the cur rent\nby keeping the magnetization in one hemisphere of the free\nlayer. The validity of the analytical solution was confirmed\nby numerical simulations. We also found that the field-like\n∗Corresponding author. Email address: h-imamura@aist.go. jppmelectron (I>0)z\nxy\nspin torquedamping damping spin torque\nFig. 1. Schematic view of the system. The directions of the sp in torque and\nthe damping during the precession around the z-axis are indicated.\ntorque breaks the energy balance, and change the oscillatio n\nfrequency. The shift direction of the frequency, high or low ,\nis determined by the sign of the field-like torque.\nThis paper is organized as follows. In Sec. II, the current\ndependence of the oscillation frequency is derived by solvi ng\nthe Landau-Lifshitz-Gilbert (LLG) equation. In Sec. III, t he\neffect of the field-like torque on the oscillation behaviour is\ninvestigated. Section IV is devoted to the conclusions.\nII. LLG STUDY OF SPIN TORQUE OSCILLATION\nThe system we consider is schematically shown in Fig.\n1. We denote the unit vectors pointing in the directions\nof the magnetization of the free and the pinned layers as\nm= (sinθcosϕ,sinθsinϕ,cosθ)andp, respectively. The\nx-axis is parallel to pwhile the z-axis is normal to the film\nplane. The variable θofmis the tilted angle from the z-axis\nwhileϕis the rotation angle from the x-axis. The current I\nflows along the z-axis, where the positive current corresponds\nto the electron flow from the free layer to the pinned layer.\nWe assume that the magnetization dynamics is well de-\nscribed by the following LLG equation:\ndm\ndt=−γm×H−γHsm×(p×m)+αm×dm\ndt.(1)\nThe gyromagneticration and the Gilbert damping constant ar e\ndenotedas γandα, respectively.The magnetic field is defined\nbyH=−∂E/∂(Mm), where the energy density Eis\nE=−MHapplcosθ−M(HK−4πM)\n2cos2θ.(2)\nHere,M,Happl, andHKare the saturation magnetization,the\napplied field along the z-axis, and the crystalline anisotropy\nfield along the z-axis, respectively. Because we are interested\nin the perpendicularly magnetized system, the crystalline\nanisotropy field, HK, should be larger than the demagneti-\nzation field, 4πM. Since the LLG equation conserves the2\nI = 1.2 ~ 2.0 (mA)\nmz\nmxmy1\n-1 0\n-1 \n-1 1\n100(a)\ncurrent (mA)34567(b)\nfrequency (GHz) \n1.2 1.4 1.6 1.8 2.0\nFig. 2. (a) The trajectories of the steady state precession o f the magne-\ntization in the free layer with various currents. (b) The dot s represent the\ndependence of the oscillation frequency obtained by numeri cally solving the\nLLG equation. The solid line is obtained by Eqs. (8) and (9).\nnormofthe magnetization,the magnetizationdynamicscan b e\ndescribed by a trajectory on an unit sphere. The equilibrium\nstates of the free layer correspond to m=±ez. In following,\nthe initial state is taken to be the north pole, i.e., m=ez. It\nshould be noted that a plane normal to the z-axis, in which θ\nis constant, corresponds to the constant energy surface.\nThe spin torque strength, Hsin Eq. (1), is [18]-[20]\nHs=/planckover2pi1ηI\n2e(1+λmx)MSd, (3)\nwhereSanddare the cross section area and the thickness\nof the free layer. Two dimensionless parameters, ηandλ\n(−1< λ <1), determine the magnitude of the spin polariza-\ntion and the angle dependence of the spin torque, respective ly.\nAlthough the relation among η,λ, and the material parameters\ndepends on the theoretical models [20]-[22], the form of Eq.\n(3) is applicable to both GMR system and MTJs. In particular,\nthe angle dependence of the spin torque characterized by λis\na key to induce the self-oscillation in this system.\nFigure 2 (a) shows the steady state precession of the mag-\nnetization in the free layer obtained by numerically solvin g\nEq. (1). The values of the parameters are M= 1313emu/c.c.,\nHK= 17.9kOe,Happl= 1.0kOe,S=π×50×50nm2,\nd= 2.0nm,γ= 17.32MHz/Oe, α= 0.005,η= 0.33,\nandλ= 0.38, respectively [17]. The self-oscillation was\nobservedforthe current I≥1.2mA.Althoughthe spintorque\nbreaks the axial symmetry of the free layer along the z-axis,\nthe magnetization precesses around the z-axis with an almost\nconstanttilted angle. Thetilted angle fromthe z-axisincreases\nwith increasing the current; however, the magnetization st ays\nin the northsemisphere( θ < π/2). The dotsin Fig. 2 (b) show\nthe dependence of the oscillation frequency on the current. As\nshown, the oscillation frequencymonotonicallydecreases with\nincreasing the current magnitude.\nLet us analytically derive the relation between the current\nand the oscillation frequency. Since the self-oscillation occurs\ndue to the energysupply into the free layer by the spin torque ,\nthe energy balance between the spin torque and the damping\nshould be investigated. By using the LLG equation, the time\nderivative of the energy density Eis given by dE/dt=Ws+\nWα, where the work done by spin torque, Ws, and the energydissipation due to the damping, Wα, are respectively given by\nWs=γMHs\n1+α2[p·H−(m·p)(m·H)−αp·(m×H)],\n(4)\nWα=−αγM\n1+α2/bracketleftBig\nH2−(m·H)2/bracketrightBig\n. (5)\nBy assuming a steady precession around the z-axis with a\nconstant tilted angle θ, the time averages of WsandWαover\none precession period are, respectively, given by\nWs=γM\n1+α2/planckover2pi1ηI\n2eλMSd/parenleftBigg\n1/radicalbig\n1−λ2sin2θ−1/parenrightBigg\n×[Happl+(HK−4πM)cosθ]cosθ,(6)\nWα=−αγM\n1+α2[Happl+(HK−4πM)cosθ]2sin2θ.(7)\nThe magnetization can move from the initial state to a point\nat which dE/dt= 0. Then, the current at which a steady\nprecession with the angle θcan be achieved is given by\nI(θ) =2αeλMSd\n/planckover2pi1ηcosθ/parenleftBigg\n1/radicalbig\n1−λ2sin2θ−1/parenrightBigg−1\n×[Happl+(HK−4πM)cosθ]sin2θ.(8)\nThe corresponding oscillation frequency is given by\nf(θ) =γ\n2π[Happl+(HK−4πM)cosθ].(9)\nEquations (8) and (9) are the main results in this section.\nThe solid line in Fig. 2 (b) shows the current dependence of\nthe oscillation frequency obtained by Eqs. (8) and (9), wher e\nthe good agreement with the numerical results confirms the\nvalidity of the analytical solution. The critical current f or the\nself-oscillation, Ic= limθ→0I(θ), is given by\nIc=4αeMSd\n/planckover2pi1ηλ(Happl+HK−4πM).(10)\nThe value of Icestimated by using the aboveparametersis 1.2\nmA, showing a good agreement with the numerical simulation\nshown in Fig. 2 (a). The sign of Icdepends on that of λ,\nand the self-oscillation occurs only for the positive (nega tive)\ncurrent for the positive (negative) λ. This is because a finite\nenergy is supplied to the free layer for λ/negationslash= 0, i.e.,Ws>0. In\nthe case of λ= 0, the average of the work done by the spin\ntorque is zero, and thus, the self-oscillation does not occu r.\nIt should be noted that I(θ)→ ∞in the limit of θ→π/2.\nThis means the magnetization cannot cross over the xy-plane,\nand stays in the north hemisphere ( θ < π/2). The reason\nis as follows. The average of the work done by spin torque\nbecomes zero in the xy-plane (θ=π/2) because the direction\nof the spin torque is parallel to the constant energy surface .\nOn the other hand, the energy dissipation due to the damping\nis finite in the presence of the applied field [21]. Then,\ndE/dt(θ=π/2) =−αγMH2\nappl/(1 +α2)<0, which\nmeansthe dampingmovesthe magnetizationto the northpole.\nThus, the magnetization cannot cross over the xy-plane. The\ncontrollable range of the oscillation frequency by the curr ent\nisf(θ= 0)−f(θ=π/2) =γ(HK−4πM)/(2π), which is\nindependent of the magnitude of the applied field.3\nSince the spin torque breaks the axial symmetry of the\nfree layer along the z-axis, the assumption that the tilted\nangle is constant used above is, in a precise sense, not valid ,\nand thez-component of the magnetization oscillates around\na certain value. Then, the magnetization can reach the xy-\nplane and stops its dynamics when a large current is applied.\nHowever,thevalue ofsuch currentis morethan15mA forour\nparameter values, which is much larger than the maximum of\nthe experimentallyavailable current. Thus, the above form ulas\nwork well in the experimentally conventional current regio n.\nContrary to the system considered here, the oscillation be-\nhaviour of an MTJ with an in-plane magnetized free layer and\na perpendicularly magnetized pinned layer has been widely\ninvestigated [23]-[26]. The differences of the two systems\nare as follows. First, the oscillation frequency decreases with\nincreasing the current in our system while it increases in\nthe latter system. Second, the oscillation frequency in our\nsystem in the large current limit becomes independent of the\nz-component of the magnetization while it is dominated by\nmz= cosθin the latter system. The reasons are as follows.\nIn our system, by increasing the current, the magnetization\nmoves away from the z-axis due to which the effect of the\nanisotropy field on the oscillation frequency decreases, an d\nthe frequency tends to γHappl/(2π), which is independent\nof the anisotropy. On the other hand, in the latter system,\nthe magnetization moves to the out-of-plane direction, due\nto which the oscillation frequency is strongly affected by t he\nanisotropy (demagnetization field).\nThe macrospin model developed above reproduces the ex-\nperimentalresultswith the freelayerof2nmthick[17],for ex-\nample the current-frequency relation, quantitatively. Al though\nonlythezero-temperaturedynamicsisconsideredinthispa per,\nthe macrospin LLG simulation at a finite temperature also\nreproduces other properties, such as the power spectrum and\nits linewidth, well. However, when the free layer thickness\nfurther decreases, an inhomogeneousmagnetization due to t he\nroughnessattheMgOinterfacesmayaffectsthemagnetizati on\ndynamics: for example, a broadening of the linewidth.\nIII. EFFECT OF FIELD -LIKE TORQUE\nThe field-like torque arises from the spin transfer from the\nconductionelectrons to the local magnetizations,as is the spin\ntorque. When the momentum average of the transverse spin of\ntheconductionelectronsrelaxesinthefreelayerveryfast ,only\nthe spin torque acts on the free layer [19]. On the other hand,\nwhen the cancellation of the transverse spin is insufficient ,\nthe field-like torque appears. The field-like torque added to\nthe right hand side of Eq. (1) is\nTFLT=−βγHsm×p, (11)\nwhere the dimensionless parameter βcharacterizes the ratio\nbetween the magnitudes of the spin torque and the field-like\ntorque. The value and the sign of βdepend on the system\nparameters such as the band structure, the thickness, the\nimpurity density, and/or the surface roughness [22],[27]- [29].\nThe magnitude of the field-like torque in MTJ is much larger\nthan that in GMR system [30],[31] because the band selectionmz\nmxmy1\n-1 0\n-1 \n-1 1\n100(a)\nmz\nmxmy1\n-1 0\n-1 \n-1 1\n100(b)\nmz\nmxmy1\n-1 0\n-1 \n-1 1\n100(c) (d)\n0 0.2 0.1\ntime (μs)0\n-0.5 \n-1.00.51.0mzβ=0 β=0.5\nβ=-0.5\nβ=-0.5\nβ=0.5β=0\nFig. 3. The magnetization dynamics from t= 0with (a) β= 0, (b)\nβ= 0.5, andβ=−0.5. The current magnitude is 2.0mA. (d) The time\nevolutions of mzfor various β.\nduringthe tunnelingleads to an insufficient cancellation o f the\ntransverse spin by the momentum average.\nIt should be noted that the effective energy density,\nEeff=E−βM/planckover2pi1ηI\n2eλMSdlog(1+λmx),(12)\nsatisfying −γm×H+TFLT=−γm×[−∂Eeff/(Mm)],\ncan be introduce to describe the field-like torque. The time\nderivative of the effective energy, Eeff, can be obtained by\nreplacing the magnetic field, H, in Eqs. (4) and (5) with\nthe effective field −∂Eeff/∂(Mm) =H+βHsp. Then, the\naverage of dEeff/dtover one precession period around the\nz-axis consists of Eq. (6), (7), and the following two terms:\nW′\ns=βγM\n1+α2/parenleftbigg/planckover2pi1ηI\n2eλMSd/parenrightbigg2/bracketleftbigg1+λ2cos2θ\n(1−λ2sin2θ)3/2−1/bracketrightbigg\n,\n(13)\nW′\nα=−αγM\n1+α2/parenleftbiggβ/planckover2pi1ηI\n2eλMSd/parenrightbigg2/bracketleftbigg1+λ2cos2θ\n(1−λ2sin2θ)3/2−1/bracketrightbigg\n−2αβγM\n1+α2/planckover2pi1ηI\n2eλMSd/parenleftBigg\n1/radicalbig\n1−λ2sin2θ−1/parenrightBigg\n×[Happl+(HK−4πM)cosθ]cosθ.\n(14)\nThe constant energy surface of Eeffshifts from the xy-plane\ndue to a finite |β|(≃1), leading to an inaccuracy of the\ncalculation of the time average with the constant tilted ang le\nassumption. Thus, Eqs. (13) and (14) are quantitatively val id\nfor only |β| ≪1. However, predictions from Eqs. (13) and\n(14) qualitatively show good agreements with the numerical\nsimulations, as shown below.\nFor positive β,W′\nsis also positive, and is finite at θ=π/2.\nThus,dEeff/dt(θ=π/2)can be positive for a sufficiently\nlarge current. This means, the magnetization can cross over\nthexy-plane, and move to the south semisphere ( θ > π/2).\nOn the other hand, for negative β,W′\nsis also negative. Thus,\ntheenergysupplybythespintorqueissuppressedcomparedt o4\ncurrent (mA)34567frequency (GHz) \n1.2 1.4 1.6 1.8 2.02\n1\n0β=0\nβ=0.5β=-0.5\nFig. 4. The dependences of the oscillation frequency on the c urrent for\nβ= 0(red),β= 0.5(orange), and β=−0.5(blue), respectively.\nthecaseof β= 0.Then,arelativelylargecurrentisrequiredto\ninduce the self-oscillation with a certain oscillation fre quency.\nAlso, the magnetization cannot cross over the xy-plane.\nWe confirmed these expectations by the numerical simu-\nlations. Figures 3 (a), (b) and (c) show the trajectories of\nthe magnetization dynamics with β= 0,0.5, and−0.5\nrespectively,while thetime evolutionsof mzareshownin Fig.\n3 (d). The current value is 2.0 mA. The current dependences\nof the oscillation frequency are summarized in Fig. 4.\nIn the case of β= 0.5>0, the oscillation frequency is low\ncompared to that for β= 0because the energy supply by the\nspin torque is enhanced by the field-like torque, and thus, th e\nmagnetization can largely move from the north pole. Above\nI= 1.9mA, the magnetizationmovesto the southhemisphere\n(θ > π/2), and stops near θ≃cos−1[−Happl/(HK−4πM)]\nin the south hemisphere, which corresponds to the zero fre-\nquency in Fig. 4.\nOn the other hand, in the case of β=−0.5<0, the\nmagnetization stays near the north pole compared to the case\nofβ= 0, because the energy supply by the spin torque is\nsuppressed by the field-like torque. The zero frequency in\nFig. 4 indicates the increase of the critical current of the s elf-\noscillation. Compared to the case of β= 0, the oscillation\nfrequency shifts to the high frequency region because the\nmagnetization stays near the north pole.\nIV. CONCLUSIONS\nIn conclusion, we derived the theoretical formula of the\nrelation between the current and the oscillation frequency of\nSTO consisting of the perpendicularly magnetized free laye r\nand the in-plane magnetized pinned layer. The derivation is\nbased on the analysis of the energy balance between the work\ndone by the spin torque and the energy dissipation due to the\ndamping.The validityof the analyticalsolutionwas confirm ed\nby numerical simulation. We also found that the field-like\ntorque breaks the energy balance, and changes the oscillati on\nfrequency. The shift direction of the frequency, high or low ,\ndepends on the sign of the field-like torque ( β).ACKNOWLEDGMENT\nThe authors would like to acknowledge T. 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Buhrman,\nand D. C. Ralph, Nat. Phys. , vol.4, pp.67-71, 2008."    },    {        "title": "2204.10596v2.A_short_circuited_coplanar_waveguide_for_low_temperature_single_port_ferromagnetic_resonance_spectroscopy_set_up_to_probe_the_magnetic_properties_of_ferromagnetic_thin_films.pdf",        "content": "arXiv:2204.10596v2  [cond-mat.mtrl-sci]  19 Jul 2022A short-circuited coplanar waveguide for low-temperature single-port ferromagnetic\nresonance spectroscopy set-up to probe the magnetic proper ties of ferromagnetic thin\nfilms\nSayani Pal, Soumik Aon, Subhadip Manna and Chiranjib Mitra∗\nIndian Institute of Science Education and Research Kolkata ,\nWest Bengal, India\nA coplanar waveguide shorted in one end is proposed, designe d, and implemented successfully to\nmeasure the properties of magnetic thin films as a part of the v ector network analyser ferromag-\nnetic resonance (VNA-FMR) spectroscopy set-up. Its simple structure, potential applications and\neasy installation inside the cryostat chamber made it advan tageous especially for low-temperature\nmeasurements. It provides a wide band of frequencies in the g igahertz range essential for FMR\nmeasurements. Our spectroscopy set-up with short-circuit ed coplanar waveguide has been used to\nextract Gilbert damping coefficient and effective magnetizat ion values for standard ferromagnetic\nthin films like Py and Co. The thickness and temperature depen dent studies of those magnetic\nparameters have also been done here for the afore mentioned m agnetic samples.\nINTRODUCTION\nIn recent years, extensive research on microwave mag-\nnetization dynamics in magnetic thin films[1–3], planar\nnanostructures[4–6] and multi-layers[7–9] havebeen per-\nformedduetotheirpotentialapplicationsinvariousfields\nof science and technology. Spintronics is one such emerg-\ning discipline that encompasses the interplay between\nmagnetization dynamics and spin transport. It also in-\ncludes fields like spin-transfer torque [10–13], direct and\ninversespin hall effect [14–18], spin pumping [19, 20] etc.,\nwhich are crucial in industrial applications for develop-\ning devices like magnetic recording head[21], magnetic\ntunnel junction(MTJ) sensors [22, 23], magnetic memory\ndevices[24, 25] andspin-torquedevices[26, 27]. Thus ex-\nploring more about the static and dynamic properties of\nmagnetic materials in itself is an interesting subject. Fer-\nromagnetic resonance spectroscopy(FMR) is a very ba-\nsic and well-understood technique that is used to study\nthe magnetization dynamics of ferromagnets[28, 29, 31].\nNowadays, most advanced FMR spectroscopy methods\nuse a vector network analyzer (VNA)[30, 31] as the mi-\ncrowave source and detector. We have used VNA in our\nset-up too.\nTo determine the magnetic parameters of the ferromag-\nnetic materials using the VNA-FMR spectroscopy, one\nneeds to carry out the measurements at a wide range of\nfrequencies. Since the microwave magnetic field in the\ncoplanar waveguide (CPW) is parallel to the plane, it\nservesthepurposeofexploringthemagneticpropertiesof\nthe concernedsystem overabroadfrequencyrangein the\nGHz region. The advantage of using CPW in the spec-\ntroscopy system lies in the fact that we no longer need\nto remount samples at different waveguides or cavities\nforeveryotherfrequency measurements, which consumes\n∗Corresponding author:chiranjib@iiserkol.ac.ina lot of time and effort in an experiment[32, 33]. Re-\nsearchers design and use different types of CPW for vari-\nous other purposes like micron-sized CPW in microwave-\nassisted magnetic recording; two-port CPW in antenna;\nshorted CPW in ultra-wideband bandpass-filter and per-\nmeability measurements [34–36]. However, in broadband\nFMR spectroscopy two-port CPW jigs have most com-\nmonly been used till date. Using two-port CPW in FMR\nspectroscopy, one gets absorption spectra in terms of\nthe transmissioncoefficient of scatteringparameters, and\nfrom there magnetic parameters of the samples can be\ndetermined. The use of two-port CPW in VNA-FMR\ncan be replaced by one-port CPW where the reflection\ncoefficient of scattering parameters of the FMR spectra\ncan be used to determine the magnetic parameters of\nthe sample. One port reflection geometry is a lot more\nconvenient in terms of easy design, calibration, installa-\ntion, and sample loading. This is especially true when\nthe whole CPW arrangement is kept inside the cryostat\nchamber for low-temperature measurements and the sys-\ntem becomes very sensitive to vibration and any kind\nof magnetic contacts, one port CPW seems very con-\nvenient to operate rather than the two-port one. Previ-\nously, manyhavedesignedandusedshort-circuitedCPW\njigs for other purposes but to the best our knowledge it\nhas not been used for low-temperature VNA-FMR spec-\ntroscopy measurements before.\nIn this work, we report the development of short-\ncircuited CPW based low-temperature broadband VNA-\nFMR spectroscopy set-up to study the magnetic param-\neters of standard ferromagnetic samples. For measure-\nments, we chose the permalloy(Py) thin films as ferro-\nmagnetic (FM) material which has greatly been used in\nresearchfields like spintronics and industrial applications\ndue to its interesting magnetic properties like high per-\nmeability, large anisotropy magnetoresistance, low coer-\ncivity, and low magnetic anisotropy. We have also con-\nsidered another standard magnetic thin film, Co of thick-\nness 30nm as a standard for ascertaining the measure-2\nment accuracy. In our system, we swept the magnetic\nfield keeping the frequencies constant, and got the FMR\nspectra for several frequencies. From there we found the\nvariation of resonance fields and field linewidths with\nthe resonance frequencies. We have used the linear fit\nfor resonance frequencies vs field line-widths data to\ncalculate the Gilbert damping coefficient( α). We fit-\nted the set of resonance frequencies vs resonance fields\ndata to the Kittel formula [59] to obtain the effec-\ntive magnetization(4 πMeff). Subsequently, we investi-\ngated the thickness and temperature-dependent studies\nof 4πMeffandαfor FM thin films of different thickness\ninthetemperaturerangeof7.5Kto300K.Tocharacterise\nthe measurement set-up using short-circuited CPW, we\ncompared the previous measurements in the literature\nwith ourresults and there wasa good agreementbetween\nthe two[36, 41].\nEXPERIMENTAL DETAILS\nA short-circuited CPW has been designed and fab-\nricated as a part of our low-temperature VNA-FMR\nspectroscopy set-up. To make the CPW we have used\nRogers AD1000, a laminated PCB substrate with copper\ncladding on both sides of the dielectric. The thickness of\nthe dielectric and the copper layer are 1.5 mm and 17.5\nmicrons respectively and the dielectric constant of the\nsubstrate is 10.7. The main concern about the design of\nthe CPW is to match its characteristic impedance with\nthe impedance of the microwave transmission line con-\nnected to it. We haveused the line calculatorto calculate\nthe dimensions of CPW. For a CPW with a characteris-\ntic impedance of 50 ohms, the line calculator calculated\nthe width of the signal line and the gap to be 900 mi-\ncrons and 500 microns respectively. The fabrication is\ndone using optical lithography which is described in de-\ntail in the literature[49]. Other components of our mea-\nCryostatVNA\nElectromagnetSample\nCPWCoaxial Transmission Line\nFIG. 1. The schematic diagram of measurement system and\nthe arrangement inside the cryostat with the sample on top\nof the CPW\nsurement system are a)Vector Network Analyser(VNA),\nwhich is a microwave source as well as a detector, b)theelectromagnet that generates the external magnetic field,\ni.e., Zeemanfieldand, c)optistatdrycryogen-freecooling\nsystem from Oxford instruments which is used for low-\ntemperature measurements. One end of the CPW signal\nline is shorted to the ground, and the other end is con-\nnected to the VNA through a SMA connector and coax-\nial cable (fig 3b). On top of the CPW, thin-film samples\nhave been placed face down after wrapping them with\nan insulating tape to electrically isolate them. For low-\ntemperature measurements, the sample has been glued\nto the CPW using a low-temperature adhesive to ensure\ncontact of sample and resonator at all times, in spite of\nthe vibration caused by the cryostat unit. This whole ar-\nrangementis then placed inside the twopole pieces of the\nelectromagnet as we can see from the diagram in fig 1.\nTherearetwostandardmethods ofgettingFMR spectra:\nsweeping the frequency keeping the field constant and\nsweeping the magnetic field while keeping the frequency\nconstant. We have adopted the second method. We have\nworked in the frequency range from 2.5GHz to 5.5GHz\nand in the magnetic field range from 0 Oe to roughly\naround 500 Oe. We have used 1mW of microwave power\nthroughout the experiment. From the FMR spectra, we\nhavedeterminedeffectivemagnetizationanddampingco-\nefficient of FM thin films and studied their variation with\ntemperature and sample thickness.\nSAMPLE PREPARATION AND\nCHARACTERIZATION\nPy (Ni80Fe20) and Co thin films were fabricated by\nthermal evaporation technique on Si/SiO 2substrates,\nfrom commercially available pellets (99 .995%pure) at\nroom temperature. The substrates were cleaned with\nacetone, IPA and DI water respectively in ultrasonica-\ntor and dried with a nitrogen gun. The chamber was\npumped down to 1 ×10−7torr using a combination of\na scroll pump and turbo pump. During the deposition,\npressure reached upto 1 ×10−6torr. Thin films were fab-\nricated at a rate of 1 .2˚A/swhere thickness can be con-\ntrolled by Inficon SQM 160 crystal monitor. For our\nexperiments a series of Py thin films of different thick-\nnesses were fabricated by keeping the other parameters\nlike base pressure, deposition pressure and growth rate\nconstant. Film thickness and morphology was measured\nby using atomic force microscopy technique as shown in\nfig 2(a). We have used Py films with thicknesses 10nm,\n15nm, 34nm, 50nm, and 90nm with a surface roughness\nof around 1nm and one Co film of thickness 30nm. X-ray\ndiffraction experiment confirms the polycrystalline struc-\nture of the samples as shown in fig 2b and fig 2c for Py\nand Co respectively.3\n2µm\n2µm\n(a)\n/s51/s53 /s52/s48 /s52/s53 /s53/s48 /s53/s53 /s54/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48\n/s52/s52/s46/s51/s54/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s105/s116/s97/s114/s121/s32/s117/s110/s105/s116/s41\n/s50 /s113 /s32/s40/s100/s101/s103/s114/s101/s101/s41/s80/s121/s32/s40/s49/s53/s110/s109/s41\n(b)\n/s51/s53 /s52/s48 /s52/s53 /s53/s48 /s53/s53/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s65/s114/s98/s105/s116/s97/s114/s121/s32/s117/s110/s105/s116/s41\n/s50 /s113 /s32/s40/s100/s101/s103/s114/s101/s101/s41/s67/s111/s32/s40/s51/s48/s110/s109/s41\n(c)\nFIG. 2. (a)Atomic force microscope (AFM) image of 30 nm\nthick Py thin film with a surface roughness of 1 nm . X-ray\ndiffraction peak of (b)15nm thick Py film and (c)30nm Co\nprepared by thermal evaporation.\nRESULTS AND DISCUSSION\nWe have calculated the dimensions of the short-\ncircuited CPW using the line calculator of the CST Stu-\ndio Suite software as mentioned in the experimental de-\ntails section. Using those dimensions we have also done\nthe full-waveelectromagneticsimulation in CST software\nto get the electric and magnetic field distribution of the\nCPW. One can see from the simulation result displayed\nin figure 3a that the farther it is from the gap, the weaker\nthe intensity of the magnetic field, and the magnitude of\nthe field in the gap area is one order of greater than that\non the signal line. When placing the thin film sample\non top of the CPW, the dimension of the sample shouldDielectricSampleSignal  Line\nGap\nMagnetic field lines\nElectric field lines\na) b)\nc)\nFIG. 3. (a) Schematic diagram of the cross-sectional view of\nCPW. (b) Top view of the short-circuited CPW after fabri-\ncation. (c)Intensity distribution of microwave magnetic fi eld\nin the one end shorted CPW at 5GHz (top view)\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s45/s48/s46/s53/s45/s48/s46/s52/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s83\n/s49/s49/s40/s100/s66/s41\n/s72/s32/s40/s79/s101/s41/s32/s32/s32 /s102/s114/s101/s113/s117/s101/s110/s99/s121\n/s32/s50/s46/s53/s71/s72/s122\n/s32/s51/s46/s53/s71/s72/s122\n/s32/s52/s46/s53/s71/s72/s122\n/s32/s53/s46/s53/s71/s72/s122/s49/s53/s110/s109/s32/s80/s121\n/s84/s61/s51/s48/s48/s75\nFIG. 4. Ferromagnetic Resonance spectra of absorption at\nfrequencies 2.5 GHz, 3.5 GHz, 4.5 GHz, 5.5 GHz for 15nm Py\nthin films at room temperature after background subtraction\nbe such that it can cover the gap area on both sides of\nthe signal line of the CPW because the magnetic field is\nmost intense in that area. This microwave magnetic field\ncirculatingthe signal line ofthe CPW is perpendicular to4\n/s50 /s51 /s52 /s53 /s54 /s55/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s84/s61/s51/s48/s48/s75\n/s32/s67/s111/s32/s40/s51/s48/s110/s109/s41\n/s32/s80/s121/s32/s40/s51/s52/s110/s109/s41/s68 /s72/s32/s40/s79/s101/s41\n/s102/s32/s40/s71/s72/s122/s41\n(a)/s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48/s46/s48/s48/s53/s48/s46/s48/s48/s54/s48/s46/s48/s48/s55/s48/s46/s48/s48/s56/s48/s46/s48/s48/s57/s97\n/s116/s32\n/s80/s121 /s32/s40/s110/s109/s41/s32/s84/s61/s51/s48/s48/s75\n(b)\n/s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s50/s51/s52/s53/s54/s55\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s84/s61/s32/s51/s48/s48/s75\n/s32/s67/s111/s32/s40/s51/s48/s110/s109/s41\n/s32/s80/s121/s32/s40/s51/s52/s110/s109/s41/s102/s32/s40/s71/s72/s122/s41\n/s72/s32/s40/s79/s101/s41\n(c)/s48/s46/s48/s48 /s48/s46/s48/s50 /s48/s46/s48/s52 /s48/s46/s48/s54 /s48/s46/s48/s56 /s48/s46/s49/s48/s56/s57/s49/s48/s49/s49/s52 /s112 /s77\n/s101/s102/s102/s32/s40/s107/s71/s41\n/s116/s32/s45/s49\n/s32/s80/s121/s32/s40/s110/s109/s45/s49\n/s41/s32/s84/s61/s51/s48/s48/s75\n(d)\nFIG. 5. a)Field linewidth variation with resonance frequen cies at 300K for 34nm Py and 30nm Co thin films. Equation 1 has\nbeen used for fitting the curve and to determine the Gilbert da mping coefficient; b)thickness dependence of Gilbert dampin g\ncoefficient at room temperature for Py thin films; c)resonance field variation with resonance frequencies at 300K for 34 nm P y\nand 30 nm Co thin films. Kittel formula (eqn-3)has been used fo r fitting the curve and to determine the effective magnetizati on;\nd)thickness dependence of effective magnetization for Py th in films at room temperature.\nthe external magnetic field and both the magnetic fields\nare parallel to the film surface as can be seen from fig\n3a and 3b. On account of the static magnetic field, the\nmagnetic moment will undergo a precession around the\nstatic magnetic field at a frequency called the Larmor\nprecession frequency. Absorption of electromagnetic en-\nergy happens when the frequency of the transverse mag-\nnetic field (microwave) is equal to the Larmor frequency.\nFig4exhibitsthe absorptionspectrafor15nmbarePy\nfilm after subtraction of a constant background for four\ndifferent frequencies, 2.5 GHz, 3.5 GHz, 4.5 GHz and 5.5\nGHz at room temperature in terms of S-parameter re-\nflection coefficient ( S11) vs. external magnetic field. We\nfitted these experimental results to the Lorentz equation\n[56]. We extracted the field linewidth at half maxima\nfrom the FMR spectra at different frequencies and fitted\nthem using equation 1 to obtain αas one can see from\nfig 5a and fig 6a. The experimental values of the absorp-tion linewidth (∆ H) contains both the effect of intrinsic\nGilbert damping and the extrinsic contribution to the\ndamping. Linewidth due to Gilbert damping is directly\nproportional to the resonance frequency and follows the\nequation:\n∆H= (2π\nγ)αf+∆H0 (1)\nwhereγis the gyromagneticratio, αis the Gilbert damp-\ning coefficient and ∆ H0is the inhomogeneous linewidth.\nA number of extrinsic contributions to the damping coef-\nficient like magnetic inhomogeneities, surface roughness,\ndefects of the thin films bring about the inhomogeneous\nlinewidth broadening [55]. αhas been determined using\nthe above equation only. Damping coefficient values ob-\ntainedhereareintherangeofabout0 .005to0.009forPy\nsamplesofthicknessescoveringthe whole thin film region\ni.e., 10nm to 90nm at room temperature. These values5\n/s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48 /s53/s46/s53 /s54/s46/s48/s51/s48/s51/s53/s52/s48/s52/s53/s53/s48/s53/s53/s54/s48\n/s32/s49/s48/s110/s109/s32/s84/s61/s51/s48/s48/s75\n/s32/s49/s48/s110/s109/s32/s84/s61/s52/s53/s75/s80/s121/s68 /s72/s32/s40/s79/s101/s41\n/s102/s32/s40/s71/s72/s122/s41/s32/s49/s53/s110/s109/s32/s84/s61/s51/s48/s48/s75\n/s32/s49/s53/s110/s109/s32/s84/s61/s52/s53/s75/s80/s121\n(a)/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s48/s48/s54/s48/s46/s48/s48/s56/s48/s46/s48/s49/s48/s48/s46/s48/s49/s50\n/s84/s32/s40/s75/s41/s32/s32/s32/s32/s32/s32/s32 /s32/s32/s116 /s32\n/s80/s121 \n/s32/s49/s53/s110/s109\n/s32/s49/s48/s110/s109\n(b)\n/s54/s48 /s49/s50/s48 /s49/s56/s48 /s50/s52/s48 /s51/s48/s48 /s51/s54/s48 /s52/s50/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53/s53/s46/s48/s53/s46/s53/s54/s46/s48\n/s32/s49/s53/s110/s109/s32/s84/s61/s51/s48/s48/s75\n/s32/s49/s53/s110/s109/s32/s84/s61/s32/s52/s53/s75/s80/s121/s102/s32/s40/s71/s72/s122/s41\n/s72/s32/s40/s79/s101/s41/s32/s49/s48/s110/s109/s32/s84/s61/s51/s48/s48/s75\n/s32/s49/s48/s110/s109/s32/s84/s61/s52/s53/s75/s80/s121\n(c)/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s56/s46/s48/s56/s46/s50/s56/s46/s52/s56/s46/s54/s56/s46/s56/s57/s46/s48/s57/s46/s50/s57/s46/s52/s57/s46/s54/s52 /s77\n/s101/s102/s102/s32/s40/s107/s71/s41\n/s84/s32/s40/s75/s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s116/s32\n/s80/s121 \n/s32/s49/s53/s110/s109\n/s32/s49/s48/s110/s109\n(d)\nFIG. 6. a)Field linewidth variation with resonance frequen cies at 300K and 45K for 10 nm and 15nm Py films. Equation 1\nhas been used for fitting the curve and to determine the Gilber t damping coefficient; b)temperature dependence of damping\ncoefficient for 10nm and 15nm Py thin films; c)resonance field va riation with resonance frequencies at 300K and 45K for 10nm\nand 15nm Py thin films. Kittel formula (eqn-3) has been used fo r fitting the curve and to determine the 4 πMeff; d)temperature\ndependence of 4 πMefffor 10nm and 15nm Py thin films.\nare pretty close to the values previously reported in the\nliterature [39–41, 43, 44]. For the Co film of thickness 30\nnm we have obtained the value of αto be 0.008 ±0.0004.\nBaratiet al.measured the damping value of 30nm Co\nfilm to be 0.004 [37, 38]. There are other literature also\nwhere Co multilayers have been studied where damping\ncoefficient value increasesbecause ofspin pumping effect.\nαis a veryinterestingparameterto investigatebecause it\nis used in the phenomenological LLG equation [57], [58]\nto describe magnetization relaxation:\nd/vectorM\ndt=−γ/vectorM×/vectorHeff+α\nMS/vectorM×d/vectorM\ndt(2)\nwhere,µBisBohrmagneton, /vectorMisthemagnetizationvec-\ntor,MSis the saturation magnetization and Heffis the\neffectve magnetic field which includes the external field,\ndemagnetization and crystalline anisotropy field. The in-troduction of the Damping coefficient in LLG equation is\nphenomenological in nature and the question of whether\nit has a physical origin or not has not been fully under-\nstood till date. We have measured 4 πMeffalso from\nthe absorption spectra. We have fitted the Kittel for-\nmula (equation 3) into resonance field vs. the resonance\nfrequency ( fres) data as shown in fig 5c and fig 6c.\nfres= (γ\n2π)[(H+4πMeff)H]1\n2 (3)\nwhere,His the applied magnetic field, and Meffis the\neffective magnetization which contains saturation mag-\nnetization and other anisotropic contributions. We ob-\ntained the 4 πMeffvalue for 30nm thick Co and 34nm Py\nto be 17.4 ±0.2kG and 9.6 ±0.09kG respectively at room\ntemperature. These values also agree quite well with the\nliterature. For a 10nm Co film, Beaujour et al.measured\nthe value to be around 16 kG[45] and for a 30nm Py the6\nvalue is 10 .4kGas measured by Zhao et al[41].\nWe tried to address here the thickness and tempera-\nture dependence of αand 4πMeffusing our measure-\nment set-up. The variation of the αwith thickness is\nshown here in figure 5b. It increases smoothly as film\nthickness decreases and then shows a sudden jump below\n15nm. Increased surface scattering could be the reason\nbehind this enhanced damping for thinner films. It has\nbeen previously observed [60] that damping coefficient\nand electrical resistivity follows a linear relation at room\ntemperature for Py thin film. It suggests a strong corre-\nlation between magnetization relaxation( α) and electron\nscattering. Magnetization relaxation could be explained\nby electron scattering by phonons and magnons. In the\nformer case, αis proportional to the electron scatter-\ning rate, τ−1and in the later case, α∼τ. Theoretical\npredictions by Kambersky [61] suggests that at higher\ntemperature α∼τ−1as electron scattering by phonons\nare predominant there. So, here in our case we can elim-\ninate the possibility of electron scattering by magnons as\nthickness dependent study has only been done at room\ntemperature where phonon scattering is prevalent. Ing-\nvasson et.al in their paper[60] also suggests that the re-\nlaxation of magnetization is similar to bulk relaxation\nwhere phonon scattering in bulk is replaced by surface\nand defect scattering in thin films.\nThicknessdependent studyof4 πMeffalsohasbeen done\nfor Py thin films at room temperature. As we can see\nfrom fig 5d, Meffis linear for thinner films and becomes\nalmost independent of thickness for thicker films. The\nchange in Meffwith thickness mainly comes from the\nsurface anisotropy,\nµ0Meff=µ0Ms−2Ks\nMsd(4)\nwhereMsis the saturation magnetization and2Ks\nMsdis\nthe surface anisotropy field. Surface anisotropy is higher\nfor thinner films and the anisotropy reduces as one in-\ncreases the film thickness. We have obtained saturation\nmagnetization(4 πMs) value of Py to be 10 .86kGusing\nthe linear fit (equation 4). Previously Chen et al.has re-\nported the 4 πMeffvalue for a 30nm Py film to be 12 kG\n[54] which includes both 4 πMsand anisotropy field.\nTemperature dependence of αfor 15nm and 10nm Py\nfilm is represented in figure 6b. The αvalue decreases\nmonotonically from room temperature value and reaches\na minimum value at around 100K and then starts to in-\ncrease with further decrease of temperature and reaches\na maximum value at 45K. Zhao et al.have seen this\nkind of damping enhancement at around50Kin their low\ntemperature experiment with Py thin films with differ-\nent types of capping layers and Rio et al.observed the\ndamping anomaly at temperature 25K when they have\nusedPtas a capping layer on Py thin film.[39, 41]. We\ndid not use any capping layer on Py film in our mea-\nsurement. So there is no question of interface effect for\nthe enhanced damping at 45K. A possible reason for the\nstrong enhancement of damping at 45K could be the/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48/s49/s50/s48/s49/s52/s48/s49/s54/s48/s49/s56/s48\n/s32/s57/s71/s72/s122\n/s32/s52/s71/s72/s122/s51/s48/s110/s109/s32/s67/s111/s68 /s72 /s32/s40/s79/s101/s41\n/s84/s32/s40/s75/s41\nFIG. 7. Temperature induced linewidth variation of 30nm Co\nthin film at two different frequencies 4GHz and 9GHz\nspin reorientation transition(SRT) on the Py surface at\nthat particular temperature [41, 42]. Previously it has\nbeen established that the competition between different\nanisotropy energies: magnetocrystalline anisotropy, sur-\nface anisotropy, shape anisotropy decides the magnetiza-\ntion direction in magnetic films. For thin films, the vari-\nation of temperature, film thickness, strain can alter the\ncompetition between shape and surface anisotropy. In\nour case, temperature variation could be the reason for\nthe spin reorientation transition on Py surface at around\n45K.Foradeeperunderstandingofthespinreorientation\nwe investigated the temperature dependence of 4 πMeff\nfor 15nm and 10nm Py film as shown in fig 6d. There\nMeffis showing an anomaly at around 45K, otherwise it\nis increasing smoothly with the decrease of temperature.\nSince there is no reason of sudden change in saturation\nmagnetization at this temperature, the possible reason\nfor the anomaly in Meffshould come from any change\ninmagneticanisotropy. Thatchangeofanisotropycanbe\nrelated to a spin reorientation at that particular temper-\nature value. Sierra et.al., [42], have also argued that in\nthe temperature dependent spin re-orientation (T-SRT),\nthe central effect of temperature on the magnetic prop-\nerties of Py films was to increase the in-plane uniax-\nial anisotropy and to induce a surface anisotropy which\norients the magnetization out of plane in the Py sur-\nface. They have verified this using X-Ray diffraction\nexperiments and high resolution transmission electron\nmicroscopy images. This establishes reasonably enough\nthat it is a spin re-orientation transition around 45K.\nLastly, for a 30nm Co thin film we have studied the\ntemperature variation of FMR linewidth(∆ H) at mi-\ncrowave frequencies 9GHz and 4GHz. One can see from\nfig7 that the linewidth does not change much in the tem-\nperature range 100 <T<300 but below 100K, ∆ Hstarts\nto increase significantly. This behaviour of ∆ Hhas been7\nobserved previously by Bhagat et al.[62]. They sug-\ngested that the increase of relaxation frequency at low\ntemp might be related to the rapid variation in the elec-\ntronicmeanfreepath. Thiscomparisonsuggeststhatour\nset-up with a single port CPW has the same sensitivity\nof the previously reported set-up.\nCONCLUSIONS\nShort-circuited coplanar waveguide can successfully be\nused for low-temperature single-port broadband FMR\nspectroscopy measurements. The magnetic parameters\nobtained here for the standard ferromagnetic materials,\nPy and Co are in good agreement with other experimen-\ntal works and theoretical predictions. Temperature de-\npendentstudiesof αand4πMeffforPyfilmshereexhibit\nspin reorientation phenomenon at low temperatures. We\nbelieve that the findings with Py sample will help in bet-\nter understanding of magnetic phenomena for other fer-\nromagnetic materials at low temperature. Though this\nsetup has limitations for angle-dependent FMR measure-\nments, itcanreadilybe usedforstudiesonmagnetization\ndynamics for multi-layered films and planar nanostruc-\ntures and is currently under way. In future we aim to in-\ntegrate electrical measurement facilities with the existing\nset-up thereby extending the possibilities of inverse spinhall(ISHE) measurements and spin-torque ferromagnetic\nresonance(ST-FMR)spectroscopyexperimentswhich are\nvery relevant for current scientific interests. We believe\nthe short-circuited CPW will serve its purpose conve-\nniently there too.\nACKNOWLEDGEMENTS\nTheauthorssincerelyacknowledgesMinistryofEduca-\ntion, Government of India and Science and Engineering\nResearch Board (SERB) (grant no:EMR/2016/007950)\nand Department of Science and Technology (grant\nno. DST/ICPS/Quest/2019/22) for financial sup-\nport. S.P. acknowledges Department of Science and\nTechnology(DST)-INSPIRE fellowship India, S. A. ac-\nknowledges Ministry of Education of Government of In-\ndia and S.M. acknowledges Council Of Scientific and\nIndustrial Research(CSIR),India for research fellowship.\nThe authors would like to thank Dr. Partha Mitra of the\nDepartment of Physical Sciences, Indian Institute of Sci-\nence Education and Research Kolkata for providing the\nlab facilities for sample deposition. The authors would\nalso like to thank Mr. Subhadip Roy, IISER Kolkata for\nhis help in fabricating the CPW structure using home-\nbuilt optical lithography set-up.\n[1]Barman, A. and Haldar, A., 2014. Time-domain study\nof magnetization dynamics in magnetic thin films and\nmicro-and nanostructures. Solid State Physics, 65, pp.1-\n108.\n[2]Silva, T.J., Lee, C.S., Crawford, T.M. and Rogers, C.T.,\n1999. Inductive measurement of ultrafast magnetization\ndynamics in thin-film Py. 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Ferro-\nmagnetic resonance studyof thickness-dependentmagne-tization precession in Ni80Fe20films. Journal ofapplied\nphysics. 2007 May 1;101(9):09C104.\n[55]Jiang, S., Sun, L., Yin, Y., Fu, Y., Luo, C., Zhai, Y.\nand Zhai, H., 2017. Ferromagnetic resonance linewidth\nand two-magnon scattering in Fe1-x Gd x thin films. AIP\nAdvances, 7(5), p.056029.\n[56]Celinski, Z., Urquhart, K.B. and Heinrich, B., 1997. Us-\ningferromagnetic resonance tomeasurethemagneticmo-\nments of ultrathin films. Journal of Magnetism and Mag-\nnetic Materials, 166(1-2), pp.6-26.\n[57]Landau LA, Lifshitz E. On the theory of the dispersion\nof magnetic permeability in ferromagnetic bodies. InPer-\nspectives in Theoretical Physics 1992 Jan 1 (pp. 51-65).\nPergamon.\n[58]Gilbert, T.L., 2004. A phenomenological theory of damp-\ning in ferromagnetic materials. IEEE transactions on\nmagnetics, 40(6), pp.3443-3449.\n[59]Kittel, C., 1948. On the theory of ferromagnetic reso-\nnance absorption. Physical review, 73(2), p.155.\n[60]Ingvarsson, S., Ritchie, L., Liu, X.Y., Xiao, G., Slon-\nczewski, J.C., Trouilloud, P.L. and Koch, R.H., 2002.\nRole of electron scattering in the magnetization relax-\nation ofthinNi81Fe19films.Physical ReviewB,66(21),\np.214416.\n[61]Kambersk´ y, V., 1976. Onferromagnetic resonance damp-\ning in metals. Czechoslovak Journal of Physics B, 26(12),\npp.1366-1383.\n[62]Bhagat SM, Lubitz P. Temperature variation of ferro-\nmagnetic relaxation in the 3 d transition metals. Physical\nReview B. 1974 Jul 1;10(1):179."    },    {        "title": "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. 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Tokura,\nPhys. Rev. Lett. 94, 087202 (2005).\n[17] T. Kise, T. Ogasawara, M. Ashida, Y. Tomioka, Y. Tokura, and M. Kuwata-Gonokami, Phys. Rev.\nLett.85, 1986 (2000).\n[18] W.F.Brown, Micromagnetics (Krieger, 1963).\n[19] B.Koopmans,M.vanKampen,J.T.Kohlhepp,andW.J.M.de Jonge,Phys.Rev.Lett. 85,844(2000).\n[20] R. Mathieu, A. Asamitsu, H. Yamada, K. S. Takahashi, M. K awasaki, Z. Fang, N. Nagaosa, and\nY. Tokura, Phys. Rev. Lett. 93, 016602 (2004).\n[21] L. Klein, J. R. Reiner, T. H. Geballe, M. R. Beasley, and A . Kapitulnik, Phys. Rev. B 61, R7842\n(2000).\n[22] Z. Fang, N. Nagaosa, K. Takahashi, A. Asamitsu, R. Mathi eu, T. Ogasawara, H. Yamada,\nM. Kawasaki, Y. Tokura, and K.Terakura, Science 302, 92(2003).\n[23] M. Onoda, A.S.Mishchenko, and N. Nagaosa, J.Phys. Soc. Jap.77, 013702 (2008).\n[24] M. Onoda and N.Nagaosa, J. Phys.Soc. Jap. 71, 19 (2002).\n[25] X.Wang, J.R. Yates, I. Souza, and D.Vanderbilt, Phys.R ev. B.74, 195118 (2006).\n10"    },    {        "title": "2102.01137v2.Blow_up_and_lifespan_estimates_for_a_damped_wave_equation_in_the_Einstein_de_Sitter_spacetime_with_nonlinearity_of_derivative_type.pdf",        "content": "arXiv:2102.01137v2  [math.AP]  20 Jan 2022BLOW-UP AND LIFESPAN ESTIMATES FOR A DAMPED WAVE\nEQUATION IN THE EINSTEIN-DE SITTER SPACETIME WITH\nNONLINEARITY OF DERIVATIVE TYPE\nMAKRAM HAMOUDA1, MOHAMED ALI HAMZA1AND ALESSANDRO PALMIERI2,3\nAbstract. Inthis article, weinvestigatethe blow-upforlocalsolutionstoasem ilinear\nwave equation in the generalized Einstein - de Sitter spacetime with no nlinearity of\nderivative type. More precisely, we consider a semilinear damped wav e equation with\na time-dependent and not summable speed of propagation and with a time-dependent\ncoefficient for the linear damping term with critical decay rate. We pr ove in this work\nthat the results obtained in a previous work, where the damping coe fficient takes two\nparticular values 0 or 2, can be extended for any positive damping co efficient. We\nshow the blow-up in finite time of local in time solutions and we establish u pper bound\nestimates for the lifespan, provided that the exponent in the nonlin ear term is below a\nsuitable threshold and that the Cauchydata are nonnegativeand c ompactlysupported.\n1.Introduction\nWe are interested in the semilinear damped wave equation when the sp eed of propa-\ngation is depending on time, namely the damped wave equations in Einst ein - de Sitter\nspacetime, with time derivative nonlinearity which reads as follows:\n(1.1)/braceleftBigg\nutt−t−2k∆u+µ\ntut=|ut|p,inRN×[1,∞),\nu(x,1) =εf(x), ut(x,1) =εg(x), x∈RN,\nwherek∈[0,1),µ≥0,p >1,N≥1 is the space dimension, ε >0 is a parameter\nillustrating the size of the initial data, and f,gare supposed to be positive functions.\nFurthermore, we consider fandgwith compact support on B(0RN,R),R>0.\nThe problem ( 1.1) with time derivative nonlinearity being replaced by power non-\nlinearity is well understood in terms of blow-up phenomenon. Let us fi rst recall the\nequation in this case. Under the usual Cauchy conditions, the semilin ear wave equation\nwith power nonlinearity is\n(1.2) utt−t−2k∆u+µ\ntut=|u|q,inRN×[1,∞).\n2010Mathematics Subject Classification. 35L15, 35L71, 35B44.\nKey words and phrases. Blow-up, Einsten-de Sitter spacetime, Glassey exponent, Lifespa n, Critical\ncurve, Nonlinear wave equations, Time-derivative nonlinearity.\n1The blow-up phenomenon for ( 1.2) is related to two particular exponents. The first\nexponent,q0(N,k), is the positive root of\n((1−k)N−1)q2−((1−k)N+1+2k)q−2(1−k) = 0,\nand the second exponent is given by\nq1(N,k) = 1+2\nN(1−k).\nHence, the positive number max/parenleftbigg\nq0(N+µ\n1−k,k),q1(N,k)/parenrightbigg\nseems to be a serious\ncandidate for the critical power stating thus the threshold betwe en the global existence\nand the blow-up regions, see e.g. [ 7,22,23,27,29].\nLet us go back to ( 1.1) withk=µ= 0. This case is in fact connected to the Glassey\nconjecture in which the critical exponent pGis given by\n(1.3) pG=pG(N) := 1+2\nN−1.\nThe above value pGis creating a threshold (depending on p) between the region where\nwe have the global existence of small data solutions (for p>pG) and another where the\nblow-up of the solutions under suitable sign assumptions for the Cau chy data occurs (for\np≤pG); see e.g. [ 14,15,17,26,32,35].\nNow, fork<0 andµ= 0, it is proven in [ 20] that the solution of ( 1.1), in the subcrit-\nical case (1 <p≤pG(N(1−k))), blows up in finite time giving hence a lifespan estimate\nof the maximal existence time. This is equivalent to say that, for 1 <p≤pG(N(1−k)),\nwe have the nonexistence of the solution of ( 1.1). However, the aforementioned result\nwas recently improved in [ 18] thanks to the construction of adequate test functions. The\nnew region obtained in [ 18] gives a plausible characterization of the critical exponent,\nnamely\n(1.4) p≤pT(N,k) := 1+2\n(1−k)(N−1)+k.\nVery recently, it is proved in [ 12] with different approaches, as an application of the\ncase of mixed nonlinearities, that results similar to the above for the problem ( 1.1) with\nk<0 andµ= 0 hold.\nWe consider now the case µ>0 andk= 0 in (1.2). Hence, for a small µ, the solution\nof (1.2) behaves like a wave. In fact, the damping produces a shifting by µ>0 on the\ndimensionNfor the value of the critical power, see e.g. [ 16,24,30,31], and [5,6] for\nthe caseµ= 2 andN= 2,3. The global existence for µ= 2 is proven in [ 5,6,21].\n2However, for µlarge, the equation ( 1.2) is of a parabolic type and the behavior is like a\nheat-type equation; see e.g. [ 3,4,33].\nOn the other hand, for the solution of ( 1.1) withµ>0 andk= 0, in [19] a blow-up\nresult is proved for 1 <p≤pG(N+2µ) and upper bound estimates for the lifespan are\ngiven as well. Later, this result was improved in [ 25], wherepG(N+µ) is found as upper\nbound forµ≥2. Recently, an improvement is obtained in [ 10] stating that the critical\nvalue forpis given by pG(N+µ) for allµ >0. This should be the optimal threshold\nvalue that needs to be rigorously proved by completing the present blow-up result with\na global existence one when the exponent pis beyond the critical value.\nWe focus in this article on the blow-up of the solution of ( 1.1) fork∈[0,1). Our\ntarget is to give the upper bound, denoted here by pE=pE(N,k,µ), delimiting a new\nblow-up region for the Einstein - de Sitter spacetime equation ( 1.1).\nFirst, as observed for the equation ( 1.2), where the damping produces a shift in q0in\nthe dimensional parameter of magnitudeµ\n1−k, we expect that the same phenomenon\nholds for ( 1.1). In other words, we predict that the upper bound pE=pE(N,k,µ)\nsatisfies\n(1.5) pE(N,k,µ) =pE(N+µ\n1−k,k,0).\nUsing an explicit representation formula and Zhou’s approach to pro ving the blow-up\non a certain characteristic line, in [ 13], we proved that\n(1.6) pE(N,k,0) =pT(N,k),\nwherepTis defined by ( 1.4).\nNow, in view of ( 1.5) and (1.6), we await, for the solution of ( 1.1) withk∈[0,1) and\nµ>0, that\n(1.7) pE=pE(N,k,µ) := 1+2\n(1−k)(N−1)+k+µ.\nAs we have mentioned, in [ 13] we proved that ( 1.7) holds true under some sign\nassumptions for the data for µ= 0, but also for µ= 2 (cf. Theorems 1.1 and 1.2).\nWe aim in the present work to extend this result for all µ>0, and show that the upper\nbound value for pis in fact given by ( 1.7). We think that pE(N,k,µ), forksmall,\ncharacterizes the limiting value between the existence and nonexist ence regions of the\nsolution of ( 1.1). However, it is clear that this limiting exponent does not reach the\noptimal one in view of the very recent results in [ 28].\nFinally, we recall here that the wave in ( 1.1) has a speed of propagation dependent\nof time. Therefore, this time-dependent speed of propagation te rm can be seen, after\n3rescaling (see ( 1.9) below), as a scale-invariant damping. Let v(x,τ) =u(x,t), where\n(1.8) τ=φk(t) :=t1−k\n1−k.\nHence, we can easily see that v(x,τ) satisfies the following equation:\nvττ−∆v+µ−k\n(1−k)τ∂τv=Ck,pτµk(p−2)|∂τv|p,inRN×[1/(1−k),∞), (1.9)\nwhereµk:=−k\n1−kandCk,p= (1−k)µk(p−2). Moreover, thanks to the above transforma-\ntion, we can use the methods carried out in some earlier works [ 2,9,10,11,12] to build\nthe proof of our main result.\nThe rest of the paper is arranged as follows. First, we state in Sect ion2the weak\nformulation of ( 1.1) in the energy space, and then we give the main theorem. Section\n3is concerned with some technical lemmas that we will use to prove the main result.\nFinally, Section 4is assigned to the proof of Theorem 2.2which constitutes the main\nresult of this article.\n2.Nonexistence Result\nFirst, we define in the sequel the energy solution associated with ( 1.1).\nDefinition 2.1. Letf∈H1(RN)andg∈L2(RN). The function uis said to be an\nenergy solution of ( 1.1) on[1,T)if\n/braceleftBigg\nu∈ C([1,T),H1(RN))∩C1([1,T),L2(RN)),\nut∈Lp\nloc((1,T)×RN),\nsatisfies, for all Φ∈ C∞\n0(RN×[1,T))and allt∈[1,T), the following equation:\n(2.1)/integraldisplay\nRNut(x,t)Φ(x,t)dx−ε/integraldisplay\nRNg(x)Φ(x,1)dx\n−/integraldisplayt\n1/integraldisplay\nRNut(x,s)Φt(x,s)dxds+/integraldisplayt\n1s−2k/integraldisplay\nRN∇u(x,s)·∇Φ(x,s)dxds\n+/integraldisplayt\n1/integraldisplay\nRNµ\nsut(x,s)Φ(x,s)dxds=/integraldisplayt\n1/integraldisplay\nRN|ut(x,s)|pΦ(x,s)dxds,\n4and the condition u(x,1) =εf(x)is fulfilled in H1(RN).\nA straightforward computation shows that (2.1)is equivalent to\n(2.2)/integraldisplay\nRN/bracketleftbig\nut(x,t)Φ(x,t)−u(x,t)Φt(x,t)+µ\ntu(x,t)Φ(x,t)/bracketrightbig\ndx\n/integraldisplayt\n1/integraldisplay\nRNu(x,s)/bracketleftbigg\nΦtt(x,s)−s−2k∆Φ(x,s)−∂\n∂s/parenleftBigµ\nsΦ(x,s)/parenrightBig/bracketrightbigg\ndxds\n=/integraldisplayt\n1/integraldisplay\nRN|ut(x,s)|pψ(x,s)dxds+ε/integraldisplay\nRN/bracketleftbig\n−f(x)Φt(x,1)+(µf(x)+g(x))Φ(x,1)/bracketrightbig\ndx.\nRemark 2.1.Obviously, we can choose a test function Φ which is not compactly sup -\nported in view of the fact that the initial data fandgare supported on BRN(0,R). In\nfact, we have supp( u)⊂ {(x,t)∈RN×[1,∞) :|x| ≤φk(t)+R}.\nThe blow-up region and the lifespan estimate of the solutions of ( 1.1) constitute the\nobjective of our main result which is the subject of the following theo rem.\nTheorem 2.2. Letµ >0,p∈(1,pE(N,k,µ)],N≥1andk∈[0,1). Suppose\nthatf∈H1(RN)andg∈L2(RN)are functions which are non-negative, with com-\npact support on B(0RN,R), and non-vanishing everywhere. Then, there exists ε0=\nε0(f,g,N,R,p,k,µ )>0such that for any 0< ε≤ε0the solution uto(1.1)which\nsatisfies\nsupp(u)⊂ {(x,t)∈RN×[1,∞) :|x| ≤φk(t)+R},\nblows up in finite time Tε, and\nTε≤/braceleftBigg\nCε−2(p−1)\n2−((1−k)(N−1)+k+µ)(p−1)for1<p<p E(N,k,µ),\nexp/parenleftbig\nCε−(p−1)/parenrightbig\nforp=pE(N,k,µ),\nwherepE(N,k,µ)is given by (1.7)andCis a positive constant independent of ε.\nRemark 2.2.The results stated in Theorem 2.2hold true for k <0 andµ >0; see [1]\nwhere a more general model with mass term is studied.\nRemark2.3.After completing the first version of the present manuscript, we r eceived a\ndraft version of [ 28], where problem ( 1.1) is studied, among other things. In particular,\nforn+1\nn+2< k <1 andµ∈[0,(n+2)k−(n+1)) the upper bound for pin the blow-up\nresult is improved in [ 28] by proving the nonexistence of global solutions to ( 1.1) for\n1<p<1+1\n(1−k)n+µ.\n3.Auxiliary results\nIt is worth mentioning that the choice of the test function, that we will use in the\nfunctionals that will be introduced later on, is crucial here. Natura lly, in terms of\ndynamics of the solution of ( 1.1), the more accurate the choice of the test function is,\n5the better lifespan estimate we obtain. This is why we choose in the fo llowing to include\nall the linear terms inherited from ( 1.1). First, we introduce the function ρ(t) [22] given\nby\n(3.1) ρ(t) :=t1+µ\n2Kµ−1\n2(1−k)/parenleftbiggt1−k\n1−k/parenrightbigg\n,∀t≥1,\nwhereKν(t) is the modified Bessel function of second kind defined as\n(3.2) Kν(t) =/integraldisplay∞\n0exp(−tcoshζ)cosh(νζ)dζ, ν∈R.\nIt is easy to see that ρ(t) satisfies\n(3.3)d2ρ(t)\ndt2−t−2kρ(t)−d\ndt/parenleftBigµ\ntρ(t)/parenrightBig\n= 0,∀t≥1.\nSecond, we define the function ϕ(x) by\n(3.4) ϕ(x) :=\n\n/integraldisplay\nSN−1ex·ωdωforN≥2,\nex+e−xforN= 1;\nnote thatϕ(x) is introduced in [ 34] and satisfies ∆ ϕ=ϕ.\nHence, the function ψ(x,t) :=ϕ(x)ρ(t) verifies the following equation:\n(3.5) ∂2\ntψ(x,t)−t−2k∆ψ(x,t)−∂\n∂t/parenleftBigµ\ntψ(x,t)/parenrightBig\n= 0.\nIn the following we enumerate some properties of the function ρ(t) that we will use\nlater on in the proof of our main result.\nLemma 3.1. The next properties hold true for the function ρ(t).\n(i)The function ρ(t)is positive on [1,∞). Moreover, for all t≥1, there exists a\nconstantC1such thatρ(t)satisfies\n(3.6) C−1\n1tk+µ\n2exp(−φk(t))≤ρ(t)≤C1tk+µ\n2exp(−φk(t)),\nwhereφk(t)is given by (1.8).\n(ii)We have\n(3.7) lim\nt→+∞/parenleftbiggtkρ′(t)\nρ(t)/parenrightbigg\n=−1.\nProof.First, we recall here the definition of ρ(t), as in (3.1), and (1.8)\n(3.8) ρ(t) =t1+µ\n2Kµ−1\n2(1−k)(φk(t)),∀t≥1.\n6Hence, the positivity of ρ(t) is straightforward thanks to ( 3.2). On the other hand, from\n[8], we have the following property for the function Kµ(t):\n(3.9) Kµ(t) =/radicalbiggπ\n2te−t(1+O(t−1)),ast→ ∞.\nCombining ( 3.8) and(3.9), andagainremembering thedefinition of φk(t), given by ( 1.8),\nand the fact that k<1, we conclude ( 3.6). The assertion (i)is thus proven.\nNow, to prove (ii), using (3.8) we observe that\n(3.10)ρ′(t)\nρ(t)=µ+1\n2t+t−kK′\nµ−1\n2(1−k)(φk(t))\nKµ−1\n2(1−k)(φk(t)).\nExploiting the well-known identity for the modified Bessel function,\n(3.11)d\ndzKν(z) =−Kν+1(z)+ν\nzKν(z),\nand combining ( 3.10) and (3.11) yields\n(3.12)ρ′(t)\nρ(t)=µ\nt−t−kK1+µ−1\n2(1−k)(φk(t))\nKµ−1\n2(1−k)(φk(t)).\nFrom (3.9) and (3.12), and using the fact that k∈[0,1), we deduce ( 3.7).\nThis ends the proof of Lemma 3.1. /square\nThroughout this article, the use of a generic parameter Cis designed to denote a\npositive constant that might be dependent on p,q,k,N,R,f,g,µ but independent of ε.\nThe value of the constant Cmay change from line to line. Nevertheless, when it is nec-\nessary, we will clearly mention the expression of Cin terms of the parameters involved\nin our problem.\nA classical estimate result for the function ψ(x,t) is stated in the next lemma.\nLemma 3.2 ([34]).Letr>1. Then, there exists a constant C=C(N,µ,R,p,k,r )>0\nsuch that\n(3.13)/integraldisplay\n|x|≤φk(t)+R/parenleftBig\nψ(x,t)/parenrightBigr\ndx≤Cρr(t)erφk(t)(1+φk(t))(2−r)(N−1)\n2,∀t≥1.\nLetube a solution to ( 1.1) for which we introduce the following functionals:\n(3.14) U(t) :=/integraldisplay\nRNu(x,t)ψ(x,t)dx,\nand\n(3.15) V(t) :=/integraldisplay\nRNut(x,t)ψ(x,t)dx.\n7The first lower bounds for U(t) andV(t) are respectively given by the following two\nlemmas where, for tlarge enough, we will prove that ε−1t−kU(t) andε−1V(t) are two\nbounded from below functions by positive constants.\nLemma 3.3. Letube a solution of (1.1). Assume in addition that the corresponding\ninitial data satisfythe assumptionsas in Theorem 2.2. Then, there exists T0=T0(k,µ)>\n2such that\n(3.16) U(t)≥CUεtk,for allt≥T0,\nwhereCUis a positive constant that may depend on f,g,N,µ,Randk, but not on ε.\nProof.Lett∈(1,T). Substituting in ( 2.2) Φ(x,t) byψ(x,t), we obtain\n(3.17)/integraldisplay\nRN/bracketleftbig\nut(x,t)ψ(x,t)−u(x,t)ψt(x,t)+µ\ntu(x,t)ψ(x,t)/bracketrightbig\ndx\n=/integraldisplayt\n1/integraldisplay\nRN|ut(x,s)|pψ(x,s)dxds+εC(f,g),\nwhere\n(3.18) C(f,g) :=ρ(1)/integraldisplay\nRN/bracketleftbig/parenleftbig\nµ−ρ′(1)\nρ(1)/parenrightbig\nf(x)+g(x)/bracketrightbig\nφ(x)dx.\nNote thatC(f,g) is positive thanks to the fact that ρ(1) andµ−ρ′(1)\nρ(1)are positive as\nwell (in view of ( 3.12)) and the sign of the initial data. Hence, recall the definition of\nU, as in (3.14), and (3.4), (3.17) gives\n(3.19) U′(t)+Γ(t)U(t) =/integraldisplayt\n1/integraldisplay\nRN|ut(x,s)|pψ(x,s)dxds+εC(f,g),\nwhere\n(3.20) Γ( t) :=µ\nt−2ρ′(t)\nρ(t).\nNeglecting the nonlinear term in ( 3.19), then multiplying the resulting equation from\n(3.19) bytµ\nρ2(t)and integrating on (1 ,t), we get\nU(t)≥ U(1)ρ2(t)\ntµρ2(1)+εC(f,g)ρ2(t)\ntµ/integraldisplayt\n1sµ\nρ2(s)ds. (3.21)\nFrom (3.1), the definition of φk(t), given by ( 1.8), and using the fact that U(1)>0, the\nestimate ( 3.21) implies that\nU(t)≥εC(f,g)tK2\nµ−1\n2(1−k)(φk(t))/integraldisplayt\nt/21\nsK2\nµ−1\n2(1−k)(φk(s))ds,∀t≥2. (3.22)\n8In view of ( 3.9), we deduce the existence of T0=T0(k,µ)>2 such that\nφk(t)K2\nµ−1\n2(1−k)(φk(t))>π\n4e−2φk(t)andφk(t)−1K−2\nµ−1\n2(1−k)(φk(t))>1\nπe2φk(t),∀t≥T0/2.(3.23)\nInserting ( 3.23) in (3.22) and using ( 1.8), we obtain that\nU(t)≥εC(f,g)\n4tke−2φk(t)/integraldisplayt\nt/2φ′\nk(s)e2φk(s)ds (3.24)\n≥εC(f,g)\n8tk[1−e−2(φk(t)−φk(t/2))],∀t≥T0.\nThanks to ( 1.8) and the fact that k <1, we observe that t/ma√sto→1−e−2(φk(t)−φk(t/2))is an\nincreasing function on ( T0,∞), hence, its minimum is achieved at t=T0. Therefore we\ndeduce that\nU(t)≥εκC(f,g)tk,∀t≥T0, (3.25)\nwhere\nκ:=1\n8/parenleftbigg\n1−exp/parenleftbigg\n−(2−2k)T1−k\n0\n1−k/parenrightbigg/parenrightbigg\n.\nHence, Lemma 3.3is now proved. /square\nThe next lemma gives the lower bound of the functional V(t).\nLemma 3.4. Assume that the initial data are as in Theorem 2.2. Foruan energy\nsolution of (1.1), there exists T1=T1(k,µ)>T0such that\n(3.26) V(t)≥CVε,for allt≥T1,\nwhereCVis a positive constant depending on f,g,N,µ,Randk, but not on ε.\nProof.Lett∈[1,T). Recall the definitions of UandV, given respectively by ( 3.14) and\n(3.15), (3.4) and the identity\n(3.27) U′(t)−ρ′(t)\nρ(t)U(t) =V(t).\nHence, the equation ( 3.19) yields\n(3.28) V(t)+/bracketleftbiggµ\nt−ρ′(t)\nρ(t)/bracketrightbigg\nU(t) =/integraldisplayt\n1/integraldisplay\nRN|ut(x,s)|pψ(x,s)dxds+εC(f,g).\nA differentiation in time of the equation ( 3.28) gives\nV′(t)+/bracketleftbiggµ\nt−ρ′(t)\nρ(t)/bracketrightbigg\nU′(t)−/parenleftbiggµ\nt2+ρ′′(t)ρ(t)−(ρ′(t))2\nρ2(t)/parenrightbigg\nU(t) =/integraldisplay\nRN|ut(x,t)|pψ(x,t)dx.(3.29)\n9Now, thanks to ( 3.3) and (3.27), we deduce from ( 3.29) that\nV′(t)+/bracketleftbiggµ\nt−ρ′(t)\nρ(t)/bracketrightbigg\nV(t) =t−2kU(t)+/integraldisplay\nRN|ut(x,t)|pψ(x,t)dx, (3.30)\nthat we rewrite as\n/parenleftbigg\ntµV(t)\nρ(t)/parenrightbigg′\n=tµ\nρ(t)/parenleftbigg\nt−2kU(t)+/integraldisplay\nRN|ut(x,t)|pψ(x,t)dx/parenrightbigg\n,∀t≥1. (3.31)\nAn integration of ( 3.31) over (1,t) implies that\ntµV(t)\nρ(t)=V(1)\nρ(1)+/integraldisplayt\n1sµ\nρ(s)/parenleftbigg\ns−2kU(s)+/integraldisplay\nRN|ut(x,s)|pψ(x,s)dx/parenrightbigg\nds,∀t≥1. (3.32)\nThanks to the fact that V(1)≥0,ρ(1)>0 and using the lower bound of Uas in (3.16),\nwe infer that\nV(t)≥ρ(t)\ntµ/integraldisplayt\n1sµ\nρ(s)/parenleftbigg\nCUεs−k+/integraldisplay\nRN|ut(x,s)|pψ(x,s)dx/parenrightbigg\nds,∀t≥T0. (3.33)\nTherefore the estimate ( 3.33) gives\nV(t)≥CUερ(t)\ntµ/integraldisplayt\nt/2s−k+µ\nρ(s)ds,∀t≥2T0. (3.34)\nFor convenience, we rewrite ( 3.23) as follows:\n/radicalbig\nφk(t)Kµ−1\n2(1−k)(φk(t))>√π\n2e−φk(t)and1/radicalbig\nφk(t)K−1\nµ−1\n2(1−k)(φk(t))>1√πeφk(t),∀t≥T0/2.(3.35)\nUsing the expressions of ρ(t) andφk(t), given respectively by ( 3.1) and (1.8), we deduce\nthat\nV(t)≥εCU/parenleftbigg1\n2/parenrightbiggµ\n2+1\ne−φk(t)/integraldisplayt\nt/2φ′\nk(s)eφk(s)ds (3.36)\n≥εCU/parenleftbigg1\n2/parenrightbiggµ\n2+1\n[1−e−(φk(t)−φk(t/2))],∀t≥2T0.\nAnalogously as in Lemma 3.3, we have\nV(t)≥CVε,∀t≥T1:= 2T0, (3.37)\nwhere\nCV:=CU/parenleftbigg1\n2/parenrightbiggµ\n2+1/parenleftbigg\n1−exp/parenleftbigg\n−(1−2k−1)(2T0)1−k\n1−k/parenrightbigg/parenrightbigg\n.\nThis completes the proof of Lemma 3.4. /square\n104.Proof of Theorem 2.2.\nThis section is dedicated to proving the main result in Theorem 2.2which exposes\nthe blow-up dynamics of the solution of ( 1.1). Hence, to prove the blow-up result for\n(1.1) we will use ( 3.28) and (3.30). For this purpose, we multiply ( 3.28) byαρ′(t)\nρ(t), and\nsubtract the resulting equation from ( 3.30). Therefore we obtain for a certain α≥0,\nwhose range will be fixed afterward,\n(4.1)\nV′(t)+/bracketleftbiggµ\nt−(1+α)ρ′(t)\nρ(t)/bracketrightbigg\nV(t) =−εαρ′(t)\nρ(t)C(f,g)+/bracketleftbigg\nt−2k+αρ′(t)\nρ(t)/parenleftbiggµ\nt−ρ′(t)\nρ(t)/parenrightbigg/bracketrightbigg\nU(t)\n+/integraldisplay\nRN|ut(x,t)|pψ(x,t)dx−αρ′(t)\nρ(t)/integraldisplayt\n1/integraldisplay\nRN|ut(x,s)|pψ(x,s)dxds,∀t≥1.\nUsing (3.7), we can choose ˜T2≥T1(T1is given in Lemma 3.4) such that\n(4.2)V′(t)+/bracketleftbiggµ\nt−(1+α)ρ′(t)\nρ(t)/bracketrightbigg\nV(t)≥εαt−k\n2C(f,g)+(1−4α)t−2kU(t)\n+/integraldisplay\nRN|ut(x,t)|pψ(x,t)dx+αt−k\n2/integraldisplayt\n1/integraldisplay\nRN|ut(x,s)|pψ(x,s)dxds,∀t≥˜T2.\nFrom now on the parameter αis chosen in (1 /7,1/4). Thanks to ( 3.16), the estimate\n(4.2) leads to the following lower bound:\n(4.3)V′(t)+/bracketleftbiggµ\nt−(1+α)ρ′(t)\nρ(t)/bracketrightbigg\nV(t)≥εαt−k\n2C(f,g)+/integraldisplay\nRN|ut(x,t)|pψ(x,t)dx\n+αt−k\n2/integraldisplayt\n1/integraldisplay\nRN|ut(x,s)|pψ(x,s)dxds,∀t≥˜T2.\nNow, we introduce the following functional:\nH(t) :=C2ε+1\n16/integraldisplayt\n˜T3/integraldisplay\nRN|ut(x,s)|pψ(x,s)dxds,\nwhereC2:= min(αC(f,g)/4(1 +α),CV) (CVis given by Lemma 3.4) and we choose\n˜T3>˜T2such that\n(4.4)α\n2C(f,g)−C2tk/parenleftbiggµ\nt−(1+α)ρ′(t)\nρ(t)/parenrightbigg\n≥0,\nand\n(4.5)α\n2−1\n16tk/parenleftbiggµ\nt−(1+α)ρ′(t)\nρ(t)/parenrightbigg\n≥0,\nfor allt≥˜T3(this is possible thanks to ( 3.7), the definition of C2and the fact that\nα∈(1/7,1/4)).\nLet\nF(t) :=V(t)−H(t),\n11which satisfies\n(4.6)F′(t)+/bracketleftbiggµ\nt−(1+α)ρ′(t)\nρ(t)/bracketrightbigg\nF(t)≥15\n16/integraldisplay\nRN|ut(x,t)|pψ(x,t)dx\n+/bracketleftbiggα\n2−1\n16/parenleftbiggµ\nt1−k−(1+α)tkρ′(t)\nρ(t)/parenrightbigg/bracketrightbigg\nt−k/integraldisplayt\n˜T3/integraldisplay\nRN|ut(x,s)|pψ(x,s)dxds\n+/bracketleftbiggα\n2C(f,g)−C2/parenleftbiggµ\nt1−k−(1+α)tkρ′(t)\nρ(t)/parenrightbigg/bracketrightbigg\nεt−k,∀t≥˜T3.\nThanks to ( 4.4) and (4.5), we easily conclude that\n(4.7) F′(t)+/bracketleftbiggµ\nt−(1+α)ρ′(t)\nρ(t)/bracketrightbigg\nF(t)≥0,∀t≥˜T3.\nMultiplying ( 4.7) bytµ\nρ1+α(t)and integrating over ( ˜T3,t), we get\nF(t)≥ F(˜T3)˜Tµ\n3ρ1+α(t)\ntµρ1+α(˜T3),∀t≥˜T3. (4.8)\nHence, we see that F(˜T3) =V(˜T3)−C2ε≥ V(˜T3)−CVε≥0 in view of Lemma 3.4and\nthe definition of C2implying that C2≤CV.\nTherefore we deduce that\n(4.9) V(t)≥H(t),∀t≥˜T3.\nNow, employing the H¨ older inequality and the estimates ( 3.13) and (3.15), we obtain\n(4.10)H′(t)≥1\n16Vp(t)/parenleftbigg/integraldisplay\n|x|≤φk(t)+Rψ(x,t)dx/parenrightbigg−(p−1)\n≥CVp(t)ρ−(p−1)(t)e−(p−1)φk(t)(φk(t))−(N−1)(p−1)\n2.\nIn view of ( 3.6), we see that\n(4.11) H′(t)≥CVp(t)t−[(N−1)(1−k)+k+µ](p−1)\n2,∀t≥˜T3.\nFrom the above estimate and ( 4.9), we have\n(4.12) H′(t)≥CHp(t)t−[(N−1)(1−k)+k+µ](p−1)\n2,∀t≥˜T3.\nSinceH(˜T3) =C2ε >0, we easily obtain the blow-up in finite time for the functional\nH(t), and consequently the one for V(t) due to ( 4.9).\nThe proof of Theorem 2.2is now achieved.\naknowledgments\nThe authors are deeply thankful to the anonymous reviewer for t he valuable re-\nmarks that improved the paper. A. Palmieri is supported by the Jap an Society for the\n12Promotion of Science (JSPS) – JSPS Postdoctoral Fellowship for Re search in Japan\n(Short-term) (PE20003).\nReferences\n[1] M.F.BenHassen,M.Hamouda,M.A.HamzaandH.K.Teka Nonexistence result for the generalized\nTricomi equation with the scale-invariant damping, mass te rm and time derivative nonlinearity.\nTo appear in Asymptotic Analysis, (2021), pp. 1–21. 5\n[2] W. Chen, S. Lucente and A. Palmieri, Nonexistence of global solutions for generalized Tricomi\nequations with combined nonlinearity. Nonlinear Analysis: Real World Applications, 61(2021),\n103354.4\n[3] M. D’Abbicco, The threshold of effective damping for semilinear wave equat ions.Math. Methods\nAppl. Sci. 38(6) (2015), 1032–1045. 3\n[4] M. D’Abbicco and S. Lucente, A modified test function method for damped wave equations. 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Ann.\nMath.,22B(3) (2001), 275–280. 2\n141Department of Basic Sciences, Deanship of Preparatory Year and Supporting Stud-\nies, Imam Abdulrahman Bin Faisal University, P.O. Box 1982, 34212 Dammam, Saudi Ara-\nbia.\n2Department of Mathematics, University of Pisa, Largo B. Pon tecorvo 5, 56127\nPisa, Italy.\n3Current address: Mathematical Institute, Tohoku University, Aoba, Sendai 9 80-8578,\nJapan.\nEmail address :mmhamouda@iau.edu.sa (M. Hamouda)\nEmail address :mahamza@iau.edu.sa (M.A. Hamza)\nEmail address :alessandro.palmieri.math@gmail.com (A. Palmieri)\n15"    },    {        "title": "2211.13486v1.Influence_of_non_local_damping_on_magnon_properties_of_ferromagnets.pdf",        "content": "In\ruence of non-local damping on magnon properties of ferromagnets\nZhiwei Lu,1,\u0003I. P. Miranda,2,\u0003Simon Streib,2Manuel Pereiro,2Erik Sj oqvist,2\nOlle Eriksson,2, 3Anders Bergman,2Danny Thonig,3, 2and Anna Delin1, 4\n1Department of Applied Physics, School of Engineering Sciences, KTH Royal\nInstitute of Technology, AlbaNova University Center, SE-10691 Stockholm, Swedeny\n2Department of Physics and Astronomy, Uppsala University, Box 516, SE-75120 Uppsala, Sweden\n3School of Science and Technology, Orebro University, SE-701 82, Orebro, Sweden\n4SeRC (Swedish e-Science Research Center), KTH Royal Institute of Technology, SE-10044 Stockholm, Sweden\n(Dated: November 28, 2022)\nWe study the in\ruence of non-local damping on magnon properties of Fe, Co, Ni and Fe 1\u0000xCox\n(x= 30%;50%) alloys. The Gilbert damping parameter is typically considered as a local scalar\nboth in experiment and in theoretical modelling. However, recent works have revealed that Gilbert\ndamping is a non-local quantity that allows for energy dissipation between atomic sites. With\nthe Gilbert damping parameters calculated from a state-of-the-art real-space electronic structure\nmethod, magnon lifetimes are evaluated from spin dynamics and linear response, where a good\nagreement is found between these two methods. It is found that non-local damping a\u000bects the\nmagnon lifetimes in di\u000berent ways depending on the system. Speci\fcally, we \fnd that in Fe, Co,\nand Ni the non-local damping decreases the magnon lifetimes, while in Fe 70Co30and Fe 50Co50an\nopposite, non-local damping e\u000bect is observed, and our data show that it is much stronger in the\nformer.\nINTRODUCTION\nIn recent years, there has been a growing interest in\nmagnonics, which uses quasi-particle excitations in mag-\nnetically ordered materials to perform information trans-\nport and processing on the nanoscale. Comparing to the\nconventional information device, the magnonics device\nexhibits lower energy consumption, easier integrability\nwith complementary metal-oxide semiconductor (CMOS)\nstructure, anisotropic properties, and e\u000ecient tunability\nby various external stimuli to name a few [1{10]. Yttrium\niron garnet (YIG) [11] as well as other iron garnets with\nrare-earth elements (Tm, Tb, Dy, Ho, Er) [12] are very\npromising candidates for magnonics device applications\ndue to their low energy dissipation properties and, thus,\nlong spin wave propagation distances up to tens of \u0016m.\nContrary, the damping of other materials for magnonics,\nlike CoFeB, is typically two orders of magnitude higher\ncompared to YIG [12], leading to much shorter spin wave\npropagation distances. A clear distinction can be made\nbetween materials with an ultra-low damping parame-\nter, like in YIG, and those with a sign\fciantly larger,\nbut still small, damping parameter. Materials like YIG\nare insulating, which hinders many of the microscopic\nmechanisms for damping, resulting in the low observed\ndamping parameter. In contrast, materials like CoFeB\nare metallic. In research projects that utilize low damp-\ning materials, YIG and similar non-metallic low damping\nsystems are typically favored. However, metallic systems\nhave an advantage, since magnetic textures can easily by\nin\ruenced by electrical currents. Hence, there is good\n\u0003These two authors contributed equally\nyCorresponding author: zhiweil@kth.sereason to consider metallic systems for low damping ap-\nplications, even though their damping typically is larger\nthan in YIG. One can conclude that Gilbert damping\nis one of the major bottlenecks for the choice of mate-\nrial in magnonics applications and a detailed experimen-\ntal as well as theoretical characterisation is fundamen-\ntal for this \feld of research, especially for metallic sys-\ntems. Thus, a more advanced and detailed understand-\ning of Gilbert damping is called for, in order to overcome\nthis obstacle for further development of magnonics-based\ntechnology.\nWhereas most studies consider chemical modi\fcations\nof the materials in order to tune damping [13, 14], only a\nfew focus on the fundamental physical properties as well\nas dependencies of the Gilbert damping. Often Gilbert\ndamping is considered as a phenomenological scalar pa-\nrameter in the equation of motion of localized atom-\nistic magnetic moments, i.e. the Landau-Lifshitz-Gilbert\n(LLG) equation [15]. However, from using the general\nRayleigh dissipation function in the derivation proposed\nby Gilbert [16], it was theoretically found that the Gilbert\ndamping should be anisotropic, a tensor, and non-local.\nFurthermore, it depends on the temperature and, thus,\non underlying magnon as well as phonon con\fgurations\n[17{20]. This is naturally built into the multiple theoret-\nical methods developed to predict the damping parame-\nter, including breathing Fermi surface model [21], torque\ncorrelation model [22], and linear response formulation\n[23]. For instance, the general Gilbert damping tensor\nas a function of the non-collinear spin con\fguration has\nbeen proposed in Ref. 24.\nNonetheless, an experimental veri\fcation is still miss-\ning due to lacking insights into the impact of the gen-\neralised damping on experimental observables. In a re-\ncent experiment, however, the anisotropic behavior of the\ndamping has been con\frmed for Co 50Fe50thin \flms andarXiv:2211.13486v1  [cond-mat.mtrl-sci]  24 Nov 20222\nwas measured to be of the order of 400% [25], with respect\nto changing the magnetization direction. Changes of\nGilbert damping in a magnetic domain wall and, thus, its\ndependency on the magnetic con\fguration was measured\nin Ref. [26] and \ftted to the Landau-Lifshitz-Baryakhtar\n(LLBar) equation, which includes non-locality of the\ndamping by an additional dissipation term proportional\nto the gradient of the magnetisation [27{29]. However,\nthe pair-wise non-local damping \u000bijhas not yet been\nmeasured.\nThe most common experimental techniques of evaluat-\ning damping are ferromagnetic resonance (FMR) [30] and\ntime-resolved magneto-optical Kerr e\u000bect (TR-MOKE)\n[31]. In these experiments, Gilbert damping is related\nto the relaxation rate when (i)slightly perturbing the\ncoherent magnetic moment out of equilibrium by an ex-\nternal magnetic \feld [32] or (ii)when disordered mag-\nnetic moments remagnetise after pumping by an ultrafast\nlaser pulse [33]. Normally, in case (i)the non-locality is\nsuppressed due to the coherent precession of the atomic\nmagnetic moments. However, this coherence can be per-\nturbed by temperature, making non-locality in principle\nmeasurable. One possible other path to link non-local\ndamping with experiment is magnon lifetimes. Theoret-\nically, the magnon properties as well as the impact of\ndamping on these properties can be assessed from the\ndynamical structure factor, and atomistic spin-dynamics\nsimulations have been demonstrated to yield magnon dis-\npersion relations that are in good agreement with exper-\niment [34]. In experiment, neutron scattering [35] and\nelectron scattering [36] are the most common methods for\nprobing magnon excitations, where the linewidth broad-\nening of magnon excitations is related to damping and\nprovides a way to evaluate the magnon lifetimes [37]. It is\nfound in ferromagnets that the magnon lifetimes is wave\nvector (magnon energy) dependent [38{40]. It has been\nreported that the magnon energy in Co \flms is nearly\ntwice as large as in Fe \flms, but they have similar magnon\nlifetimes, which is related to the intrinsic damping mech-\nanism of materials [41]. However, this collective e\u000bect of\ndamping and magnon energy on magnon lifetimes is still\nan open question. The study of this collective e\u000bect is of\ngreat interest for both theory and device applications.\nHere, we report an implementation for solving the\nstochastic Landau-Lifshitz-Gilbert (SLLG) equation in-\ncorporating the non-local damping. With the dynamical\nstructure factor extracted from the spin dynamics sim-\nulations, we investigate the collective e\u000bect of non-local\ndamping and magnon energy on the magnon lifetimes.\nWe propose an e\u000ecient method to evaluate magnon life-\ntimes from linear response theory and verify its validity.\nThe paper is organized as follows. In Sec. I, we give\nthe simulation details of the spin dynamics, the adiabatic\nmagnon spectra and dynamical structure factor, and the\nmethodology of DFT calculations and linear response.\nSec. II presents the non-local damping in real-space, non-\nlocal damping e\u000bects on the spin dynamics and magnon\nproperties including magnon lifetimes of pure ferromag-nets (Fe, Co, Ni), and Fe 1\u0000xCox(x= 30%;50%) alloys.\nIn Sec. III, we give a summary and an outlook.\nI. THEORY\nA. Non-local damping in atomistic spin dynamics\nThe dynamical properties of magnetic materials at \f-\nnite temperature have been so far simulated from atom-\nistic spin dynamics by means of the stochastic Landau-\nLifshitz-Gilbert equation with scalar local energy dissipa-\ntion. Here, the time evolution of the magnetic moments\nmi=mieiat atom site iis well described by:\n@mi\n@t=mi\u0002\u0012\n\u0000\r[Bi+bi(t)] +\u000b\nmi@mi\n@t\u0013\n;(1)\nwhere\ris the gyromagnetic ratio. The e\u000bective \feld Bi\nacting on each magnetic moment is obtained from:\nBi=\u0000@H\n@mi: (2)\nThe here considered spin-Hamiltonian Hconsists of a\nHeisenberg spin-spin exchange:\nH=\u0000X\ni6=jJijei\u0001ej: (3)\nHere,Jij{ the Heisenberg exchange parameter { cou-\nples the spin at site iwith the spin at site jand is cal-\nculated from \frst principles (see Section I C). Further-\nmore,\u000bis the scalar phenomenological Gilbert damp-\ning parameter. Finite temperature Tis included in\nEq. (1) via the \ructuating \feld bi(t), which is modeled\nby uncorrelated Gaussian white noise: hbi(t)i= 0 and\nb\u0016\ni(t)b\u0017\nj(t0)\u000b\n= 2D\u000eij\u000e\u0016\u0017\u000e(t\u0000t0), where\u000eis the Kro-\nnecker delta, i;jare site and \u0016;\u0017=fx;y;zgCartesian\nindices. Furthermore, the \ructuation-dissipation theo-\nrem givesD=\u000bkBT\n\rmi[42], with the Boltzman constant\nkB.\nA more generalized form of the SLLG equation that\nincludes non-local tensorial damping has been reported\nin previous studies [20, 43, 44] and is:\n@mi\n@t=mi\u00020\n@\u0000\r[Bi+bi(t)] +X\nj\u000bij\nmj@mj\n@t1\nA;(4)\nwhich can be derived from Rayleigh dissipation func-\ntional in the Lagrange formalism used by Gilbert [16].\nIn the presence of non-local damping, the Gaussian \ruc-\ntuating \feld ful\flls [43, 45, 46]\n\nb\u0016\ni(t)b\u0017\nj(t0)\u000b\n= 2D\u0016\u0017\nij\u000e(t\u0000t0); (5)\nwithD\u0016\u0017\nij=\u000b\u0016\u0017\nijkBT\n\rmi. The damping tensor \u000b\u0016\u0017\nijmust be\npositive de\fnite in order to be physically-de\fned. Along3\nwith spatial non-locality, the damping can also be non-\nlocal in time, as discussed in Ref. [47]. To prove the\n\ructuation-dissipation theorem in Eq. (5), the Fokker-\nPlanck equation has to be analysed in the presence of\nnon-local damping, similar to Ref. [15]. This is, however,\nnot the purpose of this paper. Instead, we will use the\napproximation \u000b\u0016\u0017\nij=1\n3Trf\u000biig\u000eij\u000e\u0016\u0017within the di\u000busion\nconstantD. Such an approximation is strictly valid only\nin the low temperature limit.\nTo solve this SLLG equation incorporating the non-\nlocal damping, we have implemented an implicit mid-\npoint solver in the UppASD code [48]. This iterative\n\fx-point scheme converges within an error of 10\u000010\u0016B,\nwhich is typically equivalent to 6 iteration steps. More\ndetails of this solver are provided in Appendix A. The\ninitial spin con\fguration in the typical N= 20\u000220\u000220\nsupercell with periodic boundary conditions starts from\ntotally random state. The spin-spin exchange interac-\ntions and non-local damping parameters are included up\nto at least 30 shells of neighbors, in order to guarantee\nthe convergence with respect to the spatial expansion of\nthese parameters (a discussion about the convergence is\ngiven in Section II A). Observables from our simulations\nare typically the average magnetisation M=1\nNPN\nimi\nas well as the magnon dispersion.\nB. Magnon dispersion\nTwo methods to simulate the magnon spectrum are\napplied in this paper: i)the dynamical structure factor\nandii)frozen magnon approach.\nFor the dynamical structure factor S(q;!) at \fnite\ntemperature and damping [34, 49], the spatial and time\ncorrelation function between two magnetic moments iat\npositionrandjat positionr0as well as di\u000berent time 0\nandtis expressed as:\nC\u0016(r\u0000r0;t) =hm\u0016\nr(t)m\u0016\nr0(0)i\u0000hm\u0016\nr(t)ihm\u0016\nr0(0)i:(6)\nHereh\u0001idenotes the ensemble average and \u0016are Carte-\nsian components. The dynamical structure factor can be\nobtained from the time and space Fourier transform of\nthe correlation function, namely:\nS\u0016(q;!) =1p\n2\u0019NX\nr;r0eiq\u0001(r\u0000r0)Z1\n\u00001ei!tC\u0016(r\u0000r0;t)dt:\n(7)\nThe magnon dispersion is obtained from the peak\npositions of S(q;!) along di\u000berent magnon wave vectors\nqin the Brillouin zone and magnon energies !. It\nshould be noted that S(q;!) is related to the scattering\nintensity in inelastic neutron scattering experiments [50].\nThe broadening of the magnon spectrum correlates to\nthe lifetime of spin waves mediated by Gilbert damping\nas well as intrinsic magnon-magnon scattering processes.\nGood agreement between S(q;!) and experiment hasbeen found previously [34].\nThe second method { the frozen magnon approach\n{ determines the magnon spectrum directly from the\nFourier transform of the spin-spin exchange parameters\nJij[51, 52] and non-local damping \u000bij. At zero tempera-\nture, a time-dependent external magnetic \feld is consid-\nered,\nB\u0006\ni(t) =1\nNX\nqB\u0006\nqeiq\u0001Ri\u0000i!t; (8)\nwhereNis the total number of lattice sites and B\u0006\nq=\nBx\nq\u0006iBy\nq. The linear response to this \feld is then given\nby\nM\u0006\nq=\u001f\u0006(q;!)B\u0006\nq: (9)\nWe obtain for the transverse dynamic magnetic suscep-\ntibility [53, 54]\n\u001f\u0006(q;!) =\u0006\rMs\n!\u0006!q\u0007i!\u000bq; (10)\nwith saturation magnetization Ms, spin-wave frequency\n!q=E(q)=~and damping\n\u000bq=X\nj\u000b0je\u0000iq\u0001(R0\u0000Rj): (11)\nWe can extract the spin-wave spectrum from the imagi-\nnary part of the susceptibility,\nIm\u001f\u0006(q;!) =\rMs\u000bq!\n[!\u0006!q]2+\u000b2q!2; (12)\nwhich is equivalent to the correlation function S\u0006(q;!)\ndue to the \ructuation-dissipation theorem [55]. We\n\fnd that the spin-wave lifetime \u001cqis determined by the\nFourier transform of the non-local damping (for \u000bq\u001c1),\n\u001cq=\u0019\n\u000bq!q: (13)\nThe requirement of positive de\fniteness of the damping\nmatrix\u000bijdirectly implies \u000bq>0, since\u000bijis diago-\nnalized by Fourier transformation due to translational\ninvariance. Hence, \u000bq>0 is a criterion to evaluate\nwhether the damping quantity in real-space is physically\nconsistent and whether \frst-principles calculations are\nwell converged. If \u000bq<0 for some wave vector q, energy\nis pumped into the spin system through the correspon-\ndent magnon mode, preventing the system to fully reach\nthe saturation magnetization at su\u000eciently low temper-\natures.\nThe e\u000bective damping \u000b0of the FMR mode at q=\n0 is determined by the sum over all components of the\ndamping matrix, following Eqn.11,\n\u000btot\u0011\u000b0=X\nj\u000b0j: (14)\nTherefore, an e\u000bective local damping should be based\non\u000btotif the full non-local damping is not taken into\naccount.4\nC. Details of the DFT calculations\nThe electronic structure calculations, in the framework\nof density functional theory (DFT), were performed us-\ning the fully self-consistent real-space linear mu\u000en-tin\norbital in the atomic sphere approximation (RS-LMTO-\nASA) [56, 57]. The RS-LMTO-ASA uses the Haydock\nrecursion method [58] to solve the eigenvalue problem\nbased on a Green's functions methodology directly in\nreal-space. In the recursion method, the continued frac-\ntions have been truncated using the Beer-Pettifor termi-\nnator [59], after a number LLof recursion levels. The\nLMTO-ASA [60] is a linear method which gives precise\nresults around an energy E\u0017, usually taken as the center\nof thes,panddbands. Therefore, as we calculate \fne\nquantities as the non-local damping parameters, we here\nconsider an expression accurate to ( E\u0000E\u0017)2starting\nfrom the orthogonal representation of the LMTO-ASA\nformalism [61].\nFor bcc FeCo alloys and bcc Fe we considered LL= 31,\nwhile for fcc Co and fcc Ni much higher LLvalues (51 and\n47, respectively), needed to better describe the density of\nstates and Green's functions at the Fermi level.\nThe spin-orbit coupling (SOC) is included as a l\u0001s\n[60] term computed in each variational step [62]. All\ncalculations were performed within the local spin den-\nsity approximation (LSDA) exchange-functional (XC) by\nvon Barth and Hedin [63], as it gives general magnetic\ninformation with equal or better quality as, e.g., the\ngeneralized gradient approximation (GGA). Indeed, the\nchoice of XC between LSDA and GGA [64] have a mi-\nnor impact on the onsite damping and the shape of the\n\u000bqcurves, when considering the same lattice parame-\nters (data not shown). No orbital polarization [65] was\nconsidered here. Each bulk system was modelled by a\nbig cluster containing \u001855000 (bcc) and\u0018696000 (fcc)\natoms located in the perfect crystal positions with the re-\nspective lattice parameters of a= 2:87\u0017A (bcc Fe and bcc\nFe1\u0000xCox, su\u000eciently close to experimental observations\n[66]),a= 3:54\u0017A (fcc Co [20, 67]), and a= 3:52\u0017A (fcc\nNi [68]). To account for the chemical disorder in the\nFe70Co30and Fe 50Co50bulks, the electronic structure\ncalculated within the simple virtual crystal approxima-\ntion (VCA), which has shown to work well for the fer-\nromagnetic transition metals alloys (particularly for el-\nements next to each other in the Periodic Table, such\nas FeCo and CoNi) [69{76], and also describe in a good\nagreement the damping trends in both FeCo and CoNi\n(see Appendix C).\nAs reported in Ref. [77], the total damping of site\ni, in\ruenced by the interaction with neighbors j, can\nbe decomposed in two main contributions: the onsite\n(fori=j), and the non-local (for i6=j). Both can be\ncalculated, in the collinear framework, by the followingexpression,\n\u000b\u0016\u0017\nij=\u000bCZ1\n\u00001\u0011(\u000f)Tr\u0010\n^T\u0016\ni^Aij(^T\u0017\nj)y^Aji\u0011\nd\u000fT!0K\u0000\u0000\u0000\u0000!\n\u000bCTr\u0010\n^T\u0016\ni^Aij(\u000fF+i\u000e)(^T\u0017\nj)y^Aji(\u000fF+i\u000e)\u0011\n;\n(15)\nwhere we de\fne ^Aij(\u000f+i\u000e) =1\n2i(^Gij(\u000f+i\u000e)\u0000^Gy\nji(\u000f+i\u000e))\nthe anti-Hermitian part of the retarded physical Green's\nfunctions in the LMTO formalism, and \u000bC=g\nmti\u0019a\npre-factor related to the i-th site magnetization. The\nimaginary part, \u000e, is obtained from the terminated con-\ntinued fractions. Also in Eq. 15, ^T\u0016\ni= [\u001b\u0016\ni;Hso] is the\nso-called torque operator [20] evaluated in each Cartesian\ndirection\u0016;\u0017=fx;y;zgand at site i,\u0011(\u000f) =\u0000@f(\u000f)\n@\u000fis\nthe derivative of the Fermi-Dirac distribution f(\u000f) with\nrespect to the energy \u000f,g= 2\u0010\n1 +morb\nmspin\u0011\ntheg-factor\n(not considering here the spin-mixing parameter [78]),\n\u001b\u0016are the Pauli matrices, and mtiis the total magnetic\nmoment of site i(mti=morbi+mspini). This results\nin a 3\u00023 tensor with terms \u000b\u0016\u0017\nij. In the real-space bulk\ncalculations performed in the present work, the \u000bij(with\ni6=j) matrices contain o\u000b-diagonal terms which are can-\ncelled by the summation of the contributions of all neigh-\nbors within a given shell, resulting in a purely diagonal\ndamping tensor, as expected for symmetry reasons [15].\nTherefore, as in the DFT calculations the spin quanti-\nzation axis is considered to be in the z([001]) direction\n(collinear model), we can ascribe a scalar damping value\n\u000bijas the average \u000bij=1\n2(\u000bxx\nij+\u000byy\nij) =\u000bxx\nijfor the\nsystems investigated here. This scalar \u000bijis, then, used\nin the SLLG equation (Eq. 1).\nThe exchange parameters Jijin the Heisenberg\nmodel were calculated by the Liechtenstein-Katsnelson-\nAntropov-Gubanov (LKAG) formalism [79], according to\nthe implementation in the RS-LMTO-ASA method [61].\nHence all parameters needed for the atomistic LLG equa-\ntion have been evaluated from ab-initio electronic struc-\nture theory.\nII. RESULTS\nA. Onsite and non-local dampings\nTable I shows the relevant ab-initio magnetic prop-\nerties of each material; the TCvalues refer to the Curie\ntemperature calculated within the random-phase approx-\nimation (RPA) [80], based on the computed Jijset. De-\nspite the systematic \u000btotvalues found in the lower limit\nof available experimental results (in similar case with,\ne.g., Ref. [81]), in part explained by the fact that we\nanalyze only the intrinsic damping, a good agreement\nbetween theory and experiment can be seen. When the\nwhole VCA Fe 1\u0000xCoxseries is considered (from x= 0%\ntox= 60%), the expected Slater-Pauling behavior of5\nthe total magnetic moment [73, 82] is obtained (data not\nshown).\nFor all systems studied here, the dissipation is domi-\nnated by the onsite ( \u000bii) term, while the non-local pa-\nrameters (\u000bij,i6=j) exhibit values at least one order of\nmagnitude lower; however, as it will be demonstrated in\nthe next sections, these smaller terms still cause a non-\nnegligible impact on the relaxation of the average magne-\ntization as well as magnon lifetimes. Figure 1 shows the\nnon-local damping parameters for the investigated ferro-\nmagnets as a function of the ( i;j) pairwise distance rij=a,\ntogether with the correspondent Fourier transforms \u000bq\nover the \frst Brillouin Zone (BZ). The \frst point to no-\ntice is the overall strong dependence of \u000bon the wave\nvectorq. The second point is the fact that, as also re-\nported in Ref. [20], \u000bijcan be an anisotropic quantity\nwith respect to the same shell of neighbors, due to the\nbroken symmetry imposed by a preferred spin quantiza-\ntion axis. This means that, in the collinear model and for\na given neighboring shell, \u000bijis isotropic only for equiva-\nlent sites around the magnetization as a symmetry axis.\nAnother important feature that can be seen in Fig. 1\nis the presence of negative \u000bijvalues. Real-space neg-\native non-local damping parameters have been reported\npreviously [20, 77, 97]. They are related to the decrease\nof damping at the \u0000-point, but may also increase \u000bqfrom\nthe onsite value in speci\fc qpoints inside the BZ; there-\nfore, they cannot be seen as ad hoc anti-dissipative con-\ntributions. In the ground-state, these negative non-local\ndampings originate from the overlap between the anti-\nHermitian parts of the two Green's functions at the Fermi\nlevel, each associated with a spin-dependent phase factor\n\b\u001b(\u001b=\";#) [20, 80].\nFinally, as shown in the insets of Fig. 1, a long-range\nconvergence can be seen for all cases investigated. An\nillustrative example is the bcc Fe 50Co50bulk, for which\nthe e\u000bective damping can be \u001860% higher than the con-\nverged\u000btotif only the \frst 7 shells of neighbors are con-\nsidered in Eq. 14. The non-local damping of each neigh-\nboring shell is found to follow a1\nr2\nijtrend, as previously\nargued by Thonig et al. [20] and Umetsu et al. [97].\nExplicitly,\n\u000bij/sin(k\"\u0001rij+ \b\") sin(k#\u0001rij+ \b#)\njrijj2; (16)\nwhich also qualitatively justi\fes the existence of negative\n\u000bij's. Thus, the convergence in real-space is typically\nslower than other magnetic quantities, such as exchange\ninteractions ( Jij/1\njrijj3) [80], and also depends on the\nimaginary part \u000e(see Eq. 15) [20]. The di\u000berence in the\nasymptotic behaviour of the damping and the Heisenberg\nexchange is distinctive; the \frst scales with the inverse of\nthe square of the distance while the latter as the inverse\nof the cube of the distance. Although this asymptotic\nbehaviour can be derived from similar arguments, both\nusing the Greens function of the free electron gas, the\nresults are di\u000berent. The reason for this di\u000berence issimply that the damping parameter is governed by states\nclose to the Fermi surface, while the exchange parameter\ninvolves an integral over all occupied states [20, 79].\nFrom bcc Fe to bcc Fe 50Co50(Fig. 1(a-f)), with in-\ncreasing Co content, the average \frst neighbors \u000bijde-\ncreases to a negative value, while the next-nearest neigh-\nbors contributions reach a minimum, and then increase\nagain. Similar oscillations can be found in further shells.\nAmong the interesting features in the Fe 1\u0000xCoxsystems\n(x= 0%;30%;50%), we highlight the low \u000bqaround the\nhigh-symmetry point H, along the H\u0000PandH\u0000N\ndirections, consistently lower than the FMR damping.\nBoth\u000bvalues are strongly in\ruenced by non-local con-\ntributions &5 NN. Also consistent is the high \u000bqob-\ntained forq=H. For long wavelengths in bcc Fe, some\n\u000bqanisotropy is observed around \u0000, which resembles the\nsame trait obtained for the corresponding magnon dis-\npersion curves [80]. This anisotropy changes to a more\nisotropic behavior by FeCo alloying.\nFar from the more noticeable high-symmetry points,\n\u000bqpresents an oscillatory behavior along BZ, around the\nonsite value. It is noteworthy, however, that these oscil-\nlatory\u000bqparameters exhibit variations up to \u00182 times\n\u000bii, thus showing a pronounced non-local in\ruence in\nspeci\fcqpoints.\nIn turn, for fcc Co (Fig. 1(g,h)) the \frst values are\ncharacterized by an oscillatory behavior around zero,\nwhich also re\rects on the damping of the FMR mode,\n\u000bq=0. In full agreement with Ref. [20], we compute a\npeak of\u000bijcontribution at rij\u00183:46a, which shows\nthe long-range character that non-local damping can ex-\nhibit for speci\fc materials. Despite the relatively small\nmagnitude of \u000bij, the multiplicity of the nearest neigh-\nbors shells drives a converged \u000bqdispersion with non-\nnegligible variations from the onsite value along the BZ,\nspecially driven by the negative third neighbors. The\nmaximum damping is found to be in the region around\nthe high-symmetry point X, where thus the lifetime of\nmagnon excitations are expected to be reduced. Simi-\nlar situation is found for fcc Ni (Fig. 1(i,j)), where the\n\frst neighbors \u000bijare found to be highly negative, con-\nsequently resulting in a spectrum in which \u000bq> \u000bq=0\nfor everyq6= 0. In contrast with fcc Co, however, no\nnotable peak contributions are found.\nB. Remagnetization\nGilbert damping in magnetic materials determines the\nrate of energy that dissipates from the magnetic to other\nreservoirs, like phonons or electron correlations. To ex-\nplore what impact non-local damping has on the energy\ndissipation process, we performed atomistic spin dynam-\nics (ASD) simulations for the aforementioned ferromag-\nnets: bcc Fe 1\u0000xCox(x= 0%;30%;50%), fcc Co, and\nfcc Ni, for the (i)fully non-local \u000bijand (ii)e\u000bective\n\u000btot(de\fned in 14) dissipative case. We note that, al-\nthough widely considered in ASD calculations, the adop-6\nTABLE I. Spin ( mspin) and orbital ( morb) magnetic moments, onsite ( \u000bii) damping, total ( \u000btot) damping, and Curie temper-\nature (TC) of the investigated systems. The theoretical TCvalue is calculated within the RPA. In turn, mtdenotes the total\nmoments for experimental results of Ref. [82].\nmspin(\u0016B)morb(\u0016B)\u000bii(\u000210\u00003) \u000btot(\u000210\u00003) TC(K)\nbcc Fe (theory) 2.23 0.05 2.4 2.1 919\nbcc Fe (expt.) 2.13 [68] 0 :08 [68] \u0000 1:9\u00007:2 [33, 83{89] 1044\nbcc Fe 70Co30(theory) 2.33 0.07 0.5 0.9 1667\nbcc Fe 70Co30(expt.) mt= 2:457 [82] \u0000 0:5\u00001:7a[33, 83, 90] 1258 [92]\nbcc Fe 50Co50(theory) 2.23 0.08 1.5 1.6 1782\nbcc Fe 50Co50(expt.) mt= 2:355 [82] \u0000 2:0\u00003:2b[25, 33, 83] 1242 [93]\nfcc Co (theory) 1.62 0 :08 7.4 1.4 1273\nfcc Co (expt.) 1 :68(6) [94] \u0000 \u0000 2:8(5) [33, 89] 1392\nfcc Ni (theory) 0 :61 0 :05 160.1 21.6 368\nfcc Ni (expt.) 0 :57 [68] 0 :05 [68] \u0000 23:6\u000064 [22, 83, 87{89, 95, 96] 631\naThe lower limit refers to polycrystalline Fe 75Co2510 nm-thick \flms from Ref. [33]. Lee et al. [90] also found a low Gilbert damping in\nan analogous system, where \u000btot<1:4\u000210\u00003. For the exact 30% of Co concentration, however, previous results [33, 84, 91] indicate\nthat we should expect a slightly higher damping than in Fe 75Co25.\nbThe upper limit refers to the approximate minimum intrinsic value for a 10 nm-thick \flm of Fe 50Co50jPt (easy magnetization axis).\ntion of a constant \u000btotvalue (case (ii)) is only a good ap-\nproximation for long wavelength magnons close to q= 0.\nFirst, we are interested on the role of non-local damp-\ning in the remagnetization processes as it was already\ndiscussed by Thonig et al. [20] and as it is important\nfor,e.g., ultrafast pump-probe experiments as well as all-\noptical switching. In the simulations presented here, the\nrelaxation starts from a totally random magnetic con-\n\fguration. The results of re-magnetization simulations\nare shown in Figure 2. The fully non-local damping (i)\nin the equation of motion enhances the energy dissipa-\ntion process compared to the case when only the e\u000bective\ndamping (ii)is used. This e\u000bect is found to be more pro-\nnounced in fcc Co and fcc Ni compared to bcc Fe and bcc\nFe50Co50. Thus, the remagnetization time to 90% of the\nsaturation magnetisation becomes \u00185\u00008 times faster\nfor case (i)compared to the case (ii). This is due to\nthe increase of \u000bqaway from the \u0000 point in the whole\nspectrum for Co and Ni (see Fig. 1), where in Fe and\nFe50Co50it typically oscillates around \u000btot.\nFor bcc Fe 70Co30, the e\u000bect of non-local damping on\nthe dynamics is opposite to the data in Fig. 2; the re-\nlaxation process is decelerated. In this case, almost the\nentire\u000bqspectrum is below \u000bq=0, which is an interest-\ning result given the fact that FMR measurements of the\ndamping parameter in this system is already considered\nan ultra-low value, when compared to other metallic fer-\nromagnets [33]. Thus, in the remagnetization process of\nFe70Co30, the majority of magnon modes lifetimes is un-\nderestimated when a constant \u000btotis considered in the\nspin dynamics simulations, which leads to a faster overall\nrelaxation rate.\nAlthough bcc Fe presents the highest Gilbert damp-ing obtained in the series of the Fe-Co alloys (see Table\nI) the remagnetization rate is found to be faster in bcc\nFe50Co50. This can be explained by the fact that the ex-\nchange interactions for this particular alloy are stronger\n(\u001880% higher for nearest-neighbors) than in pure bcc\nFe, leading to an enhanced Curie temperature (see Table\nI). In view of Eq. 13 and Fig. 1, the di\u000berence in the\nremagnetization time between bcc Fe 50Co50and elemen-\ntal bcc Fe arises from \u000bqvalues that are rather close,\nbut where the magnon spectrum of Fe 50Co50has much\nhigher frequencies, with corresponding faster dynamics\nand hence shorter remagnetization times.\nFrom our calculations we \fnd that the sum of non-local\ndamping\u0010P\ni6=j\u000bij\u0011\ncontributes with \u000013%,\u000081%,\n\u000087%, +80%, and +7% to the local damping in bcc Fe,\nfcc Co, fcc Ni, bcc Fe 70Co30, and bcc Fe 50Co50, respec-\ntively. The high positive ratio found in Fe 70Co30indi-\ncates that, in contrast to the other systems analyzed, the\nnon-local contributions act like an anti-damping torque,\ndiminishing the local damping torque. A similar anti-\ndamping e\u000bect in antiferromagnetic (AFM) materials\nhave been reported in theoretical and experimental in-\nvestigations ( e.g., [98, 99]), induced by electrical current.\nHere we \fnd that an anti-damping torque e\u000bect can have\nan intrinsic origin.\nTo provide a deeper understanding of the anti-damping\ne\u000bect caused by a positive non-local contribution, we an-\nalytically solved the equation of motion for a two spin\nmodel system, e.g. a dimer. In the particular case when\nthe onsite damping \u000b11is equal to the non-local con-\ntribution\u000b12, we observed that the system becomes un-\ndamped (see Appendix B). As demonstrated in Appendix\nB, ASD simulations of such a dimer corroborate the re-7\nFIG. 1. Non-local damping ( \u000bij) as a function of the nor-\nmalized real-space pairwise ( i;j) distance computed for each\nneighboring shell, and corresponding Fourier transform \u000bq\n(see Eq. 11) from the onsite value ( \u000bii) up to 136 shells of\nneighbors (136 NN) for: (a,b) bcc Fe; (c,d) bcc Fe 70Co30;\n(e,f) bcc Fe 50Co50in the virtual-crystal approximation; and\nup to 30 shells of neighbors (30 NN) for: (g,h) fcc Co; (i,j) fcc\nNi. The insets in sub\fgures (a,c,e,g,i) show the convergence\nof\u000btotin real-space. The obtained onside damping values are\nshown in Table I. In the insets of the left panel, green full\nlines are guides for the eyes.\nsult of undamped dynamics. It should be further noticed\nthat this proposed model system was used to analyse\nthe stability of the ASD solver, verifying whether it can\npreserve both the spin length and total energy. Full de-\ntail of the analytical solution and ASD simulation of a\nspin-dimer and the anti-damping e\u000bect are provided inAppendix B.\nFIG. 2. Remagnetization process simulated with ASD, con-\nsidering fully non-local Gilbert damping ( \u000bij, blue sold lines),\nand the e\u000bective damping ( \u000btot, red dashed lines), for: (a) fcc\nNi; (b) fcc Co; and (c) bcc Fe 1\u0000xCox(x= 0%;30%;50%).\nThe dashed gray lines indicate the stage of 90% of the satu-\nration magnetization.\nC. Magnon spectra\nIn order to demonstrate the in\ruence of damping on\nmagnon properties at \fnite temperatures, we have per-\nformed ASD simulations to obtain the excitation spectra\nfrom the dynamical structure factor introduced in Sec-\ntion I. Here, we consider 16 NN shells for S(q;!) calcula-\ntions both from simulations that include non-local damp-\ning as well as the e\u000bective total damping (see Appendix\nD for a focused discussion). In Fig. 3, the simulated\nmagnon spectra of the here investigated ferromagnets are\nshown. We note that a general good agreement can be\nobserved between our computed magnon spectra (both\nfrom the the frozen magnon approach as well as from the\ndynamical structure factor) and previous theoretical as\nwell as experimental results [34, 52, 80, 100{103], where\ndeviations from experiments is largest for fcc Ni. This\nexception, however, is well known and has already been\ndiscussed elsewhere [104].\nThe main feature that the non-local damping causes to\nthe magnon spectra in all systems investigated here, is in\nchanges of the full width at half maximum (FWHM) 4q\nofS(q;!). Usually,4qis determined from the super-\nposition of thermal \ructuations and damping processes.\nMore speci\fcally, the non-local damping broadens the\nFWHM compared to simulations based solely on an e\u000bec-\ntive damping, for most of the high-symmetry paths in all\nof the here analyzed ferromagnets, with the exception of\nFe70Co30. The most extreme case is for fcc Ni, as \u000bqex-\nceeds the 0:25 threshold for q=X, which is comparable\nto the damping of ultrathin magnetic \flms on high-SOC\nmetallic hosts [105]. As a comparison, the largest di\u000ber-\nence of FWHM between the non-local damping process\nand e\u000bective damping process in bcc Fe is \u00182 meV, while\nin fcc Ni the largest di\u000berence can reach \u0018258 meV. In\ncontrast, the di\u000berence is \u0018\u00001 meV in Fe 70Co30and the8\nlargest non-local damping e\u000bect occurs around q=N\nand in the H\u0000Pdirection, corroborating with the dis-\ncussion in Section II A. At the \u0000 point, which corresponds\nto the mode measured in FMR experiments, all spins in\nthe system have a coherent precession. This implies that\n@mj\n@tin Eq. 4 is the same for all moments and, thus, both\ndamping scenarios discussed here (e\u000becive local and the\none that also takes into account non-local contributions)\nmake no di\u000berence to the spin dynamics. As a conse-\nquence, only a tiny (negligible) di\u000berence of the FWHM\nis found between e\u000bective and non-local damping for the\nFMR mode at low temperatures.\nThe broadening of the FWHM on the magnon spec-\ntrum is temperature dependent. Thus, the e\u000bect of non-\nlocal damping to the width near \u0000 can be of great in-\nterest for experiments. More speci\fcally, taking bcc Fe\nas an example, the di\u000berence between width in e\u000bective\ndamping and non-local damping process increases with\ntemperature, where the di\u000berence can be enhanced up to\none order of magnitude from T= 0:1 K toT= 25 K.\nNote that this enhancement might be misleading due to\nthe limits of \fnite temperature assumption made here.\nThis temperature dependent damping e\u000bect on FWHM\nsuggests a path for the measurement of non-local damp-\ning in FMR experiments.\nWe have also compared the di\u000berence in the imaginary\npart of the transverse dynamical magnetic susceptibility\ncomputed from non-local and e\u000bective damping. De\fned\nby Eq. 12, the imaginary part of susceptibility is re-\nlated to the FWHM [15]. Similar to the magnon spectra\nshown in Fig. 3, the susceptibility di\u000berence is signi\f-\ncant at the BZ boundaries. Taking the example of fcc\nCo, Im\u001f\u0006(q;!) for e\u000bective damping processes can be\n11:8 times larger than in simulations that include non-\nlocal damping processes, which is consistent to the life-\ntime peak that occurs at high the symmetry point, X,\ndepicted in Fig. 4. In the Fe 1\u0000xCoxalloy, and Fe 70Co30,\nthe largest ratio is 1 :7 and 2:7 respectively. The intensity\nat \u0000 point is zero since \u000bqis independent on the coupling\nvector and equivalent in both damping modes. The ef-\nfect of non-local damping on susceptibility coincides well\nwith the magnon spectra from spin dynamics. Thus, this\nmethod allows us to evaluate the magnon properties in a\nmore e\u000ecient way.\nD. Magnon lifetimes\nBy \ftting the S(q;!) curve at each wave vector with\na Lorentzian curve, the FWHF and hence the magnon\nlifetimes,\u001cq, can be obtained from the simple relation\n[15]\n\u001cq=2\u0019\n4q: (17)\nFigure 4 shows the lifetimes computed in the high-\nsymmetry lines in the BZ for all ferromagnets here in-vestigated. As expected, \u001cqis much lower at the qvec-\ntors far away from the zone center, being of the order\nof 1 ps for the Fe 1\u0000xCoxalloys (x= 0%;30%;50%),\nand from\u00180:01\u00001 ps in fcc Co and Ni. In view of\nEq. 13, the magnon lifetime is inversely proportional to\nboth damping and magnon frequency. In the e\u000bective\ndamping process, \u000bqis a constant and independent of\nq; thus, the lifetime in the entire BZ is dictated only by\n!q. The situation becomes more complex in the non-\nlocal damping process, where the \u001cqis in\ruenced by the\ncombined e\u000bect of changing damping and magnon fre-\nquency. Taking Fe 70Co30as an example, even though\nthe\u000bqis higher around the \u0000, the low magnon frequency\ncompensates the damping e\u000bect, leading to an asymp-\ntotically divergent magnon lifetime as !q!0. However,\nthis divergence becomes \fnite when including e.g. mag-\nnetocrystalline anisotropy or an external magnetic \feld\nto the spin-Hamiltonian. In the H\u0000Npath, the magnon\nenergy of Fe 70Co30is large, but \u000bqreaches\u00184\u000210\u00004\natq=\u00001\n4;1\n4;1\n2\u0001\n, resulting in a magnon lifetime peak of\n\u001810 ps. This value is not found for the e\u000bective damping\nmodel.\nIn the elemental ferromagnets, as well as for Fe 50Co50,\nit is found that non-local damping decreases the magnon\nlifetimes. This non-local damping e\u000bect is signi\fcant in\nboth Co and Ni, where the magnon lifetimes from the \u000bij\nmodel di\u000ber by an order of magnitude from the e\u000bective\nmodel (see Fig. 4). In fact, considering \u001cqobtained from\nEq. 13, the e\u000bective model predicts a lifetime already\nhigher by more than 50% when the magnon frequencies\nare\u001833 meV and\u001814 meV in the K\u0000\u0000 path ( i.e.,\nnear \u0000) of Ni and Co, respectively. This di\u000berence mainly\narises, in real-space, from the strong negative contriu-\ntions of\u000bijin the close neighborhood around the refer-\nence site, namely the NN in Ni and third neighbors in Co.\nIn contrast, due to the \u000bqspectrum composed of almost\nall dampings lower than \u000btot, already discussed in Section\nII A, the opposite trend on \u001cqis observed for Fe 70Co30:\nthe positive overall non-local contribution guide an anti-\ndamping e\u000bect, and the lifetimes are enhanced in the\nnon-local model.\nAnother way to evaluate the magnon lifetimes is from\nthe linear response theory. As introduced in Section I B,\nwe have access to magnon lifetimes at low temperatures\nfrom the imaginary part of the susceptibility. The \u001cq\ncalculated from Eq. 13 is also displayed in Fig. 4. Here\nthe spin-wave frequency !qis from the frozen magnon\nmethod. The magnon lifetimes from linear response have\na very good agreement with the results from the dynam-\nical structure factor, showing the equivalence between\nboth methods. Part of the small discrepancies are re-\nlated to magnon-magnon scattering induced by the tem-\nperature e\u000bect in the dynamical structure factor method.\nWe also \fnd a good agreement on the magnon lifetimes\nof e\u000bective damping in pure Fe with previous studies\n[106]. They are in the similar order and decrease with\nthe increasing magnon energy. However, their results\nare more di\u000bused since the simulations are performed at9\nFIG. 3. Magnon spectra calculated with non-local Gilbert damping and e\u000bective Gilbert damping in: (a) bcc Fe; (b) bcc\nFe70Co30; (c) bcc Fe 50Co50; (d) fcc Co; and (e) fcc Ni. The black lines denote the adiabatic magnon spectra calculated from\nEq. 7. Full red and open blue points denote the peak positions of S(q;!) at each qvector for\u000btotand\u000bijcalculations,\nrespectively, at T= 0:1 K. The width of transparent red and blue areas corresponds to the full width half maximum (FWHM)\non the energy axis \ftted from a Lorentzian curve, following the same color scheme. To highlight the di\u000berence of FWHM\nbetween the two damping modes, the FWHMs shown in the magnon spectrum of Fe 1\u0000xCox, Co, and Ni are multiplied by 20,\n5, and1\n2times, in this order. The triangles represent experimental results: in (a), Fe at 10 K [102] (yellow up) and Fe with\n12% Si at room-temperature [101] (green down); in (d), Co(9 ML)/Cu(100) at room-temperature [103] (green down); in (e) Ni\nat room-temperature (green down) [100]. The standard deviation of the peaks are represented as error bars.\nroom-temperature.\nIII. CONCLUSION\nWe have presented the in\ruence of non-local damping\non spin dynamics and magnon properties of elemental fer-\nromagnets (bcc Fe, fcc Co, fcc Ni) and the bcc Fe 70Co30\nand bcc Fe 50Co50alloys in the virtual-crystal approxima-\ntion. It is found that the non-local damping has impor-\ntant e\u000bects on relaxation processes and magnon prop-\nerties. Regarding the relaxation process, the non-local\ndamping in Fe, Co, and Ni has a negative contribution\nto the local (onsite) part, which accelerates the remagne-\ntization. Contrarily, in\ruenced by the positive contribu-\ntion of\u000bij(i6=j), the magnon lifetimes of Fe 70Co30and\nFe50Co50are increased in the non-local model, typically\nat the boundaries of the BZ, decelerating the remagneti-\nzation.\nConcerning the magnon properties, the non-local\ndamping has a signi\fcant e\u000bect in Co and Ni. More\nspeci\fcally, the magnon lifetimes can be overestimated\nby an order of magnitude in the e\u000bective model for these\ntwo materials. In real-space, this di\u000berence arises as a\nresult of strong negative non-local contributions in theclose neighborhood around the reference atom, namely\nthe NN in Ni and the third neighbors in Co.\nAlthough the e\u000bect of non-local damping to the\nstochastic thermal \feld in spin dynamics is not included\nin this work, we still obtain coherent magnon lifetimes\ncomparing to the analytical solution from linear response\ntheory. Notably, it is predicted that the magnon lifetimes\nat certain wave vectors are higher for the non-local damp-\ning model in some materials. An example is Fe 70Co30, in\nwhich the lifetime can be \u00183 times higher in the H\u0000N\npath for the non-local model. On the other hand, we\nhave proposed a fast method based on linear response\nto evaluate these lifetimes, which can be used to high-\nthroughput computations of magnonic materials.\nFinally, our study provides a link on how non-local\ndamping can be measured in FMR and neutron scat-\ntering experiments. Even further, it gives insight into\noptimising excitation of magnon modes with possible\nlong lifetimes. This optimisation is important for any\nspintronics applications. As a natural consequence of\nany real-space ab-initio formalism, our methodology and\n\fndings also open routes for the investigation of other\nmaterials with preferably longer lifetimes caused by non-\nlocal energy dissipation at low excitation modes. Such\nmaterials research could also include tuning the local10\nFIG. 4. Magnon lifetimes \u001cqof: (a) bcc Fe; (b) bcc Fe 70Co30; (c) bcc Fe 50Co50; (d) fcc Co; and (e) fcc Ni as function of q,\nshown in logarithmic scale. The color scheme is the same of Fig. 3, where blue and red represents \u001cqcomputed in the e\u000bective\nand non-local damping models. The transparent lines and opaque points depict the lifetimes calculated with Eq. 13 and by\nthe FWHM of S(q;!) atT= 0:1 K (see Eq. 17). The lifetime asymptotically diverges around the \u0000-point due to the absence\nof anisotropy e\u000bects or external magnetic \feld in the spin-Hamiltonian.\nchemical environments by doping or defects.\nIV. ACKNOWLEDGMENTS\nFinancial support from Vetenskapsr\u0017 adet (grant num-\nbers VR 2016-05980 and VR 2019-05304), and the\nKnut and Alice Wallenberg foundation (grant number\n2018.0060) is acknowledged. Support from the Swedish\nResearch Council (VR), the Foundation for Strategic Re-search (SSF), the Swedish Energy Agency (Energimyn-\ndigheten), the European Research Council (854843-\nFASTCORR), eSSENCE and STandUP is acknowledged\nby O.E. . Support from the Swedish Research Coun-\ncil (VR) is acknowledged by D.T. and A.D. . The\nChina Scholarship Council (CSC) is acknowledged by\nZ.L.. The computations/data handling were enabled by\nresources provided by the Swedish National Infrastruc-\nture for Computing (SNIC) at the National Supercom-\nputing Centre (NSC, Tetralith cluster), partially funded\nby the Swedish Research Council through grant agree-\nment No. 2016-07213.\n[1] A. Barman, G. Gubbiotti, S. Ladak, A. O. Adeyeye,\nM. Krawczyk, J. Gr afe, C. Adelmann, S. Cotofana,\nA. Naeemi, V. I. Vasyuchka, et al. , J. Phys. Condens.\nMatter 33, 413001 (2021).\n[2] P. Pirro, V. I. Vasyuchka, A. A. Serga, and B. Hille-\nbrands, Nat. Rev. Mater 6, 1114 (2021).\n[3] B. Rana and Y. 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Sakuma, J. Appl. Phys. 117, 013912 (2015).13\nAppendix A: Numerical solver\nIn this Appendix, the numerical method to solve Eq.\n1 is described. In previous studies, several numerical\napproaches have been proposed to solve the local LLG\nequations, including HeunP method, implicit midpoint\nmethod, Depondt-Merten's method [107], semi-implicit\nA (SIA) and semi-implicit B (SIB) methods [42]. To solve\nthis non-local LLG equation, we use the \fxed-point iter-\nation midpoint method. We have done convergence tests\non this method and \fnd that it preserve the energy and\nspin length of the system, which is demonstrated in Fig.\n5 for the case of a dimer. With stable outputs, the solver\nallows for a relatively large time step size, typically of\nthe order of \u0001 t\u00180:1\u00001 fs.\nFollowing the philosophy of an implicit midpoint\nmethod, the implemented algorithm can be described as\nfollows. Let mt\nibe the magnetic moment of site iat a\ngiven time step t. Then we can de\fne the quantity mmidand the time derivative of mi, respectively, as\nmmid=mt+1\ni+mt\ni\n2;\n@mi\n@t=mt+1\ni\u0000mt\ni\n\u0001t:(A1)\nUsing this de\fnition in Eq. 4, the equation of motion\nof thei-th spin becomes:\n@mi\n@t=mmid\u00020\n@\u0000\r[Bi(mmid) +bi(t)] +X\nj\u000bij\nmj@mj\n@t1\nA:\n(A2)\nThus, with a \fxed-point scheme, we can do the follow-\ning iteration\nmt+1(k+1)\ni =mt\ni+ \u0001t0\n@ \nmt+1(k)\ni +mt\ni\n2!\n\u00020\n@\u0000\r\"\nBi \nmt+1(k)\ni +mt\ni\n2!\n+bi(t)#\n+X\nj\u000bij\nmjmt+1(k)\nj\u0000mt\nj\n\u0001t1\nA1\nA:\n(A3)\nIfmt+1(k+1)\ni\u0019mt+1(k)\ni , the self-consistency con-\nverges. Typically, about 6 iteration steps are needed.\nThis solver was implemented in the software package Up-\npASD [48] for this work.\nAppendix B: Analytical model of anti-damping in\ndimers\nIn the dimer model, there are two spins on site 1 and\nsite 2 denoted by m1andm2, which are here supposed\nto be related to the same element { so that, naturally,\n\u000b11=\u000b22>0. Also, let's consider a su\u000eciently low\ntemperature so that bi(t)!0, which is a reasonable\nassumption, given that damping has an intrinsic origin\n[108]. This simple system allows us to provide explicit\nexpressions for the Hamiltonian, the e\u000bective magnetic\n\felds and the damping term. From the analytical solu-\ntion, it is found that the dimer spin system becomes an\nundamped system when local damping is equal to non-\nlocal damping, i.e.the e\u000bective damping of the system\nis zero.\nFollowing the de\fnition given by Eq. 4 in the main\ntext, the equation of motion for spin 1 reads:\n@m1\n@t=m1\u0002\u0012\n\u0000\rB1+\u000b11\nm1@m1\n@t+\u000b12\nm2@m2\n@t\u0013\n;(B1)\nand an analogous expression can be written for spin 2.\nFor sake of simplicity, the Zeeman term is zero and thee\u000bective \feld only includes the contribution from Heisen-\nberg exchange interactions. Thus, we have B1= 2J12m2\nandB2= 2J21m1. Withj\u000bijj\u001c 1, we can take the\nLL form@mi\n@t=\u0000\rmi\u0002Bito approximate the time-\nderivative on the right-hand side of the LLG equation.\nLetm1=m2and\u000b12=\u0015\u000b11. SinceJ12=J21and\nm1\u0002m2=\u0000m2\u0002m1, then we have\n@m1\n@t=\u00002\rJ12m1\u0002\u0014\nm2+ (1\u0000\u0015)\u000b11\nm1(m1\u0002m2)\u0015\n:\n(B2)\nTherefore, when \u000b12=\u000b21=\u000b11(i.e.,\u0015= 1), Eq. B1\nis reduced to:\n@m1\n@t=\u00002\rJ12m1\u0002m2; (B3)\nand the system becomes undamped. It is however\nstraightforward that, for the opposite case of a strong\nnegative non-local damping ( \u0015=\u00001), Eq. B2 describes\na common damped dynamics. A side (and related) con-\nsequence of Eq. B2, but important for the discussion in\nSection II B, is the fact that the e\u000bective onsite damp-\ning term\u000b\u0003\n11= (1\u0000\u0015)\u000b11becomes less relevant to the\ndynamics as the positive non-local damping increases\n(\u0015!1), or, in other words, as \u000btot= (\u000b11+\u000b12) strictly\nincreases due to the non-local contribution. Exactly the\nsame reasoning can be made for a trimer, for instance,\ncomposed by atoms with equal moments and exchange\ninteractions ( m1=m2=m3,J12=J13=J23), and\nsame non-local dampings ( \u000b13=\u000b12=\u0015\u000b11).14\nThe undamped behavior can be directly observed from\nASD simulations of a dimer with \u000b12=\u000b11, as shown in\nFig. 5. Here the magnetic moment and the exchange are\ntaken the same of an Fe dimer, m1= 2:23\u0016BandJ12=\n1:34 mRy. Nevertheless, obviously the overall behavior\ndepicted in Fig. 5 is not dependent on the choice of\nm1andJ12. Thezcomponent is constant, while the x\nandycomponents of m1oscillate in time, indicating a\nprecessing movement.\nIn a broader picture, this simple dimer case exempli\fes\nthe connection between the eigenvalues of the damping\nmatrix\u000b= (\u000bij) and the damping behavior. The occur-\nrence of such undamped dynamics has been recently dis-\ncussed in Ref. [109], where it is shown that a dissipation-\nfree mode can occur in a system composed of two sub-\nsystems coupled to the same bath.\n0.00 0.02 0.04 0.06 0.08 0.10\nt(ps)0.2\n0.00.20.40.60.81.0Magnetization\n3.0\n2.5\n2.0\n1.5\n1.0\n0.5\n0.0\nEnergy(mRy)mxmymzmEnergy\nFIG. 5. Spin dynamics at T= 0 K of an undamped dimer\nin which\u000b12=\u000b21=\u000b11(see text). The vector m1is\nnormalized and its Cartesian components are labeled in the\n\fgure asmx,myandmz. The black and grey lines indicate\nthe length of spin and energy (in mRy), respectively.\nAppendix C: E\u000bective and onsite damping in the\nFeCo and CoNi alloys\nAs mentioned in Section I, the simple VCA model al-\nlows us to account for the disorder in 3 d-transition-metal\nalloys in a crude but e\u000ecient way which avoids the use\nof large supercells with random chemical distributions.\nWith exactly the same purpose, the coherent potential\napproximation (CPA) [110] has also been employed to\nanalyze damping in alloys ( e.g., in Refs. [84, 111, 112]),\nshowing a very good output with respect to trends, when\ncompared to experiments [33, 81]. In Fig. 6 we show\nthe normalized calculated local (onsite, \u000bii) and e\u000bec-\ntive damping ( \u000btot) parameters for the zero-temperature\nVCA Fe 1\u0000xCoxalloy in the bcc structure, consistent with\na concentration up to x\u001960% of Co [33]. The computed\nvalues in this work (blue, representing \u000bii, and red points,\nrepresenting \u000btot) are compared to previous theoretical\nCPA results and room-temperature experimental data.\nThe trends with VCA are reproduced in a good agree-ment with respect to experiments and CPA calculations,\nshowing a minimal \u000btotwhen the Co concentration is\nx\u001930%. This behavior is well correlated with the local\ndensity of states (LDOS) at the Fermi level, as expected\nby the simpli\fed Kambersk\u0013 y equation [113], and the on-\nsite contribution. Despite the good agreement found, the\nvalues we have determined are subjected to a known error\nof the VCA with respect to the experimental results.\nThis discrepancy can be partially explained by three\nreasons: ( i) the signi\fcant in\ruence of local environ-\nments (local disorder and/or short-range order) to \u000btot\n[25, 77]; ( ii) the fact that the actual electronic lifetime\n(i.e., the mean time between two consecutive scattering\nevents) is subestimated by the VCA average for random-\nness in the FeCo alloy, which can have a non-negligible\nimpact in the damping parameter [22, 114]; and ( iii) the\nin\ruence on damping of noncollinear spin con\fgurations\nin \fnite temperature measurements [54, 115]. On top of\nthat, it is also notorious that damping is dependent on\nthe imaginary part of the energy (broadening) [22, 114],\n\u000e, which can be seen as an empirical quantity, and ac-\ncounts for part of the di\u000berences between theory and ex-\nperiments.\n 0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035\n 0  10  20  30  40  50  60 0 5 10 15 20 25Damping value\nDOS at EF (states/Ry−atom)\nCo concentration (%)onsite  (αii)\ntotal  (αtot)\nTurek et al.  (αtot)\nMankovsky  et al.  [2013]  (αtot)\nMankovsky  et al.  [2018]  (αtot)\nSchoen et al.\nn(EF)\nFIG. 6. (Color online) Left scale : Computed Gilbert e\u000bec-\ntive (\u000btot, red circles) and onsite ( \u000bii, blue squares) damping\nparameters as a function of Co the concentration ( x) for bcc\nFe1\u0000xCoxbinary alloy in the virtual-crystal approximation.\nThe values are compared with previous theoretical results us-\ning CPA, from Ref. [84] (gray full triangles), Ref. [54] (black\nopen rhombus), Ref. [112] (yellow open triangles), and room-\ntemperature experimental data [33]. Right scale : The calcu-\nlated density of states (DOS) at the Fermi level as a function\nofx, represented by the black dashed line.\nIn the spirit of demonstrating the e\u000bectiveness of the\nsimple VCA to qualitatively (and also, to some extent,\nquantitatively) describe the properties of Gilbert damp-\ning in suitable magnetic alloys, we also show in Fig. 7 the\nresults obtained for Co xNi1\u0000xsystems. The CoNi alloys15\nare known to form in the fcc structure for a Ni concen-\ntration range of 10% \u0000100%. Therefore, here we mod-\neled CoxNi1\u0000xby a big fcc cluster containing \u0018530000\natoms in real-space with the equilibrium lattice parame-\nter ofa= 3:46\u0017A. The number of recursion levels consid-\nered isLL= 41. A good agreement with experimental\nresults and previous theoretical calculations can be no-\nticed. In particular, the qualitative comparison with the-\nory from Refs. [81, 84] indicates the equivalence between\nthe torque correlation and the spin correlation models\nfor calculating the damping parameter, which was also\ninvestigated by Sakuma [116]. The onsite contribution\nfor each Co concentration, \u000bii, is omitted from Fig. 7\ndue to an absolute value 2 \u00004 times higher than \u000btot,\nbut follows the same decreasing trend. Again, the over-\nall e\u000bective damping values are well correlated with the\nLDOS, and re\rect the variation of the quantity1\nmtwith\nCo concentration (see Eq. 15).\n 0 0.005 0.01 0.015 0.02 0.025\n 0  10  20  30  40  50  60  70 10 15 20 25 30Damping value\nDOS at EF (states/Ry−atom)\nCo concentration (%)total  (αtot)\nMankovsky  et al.  [2013]  (αtot)\nStarikov et al.  (αtot)\nSchoen [2017] et al.\nn(EF)\nFIG. 7. (Color online) Left scale : Computed Gilbert e\u000bective\n(\u000btot, red circles) damping parameters as a function of the Co\nconcentration ( x) for fcc Co xNi1\u0000xbinary alloy in the virtual-\ncrystal approximation. The values are compared with previ-\nous theoretical results using CPA, from Ref. [84] (gray full\ntriangles), Ref. [81] (gold full circles), and room-temperature\nexperimental data [89]. Right scale : The calculated density of\nstates (DOS) at the Fermi level as a function of x, represented\nby the black dashed line.\nAppendix D: E\u000bect of further neighbors in the\nmagnon lifetimes\nWhen larger cuto\u000b radii ( Rcut) of\u000bijparameters are\nincluded in ASD, Eq. A3 takes longer times to achieve a\nself-consistent convergence. In practical terms, to reach a\nsizeable computational time for the calculation of a given\nsystem,Rcutneeds to be chosen in order to preserve the\nmain features of the magnon properties as if Rcut!1 .\nA good quantity to rely on is the magnon lifetime \u001cq,as it consists of both magnon frequency and q-resolved\ndamping (Eq. 13). In Section II C, we have shown the\nequivalence between Eq. 13 and the inverse of FWHM\non the energy axis of S(q;!) for the ferromagnets inves-\ntigated here. Thus, the comparison of two \u001cqspectra for\ndi\u000berentRcutcan be done directly and in an easier way\nusing Eq. 13.\nFIG. 8. (Color online) Magnon lifetimes calculated using Eq.\n13 for: (a) bcc Fe; and (b) bcc Fe 50Co50, using a reduced set\nof 16 NN shells (opaque lines), and the full set of 136 NN\nshells (transparent lines).\nAn example is shown in Figure 8 for bcc Fe and bcc\nFe50Co50. Here we choose the \frst 16 NN ( Rcut\u00183:32a)\nand compare the results with the full calculated set of\n136 NN (Rcut= 10a). It is noticeable that the reduced\nset of neighbors can capture most of the features of the\n\u001cqspectrum for a full NN set. However, long-range in-\n\ruences of small magnitudes, such as extra oscillations\naround the point q=Hin Fe, can occur. In particu-\nlar, these extra oscillations arise mainly due to the pres-\nence of Kohn anomalies in the magnon spectrum of Fe,\nalready reported in previous works [52, 80]. In turn, for\nthe case of Fe 50Co50, the long-range \u000bijreduces\u000btot, and\ncauses the remagnetization times for non-local and e\u000bec-16\ntive dampings to be very similar (see Fig. 2). For the\nother ferromagnets considered in the present research,comparisons of the reduced Rcutwith analogous quality\nwere reached."    },    {        "title": "1012.1371v1.Turbulence_damping_as_a_measure_of_the_flow_dimensionality.pdf",        "content": "Turbulence damping as a measure of the \row dimensionality\nM. Shats,\u0003D. Byrne, and H. Xia\nResearch School of Physics and Engineering, The Australian National University, Canberra ACT 0200, Australia\n(Dated: October 23, 2018)\nThe dimensionality of turbulence in \ruid layers determines their properties. We study electro-\nmagnetically driven \rows in \fnite depth \ruid layers and show that eddy viscosity, which appears as\na result of three-dimensional motions, leads to increased bottom damping. The anomaly coe\u000ecient,\nwhich characterizes the deviation of damping from the one derived using a quasi-two-dimensional\nmodel, can be used as a measure of the \row dimensionality. Experiments in turbulent layers show\nthat when the anomaly coe\u000ecient becomes high, the turbulent inverse energy cascade is suppressed.\nIn the opposite limit turbulence can self-organize into a coherent \row.\nPACS numbers: 47.27.Rc, 47.55.Hd, 42.68.Bz\nFluid layers represent a broad class of \rows whose\ndepths are much smaller than their horizontal extents,\nfor example, planetary atmospheres and oceans. A dis-\ncovery of the upscale energy transfer in two-dimensional\n(2D) turbulence [1] gave new insight into the energy bal-\nance in turbulent layers. The inverse cascade transfers\nenergy from smaller to larger scales thus allowing for tur-\nbulence self-organization. This is in contrast with three-\ndimensional (3D) turbulence where energy is nonlinearly\ntransferred towards small scales (direct cascade).\nReal physical layers di\u000ber from the ideal 2D model\nsince they have \fnite depths and non-zero dissipation.\nThe e\u000bect of the layer thickness on turbulence driven by\n2D forcing has been studied in 3D numerical simulations\n[2, 3]. It has been shown that in \\turbulence in more\nthan two and less than three dimensions\", the injected\nenergy \rux splits into the direct and inverse parts. At\nratios of the layer depth hover the forcing scale lfabove\nh=lf\u00180:5 the inverse energy cascade is greatly reduced.\nWhen the inverse energy \rux is suppressed, the energy\ninjected into the \row is transferred towards small scales\nby the direct cascade, developing the Kolmogorov k\u00005=3\nspectrum at k >k f. This result illustrates that 2D and\n3D turbulence may coexist.\n2D/3D e\u000bects have been studied in electromagnetically\ndriven \rows using two main schemes to force the \ruid mo-\ntion. In liquid metals placed in the vertical homogeneous\nmagnetic \feld the \row is forced by applying spatially\nvarying electric \feld which generates J\u0002Bforces. In\nsuch magnetohydrodynamic (MHD) \rows 2D properties\nare enforced by the magnetic \feld and the 3D behav-\nior is restricted to a very thin Hartmann layer [4]. The\ndeviations from 2D in such \rows may be due to the \f-\nnite resistivity in very thick layers [5, 6]. Another class\nof experiments employs spatially periodic magnetic \feld\ncrossed with the constant horizontal electric current to\nproduce interacting vortices [7{9]. In this case the thick-\n\u0003Electronic address: Michael.Shats@anu.edu.auness of the Hartmann layer exceeds the layer depth and\n2D/3D e\u000bects are determined by the factors which are\ndi\u000berent from those in MHD \rows, for example, by a\ndensity strati\fcation.\nThe 3D e\u000bects are closely related to the energy dissi-\npation in the layers. This connection however is not fully\nunderstood in experiments. The measured \row damp-\ning rates are often compared with those derived from a\nquasi-2D model [10, 11] which assumes no vertical mo-\ntions within the layer. In thin layers, the agreement is\nusually within a factor of 2 [8, 12]. However in some\nexperiments a much better agreement with the quasi-2D\nmodel was observed [13]. This contradicts recent claims\nabout the intrinsic three-dimensionality of the \rows in\nthin layers of electrolytes [14, 15]. There is a need to\nclarify this.\nPhysical three-dimensionality of the \row is determined\nby the amount of 3D motion in the layer. This motion\nmay naturally develop in the layer, as in [3], but it can\nalso be injected into the \row by non-2D forcing or it\ncan be generated by the shear-driven instabilities in the\nboundary layer. In this case, the critical layer thickness\ncannot be used as a practical criterion of the 2D/3D tran-\nsition since it will vary depending on the source of 3D\nmotion. The transition from 2D to 3D, which marks a\nfundamental change in the energy transfer, needs to be\ncharacterized quantitatively, in other words, it is neces-\nsary to \fnd a measure of the \row dimensionality which\nwould help to predict turbulence behavior.\nIn this Letter we show that eddy viscosity increases\ndamping in \fnite-depth \ruid layers compared with the\nquasi-2D model prediction. This increase can be used\nas the measure of the \row dimensionality which allows\nto evaluate the likelihood of the inverse energy cascade\nand of turbulence self-organization. We also show that\nthe increased degree of three-dimensionality leads to the\nsuppression of the turbulent cascades.\nIn these experiments turbulence is generated via the in-\nteraction of a large number of electromagnetically driven\nvortices [9, 16, 17]. The electric current \rowing through a\nconducting \ruid layer interacts with the spatially variablearXiv:1012.1371v1  [physics.flu-dyn]  7 Dec 20102\nvertical magnetic \feld produced by arrays of magnets\nplaced under the bottom. In this paper we use a 30 \u000230\narray of magnetic dipoles (8 mm apart) for the turbu-\nlence studies requiring large statistics. For the studies of\nvertical motions, a 6 \u00026 array of larger magnets (25 mm\nseparation) is used. The \row is visualized using seeding\nparticles, which are suspended in the \ruid, illuminated\nusing a horizontal laser slab and \flmed from above. Par-\nticle image velocimetry (PIV) is used to derive turbulent\nvelocity \felds. The \row is generated either in a single\nlayer of electrolyte ( Na2SO4water solution), or in two\nimmiscible layers of \ruids (electrically neutral heavier liq-\nuid at the bottom, electrolyte on top). Shortly after the\ncurrent is switched on, J\u0002Bdriven vortices interact with\neach other forming complex turbulent motion character-\nized by a broad wave number spectrum. The steady state\nis reached within tens of seconds.\nTo study vertical motions in single electrolyte layers,\nvertical laser slabs are used to illuminate the \row in the\ny\u0000zplane. Streaks of the seeding particles within the\nslab are \flmed with the exposure time of 1 s. Quantita-\ntive measurements of the horizontal and vertical veloci-\nties are performed using defocusing PIV technique. This\ntechnique, was \frst described in [18], but had never been\nused in turbulence studies. It allows measurements of 3D\nvelocity components of seeding particles using a single\nvideo camera with a multiple pinhole mask (three pin-\nholes constituting a triangle are used here). A schematic\nof the method is shown in Fig. 1. An image of a particle\nplaced in the reference plane at z=0 (where the parti-\ncle is in focus) corresponds to a single dot in the image\nplane. As the particle moves vertically away from the\nreference plane, the light passes through each pinhole in\nthe mask and reaches three di\u000berent positions on the im-\nage plane. The distances between the triangle vertices\nin the image plane are used to decode z-positions of the\nparticles. The xy-components of velocity are determined\nusing a PIV/PTV hybrid algorithm to match particle\npairs from frame to frame. This process is illustrated in\nFig. 1. The technique allows to resolve vertical veloci-\nties above<Vz>RMS\u00150:5 mm/s. The imaged area in\nthis experiment is 5 \u00025 cm2. On average about 50 par-\nticles (triangles) are tracked in two consecutive frames.\nDerived velocities are then averaged over about 100 of\nthe frame pairs to generate converged statistics of the\nmean-square-root velocities <Vx;y;z>RMS.\nFigures 2(a-c) show particle streaks and corresponding\nvertical velocity pro\fles Vz(z) for di\u000berent layer depths.\nTo keep forcing approximately constant, the electric cur-\nrent is increased proportionally to the layer thickness\n(constant current density). To obtain better vertical spa-\ntial resolution, a 6 \u00026 array of larger magnets is used.\nFor the layers thicknesses of up to 30 mm, a range of\nh=lf= 0:2\u00001:2 is achieved. Particle streaks show reason-\nably 2D motion in a thin (5 mm) layer, Fig. 2(a). Vertical\nvelocity is small over most of the layer thickness and is\nz=zImage Plane\nLensMask\nLf\nReference Plane\nz=0m m\nz=3m m\nz=5m mz=0\n026810\n0 2 4 6 81 04\nx(mm)y(mm)(a)\n(b)FIG. 1: Schematic of the defocusing particle image velocime-\ntry technique.\nclose to the resolution of the technique, <Vz>RMS\u00180:5\nmm/s. As the layer thickness is increased, 3D motions\ndevelop. The corresponding vertical velocities increase\nup to\u00184 mm/s, Figs. 2(b,c). Fig. 2(d) shows the ra-\ntio of vertical to horizontal velocities as a function of\nthe normalized layer thickness. In single layers this ra-\ntio increases approximately linearly with h=lfreaching\nover< V z> = < V x;y>= 0:3 ath=lf= 0:8. In\nstrati\fed double layers this ratio is substantially smaller,\n< Vz> = < V x;y>\u00140:08 (solid squares in Fig. 2(d)),\nsuggesting that the \row in a double layer con\fguration\nis much closer to 2D.\nIn the absence of 3D motions, the \row in the layer\nis damped due to molecular viscosity. A decay of hori-\nzontal velocity Vx;y(z;t) in the quasi-2D \row due to the\nbottom friction is described by the di\u000busive type equa-\ntion@Vx;y=@t=\u0017@2Vx;y=@z2, which together with the\nboundary conditions Vx;y(z= 0;t) = 0 and @Vx;y(z=\nh;t)=@z= 0 gives the characteristic inverse time of the\nenergy decay, e.g. [10]:\n\u000bL=\u0017\u00192=2h2: (1)\nHere\u0017is the kinematic viscosity.\nThe onset of 3D turbulent eddies in thicker layers\nshould lead to a vertical \rux of horizontal momentum\nand faster dissipation of the \row. Such a \rux is related\nto the mean vertical velocity gradient @Vx;y=@z[19]:\n<~Vx;y~Vz>=\u0000K@Vx;y\n@z: (2)\nHereKis the eddy (turbulent) viscosity coe\u000ecient. By\nassuming that \ructuations of vertical and horizontal ve-3\n012345\n0123Vz(mm s )-1h(mm)hl/ = 0.2f(a)\n05101520\n0123hl/ = 0.8fh(mm)\nVz(mm s )-1(c)\n051015\n0123hl/ = 0.6f(b)\nVz(mm s )-1h(mm)\n00.10.20.30.4\n0.0 0.5 1.0 1.5h/lf(d) V/zVx,y\n01234\n0 0.5 1 1.5h/lf(e) /c97/c97t/L\nFIG. 2: Particle streaks \flmed with an exposure time of\n1 s (top panels) and the distribution of the vertical velocity\n\ructuations (rms) over the layer thickness (bottom panels) in\nsingle layers: (a) h= 5 mm; (b) h= 15 mm; (c) h= 20\nmm. (d) Ratio of rms vertical to the rms horizontal veloc-\nity as a function of the normalized layer thickness h=lfin\na single (open circles) and in a double (solid squares) layer\ncon\fgurations. (e) \u000bt=\u000bLversush=lf.\nlocities are well correlated, we can estimate the eddy\nviscosity coe\u000ecient using the defocusing PIV data as\nK\u0019<~Vx;y>< ~Vz>(@Vx;y=@z)\u00001. Then the damp-\ning rate can be estimated using the contribution of both\nmolecular and the eddy viscosities, \u000bt= (\u0017+K)\u00192=2h2.\nThe ratio of thus calculated damping rate to the linear\ndamping\u000bL(1) is shown in Fig. 2(e).\nThe damping should become anomalous ( \u000bt=\u000bL>1)\nabove some critical layer thickness of h=lf\u00190:3. Accord-\ning to Fig. 2(e) this anomaly should increase linearly with\nthe increase in h=lf.\nDirect measurements of damping were performed to\ntest that eddy viscosity increases the dissipation above\nits quasi-2D value (1) in layers thicker than h=lf>0:2.\nThe \row is forced by a 30 \u000230 magnet array . The bot-\ntom drag is derived from the energy decay of the steady\n\row. After forcing is switched o\u000b, the mean \row energy\nexponentially decays in time with a characteristic time\nconstant\u000b, as shown in Fig. 3(a). We compare the en-\n02468\n0 5 10 15 20t(s)E0(10 m s )-4 2 -2(a)\nt0E=E t () e0-( t - 0)/c97t\n01234567\n0 0.5 1 1.5hl/fa/D=/c97/c97L(b)\n012345\n0 0.06 0.12hl/f(c) a/D=/c97/c97LFIG. 3: (a) Decay of the \row energy in a single layer, h=\n10 mm; (b) Energy damping rate normalized by the viscous\nquasi-2D damping rate aD=\u000b=\u000bL, as a function of h=lf.\nOpen circles refer to single layers, solid squares were obtained\nin the double layer con\fgurations. (c) The damping anomaly\ncoe\u000ecientaDversush=lffor the case of a strong large-scale\nvortex (100 mm diameter, Vmax\nx;y = 16 mm/s).\nergy damping rate measured in a single layer of di\u000berent\ndepths with the linear damping rate. Fig. 3(b) shows\nthe anomaly coe\u000ecient aD=\u000b=\u000bLas a function of the\nnormalized layer thickness h=lf. In the thinnest layer\n(h\u00191:7 mm,h=lf\u00190:21) the damping rate coincides\nwith the linear damping rate (1). However for thicker\nlayers the damping anomaly is higher, such that aDin-\ncreases linearly with hreachingaD= 6 ath=lf= 1:25.\nMeasurements of the damping show that the anomaly\ncoe\u000ecient aDin Fig. 3(b) agrees very well with the\nanomaly estimated using the eddy viscosity derived from\n(2), Fig. 2(e). In the double layer experiments however,\naDis substantially lower, as shown by the solid squares in\nFig. 3(b). This is not surprising in the light of the result\nof Fig. 2(d) (solid squares) which shows substantially less\n3D motion in double layers.\nThe above results are related to low forcing levels,\nwhen 3D eddies are generated due to the \fnite layer\nthickness, as in [3]. However, electromagnetic forcing,\nwhich is maximum near the bottom in the single layer\nexperiments (magnets underneath the \ruid cell), may\ninject 3D eddies into the \row from the bottom bound-\nary layer at higher forcing levels. Figure 3(c) shows the\ndamping anomaly coe\u000ecient aDmeasured in the \row\ndriven by a single strong large magnetic dipole. A single4\nS3(10 m s )-7 3 -3\n-1123\n0\n0.02 0.04 0.06l(m)h=3m m\nh=1 0m m\nFIG. 4: Third-order structure functions measured in a thin\nlayer,h= 3 mm (solid squares), and in a thick layer h= 10\nmm (open diamonds). The forcing scale lf\u00198 mm.\nlarge-scale vortex is produced, whose diameter is about\n100 mm and the maximum horizontal velocity is about\n16 mm/s. As the layer thickness is increased from 2 to\n10 mm (h=lf= 0:02\u00000:1) while keeping the current\ndensity constant, the anomaly coe\u000ecient increases up to\naD= 3:6 due to the increase in the vertical velocity \ruc-\ntuations. Thus, turbulent bottom drag may occur in rel-\natively thin layers at stronger forcing.\nNow we test if the increased three-dimensionality, as\ncharacterized by aD, leads to the suppression of the in-\nverse energy cascade. The inverse energy cascade can be\ndetected by measuring the third-order structure function\nS3and by using the Kolmogorov \rux relation which pre-\ndicts linear dependence of S3on the separation distance\nl,S3=\u000fl. Here\u000fis the energy \rux in k-space. It has\nbeen shown that in thin strati\fed layers S3is positive\nand it is a linear function of l, as expected for 2D turbu-\nlence [9]. Figure 4 shows third-order structure functions\nmeasured in a single layer of electrolyte for two layer\ndepths,h= 3 and 10 mm. In the 3 mm layer, S3is a\npositive linear function of l, while in the 10 mm layer\nS3is much smaller, indicating very low energy \rux in\nthe inverse energy cascade. The damping anomaly in the\n3 mm layer is aD\u00192, while for the 10 mm layer it is\nhigh,aD\u00195. Since in this experiment, the forcing is\n2D and it is relatively weak (no secondary instabilities in\nthe boundary layer), this result is in agreement with nu-\nmerical simulations [3] which show strong suppression of\nthe inverse energy cascade above h=lf\u00150:5. The 3 mm\nlayer corresponds to h=lf\u00190:38, while for the 10 mm\nlayerh=lf\u00191:25. We do not observe however any sig-\nnatures of the direct energy cascade range, Ek/k\u00005=3\natk > k fin the 10 mm layer. Instead, the spectrum is\nmuch steeper than the usual k\u00003enstrophy range. This\nis probably due to the fact that the Reynolds numberin this experiment is not su\u000ecient to sustain 3D direct\nturbulent cascade.\nSummarizing, we demonstrate for the \frst time that\nincreased three-dimensionality of \rows in layers can be\ncharacterized by the anomalous damping coe\u000ecient aD.\nWe show that the increase in aDcorrelates with the sup-\npression of the inverse energy cascade. On the other\nhand, a strong reduction in aD, which can be achieved in\nthe double layer con\fguration, correlates well with the\nobservation of the inverse energy cascade and spectral\ncondensation of turbulence into a \row coherent over the\nentire domain [7, 9, 16].\nThe authors are grateful to H. Punzmann and V. Stein-\nberg for useful discussions. This work was supported\nby the Australian Research Council's Discovery Projects\nfunding scheme (DP0881544).\n[1] R. Kraichnan, Phys. Fluids 10, 1417 (1967).\n[2] L.M. Smith, J. R. Chasnov and F. Wale\u000be, Phys. Rev.\nLett77, 2467 (1996).\n[3] A. Celani, S. Musacchio and D. Vincenzi, Phys. Rev. Lett.\n104, 184506 (2010).\n[4] J. Sommeria, J. Fluid Mech. 170,139 (1986)\n[5] A. Thess and O. Zikanov, J. Fluid Mech. 579,383 (2007)\n[6] R. Klein and A. Potherat, Phys. Rev. Lett. 104, 34502\n(2010).\n[7] J. Paret & P. Tabeling, Phys. Fluids 10, 3126 (1998).\n[8] G. Bo\u000betta, A. Cenedese, S. Espa and S. Musacchio, Eu-\nrophys. Lett. 71, 590 (2005).\n[9] H. Xia, M. Shats and G. Falkovich, Phys. Fluids 21,\n125101 (2009).\n[10] F.V. Dolzhanskii, V.A. Krymov and D.Yu. Manin, J.\nFluid Mech 241, 705(1992).\n[11] A.E. Hansen, D. Marteau and P. Tabeling, Phys. Rev. E\n58, 7261 (1998).\n[12] S. Danilov, F.V. Dolzhanskii, V.A. Dovzhenko and V.A.\nKrymov, Phys. Rev. E 65, 036316 (2002).\n[13] H.J.H. Clercx, G.J.F. van Heijst and M.L. Zoeteweij,\nPhys. Rev. E 67, 066303 (2003).\n[14] R. A. D. Akkermans, A. R. Cieslik, L. P. J. Kamp, R. R.\nTrieling, H. J. H. Clercx, and G. J. F. van Heijst, Phys.\nFluids ,20, 116601(2008)\n[15] R. A. D. Akkermans, L. P. J. Kamp, H. J. H. Clercx, and\nG. J. F. van Heijst, Phys. Rev. E ,82, 026314 (2010)\n[16] M.G. Shats, H. Xia and H. Punzmann, Phys. Rev. E 71,\n046409 (2005).\n[17] H. Xia, H. Punzmann, G. Falkovich and M.G. Shats,\nPhys. Rev. Lett. 101, 194504 (2008).\n[18] C.E. Willert and M. Gharib, Experiments in Fluids 10,\n181 (1991).\n[19] Glossary of meteorology, ed. E. R. Huschke, Boston,\nAmerican Meteorological Society, 1959."    },    {        "title": "2206.04899v1.Spin_Pumping_into_Anisotropic_Dirac_Electrons.pdf",        "content": "Spin Pumping into Anisotropic Dirac Electrons\nTakumi Funato1;2, Takeo Kato3, Mamoru Matsuo2;4;5;6\n1Center for Spintronics Research Network, Keio University, Yokohama 223-8522, Japan\n2Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China.\n3Institute for Solid State Physics, The University of Tokyo, Kashiwa, Japan\n4CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan and\n6RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: June 13, 2022)\nWe study spin pumping into an anisotropic Dirac electron system induced by microwave irra-\ndiation to an adjacent ferromagnetic insulator theoretically. We formulate the Gilbert damping\nenhancement due to the spin current \rowing into the Dirac electron system using second-order\nperturbation with respect to the interfacial exchange coupling. As an illustration, we consider the\nanisotropic Dirac system realized in bismuth to show that the Gilbert damping varies according to\nthe magnetization direction in the ferromagnetic insulator. Our results indicate that this setup can\nprovide helpful information on the anisotropy of the Dirac electron system.\nI. INTRODUCTION\nIn spintronics, spin currents are crucial in using elec-\ntrons' charge and spin. Spin pumping, the spin current\ngeneration of conduction electrons from nonequilibrium\nmagnetization dynamics at magnetic interfaces, is a pop-\nular method for generating and manipulating spin cur-\nrents. In previous experimental reports on spin pumping,\nthe enhancement of Gilbert damping in ferromagnetic\nresonance (FMR) was observed due to the loss of angu-\nlar momentum associated with the spin current injection\ninto the nonmagnetic layer adjacent to the ferromagnetic\nlayer1{9. Mizukami et al. measured the enhancement of\nthe Gilbert damping associated with the adjacent non-\nmagnetic metal. They reported that the strong spin-orbit\ncoupling in the nonmagnetic layer strictly a\u000bected the\nenhancement of the Gilbert damping3{5. Consequently,\nelectric detection by inverse spin Hall e\u000bect, in which the\ncharge current is converted from the spin current, led to\nspin pumping being used as an essential technique for\nstudying spin-related phenomena in nonmagnetic mate-\nrials10{24. Saitoh et al. measured electric voltage in a\nbilayer of Py and Pt under microwave application. They\nobserved that charge current converted because of inverse\nspin Hall e\u000bect from spin current injected by spin pump-\ning11.\nIn the \frst theoretical report on spin pumping, Berger\npredicted an increase in Gilbert damping due to the spin\ncurrent \rowing interface between the ferromagnetic and\nnonmagnetic layers25,26. Tserkovnyak et al. calculated\nthe spin current \rowing through the interface27{29based\non the scattering-matrix theory and the picture of adi-\nabatic spin pumping30{32. They introduced a complex\nspin-mixing conductance that characterizes spin trans-\nport at the interfaces based on spin conservation and no\nspin loss. The spin mixing conductance can represent\nthe spin pumping-associated phenomena and is quanti-\ntatively evaluated using the \frst principle calculation33.\nNevertheless, microscopic analysis is necessary to under-stand the detailed mechanism of spin transport at the in-\nterface34{44. It was clari\fed that spin pumping depends\non the anisotropy of the electron band structure and spin\ntexture. Spin pumping is expected to be one of the probes\nof the electron states41{44.\nBismuth has been extensively studied because of its at-\ntractive physical properties, such as large diamagnetism,\nlargeg-factor, high e\u000ecient Seebeck e\u000bect, Subrikov-de\nHaas e\u000bect, and de Haas-van Alphen e\u000bect45,46. The\nelectrons in the conduction and valence bands near the\nL-point in bismuth, which contribute mainly to the vari-\nous physical phenomena, are expressed as e\u000bective Dirac\nelectrons. Thus, electrons in bismuth are called Dirac\nelectrons45{47. The doping antimony to bismuth is known\nto close the gap and makes it a topological insulator48,49.\nBecause of its strong spin-orbit interaction, bismuth has\nattracted broad attention in spintronics as a high e\u000ecient\ncharge-to-spin conversion material50{55. The spin current\ngeneration at the interface between the bismuth oxide\nand metal has been studied since a signi\fcant Rashba\nMicrowaveDirac electron\nsystem\nInterfacial\nexchange\nFerromagnetic\ninsulator\nFIG. 1. Schematic illustration of a bilayer system composed\nof the Dirac electron system and ferromagnetic insulator. The\napplied microwave excited precession of the localized spin in\nthe ferromagnetic insulator and spin current is injected into\nthe Dirac electron system.arXiv:2206.04899v1  [cond-mat.mes-hall]  10 Jun 20222\nspin-orbit interaction appears at the interface56. The\nspin injection into bismuth was observed due to spin\npumping from yttrium iron garnet or permalloy57{59.\nNevertheless, microscopic analysis of spin pumping into\nbismuth has not been performed. The dependence of the\nspin pumping on the crystal and band structure of bis-\nmuth remains unclear.\nThis study aims at a microscopic analysis of spin in-\njection due to spin pumping into an anisotropic Dirac\nelectron system, such as bismuth, and investigates the\ndependence of spin pumping on the band structure. We\nconsider a bilayer system comprising an anisotropic Dirac\nelectron system and a ferromagnetic insulator where a\nmicrowave is applied (see Fig. 1). The e\u000bect of the inter-\nface is treated by proximity exchange coupling between\nthe Dirac electron spins and the localized spins of the\nferromagnetic insulator34{44. We calculate the Gilbert\ndamping enhancement due to spin pumping from the fer-\nromagnetic insulator into the Dirac electron system up to\nthe second perturbation of the interfacial exchange cou-\npling. For illustarion, we calculate the enhancement of\nthe Gilbert damping for an anisotropic Dirac system in\nbismuth.\nThis paper is organized as follows: Sec. II describes\nthe model. Sec. III shows the formulation of the Gilbert\ndamping enhancement and discuss the e\u000bect of the inter-\nfacial randomness on spin pumping. Sec. IV summarizes\nthe results and demonstration of the Gilbert damping\nenhancement in bismuth. Sec. V presents the conclu-\nsion. The Appendices show the details of the calcula-\ntion. Appendix A de\fnes the magnetic moment of elec-\ntrons in a Dirac electron system. Appendix B provides\nthe detailed formulation of the Gilbert damping modu-\nlation, and Appendix C presents the detailed derivation\nof Gilbert damping modulation.\nII. MODEL\nWe consider a bilayer system composed of an\nanisotropic Dirac electron system and a ferromagnetic\ninsulator under a static magnetic \feld. We evaluate a\nmicroscopic model whose Hamiltonian is given as\n^HT=^HD+^HFI+^Hex; (1)\nwhere ^HD,^HFI, and ^Hexrepresent an anisotropic Dirac\nelectron system, a ferromagnetic insulator, and an inter-\nfacial exchange interaction, respectively.\nA. Anisotropic Dirac system\nThe following Wol\u000b Hamiltonian models the\nanisotropic Dirac electron system46,47,50:\n^HD=X\nkcy\nk(\u0000~k\u0001v\u001a2+ \u0001\u001a3)ck; (2)where 2\u0001 (6= 0) is the band gap, cy\nk(ck) is the electrons'\nfour-component creation (annihilation) operator, and v\nis the velocity operator given by vi=P\n\u000bwi\u000b\u001b\u000bwith\nwi\u000bbeing the matrix element of the velocity operator.\n\u001b= (\u001bx;\u001by;\u001bz) are the Pauli matrices in the spin space\nand\u001a= (\u001a1;\u001a2;\u001a3) are the Pauli matrices specifying the\nconduction and valence bands.\nFor this anisotropic Dirac system, the Matsubara\nGreen function of the electrons is given by\ngk(i\u000fn) =i\u000fn+\u0016\u0000~~k\u0001\u001b\u001a2+ \u0001\u001a3\n(i\u000fn+\u0016)2\u0000\u000f2\nk; (3)\nwhere\u000fn= (2n+ 1)\u0019=\fis the fermionic Matsubara fre-\nquencies with nbeing integers, \u0016(>\u0001) is the chem-\nical potential in the conduction band ~kis de\fned by\n~k\u0001\u001b=~k\u000b\u001b\u000b=k\u0001v, and\u000fkis the eigenenergy given\nby\n\u000fk=p\n\u00012+ (~kiwi\u000b)2=q\n\u00012+~2~k2: (4)\nThe density of state of the Dirac electrons per unit cell\nper band and spin is givcen by\n\u0017(\u000f) =n\u00001\nDX\nk;\u0015\u000e(\u000f\u0000\u0015\u000fk); (5)\n=j\u000fj\n2\u00192~3s\n\u000f2\u0000\u00012\n\u00013det\u000bij\u0012(j\u000fj\u0000\u0001); (6)\nwherenDis the number of unit cells in the system and \u000bij\nis the inverse mass tensor near the bottom of the band,\nwhich characterize the band structure of the anisotropic\nDirac electron system:\n\u000bij=1\n~2@2\u000fk\n@ki@kj\f\f\f\f\nk=0=1\n\u0001X\n\u000bwi\u000bwj\u000b: (7)\nThe spin operator can be de\fned as\n^sq=X\nkcy\nk\u0000q=2sck+q=2; (8)\nsi=m\n\u0001Mi\u000b\u001a3\u001b\u000b;(i=x;y;z ); (9)\nwhereMi\u000bare the matrix elements of the spin magnetic\nmoment given as50,51\nMi\u000b=\u000f\u000b\f\r\u000fijkwi\fwj\r=2: (10)\nThe detailed derivation of the spin magnetic moment can\nbe found in Appendix A.\nB. Ferromagnetic insulator\nThe bulk ferromagnetic insulator under a static mag-\nnetic \feld is described by the quantum Heisenberg model\nas\n^HFI=\u00002JX\nhi;jiSi\u0001Sj\u0000g\u0016BhdcX\niSX\ni; (11)3\nFIG. 2. Relation between the original coordinates ( x;y;z ) and\nthe magnetization-\fxed coordinates ( X;Y;Z ). The direction\nof the ordered localized spin hSi0is \fxed to the X-axis.\u0012is\nthe polar angle and \u001eis the azimuthal angle.\nwhereJis an exchange interaction, gis g-factor of the\nelectrons,\u0016Bis the Bohr magnetization, and hi;jirepre-\nsents the pair of nearest neighbor sites. Here, we have in-\ntroduced a magnetization-\fxed coordinate ( X;Y;Z ), for\nwhich the direction of the ordered localized spin hSi0is\n\fxed to the X-axis. The localized spin operators for the\nmagnetization-\fxed coordinates are related to the ones\nfor the original coordinates ( x;y;z ) as\n0\n@Sx\nSy\nSz1\nA=R(\u0012;\u001e)0\n@SX\nSY\nSZ1\nA; (12)\nwhereR(\u0012;\u001e) =Rz(\u001e)Ry(\u0012) is the rotation matrix com-\nbining the polar angle \u0012rotation around the y-axisRy(\u0012)\nand the azimuthal angle \u001erotation around the z-axis\nRz(\u001e), given by\nR(\u0012;\u001e) =0\n@cos\u0012cos\u001e\u0000sin\u001esin\u0012cos\u001e\ncos\u0012sin\u001ecos\u001esin\u0012sin\u001e\n\u0000sin\u0012 0 cos \u00121\nA:(13)\nBy applying the spin-wave approximation, the spin op-\nerators are written as S\u0006\nk=SY\nk\u0006iSZ\nk=p\n2Sbk(by\nk) and\nSX\nk=S\u0000by\nkbkusing magnon creation/annihilation op-\nerators,by\nkandbk. Then, the Hamiltonian is rewritten\nas\n^HFI=X\nk~!kby\nkbk; (14)\nwhere ~!k=Dk2+~!0withD=zJSa2being the spin\nsti\u000bness and zbeing the number of the nearest neighbor\nsites, and ~!0=g\u0016Bhdcis the Zeeman energy.C. Interfacial exchange interaction\nThe proximity exchange coupling between the electron\nspin in the anisotropic Dirac system and the localized\nspin in the ferromagnetic insulator is modeled by\n^Hex=X\nq;k(Tq;k^s+\nqS\u0000\nk+ h.c.); (15)\nwhereTq;kis a matrix element for spin transfer through\nthe interface and ^ s\u0006\nq= ^sY\nq\u0006i^sZ\nqare the spin ladder\noperators of the Dirac electrons. According to the re-\nlation between the original coordinate ( x;y;z ) and the\nmagnetization-\fxed coordinate ( X;Y;Z ), the spin oper-\nators of the Dirac electrons are expressed as\n0\n@sX\nsY\nsZ1\nA=R\u00001(\u0012;\u001e)0\n@sx\nsy\nsz1\nA; (16)\nwhereR\u00001(\u0012;\u001e) =Ry(\u0012)Rz(\u0000\u001e) is given by\nR\u00001(\u0012;\u001e) =0\n@cos\u0012cos\u001ecos\u0012sin\u001e\u0000sin\u0012\n\u0000sin\u001e cos\u001e 0\nsin\u0012cos\u001esin\u0012sin\u001ecos\u00121\nA:(17)\nThe spin ladder operators are given by\ns+=m\n\u0001aiMi\u000b\u001b\u000b; s\u0000=m\n\u0001a\u0003\niMi\u000b\u001b\u000b; (18)\nwhereai(i=x;y;z ) are de\fned by\n0\n@ax\nay\naz1\nA=0\n@\u0000sin\u001e+isin\u0012cos\u001e\ncos\u001e+isin\u0012sin\u001e\nicos\u00121\nA: (19)\nIII. FORMULATION\nApplying a microwave to the ferromagnetic insulator\nincludes the localized spin's precession. The Gilbert\ndamping constant can be read from the retarded magnon\nGreen function de\fned by\nGR\nk(!) =\u0000i\n~Z1\n0dtei(!+i\u000e)th[S+\nk(t);S\u0000\nk]i; (20)\nwithS+\nk(t) =ei^HT=~S+\nke\u0000i^HT=~being the Heisenberg\nrepresentation of the localized spin, since one can prove\nthat the absorption rate of the microwave is proportional\nto ImGR\nk=0(!) (see also Appendix B). By considering the\nsecond-order perturbation with respect to the matrix el-\nement for the spin transfer Tq;k, the magnon Green func-\ntion is given by34{44\nGR\n0(!) =2S=~\n(!\u0000!0) +i(\u000b+\u000e\u000b)!: (21)\nHere, we introduced a term, i\u000b!, in the denominator\nto express the spin relaxation within a bulk FI, where4\n\u000bindicates the strength of the Gilbert damping. The\nenhancement of the damping, \u000e\u000b, is due to the adjacent\nDirac electron system, calculated by\n\u000e\u000b=2S\n~!X\nqjTq;0j2Im\u001fR\nq(!); (22)\nwhere\u001fR\nq(!) is the retarded component of the spin sus-\nceptibility (de\fned below). We assume that the FMR\npeak described by Im GR\nk=0(!) is su\u000eciently sharp, i.e.,\n\u000b+\u000e\u000b\u001c1. Then, the enhancement of the Gilbert damp-\ning can be regarded as almost constant around the peak\n(!'!0), allowing us to replace !in\u000e\u000bwith!0.\nThe retarded component of the spin susceptibility for\nthe Dirac electrons:\n\u001fR\nq(!) =i\n~Z1\n\u00001dtei(!+i\u000e)t\u0012(t)h[s+\nq(t);s\u0000\n\u0000q]i: (23)\nThe retarded component of the spin susceptibility is\nderived from the following Matsubara Green function\nthrough analytic continuation i!l!~!+i\u000e:\n\u001fq(i!l) =Z\f\n0d\u001cei!l\u001ch^s+\nq(\u001c)^s\u0000\n\u0000qi; (24)\nwhere!l= 2\u0019l=\f is the bosonic Matsubara frequency\nwithlbeing integers. According to Wick's theorem,\nthe Matsubara representation of the spin susceptibility\nis given by\n\u001fq(i!l)\n=\u0000\f\u00001X\nk;i\u000fntr[s+gk+q(i\u000fn+i!l)s\u0000gk(i\u000fn)];(25)\nwhereP\ni\u000fnindicates the sum with respect to the\nfermionic Matsubara frequency, \u000fn= (2\u0019+ 1)n=\f. The\nimaginary part of the spin susceptibility is given by\nIm\u001fR\nq(!) =\u0000\u0019F(\u0012;\u001e)X\nkX\n\u0015;\u00150=\u0006\u00141\n2+\u0015\u00150\n62\u00012+\u000f2\nk\n\u000fk\u000fk+q\u0015\n\u0002h\nf(\u00150\u000fk+q)\u0000f(\u0015\u000fk)i\n\u000e(~!\u0000\u00150\u000fk+q+\u0015\u000fk);(26)\nwheref(\u000f) = (e\f(\u000f\u0000\u0016)+ 1)\u00001is the Fermi distribution\nfunction,\u0015=\u0006is a band index (see Fig. 3), and F(\u0012;\u001e)\nis the dimensionless function which depends on the di-\nrection of the ordered localized spin, de\fned by\nF(\u0012;\u001e) =\u00122m\n\u0001\u00132X\n\u000baiMi\u000ba\u0003\njMj\u000b: (27)\nFor detailed derivation, see Appendix C.\nIn this paper, we model the interfacial spin transfer as a\ncombination of the clean and dirty processes. The former\ncorresponds to the momentum-conserved spin transfer\nand the latter to the momentum-nonconserved one41,44.\nBy averaging over the position of the localized spin at\nFIG. 3. Schematic illustration of the band structure of the\nanisotropic Dirac electron system. The red band represents\nthe conduction band with \u0015= +, and the blue band repre-\nsents the valence band with \u0015=\u0000. The chemical potential\nis in the conduction band.\nthe interface, we can derive the matrix elements of the\ninterfacial spin-transfer process as\njTq;0j2=T2\n1\u000eq;0+T2\n2; (28)\nwhereT1andT2are the averaged matrix elements con-\ntributing to the clean and dirty processes, respectively.\nThen, the enhancement of the Gilbert damping is given\nby\n\u000e\u000b=2S\n~!F(\u0012;\u001e)n\nT1Im ~\u001fR\nuni(!0) +T2Im ~\u001fR\nloc(!0)o\n;\n(29)\nwhere\u001fR\nuni(!) and\u001fR\nuni(!) are the local and uniform spin\nsusceptibilities de\fned by\n~\u001fR\nloc(!0) =F\u00001(\u0012;\u001e)X\nq\u001fR\nq(!0); (30)\n~\u001fR\nuni(!0) =F\u00001(\u0012;\u001e)\u001fR\n0(!0); (31)\nrespectively. From Eq. (26), their imaginary parts are\ncalculated as\nIm ~\u001fR\nloc(!0) =\u0000\u0019n2\nDZ\nd\u000f\u0017(\u000f)\u0017(\u000f+~!0)\n\u0002\u00141\n2+2\u00012+\u000f2\n6\u000f(\u000f+~!0)\u0015h\nf(\u000f+~!0)\u0000f(\u000f)i\n;\n(32)\nIm ~\u001fR\nuni(!0) =\u0000\u0019nD\u0017\u0000~!0\n2\u0001~2!2\n0\u00004\u00012\n3~2!2\n0\n\u0002h\nf(~!0\n2)\u0000f(\u0000~!0\n2)i\n: (33)\nThe enhancement of the Gilbert damping, \u000e\u000b, depends\non the direction of the ordered localized spin through the5\nFIG. 4. FMR frequency dependence of the (a) local\nand (b) uniform spin susceptibilities. The local spin sus-\nceptibility is normalized by \u0019n2\nD\u00172\n0and scaled by 106, and\nthe uniform spin susceptibility is normalized by \u0019nD\u00170with\n\u00170\u00111=2\u00192~3p\ndet\u000bij. Note that kBis the Boltzmann con-\nstant. The line with kBT=\u0001 = 0:001 is absent in (a) because\nthe local spin susceptibility approaches zero at low tempera-\nture.\ndimensionless function F(\u0012;\u001e) regardless of the interfa-\ncial condition.\nBy contrast, the FMR frequency dependence of \u000e\u000bre-\n\rects the interfacial condition; for a clean interface, it is\ndetermined mainly by Im \u001fR\nuni(!0), whereas for a dirty\ninterface, it is determined by Im \u001fR\nloc(!0). The FMR fre-\nquency dependence of the local and uniform spin sus-\nceptibilities, Im \u001fR\nloc(!0) and Im\u001fR\nuni(!0), are plotted in\nFigs. 4 (a) and (b), respectively. The local and uniform\nspin susceptibilities are normalized by \u0019n2\nD\u00172\n0and\u0019nD\u00170,\nrespectively, where \u00170\u00111=2\u00192~3p\ndet\u000bijis de\fned. In\nthe calculation, the ratio of the chemical potential to the\nenergy gap was set to \u0016=\u0001'4:61, which is the value in\nthe bismuth46. According to Fig. 4 (a), the local spin sus-\nceptibility increases linearly with the frequency !in the\nlow-frequency region. This !-linear behavior can be re-\nproduced analytically for low temperatures and ~!\u001c\u0016:\nIm ~\u001floc(!0)'~!0\u0019\n2n2\nD[\u0017(\u0016)]2\u0014\n1 +2\u00012+\u00162\n3\u00162\u0015\n:(34)Fig. 4 (b) indicates a strong suppression of the uniform\nspin susceptibility below a spin-excitation gap ( !0<2\u0016).\nThis feature can be checked by its analytic form at zero\ntemperature:\nIm ~\u001fR\nuni(!0) =\u0019nD\u0017\u0000~!0\n2\u0001~2!2\n0\u00004\u00012\n3~2!2\n0\u0012(~!0\u00002\u0016):\n(35)\nThus, the FMR frequency dependence of the enhance-\nment of the Gilbert damping depends on the interfacial\ncondition. This indicates that the measurement of the\nFMR frequency dependence may provide helpful infor-\nmation on the randomness of the junction.\nIV. RESULT\nWe consider bismuth, which is one of the anisotropic\nDirac electron systems45,46,52,60,61. The crystalline struc-\nture of pure bismuth is a rhombohedral lattice with the\nspace group of R\u00163msymmetry, see Figs. 5 (a) and (b).\nIt is reasonable to determine the Cartesian coordinate\nsystem in the rhombohedral structure using the trigonal\naxis withC3symmetry, the binary axis with C2symme-\ntry, and the bisectrix axis, which is perpendicular to the\ntrigonal and binary axes. Hereafter, we choose the x-axis\nas the binary axis, the y-axis as the bisectrix axis, and\nthez-axis as the trigonal axis. Note that the trigonal, bi-\nnary, and bisectrix axes are denoted as [0001], [1 \u0016210], and\n[10\u001610], respectively, where the Miller-Bravais indices are\nused. The bismuth's band structure around the Fermi\nsurface consists of three electron ellipsoids at L-points\nand one hole ellipsoid at the T-point. It is well known\nthat the electron ellipsoids are the dominant contribu-\ntion to the transport phenomena since electron's mass\nis much smaller than that of the hole, see Fig. 5 (c).\nTherefore, the present study considers only the electron\nsystems at the L-points. The electron ellipsoids are sig-\nni\fcantly elongated, with the ratio of the major to minor\naxes being approximately 15 : 1. Each of the three elec-\ntron ellipsoids can be converted to one another with 2 \u0019=3\nrotation around the trigonal axis. The electron ellipsoid\nalong the bisectrix axis is labeled as e1, and the other\ntwo-electron ellipsoids are labeled e2 ande3. The in-\nverse mass tensor for the e1 electron ellipsoids is given\nby\n\u000b$\ne1=0\nB@\u000b10 0\n0\u000b2\u000b4\n0\u000b4\u000b31\nCA: (36)\nThe inverse mass tensor of the electron ellipsoids e2 and\ne3 are obtained by rotating that of e1 by 2\u0019=3 rotation6\nas below:\n\u000b$\ne2;e3=1\n40\nBB@\u000b1+ 3\u000b2\u0006p\n3(\u000b1\u0000\u000b2)\u00062p\n3\u000b4\n\u0006p\n3(\u000b1\u0000\u000b2) 3\u000b1+\u000b2\u00002\u000b4\n\u00062p\n3\u000b4\u00002\u000b4 4\u000b31\nCCA:\n(37)\nLet us express the dimensionless function F(\u0012;\u001e) rep-\nresenting the localized spin direction dependence of the\ndamping enhancement on the inverse mass tensors.\nF(\u0012;\u001e) =\u00122m\n\u0001\u00132X\n\u000bh\n(sin2\u001e+ sin2\u0012cos2\u001e)M2\nx\u000b\n+(cos2\u001e+ sin2\u0012sin2\u001e)M2\ny\u000b\n+ cos2\u0012(M2\nz\u000b\u0000sin 2\u001eMx\u000bMy\u000b)\n+ sin 2\u0012Mz\u000b(Mx\u000bcos\u001e+My\u000bsin\u001e)i\n: (38)\nHere, we use the following calculations:\nX\n\u000bM2\nx\u000b=\u00012\n4(\u000byy\u000bzz\u0000\u000b2\nyz)total=\u00012\n4m2\u0016\u0014?;(39)\nX\n\u000bM2\ny\u000b=\u00012\n4(\u000bzz\u000bxx\u0000\u000b2\nzx)total=\u00012\n4m2\u0016\u0014?;(40)\nX\n\u000bM2\nz\u000b=\u00012\n4(\u000bxx\u000byy\u0000\u000b2\nxy)total=\u00012\n4m2\u0016\u0014k;(41)\nX\n\u000bMi\u000bMj\u000b=\u00012\n4(\u000bik\u000bjk\u0000\u000bij\u000bkk)total= 0;(42)\nwherei;j;k are cyclic. (\u0001\u0001\u0001)totalrepresents the summa-\ntion of the contributions of the three electron ellipsoids,\nand \u0016\u0014k, \u0016\u0014?(>0) are the total Gaussian curvature of the\nthree electron ellipsoids normalized by the electron mass\nm, given by\n\u0016\u0014k= 3m2\u000b1\u000b2; (43)\n\u0016\u0014?=3\n2m2[(\u000b1+\u000b2)\u000b3\u0000\u000b2\n4]: (44)\nHence, the dimensionless function Fis given by\nF(\u0012) = (1 + sin2\u0012)\u0016\u0014?+ cos2\u0012\u0016\u0014k: (45)\nThe results suggest that the variation of the damping\nenhancement depends only on the polar angle \u0012, which is\nthe angle between the direction of the ordered localized\nspinhSi0and the trigonal axis. It is also found that the \u0012\ndependence of the damping enhancement originates from\nthe anisotropy of the band structure. The dimensionless\nfunctionF(\u0012) is plotted in Fig. 6 by varying the ratio\nof the total Gaussian curvatures x= \u0016\u0014?=\u0016\u0014k, which cor-\nresponds to the anisotropy of the band structure. Fig-\nure 6 shows that the \u0012-dependence of the damping en-\nhancement decreases with smaller xand the angular de-\npendence vanishes in an isotropic Dirac electron system\nBinaryBisectrixTrigonal\ne�e�e�(c)\nBinary(x)(a)\nBisectrix(y)Trigonal(z)\nBinaryBisectrixTrigonal(b)FIG. 5. (a) The rhombohedral lattice structure of bismuth.\nThex-axis,y-axis, andz-axis are chosen as the binary axis\nwithC2symmetry, the bisectrix axis, and the trigonal axis\nwithC3symmetry, respectively. The yellow lines represents\nthe unit cell of the rhombohedral lattice. (b) The rhombohe-\ndral structure viewed from the trigonal axis. (c) Schematic\nillustration of the band structure at the Fermi surface. The\nthree electron ellispoids at L-points are dominant contribu-\ntion to the spin transport.\nx= 1. Bismuth is known to have a strongly anisotropic\nband structure. The magnitude of the matrix elements of\nthe inverse mass \u000b1-\u000b4was experimentally determined as\nm\u000b1= 806,m\u000b2= 7:95,m\u000b3= 349, and m\u000b4= 37:6.\nThe total Gaussian curvatures are evaluated as46\n\u0016\u0014k'1:92\u0002104; (46)\n\u0016\u0014?'4:24\u0002105: (47)\nThe ratio of the total Gaussian curvature is estimated\nasx'22:1. Therefore, the damping enhancement is\nexpected to depend strongly on the polar angle \u0012in a bi-\nlayer system composed of single-crystalline bismuth and\nferromagnetic insulator. Conversely, the \u0012-dependence of\nthe damping enhancement is considered to be suppressed\nfor polycrystalline bismuth.\nThe damping enhancement is independent of the az-\nimuthal angle \u001e. Therefore, it is invariant even on ro-\ntating the spin orientation around the trigonal axis. The\nreason is that the azimuthal angular dependence of the\ndamping enhancement cancels out when the contribu-\ntions of the three electron ellipsoids are summed over,\nalthough each contribution depends on the azimuthal an-\ngle. The azimuthal angular dependence of the damping\nenhancement is expected to remain when strain breaks\nthe in-plane symmetry. Additionally, suppose the spin\ncan be injected into each electron ellipsoid separately,\ne.g., by interfacial manipulation of the bismuth atoms.\nIn that case, the damping enhancement depends on the\nazimuthal angle of the spin orientation of the ferromag-\nnetic insulator39. This may be one of the probes of the\nelectron ellipsoidal selective transport phenomena.7\n- /2\n0 /2\ntheta1.01.52.0damping_modulation\nFIG. 6. The \u0012-dependence of the damping enhancement\nfor di\u000berent x. The ratio of the total Gaussian curvatures\nx= \u0016\u0014?=\u0016\u0014krepresents the anisotropy of the band structure.\nThe blue line with x= 22:1 corresponds to the damping en-\nhancement in single-crystalline bismuth, and the other lines\ncorrespond to that in the weakly anisotropic band structure.\nAs can be seen from the graph, the \u0012-dependence of the damp-\ning enhancement decreases as the more weakly anisotropic\nband structure, and the angular dependence turns out to van-\nish in an isotropic Dirac electron system with x= 1.\nIt is also noteworthy that the damping enhancement\nvaries according to the ordered localized spin direction\nwith both clean and dirty interfaces; that is independent\nof whether momentum is conserved in interfacial spin\ntransport. Conversely, it was reported that the spin ori-\nentation dependence of the damping enhancement due to\nthe Rashba and Dresselhaus spin-orbit interaction turned\nout to vanish by interfacial inhomogeneity42,43.\nV. CONCLUSION\nWe theoretically studied spin pumping from a ferro-\nmagnetic insulator to an anisotropic Dirac electron sys-\ntem. We calculated the enhancement of the Gilbert\ndamping in the second perturbation concerning the prox-\nimity interfacial exchange interaction by considering\nthe interfacial randomness. For illustration, we calcu-\nlated the enhancement of the Gilbert damping for an\nanisotropic Dirac system realized in bismuth. We showed\nthat the Gilbert damping varies according to the polar\nangle between the ordered spin hSi0and the trigonal axis\nof the Dirac electron system whereas it is invariant in its\nrotation around the trigonal axis. Our results indicate\nthat the spin pumping experiment can provide helpful in-\nformation on the anisotropic band structure of the Dirac\nelectron system.\nThe Gilbert damping is invariant in the rotation\naround the trigonal axis because the contributions of each\nelectron ellipsoid depend on the in-plane direction of theordered spinhSi0. Nevertheless, the total contribution\nbecomes independent of the rotation of the trigonal axis\nafter summing up the contributions from the three elec-\ntron ellipsoids that are related to each other by the C3\nsymmetry of the bismuth crystalline structure. If the spin\ncould be injected into each electron ellipsoid separately,\nit is expected that the in-plane direction of the ordered\nlocalized spin would in\ruence the damping enhancement.\nThis may be one of the electron ellipsoid selective spin in-\njection probes. The in-plane direction's dependence will\nalso appear when a static strain is applied. A detailed\ndiscussion of these e\u000bects is left as a future problem.\nACKNOWLEDGMENTS\nThe authors would like to thank A. Yamakage and Y.\nOminato for helpful and enlightening discussions. The\ncontinued support of Y. Nozaki is greatly appreciated.\nWe also thank H, Nakayama for the daily discussions.\nThis work was partially supported by JST CREST Grant\nNo. JPMJCR19J4, Japan. This work was supported by\nJSPS KAKENHI for Grants (Nos. 20H01863, 20K03831,\n21H04565, 21H01800, and 21K20356). MM was sup-\nported by the Priority Program of the Chinese Academy\nof Sciences, Grant No. XDB28000000.\nAppendix A: Magnetic moment of electrons in Dirac\nelectron system\nIn this section, we de\fne the spin operators in the\nDirac electron systems. The Wol\u000b Hamiltonian around\nthe L point is given by HD=\u001a3\u0001\u0000\u001a2\u0019\u0001v, where\nvi=P\n\u000bwi\u000b\u001b\u000bwithwi\u000bbeing the matrix component\nof the velocity vectors and \u0019=p+e\ncAis the momen-\ntum operator including the vector potential. It is rea-\nsonable to determine the magnetic moment of electrons\nin an e\u000bective Dirac system as the coe\u000ecient of the Zee-\nman term. The Wol\u000b Hamiltonian is diagonalized by the\nSchrie\u000ber-Wol\u000b transformation up to v=\u0001 as below:\nei\u0018HDe\u0000i\u0018'\u0014\n\u0001 +1\n2\u0001(\u0019\u0001v)2\u0015\n\u001a3; (A1)\nwhere\u0018=\u001a1\n2\u0001\u0019\u0001vis chosen to erase the o\u000b-diagonal\nmatrix for the particle-hole space. We can proceed cal-\nculation as follows:\n(\u0019\u0001v)2=\u0019i\u0019jwi\u000bwj\f(\u000e\u000b\f+i\u000f\u000b\f\r\u001b\r);\n= (\u0019iwi\u000b)2+i\n2\u000f\u000b\f\r\u001b\r[\u0019\u0002\u0019]i\u000fijkwj\u000bwk\f;\n= \u0001\u0012\n\u0019\u0001\u000b\u0001\u0019+~e\nc\u0001Mi\u000b\u001b\u000bBi\u0013\n; (A2)\nwhere we used ( \u0019\u0002\u0019) =e~\ncir\u0002AandMi\u000bis de\fned as\nMi\u000b=1\n2\u000f\u000b\f\r\u000fijkwj\fwk\r: (A3)8\nFinally, we obtain\nei\u0018HDe\u0000i\u0018'\u0014\n\u0001 +\u0019\u0001\u000b$\u0001\u0019\n2\u0015\n\u0000Bi\u0016s;i; (A4)\nwhere\u0016s;iis a magnetic moment of the Dirac electrons\nde\fned as\n\u0016s;i=\u0000~e\n2c\u0001Mi\u000b\u001a3\u001b\u000b=\u0000~e\n2c\u0001Mi\u000b\u0012\n\u001b\u000b0\n0\u0000\u001b\u000b\u0013\n:\n(A5)\nIn the main text, we de\fned the spin operator sas the\nmagnetic moment \u0016sdivided by the Bohr magnetization\n\u0016B=~e=2mc, i.e.,\nsi=\u0000\u0016s;i\n\u0016B=m\n\u0001Mi\u000b\u0012\n\u001b\u000b0\n0\u0000\u001b\u000b\u0013\n: (A6)\nFor an isotropic Dirac system, the matrix component is\ngiven bywi\u000b=v\u000ei\u000band Eq. (A6) reproduces the well-\nknown form of the spin operator\ns=g\u0003\n2\u0012\n\u001b0\n0\u0000\u001b\u0013\n; (A7)\nwhereg\u0003= 2m=m\u0003is the e\u000bective g-factor with m\u0003=\n\u0001=v2being e\u000bective mass.\nAppendix B: Linear Response Theory\nIn this section, we brie\ry explain how the microwave\nabsorption rate is written in terms of the uniform spincorrelation function. The Hamiltonian of an external\ncircular-polarized microwave is written as\n^Hrf=\u0000g\u0016Bhrf\n2X\ni(S\u0000\nie\u0000i!t+S+\niei!t)\n=\u0000g\u0016BhrfpnF\n2(S\u0000\n0e\u0000i!t+S+\n0ei!t); (B1)\nwherehrfis an amplitude of the magnetic \feld of the\nmicrowave, S\u0006\nkare the Fourier transformations de\fned\nas\nS\u0006\nk=1pnFX\niS\u0006\nie\u0000ik\u0001Ri; (B2)\nandRiis the position of the locazed spin i. Using the lin-\near response theory with respect to ^Hrf, the expectation\nvalue of the local spin is calculated as\nhS+\n0i!=GR\n0(!)\u0002g\u0016BhrfpnF\n2; (B3)\nwhereGR\nk(!) is the spin correlation function de\fned in\nEq. (20). Since the microwave absorption is determined\nby the dissipative part of the response function, it is\nproportional to Im GR\n0(!), that reproduces a Lorentzian-\ntype FMR lineshape. As explained in the main text, the\nchange of the linewidth of the microwave absorption, \u000e\u000b,\ngives information on spin excitation in the Dirac system\nvia the spin susceptibility as shown in Eq. (22).\nAppendix C: Spin susceptibility of Dirac electrons\nIn this section, we give detailed derivation of Eq. (26). The trace part in Eq. (25) is calculated as\ntr[s+gk+q(i\u000fn+i!l)s\u0000gk(i\u000fn)] =[(i\u000fn+i!l+\u0016)(i\u000fn+\u0016) + \u00012]tr[s+s\u0000]\u0000tr[s+~(~k+~q)\u0001\u001bs\u0000~~k\u0001\u001b]\n[(i\u000fn+i!l+\u0016)2\u0000\u000f2\nk+q][(i\u000fn+\u0016)2\u0000\u000f2\nk]; (C1)\nwhere ( ~k+~q)\u0001\u001b= (k+q)\u0001v. Using the following relations\ntr[s+s\u0000] =\u00122m\n\u0001\u00132X\n\u000baiMi\u000ba\u0003\njMj\u000b; (C2)\ntr[s+~(~k+~q)\u0001\u001bs\u0000~~k\u0001\u001b] =\u00122m\n\u0001\u00132X\n\u000b(2aiMi\u000b~~k\u000ba\u0003\njMj\f~~k\f\u0000~2~k2aiMi\u000ba\u0003\njMj\u000b); (C3)\nthe spin susceptibility is given by\n\u001fq(i!l) =\u00002F(\u0012;\u001e)X\nk\f\u00001X\ni\u000fn(i\u000fn+i!l+\u0016)(i\u000fn+\u0016) + \u00012+~2~k2=3\n[(i\u000fn+i!l+\u0016)2\u0000\u000f2\nk+q][(i\u000fn+\u0016)2\u0000\u000f2\nk]; (C4)\nwhere we dropped the terms proportional to ~k\u000b~k\f(\u000b6=\f) because they vanish after the summation with respect\nto the wavenumber k. Here, we introduced a dimensionless function, F(\u0012;\u001e) = (2m=\u0001)2P\n\u000baiMi\u000ba\u0003\njMj\u000b, which9\ndepends on the direction of the magnetization of the FI. Representing the Matsubara summation as the following\ncontour integral, we derive\n\u001fq(i!l) =\u00002F(\u0012;\u001e)X\nkIdz\n4\u0019itanh\u0012\f(z\u0000\u0016)\n2\u0013z(z+i!l) + \u00012+~2~k2=3\n[(z+i!l)2\u0000\u000f2\nk+q][z2\u0000\u000f2\nk]; (C5)\n= 2F(\u0012;\u001e)X\nkIdz\n2\u0019if(z)z(z+i!l) + \u00012+~2~k2=3\n[(z+i!l)2\u0000\u000f2\nk+q][z2\u0000\u000f2\nk]; (C6)\nWe note that tanh( \f(z\u0000\u0016)=2) has poles at z=i\u000fn+\u0016and is related to the Fermi distribution function f(z) as\ntanh[\f(z\u0000\u0016)=2] = 1\u00002f(z). Using the following identities\n1\nz2\u0000\u000f2\nk=1\n2\u000fkX\n\u0015=\u0006\u0015\nz\u0000\u0015\u000fk; (C7)\nz\nz2\u0000\u000f2\nk=1\n2X\n\u0015=\u00061\nz\u0000\u0015\u000fk; (C8)\nthe spin susceptibility is given by\n\u001fq(i!l) =F(\u0012;\u001e)X\nkIdz\n2\u0019if(z)X\n\u0015;\u00150=\u0006\"\n1\n2+(\u00012+~2~k2=3)\u0015\u00150\n2\u000fk\u000fk+q#\n1\nz\u0000\u0015\u000fk1\nz+i!l\u0000\u00150\u000fk+q; (C9)\n=F(\u0012;\u001e)X\nkX\n\u0015;\u00150=\u0006\u00141\n2+\u0015\u00150\n62\u00012+\u000f2\nk\n\u000fk\u000fk+q\u0015f(\u00150\u000fk+q)\u0000f(\u0015\u000fk)\ni!l\u0000\u00150\u000fk+q+\u0015\u000fk: (C10)\nBy the analytic continuation i!l=~!+i\u000e, we derive the retarded spin susceptibility as below:\n\u001fR\nq(!) =F(\u0012;\u001e)X\nkX\n\u0015;\u00150=\u0006\u00141\n2+\u0015\u00150\n62\u00012+\u000f2\nk\n\u000fk\u000fk+q\u0015f(\u00150\u000fk+q)\u0000f(\u0015\u000fk)\n~!+i\u000e\u0000\u00150\u000fk+q+\u0015\u000fk: (C11)\nThe imaginary part of the spin susceptibility is given by\nIm\u001fR\nq(!) =\u0000\u0019F(\u0012;\u001e)X\nkX\n\u0015;\u00150=\u0006\u00141\n2+\u0015\u00150\n62\u00012+\u000f2\nk\n\u000fk\u000fk+q\u0015h\nf(\u00150\u000fk+q)\u0000f(\u0015\u000fk)i\n\u000e(~!\u0000\u00150\u000fk+q+\u0015\u000fk): (C12)\nFrom this expression, Eqs. 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Rev.\nB84, 115137 (2011)."    },    {        "title": "2009.14143v1.Structural_Phase_Dependent_Giant_Interfacial_Spin_Transparency_in_W_CoFeB_Thin_Film_Heterostructure.pdf",        "content": "  \n1 \n Structural Phase Dependent Giant Interfacial Spin Transparency in \nW/CoFeB Thin Film Heterostructure \n \nSurya Narayan Panda, Sudip Majumder, Arpan Bhattacharyya, Soma Dutta, Samiran \nChoudhury and Anjan Barman* \n \nDepartment of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre \nfor Basic Sciences, Block JD, Sector-III, Salt Lake, Kolkata 700 106, India \n \n*E-mail: abarman@bose.res.in \n \n \nKeywords: (Thin Film Heterostructures, Interface Properties, Spin Pumping, Spin \nTransparency, Spin-Mixing Conductance, Gilbert Damping, Time-resolved Magneto-optical \nKerr Effect) \n \n \nAbstract \nPure spin current has transfigured the energy-efficient spintronic devices and it has the salient \ncharacteristic of transport of the spin angular momentum. Spin pumping is a potent method to \ngenerate pure spin current and for its increased efficiency high effective spin-mixing \nconductance ( Geff) and interfacial spin transparency ( T) are essential. Here, a giant T is reported \nin Sub/W( t)/Co20Fe60B20(d)/SiO2(2 nm) heterostructures in beta-tungsten (β-W) phase by \nemploying all-optical time-resolved magneto-optical Kerr effect technique. From the variation \nof Gilbert damping with W and CoFeB thicknesses, the spin diffusion length of W and spin-\nmixing conductances are extracted. Subsequently, T is derived as 0.81 ± 0.03 for the β-\nW/CoFeB interface. A sharp variation of Geff and T with W thickness is observed in consonance \nwith the thickness-dependent structural phase transition and resistivity of W. The spin memory \nloss and two-magnon scattering effects are found to have negligible contributions to damping \nmodulation as opposed to spin pumping effect which is reconfirmed from the invariance of \ndamping with Cu spacer layer thickness inserted between W and CoFeB. The observation of \ngiant interfacial spin transparency and its strong dependence on crystal structures of W will be \nimportant for pure spin current based spin-orbitronic devices. \n    \n2 \n 1. Introduction \nThe rapid emergence of spintronics has promised a new paradigm of electronics based on the \nspin degree of freedom either associated with the charge or by itself.[1-3]  This has potential \nadvantages of non-volatility, reduced electrical power consumption, increased data processing \nspeed, and increased integration densities as opposed to its semiconductor counterpart.[4] A \nmajor objective of modern spintronics is to harness pure spin current, which comprises of flow \nof spins without any net flow of charge current.[5, 6] This has the inherent benefit of reduced \nJoule heating and Oersted fields together with the ability to manipulate magnetization . Three \nmajor aspects of spin current are its generation, transport, and functionalization. Pure spin \ncurrent can be generated by spin-Hall effect,[7,8] Rashba-Edelstein effect,[9,10] spin pumping,[11-\n13] electrical injection in a lateral spin valve using a non-local geometry,[14,15] and spin \ncaloritronic effects.[16,17] Among these, spin pumping is an efficient and extensively used \nmethod of spin injection from ferromagnet (FM) into normal metal (NM) where the precessing \nspins from FM transfer spin angular momentum to the conduction electrons of adjacent NM \nlayer in NM/FM heterostructure, which gets dissipated by spin-flip scattering. The efficiency \nof spin pumping is characterized by spin-mixing conductance and spin diffusion length. The \ndissipation of spin current into the NM layer results in loss of spin angular momentum in the \nFM layer leading to an increase in its effective Gilbert damping parameter ( αeff). Thus, spin \npumping controls the magnetization dynamics in NM/FM heterostructures, which is crucial for \ndetermining the switching efficiency of spin-torque based spintronic devices. The enhancement \nin αeff is more prominent in heavy metals (HM) with high spin-orbit coupling (SOC) due to \nstronger interaction between electron spin and lattice. Intense research in the field of spin-\norbitronics has revealed that interface dependent spin transport is highly influenced by the spin \ntransparency, which essentially determines the extent of spin current diffused through the \nNM/FM interface.[18,19]     \n3 \n The highly resistive β-W, which shows a distorted tetragonal phase commonly referred to as \nA15 structure, is well known for exhibiting large spin Hall angle (SHA) (up to ~0.50) [20]  as \ncompared to other transition metal elements such as Pt (0.08) [21]  and β-Ta (0.12).[7] Besides, \nin W/FM heterostructures, W leads to highly stable perpendicular magnetic anisotropy[22]  and \ninterfacial Dzyaloshinskii-Moriya interaction.[23] Another important characteristic associated \nwith W is that it shows a thickness-dependent phase transition in the sub-10 nm thickness \nregime.[24,25] In general, sputter-deposited W films with thickness well below 10 nm are found \nto have β phase with high resistivity, whereas the films with thickness above 10 nm possess \npredominantly α phase (bcc structure) with low resistivity. A small to moderate SHA has been \nreported for the α and mixed (α + β) phase (<0.2) of W.[24] As SHA and effective spin-mixing \nconductance ( Geff) are correlated, one would expect that interfacial spin transparency ( T), which \nis also a function of Geff, should depend on the structural phase of W thin films. Furthermore, \nthe magnitude of the spin-orbit torque (SOT) depends on the efficiency of spin current \ntransmission (i.e. T) across the NM/FM interface. It is worth mentioning that due to high SOC \nstrength, W is a good spin-sink material and also cost-effective in comparison with the widely \nused NM like Pt. On the other hand, CoFeB due to its notable properties like high spin \npolarization, large tunnel magnetoresistance, and low intrinsic Gilbert damping, is used as FM \nelectrode in magnetic tunnel junctions. The presence of Boron at the NM/CoFeB interface \nmakes this system intriguing as some recent studies suggest that a small amount of boron helps \nin achieving a sharp interface and increases the spin polarization, although an excess of it causes \ncontamination of the interface. To this end, determination of T of the technologically important \nW/CoFeB interface and its dependence on the W-crystal phase are extremely important but still \nabsent in the literature.  \nBesides spin pumping, there are different mechanisms like spin memory loss (SML),[26] Rashba \neffect,[10] two-magnon scattering (TMS),[27] and interfacial band hybridization[28] which may \nalso cause loss of spin angular momentum at NM/FM interface, resulting in increase of αeff and   \n4 \n decrease of the spin transmission probability. However, for improved energy efficiency, the \nNM/FM interface in such engineered heterostructures must possess high spin transmission \nprobability. Consequently, it is imperative to get a deeper insight into all the mechanisms \ninvolved in generation and transfer of spin current for optimizing its efficiency. Here, we \ninvestigate the effects of spin pumping on the Gilbert damping in W/CoFeB bilayer system as \na function of W-layer thickness using recently developed all-optical technique, which is free \nfrom delicate micro-fabrication and electrical excitation and detection.[29] This is a local and \nnon-invasive method based on time-resolved magneto-optical Kerr effect (TR-MOKE) \nmagnetometry. Here, the damping is directly extracted from the decaying amplitude of time-\nresolved magnetization precession, which is free from experimental artifacts stemming from \nmultimodal oscillation, sample inhomogeneity, and defects. From the modulation of damping \nwith W layer thickness, we have extracted the intrinsic spin-mixing conductance ( G↑↓) of the \nW/CoFeB interface which excludes the backflow of spin angular momentum and spin diffusion \nlength(𝜆௦ௗ) of W. Furthermore, we have modeled the spin transport using both the ballistic \ntransport model[30, 31] and the model based on spin diffusion theory[32,33]. Subsequently, Geff, \nwhich includes the backflow of spin angular momentum, is estimated from the dependence of \ndamping on the CoFeB layer thicknesses. By using both the spin Hall magnetoresistance \nmodel[34] and spin transfer torque based model utilizing the drift-diffusion approximation[35], \nwe have calculated the T of W/CoFeB interface. The spin Hall magnetoresistance model gives \nlower value of T than the drift-diffusion model, but the former is considered more reliable as \nthe latter ignores the spin backflow. We found a giant value of T exceeding 0.8 in the β phase \nof W, which exhibits a sharp decrease to about 0.6 in the mixed (α+β) phase using spin Hall \nmagnetoresistance model. We have further investigated the other possible interface effects in \nour W/CoFeB system, by incorporating a thin Cu spacer layer of varying thickness between the \nW and CoFeB layers. Negligible modulation of damping with Cu thickness confirms the   \n5 \n dominance of spin pumping generated pure spin current and its transport in the modulation of \ndamping in our system.   \n \n2. Results and Discussion \nFigure 1 (a) shows the grazing incidence x-ray diffraction (GIXRD) patterns of \nSub/W(t)/Co20Fe60B20(3 nm)/SiO 2(2 nm) heterostructures at the glancing angle of 2o. In these \nplots, the peaks corresponding to α and β phase of W are marked. The high-intensity GIXRD \npeak at ∼44.5° and low intensity peak at ∼64° correspond primarily to the β phase (A15 \nstructure) of W (211) and W(222) orientation, respectively. Interestingly, we find these peaks \nto be present for all thicknesses of W, but when t > 5 nm, then an additional peak at ∼40.1° \ncorresponding to α-W with (110) crystal orientation appears. Consequently, we understand that \nfor t ≤ 5 nm, W is primarily in β-phase, while for t > 5 nm a fraction of the α phase appears, \nwhich we refer to as the mixed (α+β) phase of W. These findings are consistent with some \nexisting literature.[24,25]  Some other studies claimed that this transition thickness can be tuned \nby carefully tuning the deposition conditions of the W thin films.[36] The average lattice \nconstants obtained from the β-W peak at 44.5o and α-W peak at 40.1o correspond to about 4.93 \nand 3.15 Å, respectively. By using the Debye-Scherrer formula, we find the average crystallite \nsize in β and α phase of W to be about 14 and 7 nm, respectively. \nIt is well known that the formation of β-W films is characterized by large resistivity due to its \nA-15 structure which is associated with strong electron-phonon scattering, while the α-W \nexhibits comparatively lower resistivity due to weak electron-phonon scattering. We measured \nthe variation of resistivity of W with its thickness across the two different phases, using the \nfour-probe method. The inverse of sheet resistance ( Rs) of the film stack as a function of W \nthickness is plotted in Figure 1(b). A change of the slope is observed beyond 5 nm, which \nindicates a change in the W resistivity. The data have been fitted using the parallel resistors \nmodel[24] (shown in Figure S1  of the Supporting Information). [37] We estimate the average   \n6 \n resistivity of W ( ρW) in β and mixed (α+β) phase to be about 287 ± 19 and 112 ± 14 µΩ.cm, \nrespectively, while the resistivity of CoFeB (ρCoFeB) is found to be 139 ± 16 μΩ.cm. Thus, the \nresistivity results corroborate well with those of the XRD measurement. \nThe AFM image of Sub/W ( t)/Co20Fe60B20 (3 nm)/SiO 2 (2 nm) (t = 1, 5 and 10 nm) samples in \nFigure 1(c) revealed the surface topography. We have used WSxM software to process the \nimages.[38] The variation in the average surface roughness of the films with W thickness is listed \nin Table 1. The roughness varies very little when measured at various regions of space of the \nsame sample. The surface roughness in all samples is found to be small irrespective of the \ncrystal phase of W. Due to the small thicknesses of various layers in the heterostructures, the \ninterfacial roughness is expected to show its imprint on the measured topographical roughness. \nWe thus understand that the interfacial roughness in these heterostructures is very small and \nsimilar in all studied samples. Details of AFM characterization is shown in Figure S2  of the \nSupporting Information.[37] \n2.1. Principles behind the modulation of Gilbert damping with layer thickness: \nIn an NM/FM bilayer magnetic damping can have various additional contributions, namely \ntwo-magnon scattering, eddy current, and spin pumping in addition to intrinsic Gilbert damping. \nAmong these, the spin pumping effect is a non-local effect, in which an external excitation \ninduces magnetization precession in the FM layer. The magnetization precession causes a spin \naccumulation at the NM/FM interface. These accumulated spins carry angular momentum to \nthe adjacent NM layer, which acts as a spin sink by absorbing the spin current by spin-flip \nscattering, leading to an enhancement of the Gilbert damping parameter of FM. In 2002, \nTserkovnyak and Brataas theoretically demonstrated the spin pumping induced enhancement \nin Gilbert damping in NM/FM heterostructures using time-dependent adiabatic scattering \ntheory where magnetization dynamics in the presence of spin pumping can be described by a \nmodified Landau-Lifshitz-Gilbert (LLG) equation as: [11-13]   \n7 \n ௗ𝒎\nௗ௧= −𝛾(𝒎×𝑯eff)+𝛼0(𝒎×ௗ𝒎\nௗ௧)+ఊ\nVMೞ𝑰௦                          (1) \nwhere γ is the gyromagnetic ratio,  Is is the total spin current, Heff is the effective magnetic field, \nα0 is intrinsic Gilbert damping constant, V is the volume of ferromagnet and Ms is saturation \nmagnetization of the ferromagnet. As shown in equation (2), Is generally consists of a direct \ncurrent contribution 𝑰𝒔𝟎 which is nonexistent in our case as we do not apply any charge current, \n𝑰𝒔𝒑𝒖𝒎𝒑, i.e. spin current due to pumped spins from the FM to NM and 𝑰𝒔𝒃𝒂𝒄𝒌, i.e. a spin current \nbackflow to the FM reflecting from the NM/substrate interface which is assumed to be a perfect \nreflector. \n𝑰𝒔=𝑰𝒔𝟎+𝑰𝒔pump+𝑰𝒔back                                                    (2) \nHere, 𝑰𝒔𝒃𝒂𝒄𝒌 is determined by the spin diffusion length of the NM layer. Its contribution to \nGilbert damping for most metals with a low impurity concentration is parametrized by a \nbackflow factor β which can be expressed as:[39] \n𝛽=൭2𝜋𝐺↑↓ටఌ\nଷtanhቀ௧\nఒೞቁ൱ିଵ\n                                      (3) \nwhere ε is the material-dependent spin-flip probability, which is the ratio of the spin-conserved \nto spin-flip scattering time. It can be expressed as: [40]                                                                \n 𝜀= (𝜆𝜆௦ௗ⁄)ଶ3⁄                                                                                     (4) \nwhere λel and λsd are the electronic mean free path and spin diffusion length of NM, respectively.  \nThe spin transport through NM/FM interface directly depends on the spin-mixing conductance, \nwhich is of two types: (a) G↑↓, which ignores the contribution of backflow of spin angular \nmomentum, and (b) Geff, which includes the backflow contribution. Spin-mixing conductance  \ndescribes the conductance property of spin channels at the interface between NM and FM. Also, \nspin transport across the interface affects the damping parameter giving rise to αeff of the system   \n8 \n that can be modeled by both ballistic and diffusive transport theory. In the ballistic transport \nmodel, the αeff is fitted with the following simple exponential function:[30,31,39] \n𝐺eff=𝐺↑↓൬1−𝑒ିమ\nഊೞ൰=ସగdM\nఓಳ(𝛼eff−𝛼)                            (5) \n𝛥𝛼=𝛼eff−𝛼=ఓಳீ↑↓൭ଵିషమ\nഊೞ൱\nସdMeff                                         (6) \nHere, the exponential term signifies backflow spin current contribution and a factor of 2 in the \nexponent signifies the distance traversed by the spins inside the NM layer due to reflection from \nthe NM/substrate interface.  \nIn the ballistic approach, the resistivity of NM is not considered while the NM thickness is \nassumed to be less than the mean free path. To include the effect of the charge properties of \nNM on spin transport, the model based on spin diffusion theory is used to describe αeff (t). \nWithin this model, the additional damping due to spin pumping is described as:[32,33,36]  \n            𝐺eff=ீ↑↓\nቆଵାమഐഊೞಸ↑↓\nୡ୭୲୦ቀ௧ఒೞൗቁቇ=ସగdM\nఓಳ(𝛼eff−𝛼)                                         (7)                                                                                       \n                                                                                        \n           ∆𝛼=𝛼eff−𝛼=ఓಳீ↑↓\nସగdMቆଵା మഐഊೞಸ↑↓\nୡ୭୲୦ቀ௧ఒೞൗቁቇ (8)                                                                                            \n     \nwhere ρ is the electrical resistivity of the W layer. Here the term  మఘఒೞீ↑↓\ncothቀ𝑡𝜆௦ௗൗቁ account \nfor the back-flow of pumped spin current into the ferromagnetic layer. \nThe reduction of spin transmission probability implies a lack of electronic band matching, \nintermixing, and disorder at the interface. The spin transparency, T of an NM/FM interface \ntakes into account all such effects that lead to the electrons being reflected from the interface \ninstead of being transmitted during transport. Further, T depends on both intrinsic and extrinsic \ninterfacial factors, such as band-structure mismatch, Fermi velocity, interface imperfections, \netc.[19,39] According to the spin Hall magnetoresistance model, the spin current density that   \n9 \n diffuses into the NM layer is smaller than the actual spin current density generated via the spin \npumping in the FM layer. This model linked T with 𝐺eff  by the following relation:[34,39] \n𝑇=ீeff  tanh൬\nమഊೞ൰\nீeff  coth൬\nഊೞ൰ା\nమഊೞమഐ                              (9) \nThe interfacial spin transparency was also calculated by Pai et al. in the light of damping-like \nand field-like torques utilizing the drift-diffusion approximation. Here, the effects of spin \nbackflow are neglected as it causes a reduction in the spin torque efficiencies. Assuming t ≫ λ \nand a very high value of d, T can be expressed as:[35] \n𝑇=ଶீ↑↓ீಿಾ⁄\nଵାଶீ↑↓ீಿಾ⁄                                                                                                               (10) \nwhere, 𝐺ேெ=\nఘఒೞమ  is the spin conductance of the NM layer. \nIn an NM/FM heterostructure, other than spin pumping, there is a finite probability to have \nsome losses of spin angular momentum due to interfacial depolarization and surface \ninhomogeneities, known as SML and TMS, respectively. In SML, loss of spin angular \nmomentum occurs when the atomic lattice at the interface acts as a spin sink due to the magnetic \nproximity effect or due to the interfacial spin-orbit scattering which could transfer spin \npolarization to the atomic lattice.[26] The TMS arises when a uniform FMR mode is destroyed \nand a degenerate magnon of different wave vector is created.[27] The momentum non-\nconservation is accounted for by considering a pseudo-momentum derived from internal field \ninhomogeneities or secondary scattering. SML and TMS may contribute to the enhancement of \nthe Gilbert damping parameter considerably. Recently TMS is found to be the dominant \ncontribution to damping for Pt-FM heterostructures.[41] In the presence of TMS and SML \neffective Gilbert damping can be approximated as:[41] \n    αeff = α0 + αSP + αSML + αTMS    \n      ∆𝛼=𝛼eff−𝛼= 𝑔𝜇ீeff  ା ீೄಾಽ\nସdM+𝛽்ெௌ𝑑ିଶ                                                      (11) \n   \n10 \n where 𝐺ௌெ is the “effective SML conductance”, and βTMS is a “coefficient of TMS” that \ndepends on both interfacial perpendicular magnetic anisotropy field and the density of magnetic \ndefects at the FM surfaces.  \n2.2. All-optical measurement of magnetization dynamics: \nA schematic of the spin pumping mechanism along with the experimental geometry is shown \nin Figure 2(a). A typical time-resolved Kerr rotation data for the Sub/Co 20Fe60B20(3 nm)/SiO 2(2 \nnm) sample at a bias magnetic field, H = 2.30 kOe is shown in Figure 2(b) which consists of \nthree different temporal regimes. The first regime is called ultrafast demagnetization, where a \nsharp drop in the Kerr rotation (magnetization) of the sample is observed immediately after \nfemtosecond laser excitation. The second regime corresponds to the fast remagnetization where \nmagnetization recovers to equilibrium by spin-lattice interaction. The last regime consists of \nslower relaxation due to heat diffusion from the lattice to the surrounding (substrate) superposed \nwith damped magnetization precession. The red line in Figure 2(b) denotes the bi-exponential \nbackground present in the precessional data. We are mainly interested here in the extraction of \ndecay time from the damped sinusoidal oscillation about an effective magnetic field and its \nmodulation with the thickness of FM and NM layers. We fit the time-resolved precessional data \nusing a damped sinusoidal function given by: \n𝑀(𝑡)=𝑀(0)𝑒ିቀ\nഓቁsin(2π𝑓𝑡+𝜑)                               (12) \nwhere τ is the decay time, φ is the initial phase of oscillation and f  is the precessional frequency. \nThe bias field dependence of precessional frequency can be fitted using the Kittel formula given \nbelow to find the effective saturation magnetization ( Meff): \n𝑓=ఊ\nଶ(𝐻(𝐻+4π𝑀eff))ଵ/ଶ                                                              (13)                       \nwhere γ = gµB/ħ, g is the Landé g-factor and ћ is the reduced Planck’s constant. From the fit, \nMeff and g are determined as fitting parameters. For these film stacks, we obtained effective   \n11 \n magnetization, Meff ≈ 1200 ± 100 emu/cc, and g = 2.0 ± 0.1. The comparison between Meff \nobtained from the magnetization dynamics measurement and Ms from VSM measurement for \nvarious thickness series are presented systematically in Figures S3-S5  of the Supporting \nInformation.[37] For almost all the film stacks investigated in this work, Meff  is found to be close \nto Ms, which indicates that the interface anisotropy is small in these heterostructures. We \nestimate αeff using the expression: [42]  \n𝛼eff=1\nγτ(𝐻+2π𝑀eff)                                           (14) \nwhere τ is the decay time obtained from the fit of the precessional oscillation with equation (12). \nWe have plotted the variation of time-resolved precessional oscillation with the bias magnetic \nfield and the corresponding fast Fourier transform (FFT) power spectra in Figure S6  of the \nSupporting Information.[37] The extracted values of αeff are found to be independent of the \nprecession frequency f. Recent studies show that in presence of extrinsic damping contributions \nlike TMS, αeff should increase with f, while in presence of inhomogeneous anisotropy in the \nsystem αeff should decrease with f.[43] Thus, frequency-independent αeff rules out any such \nextrinsic contributions to damping in our system. \n2.3. Modulation of the Gilbert damping parameter: \nIn Figure 3 (a) we have presented time-resolved precessional dynamics for \nSub/W(t)/Co20Fe60B20(3 nm)/SiO 2(2 nm) samples with 0 ≤ t ≤ 15 nm at H = 2.30 kOe. The \nvalue of α0 for the 3-nm-thick CoFeB layer without the W underlayer is found to be 0.006 ± \n0.0005. The presence of W underlayer causes αeff to vary non monotonically over the whole \nthickness regime as shown by the αeff vs. t plot in Figure 3(b). In the lower thickness regime, \ni.e.  0 ≤ t ≤ 3 nm, Δ α increases sharply by about 90% due to spin pumping but it saturates for t \n≥ 3 nm. However, for t > 5 nm, Δ α drops by about 30% which is most likely related to due to \nthe thickness-dependent phase transition of W.  At first, we have fitted our result for t ≤ 5 nm \nwith equation (6) of the ballistic transport model and determined G↑↓ = (1.46 ± 0.01) × 1015 cm-  \n12 \n 2 and λsd = 1.71 ± 0.10 nm as fitting parameters. Next, we have also fitted our results with \nequation (8) based on spin diffusion theory, where we have obtained G↑↓ = (2.19 ± 0.02) × 1015 \ncm-2 and λsd = 1.78 ± 0.10 nm. The value of G↑↓ using spin diffusion theory is about 28% higher \nthan that of ballistic model while the value of λsd is nearly same in both models. Using values \nfor λel (about 0.45 nm for W) from the literature[44]  and λsd derived from our experimental data, \nwe have determined the spin-flip probability parameter, ε = 2.30 × 10−2 from equation (4). To \nbe considered as an efficient spin sink, a nonmagnetic metal must have ε  ≥ 1.0 × 10-2 and hence \nwe can infer that the W layer acts as an efficient spin sink here.[13] The backflow factor β can \nbe extracted from equation (3). We have quantified the modulation of the backflow factor (Δ β) \nto be about 68% within the experimental thickness regime. \nTo determine the value of 𝐺eff  directly from the experiment, we have measured the time-\nresolved precessional dynamics for Sub/W (4 nm)/Co 20Fe60B20 (d)/SiO2 (2 nm) samples with 1 \nnm ≤ d ≤ 10 nm at H = 2.30 kOe as shown in Figure 4(a). The αeff  is found to increase with the \ninverse of FM layer thickness ( Figure 4(b)). We have fitted our results first with equation (5), \nfrom which we have obtained 𝐺eff  and 𝛼 to be (1.44 ± 0.01) × 1015 cm-2 and 0.006 ± 0.0005, \nrespectively.  \nBy modelling the W thickness dependent modulation of damping of Figure 3(b) using equation \n(5), we have obtained 𝐺eff  of W/CoFeB in β-phase (where ∆𝛼 ≈  0.006) and α+β-mixed phase \n(where ∆𝛼  ≈ 0.004) of W to be (1.44 ± 0.01) × 1015 cm-2 and (1.07 ± 0.01) × 1015 cm-2, \nrespectively. From these, we conclude that β-phase of W has higher conductance of spin \nchannels in comparison to the α+β-mixed phase. The variation of 𝐺eff  with W layer thickness \nis presented in Figure 5(a), which shows that 𝐺eff  increases non monotonically and nearly \nsaturates for t ≥ 3 nm. For t > 5 nm, 𝐺eff  shows a sharp decrease in consonance with the variation \nof αeff.  \nWe have further fitted the variation of αeff  with the inverse of FM layer thickness ( Figure 4(b)) \nusing with equation (11) to isolate the contributions from SML, TMS and spin pumping (SP).   \n13 \n The values of  𝐺ௌெ , and βTMS are found to be (2.45 ± 0.05) × 1013 cm-2 and (1.09 ± 0.02) × 10-\n18 cm2, respectively. 𝐺ௌெ is negligible in comparison with 𝐺eff  which confirms the absence of \nSML contribution in damping. Contribution of TMS to damping modulation ( 𝛽்ெௌ𝑑ଶ) is also \nbelow 2% for all the FM thicknesses. The relative contributions are plotted in Figure 5(b). It is \nclear that spin pumping contribution is highly dominant over the SML and TMS for our studied \nsamples. The value of our 𝐺eff  in β-W/CoFeB is found to be much higher than that obtained for \nβ-Ta/CoFeB[39] measured by all-optical TRMOKE technique as well as various other NM/FM \nheterostructures measured by conventional techniques as listed in Table 2. This provides \nanother confirmation of W being a good spin sink material giving rise to strong spin pumping \neffect. \nWe subsequently investigate the value of T for W/CoFeB interface, which is associated with \nthe spin-mixing conductances of interface, spin diffusion length, and resistivity of NM as \ndenoted in equations (9) and (10). T is an electronic property of a material that depends upon \nelectronic band matching of the two materials on either side of the interface. After determining \nthe resistivity, spin diffusion length and spin-mixing conductances experimentally, we have \ndetermined the value of T which depends strongly on the structural phase of W. Using equation \n(9) based on the spin-Hall magnetoresistance model, Tβ-W and T(α+β)-W are found to be 0.81 ± \n0.03 and 0.60 ± 0.02, respectively. On the other hand, equation (10) of spin transfer torque \nbased model utilizing the drift-diffusion approximation gives Tβ-W and T(α+β)-W to be 0.85 ± 0.03 \nand 0.63 ± 0.02, respectively, which are slightly higher than the values obtained from spin-Hall \nmagnetoresistance model. However, we consider the values of T obtained from the spin-Hall \nmagnetoresistance model to be more accurate as it includes the mandatory contribution of spin \ncurrent backflow from W layer into the CoFeB layer. Nevertheless, our study clearly \ndemonstrates that the value of spin transparency of the W/CoFeB interface is the highest \nreported among the NM/FM heterostructures as listed in Table 2. This high value of T, \ncombined with the high spin Hall angle of β-W makes it an extremely useful material for pure   \n14 \n spin current based spintronic and spin-orbitronic devices. The structural phase dependence of \nT for W also provides a particularly important guideline for choosing the correct thickness and \nphase of W for application in the above devices.  \nFinally, to directly examine the additional possible interfacial effects present in the W/CoFeB \nsystem, we have introduced a copper spacer layer of a few different thicknesses between the W \nand CoFeB layers. Copper has very small SOC and spin-flip scattering parameters and it shows \na very high spin diffusion length. Thus, a thin copper spacer layer should not affect the damping \nof the FM layer due to the spin pumping effect but can influence the other possible interface \neffects. Thus, if other interface effects are substantial in our samples, the introduction of the \ncopper spacer layer would cause a notable modulation of damping with the increase of copper \nspacer layer thickness ( c).[19,39] The time-resolved Kerr rotation data for the Sub/W(4 \nnm)/Cu(c)/Co20Fe60B20(3 nm)/SiO 2(2 nm) heterostructures with 0 ≤ c ≤ 1 nm are presented in \nFigure 6(a) at H = 2.30 kOe and Figure 6(b) shows the plot of αeff as a function of c. The \ninvariance of αeff with c confirms that the interface of Cu/CoFeB is transparent for spin transport \nand possible additional interfacial contribution to damping is negligible, which is in agreement \nwith our modelling as shown in Figure 5(b).  \n \n3. Conclusion \nIn summary, we have systematically investigated the effects of thickness-dependent structural \nphase transition of W in W( t)/CoFeB( d) thin film heterostructures and spin pumping induced \nmodulation of Gilbert damping by using an all-optical time-resolved magneto-optical Kerr \neffect magnetometer. The W film has exhibited structural phase transition from a pure β phase \nto a mixed (α + β) phase for t > 5 nm. Subsequently, β-W phase leads to larger modulation in \neffective damping ( αeff) than (α+β)-W. The spin diffusion length of W is found to be 1.71 ± \n0.10 nm, while the spin pumping induced effective spin-mixing conductance  𝐺eff  is found to be  \n(1.44 ± 0.01) × 1015 cm-2 and (1.07 ± 0.01) × 1015 cm-2 for β and mixed (α+β) phase of W,   \n15 \n respectively. This large difference in 𝐺eff  is attributed to different interface qualities leading \ntowards different interfacial spin-orbit coupling. Furthermore, by analyzing the variation of αeff \nwith CoFeB thickness in W (4 nm)/CoFeB  (d)/SiO2 (2 nm), we have isolated the contributions \nof spin memory loss and two-magnon scattering from spin pumping, which divulges that spin \npumping is the dominant contributor to damping. By modeling our results with the spin Hall \nmagnetoresistance model, we have extracted the interfacial spin transparency ( T) of β-\nW/CoFeB and (α + β)-W/CoFeB as 0.81 ± 0.03 and 0.60 ± 0.02, respectively. This structural \nphase-dependent T value will offer important guidelines for the selection of material phase for \nspintronic applications. Within the framework of ballistic and diffusive spin transport models, \nthe intrinsic spin-mixing conductance ( G↑↓) and spin-diffusion length ( λsd) of β-W are also \ncalculated by studying the enhancement of αeff as a function of β-W thickness. Irrespective of \nthe used model, the value of T for W/CoFeB interface is found to be highest among the NM/FM \ninterfaces, including the popularly used Pt/FM heterostructures. The other possible interface \neffects on the modulation of Gilbert damping are found to be negligible as compared to the spin \npumping effect. Thus, our study helps in developing a deep understanding of the role of W thin \nfilms in NM/FM heterostructures and the ensuing spin-orbit effects. The low intrinsic Gilbert \ndamping parameter, high effective spin-mixing conductance combined with very high interface \nspin transparency and spin Hall angle can make the W/CoFeB system a key material for spin-\norbit torque-based magnetization switching, spin logic and spin-wave devices.  \n \n4. Experimental Section/Methods \n4.1. Sample Preparation \nThin films of Sub/W( t)/Co20Fe60B20(d)/SiO2(2 nm) were deposited by using RF/DC magnetron \nsputtering system on Si (100) wafers coated with 285 nm-thick SiO 2. We varied the W layer \nthickness as t = 0, 0.5, 1, 1.5, 2, 3, 4, 5, 8, 10 and 15 nm and CoFeB layer thickness as d = 1, 2, \n3, 5 and 10 nm. The depositions were performed at an average base pressure of 1.8  × 10-7 Torr   \n16 \n and argon pressure of about 0.5 mTorr at a deposition rate of 0.2 Å/s. Very slow deposition \nrates were chosen for achieving a uniform thickness of the films even at a very thin regime \ndown to sub-nm. The W and CoFeB layers were deposited using average DC voltages of 320 \nand 370 V, respectively, while SiO 2 was deposited using average RF power of 55 watts. All \nother deposition conditions were carefully optimized and kept almost identical for all samples. \nIn another set of samples, we introduced a thin Cu spacer layer in between the CoFeB and W \nlayers and varied its thickness from 0 nm to 1 nm. The Cu layer was deposited at a DC voltage \nof 350 V, argon pressure of 0.5 mTorr and deposition rate of 0.2 Ǻ/s. \n4.2. Characterization \nAtomic force microscopy (AFM) was used to investigate the surface topography and vibrating \nsample magnetometry (VSM) was used to characterize the static magnetic properties of these \nheterostructures. Using a standard four-probe technique the resistivity of the W films was \ndetermined and grazing incidence x-ray diffraction (GIXRD) was used for investigating the \nstructural phase of W. To study the magnetization dynamics, we used a custom-built TR-\nMOKE magnetometer based on a two-color, collinear optical pump-probe technique. Here, the \nsecond harmonic laser pulse (λ = 400 nm, repetition rate = 1 kHz, pulse width >40 fs) of an \namplified femtosecond laser, obtained using a regenerative amplifier system (Libra, Coherent) \nwas used to excite the magnetization dynamics, while the fundamental laser pulse (λ = 800 nm, \nrepetition rate = 1 kHz, pulse width ~40 fs) was used to probe the time-varying polar Kerr \nrotation from the samples. The pump laser beam was slightly defocused to a spot size of about \n300 µm and was obliquely (approximately 30° to the normal on the sample plane) incident on \nthe sample. The probe beam having a spot size of about 100 µm was normally incident on the \nsample, maintaining an excellent spatial overlap with the pump spot to avoid any spurious \ncontribution to the Gilbert damping due to the dissipation of energy of uniform precessional \nmode flowing out of the probed area. A large enough magnetic field was first applied at an \nangle of about 25° to the sample plane to saturate its magnetization. This was followed by a   \n17 \n reduction of the magnetic field to the bias field value ( H = in-plane component of the bias field) \nto ensure that the magnetization remained saturated along the bias field direction. The tilt of \nmagnetization from the sample plane ensured a finite demagnetizing field along the direction \nof the pump pulse, which was modified by the pump pulse to induce a precessional \nmagnetization dynamics in the sample. The pump beam was chopped at 373 Hz frequency and \nthe dynamic Kerr signal in the probe pulse was detected using a lock-in amplifier in a phase-\nsensitive manner. The pump and probe fluences were kept constant at 10 mJ/cm2 and 2 mJ/cm2, \nrespectively, during the measurement. All the experiments were performed under ambient \nconditions at room temperature. \n \nAcknowledgements \n \nAB gratefully acknowledges the financial assistance from the S. N. Bose National Centre for \nBasic Sciences (SNBNCBS), India under Project No. SNB/AB/18-19/211.  SNP, SM and SC \nacknowledge SNBNCBS for senior research fellowship. ArB acknowledges SNBNCBS for \npostdoctoral research associateship. 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Heinrich, C. H. Back, Phys. Rev. Lett.,  2005, 95, 037401.  \n[44]    Q. Hao, W. Chen, G. Xiao, Appl. Phys. Lett ., 2015, 106, 182403. \n[45] A. K. Patra, S. Singh, B. Barin, Y. Lee, J.-H. Ahn, E. D. Barco, E. R. Mucciolo, B. \nÖzyilmaz, Appl. Phys. Lett ., 2012,  101, 162407. \n[46] N. Behera, M. S. Singh, S. Chaudhary, D. K. Pandya, P. K. Muduli, J. Appl. Phys ., 2015, \n117, 17A714. \n[47] M. Haertinger, C. H. Back, J. Lotze, M. Weiler, S. Geprägs, H. Huebl, S. T. B. \nGoennenwein, G. Woltersdorf, Phys. Rev. B , 2015, 92, 054437.   \n21 \n [48] S. Husain, A. Kumar, P. Kumar, A. Kumar, V. Barwal, N. Behera, S. Choudhary, P. \nSvedlindh, S. Chaudhary, Phys. Rev. B , 2018, 98, 180404(R). \n[49] A. Kumar, S. Akansel, H. Stopfel, M. Fazlali, J. Åkerman, R. Brucas, P. Svedlindh, \nPhys. Rev. B , 2017,  95, 064406. \n[50] M. Caminale, A. Ghosh, S. Auffret, U. Ebels, K. Ollefs, F. Wilhelm, A. Rogalev, W. E. \nBailey, Phys. Rev. B , 2016, 94, 014414. \n[51] S. Husain, A. Kumar, V. Barwal, N. Behera, S. Akansel, P. Svedlindh, S. Chaudhary, \nPhys. Rev. B , 2018, 97, 064420. \n[52] G. Wu, Y. Ren, X. He, Y. Zhang, H. Xue, Z. Ji, Q. Y. Jin, Z. Zhang, Phys. Rev. Appl ., \n2020,  13, 024027. \n[53] H. L. Wang, C. H. Du, Y. Pu, R. Adur, P. C. Hammel, F. Y. Yang, Phys. Rev. Lett ., \n2014, 112, 197201. \n \n \n    \n22 \n  \n  \n \nFigure 1.  (a) X-ray diffraction patterns measured at 2° grazing angle incidence for different W \nthickness. (b) Variation of inverse sheet resistance with W thickness. (c) AFM images of the \nsamples showing the surface topography.  \n \n  \n23 \n   \nFigure 2. (a) Schematic of experimental geometry and (b) typical TR-MOKE data from \nCo20Fe60B20(3 nm)/SiO 2(2 nm) heterostructure at an applied bias magnetic field of 2.30 kOe. \nThe three important temporal regimes are indicated in the graph. The solid red line shows a \nbiexponential fit to the decaying background of the time-resolved Kerr rotation data. \n  \n24 \n   \nFigure 3. (a) Background subtracted time-resolved Kerr rotation data showing precessional \noscillation for Sub/W( t)/ Co20Fe60B20(3 nm)/SiO 2(2 nm) as function of W thickness at an \napplied bias magnetic field of 2.30 kOe. (b) Experimental result of variation damping with t \n(symbol) fitted with theoretical models (solid and dashed lines) of spin pumping. Two different \nregions corresponding to W crystal phase, namely β and α+β are shown.  \n \n \n \n \n  \n25 \n  \n  \nFigure 4. (a) Background subtracted time-resolved Kerr rotation data showing precessional \noscillation for Sub/W (4 nm)/Co 20Fe60B20 (d)/SiO2 (2 nm) as function of Co 20Fe60B20 thickness \nd at an applied bias magnetic field of 2.30 kOe. (b) Experimental result of variation of damping \nvs 1/d (symbol) fitted with theoretical models (solid and dashed lines).  \n \n  \n26 \n   \nFigure 5.  (a) Variation of effective spin-mixing conductance( 𝐺eff ) with W layer thickness t \n(symbol). The solid line is guide to the eye. (b) Contributions of SP, SML and TMS to the \nmodulation of damping for different Co 20Fe60B20 layer thickness d (symbol). The solid line is \nguide to the eye. 0 2 4 6 8 10039095100 \n  SP\n TMS\n SML\n  Damping (%) \n                d  (nm) 0 2 4 8 12 160.00.51.01.5 \n \n  Geff (1015 cm-2)\nt (nm)(a)\n(b)  \n27 \n   \nFigure 6. (a) Background subtracted time-resolved Kerr rotation data showing precessional \noscillation for Sub/W(4 nm)/Cu( c)/Co20Fe60B20(3 nm)/SiO 2(2 nm) as function of Cu layer \nthickness c at an applied bias magnetic field of 2.30 kOe. (b) Experimental result of variation \nof damping vs c. The dotted line is guide to the eye, showing very little dependence of damping \non Cu layer thickness. \n  \n28 \n Table 1. The average surface roughness values of Sub/W ( t)/Co20Fe60B20 (3 nm)/SiO 2 (2 nm) \nsamples obtained using AFM. \n \nTable 2. Comparison of the effective spin-mixing conductance and interfacial spin \ntransparency of the W/CoFeB samples studied here with the important NM/FM interfaces taken \nfrom the literature. \nMaterial  \nInterface Effective Spin-Mixing \nConductance  (×1015 cm-2) Interfacial Spin \nTransparency  \nPt/Py 1.52 [19] 0.25 [19] \nPt/Co 3.96 [19] 0.65 [19] \nPd/CoFe 1.07 [31] N.A. \nPt/FM 0.6-1.2 [35] 0.34-0.67 [35] \nβ-Ta/CoFeB  0.69 [39] 0.50 [39] \nβ-Ta/ CFA 2.90 [40] 0.68 [40] \nPd0.25Pt0.75/Co 9.11 [41] N.A. \nAu0.25Pt0.75/Co 10.73 [41] N.A. \nPd/Co 4.03 [41] N.A. \nPd0.25Pt0.75/FeCoB 3.35 [41] N.A. \nAu0.25Pt0.75/ FeCoB 3.64 [41] N.A. \nGr/Py 5.26 [45] N.A. \nRu/Py 0.24 [46] N.A. \nPt/YIG 0.3-1.2 [47] N.A. \nMoS2/CFA 1.49 [48] 0.46 [48] \nPd/Fe 0.49-1.17 [49] 0.04-0.33 [49] \nPd/Py 1.40 [50] N.A. \nMo/CFA 1.56 [51] N.A. \nMoS2/CoFeB 16.11 [52] N.A. \nTa/YIG 0.54 [53] N.A. \nW/YIG 0.45 [53] N.A. \nCu/YIG 0.16 [53] N.A. \nAg/YIG 0.05 [53] N.A. \nAu/YIG 0.27 [53] N.A. \nβ-W/CoFeB  1.44 (This work)  0.81 (This work)  \nMixed(α+β)-W/CoFeB  1.07 (This work)  0.60 (This work)  \n  \n((N.A. =  Not available)) \n \n t (nm) 0 0.5 1.0 1.5 2 3 5 8 10 15 \nRoughness \n(nm) 0.23 0.21 0.32 0.28 0.25 0.21 0.19 0.29 0.28 0.22 \n   \n29 \n Supporting Information \n \n \nStructural Phase Dependent Giant Interfacial Spin Transparency in W/CoFeB Thin \nFilm Heterostructure \n \nSurya Narayan Panda, Sudip Majumder, Arpan Bhattacharyya, Soma Dutta, Samiran \nChoudhury and Anjan Barman* \n \nDepartment of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre \nfor Basic Sciences, Block JD, Sector-III, Salt Lake, Kolkata 700 106, India \n \nE-mail: abarman@bose.res.in \n \n \nThis file includes: \n1. Determination of resistivity of W and Co 20Fe60B20 layers. \n2. Measurement of surface roughness of the sample using AFM. \n3. Determination of saturation magnetization of the samples from static and dynamic \nmeasurements.  \n4. Variation of effective damping with precessional frequency. \n \n \n \n1. Determination of resistivity of W and CoFeB layers :  \n \nThe variation of sheet resistance ( Rs) of the W( t)/Co20Fe60B20(3 nm) film stack with W layer \nthickness, t is shown in Figure S1 . The data is fitted with a parallel resistor model (Ref. 24 of \nthe article) by the formula given in the inset of the figure. This yields the resistivity of W in its \nβ and (α+β) phase as: 287 ± 19 µΩ.cm and 112 ± 14 µΩ.cm, respectively. On the other hand, \nthe resistivity of Co 20Fe60B20 is found to be 139 ± 16 µΩ.cm. \n   \n30 \n   \n \nFigure S1.  Variation of sheet resistance ( Rs) of the W ( t)/ Co20Fe60B20(3 nm) film stack vs. W \nthickness t used for the determination of resistivity of the W and Co 20Fe60B20 layers. \n \n2. Measurement of surface roughness of the sample using AFM:  \nWe have measured the surface topography of Sub/W ( t)/Co20Fe60B20 (3 nm)/SiO 2 (2 nm) thin \nfilms by atomic force microscopy (AFM) in dynamic tapping mode by taking scan over 10 μm \n× 10 μm area. We have analyzed the AFM images using WSxM software. Figures S2 (a) and \nS2(d) show two-dimensional planar AFM images for t = 1 nm and 10 nm, respectively. Figures \nS2(b) and S2(e) show the corresponding three-dimensional AFM images for t = 1 nm and 10 \nnm, respectively. The dotted black lines on both images show the position of the line scans to \nobtain the height variation. Figures S2 (c) and S2(f) show the surface roughness profile along \nthat dotted lines, from which the average roughness ( Ra) is measured as 0.32 ± 0.10  nm and \n0.28 ± 0.12 nm for t = 1 nm and 10 nm, respectively. Topographical roughness is small and \nconstant within the error bar in all samples irrespective of the crystal phase of W. Furthermore, \nsurface roughness varies very little when measured at different regions of same sample. The \ninterfacial roughness is expected to show its imprint on the measured topographical roughness 0 3 10 150200400\n  Rs(Ω)\nt(nm)(ρW)β= 287 µΩ.cm\n(ρW)α+β= 112 µΩ.cm\n= 139 µΩ.cm  \n31 \n due to the small thickness of our thin films. Small and constant surface roughness in these \nheterostructures proves the high quality of the thin films.  \n \n  \nFigure S2.  (a) The two-dimensional AFM image, (b) the three-dimensional AFM image, and \n(c) the line scan profile along the black dotted line for W(1 nm)/ Co 20Fe60B20(3 nm) /SiO 2(2 \nnm) sample. (d) The two-dimensional AFM image, (e) the three-dimensional AFM image, and \n(f) the line scan profile along the black dotted line for W(10 nm)/ Co 20Fe60B20(3 nm) /SiO 2(2 \nnm) sample. \n \n3. Determination of saturation magnetization of the samples from static and dynamic \nmagnetic measurements :  \nWe have measured the in-plane saturation magnetization ( Ms) of all the W( t)/ \nCo20Fe60B20(d)/SiO2(2 nm) samples using vibrating sample magnetometry (VSM). Typical \nmagnetic hysteresis loops (magnetization vs. magnetic field) for W( t)/ Co20Fe60B20(3 \nnm)/SiO 2(2 nm), W(4 nm)/ Co 20Fe60B20(d)/SiO2(2 nm) and W(4 nm)/Cu( c)/ Co20Fe60B20(3 \n  \n32 \n nm)/SiO 2(2 nm) series are plotted in Figures S3 (a), S4(a) and S5(a), respectively. Here, Ms is \ncalculated from the measured magnetic moment divided by the total volume of the Co 20Fe60B20 \nlayer. These films have very small coercive field (~5 Oe).  The effective magnetization Meff of \nthe samples are obtained by fitting the bias magnetic field ( H) dependent precessional frequency \n(f) obtained from the TR-MOKE measurements, with the Kittel formula (equation (13) of the \narticle) (see Figures S3 (b), S4(b) and S5(b)). We have finally plotted the variation of Meff and \nMs with W, Co 20Fe60B20, and Cu thickness in Figures S3 (c), S4(c), and S5(c), respectively. The \nMeff and Ms values are found to be in close proximity with each other, indicating that the \ninterfacial anisotropy is small for all these samples. Since these films were not annealed post-\ndeposition, the interfacial anisotropy stays small and plays only a minor role in modifying the \nmagnetization dynamics for these heterostructures.  \n \n \nFigure S3.     (a) VSM loops for W( t)/ Co20Fe60B20(3 nm)/SiO 2(2 nm). (b) Kittel fit (solid line) \nto experimental data (symbol) of precessional frequency vs. magnetic field for W( t)/ \nCo20Fe60B20(3 nm)/SiO 2( 2 nm) samples. (c) Comparison of variation of Ms from VSM and  Meff \nfrom TR-MOKE as a function of W layer thickness. \n \n \n 0 4 8 12 16500100015002000\n  \nt (nm)Ms (emu/cc)\n500100015002000 Meff (emu/cc)-100001000\n-100001000\n-0.4 0.0 0.4-100001000  \n  \nt =1 nm\nt = 8 nm\n  \n  \nH (kOe)t = 15 nm\n  \n  M (emu/cc)141618\n141618\n1.5 2.0 2.5141618  \n  \nt = 1 nm\nt = 8 nm\nt = 15 nm  \n  f (GHz)\nH (kOe)  \n  (a) (b)\n(c)  \n33 \n  \n \n \nFigure S4.     (a) VSM loops for W(4 nm)/ Co 20Fe60B20(d)/SiO2(2 nm). (b) Kittel fit (solid line) \nto experimental data (symbol) of precessional frequency vs. magnetic field for W(4 nm)/ \nCo20Fe60B20(d)/SiO2(2 nm) samples. (c) Comparison between variation of Ms from VSM and  \nMeff from TR-MOKE as a function of Co 20Fe60B20 layer thickness. \n \n \nFigure S5.     (a) VSM loops for W(4 nm)/Cu( c)/ Co20Fe60B20(3 nm)/SiO 2(2 nm). (b) Kittel fit \n(solid line) to experimental data (symbol) of precessional frequency vs. magnetic field for W(4 \nnm)/ Cu(c)/ Co20Fe60B20(3 nm)/SiO 2(2 nm) samples. (c) Comparison between variation of Ms \nfrom VSM and  Meff from TR-MOKE as a function of Cu layer thickness.  \n \n 0 3 6 9500100015002000\n  \nd (nm)Ms (emu/cc)\n500100015002000 Meff (emu/cc)141618\n141618\n1.5 2.0 2.514161820  \n  \n  \n  \n  \n  f  (GHz)\nH (kOe)d = 10 nmd = 5 nmd = 2 nm\n-100001000\n-100001000\n-0.3 0.0 0.3-100001000  \n  \nd = 5 nmd = 2 nm\nd = 10 nmM (emu/cc)\n  \n  \nH (kOe)  \n  (a) (b)\n(c)\n0.00 0.25 0.50 0.75 1.00500100015002000\n  \nc (nm)Ms (emu/cc)\n500100015002000 Meff (emu/cc)141618\n141618\n1.5 2.0 2.5141618  \n  \n  \n  \n  \n  f (GHz)\nH (kOe)c = 1 nmc = 0.5 nmc = 0 nm\n-100001000\n-100001000\n-0.4 0.0 0.4-100001000\nH (kOe)M (emu/cc)  \n  \nc = 0.5 nmc = 0 nm \nc = 1 nm  \n  \n  \n  (a) (b)\n(c)  \n34 \n 4. Variation of effective damping with precessional frequency:   \nFor all the sample series the time-resolved precessional oscillations have been recorded at \ndifferent bias magnetic field strength. The precessional frequency has been extracted by taking \nthe fast Fourier transform (FFT) of the background-subtracted time-resolved Kerr rotation. \nSubsequently, the time-resolved precessional oscillations have also been fitted with a damped \nsinusoidal function given by equation (12) of the article to extract the decay time τ. The value \nof effective Gilbert damping parameter ( αeff) have then been extracted using equation (14). \nVariation of this αeff with precessional frequency ( f) is plotted to examine the nature of the \ndamping. Here, we have plotted the time-resolved precessional oscillations ( Figure S6( a)), FFT \npower spectra ( Figure S6( b)) and αeff vs. f (Figure S6( c)) for Sub/W(0.5 nm)/Co 20Fe60B20(3 \nnm)/SiO 2(2 nm) sample.  It is clear from this data that damping is frequency independent, which \nrules out the contribution of various extrinsic factors such as two-magnon scattering, \ninhomogeneous anisotropy, eddy current in the damping for our samples.  \n \n \nFigure S6.     (a) Background subtracted time-resolved precessional oscillations at different bias \nmagnetic fields for Sub/W(0.5 nm)/Co 20Fe60B20(3 nm)/SiO 2(2 nm) sample, where symbols \nrepresent the experimental data points and solid lines represent fits using equation (12) of the \narticle. (b) The FFT power spectra of the time-resolved precessional oscillations showing the 0.0 0.3 0.6 0.9 1.2 1.5  \n  \n  \n  \n  \n  \nH = 1.50 kOeH = 1.80 kOeH = 2.10 kOe\n  \n  Kerr Rotation (arb. units)\nTime (ns)H = 2.30 kOe\n0 10 20 30  \n  \n  \n  \n  \n  \n  \n  Power (arb. units)\nf (GHz)12 14 16 180.0000.0080.016\n  \n \nf (GHz)eff\n(c)\n(b) (a)  \n35 \n precessional frequency. (c) Variation of effective damping with precessional frequency is \nshown by symbol and the dotted line is guide to the eye. "    },    {        "title": "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": "1507.03075v1.Realization_of_the_thermal_equilibrium_in_inhomogeneous_magnetic_systems_by_the_Landau_Lifshitz_Gilbert_equation_with_stochastic_noise__and_its_dynamical_aspects.pdf",        "content": "Realization of the thermal equilibrium in inhomogeneous\nmagnetic systems by the Landau-Lifshitz-Gilbert equation with\nstochastic noise, and its dynamical aspects\nMasamichi Nishino1\u0003and Seiji Miyashita2;3\n1Computational Materials Science Center,\nNational Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan\n2Department of Physics, Graduate School of Science,\nThe University of Tokyo, Bunkyo-Ku, Tokyo, Japan\n3CREST, JST, 4-1-8 Honcho Kawaguchi, Saitama, 332-0012, Japan\n(Dated: July 14, 2015)\n1arXiv:1507.03075v1  [cond-mat.mtrl-sci]  11 Jul 2015Abstract\nIt is crucially important to investigate e\u000bects of temperature on magnetic properties such as\ncritical phenomena, nucleation, pinning, domain wall motion, coercivity, etc. The Landau-Lifshitz-\nGilbert (LLG) equation has been applied extensively to study dynamics of magnetic properties.\nApproaches of Langevin noises have been developed to introduce the temperature e\u000bect into the\nLLG equation. To have the thermal equilibrium state (canonical distribution) as the steady state,\nthe system parameters must satisfy some condition known as the \ructuation-dissipation relation.\nIn inhomogeneous magnetic systems in which spin magnitudes are di\u000berent at sites, the condition\nrequires that the ratio between the amplitude of the random noise and the damping parameter\ndepends on the magnitude of the magnetic moment at each site. Focused on inhomogeneous mag-\nnetic systems, we systematically showed agreement between the stationary state of the stochastic\nLLG equation and the corresponding equilibrium state obtained by Monte Carlo simulations in\nvarious magnetic systems including dipole-dipole interactions. We demonstrated how violations of\nthe condition result in deviations from the true equilibrium state. We also studied the characteris-\ntic features of the dynamics depending on the choice of the parameter set. All the parameter sets\nsatisfying the condition realize the same stationary state (equilibrium state). In contrast, di\u000berent\nchoices of parameter set cause seriously di\u000berent relaxation processes. We show two relaxation\ntypes, i.e., magnetization reversals with uniform rotation and with nucleation.\nPACS numbers: 75.78.-n 05.10.Gg 75.10.Hk 75.60.Ej\n2|||||||||||||||||||||||||-\nI. INTRODUCTION\nThe Landau-Lifshitz-Gilbert (LLG) equation1has been widely used in the study of dy-\nnamical properties of magnetic systems, especially in micromagnetics. It contains a relax-\nation mechanism by a phenomenological longitudinal damping term. The Landau-Lifshitz-\nBloch (LLB) equation2contains, besides the longitudinal damping, a phenomenological\ntransverse damping and the temperature dependence of the magnetic moment are taken\ninto account with the aid of the mean-\feld approximation. Those equations work well in\nthe region of saturated magnetization at low temperatures.\nThermal e\u000bects are very important to study properties of magnets, e.g., the amount of\nspontaneous magnetization, hysteresis nature, relaxation dynamics, and the coercive force in\npermanent magnets. Therefore, how to control temperature in the LLG and LLB equations\nhas been studied extensively. To introduce temperature in equations of motion, a coupling\nwith a thermal reservoir is required. For dynamics of particle systems which is naturally\nexpressed by the canonical conjugated variables, i.e., ( q;p), molecular dynamics is performed\nwith a Nose-Hoover (NH) type reservoir3{5or a Langevin type reservoir6. However, in the\ncase of systems of magnetic moments, in which dynamics of angular momenta is studied, NH\ntype reservoirs are hardly used due to complexity7. On the other hand, the Langevin type\nreservoirs have been rather naturally applied2,8{18although multiplicative noise19requires the\nnumerical integration of equations depending on the interpretation, i.e., Ito or Stratonovich\ntype.\nTo introduce temperature into a LLG approach by a Langevin noise, a \ructuation-\ndissipation relation is used, where the temperature is proportional to the ratio between\nthe strength of the \ructuation (amplitude of noise) and the damping parameter of the\nLLG equation. For magnetic systems consisting of uniform magnetic moments, the ratio is\nuniquely given at a temperature and it has been often employed to study dynamical prop-\nerties, e.g., trajectories of magnetic moments of nano-particles8, relaxation dynamics in a\nspin-glass system20or in a semiconductor21. The realization of the equilibrium state by\nstochastic LLG approaches by numerical simulations is an important issue, and it has been\ncon\frmed in some cases of the Heisenberg model for uniform magnetic moments.22,23\n3In general cases, however, magnetic moments in atomic scale have various magnitudes of\nspins. This inhomogeneity of magnetization is important to understand the mechanisms of\nnucleation or pinning.24{28To control the temperature of such systems, the ratio between the\namplitude of noise and the damping parameter depends on the magnetic moment at each\nsite. In order to make clear the condition for the realization of the canonical distribution\nas the stationary state in inhomogeneous magnetic systems, we review the guideline of the\nderivation of the condition in the Fokker-Planck equation formalism in the Appendix A.\nSuch a generalization of the LLG equation with a stochastic noise was performed to study\nproperties of the alloy magnet GdFeCo29, in which two kinds of moments exist. They ex-\nploited a formula for the noise amplitude, which is equivalent to the formula of our condition\nA (see Sec II). They found surprisingly good agreements of the results between the stochas-\ntic LLG equation and a mean-\feld approximation. However, the properties in the true\ncanonical distribution is generally di\u000berent from those obtained by the mean-\feld analysis.\nThe LLG and LLB equations have been often applied for continuous magnetic systems or\nassemblies of block spins in the aim of simulation of bulk systems, but such treatment of the\nbulk magnets tend to overestimate the Curie temperature11, and it is still under develop-\nment to obtain properly magnetization curves in the whole temperature region2,11,17,18. The\nin\ruence of coarse graining of block spin systems on the thermal properties is a signi\fcant\ntheme, which should be clari\fed in the future. To avoid such a di\u000eculty, we adopt a lattice\nmodel, in which the magnitude of the moment is given at each magnetic site.\nWithin the condition there is some freedom of the choice of parameter set. In the present\npaper, in particular, we investigate the following two cases of parameter sets, i.e., case A,\nin which the LLG damping constant is the same in all the sites and the amplitude of the\nnoise depends on the magnitude of the magnetic moment at each site, and case B, in which\nthe amplitude of the noise is the same in all the sites and the damping constant depends on\nthe magnitude of the moment. (see Sec II.). We con\frm the realization of the equilibrium\nstate, i.e., the canonical distribution in various magnetic systems including critical region by\ncomparison of magnetizations obtained by the LLG stochastic approach with those obtained\nby standard Monte Carlo simulations, not by the mean-\feld analysis. We study systems\nwith not only short range interactions but also dipole-dipole interactions, which causes\nthe demagnetizing \feld statically. We \fnd that di\u000berent choices of the parameter set which\nsatis\fes the \ructuation-dissipation relation give the same stationary state (equilibrium state)\n4even near the critical temperature. We also demonstrate that deviations from the relation\ncause systematic and signi\fcant deviations of the results.\nIn contrast to the static properties, we \fnd that di\u000berent choices of parameter set cause\nserious di\u000berence in the dynamics of the relaxation. In particular, in the rotation type\nrelaxation in isotropic spin systems, we \fnd that the dependences of the relaxation time on\nthe temperature in cases A and B show opposite correlations as well as the dependences of the\nrelaxation time on the magnitude of the magnetic moment. That is, the relaxation time of\nmagnetization reversal under an unfavorable external \feld is shorter at a higher temperature\nin case A, while it is longer in case B. On the other hand, the relaxation time is longer for\na larger magnetic moment in case A, while it is shorter in case B. We also investigate the\nrelaxation of anisotropic spin systems and \fnd that the metastability strongly a\u000bects the\nrelaxation at low temperatures in both cases. The system relaxes to the equilibrium state\nfrom the metastable state by the nucleation type of dynamics. The relaxation time to the\nmetastable state and the decay time of the metastable state are a\u000bected by the choice of\nthe parameter set.\nThe outline of this paper is as follows. The model and the method in this study are ex-\nplained in Sec II. Magnetization processes as a function of temperature in uniform magnetic\nsystems are studied in Sec III. Magnetizations as a function of temperature for inhomoge-\nneous magnetic systems are investigated in Sec. IV, in which not only exchange interactions\n(short-range) but also dipole interactions (long-range) are taken into account. In Sec. V\ndynamical aspects with the choice of the parameter set are considered, and the dependences\nof the relaxation process on the temperature and on the magnitude of magnetic moments\nare also discussed. The relaxation dynamics via a metastable state is studied in Sec. VI.\nSec. VII is devoted to summary and discussion. In Appendix A the Fokker-Planck equation\nfor inhomogeneous magnetic systems is given both in Stratonovich and Ito interpretations,\nand Appendix B presents the numerical integration scheme in this study.\nII. MODEL AND METHOD\nAs a microscopic spin model, the following Hamiltonian is adopted,\nH=\u0000X\nhi;jiJi;jSi\u0001Sj\u0000X\niDA\ni(Sz\ni)2\u0000X\nihi(t)Sz\ni+X\ni6=kC\nr3\nik\u0010\nSi\u0001Sk\u00003(rik\u0001Si)(rik\u0001Sk)\nr2\nik\u0011\n:(1)\n5Here we only consider a spin angular momentum Sifor a magnetic moment Miat each\nsite (iis the site index) and regard Mi=Siignoring the di\u000berence of the sign between\nthem and setting a unit: g\u0016B= 1 for simplicity, where gis the g-factor and \u0016Bis the Bohr\nmagneton30. Interaction Ji;jbetween the ith andjth magnetic sites indicates an exchange\ncoupling,hi;jidenotes a nearest neighbor pair, DA\niis an anisotropy constant for the ith\nsite,hiis a magnetic \feld applied to the ith site, and the \fnal term gives dipole interactions\nbetween the ith andkth sites whose distance is ri;k, whereC=1\n4\u0019\u00160is de\fned using the\npermeability of vacuum \u00160.\nThe magnitude of the moment Miis de\fned as Mi\u0011jMij, which is not necessarily\nuniform but may vary from site to site. In general, the damping parameter may also have\nsite dependence, i.e., \u000bi, and thus the LLG equation at the ith site is given by\nd\ndtMi=\u0000\rMi\u0002He\u000b\ni+\u000bi\nMiMi\u0002dMi\ndt; (2)\nor in an equivalent formula:\nd\ndtMi=\u0000\r\n1 +\u000b2\niMi\u0002He\u000b\ni\u0000\u000bi\r\n(1 +\u000b2\ni)MiMi\u0002(Mi\u0002He\u000b\ni); (3)\nwhere\ris the gyromagnetic constant. Here He\u000b\niis the e\u000bective \feld at the ith site and\ndescribed by\nHe\u000b\ni=\u0000@\n@MiH(M1;\u0001\u0001\u0001;MN;t) (4)\n, which contains \felds from the exchange and the dipole interactions, the anisotropy, and\nthe external \feld.\nWe introduce a Langevin-noise formalism for the thermal e\u000bect. There have been several\nways for the formulation to introduce a stochastic term into the LLG equation. The stochas-\ntic \feld can be introduced into the precession term and/or damping term8,9,11. Furthermore,\nan additional noise term may be introduced10,12. In the present study we add the random\nnoise to the e\u000bective \feld He\u000b\ni!He\u000b\ni+\u0018iand we have\nd\ndtMi=\u0000\r\n1 +\u000b2\niMi\u0002(He\u000b\ni+\u0018i)\u0000\u000bi\r\n(1 +\u000b2\ni)MiMi\u0002(Mi\u0002(He\u000b\ni+\u0018i)); (5)\nwhere\u0018\u0016\niis the\u0016(=1,2 or 3 for x,yorz) component of the white Gaussian noise applied at\ntheith site and the following properties are assumed:\nh\u0018\u0016\nk(t)i= 0;h\u0018\u0016\nk(t)\u0018\u0017\nl(s)i= 2Dk\u000ekl\u000e\u0016\u0017\u000e(t\u0000s): (6)\n6We call Eq. (5) stochastic LLG equation. We derive a Fokker-Planck equation6,8for\nthe stochastic equation of motion in Eq. (5) in Stratonovich interpretation, as given in\nappendix A,\n@\n@tP(M1;\u0001\u0001\u0001;MN;t) =X\ni\r\n1 +\u000b2\ni@\n@Mi\u0001\u001a\u0014\u000bi\nMiMi\u0002(Mi\u0002He\u000b\ni) (7)\n\u0000\rDiMi\u0002(Mi\u0002@\n@Mi)\u0015\nP(M1;\u0001\u0001\u0001;MN;t)\u001b\n:\nHere we demand that the distribution function at the stationary state ( t!1 ) of the\nequation of motion (Eq. (7)) agrees with the canonical distribution of the system (Eq. (1))\nat temperature T, i.e.,\nPeq(M1;\u0001\u0001\u0001;MN)/exp\u0010\n\u0000\fH(M1;\u0001\u0001\u0001;MN)\u0011\n; (8)\nwhere\f=1\nkBT.\nConsidering the relation\n@\n@MiPeq(M1;\u0001\u0001\u0001;MN) =\fHe\u000b\niPeq(M1;\u0001\u0001\u0001;MN); (9)\nwe \fnd that if the following relation\n\u000bi\nMi\u0000\rDi\f= 0 (10)\nis satis\fed at each site i, the canonical distribution in the equilibrium state is assured.\nWhen the magnetic moments are uniform, i.e., the magnitude of each magnetic moment\nis the same and Mi=jMij=M, the parameters \u000biandDiare also uniform \u000bi=\u000band\nDi=Dfor a given T. However, when Miare di\u000berent at sites, the relation (10) must be\nsatis\fed at each site independently. There are several ways of the choice of the parameters\n\u000biandDito satisfy this relation. Here we consider the following two cases: A and B.\nA: we take the damping parameter \u000bito be the same at all sites, i.e., \u000b1=\u000b2=\u0001\u0001\u0001=\u000bN\u0011\n\u000b. In this case the amplitude of the random \feld at the ith site should be\nDi=\u000b\nMikBT\n\r/1\nMi: (11)\nB: we take the amplitude of the random \feld to be the same at all sites, i.e., D1=D2=\n\u0001\u0001\u0001=DN\u0011D. In this case the damping parameter at the ith site should be\n\u000bi=D\rMi\nkBT/Mi: (12)\n700.20.40.60.81\n0123456m\nTFIG. 1: (color online) Comparison of the temperature dependence of min the stationary state\nbetween the stochastic LLG method and the Langevin function (green circles). Crosses and boxes\ndenotemin case A ( \u000b= 0:05) and case B ( D= 1:0), respectively. In the stochastic LLG\nsimulation \u0001 t= 0:005 was set and 80000 time steps (40,000 steps for equilibration and 40,000\nsteps for measurement) were employed. The system size N=L3= 103was adopted.\nWe study whether the canonical distribution is realized in both cases by comparing data\nobtained by the stochastic LLG method with the exact results or with corresponding data\nobtained by Monte Carlo simulations. We set the parameters \r= 1 andkB= 1 hereafter.\nIII. REALIZATION OF THE THERMAL EQUILIBRIUM STATE IN HOMOGE-\nNEOUS MAGNETIC SYSTEMS\nA. Non-interacting magnetic moments\nAs a \frst step, we check the temperature e\u000bect in the simplest case of non-interacting\nuniform magnetic moments, i.e., Ji;j= 0,DA\ni= 0,C= 0 in Eq. (1) and Mi=M(or\nSi=S), where\u000bandDhave no site i-dependence. In this case the magnetization in a\nmagnetic \feld ( h) at a temperature ( T) is given by the Langevin function:\nm=1\nNhNX\ni=1Sz\nii=M \ncoth\u0010hM\nkBT\u0011\n\u0000kBT\nhM!\n: (13)\nWe compare the stationary state obtained by the stochastic LLG method and Eq. (13).\n8We investigate m(T) ath= 2 forM= 1. Figure 1 shows m(T) when\u000b= 0:05 is \fxed (case\nA) and when D= 1:0 is \fxed (case B). We \fnd a good agreement between the results of\nthe stochastic LLG method and the Langevin function in the whole temperature region as\nlong as the relation (10) is satis\fed. Numerical integration scheme is given in Appendix B.\nThe time step of \u0001 t= 0:005 and total 80000 time steps (40000 steps for equilibration and\n40000 steps for measurement) were adopted.\nB. Homogeneous magnetic moments with exchange interactions\nNext, we investigate homogenous magnetic moments ( Mi=jMij=M) in three di-\nmensions. The following Hamiltonian ( C= 0,Ji;j=J,DA\ni=DA, andh(t) =hin Eq.\n(1)):\nH=\u0000X\nhi;jiJSi\u0001Sj\u0000X\niDA(Sz\ni)2\u0000X\nihSz\ni (14)\nis adopted.\nThere is no exact formula for magnetization ( m) as a function of temperature for this\nsystem, and thus a Monte Carlo (MC) method is applied to obtain reference magnetization\ncurves for the canonical distribution because MC methods have been established to obtain\n\fnite temperature properties for this kind of systems in the equilibrium state. Here we\nemploy a MC method with the Metropolis algorithm to obtain the temperature dependence\nof magnetization.\nIn order to check the validity of our MC procedure, we investigated magnetization\ncurves as functions of temperature (not shown) with system-size dependence for the three-\ndimensional classical Heisenberg model ( DA= 0 andh= 0 in Eq. (14)), and con\frmed that\nthe critical temperature agreed with past studies31, wherekBTc= 1:443Jfor the in\fnite\nsystem size with M= 1.\nWe givem(T) for a system of M= 2 with the parameters J= 1,h= 2 andDA= 1:0\nfor cases A and B in Fig 2. The system size was set N=L3= 103and periodic boundary\nconditions (PBC) were used. Green circles denote mobtained by the Monte Carlo method.\nAt each temperature ( T) 10,000 MC steps (MCS) were applied for the equilibration and\nfollowing 10,000\u000050,000 MCS were used for measurement to obtain m. Crosses and boxes\ndenotemin the stationary state of the stochastic LLG equation in case A ( \u000b= 0:05) and\nin case B ( D= 1:0), respectively. Here \u0001 t= 0:005 was set and 80000 steps (40000 for\n900.511.52\n0 5 10 15 20 25m\nTFIG. 2: (color online) Comparison of temperature ( T) dependence of mbetween the Monte Carlo\nmethod (green circles) and the stochastic LLG method in the homogeneous magnetic system with\nM= 2. Crosses and boxes denote case A with \u000b= 0:05 and case B with D= 1:0, respectively.\ntransient and 40000 for measurement) were used to obtain the stationary state of m. The\nm(T) curves show good agreement between the MC method and the stochastic LLG method\nin both cases. We checked that the choice of the initial state for the MC and the stochastic\nLLG method does not a\u000bect the results. The dynamics of the stochastic LLG method leads\nto the equilibrium state at temperature T.\nIV. REALIZATION OF THE THERMAL EQUILIBRIUM STATE IN INHOMO-\nGENEOUS MAGNETIC SYSTEMS\nA. Inhomogeneous magnetic moments with exchange interactions\nHere we study a system which consists of two kinds of magnitudes of magnetic moments.\nThe Hamiltonian (14) is adopted but the moment Mi=jMijhasi-dependence. We investi-\ngate a simple cubic lattice composed of alternating M= 2 andM= 1 planes (see Fig. 3 (a)),\nwhereJ= 1,h= 2 andDA= 1:0 are applied. We consider two cases A and B mentioned\nin Sec. II.\nThe reference of m(T) curve was obtained by the MC method and is given by green\ncircles in Figs. 3 (b) and (c). In the simulation, at each temperature ( T) 10,000 MCS were\napplied for the equilibration and following 10,000 \u000050,000 MCS were used for measurement.\n10(a)\n00.511.5\n05 1 0 1 5 2 0m\nT(b)\n00.511.5\n05 1 0 1 5 2 0m\nT(c)FIG. 3: (color online) (a) A part of the system composed of alternating M= 2 (red long\narrows) and M= 1 (short blue arrows) layers. (b) Comparison of temperature ( T) dependence of\nmbetween the Monte Carlo method (green circles) and the stochastic LLG method for \u000b= 0:05.\n\u0001t= 0:005 and 80,000 steps (40,000 for transient time and 40,000 for measurement) were employed.\nCrosses denote mwhenDi=D(Mi)\u0011\u000b\nMikBT\n\rwas used. Triangles and Diamonds are mfor\nDi=D(1) =\u000bkBT\n\rfor alliandDi=D(2) =\u000b\n2kBT\n\rfor alli, respectively. (c) Comparison\nof temperature ( T) dependence of mbetween the Monte Carlo method (green circles) and the\nstochastic LLG method for D= 1:0. \u0001t= 0:005 and 80,000 steps (40,000 for transient time and\n40,000 for measurement) were employed. Crosses denote mwhen\u000bi=\u000b(Mi)\u0011D\rMi\nkBTwas used.\nTriangles are mfor\u000bi=\u000b(Mi= 1) =D\r\u00021\nkBTfor alliand Diamonds are mfor\u000bi=\u000b(2) =D\r\u00022\nkBT\nfor alli.\n11The system size N=L3= 103was adopted with PBC. In case A, \u000b(= 0:05) is common for\nall magnetic moments in the stochastic LLG method and Mi(orSi) dependence is imposed\nonDiasDi=D(Mi)\u0011\u000b\nMikBT\n\r. In case B, D= 1:0 is common for all magnetic moments in\nthe stochastic LLG method and \u000bi=\u000b(Mi)\u0011D\rMi\nkBT. Crosses in Figs. 3 (b) and (c) denote\nmby the stochastic LLG method for cases A and B, respectively. For those simulations\n\u0001t= 0:005 and 80,000 steps (40,000 for transient time and 40,000 for measurement) were\nemployed at each temperature. In both Figs. 3 (b) and (c), we \fnd good agreement between\nm(T) by the stochastic LLG method (crosses) and m(T) by the MC method (green circles).\nNext, we investigate how the results change if we take wrong choices of parameters. We\nstudym(T) when a uniform value Di=Dfor case A ( \u000bi=\u000bfor case B) is used for all\nspins, i.e., for both Mi= 1 andMi= 2. IfD(Mi= 2) =\u000b\n2kBT\n\ris used for all spins, m(T)\nis shown by Diamonds in Fig. 3 (b), while if D(Mi= 1) =\u000bkBT\n\ris applied for all spins,\nm(T) is given by triangles in Fig. 3 (b). In the same way, we study m(T) for a uniform\nvalue of\u000b. In Fig. 3 (c) triangles and diamonds denote m(T) when\u000bi=\u000b(Mi= 1) and\n\u000bi=\u000b(Mi= 2) are used, respectively. We \fnd serious di\u000berence in m(T) when we do not\nuse correct Mi-dependent choices of the parameters. The locations of triangle (diamond) at\neach temperature Tare the same in Figs. 3 (a) and (b), which indicates that if the ratio\n\u000b=D is the same in di\u000berent choices, the same steady state is realized although this state is\nnot the true equilibrium state for the inhomogeneous magnetic system. Thus we conclude\nthat to use proper relations of Mi-dependence of Dior\u000biis important for m(T) curves of\ninhomogeneous magnetic systems and wrong choices cause signi\fcant deviations.\nB. Critical behavior of Inhomogeneous magnetic moments\nIn this subsection, we examine properties near the critical temperature. Here we adopt\nthe case ofh= 0 andDA= 0 in the same type of lattice with M= 1 and 2 as Sec. IV A. We\ninvestigate both cases of the temperature control (A and B). The Hamiltonian here has O(3)\nsymmetry and mis not a suitable order parameter. Thus we de\fne the following quantity\nas the order parameter31:\nma=q\nm2\nx+m2\ny+m2\nz; (15)\n1200.511.5\n0123456ma\nTFIG. 4: (color online) Comparison of temperature ( T) dependence of mabetween the MC method\n(green circles) and the stochastic LLG method for the system of inhomogeneous magnetic moments.\nN=L3= 203. PBC were used. In the MC method 10,000 MCS and following 50,000 MCS were\nused for equilibration and measurement at each temperature, respectively. The stochastic LLG\nmethod was performed in case A with \u000b= 0:05 (croses) and in case B with D= 1:0 (diamonds).\nHere \u0001t= 0:005 was applied and 240,000 steps were used (40,000 for transient and 200,000 for\nmeasurement).\nwhere\nmx=1\nNhNX\ni=1Sx\nii; my=1\nNhNX\ni=1Sy\nii;andmz=m=1\nNhNX\ni=1Sz\nii: (16)\nIn Fig. 4, green circles denote temperature ( T) dependence of magiven by the MC\nmethod. The system size N=L3= 203with PBC was adopted and in MC simulations\n10,000 MCS and following 50,000 MCS were employed for equilibration and measurement,\nrespectively at each temperature. The magnetizations of maobtained by the stochastic LLG\nmethod for case A (crosses) and case B (diamonds) are given in Fig. 4. Here \u000b= 0:05 and\nD= 1:0 were used for (a) and (b), respectively. \u0001 t= 0:005 was set and 240,000 steps\n(40,000 for transient and 200,000 for measurement) were applied.\nIn both cases ma(T) curve given by the stochastic LLG method shows good agreement\nwith that obtained by the MC method. Thus, we conclude that as long as the relation (10)\nis satis\fed, the temperature dependence of the magnetization is reproduced very accurately\neven around the Curie temperature, regardless of the choice of the parameter set.\n1300.511.5\n0123456m\nTFIG. 5: (color online) Comparison of temperature ( T) dependence of mbetween the Monte Carlo\nmethod (green circles) and the stochastic LLG method. Crosses and diamonds denote case A\nwith\u000b= 0:05 and case B with D= 1:0, respectively. A reduction of mfrom fully saturated\nmagnetization is observed at around T= 0 due to the dipole interactions. As a reference, mby\nthe MC method without the dipole interactions ( C= 0) is given by open circles.\nC. Inhomogeneous magnetic moments with exchange and dipole interactions\nWe also study thermal e\u000bects in a system with dipole interactions. We use the same\nlattice as in the previous subsections. The system is ( Ji;j=J,DA\ni=DA, andhi(t) =hin\nEq. (1)) given by\nH=\u0000X\nhi;jiJSi\u0001Sj\u0000X\niDA(Sz\ni)2\u0000X\nihSz\ni+X\ni6=kC\nr3\nik\u0010\nSi\u0001Sk\u00003(rik\u0001Si)(rik\u0001Sk)\nr2\nik\u0011\n:(17)\nHere a cubic lattice with open boundary conditions (OBC) is used. Since Jis much larger\nthanC=a3(J\u001dC=a3) for ferromagnets, where ais a lattice constant between magnetic\nsites. However, we enlarge dipole interaction as C= 0:2 witha= 1 forJ= 1 to highlight\nthe e\u000bect of the noise on dipole interactions. We set other parameters as h= 0:1,DA= 0:1.\nStudies with realistic situations will be given separately.\nWe study cases A ( \u000b= 0:05) and B ( D= 1:0) for this system. We depict in Fig. 5\nthe temperature ( T) dependences of mwith comparison between the MC (green circles) and\nstochastic LLG methods. Crosses and diamonds denote m(T) for cases A and B, respectively.\nDipole interactions are long-range interactions and we need longer equilibration steps, and\n14we investigate only a small system with N=L3= 63. In the MC method 200,000 MCS\nwere used for equilibration and 600,000 steps were used for measurement of m, and for\nthe stochastic LLG method \u0001 t= 0:005 was set and 960,000 steps (160,000 and 800,000\ntime steps for equilibration and measurement, respectively) were consumed. A reduction of\nmfrom fully saturated magnetization is observed. As a reference, mby the MC method\nwithout the dipole interactions ( C= 0) is given by open circles in Fig. 5. This reduction of\nmis caused by the dipole interactions.\nWe \fnd that even when dipole interactions are taken into account in inhomogeneous\nmagnetic moments, suitable choices of the parameter set leads to the equilibrium state.\nFinally, we comment on the comparison between the LLG method and the Monte Carlo\nmethod. To obtain equilibrium properties of spin systems, the Monte Carlo method is more\ne\u000ecient and powerful in terms of computational cost. It is much faster than the stochastic\nLLG method to obtain the equilibrium m(T) curves, etc. For example, it needs more than\n10 times of CPU time of the MC method to obtain the data for Fig. 5. However, the MC\nmethod has little information on the dynamics and the stochastic LLG method is used to\nobtain dynamical properties because it is based on an equation of motion of spins. Thus, it\nis important to clarify the nature of stochastic LLG methods including the static properties.\nFor static properties, as we saw above, the choice of the parameter set, e.g., cases A and\nB, did not give di\u000berence. However, the choice gives signi\fcant di\u000berence in dynamical\nproperties, which is studied in the following sections.\nV. DEPENDENCE OF DYNAMICS ON THE CHOICE OF THE PARAMETER\nSET IN ISOTROPIC SPIN SYSTEMS ( DA=0)\nNow we study the dependence of dynamics on the choice of parameter set. The temper-\nature is given by\nkBT=\rDiMi\n\u000bi; (18)\nwhich should be the same for all the sites. In general, if the parameter D(amplitude of\nthe noise) is large, the system is strongly disturbed, while if the parameter \u000b(damping\nparameter) is large, the system tends to relax fast. Therefore, even if the temperature is\nthe same, the dynamics changes with the values of Dand\u000b. When the anisotropy term\nexists, i.e.,DA6= 0, in homogeneous systems ( Mi=M) given by Eq. (14), the Stoner-\n15Wohlfarth critical \feld is hc= 2MDAatT= 0. If the temperature is low enough, the\nmetastable nature appears in relaxation. On the other hand, if Tis rather high orDA= 0,\nthe metastable nature is not observed. In this section we focus on dynamics of isotropic spin\nsystems, i.e.,DA=0.\nA. Relaxation with temperature dependence\nIn this subsection we investigate the temperature dependence of magnetization relaxation\nin cases A and B. We adopt a homogeneous system ( Mi=M= 2) withDA= 0 in Eq. (14).\nInitially all spins are in the spin down state and they relax under a unfavorable external\n\feldh= 2. The parameter set M= 2,\u000b= 0:05,D= 0:05 givesT= 2 by the condition\n(Eq. (10)). Here we study the system at T= 0:2;1;2, and 10. We set \u000b= 0:05 in case A\nand the control of the temperature is performed by D, i.e.D= 0:005;0:025;0:05, and 0:25,\nrespectively. In case B we set D= 0:05, and the control of the temperature is realized by\n\u000b, i.e.,\u000b= 0:5;0:1;0:05, and 0:01, respectively.\nWe depict the temperature dependence of m(t) for cases A and B in Figs. 6 (a) and (b),\nrespectively. Here the same random number sequence was used for each relaxation curve.\nRed dash dotted line, blue dotted line, green solid line, and black dashed line denote T= 0:2,\nT= 1,T= 2 andT= 10, respectively. Relaxation curves in initial short time are given in\nthe insets.\nIn case A, as the temperature is raised, the initial relaxation speed of mbecomes faster\nand the relaxation time to the equilibrium state also becomes shorter. This dependence is\nascribed to the strength of the noise with the dependence D/T, and a noise with a larger\namplitude disturbs more the precession of each moment, which causes faster relaxation.\nOn the other hand, in case B, the relaxation time to the equilibrium state is longer at\nhigher temperatures although the temperature dependence of the initial relaxation speed of\nmis similar to the case A. In the initial relaxation process all the magnetic moments are\nin spin-down state ( Sz\ni'\u00002). There the direction of the local \feld at each site is given\nbyHe\u000b\ni'JP\njSz\nj+h=\u00002\u00026 + 2 =\u000010, which is downward and the damping term\ntends to \fx moments to this direction. Thus, a large value of the damping parameter at a\nlow temperature T(\u000b/1\nT) suppresses the change of the direction of each moment and the\ninitial relaxation speed is smaller. However, in the relaxation process thermal \ructuation\n16-2-1012\n0 50 100 150 200m\ntime(a)\n-2.2-2-1.8-1.6-1.4-1.2-1\n012345678\n-2-1012\n0 50 100 150 200m\ntime-2.2-2-1.8-1.6-1.4-1.2-1\n012345678(b)FIG. 6: (color online) (a) Time dependence of the magnetization ( m(t)) in case A, where \u000b= 0:05\nfor a homogeneous system with M= 2. Red dash dotted line, blue dotted line, green solid line,\nand black dashed line denote T= 0:2,T= 1,T= 2 andT= 10, respectively. Inset shows the time\ndependence of m(t) in the initial relaxation process. (b) Time dependence of the magnetization\n(m(t)) in case B, where D= 0:05 for a homogeneous system with M= 2. Correspondence between\nlines and temperatures is the same as (a).\ncauses a deviation of the local \feld and then a rotation of magnetic moments from \u0000z\ntozdirection advances (see also Fig. 11 ). Once the rotation begins, the large damping\nparameter accelerates the relaxation and \fnally the relaxation time is shorter.\nB. Relaxation with spin-magnitude dependence\nNext we study the dependence of relaxation on the magnitude of magnetic moments\nin cases A and B. Here we adopt a homogeneous system ( Mi=M) without anisotropy(\nDA= 0) atT= 2 andh= 2. The initial spin con\fguration is the same as the previous\nsubsection. Because\nD/T\nM;and\u000b/M\nT; (19)\nraising the value of Mis equivalent to lowering temperature in both cases A and B and it\ncauses suppression of relaxation in case A, while it leads to acceleration of relaxation in case\nB. Because Ma\u000bects the local \feld from the exchange energy at each site, changing the\nvalue ofMunder a constant external \feld his not the same as changing Tand it may show\n17-1.5-1-0.500.511.5\n01 0 2 0 3 0 4 0 5 0m\ntime(a)\n-1.5-1-0.500.511.5\n02468 1 0m\ntime(b)FIG. 7: (color online) Comparison of the time dependence of mbetween cases A and B by the\nstochastic LLG method. Red and blue lines denote cases A and B, respectively. (a) \u000b= 0:05 for\ncase A and D= 1:0 for case B, (b) \u000b= 0:2 for case A and D= 1:0 for case B.\nsome modi\fed features.\nIn the relation (19), T= 0:2, 1, 2, 10 at M= 2 (Fig.6 (a) and (b)) are the same as\nM= 20, 4, 2, 0.4 at T= 2, respectively. We studied the relaxation ratio de\fned as m(t)=M\nwithMdependence at T= 2 for these four values of M, and compared with the relaxation\ncurves of Fig.6 (a) and (b). We found qualitatively the same tendency between relaxation\ncurves with Mdependence and those with 1 =Tdependence in both cases. A di\u000berence was\nfound in the initial relaxation speed (not shown). When M > 2, the initial relaxation at\nT= 2 is slower than that of the corresponding TatM= 2. The downward initial local\n\feld at each site is stronger for larger Mdue to a stronger exchange coupling, which also\nassist the suppression of the initial relaxation.\nIt is found that the relaxation time under a constant external \fled becomes longer as\nthe value of Mis raised in case A, while it becomes shorter in case B. This suggests that\ndi\u000berent choices of the parameter set lead to serious di\u000berence in the relaxation dynamics\nwithMdependence.\n18VI. DEPENDENCE OF DYNAMICS ON THE CHOICE OF THE PARAMETER\nSET IN ANISOTROPIC SPIN SYSTEMS ( DA6= 0)\nA. Di\u000berent relaxation paths to the equilibrium in magnetic inhomgeneity\nIf the anisotropy term exists DA6= 0 but the temperature is relatively high, metastable\nnature is not observed in relaxation. We consider the relaxation dynamics when Mihas\nidependence in this case. We study the system (alternating M= 2 andM= 1 planes)\ntreated in Sec. IV A. We set a con\fguration of all spins down as the initial state and observe\nrelaxation of min cases A and B. In Sec. IV A we studied cases A ( \u000b=0.05) and B ( D=1.0)\nfor the equilibrium state and the equilibrium magnetization is m'0:95 atT= 5. We\ngive comparison of the time dependence of mbetween the two cases in Fig. 7 (a), with the\nuse of the same random number sequence. The red and blue curves denote cases A and\nB, respectively. We \fnd a big di\u000berence in the relaxation time of mand features of the\nrelaxation between the two cases.\nThe parameter values of \u000bandDare not so close between the two cases at this tempera-\nture (T= 5), i.e.,D(M= 1) = 0:25 andD(M= 2) = 0:125 for case A and \u000b(M= 1) = 0:2\nand\u000b(M= 2) = 0:4 for case B. Thus, to study if there is a di\u000berence of dynamics even\nin close parameter values of \u000bandDbetween cases A and B at T= 5, we adopt common\n\u000b= 0:2, whereD(M= 1) = 1 and D(M= 2) = 0:5, as case A and common D= 1:0, where\n\u000b(M= 1) = 0:2 and\u000b(M= 2) = 0:4, as case B. We checked that this case A also gives the\nequilibrium state. In Fig. 7 (b), the time dependence of mfor both cases is given. The red\nand blue curves denote cases A and B, respectively. There is also a di\u000berence (almost twice)\nof the relaxation time of mbetween cases A and B. Thus, even in close parameter region of\n\u000bandD, dynamical properties vary depending on the choice of the parameters.\nB. Relaxation with nucleation mechanism\nIn this subsection we study a system with metastability. We adopt a homogeneous\nsystem (M= 2) withJ= 1,DA= 1 andh= 2. Here the Stoner-Wohlfarth critical \feld\nishc= 2MDA=4, and if the temperature is low enough, the system has a metastable state\nunderh= 2.\nAt a high temperature, e.g., T= 10 (\u000b= 0:05,D= 0:25), the magnetization relaxes\n19(a)\n-2-1012\n0 80 160 240 320m\ntime\n-2-1012\n0 50 100 150 200 250 300 350m\ntime(b)\n-2-1012\n0 50 100 150 200 250 300 350m\ntime(c)FIG. 8: (color online) (a) Dashed line shows m(t) for\u000b= 0:05,D= 0:25, andT= 10. Blue\nand green solid lines give m(t) for\u000b= 0:05 atT= 3:5 (case A) and D= 0:25 atT= 3:5 (case\nB), respectively. These two lines were obtained by taking average over 20 trials with di\u000berent\nrandom number sequences. The 20 relaxation curves for cases A and B are given in (b) and (c),\nrespectively.\nwithout being trapped as depicted in Fig 8(a) with a black dotted line. When the tempera-\nture is lowered, the magnetization is trapped at a metastable state. We observe relaxations\nin cases A and B, where \u000b= 0:05 for case A and D= 0:25 for case B are used. In Figs. 8(b)\nand (c), we show 20 samples (with di\u000berent random number sequences) of relaxation pro-\ncesses atT= 3:5 for case A ( \u000b= 0:05,D= 0:0875) and case B ( D= 0:25,\u000b= 0:143),\nrespectively. The average lines of the 20 samples are depicted in Fig 8(a) by blue and green\nsolid lines for cases A and B, respectively. In both cases, magnetizations are trapped at a\nmetastable state with the same value of m(m'\u00001:55). This means that the metastabil-\nity is independent of the choice of parameter set. Relaxation from the metastable state to\nthe equilibrium is the so-called stochastic process and the relaxation time distributes. The\nrelaxation time in case A is longer. If the temperature is further lowered, the escape time\nfrom the metastable state becomes longer. In Figs. 9 (a) and (b), we show 20 samples of\nrelaxation at T= 3:1 for cases A and B, respectively. There we \fnd the metastable state\nmore clearly.\nHere we investigate the initial relaxation to the metastable state at a relatively low\ntemperature. In Figs. 10 (a) and (b), we depict the initial short time relaxation of 20\nsamples at T= 2 in cases A ( \u000b= 0:05,D= 0:05) and B(D= 0:25,\u000b= 0:25), respectively.\nThe insets show the time dependence of the magnetization in the whole measurement time.\n20-2-1012\n0 200 400 600 800m\ntime(a)\n-2-1012\n0 200 400 600 800m\ntime(b)FIG. 9: (a) and (b) illustrate 20 relaxation curves for \u000b= 0:05 atT= 3:1 (case A) and D= 0:25\natT= 3:1 (case B), respectively. Metastability becomes stronger than T= 3:5. No relaxation\noccurs in all 20 trials in (a), while \fve relaxations take place in 20 trials in (b).\nWe \fnd that the relaxation is again faster in case B.\nThe metastability also depends on Mas well asDAand largeMgives a strong metastabil-\nity. Here we conclude that regardless of the choice of the parameter set, as the temperature\nis lowered, the relaxation time becomes longer due to the stronger metastability, in which\nlargerD(larger\u000b) gives faster relaxation from the initial to the metastable state and faster\ndecay from the metastable state.\nFinally we show typical con\fgurations in the relaxation process. When the anisotropy\nDAis zero or weak, the magnetization relaxation occurs with uniform rotation from \u0000z\ntozdirection, while when the anisotropy is strong, the magnetization reversal starts by a\nnucleation and inhomogeneous con\fgurations appear with domain wall motion. In Figs. 11\nwe give an example of the magnetization reversal of (a) the uniform rotation type (magneti-\nzation reversal for DA= 0 withD= 0:05,T= 2,\u000b= 0:1,M= 4) and of (b) the nucleation\ntype (magnetization reversal for DA= 1 withD= 0:25,T= 3:1,\u000b= 0:161,M= 2 ).\nVII. SUMMARY AND DISCUSSION\nWe studied the realization of the canonical distribution in magnetic systems with the\nshort-range (exchange) and long-range (dipole) interactions, anisotropy terms, and magnetic\n\felds by the Langevin method of the LLG equation. Especially we investigated in detail the\n21-2.2-2-1.8-1.6-1.4-1.2-1.0\n012345678m\ntime(a)\n-2-1012\n0 200 400 600 800\ntime\n-2.2-2-1.8-1.6-1.4-1.2-1\n012345678m\ntime(b)\n-2-1012\n0 200 400 600 800\ntimeFIG. 10: Initial relaxation curves of magnetization. Insets show m(t) in the whole measurement\ntime. (a) and (b) illustrate 20 relaxation curves for \u000b= 0:05 atT= 2 (case A) and D= 0:25 at\nT= 2 (case B), respectively.\n(b)(a)\nFIG. 11: (a) Typical uniform rotation type relaxation observed in the isotropic spin system. (b)\nTypical nucleation type relaxation observed in the anisotropic spin system.\nthermal equilibration of inhomogeneous magnetic systems. We pointed out that the spin-\nmagnitude dependent ratio between the strength of the random \feld and the coe\u000ecient of the\ndamping term must be adequately chosen for all magnetic moments satisfying the condition\n(10). We compared the stationary state obtained by the present Langevin method of the\n22LLG equation with the equilibrium state obtained by the standard Monte Carlo simulation\nfor given temperatures. There are several choices for the parameter set, e.g., A and B. We\nfound that as long as the parameters are suitably chosen, the equilibrium state is realized as\nthe stationary state of the stochastic LLG method regardless of the choice of the parameter\nset, and the temperature dependence of the magnetization is accurately produced in the\nwhole region, including the region around the Curie temperature.\nWe also studied dynamical properties which depend on the choice of the parameters. We\nshowed that the choice of the parameter values seriously a\u000bects the relaxation process to\nthe equilibrium state. In the rotation type relaxation in isotropic spin systems under an\nunfavorable external \feld, the dependences of the relaxation time on the temperature in\ncases A and B exhibited opposite correlations as well as the dependences of the relaxation\ntime on the magnitude of the magnetic moment. The strength of the local \feld in the initial\nstate strongly a\u000bects the speed of the initial relaxation in both cases.\nWe also found that even if close parameter values are chosen in di\u000berent parameter sets\nfor inhomogeneous magnetic systems, these parameter sets cause a signi\fcant di\u000berence of\nrelaxation time to the equilibrium state. In the nucleation type relaxation, the metastability,\nwhich depends on DAandM, strongly a\u000bects the relaxation in both cases A and B. Lowering\ntemperature reinforces the metastability of the system and causes slower relaxation. The\nrelaxation to the metastable state and the decay to the metastable state are a\u000bected by the\nchoice of the parameter set, in which larger Dcauses fast relaxation at a \fxed T.\nIn this study we adopted two cases, i.e., A and B in the choice of the parameter set.\nGenerally more complicated dependence of MiorTon the parameters is considered. How\nto chose the parameter set is related to the quest for the origin of these parameters. It\nis very important for clari\fcation of relaxation dynamics but also for realization of a high\nspeed and a low power consumption, which is required to development of magnetic devices.\nStudies of the origin of \u000bhave been intensively performed32{41. To control magnetization\nrelaxation at \fnite temperatures, investigations of the origin of Das well as\u000bwill become\nmore and more important. We hope that the present work gives some useful insight into\nstudies of spin dynamics and encourages discussions for future developments in this \feld.\n23Acknowledgments\nThe authors thank Professor S. Hirosawa and Dr. S. Mohakud for useful discussions.\nThe present work was supported by the Elements Strategy Initiative Center for Magnetic\nMaterials under the outsourcing project of MEXT and Grant-in-Aid for Scienti\fc Research\non Priority Areas, KAKENHI (C) 26400324.\n24Appendix A: Fokker-Planck equation\nThe LLG equation with a Langevin noise (Eq. (5)) is rewritten in the following form for\n\u0016component ( \u0016= 1;2 or 3 forx;yorz) of theith magnetic moment,\ndM\u0016\ni\ndt=f\u0016\ni(M1;\u0001\u0001\u0001;MN;t) +g\u0016\u0017\ni(Mi)\u0018\u0017\ni(t): (A1)\nHeref\u0016\niandg\u0016\u0017\niare given by\nf\u0016\ni=\u0000\r\n1 +\u000b2\ni\u0014\n\u000f\u0016\u0017\u0015M\u0017\niHe\u000b;\u0015\ni+\u000bi\nMi\u000f\u0016\u0017\u0015\u000f\u0015\u001a\u001bM\u0017\niM\u001a\niHe\u000b;\u001b\ni\u0015\n(A2)\nand\ng\u0016\u0015\ni=\u0000\r\n1 +\u000b2\ni\u0014\n\u000f\u0016\u0017\u0015M\u0017\ni+\u000bi\nMi(\u0000M2\ni\u000e\u0016\n\u0015+M\u0016\niM\u0015\ni)\u0015\n; (A3)\nwhereHe\u000b;\u0015\nican have an explicit time ( t) dependence, and \u000f\u0016\u0017\u0015denotes the Levi-Civita\nsymbol. We employ the Einstein summation convention for Greek indices ( \u0016,\u0017\u0001\u0001\u0001).\nWe consider the distribution function F\u0011F(M1;\u0001\u0001\u0001;MN;t) in the 3N-dimensional\nphase space ( M1\n1;M2\n1;M3\n1;\u0001\u0001\u0001;M1\nN;M2\nN;M3\nN). The distribution function F(M1;\u0001\u0001\u0001;MN;t)\nsatis\fes the continuity equation of the distribution:\n@\n@tF(M1;\u0001\u0001\u0001;MN;t) +NX\ni=1@\n@M\u000b\ni\u001a\u0000d\ndtM\u000b\ni\u0001\nF\u001b\n= 0: (A4)\nSubstituting the relation (A1), the following di\u000berential equation for the distribution func-\ntionFis obtained.\n@\n@tF(M1;\u0001\u0001\u0001;MN;t) =\u0000NX\ni=1@\n@M\u000b\nin\u0000\nfi+g\u000b\f\ni\u0018\f\ni\u0001\nFo\n: (A5)\nRegarding the stochastic equation (A1) as the Stratonovich interpretation, making use\nof the stochastic Liouville approach42, and taking average for the noise statistics (Eq. (6)),\nwe have a Fokker-Planck equation.\n@\n@tP(M1;\u0001\u0001\u0001;MN;t) =\u0000NX\ni=1@\n@M\u000b\ni\u001a\nf\u000b\niP\u0000Dig\u000b\f\ni@\n@M\u001b\ni(g\u001b\f\niP)\u001b\n; (A6)\nwhereP\u0011P(M1;\u0001\u0001\u0001;MN;t) is the averaged distribution function hFi.\nSubstituting the relation\n@\n@M\u001b\nig\u001b\f\ni=\u0000\r\u000bi\nMi(1 +\u000b2\ni)4M\f\ni (A7)\n25and Eq. (A3) into g\u000b\f\ni(@\n@M\u001b\nig\u001b\f\ni), we \fnd\ng\u000b\f\ni(@\n@M\u001b\nig\u001b\f\ni) = 0: (A8)\nThus Eq.(A6) is simpli\fed to\n@\n@tP(M1;\u0001\u0001\u0001;MN;t) =\u0000NX\ni=1@\n@M\u000b\ni\u001a\u0000\nf\u000b\ni\u0000Dig\u000b\f\nig\u001b\f\ni@\n@M\u001b\ni\u0001\nP\u001b\n: (A9)\nSubstituting Eqs. (A2) and (A3), we have a formula in the vector representation.\n@\n@tP(M1;\u0001\u0001\u0001;MN;t) = (A10)\nX\ni\r\n1 +\u000b2\ni@\n@Mi\u0001\u001a\u0014\nMi\u0002He\u000b\ni+\u000bi\nMiMi\u0002(Mi\u0002He\u000b\ni)\n\u0000\rDiMi\u0002(Mi\u0002@\n@Mi)\u0015\nP(M1;\u0001\u0001\u0001;MN;t)\u001b\n:\nSince@\n@Mi\u0001(Mi\u0002He\u000b\ni) = 0, it is written as\n@\n@tP(M1;\u0001\u0001\u0001;MN;t) =X\ni\r\n1 +\u000b2\ni@\n@Mi\u0001\u001a\u0014\u000bi\nMiMi\u0002(Mi\u0002He\u000b\ni) (A11)\n\u0000\rDiMi\u0002(Mi\u0002@\n@Mi)\u0015\nP(M1;\u0001\u0001\u0001;MN;t)\u001b\n:\nIn the case that Eq. (A1) is given under Ito de\fnition, we need Ito-Stratonovich trans-\nformation, and the corresponding equation of motion in Stratonovich interpretation is\ndM\u0016\ni\ndt=f\u0016\ni(M1;\u0001\u0001\u0001;MN;t)\u0000Dig\u0015\u0017\ni(Mi)@g\u0016\u0017\ni(Mi)\n@M\u0015\ni+g\u0016\u0017\ni(Mi)\u0018\u0017\ni(t): (A12)\nThen the Fokker-Planck equation in Ito interpretation is\n@\n@tP(M1;\u0001\u0001\u0001;MN;t) =\u0000NX\ni=1@\n@M\u000b\ni\u001a\u0000\nf\u000b\ni\u0000Dig\u0015\u0017\ni@g\u000b\u0017\ni\n@M\u0015\ni\u0000Dig\u000b\f\nig\u001b\f\ni@\n@M\u001b\ni\u0001\nP\u001b\n:\nSinceg\u0015\u0017\ni@g\u000b\u0017\ni\n@M\u0015\ni=\u00002\r2\n1+\u000b2\niM\u000b\ni, the vector representation is given by\n@\n@tP(M1;\u0001\u0001\u0001;MN;t) =X\ni\r\n1 +\u000b2\ni@\n@Mi\u0001\u001a\u0014\u000bi\nMiMi\u0002(Mi\u0002He\u000b\ni)\n\u00002\rDiMi\u0000\rDiMi\u0002(Mi\u0002@\n@Mi)\u0015\nP(M1;\u0001\u0001\u0001;MN;t)\u001b\n:\n(A13)\n26Appendix B: Numerical integration for stochastic di\u000berential equations\nIn stochastic di\u000berential equations, we have to be careful to treat the indi\u000berentiability\nof the white noise. In the present paper we regard the stochastic equation, e.g., Eq. (5), as\na stochastic di\u000berential equation in Stratonovich interpretation:\ndM\u0016\ni=f\u0016\ni(M1;\u0001\u0001\u0001;MN;t)dt+g\u0016\u0017\ni\u00101\n2\u0000\nMi(t) +Mi(t+dt)\u0001\u0011\ndW\u0017\ni(t); (B1)\nwheredW\u0017\ni(t) =Rt+dt\ntds\u0018\u0017\ni(s), which is the Wiener process. This equation is expressed by\ndM\u0016\ni=f\u0016\ni(M1;\u0001\u0001\u0001;MN;t)dt+g\u0016\u0017\ni(Mi(t))\u000edW\u0017\ni(t); (B2)\nwhere\u000eindicates the usage of the Stratonovich de\fnition.\nA simple predictor-corrector method called the Heun method8,19, superior to the Euler\nmethod, is given by\nM\u0016\ni(t+ \u0001t) =M\u0016\ni(t)\n+1\n2[f\u0016\ni(^M1(t+ \u0001t);\u0001\u0001\u0001;^MN(t+ \u0001t);t+ \u0001t) +f\u0016\ni(M1(t);\u0001\u0001\u0001;MN(t);t)]\u0001t\n+1\n2[g\u0016\u0017\ni(^Mi(t+ \u0001t)) +g\u0016\u0017\ni(Mi(t))]\u0001W\u0017\ni; (B3)\nwhere \u0001W\u0017\ni\u0011W\u0017\ni(t+ \u0001t)\u0000W(t) and ^M\u0016\ni(t+ \u0001t) is chosen in the Euler scheme:\n^M\u0016\ni(t+ \u0001t) =M\u0016\ni(t) +f\u0016\ni(M1(t);\u0001\u0001\u0001;MN(t);t)\u0001t+g\u0016\u0017\ni(Mi(t))\u0001W\u0017\ni: (B4)\nThis scheme assures an approximation accuracy up to the second order of \u0001 Wand \u0001t. Sev-\neral numerical di\u000berence methods19for higher-order approximation, which are often compli-\ncated, have been proposed.\nHere we adopt a kind of middle point method equivalent to the Heun method.\nM\u0016\ni(t+ \u0001t) =M\u0016\ni(t)\n+f\u0016\ni(M1(t+ \u0001t=2);\u0001\u0001\u0001;MN(t+ \u0001t=2);t+ \u0001t=2)\u0001t\n+g\u0016\u0017\ni(Mi(t+ \u0001t=2))\u0001W\u0017\ni; (B5)\nwhereM\u0016\ni(t+ \u0001t=2) is chosen in the Euler scheme:\nM\u0016\ni(t+ \u0001t=2) =M\u0016\ni(t) +f\u0016\ni(M1(t);\u0001\u0001\u0001;MN(t);t)\u0001t=2 +g\u0016\u0017\ni(Mi(t))\u0001~Wi\u0017; (B6)\n27where \u0001 ~Wi\u0017\u0011W\u0017\ni(t+ \u0001t=2)\u0000W\u0017\ni(t). Considering the following relations,\nh\u0001~Wi\u0017\u0001W\u0017\nii=\n[W\u0017\ni(t+ \u0001t=2)\u0000W\u0017\ni(t)][W\u0017\ni(t+ \u0001t)\u0000W\u0017\ni(t)]\u000b\n=Di\u0001t; (B7)\nh\u0001W\u0017\nii= 0 andh\u0001~Wi\u0017i= 0, this method is found equivalent to the Heun method. 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We explore the conditions under which relaxa tion to local\nequilibrium occurs on time scales much shorter than that of binary co llisions, which\ncharacterize the transport of energy, and subsequent relaxat ion to local thermal\nequilibrium. Starting from the pseudo-Liouville equation for the time e volution of\nphase-space distributions, we derive a master equation which gove rns the energy\nexchange between the system constituents. We thus obtain analy tical results relating\nthe transport coefficient of thermal conductivity to the frequen cy of collision events\nand compute these quantities. We also provide estimates of the Lya punov exponents\nand Kolmogorov-Sinai entropy under the assumption of scale sepa ration. The validity\nof our results is confirmed by extensive numerical studies.\nSubmitted to: New J. Phys.\nPACS numbers: 05.20.Dd,05.45.-a,05.60.-k,05.70.Ln\nE-mail:†gaspard@ulb.ac.be, ‡thomas.gilbert@ulb.ac.beFourier’s law in many particle dispersing billiards 2\n1. Introduction\nUnderstanding the dynamical origin of the mechanisms which underly the phenomenol-\nogy of heat conduction has remained one of the major open problem s of statistical\nmechanics ever since Fourier’s seminal work [1]. Fourier himself actua lly warned his\nreader that the effects of heat conduction “make up a special ran ge of phenomena\nwhich cannot be explained by the principles of motion and equilibrium,” th us seemingly\nrejecting the possibility of a fundamental level of description. Of c ourse, with the sub-\nsequent developments of the molecular kinetic theory of heat, sta rting with the works\nof the founding-fathers of statistical mechanics, Boltzmann, Gib bs and Maxwell, and\nlater with the definite triumph of atomism, thanks to Perrin’s 1908 me asurement of the\nAvogadro number in relation to Einstein’s work on Brownian motion, Fo urier’s earlier\nperception was soon discarded as it became clear that heat transp ort was indeed the\neffect of mechanical causes.\nYet, after well over a century of hard labor, the community is still a ctively hunting\nfor a first principles based derivation of Fourier’s law, some authors going so far as to\npromise “a bottle of very good wine to anyone who provides [a satisfa ctory answer to\nthis challenge]” [2]. Thus the challenge is, starting from the Hamiltonian time evolution\nof a system of interacting particles which models a fluid or a crystal, t o derive the\nconditions under which the heat flux and temperature gradients ar e linearly related by\nthe coefficient of heat conductivity. This is the embodiment of Fourie r’s law.\nAs outlined in [2], it is necessary, in order to achieve this, that the mod el statistical\nproperties be fully determined in terms of the local temperature, a notionwhich involves\nthat of local thermalization. To visualize this, we imagine that the sys tem is divided\nup into a large number of small volume elements, each large enough to contain a\nnumber of particles whose statistics is accurately described by the equilibrium statistics\nat the local temperature associated to the volume element under c onsideration. To\nestablish this property, one must ensure that time scales separat e, which implies that\nthe volume elements settle to local thermal equilibrium on time scales m uch shorter\nthan the ones which characterize the transport of heat at the ma croscopic scale. Local\nthermal equilibrium therefore relies on very strong ergodic proper ties of the model.\nThe natural framework to apply this programme is that of chaotic b illiards. Indeed\nnon-interacting particle billiards, where individual tracers move and make specular\ncollisions among a periodic array of fixed convex scatterers (the pe riodic Lorentz gases)\naretheonlyknownHamiltoniansystems forwhichmasstransport[3 ]andshearandbulk\nviscosities [4] have been established in a rigorous way. Here, rather than local thermal\nequilibrium, itisarelaxationtolocalequilibrium, occurringontheconst antenergysheet\nof the individual tracer particles, which allows to model the mass tra nsport by a random\nwalk and yields an analytic estimate of the diffusion coefficient [5]. Argua bly Lorentz\ngases, whether periodic or disordered, in or out of equilibrium, have played, over more\nthanacentury, aprivilegedandmostimportantroleinthedevelopme nt oftransportand\nkinetic theories [6]. However, in the absence of interaction among th e tracer particles,Fourier’s law in many particle dispersing billiards 3\nthere is no mechanism for energy exchange and therefore no proc ess of thermalization\nbefore heat is conducted. If, on the other hand, one adds some in teraction between\nthe tracers, say, as suggested in [2], by assigning them a size, the r esulting system will\ntypically have transport equations for the diffusion of both mass an d energy. Though\nthese systems may be conceptually simple, they are usually mathema tically too difficult\nto handle with the appropriate level or rigor.\nAn example of billiard with interacting tracer particles is the modified Lo rentz gas\nwithrotatingdiscsproposedbyMej´ ıa–Monasterio etal.[7]. Thissystemexhibitsnormal\ntransport with non-trivial coupled equations for heat and mass tr ansport. Though this\nmodel has attracted much attention among mathematicians [8, 9, 10, 11, 12, 13], the\nrigorous proof of Fourier’s law for such a system of arbitrary size a nd number of tracers\nrelies on assumptions which, though they are plausible, are themselv es not resolved.\nMoreover the equations which govern the collisions and the discs rot ations do not, as\nfar as we know, derive from a Hamiltonian description.\nYet a remedy to such limitations with respect to the number of partic les in\ninteraction that a rigorous treatment will handle may have been jus t around the corner\n[14], and, this, within a Hamiltonian framework. Indeed in [15], Bunimov ichet al.\nintroduced a class of dispersing billiard tables with particles that are in geometric\nconfinement – i. e.trapped within cells– but that can nevertheless interact among\nparticles belonging to neighbouring cells. The authors proved ergod icity and strong\nchaotic properties of such systems with arbitrary number of part icles.\nMore recently, the idea that energy transport can be modeled as a slow diffusion\nprocess resulting from the coupling of fast energy-conserving dy namics has led to proofs\nof central limit theorems in the context of models of random walks an d coupled maps\nwhich describe the diffusion of energy in a strongly chaotic, fast cha nging environment\n[16, 17]. Although the extension of these results to symplectic coup led maps, let alone\nHamiltonian flows, is not yet on the horizon, it is our belief that such sy stems as the\ndispersing billiardtablesintroducedin[15]will, ifany, lendthemselves to afullyrigorous\ntreatment of heat transport within a Hamiltonian framework. As we announced in\n[18], the reason why we should be so hopeful is that the particles con finement has two\nimportant consequences : first, relaxation to local thermal equilib rium is preceded by a\nrelaxation of individual particles to local equilibrium ‡, which occurs at constant energy\nwithin each cell, and has strong ergodic properties that guarantee the rapid decay of\nstatistical correlations; and, second, heat transport, unlike in t he rotating discs model,\ncan be controlled by the mere geometry of the billiard, which also cont rols the absence\nof mass transport. As of the first property, the relaxation to a lo cal equilibrium before\nenergy exchanges take place is characterized by a fast time scale, much faster than that\nof relaxation to local thermal equilibrium among neighbouring cells, wh ich is itself much\nfaster than the hydrodynamic relaxation scale. There is therefor e a hierarchy of three\n‡Let us underline the distinction we make here and in the sequel betwe en relaxation to local\nequilibrium, which precedes energy exchanges, and relaxation to loc al thermal equilibrium, which\ninvolves energy exchanges among particles belonging to neighbourin g cells.Fourier’s law in many particle dispersing billiards 4\nseparate time scales in this system, the first accounting for relaxa tion at the microscopic\nscale of individual cells, the second one at the mesoscopic scale of ne ighbouring cells,\nand the third one at the macroscopic scale of the whole system.\nIn this paper, we achieve two important milestones towards a comple te first\nprinciples derivation of the transport properties of such models. H aving defined the\nmodel, we establish the conditions for separation of time scales and r elaxation to local\nequilibrium, identifying a critical geometry where binary collisions beco me impossible.\nAssuming relaxation to local equilibrium holds, we go on to considering t he time\nevolution of phase-space densities and derive, from it, a master eq uation which governs\ntheexchangeofenergyinthesystem[19], thusgoingfromamicrosc opicscaledescription\nof the Liouville equation to the mesoscopic scale at which energy tran sport takes place.\nWe regard this as the first milestone, namely identifying the condition s under which\none can rigorously reduce the level of descrition from the determin istic dynamics at\nthe microscopic level to a stochastic process described by a maste r equation at the\nmesoscopic level of energy exchanges.\nThis master equation is then used to compute the frequency of bina ry collisions\nand to derive Fourier’s law and the macroscopic heat equation, which results from the\napplication of a small temperature gradient between neighbouring c ells. This is our\nsecond milestone : an analytic formula for heat conductivity, exact for the stochastic\nsystem, and thus valid for the determinisitc system at the critical g eometry limit.\nThese results are then checked against numerical computations o f these quantities, with\noutstanding agreement, and shown to extend beyond the critical geometry, with very\ngood accuracy, to a wide range of parameter values.\nWe further characterize the chaotic properties of the model and offer arguments\nto account for the spectrum of Lyapunov exponents of the syst em, as well as the\nKolmogorov-Sinai entropy, expressions which are exact at the cr itical geometry. Again,\nthese results are very nicely confirmed by our numerical computat ions.\nThe paper is organized as follows. The models, which we coin lattice billiar ds, are\nintroduced in section 2. Their main geometric properties are establis hed, distinguishing\ntransitions between insulating and conducting regimes under the tu ning of a single\nparameter. The same parameter controls the time scales separat ion responsible for local\nequilibrium. Section 3 provides the derivation of the master equation which, under the\nassumption of local equilibrium, governs energy transport. The ma in observables are\ncomputed and their scaling properties discussed. In section 4 we re view the properties\nof the model and assess the validity of the results of section 3 unde r the scope of our\nnumerical computations. The chaotic properties of the models are discussed in section\n5. We use simple theoretical arguments to predict some of these pr operties and compare\nthem to numerical computations. Finally, conclusions are drawn in se ction 6.Fourier’s law in many particle dispersing billiards 5\n2. Lattice billiards\nTo introduce our model, we start by considering the uniform motion o f a point particle\nabout a dispersing billiard table, Bρ, defined by the domain exterior to four overlapping\ndiscs of radii ρ, centered at the four corners of a square of sides l. The radius is thus\nrestricted to the interval l/2≤ρ < l/√\n2, where the lower bound is the overlap (or\nbounded horizon) condition, and the upper bound is reached when Bρis empty.\nΡl\nΡfΡml\nFigure 1. Two equivalent representations of a dispersing billiard table : (left) a point\nparticle moves uniformly inside the domain Bρand performs specular collisions with\nits boundary; (right) a disc of radius ρmmoves uniformly inside the domain Bρand\nperforms specular collisions with fixed discs of radii ρf=ρ−ρm. The radius ρmis a\nfree parameter which is allowed to take any value between 0 and ρ.\nAs illustrated in figure 1, the motion of a point particle in this environme nt is\na limiting case of a class of equivalent dispersing billiards, whereby the p oint particle\nbecomes a moving disc with radius ρm, 0≤ρm≤ρ, and bounces off fixed discs of radii\nρf=ρ−ρm. In all these cases, the motion of the center of the moving disc is eq uivalent\nto that of the point particle in Bρ. The border of the domain Bρconstitutes the walls\nof the billiards.\nIn the absence of cusps (which occur at ρ=l/2), the ergodic and hyperbolic\nproperties of these billiards are well established [20]. In particular, t he long term\nstatistics of the billiard map, which takes the particle from one collision event to the\nnext, preserves the measure cos φdrdφ, whereφdenotes the angle that the particle\npost-collisional velocity makes with respect to the normal vector t o the boundary. A\ndirect consequence of this invariance is a general formula which rela tes the mean free\npath,ℓ, to the billiard table area, |Bρ|, and perimeter |∂Bρ|,ℓ=π|Bρ|/|∂Bρ|.\nThe ratio between the speed of the particle, which we denote v, and the mean free\npath gives the wall collision frequency §,νc=v/ℓ, whose computation is shown in figure\n2. With this quantity, one can relate the billiard map iterations to the t ime-continuous\ndynamics of the flow. In particular, the billiard map has two Lyapunov exponents,\nopposite in signs and equal in magnitudes, which, multiplied by the wall c ollision\n§Anticipating the more general definition of the wall collision frequenc y for interacting particle billiard\ncells, we adopt the subscript “c” in reference to “critical” for reas ons to be clarified below.Fourier’s law in many particle dispersing billiards 6\nfrequency, correspond to the two non-zero Lyapunov exponen ts of the flow (there are\ntwo additional zero exponents related to the direction of the flow a nd conservation of\nenergy). The results of numerical computations of the positive on e, denoted λ+, are\nshown in figure 2 for different values of the parameters ρ.\n0.360.380.400.420.440.460.480.501020304050\nΡΝc,Λ/PΛus\nFigure 2. Collision frequency νc(solid line) and numerical computation of the\ncorresponding positive Lyapunov exponent of the flow (dots) of d ispersing billiards\nsuch as shown in figure 1. Here we took v= 1 and the square side to be l= 1/√\n2.\nThus let Q(i,j) denote the rhombus of sides lcentered at point\n(cij,dij) =/braceleftBigg\n(√\n2li,lj/√\n2), j even,\n(√\n2li+l/√\n2,lj/√\n2), jodd.(1)\nThe rhombic billiard cell illustrated in figure 1 becomes a domain centere d at point\n(cij,dij), defined according to\nBρ(i,j) =/braceleftBig\n(x,y)∈Q(i,j)/vextendsingle/vextendsingle/vextendsingleδ[(x,y),(cij+pk,dij+qk)]≥ρ,k= 1,...,d/bracerightBig\n,(2)\nwhereδ[.,.] denotes the usual Euclidean distance between two points, and ( pk,qk) =\n(±l/√\n2,0),(0,±l/√\n2) are the coordinates of the d≡4 discs at the corners of the\nrhombus.\nNow consider a number of copies {Bρ(i,j)}i,jwhich tessellate a two-dimensional\ndomain. We define a lattice billiard as a collection of billiard cells\nLρ,ρm(n1,n2)≡/braceleftBig\n(xij,yij)∈ Bρ(i,j)/vextendsingle/vextendsingle/vextendsingleδ[(xij,yij),(xi′j′,yi′j′)]≥2ρm\n∀1≤i,i′≤n1,1≤j,j′≤n2,i∝ne}ationslash=i′,j∝ne}ationslash=j′/bracerightBig\n. (3)\nEach individual cell of this billiard table possesses a single moving partic le of radius ρm,\n0≤ρm≤ρand unit mass. All the moving particles are assumed to have independ ent\ninitial coordinates within their respective cells, with the proviso that no overlap can\noccur between any pair of moving particles. The system energy, E=NkBT(with\nN=n1×n2, the number of moving particles, Tthe system temperature, and kB\nBoltzmann’s constant) is constant and assumed to be initially random ly divided amongFourier’s law in many particle dispersing billiards 7\nthekinetic energies ofthemoving particles, E=/summationtext\ni,jǫij,ǫij=mv2\nij/2, where vijdenotes\nthe speed of particle ( i,j).\nEnergy exchanges occur when two moving particles located in neighb ouring cells\ncollide. Such events can take place provided the radii of the moving p articlesρmis\nlarge enough compared to ρ. Indeed the value of the critical radius, below which binary\ncollisions do not occur, is determined by half the separation between the corners of two\nneighbouring cells,\nρc=/radicalbigg\nρ2−l2\n4. (4)\nFigure 3. A binary collision event in the critical configuration where ρm=ρcwould\noccur only provided the colliding particles visit the corresponding cor ners of their cells\nsimultaneously. The value of ρin this figure is the same as in figures 1 and 4.\nFor the sake of illustration, the unlikely occurrence of a binary collisio n event at\nthe critical radius ρm=ρcis shown in figure 3.\nAll the collisions are elastic and conserve energy, so that the dynam ical system is\nHamiltonian with 2 Ndegrees of freedom. Its phase space of positions and velocities\nis 4N-dimensional. Accordingly, the sensitivity to initial conditions of the d ynamics is\ncharacterizedby4 NLyapunovexponents, {λi}4N\ni=1,obeyingthepairingruleofsymplectic\nsystems, λ4N−i+1=−λi,i= 1,...,2N.\nCollision events between two moving particles are referred to as binary collision\neventsand will be distinguished from wall collision events , which occur between the\nmoving particles and the walls of their respective confining cells. The o ccurrences of\nthe former are characterized by a binary collision frequency ,νb, and the latter by a wall\ncollision frequency ,νw. Both frequencies depend on the difference ρm−ρc, separating\nthe moving particles radii from the critical radius, equation (4). By definition of ρc,\nthe binary collision frequency vanishes at ρm=ρc,νb|ρm=ρc= 0, and, correspondingly,\nthe wall collision frequency at the critical radius is the collision freque ncy of the single-\ncell billiard, νw|ρm=ρc=νc. We will assume from now, unless otherwise stated, that\nthe system is globally isolated and apply periodic boundary conditions a t the borders,\nthereby identifying Bρ(i+kn1,j+ln2) withBρ(i,j) for any k,l∈Z, 1≤i≤n1,\n1≤j≤n2.\nExamples of such billiards are displayed in figure 4. Obviously the quincu nx\nrhombic lattice structure, which is generated by the rhombic cells, is but one amongFourier’s law in many particle dispersing billiards 8\nFigure 4. Examples of lattice billiards with triangular (top), rhombic (middle) and\nhexagonal (bottom) tilings. The coloured particles move among an a rray of fixed\nblack discs. The radii of both fixed and mobile discs are chosen so tha t (i) every\nmoving particle is geometrically confined to its own billiard cell (identified as the area\ndelimited by the exterior intersection of the black circles around the fixed discs), but\n(ii) can nevertheless exchange energy with the moving particles in th e neighbouring\ncells through binary collisions. The solid broken lines show the traject ories of the\nmoving particles centers about their respective cells. The colours a re coded according\nto the particles kinetic temperatures (from blue to red with increas ing temperature).Fourier’s law in many particle dispersing billiards 9\ndifferent possible structures. Triangular, upright square, or hex agonal cells can be used\nas alternative periodic structures. One might also cover the plane w ith random or\nquasi-crystalline tessellations. The only relevant assumptions in wha t follows is that\nthe moving particles must be confined to their (dispersing) billiard cells and that binary\ninteractions between neighbouring cells can be turned on and off by t uning the system\nparameters.\nThe two important features of such lattice billiards is that (i) there is no mass\ntransport across the billiard cells since the moving particles are confi ned to their\nrespective cells, and (ii) energy transport can occur through bina ry collision events\nwhich take place when the particles of two neighbouring cells come into contact. In\nperiodicstructures such asthequincunx rhombiclattice, theposs ibility ofsuch collisions\nis controlled by tuning the parameter ρmabout the critical radius ρc, keeping ρfixed.\nWe can therefore distinguish two separate regimes :\n•Insulating billiard cells : 0≤ρm< ρc\nAbsence of interaction between the moving discs. No transport pr ocess across the\nindividual cells can happen;\n•Conducting billiard cells :ρc< ρm< ρ\nBinary collision events are possible. Energy transport across the in dividual cells\ntakes place.\nThe case ρm=ρcis singular. We will refer to the critical geometry as the limit ρm>→ρc.\nIn the insulating regime, there is no interaction among moving particle s so that\nthe billiard cells are decoupled. The moving particles are independent a nd their kinetic\nenergies are individually conserved, resulting in 2 Nzero Lyapunov exponents. The\nequilibrium measure in turn has a product structure and phase-spa ce distributions are\nlocally uniform with respect to the particles positions and velocity dire ctions. The\nNpositive Lyapunov exponents of the system are all equal to the po sitive Lyapunov\nexponent of the single-cell dispersing billiard, up to a factor corres ponding to the\nparticles speeds, vij=/radicalbig\n2ǫij/m:λij+=vijλ+, whereλ+is the Lyapunov exponent of\nthe single-cell billiard measured per unit length.\nWhen particles are allowed to interact, on the other hand, local ene rgies are\nexchanged through collision events. Thus only the total energy is c onserved in the\nconducting regime. The ergodicity of such systems of geometrically confined particles\nin interaction was proven by Bunimovich et al.[15]. The resulting dynamical system,\nwhose equilibrium measure is the microcanonical one (taking into cons ideration that\nparticles are otherwise uniformly distributed within their respective cells), enjoys the\nK-property. This implies ergodicity, mixing, and strong chaotic prope rties, including\nthe positivity of the Kolmogorov-Sinai entropy. Two Lyapunov exp onents are zero, one\nassociated to the conservation of energy, the other to the direc tion of the flow. The\n2(2N−1) remaining Lyapunov exponents form non-vanishing pairs of expo nents with\nopposite signs, λ1> ... > λ 2N−1>0,λ4N−i+1=−λi,i= 1,...,2N−1.\nThe regime of interest to us is that corresponding to particles inter acting rarely,Fourier’s law in many particle dispersing billiards 10\nwhich is to say, in analogy with a solid, that particles mostly vibrate insid e their cells,\nignorant of each other, and only seldom making collisions with their neig hbours, thereby\nexchanging energy. As we turn on the interaction and let ρm/greaterorsimilarρc, binary collisions,\nthough they can occur, will remain unlikely. This is to say that the bina ry collision\nfrequency, νb, will, in this regime, remain small with respect to the wall collision\nfrequency, νw, which in the absence of interaction and, in particular, at the critica l\ngeometry, we recall is equal to the wall collision frequency of the sin gle-cell billiard,\nνw|ρm=ρc=νc. When ρm/greaterorsimilarρc, we therefore expect νw≫νb, as well as νw≃νc. In\nwords :time scales separate . The consequence is that relaxation to local equilibrium\n–i. e.uniformization of the distribution of the particles positions and veloc ity directions\nat fixed speeds– occurs typically much faster than the energy exc hange which drives the\nrelaxationto theglobal equilibrium. This mechanism justifies resortin g tokinetic theory\nin order to compute the transport properties of the model.\n3. Kinetic theory\n3.1. From Liouville’s equation to the master equation\nThe phase-space probability density is specified by the N-particles distribution function\npN(r1,v1,...,rN,vN,t), whereraandva,a= 1,...,N, denote the ath particle position\nand velocity vectors. The index astands for the label ( i,j) of the cells defined by\nequation(2). Foroursystem, asiscustomaryforhardspheredy namics, thisdistribution\nsatisfies a pseudo-Liouville equation [21], which is well defined despite t he singularity\nof the hard-core interactions. This equation, which describes the time evolution of pN\nis composed of three types of terms: (i) the advection terms, whic h account for the\ndisplacement of the moving particles within their respective billiard cells ; (ii) the wall\ncollision terms, which account for the wall collision events, between t he moving particles\nand thedfixed scattering discs which form the cells walls; and (iii) the binary collis ion\nterms, which account for binary collision events, between moving pa rticles belonging to\nneighbouring billiard cells :\n∂tpN=N/summationdisplay\na=1/bracketleftBigg\n−va·∂ra+d/summationdisplay\nk=1K(a,k)/bracketrightBigg\npN+1\n2N/summationdisplay\na,b=1B(a,b)pN. (5)\nEach wall collision term involves a single moving particle with index aand one of the d\nfixed discs in the corresponding cell, with index kand position Rk. Letrak=ra−Rk\ndenote their relative position. Following [22], we have\nK(a,k)pN(...,ra,va,...) =\nρ/integraldisplay\nˆe·va>0dˆe(ˆe·va)/bracketleftBig\nδ(rak−ρˆe)pN(...,ra,va−2ˆe(ˆe·va),...)\n−δ(rak−ρˆe)pN(...,ra,va,...)/bracketrightBig\n, (6)\nwhereˆedenotes the normal unit vector to the fixed disc kin the cell of particle a.Fourier’s law in many particle dispersing billiards 11\nLikewise the binary collision operator, written in terms of the relative positions rab\nand velocities vabof particles aandb, and the unit vector ˆeabthat connects them, is\nB(a,b)pN(...,ra,va,...,rb,vb,...) =\n2ρm/integraldisplay\nˆeab·vab>0dˆeab(ˆeab·vab)/bracketleftBig\nδ(rab−2ρmˆeab)\n×pN(...,ra,va−ˆeab(ˆeab·vab),...,rb,vb+ˆeab(ˆeab·vab),...)\n−δ(rab+2ρmˆeab)pN(...,ra,va,...,rb,vb,...)/bracketrightBig\n.(7)\nWe notice that only the terms B(a,b)corresponding to first neighbours are non-vanishing\nand contribute to the double sum on the RHS of equation (5).\nProvided we have a separation of time scales between wall and binary collisions, the\nadvection and wall collision terms on the RHS of equation (5) will typica lly dominate\nthe dynamics on the short time ∼1/νw, which follows every binary collision event, thus\nensuring, thanks to the mixing within individual billiard cells, the relaxat ion of the\nphase-space distribution pNto local equilibrium well before the occurrence of the next\nbinary event, whose time scale is ∼1/νb. In other words, pN(r1,v1,...,rN,vN,t)\nquickly relaxes to a locally uniform distribution, which depends only on t he local\nenergies, justifying the introduction of\nP(leq)\nN(ǫ1,...,ǫ N,t)≡/integraldisplayN/productdisplay\na=1dradvapN(r1,v1,...,rN,vN,t)N/productdisplay\na=1δ(ǫa−mv2\na/2),(8)\nwhereva≡ |va|. On the time scale of binary collision events, this distribution\nsubsequently relaxes to the global microcanonical equilibrium distrib ution. This process\naccounts for the transport of energy, and can be characterize d by the master equation\n[19]\n∂tP(leq)\nN(ǫ1,...,ǫ N,t) =1\n2N/summationdisplay\na,b=1/integraldisplay\ndη\n×/bracketleftBig\nW(ǫa+η,ǫb−η|ǫa,ǫb)P(leq)\nN(...,ǫa+η,...,ǫ b−η,...,t)\n−W(ǫa,ǫb|ǫa−η,ǫb+η)P(leq)\nN(...,ǫa,...,ǫ b,...,t)/bracketrightBig\n, (9)\nwhereW(ǫa,ǫb|ǫa−η,ǫb+η) denotes the probability that an energy ηbe transferred\nfrom particle ato particle bas the result of a binary collision event between them.\nThis equation is a closure for the local equilibrium distribution P(leq)\nN, obtained from\nequation (5) under the assumption that νw≫νb. The first two terms on the RHS\nof equation (5) are eliminated because they leave invariant the local distribution/producttextN\na=1δ(ǫa−mv2\na/2). There remain the contributions (7) from the binary collisions,\nwhich, under the assumption that the local distibutions are uniform with respect to the\npositions and velocity directions, yield the following expression of W:\nW(ǫa,ǫb|ǫa−η,ǫb+η) =2ρmm2\n(2π)2|Lρ,ρm(2)|/integraldisplay\ndφdR/integraldisplay\nˆeab·vab>0dvadvb (10)\n׈eab·vabδ/parenleftBig\nǫa−m\n2v2\na/parenrightBig\nδ/parenleftBig\nǫb−m\n2v2\nb/parenrightBig\nδ/parenleftBig\nη−m\n2[(ˆeab·va)2−(ˆeab·vb)2]/parenrightBig\n,Fourier’s law in many particle dispersing billiards 12\nwhere the first integration is performed over the positions of the c enter of mass,\nR≡(ra+rb)/2, between the two particles aandb, given that they are in contact and\nboth located in their respective cells, and over the angle φof the unit vector connecting\naandb,ˆeab= (cosφ,sinφ). The normalizing factor |Lρ,ρm(2)|denotes the 4-volume of\nthe billiard corresponding to two neighbouring cells aandb, which, with the assumption\nthatρm/greaterorsimilarρc, can be approximated by |Lρ,ρm(2)| ≃ |B ρ|2. This substitution amounts\nto neglecting the overlap between the two particles; see equation ( 23) for a refinement\nof that approximation. We point out that the position and velocity int egrations in\nequation (10) can be formally decoupled; in this way, we can prove th at the transition\nrateWis given in terms of Jacobian elliptic functions, see Appendix A.\n3.2. Geometric factor\nAs we show in Appendix A, an important property of the master equa tion (9) is that\nthe factor which accounts for the geometry of collision events fac torizes from the part of\nthe kernel that accounts for energy exchanges. Therefore, a sρm→ρc, the critical value\nof the radius at which binary collision events become impossible, which is the regime\nwhere the billiard properties are accurately described in terms of th e master equation\nabove, the geometric factor/integraltext\ndφdRencloses the scaling properties of observables with\nrespect to the billiard geometry. We now compute this quantity.\nA binary collision occurs when particles aandbcome to a distance 2 ρmof each\nother, with ra∈ Bρ(a) andrb∈ Bρ(b). LetR= (x,y) be the center of mass coordinates\nandφbe the angle between the particles relative position and the axis conn ecting the\ncenter of the cells. Taking a reference frame centered between t he cells, we may write\nri=1\n2(x,y)+σiρm(cosφ,sinφ), (11)\nwhereσi=±1 andi=aorb. The integral to be evaluated is the volume of the triplets\n(x,y,φ) about the origin so that\n/parenleftBigx\n2+σiρmcosφ/parenrightBig2\n+/parenleftbiggy\n2+σiρmsinφ±l\n2/parenrightbigg2\n≥ρ2. (12)\nAs illustrated in figure 5, for different orientations φof the vector connecting the two\nparticles, this is a region bounded by four arc-circles, which we deno te by\nyσ,τ(x)≡ −2σρmsinφ−τl+τ/radicalBig\n4ρ2−(x+2σρmcosφ)2, (13)\nwhereσ,τ=±1.\nAs seen from figure 5, the area is connected for −φT≤φ≤φT, whereφTis the\nangleφat which opposite arcs intersect,\nφT= arcsinρ2\nm−ρ2\nc\nlρm. (14)\nBeyond that value, the area splits into two triangular areas. These areas shrink to zero\nat the angle φgiven by\nφM= arccosρmρc+l/2/radicalbig\nρ2−ρ2m\nρ2. (15)Fourier’s law in many particle dispersing billiards 13\nΦ/EquΑΛ0 Φ/EquΑΛ0.05 Φ/EquΑΛ0.10 Φ/EquΑΛ0.15 Φ/EquΑΛ0.20\nFigure 5. Possible positions of the center of mass ( x,y) for different values of φ, see\nequation (11), at a binary collision event. Here ρ= 13/25landρm= 13/50l.\nLet−φM≤φ≤φM. Wedenoteby x1< x2< x3< x4thefourcornersoftherectangular\ndomain,\nx1=−x4=−2(ρmcosφ−ρc),\nx2=−x3=−2/radicalbig\nρ2−ρ2msinφ.(16)\nForφ≥φT, the points at which the opposite arcs intersect are given by\nxi=±(l−2ρmsinφ)/bracketleftbigg−l2+4lρmsinφ+4(ρ2−ρ2\nm)\nl2−4lρmsinφ+4ρ2m/bracketrightbigg1/2\n. (17)\nCombining equations (13)-(17) together, we can make use of the s ymmetry φ→ −φ\nand write the integral to be computed as\nα(ρ,ρm)≡/integraldisplay\ndφdR= 2/bracketleftbigg/integraldisplayφM\n0A1(φ)dφ+/integraldisplayφM\nφTA2(φ)dφ/bracketrightbigg\n, (18)\nwhere\nA1(φ) =/integraldisplayx3\nx1y+1,−1(x)dx+/integraldisplayx4\nx3y−1,−1(x)dx−/integraldisplayx2\nx1y+1,+1(x)dx−/integraldisplayx4\nx2y−1,+1(x)dx,(19)\nwhich is the area bounded by the four arcs yσ,τ, and\nA2(φ) =/integraldisplayxi\n−xi[y−1,+1(x)−y+1,−1(x)]dx, (20)\nis the area of the overlapping opposite arcs y−1,+1andy+1,−1, which occurs when\nφT≤φ≤φM[it gives a negative contribution to A1(φ)].\nThecomputationoftheseexpressions iseasilyperformednumerica lly, andtheresult\nshown in figure 6.\nNear the critical geometry, we expand the quantity (18) in powers of the difference\nρm−ρc,\nα(ρ,ρm) =∞/summationdisplay\nn=1cn(ρm−ρc)n. (21)\nThe first two coefficients vanish, so that the leading term correspo nds ton= 3. TheFourier’s law in many particle dispersing billiards 14\n0.010.02 0.05 0.10.2/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt1\n100000/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt1\n10000/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt1\n1000/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt1\n100/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt/FΡΑctioΝBΑΡExt1\n101\nΡm/MiΝusΡcΑ/LParen1Ρ,Ρm/RParen1\nFigure 6. Area of binary collisions α(ρ,ρm) versus ρm−ρcfor seven values of ρ\nranging from 9 /25 to 42/100 (l= 1/√\n2).\nfirst few coefficients are derived in Appendix B and given by :\nc1=c2= 0,\nc3=128ρc\n3l2,\nc4=256ρ2\nc\n3l4,\nc5=16\nl2/parenleftbigg1\n3ρc+8ρc\n5l2+16ρ3\nc\nl4+16ρ4\n5l4ρc/parenrightbigg\n.(22)\nWe further note that the computation of the two-cell 4-volume, |Lρ,ρm(2)|, which,\nas noticed above, is approximated by the square of the single cell ar ea|Bρ|2, can\nbe improved using equation (21). Indeed it is easily seen that d |Lρ,ρm(2)|/dρm=\n−2α(ρ,ρm), which implies\n|Lρ,ρm(2)|=|Bρ|2−2∞/summationdisplay\nn=4cn−1\nn(ρm−ρc)n. (23)\nIt is immediate to check that corrections |Lρ,ρm(2)|−|Bρ|2=O(ρm−ρc)4.\n3.3. Binary collision frequency\nHaving computed the transition rates of the master equation with r espect to the billiard\ngeometry, we now turn to the computation of observables. The fir st quantity of interest,\nwhich can be readily computed from equation (10), is the binary collisio n frequency νb.\nThis is an equilibrium quantity which, in the (global) microcanonical ense mble with\nenergyE=ǫ1+...+ǫN, involves the two-particle energy distribution,\nP(eq)\n2(ǫa,ǫb) =(N−1)(N−2)\nE2/parenleftbigg\n1−ǫa+ǫb\nE/parenrightbiggN−3\n, (24)\nand can be written as\nνb=/integraldisplay\ndǫadǫbdηW(ǫa,ǫb|ǫa−η,ǫb+η)P(eq)\n2(ǫa,ǫb). (25)Fourier’s law in many particle dispersing billiards 15\nTaking the large Nlimit and letting E=NkBT≡N/β, we can write P(eq)\n2(ǫa,ǫb)≃\nβ2exp[−β(ǫa+ǫb)][1+O(N−1)]. Substituting this expression into the above equation\nandinserting theexpression of Wfromequation(10), weobtain, aftersomecalculations,\nνb≃/radicalbigg\nkBT\nπm2ρm\n|Bρ|2/parenleftbigg/integraldisplay\ndφdR/parenrightbigg\n[1+O(N−1)]. (26)\nThis expression involves the geometric factor (18) in the first brac ket. The leading term\nin the second bracket is the canonical expression of the binary collis ion frequency. The\nsecond term is a positive finite Ncorrection, which is useful in that it shows that the\nbinary collision frequency decreases to its asymptotic value as N→ ∞.\n3.4. Rescaled master equation\nThe binary collision frequency (26) defines a natural dimensionless t ime scale for the\nstochastic process described by the master equation (9) with tra nsition rates (10). The\nmaster equation can thus be converted to dimensionless form by re scaling the energies\nby a reference thermal energy and time by the corresponding asy mptotic ( N→ ∞)\nvalue of the binary collision frequency (26).\nIntroducing the variables\nea≡ǫa\nkBT, (27)\nh≡η\nkBT, (28)\nτ≡νbt, (29)\nequation (9) becomes\n∂τp(leq)\nN(e1,...,e N,τ) =1\n2N/summationdisplay\na,b=1/integraldisplay\ndh\n×/bracketleftBig\nw(ea+h,eb−h|ea,eb)p(leq)\nN(...,ea+h,...,e b−h,...,τ)\n−w(ea,eb|ea−h,eb+h)p(leq)\nN(...,ea,...,e b,...,τ)/bracketrightBig\n, (30)\nwith the transition rates\nw(ea,eb|ea−h,eb+h) =/radicalbigg\n2\nπ3/integraldisplay\nx1/bardbl−x2/bardbl>0dx1/bardbldx1⊥dx2/bardbldx2⊥(x1/bardbl−x2/bardbl) (31)\n×δ(ea−x2\n1/bardbl−x2\n1⊥)δ(eb−x2\n2/bardbl−x2\n2⊥)δ(h−x2\n1/bardbl+x2\n2/bardbl)\nThis master equation shows that all the properties of heat conduc tion are rescaled by\nthe binary collision frequency and temperature in its limit of validity whe re the collision\nfrequency vanishes. In particular, this shows that the coefficient of heat conduction\nis proportional to the binary collision frequency in this limit, as explaine d in the next\nsubsection.Fourier’s law in many particle dispersing billiards 16\n3.5. Thermal conductivity\nStarting from the master equation (9), we derive an equation for t he evolution of the\nkinetic energy of each moving particle, ∝an}bracketle{tǫa∝an}bracketri}ht ≡kBTa, which defines the local temperature,\nwhere∝an}bracketle{t.∝an}bracketri}htdenotes an average with respect to the energy distributions. By t he structure\nof equation (9), such an equation can be expressed in terms of the transfer of energy due\nto the binary collisions between neighbouring cells. The time evolution o f the average\nlocal kinetic energy is given by\n∂t∝an}bracketle{tǫa∝an}bracketri}ht=−/summationdisplay\nb∝an}bracketle{tJa,b(ǫa,ǫb)∝an}bracketri}ht, (32)\nwith the energy flux defined as\nJa,b(ǫa,ǫb)≡/integraldisplay\ndηηW(ǫa,ǫb|ǫa−η,ǫb+η),\n=−/integraldisplay\ndηηW(ǫa+η,ǫb−η|ǫa,ǫb).(33)\nThis expresses the local conservation of energy.\nOver long time scales, the probability distribution becomes controlled by this local\nconservation of energy, the slowest variables being the local kinet ic energies ∝an}bracketle{tǫa∝an}bracketri}htor,\nequivalently, the local temperatures Taas defined above. This holds even though\nstatisticalcorrelationsdevelopbetweenthelocalenergiesinthep robabilitydistributions.\nThese statistical correlations are well known for transport proc esses ruled by master\nequations such as Eq. (9) [19, 23] and are observed in the present system as well.\nTo be specific, we consider a one-dimensional chain, extending along thex-axis,\nformedwithasuccession ofpairsofrhombicbilliardcells arrangedinqu incunx, similarly\nto the middle panel of figure 4, except the vertical height is here on ly one unit of\nlength. The unit of horizontal and vertical lengths is thus l√\n2 and there are two cells\nper each unit of length. Similar results hold for different choices of ge ometry modulo\nstraightforward adaptations.\nWe imagine that the system is in a non-equilibrium state, with a small tem perature\ndifference δTabout an average temperature Tbetween neighbouring cells, and consider\nthe average heat transfered from cell aat inverse temperature βa=β+δβ/2 to cell b\nat inverse temperature βb=β−δβ/2,δβ=−δT/(kBT2), both cells being assumed to\nbe in thermal equilibrium at their respective temperatures. The sta tistical correlations\nwe observe in the present system are of the order of δβ2, as it is the case in other\nsystems [19]. In the non-equilibrium state, these statistical corre lations are controlled\nin the long-time limit by the local temperatures. Since the process is h ere ruled by a\nMarkovian master equation in the limit of small binary collision frequenc y, we get the\nequation of heat for the temperature\n∂tT(x,t) =∂x[κ∂xT(x,t)], (34)\nwhere the local temperature is here written as T(x,t) =∝an}bracketle{tǫij∝an}bracketri}ht/kB,x=√\n2l(i+j/2),\ni= 1,...,N/2,j= 0,1 [see equation (1)].Fourier’s law in many particle dispersing billiards 17\nAccording to the rescaling property of the master equation discus sed in section 3.4,\nthe heat conductivity is proportional to the binary collision frequen cy∝bardbl:\nκ\nl2=Aνb, (35)\nwith a dimensionless constant A.\nAn analytical estimation can be obtained by transforming the maste r equation into\na hierarchy of equations for all the moments of the probability distr ibution:∝an}bracketle{tǫa∝an}bracketri}ht,∝an}bracketle{tǫaǫb∝an}bracketri}ht,\n∝an}bracketle{tǫaǫbǫc∝an}bracketri}ht,... The evolution equations of these moments are coupled.\nTruncating the hierarchy at the equations for the averages ∝an}bracketle{tǫa∝an}bracketri}ht, we get the\napproximate heat conductivity :\nκ\nl2≃β4\n2/integraldisplay\ndǫadǫbdη η(ǫb−ǫa)W(ǫa,ǫb|ǫa−η,ǫb+η)exp[−β(ǫa+ǫb)],\n=/radicalbigg\nkBT\nπm2ρm\n|Bρ|2/parenleftbigg/integraldisplay\ndφdR/parenrightbigg\n. (36)\nSo thatA= 1 in this approximation. The same result holds if we include the equatio ns\nfor the moments ∝an}bracketle{tǫaǫb∝an}bracketri}htwith|a−b| ≤1.\nThough the approximate result (36) above does not rule out possib le corrections to\nA= 1, wearguethat A= 1isanexactpropertyofthemasterequation(9)intheinfinite\nsystem limit. This claim is borne out by extensive studies of the stocha stic process\ndescribed by equation (9) that will be reported in a separate publica tion [24]. The\nfocus is here on the billiard systems whose conductivity may however bear corrections\nto this identity, due we believe to lack of sufficient separation of time s cales between\nwall and binary collision events. In the following, the results of numer ical computations\nare presented which support these claims.\n4. Numerical results\nThe above formulae for the binary collision frequency, equation (26 ), and the thermal\nconductivity, equation (36), together with the expressions of th e geometric factors,\nequations (22) and (23), provide a detailed picture of the mechanis m which governs the\ntransport of heat in our model. Numerical computations of these q uantities further add\nto this picture and provide strong evidence of the validity of our the oretical approach.\n4.1. Binary and wall collision frequencies\nFor the sake of computing the binary collision frequency, we simulate the quasi-one\ndimensional channel of Ncells with rhombic shapes and apply periodic boundary\nconditions at the horizontal ends of the channel. Thus each cell ha s two neighbouring\n∝bardblWe notice that the heat capacityper particle isequal to cV= (1/N)∂E/∂T=kB, sothat the thermal\nconductivity is also equal to the thermal diffusivity in units where kB= 1.Fourier’s law in many particle dispersing billiards 18\ncells, left and right, and interactions between any two neighbouring cells can occur\nthrough both top and bottom corners ¶.\nIn figure 7, we compare the computations of νbtoνwfor a system of N= 10\ncells at unit temperature. The parameters are taken to be ρ= 9/25 andρm=\n3/25,...,17/50 by steps of 1 /50. Both collision frequencies are computed in the units\nof the microcanonical average velocity, vN≡2N/radicalbig\nN/2(N−1)!/(2N−1)!!, which, as\nN→ ∞, converges to the canonical average velocity v∞=/radicalbig\nπ/2. The wall collision\nfrequency is compared to the collision frequency of the isolated cells νc, itself measured\nin the units of the single particle velocity. As expected, νb/νw≪1 andνw≃νcfor\nρm/greaterorsimilarρc. The crossover νb≃νwoccurs at ρm≃11/50 for this value of ρ.\n0.05 0.1 0.15 0.2 0.25 0.300.20.40.60.811.21.41.61.82\nρm − ρcνb, νw\n  \nνw/νc\nνb\nνb/νw\nFigure 7. Wall and binary collision frequencies νwandνbversusρm−ρcin a one-\ndimensional channel of N= 10 rhombic cells with ρ= 9/25 and lattice spacing\nl= 1/√\n2.\n4.2. Thermal conductivity\n4.2.1. Heat flux. The thermal conductivity can be obtained by computing the heat\nflux in a nonequilibrium stationary state. Such stationary states oc cur when the two\nends of the channel are put in contact with heat baths at separat e temperatures, say\nT−andT+,T−< T+.\nLet a system of Ncells be in contact with two thermostated cells at respective\ntemperatures T±, and let these cell indices be n=±(N+1)/2 (we take Nodd for the\nsake of definiteness). Provided the difference between the bath t emperatures is small,\nT+−T−≪(T++T−)/2, a linear gradient oftemperature establishes throughthesyste m,\nwith local temperatures\nTn=1\n2(T++T−)+n\nN+1(T+−T−). (37)\n¶We mention that this set-up allowsfor re-collisionbetween two partic les (under stringent conditions),\ndue to the vertical periodic boundary conditions. This conflicts with the assumptions of [15], but does\nnot seem to affect the results as far as numerics are concerned.Fourier’s law in many particle dispersing billiards 19\nUnder these conditions, the nonequilibrium stationary state is expe cted to be locally\nwell approximated by a canonical equilibrium at temperature Tn.\nThiscanbechecked numerically. Infactthelocalthermalequilibrium isverified(by\ncomparing the moments of ǫnto their Gaussian expectation values) under the weaker\nproperty of small local temperature gradients, i. e.(T+−T−)/N≪(T++T−)/2, for\nwhich the temperature profile is generally not linear since the therma l conductivity\ndepends on the temperature. Indeed, since κ∝T1/2, we expect in that case, according\nto Fourier’s law, the profile\nTn=/bracketleftbigg1\n2(T3/2\n−+T3/2\n+)+n\nN+1(T3/2\n+−T3/2\n−)/bracketrightbigg2/3\n. (38)\nThe thermal conductivity can therefore be computed from the he at exchanges of\nthe chain in contact with the two cells at the ends of the chain, respe ctively thermalized\nat temperatures T±δT/2,δT≪T. The thermalization of the end cells is achieved by\nrandomizing the velocities of the two particles at every collision they m ake with their\ncells walls, according to the usual thermalization procedure of part icles colliding with\nthermalized walls [28].\nFirst, we consider a chain containing a single cell in order to test the v alidity of\nthe master equation. In this case, the procedure amounts to simu lating a single particle\nconfined to its cell and performing random collisions with stochastic p articles which\npenetrate the cell corners according to the statistics of binary c ollisions. For a chain\nwith a single cell, the heat conductance is given by Eq. (36) with A= 1, as there are\nno correlations with the stochastic particles.\nFigure 8 shows the results of the computations of the heat conduc tance+with this\nmethod and provides a comparison with the binary collision frequency νbon the one\nhand (left panel), as well as with the results of our kinetic theory pr edictions on the\nother hand (right panel).\nTheagreementbetweenthedataandequations(26), (36),(35) andthecomputation\nof the integrals (18)-(20), especially as ρm→ρc, demonstrates the validity of the\nstochastic description of the billiard system, equation (9).\nNext, we increase the size of the chain in order to reach the therma l conductivity\nin the limit of an arbitrarily large chain. The results are that statistica l correlations\nappear between the kinetic energies along the chain. As we show belo w, their influence\non the computed value of the conductivity diminishes as ρm→ρc.\nFixT−= 0.5 andT+= 1.5 to be the baths temperatures, and let the size of the\nsystem increase from N= 1 toN= 20asρmisprogressively decreased from ρm= 11/50\ntoρm= 13/100, with fixed ρ= 9/25. As one can see from figure 8, this range of values\nofρmcrosses over from a regime where the separation of time scales is no t effective\n+Here and in the sequel, the thermal conductance or conductivity a re further divided by l2√\nT, where\nl= 1/√\n2 is the rhombic cell size, so as to eliminate its length and temperature dependences, thus\ndefining the reduced thermal conductivity κ∗=κ/(l2√\nT). In these expressions and from here on, we\nfurther set kB≡1.Fourier’s law in many particle dispersing billiards 20\n00.05 0.10.15 0.20.25 0.30.90.9511.051.11.151.21.251.3\nρm − ρcκ/(l2νb)\n10−210−110−610−410−2100102\nρm − ρcκ/(l2 T1/2), νb//T1/2\n  \nκ\nνb\nK.T.\nFigure 8. Reduced thermal conductance κ, computed from the heat exchange in a\nchainwith asingle particlewith thermalizedneighbours, and binarycollis ionfrequency\nνb, as functions of ρm−ρc. (Left) ratio between κandνb; (Right) comparison with\nthe results of section 3. The only relevant parameter is ρ= 9/25. For each value of\nρm, several temperature differences δTwere taken, all giving consistent values of κ.\nThe solid line shows the result of kinetic theory (K.T.).\nto one where it appears to be and where the stochastic model shou ld therefore be a\nreasonable approximation to the process of energy transport in t he billiard.\nForallvaluesof ρm, wemeasuredthetemperatureprofileandheatfluxesthroughou t\nthe system and inferred the value of the reduced thermal conduc tivity by linearly\nextrapolating the ratios between the average heat flux and local t emperature gradient\ndivided by the square root of the local temperature as functions o f 1/Nto the vertical\naxis intercept, corresponding to N=∞. Given the parameter values, every realisation\nwas carried out over a time corresponding to 1,000 interactions bet ween the system and\nbaths and repeated over 104realisations. For Nup to 20, this time provided satisfying\nstationary statistics, with temperature profiles verifying equatio n (38) and statistically\nconstant heat fluxes.\nThe results of the linear regression used to compute the value of κ/νbfor selective\nvalues of ρmare shown in figures 9 and 10.\nOur results thus make it plausible that the ratio between the therma l conductivity\nand binary collision frequency approaches unity as the parameter ρmdecreases towards\nthe critical value ρc. As will be show in a separate publication [24], this is indeed\na property of the stochastic model described by equation (9) and has been verified\nnumerically by direct simulation of the master equation within an accur acy of 4 digits.\nAs of the billiard system, it is unfortunately difficult to improve the res ults beyond\nthose presented here as the CPU times necessary to either increa seNor decrease ρm\nquickly become prohibitive. Nevertheless, the data displayed in figur es 9 and 10 offer\nconvincing evidence that the thermal conductivity is well approxima ted by the binary\ncollision frequency so long as the separation of time scales between w all and binary\ncollision events is effective.Fourier’s law in many particle dispersing billiards 21\nΚ/Slash1ΝB/TildeEqual1.0272Ρm/EquΑΛ0.13\n0.00.20.40.60.81.01.01.11.21.31.41.5\n1/Slash1N/LParen1jab/Slash1∆T/RParen1/Slash1ΝBΚ/Slash1ΝB/TildeEqual1.0426Ρm/EquΑΛ0.16\n0.00.20.40.60.81.01.01.11.21.31.41.5\n1/Slash1N/LParen1jab/Slash1∆T/RParen1/Slash1ΝB\nΚ/Slash1ΝB/TildeEqual1.0811Ρm/EquΑΛ0.19\n0.00.20.40.60.81.01.01.11.21.31.41.5\n1/Slash1N/LParen1jab/Slash1∆T/RParen1/Slash1ΝBΚ/Slash1ΝB/TildeEqual1.17911Ρm/EquΑΛ0.22\n0.00.20.40.60.81.01.01.11.21.31.41.5\n1/Slash1N/LParen1jab/Slash1∆T/RParen1/Slash1ΝB\nFigure 9. Ratio between thermal conductivity and binary collision frequency,\nκ/νb, extrapolated from the computation of the average ratio betwee n heat current\nand local temperature gradients, divided by the local binary collision frequency,\n1/n/summationtext\ni[Ji,i+1/(Ti+1−Ti)]/νb(i). The system sizes are N= 1,2,5,10,15,20. The\nfour pannels correspond to different values of ρm= 13/100,4/25,19/100,11/50, with\nρ= 9/25 and thus ρc≃0.068. The red dots are the data points with corresponding\nerror bars and the black solid line shows the result of a linear regress ion performed\nwith data associated to systems of lengths N≥2.\n0.060.080.100.120.140.161.001.051.101.151.20\nΡm/MiΝusΡcΚ/Slash1ΝB\nFigure 10. Ratio between thermal conductivity and binary collision frequency, κ/νb,\ncomputated as in figure 9, here collected for a largerset of values o fρm. The horizontal\naxis shows the difference ρm−ρc. The error bars are of the same order as those in\nfigure 8, with deviations from unity of the data points of the same or der as those in\nthe latter figure.Fourier’s law in many particle dispersing billiards 22\n4.2.2. Helfand moment. The computation of the thermal conductivity can also be\nperformed in the global equilibrium microcanonical ensemble using the method of\nHelfand moments [25, 26, 27].\nThe Helfand moment has expression H(t) =/summationtext\naxa(t)ǫa(t), wherexa(t) denotes the\nhorizontal position of particle aat timetandǫa(t) =|va(t)|2/2 its kinetic energy (the\nmasses are taken to be unity). The computation of the time evolutio n of this quantity\nproceeds by discrete steps, integrating the Helfand moment from one collision event to\nthe next, whether between a particle and the walls of its cell, or betw een two particles.\nLet{τn}n∈Zdenote the times at successive collision events. In the absence of b inary\ncollisions, the energies are locally conserved and the Helfand moment changes according\ntoH(τn) =H(τn−1) +/summationtext\na[xa(τn)−xa(τn−1)]ǫa(τn−1). If, on the other hand, a binary\ncollision occurs between particles kandl, the Helfand moment changes by an additional\nterm [xk(τn)−xl(τn)][ǫk(τn+0)−ǫk(τn−0)]. Computing the time average of the squared\nHelfand moment, we obtain an expression of the thermal conductiv ity according to\nκ= lim\nL→∞1\nL(kBT)2lim\nn→∞1\n2τn/angbracketleftBig\n[H(τn)−H(τ0)]2/angbracketrightBig\n(39)\nwhereL=N/2 is the horizontal length of the system.\nFigure 11 shows the results of a computation of the thermal condu ctivity through\nequation (39) for different system sizes. Though the actual value s ofκvary wildly with\nN, it is clear that a finite asymptotic value is reached for N≃102. In this case, the\nconstant of proportionality in equation (35) takes the value A= 0.98±0.08, close to 1.\nSimilar results were obtained for other parameter values, and othe r cell geometries as\nwell.\n0 0.05 0.1 0.15 0.2 0.250.080.0850.090.0950.10.1050.110.1150.120.125\n1/Nκ/(l2 T1/2)\n  \nFigure 11. Reduced thermal conductivity, computed from the mean squared Helfand\nmoment, versus 1 /N. The parameters are ρ= 9/25,ρm= 9/50. The system sizes\nvary from N= 4 toN= 100. The dashed line shows the binary collision frequency,\nνb≃0.1225. The solid line shows a linear fit of the data, with y-intercept 0 .12±0.01,\nin agreement with the prediction (35).Fourier’s law in many particle dispersing billiards 23\n5. Lyapunov spectrum\nA key aspect of our model, which justifies the assumption of local eq uilibrium, is that\nit is strongly chaotic. This property can be illustrated through the c omputation of the\nLyapunov spectrum and Kolmogorov-Sinai entropy in equilibrium con ditions.\nAs mentioned earlier, in the absence of interaction between the cells ,ρm< ρc, The\nLyapunov spectrum of a system of Ncells has Npositive and Nnegative Lyapunov\nexponents, which, if divided by the average speed of the particle to which they are\nattached, are all equal in absolute value. This reference value we d enote by λ+. The\n2Nremaining Lyapunov exponents vanish.\nAs we increase ρmand let the particles interact, we expect that, in the regime\n0< ρm−ρc≪1, where binary collision events are rare, the Lyapunov exponents will\nessentially be determined by λ+multiplied by a factor which is specified by the particle\nvelocities. The exchange of velocities thus produces an ordering of the exponents which\ncan be computed as shown below. We note that the other half of the spectrum, which\nremains zero in this approximation, will only pick up positive values as a r esult of the\ninteractions.\nAssume for the sake of the argument that Nis large. The probability that a given\nparticle with velocity vhas exponent λ=vλ+less than a value λi=viλ+can be\napproximated by the probability that the particle velocity be less tha nvi, which, if we\nassume a canonical form of the equilibrium distribution, is\nProb(λ < λi) = Prob( v < vi),\n=β/integraldisplaymv2\ni/2\n0dǫexp(−βǫ),\n= 1−exp(−βmv2\ni/2). (40)\nBut this probability is simply ( N−i+ 1/2)/N. Therefore the half of the positive\nLyapunov exponent spectrum, which is associated to the isolated m otion of particles\nwithin their cells, becomes, in the presence of rare collision events,\nλi=λ+/radicalbigg2\nmβ/bracketleftbigg\nlnN\ni−1/2/bracketrightbigg1/2\n, i= 1,...,N, (41)\nwith ordering λ1> λ2> ... > λ N. In particular, the largest exponent λ1grows\nlike√\nlnN. The Kolmogorov-Sinai entropy on the other hand is extensive : hKS=\nNλ+/radicalbig\nπ/(2mβ).\nRefined expressions can be computed by taking the microcanonical distribution\nassociated to a finite N. In particular, the expressions of the Lyapunov exponents\nbecome\nλi=λ+/radicalBigg\n2N\nmβ/bracketleftBigg\n1−/parenleftbiggi−1/2\nN/parenrightbigg1/(N−1)/bracketrightBigg1/2\n. (42)\nWe mention in passing that similar arguments are relevant and can be u sed to\napproximate the Lyapunov spectrum (actually half of it) of other m odels, such as a\nmixture of light and heavy particles [30].Fourier’s law in many particle dispersing billiards 24\n/SoΛi∆me∆sqΡ\n/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/StΑΡ\n/StΑΡ/StΑΡ\n/StΑΡ/StΑΡ\n/StΑΡ/StΑΡ\n/StΑΡ/StΑΡ\n/StΑΡ/StΑΡ\n/StΑΡ/StΑΡ\n/StΑΡ/StΑΡ\n/StΑΡ/StΑΡ\n/StΑΡ/StΑΡ\n/StΑΡ\n0 10 20 30 40/MiΝus20/MiΝus1001020\niΛi\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛeti/EquΑΛ1,...,10\n0.000.050.100.150.200.250.3051015\nΡm/MiΝusΡcΛi/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛ��et\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛet\n/BuΛΛeti/EquΑΛ11,...,19\n0.10 0.200.150.010.1110\nΡm/MiΝusΡcΛi\nFigure 12. (Top) Lyapunov exponents λiversusicomputed with a rhombic channel\nof sizeN= 10 cells and parameter ρ= 9/25. The different curves correspond to\ndifferent values of ρm= 0.11 (bottom curve) to 0 .34 (top curve) by steps of 0 .01.\nThe lines of stars are obtained for ρm= 0.04< ρc, yielding the exponents λ+and\nλ−associated to isolated cells, with all the particles at the same speed. The squares\ncorrespond to the first half of the spectrum as predicted by equa tion (42). (Bottom)\nλiversusρm−ρc. The first of the two figures displays the first half of the positive\npart of the spectrum of exponents, λ1,...,λ Nand compares them to the asymptotic\nestimate equation (42) (straight lines). The second plot shows the second half of the\npositive part of the spectrum, λN+1,...,λ 2N−1, displaying their power-law scaling to\nzero asρm→ρc.Fourier’s law in many particle dispersing billiards 25\n/Bullet\n/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet/Bullet\n/Bullet\n/Bullet\n/Bullet\n/Bullet/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle/SolidUpTriangle/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle\n/SolidUpTriangle/SoΛi∆me∆sqΡ\n/SoΛi∆me∆sqΡ\n/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ/SoΛi∆me∆sqΡ\n/SoΛi∆me∆sqΡ\n/SoΛi∆me∆sqΡ\n/SoΛi∆me∆sqΡ/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond\n/MedSolidDiamond/SolidDownTriangle\n/SolidDownTriangle\n/SolidDownTriangle\n/SolidDownTriangle\n/SolidDownTriangle\n/SolidDownTriangle\n/SolidDownTriangle\n/SolidDownTriangle\n/SolidDownTriangle\n/SolidDownTriangle\n0.10 0.200.151.00\n0.1010.00\n0.01\nΡm/MiΝusΡchKS/Slash1N/MiΝusΛ/PΛus\nFigure 13. Extensivity of the Kolmogorov-Sinai entropy. The vertical axis sh ows\nthe difference between the Kolmogorov Sinai entropy, here compu ted from the sum of\nthe positive Lyapunov exponents for system sizes N= 3 (magenta circles), 5 (blue up\ntriangles), 10 (green squares), 15 (cyan diamonds), 20 (red dow n triangles), divided\nby the corresponding microcanonical average velocity, and the Ly apunov exponent of\nthe isolated billiard cell at unit velocity. The curves fall nicely upon eac h other and\nconverge to zero as ρm→ρc.\nLet us again insist that equations (41) and (42) account for only Nof the 2N−1\npositive Lyapunov exponents. N−1 are vanishing within this approximation. For all\nthese Lyapunov exponents, we expect the corrections to vanish with the binary collision\nfrequency as we go to the critical geometry.\nFigures 12 and 13 show the results of a numerical computation of th e whole\nspectrum of Lyapunov exponents and corresponding Kolmogorov -Sinai entropy for the\none-dimensional channel of rhombic cells. The agreement with equa tion (42) as ρm→ρc\nisverygood,attheexceptionperhapsofthelastfewamongthefir stNexponents, whose\nconvergence to the asymptotic value (42) appears to be slower. I nterestingly, the largest\nexponents have a minimum which occurs at about the value of ρmfor which the binary\nand wall collision frequencies have ratio unity, see figure 7. Indeed, for larger radii ρm,\nthe spectrum is similar to that of a channel of hard discs (without ob stacles) [29]. Note\nthat the same holds for the ratio between the thermal conductivit y and binary collision\nfrequency, as seen from figure 8. Thus one can interpret the occ urrence of a minimum\nof the largest exponent as evidence of a crossover from a near clo se-packing solid-like\nphase (ρm/lessorsimilarρ) to a gaseous-like phase trapped in a rigid structure ( ρm/greaterorsimilarρc).\n6. Conclusions\nTo summarize, lattice billiards form the simplest class of Hamiltonian mod els for which\none can observe normal transport of energy, consistent with Fo urier’s law. Geometric\nconfinement restricts the transport properties of this system t o heat conductivity alone,\nthereby avoiding the complications of coupling mass and heat transp orts, which are\ncommon to other many particle billiards.Fourier’s law in many particle dispersing billiards 26\nThe strong chaotic properties of the isolated billiard cells warrant, in a parametric\nregime where interactions among moving particles are seldom, the pr operty of local\nequilibrium. This is to say, assuming that wall collision frequencies are o rder-of-\nmagnitude larger than binary collision frequencies, that one is entitle d to making\na Markovian approximation according to which phase-space distribu tions are spread\nover individual cells. We are thus allowed to ignore the details of the dis tribution at\nthe level of individual cells and coarse-grain the phase-space distr ibutions to a many-\nparticle energy distribution function, thereby going from the pseu do-Liouville equation,\ngoverning the microscopic statistical evolution, to the master equ ation, which accounts\nfor the energy exchanges at a mesoscopic cell-size scale and local t hermalization. The\nenergy exchange process further drives the relaxation of the wh ole system to global\nequilibrium.\nThis separation of scales, from the cell scale dynamics, correspon ding to the\nmicroscopic level, to the energy exchange among neighbouring cells a t the mesoscopic\nlevel, and to the relaxation of the system to thermal equilibrium at th e macroscopic\nlevel, is characterized by three different rates. The process of re laxation to local\nequilibrium has a rate given by the wall collision frequency, much larger than the\nrate of binary collisions, which characterizes the rate of energy ex changes which\naccompany the relaxation to local thermal equilibrium, itself much lar ger than the\nhydrodynamic relaxation rate, given by the binary collision rate divide d by the square\nof the macroscopic length of the system.\nOn this basis, having reduced the deterministic dynamics of the many particles\nmotions to a stochastic process of energy exchanges between ne ighboring cells, we are\nable to derive Fourier’s law and the macroscopic heat equation.\nThe energytransport master equation canbesolved withtheresu lt, tobepresented\nelsewhere [24], that the binary collision frequency and heat conduct ivity are equal.\nUnder the assumption that the wall and binary collision time scales of t he billiard are\nwell separated, the transposition of this result to the billiard dynam ics is that :\n(i) The heat conductivity of the mechanical model is proportional t o the binary\ncollision frequency, i. e.the rate of collisions among neighbouring particles,\nκ\nl2νb=A, ν b≪νw,\nwith a constant Athat is exactly 1 at the critical geometry, ρm→ρc, where the\nevolution of probability densities is rigorously described by the maste r equation,\nand remains close to unity over a large range of parameter values fo r which we\nconclude the time scale separation is effective and the master equat ion therefore\ngives a good approximation to the energy transport process of th e billiard ;\n(ii) The heat conductivity and the binary collision frequency both van ish in the limit\nof insulating system, ρm→ρc, with (ρm−ρc)3,\nlim\nρm→ρcκ\nl2(ρm−ρc)3= lim\nρm→ρcνb\n(ρm−ρc)3=2ρm\n|Bρ|2/radicalbigg\nkBT\nπmc3,\nwhere the coefficient c3depends on the specific geometry of the binary collisions.Fourier’s law in many particle dispersing billiards 27\nThough both results are exact strictly speaking only in the limit ρm→ρc, the first\none, according to our numerical computations, is robust and holds throughout the range\nofparameters for which the binarycollision frequency ismuch less th anthe wall collision\nfrequency, νb≪νw. The deviations from A= 1 which we observed at intermediary\nvalues of ρm, where the separation of time scales is less effective, are interpret ed as\nactual deviations of the energy transport process of the billiard f rom that described by\nthe master equation, where correlations between the motions of n eighboring particles\nmust be accounted for.\nUnder the conditions of local equilibrium, the Lyapunov spectrum ha s a simple\nstructure, half of it being determined according to random velocity distributions within\nthe microcanonical ensemble, while the other half remains close to ze ro. The analytic\nexpression of the Lyapunov spectrum that we obtained is thus exa ct at the critical\ngeometry. The Kolmogorov-Sinai entropy is equal to the sum of th e positive Lyapunov\nexponents andthusdetermined byhalfofthem. Itisextensive inth enumber ofparticles\nin the system, whereas the largest Lyapunov exponent grows like t he square root of the\nlogarithm of that number. As we mentioned earlier, we believe our met hod is relevant\nto the computation of the Lyapunov spectrum of other models of in teracting particles\n[30]. The computation of the full spectrum, particularly regarding t he effect of binary\ncollisions on the exponents, remains an open problem.\nAcknowledgments\nThe authors wish to thank D. Alonso, J. Bricmont, J. R. Dorfman, M . D. Jara\nValenzuela, A. Kupiainen, R. Lefevere, C. Liverani, S. Olla and C. Mej ´ ıa-Monasterio\nfor fruitful discussions and comments at different stages of this w ork. This research\nis financially supported by the Belgian Federal Government under th e Interuniversity\nAttractionPoleprojectNOSYP06/02andtheCommunaut´ efran¸ caisedeBelgiqueunder\ncontract ARC 04/09-312. TG is financially supported by the Fonds d e la Recherche\nScientifique F.R.S.-FNRS.\nAppendix A. Computation of the Kernel\nWeprovideinthisappendix theexplicit formofthetransitionrate W, givenbyequation\n(10).\nWe first substitute the two velocity integrals by two angle integrals, eliminating the\ntwo delta functions which involve only the local energies.\n/integraldisplay\nˆeab·vab>0dvadvbˆeab·vabδ/parenleftbigg\nǫa−mv2\na\n2/parenrightbigg\nδ/parenleftbigg\nǫb−mv2\nb\n2/parenrightbigg\nδ/parenleftBig\nη−m\n2[(ˆeab·va)2−(ˆeab·vb)2]/parenrightBig\n=√\n2\nm5/2/integraldisplay\nD+dθadθb(√ǫacosθa−√ǫbcosθb)δ(η−ǫacos2θa+ǫbcos2θb),(A.1)\nwhereθa/bdenote the angles of the velocity vectors va/bwith respect to the direction\nφof the relative position vector joining particles aandb,ˆeab= (cosφ,sinφ), and theFourier’s law in many particle dispersing billiards 28\nangle integration is performed over the domain D+such that√ǫacosθa>√ǫbcosθb.\nWith the above expression (A.1), the explicit φdependence has disappeared so\nthat we have effectively decoupled the velocity integration from the integration over the\ndirection of the relative position between the two colliding particles. W e can further\ntransform this expression in terms of Jacobian elliptic functions as f ollows.\nLetxi= cosθi,i=a,b, in equation (A.1), which becomes\n4√\n2\nm5/2/integraldisplay1\n−1dxa/radicalbig\n1−x2\na/integraldisplay1\n−1dxb/radicalbig\n1−x2\nbθ(√ǫaxa−√ǫbxb)(√ǫaxa−√ǫbxb)δ(η−ǫax2\na+ǫbx2\nb),(A.2)\nwhereθ(.) is the Heaviside step function. We thus have to perform the xaandxb\nintegrations along the line defined by the argument of the delta func tion,\nη=ǫax2\na−ǫbx2\nb, (A.3)\nand that satisfies the condition\n√ǫaxa>√ǫbxb. (A.4)\nTo carry out this computation, we have to consider the following alte rnatives :\n(i)ǫa< ǫb, 0< η < ǫ a\nThe solution of equation (A.3) which is compatible with equation (A.4) is\nxa=/parenleftbiggη+ǫbx2\nb\nǫa/parenrightbigg1/2\n. (A.5)\nPlugging this solution into equation (A.2) and setting the bounds of th exb-integral\nto±/radicalbig\n(ǫa−η)/ǫb, the expression (A.2) reduces to (omitting the prefactors)\n/integraldisplay√\n(ǫa−η)/ǫb\n0dxb1/radicalbig\nǫa−η−ǫbx2\nb/radicalbig\n1−x2\nb=1√ǫbK/parenleftbiggǫa−η\nǫb/parenrightbigg\n,(A.6)\nwhereKdenotes the Jabobian elliptic function of the first kind,\nK(m) =/integraldisplayπ/2\n0(1−msin2θ)−1/2dθ(m <1). (A.7)\nThus the kernel is, in this case,\nW(ǫa,ǫb|ǫa−η,ǫb+η) =2ρm\nπ2|Lρ,ρm(2)|/radicalbigg\n2\nmǫbK/parenleftbiggǫa−η\nǫb/parenrightbigg/integraldisplay\ndφdR. (A.8)\n(ii)ǫa< ǫb,ǫa−ǫb< η <0\nThis case is similar to case (i), with equation (A.5) replaced by\nxb=−/parenleftbigg−η+ǫax2\na\nǫb/parenrightbigg1/2\n(A.9)\nand−1< xa<+1. The expression of the kernel corresponding to this case is\ntherefore\nW(ǫa,ǫb|ǫa−η,ǫb+η) =2ρm\nπ2|Lρ,ρm(2)|/radicalBigg\n2\nm(ǫb+η)K/parenleftbiggǫa\nǫb+η/parenrightbigg/integraldisplay\ndφdR.(A.10)Fourier’s law in many particle dispersing billiards 29\n(iii)ǫa< ǫb,−ǫb< η < ǫ a−ǫb<0\nThis case is similar to case (ii), with −/radicalbig\n(ǫb+η)/ǫa< xa<+/radicalbig\n(ǫb+η)/ǫa. In this\ncase, the expression of the kernel is given by\nW(ǫa,ǫb|ǫa−η,ǫb+η) =2ρm\nπ2|Lρ,ρm(2)|/radicalbigg\n2\nmǫaK/parenleftbiggǫb+η\nǫa/parenrightbigg/integraldisplay\ndφdR. (A.11)\nThe cases with ǫa> ǫbare obtained from the cases above with the roles of aandb\ninterchanged and η→ −η.\nAppendix B. Collision area near the critical geometry\nThereasonwhy c1andc2inequation(22)vanishisthattheangledifferenceis O(ρm−ρc)\nand the area A1(φ) =O[(ρm−ρc)2]. These quantities are easily computed.\nLetρm= (1+ε)ρc,ε≪1. We have\nφT=2ρc\nlε−ρc\nlε2+O(ε3), (B.1)\nφM=2ρc\nlε+4ρ3\nc\nl3ε2+O(ε3), (B.2)\nwhich indicates that the bounds of the angle integrals appearing in eq uation (18) are\nO(ε), withφMandφTdiffering only to O(ε2).\nThe leading contribution to the integral α(ρ,ρm) therefore stems only from the\nintegration of A1(φ), which we can compute explicitly by expanding ρmaboutρcand\ntaking into consideration that φisO(ε). The result is\n/integraldisplayφM\n0A1(φ)dφ≃/integraldisplayφT\n0A1(φ)dφ,\n≃/integraldisplayφT\n0/parenleftBig16ρ3\nc\nlε2−4lρcφ2/parenrightBig\ndφ,\n=64ρ4\nc\n3l2ε3, (B.3)\nwhich yields the leading coefficient c3in equation (22).\nWe can compute the coefficients of the next few powers in the expan sion (21) in a\nsimilar fashion. First we notice that A2(φ) isO(ε3) so that its integral between φTand\nφMisO(ε5). Therefore only the integral of A1(φ) contributes to c4in equation (22).\nThe computation of the next terms in the expansion is more involved s ince it\nrequires the integration of A2(φ), whose expression is :\nA2(φ) = 8ρ2arcsin/bracketleftbiggρmlsinφ−(ρ2\nm−ρ2\nc)\nρ2/bracketrightbigg1/2\n(B.4)\n−4[ρmlsinφ−(ρ2\nm−ρ2\nc)]1/2(4ρ2\nm+l2−4ρmlsinφ)1/2.\nExpanding this expression for ρmε-close to ρcandφ ε2-close to φT, we get, to leading\norder,\n/integraldisplayφM\nφTA2(φ)dφ=1024ρ4ρ4\nc\n15l6ε5. (B.5)Fourier’s law in many particle dispersing billiards 30\nCombining this expression with the 5 thorder contribution to the integral of A1(φ), we\nobtainc5, equation (22). We point out that this coefficient is actually much larg er than\nc3andc4, a reason being that negative powers of ρcappear in its expression. The same\nholds of the next few coefficients.\n[1] Fourier J 1822 Th´ eorie analytique de la chaleur (Didot, Paris); Reprinted 1988 (J Gabay, Paris);\nFacsimile available online at http://gallica.bnf.fr .\n[2] Bonetto F, Lebowitz J L, and Rey-Bellet L 2000 Fourier Law: A Challenge To Theorists in Fokas\nA, Grigoryan A, Kibble T, Zegarlinski B (Eds.) Mathematical Physics 2000 (Imperial College,\nLondon).\n[3] BunimovichLAandSinaiYaG1980 Statistical properties of lorentz gas with periodic configu ration\nof scatterers Commun. Math. Phys. 78479.\n[4] Bunimovich L A and Spohn K 1996 Viscosity for a periodic two disk fluid: An existence proof\nCommun. Math. Phys. 176661.\n[5] Machta J and Zwanzig R 1983 Diffusion in a Periodic Lorentz Gas Phys. Rev. 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Phys. 109671."    },    {        "title": "1909.04362v3.Spin_Pumping_from_Permalloy_into_Uncompensated_Antiferromagnetic_Co_doped_Zinc_Oxide.pdf",        "content": "Spin Pumping from Permalloy into Uncompensated Antiferromagnetic Co doped Zinc\nOxide\nMartin Buchner,1,\u0003Julia Lumetzberger,1Verena Ney,1Tadd aus Scha\u000bers,1,yNi\u0013 eli Da\u000b\u0013 e,2and Andreas Ney1\n1Institut f ur Halbleiter- und Festk orperphysik, Johannes Kepler Universit at, Altenberger Str. 69, 4040 Linz, Austria\n2Swiss Light Source, Paul Scherrer Institut, CH-5232 Villigen PSI, Switzerland\n(Dated: August 11, 2021)\nHeterostructures of Co-doped ZnO and Permalloy were investigated for their static and dynamic\nmagnetic interaction. The highly Co-doped ZnO is paramagentic at room temperature and becomes\nan uncompensated antiferromagnet at low temperatures, showing a narrowly opened hysteresis and\na vertical exchange bias shift even in the absence of any ferromagnetic layer. At low temperatures in\ncombination with Permalloy an exchange bias is found causing a horizontal as well as vertical shift\nof the hysteresis of the heterostructure together with an increase in coercive \feld. Furthermore,\nan increase in the Gilbert damping parameter at room temperature was found by multifrequency\nFMR evidencing spin pumping. Temperature dependent FMR shows a maximum in magnetic\ndamping close to the magnetic phase transition. These measurements also evidence the exchange\nbias interaction of Permalloy and long-range ordered Co-O-Co structures in ZnO, that are barely\ndetectable by SQUID due to the shorter probing times in FMR.\nI. Introduction\nIn spintronics a variety of concepts have been devel-\noped over the past years to generate and manipulate spin\ncurrents [1, 2]. Amongst them are the spin Hall e\u000bect\n(SHE), which originates from the spin orbit coupling [3],\nspin caloritronics [4] utilizing the spin seebeck e\u000bect [5]\nor spin transfer torque (current induced torque) due to\nangular momentum conservation [6] as examples. Spin\npumping [7], where a precessing magnetization transfers\nangular momentum to an adjacent layer, proved to be a\nvery versatile method since it has been reported for di\u000ber-\nent types of magnetic orders [8{11] or electrical properties\n[12{14] of materials. Furthermore it could also be veri-\n\fed in trilayer systems where the precessing ferromagnet\nand the spin sink, into which the angular momentum is\ntransferred, are separated by a non-magnetic spacer [15{\n18]. This is strongly dependent on the material, while\nfor Cu [15], Au [16], or Al [17] pumping through a few\nnanometers is possible an MgO barrier of 1 nm is enough\nto completely suppress spin pumping [18].\nSpintronic devices are usually based on a ferromagnet\n(FM) although antiferromagnetic spintronics [19] holds\nthe advantages of faster dynamics, less perturbation by\nexternal magnetic \felds and no stray \felds. The latter\ntwo are caused by the zero net magnetization of an an-\ntiferromagnet (AFM), which on the other hand makes\nthem harder to manipulate. One way to control an\nAFM is by using an adjacent FM layer and exploiting\nthe exchange-bias (EB) e\u000bect [20, 21]. Measuring spin-\ntransfer torque in FM/AFM bilayer structures, is possi-\n\u0003Electronic address: martin.buchner@jku.at; Phone: +43-732-\n2468-9651; FAX: -9696\nyCurrent address: NanoSpin, Department of Applied Physics,\nAalto University School of Science, P.O. Box 15100, FI-00076\nAalto, Finlandble [22, 23], but challenging due to Joule heating [24{26]\nor possible unstable antiferromagnetic orders [27]. Anti-\nferromagnets can be used either as spin source [28] or as\nspin sink [11, 29] in a spin pumping experiment. Thereby\nthe spin mixing conductance, a measure for the absorp-\ntion of angular (spin) momentum at the interface [7],\nis described by intersublattice scattering at an antiferro-\nmagnetic interface [30]. Linear response theory predicted\nan enhancement of spin pumping near magnetic phase\ntransitions [31], which could recently also be veri\fed ex-\nperimentally [29].\nIn this work we investigate the behavior of the uncom-\npensated, antiferromagnetic Co xZn1-xO with x2f0.3,\n0.5, 0.6g(in the following 30 %, 50 % and 60 % Co:ZnO)\nin contact to ferromagnetic permalloy (Py). While\nweakly paramagnetic at room temperature, Co:ZnO\nmakes a phase transition to an antiferromagnetic state at\na N\u0013 eel temperature ( TN) dependent on the Co concentra-\ntion [32]. This resulting antiferromagnetism is not fully\ncompensated which is evidenced by a narrow hysteresis\nand a non saturating magnetization up to 17 T [33]. Fur-\nthermore, Co:ZnO \flms exhibit a vertical EB in complete\nabsence of a FM layer [34]. This vertical exchange shift is\ndependent on the Co concentration [32], temperature and\ncooling \feld [35] and the \feld imprinted magnetization\npredominantly shows orbital character [36]. Note that\nbelow the coalesence limit of 20 % the vertical EB van-\nishes. Co:ZnO therefore o\u000bers to study magnetic inter-\nactions between an uncompensated AFM and a FM Py\nlayer. Static coupling, visible as EB, is investigated using\nsuper conducting quantum interference device (SQUID)\nmagnetometry. The dynamic coupling across the inter-\nface is measured using ferromagnetic resonance (FMR)\nat room temperature and around the magnetic transi-\ntion temperatures determined from M(T) SQUID mea-\nsurements. Element selective XMCD studies are carried\nout to disentangle the individual magnetic contributions.\nFinally heterostructures with an Al spacer were investi-\ngated to rule out intermixing at the interface as sourcearXiv:1909.04362v3  [cond-mat.mtrl-sci]  14 Oct 20192\nfor the coupling e\u000bect.\nII. Experimental Details\nHeterostructures consisting of Co:ZnO, Py and Al, as\nshown in Fig. 1 were fabricated on c-plane sapphire sub-\nstrates using reactive magnetron sputtering (RMS) and\npulsed laser deposition (PLD) at a process pressure of 4\n\u000210-3mbar. The di\u000berent layers of a heterostructure are\nall grown in the same UHV chamber with a base pressure\nof 2\u000210-9mbar in order to ensure an uncontaminated\ninterface. While Py and Co:ZnO are grown by magnetron\nsputtering, the Al spacer and capping layers are grown\nby PLD. Al and Py are fabricated at room temperature\nusing 10 standard cubic centimeters per minute (sccm)\nAr as a process gas.\nFor the heterostructures containing a Co:ZnO layer,\nsamples with three di\u000berent Co concentrations of 30 %,\n50 % and 60 % are grown utilizing preparation conditions\nthat yield the best crystalline quality known for Co:ZnO\nsingle layers [32, 33, 36]. For 30 % and 50 % Co:ZnO\nmetallic sputter targets of Co and Zn are used at an\nAr:O 2ratio of 10 : 1 sccm, while for 60 % Co:ZnO no oxy-\ngen and a ceramic composite target of ZnO and Co 3O4\nwith a 3 : 2 ratio is used. The optimized growth temper-\natures are 450\u000eC, 294\u000eC and 525\u000eC. Between Co:ZnO\ngrowth and the next layer a cool-down period is required,\nto minimize inter-di\u000busion between Py and Co:ZnO.\nThe static magnetic properties are investigated by\nSQUID magnetometry. M(H) curves are recorded at\n300 K and 2 K in in-plane geometry with a maximum\nmagnetic \feld of \u00065 T. During cool-down either a mag-\nnetic \feld of\u00065 T or zero magnetic \feld is applied to dif-\nferentiate between plus-\feld-cooled (pFC), minus-\feld-\ncooled (mFC) or zero-\feld-cooled (ZFC) measurements.\nAll measurements shown in this work have been corrected\nby the diamagnetic background of the sapphire substrate\nand care was taken to avoid well-known artifacts [37, 38].\nFor probing the element selective magnetic properties\nX-ray absorption (XAS) measurements were conducted\nat the XTreme beamline [39] at the Swiss Synchrotron\nLightsource (SLS). From the XAS the X-ray magnetic\ncircular dichroism (XMCD) is obtained by taking the\ndirect di\u000berence between XAS with left and right cir-\ncular polarization. The measurements were conducted\nwith total \ruoresence yield under 20\u000egrazing incidence.\nThereby, the maximum magnetic \feld of 6.8 T was ap-\nplied. Both, external magnetic \feld and photon helic-\nity have been reversed to minimize measurement arte-\nfacts. Again pFC, mFC and ZFC measurements were\nconducted applying either zero or the maximum \feld in\nthe respective direction.\nThe dynamic magnetic properties were measured us-\ning multi-frequency and temperature dependent FMR.\nMulti-frequency FMR is exclusively measured at room\ntemperature from 3 GHz to 10 GHz using a short cir-\ncuited semi-rigid cable [40]. Temperature dependentmeasurements are conducted using an X-band resonator\nat 9.5 GHz. Starting at 4 K the temperature is increased\nto 50 K in order to be above the N\u0013 eel-temperature of the\nCo:ZnO samples [32, 35]. At both FMR setups the mea-\nsurements were done in in-plane direction.\nThe measured raw data for SQUID, FMR, XAS and\nXMCD can be found in a following data repository [41].\nIII. Experimental results & Discussion\nFIG. 1: (a) shows the schematic setup of the samples. For the\nCo:ZnO layer three di\u000berent Co concentrations of 30 %, 50 %\nand 60 % are used. The cross section TEM image of the 60 %\nCo:ZnO/Py sample as well as the electron di\u000braction pattern\nof the Co:ZnO layer (b) and a magni\fcation on the interface\nbetween Co:ZnO and Py (c) are shown.\nFigure 1(a) displays the four di\u000berent types of samples:3\nCo:ZnO layers, with Co concentrations of 30 %, 50 % and\n60 %, are grown with a nominal thickness of 100 nm and\nPy with 10 nm. To prevent surface oxidation a capping\nlayer of 5 nm Al is used. For single 60 % Co:ZnO \flms\nthe vertical-exchange bias e\u000bect was largest compared to\nlower Co concentrations. Therefore, for 60 % Co:ZnO\nsamples with an additional Al layer as spacer between\nCo:ZnO and Py have been fabricated. The thickness of\nthe Al spacer (1 nm, 1.5 nm and 2 nm) is in a range where\nthe Al is reported not to suppress spin pumping e\u000bects\nitself [17].\nTEM\nTo get information about the interface between Py\nand Co:ZnO high resolution cross section transmission\nelectron microscopy (TEM) was done. In Fig. 1(b) the\ncross section TEM image of 60 % Co:ZnO/Py with the\nelectron di\u000braction pattern of the Co:ZnO is shown. A\nmagni\fcation of the interface between Co:ZnO and Py is\nshown in Fig. 1(c). From XRD measurements [32] it is\nobvious that the quality of the wurtzite crystal slightly\ndecreases for higher Co doping in ZnO. A similar be-\nhavior is observed in TEM cross section images. While\n35 % Co:ZnO shows the typical only slightly misoriented\ncolumnar grain growth [32] it is obvious from Fig. 1(b)\nthat the crystalline nanocolumns are less well ordered for\n60 % Co:ZnO. Although the electron di\u000braction pattern\ncon\frms a well ordered wurtzite structure, the misorien-\ntation of lattice plains is stronger than for 35 % Co:ZnO\n[32], even resulting in faint Moir\u0013 e fringes which stem from\ntilted lattice plains along the electron path. This cor-\nroborates previous \fndings of !-rocking curves in XRD\n[32, 36] where the increase in the full width at half maxi-\nmum also evidences a higher tilting of the crystallites, i.e.\nan increased mosaicity. The interface to the Py layer is\nsmooth, although it is not completely free of dislocations.\nAlso the interface seems to be rather abrupt within one\natomic layer, i.e. free of intermixing. A similar behavior\nis found for the interface between 50 % Co:ZnO and Py\n(not shown).\nXAS and XMCD\nFigure 2 shows XAS and XMCD spectra recorded at\n3 K and a magnetic \feld of 6.8 T at the Ni L 3/2and\nCo L 3/2edges of 60 % Co:ZnO/Py after pFC, mFC or\nZFC. For all three cooling conditions the Ni L 3/2edges\n(Fig. 2(a)) show a metallic character of the Ni XAS with-\nout any additional \fne structure characteristics for NiO\nand thus no sign of oxidation of the Py. Further, no dif-\nferences in the XAS or the XMCD of the Ni edges of\ndi\u000berent cooling conditions are found. The same is ob-\nserved for the Fe L 3/2edges, however, they are a\u000bected\ngreatly by self-absorption processes in total \ruorescence\nyield (not shown).\n/s56/s52/s48 /s56/s53/s48 /s56/s54/s48 /s56/s55/s48 /s56/s56/s48/s45/s50/s46/s53/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s40/s97/s41\n/s78/s105/s32/s76/s51/s47/s50/s32/s101/s100/s103/s101/s32/s64/s32/s51/s75/s44/s32/s84/s70/s89/s32\n/s50/s48/s176/s32/s103/s114/s97/s122/s105/s110/s103/s32/s105/s110/s99/s105/s100/s101/s110/s99/s101/s44/s32/s66/s32/s61/s32/s54/s46/s56/s84\n/s32/s88/s65/s83/s32/s112/s70/s67\n/s32/s88/s65/s83/s32/s90/s70/s67\n/s32/s88/s65/s83/s32/s109/s70/s67\n/s32/s32/s32/s32/s32 /s32/s88/s77/s67/s68/s32/s112/s70/s67\n/s32/s32/s32/s32/s32 /s32/s88/s77/s67/s68/s32/s90/s70/s67\n/s32/s32/s32/s32/s32 /s32/s88/s77/s67/s68/s32/s109/s70/s67/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s88/s65/s83/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s54/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s40/s98/s41/s45/s48/s46/s56/s45/s48/s46/s54/s45/s48/s46/s52/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s88/s77/s67/s68/s32/s40/s37/s41\n/s55/s55/s48 /s55/s56/s48 /s55/s57/s48 /s56/s48/s48 /s56/s49/s48/s45/s50/s46/s53/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s67/s111/s32/s76/s51/s47/s50/s32/s101/s100/s103/s101/s32/s64/s32/s51/s75/s44/s32/s84/s70/s89/s32\n/s50/s48/s176/s32/s103/s114/s97/s122/s105/s110/s103/s32/s105/s110/s99/s105/s100/s101/s110/s99/s101/s44/s32/s66/s32/s61/s32/s54/s46/s56/s84/s32/s88/s65/s83/s32/s112/s70/s67\n/s32/s88/s65/s83/s32/s90/s70/s67\n/s32/s88/s65/s83/s32/s109/s70/s67\n/s32/s88/s77/s67/s68/s32/s112/s70/s67\n/s32/s88/s77/s67/s68/s32/s90/s70/s67\n/s32/s88/s77/s67/s68/s32/s109/s70/s67\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s88/s65/s83/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s45/s48/s46/s48/s54/s45/s48/s46/s48/s52/s45/s48/s46/s48/s50/s48/s46/s48/s48/s48/s46/s48/s50/s48/s46/s48/s52/s48/s46/s48/s54\n/s88/s77/s67/s68/s32/s40/s37/s41FIG. 2: In (a) the XMCD at the Ni L 3/2edges after pFC,\nmFC and ZFC for 60 % Co:ZnO/Py are shown. (b) shows\nthe same for the Co L 3/2edges.\nThe Co L 3/2edges in Fig. 2(b) are also greatly af-\nfected by the self absorption of the total \ruorescence\nyield, since it is buried below 10 nm of Py and 5 nm of\nAl. In contrast to Ni the XAS and XMCD at the Co\nL3/2edges (Fig. 2(b)) are not metallic and evidence the\nincorporation of Co as Co2+in the wurtzite structure\nof ZnO [32, 36]. The overall intensity of the Co XMCD\nis strongly reduced indicating a small magnetic moment\nper Co atom well below metallic Co. This small e\u000bective\nCo moment in 60 % Co:ZnO can be understood by the\ndegree of antiferromagnetic compensation that increases\nwith higher Co doping concentrations [32]. Furthermore,\nno indications of metallic Co precipitates are visible in\nthe XAS and XMCD of the heterostructure as it would\nbe expected for a strong intermixing at the interface to\nthe Py.\nNo changes between the pFC, mFC and ZFC measure-\nments are visible also for the Co edges either in XAS or\nXMCD indicating that the spin system of the Co dopants\nis not altered in the exchange bias state. This corrob-\norates measurements conducted at the Co K-edge [36].\nAfter \feld cooling the XMCD at the Co main absorption\nincreased compared to the ZFC conditions. At the Co\nK-edge the main absorption stems from the orbital mo-\nment. The spin system is only measured indirectly at the\npre-edge feature which remained una\u000bected by the cool-\ning \feld conditions. The data of K- and L-edges com-\nbined evidences that the imprinted magnetization after\n\feld cooling is composed predominantly of orbital mo-4\nment, which is in good agreement with other EB systems\n[42, 43]\nSQUID\n/s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48/s45/s49/s48/s49\n/s45/s49/s48 /s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54 /s56 /s49/s48/s45/s49/s48/s49\n/s45/s50/s48/s48 /s45/s49/s48/s48 /s48 /s49/s48/s48 /s50/s48/s48/s45/s49/s54/s48/s45/s49/s50/s48/s45/s56/s48/s45/s52/s48/s48/s52/s48/s56/s48/s49/s50/s48/s49/s54/s48/s51/s48/s48/s75\n/s40/s98/s41/s32/s77/s47/s77/s91/s49/s48/s109/s84/s93\n/s48/s72/s32/s40/s109/s84/s41/s32/s80/s121/s32\n/s32/s51/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s53/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s54/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121/s40/s97/s41\n/s32\n/s32/s51/s48/s48/s75\n/s32/s50/s75/s77/s32/s40 /s101/s109/s117/s41\n/s48/s72/s32/s40/s109/s84/s41/s32/s32/s109/s70/s67\n/s32/s32/s112/s70/s67\n/s32/s32/s90/s70/s67/s54/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s50/s75/s32/s97/s102/s116/s101/s114/s58\nFIG. 3: At 300 K the M(H) curves of the single Py \flm al-\nmost overlaps with the M(H) curves of the heterostructures\nwith all three Co:ZnO concentrations (a). In the inset it can\nbe seen that there is no di\u000berence in coercive \feld for Py at\n300 K and 2 K. Measuring the 60 % Co:ZnO/Py heterostruc-\nture after plus, minus and zero \feld cooling, horizontal and\nvertical exchange bias shifts are visible, as well as an increase\nin the coercive \feld (b).\nThe static coupling in the heterostructures was investi-\ngated by integral SQUID magnetometry. Measurements\ndone at 300 K, as shown in Fig. 3(a), do not reveal a sig-\nni\fcant in\ruence of the Co:ZnO on the M(H) curve of\nPy. Just a slight increase in coercive \feld from 0.1 mT\nto 0.4 mT is determined. Some of the M(H) curves\nin Fig. 3(a) are more rounded than the others. This\ncan be attributed to slight variations in the aspect ra-\ntio of the SQUID pieces and thus variations in the shape\nanisotropy. The inset of Fig. 3(a) shows the hysteresis of\nthe single Py \flm at 300 K and 2 K, where no di\u000berence\nin coercivity is visible. Please note that up to now mea-\nsurements were conducted only in a \feld range of \u000610 mT\nand directly after a magnet reset. This is done to avoid\nin\ruences of the o\u000bset \feld of the SQUID [38]. At lowtemperatures, to determine the full in\ruence of Co:ZnO,\nhigh \felds need to be applied, as it has been shown in [35].\nTherefore, coercive \felds obtained from low temperature\nmeasurements are corrected by the known o\u000bset \feld of\n1.5 mT of the SQUID [38].\nSince the paramagnetic signal of Co:ZnO is close to the\ndetection limit of the SQUID and thus, orders of mag-\nnitude lower than the Py signal it has no in\ruence on\nthe room temperature M(H) curve. However, with an\nadditional Co:ZnO layer a broadening of the hysteresis,\na horizontal and a small vertical shift are measured at\n2 K as can be seen exemplary for 60 % Co:ZnO/Py in\nFig. 3(b). Similar to single Co:ZnO \flms where an open-\ning of theM(H) curve is already visible in ZFC mea-\nsurements [32, 34{36] also in the heterostructure no \feld\ncooling is needed to increase the coercive \feld.\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s48/s53/s49/s48/s49/s53/s50/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56/s50/s48/s50/s50\n/s32/s32/s99/s111/s101/s114/s99/s105/s118/s101/s32/s102/s105/s101/s108/s100/s32/s40/s109/s84/s41\n/s67/s111/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s32/s40/s37/s41\n/s40/s98/s41\n/s32/s32/s99/s111/s101/s114/s99/s105/s118/s101/s32/s102/s105/s101/s108/s100/s32/s40/s109/s84/s41\n/s116/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s54/s48/s37 /s32/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s97/s102/s116/s101/s114/s32/s90/s70/s67/s40/s97/s41\n/s51/s48 /s52/s48 /s53/s48 /s54/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s32/s112/s70/s67\n/s32/s109/s70/s67\n/s67/s111/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s32/s40/s37/s41/s118/s101/s114/s116/s105/s99/s97/s108/s32/s115/s104/s105/s102/s116/s32/s40/s37/s41\n/s45/s49/s50/s45/s49/s48/s45/s56/s45/s54/s45/s52/s45/s50/s48/s50/s52/s54/s56/s49/s48/s49/s50\n/s32/s112/s70/s67\n/s32/s109/s70/s67/s104/s111/s114/s105/s122/s111/s110/s116/s97/s108/s32/s115/s104/s105/s102/s116/s58\n/s104/s111/s114/s105/s122/s111/s110/s116/s97/s108/s32/s115/s104/s105/s102/s116/s32/s40/s109/s84/s41/s118/s101/s114/s116/s105/s99/s97/s108/s32/s115/s104/s105/s102/s116/s58\nFIG. 4: (a) At 2 K the coercivity increases with Co concen-\ntration in the heterostructure. In the inset the temperature\ndependence of the coercivity of the 60 % Co:ZnO/Py het-\nerostructure is given. (b) The vertical shift (circles) and the\nhorizontal shift (squares) depend on the Co concentration.\nBoth shifts reverse the direction when the measurement is\nchanged from pFC to mFC.\nEarlier works [32, 34] demonstrated that the hystere-\nsis opening and vertical shift in Co:ZnO are strongly de-\npendent on the Co concentration and increase with in-\ncreasing Co doping level. Furthermore, the EB e\u000bects\nare observed in the in-plane and out-of-plane direction,\nwith a greater vertical shift in the plane. Therefore,\nthe heterostructers with Py are measured with the mag-\nnetic \feld in in-plane direction. Figure 4(a) provides an\noverview of the coercive \feld after ZFC for the di\u000ber-5\nent Co concentrations. The coercive \feld increases from\n0.1 mT for single Py to 20.6 mT for 60 % Co:ZnO/Py.\nAdditionally, in the inset the temperature dependence of\nthe coercive \feld of the 60 % Co:ZnO/Py heterostructure\nis shown, since it shows the strongest increase in coercive\n\feld. From the 20.6 mT at 2 K it \frst increases slightly\nwhen warming up to 5 K. That the maximum coercivity\nis not at 2 K is in good agreement with measurements at\nsingle 60 % Co:ZnO \flms where a maximum hysteresis\nopening at 7 K was determined [35]. Afterwards the co-\nercive \feld decreases. At the N\u0013 eel temperature of 20 K a\ncoercive \feld of 11.6 mT is measured. Above T Nit de-\ncreases even further but the coercivity is still 3.65 mT at\n50 K. A coupling above T Ncould stem from long range\nmagnetic ordered structures in Co:ZnO where \frst in-\ndications are visible already in single Co:ZnO \flms [32].\nHowever, for single layers they are barely detectable with\nthe SQUID.\nThe vertical (circles) and horizontal (squares) hystere-\nsis shifts after pFC and mFC are shown in Fig. 4(b) for\nthe Py samples with Co:ZnO layers. Similar to single\nCo:ZnO \flms the vertical shift increases with rising Co\nconcentration. The shift is given in percent of the magne-\ntization at 5 T to compensate for di\u000berent sample sizes.\nDue to the overall higher magnetization at 5 T in combi-\nnation with Py this percentage for the heterostructures\nis lower than the vertical shift for single Co:ZnO \flms.\nWith increasing Co concentration the degree of antiferro-\nmagnetic compensation increases [32, 35], which in turn\nshould lead to a stronger EB coupling. This can be\nseen in the horizontal shift and thus EB \feld which is\nstrongest for 60 % Co:ZnO/Py and nearly gone for 30 %\nCo:ZnO/Py. For both kinds of shift the pFC and mFC\nmeasurements behave similar, except the change of di-\nrection of the shifts.\nMultifrequency FMR\nThe dynamic coupling between the two layers has been\ninvestigated by multifrequency FMR measured at room\ntemperature. The frequency dependence of the resonance\nposition between 3 GHz and 10 GHz of the heterostruc-\ntures is shown in Fig. 5(a). The resonance position of Py\nyields no change regardless of the Co concentration in\nthe Co:ZnO layer or its complete absence. Also in 2 nm\nAl/Py and 60 % Co:ZnO/2 nm Al/Py the resonance po-\nsition stays unchanged. The resonance position of a thin\n\flm is given by Kittel formula [44]:\nf=\r\n2\u0019p\nBres(Bres+\u00160M) (1)\nwith the gyromagnetic ratio \r=g\u0016B\n\u0016hand magnetiza-\ntionM. However, any additional anisotropy adds to Bres\nand therefore alters eq. (1) [44]. The fact that all samples\nshow the identical frequency dependence of the resonance\nposition evidences that neither the gyromagnetic ratio \rand thus the Py g-factor are in\ruenced nor any addi-\ntional anisotropy BAniso is introduced by the Co:ZnO.\nBy \ftting the frequency dependence of the resonance po-\nsition using the Kittel equation with the g-factor of 2.11\n[45] all the samples are in the range of (700 \u000615) kA/m,\nwhich within error bars is in good agreement with the\nsaturation magnetization of (670 \u000650) kA/m determined\nfrom SQUID.\n/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48 /s49/s48/s48 /s49/s49/s48/s50/s52/s54/s56/s49/s48\n/s51 /s52 /s53 /s54 /s55 /s56 /s57 /s49/s48/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s48/s46/s48/s48/s53/s48/s46/s48/s48/s54/s48/s46/s48/s48/s55/s48/s46/s48/s48/s56/s48/s46/s48/s48/s57/s48/s46/s48/s49/s48/s48 /s50/s53 /s53/s48 /s55/s53 /s49/s48/s48/s45/s49/s48/s49\n/s40/s98/s41\n/s32/s32/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41\n/s66\n/s114/s101/s115/s32/s40/s109/s84/s41/s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s65/s108/s47/s80/s121 \n/s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121 \n/s32/s53/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121 \n/s32/s51/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121 \n/s32/s65/s108/s47/s80/s121 \n/s32/s80/s121 /s40/s97/s41\n/s32/s32/s66\n/s112/s112/s32/s40/s109/s84/s41\n/s102/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s71/s72/s122/s41\n/s32/s32\n/s67/s111/s32/s99/s111/s110/s99/s101/s110/s116/s114/s97/s116/s105/s111/s110/s32/s40/s37/s41\n/s32/s32/s110/s111/s114/s109/s46/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s66/s32/s40/s109/s84/s41/s32/s53/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s32/s32/s32/s32/s102/s114/s101/s113/s46/s32/s61/s32/s54/s46/s53/s56/s71/s72/s122\n/s32/s76/s111/s114/s101/s110/s116/s105/s97/s110/s32/s102/s105/s116\nFIG. 5: The resonance \felds determined at room temperature\nwith multifrequency FMR are seen in (a). In the inset an ex-\nemplary FMR spectrum for of 50 % Co:ZnO/Py at 6.58 GHz is\nshown with the corresponding Lorentian \ft. For the linewidth\n(b) and the associated damping parameter \u000b(inset) an in-\ncrease is visible for the heterostructures with higher Co con-\ncentration in the Co:ZnO. The lines are linear \fts to the data.\nEven though the Co:ZnO layer does not in\ruence the\nresonance position of the FMR measurement the het-\nerostructures exhibit an increase in linewidth. This cor-\nresponds to a change of the damping in the system. The\nfrequency dependence of the linewidth can be used to sep-\narate the inhomogeneous from the homogeneous (Gilbert\nlike) contributions, from which the Gilbert damping pa-\nrameter\u000bcan be determined.\n\u0001B= \u0001Bhom+ \u0001Binhom (2)6\nwhere\n\u0001Bhom=4\u0019\u000b\n\rf (3)\nNo di\u000berence in linewidth between Al/Py (open stars)\nand Py (full stars) is found, as can be seen in Fig. 4(b)\nwhere the peak to peak linewidth B ppis plotted over the\nmeasured frequency range for all the heterostructures.\nWhile the heterostructure with 30 % Co:ZnO/Py (green\ntriangles) lies atop the single Py and the Al/Py \flm,\nthe linewidth increases stronger with frequency for 50 %\nCo:ZnO/Py (blue circles). The broadest FMR lines are\nmeasured for the 60 % Co:ZnO/Py heterostructure (red\nsqaures).\nUsing the Py g-factor of 2.11 [45], \u000bcan be calcu-\nlated from the slopes of the frequency dependence ex-\ntracted from the linewidths seen in Fig. 5(b): the result-\ning\u000bare shown in the inset. For the single Py layer \u000bPy\n= (5.7\u00060.3)\u000210-3which compares well to previously re-\nported values [7]. This increases to \u000b50= (8.0\u00060.3)\u000210-3\nfor 50 % Co:ZnO/Py and even \u000b60= (9.4\u00060.3)\u000210-3for\n60 % Co:ZnO/Py. So the damping increases by a factor\nof 1.64 resulting in a spin pumping contribution \u0001 \u000b=\n(3.7\u00060.5)\u000210-3that stems from the angular momentum\ntransfer at the interface of Py and Co:ZnO. By insertion\nof a 2 nm Al spacer layer \u0001 \u000breduces to (0.8\u00060.5)\u000210-3.\nDependence on the Al spacer thickness\nTo obtain information about the lengthscale of the\nstatic and dynamic coupling, heterostructures with Al\nspacer layers of di\u000berent thickness (1 nm, 1.5 nm and\n2 nm thick) between Py and the material beneath (sap-\nphire substrate or 60 % Co:ZnO) were fabricated. With-\nout a Co:ZnO layer the spacer underlying the Py layer\ndoes not exhibit any changes in either SQUID (not\nshown) or FMR (see Fig 5 (a) and (b)). The results ob-\ntained for the 60 % Co:ZnO/Al/Py heterostructure for\nthe coercive \feld, vertical and horizontal shift extracted\nfromM(H) curves are shown in Fig. 6(a), whereas the\ndamping parameter \u000bfrom room temperature multifre-\nquency FMR measurements, analogues to Fig. 5(b), are\ndepicted in Fig. 6(b).\nThe horizontal shift and the increased coercive \feld\nare caused by the coupling of FM and AFM moments in\nrange of a few \u0017Angstrom to the interface [46{48]. There-\nfore, both e\u000bects show a similar decrease by the insertion\nof an Al spacer. While the horizontal shift and coer-\ncive \feld are reduced signi\fcantly already at a spacer\nthickness of 1 nm, the vertical shift (inset of Fig. 6(a))\nis nearly independent of the Al spacer. Comparing with\nthe XMCD spectra of Fig. 2 it can be concluded that\nthe vertical shift in the uncompensated AFM/FM sys-\ntem Co:ZnO/Py stems solely from the increased orbital\nmoment of pinned uncompensated moments in Co:ZnO\nand is independent of the FM moments at the interface.Furthermore, the FM moments do not exhibit any ver-\ntical shift and the exchange between the two layers only\nresults in the horizontal shift.\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s54/s48/s46/s56/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54/s49/s46/s56/s50/s46/s48\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s48/s48/s54/s48/s46/s48/s48/s55/s48/s46/s48/s48/s56/s48/s46/s48/s48/s57/s48/s46/s48/s49/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s32/s99/s111/s101/s114/s99/s105/s118/s101/s32/s102/s105/s101/s108/s100\n/s32/s104/s111/s114/s105/s122/s111/s110/s116/s97/s108/s32/s115/s104/s105/s102/s116/s104/s111/s114/s105/s122/s111/s110/s116/s97/s108/s32/s115/s104/s105/s102/s116/s32/s40/s109/s84/s41\n/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52\n/s99/s111/s101/s114/s99/s105/s118/s101/s32/s102/s105/s101/s108/s100/s32/s40/s109/s84/s41/s32/s118/s101/s114/s116/s105/s99/s97/s108/s32/s115/s104/s105/s102/s116\n/s32/s32/s118/s101/s114/s116/s105/s99/s97/s108/s32/s115/s104/s105/s102/s116/s32/s40/s37/s41\n/s115/s112/s97/s99/s101/s114/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41\n/s40/s98/s41\n/s32/s32\n/s115/s112/s97/s99/s101/s114/s32/s116/s104/s105/s99/s107/s110/s101/s115/s115/s32/s40/s110/s109/s41/s54/s48/s37/s32/s67/s111/s58/s90/s110/s79/s47/s65/s108/s32/s115/s112/s97/s99/s101/s114/s47/s80/s121/s40/s97/s41\n/s65/s108/s47/s80/s121\nFIG. 6: When an Al spacer is inserted between the Py and\nthe Co:ZnO layer horizontal shift and coercive \feld show a\nstrong decrease already at 1 nm spacer thickness (a) while the\nvertical shift (inset) is not dependent on the spacer thickness.\n(b) shows the e\u000bect of the Al spacer on the Gilbert damping\nparameter\u000b, which also decreases if the spacer gets thicker\nthan 1 nm. As shaded region the Gilbert damping parameter\nof a Al/Py \flm is indicated within error bars.\nFor the FMR measurments after inserting an Al spacer\nno e\u000bect on the resonance position is found, as was shown\nalready in Fig. 5(a). For a 1 nm thick Al spacer the damp-\ning results in \u000b= (8.8\u00060.3)\u000210-3, which gives a \u0001 \u000b=\n(3.1\u00060.5)\u000210-3. This is only a slight decrease compared\nto the sample without Al spacer. By increasing the spacer\nthickness\u000breduces to values just above the damping ob-\ntained for pure Py or Al/Py, shown as shaded region in\nFig. 6(b). The 1 nm thick Al layer is thick enough to sup-\npress intermixing between the Co:ZnO and the Py layer\nas can be seen in Fig. 1(b). Together with the unchanged\nbehavior of Al/Py without Co:ZnO damping e\u000bects due\nto intermixing between Al and Py can be excluded. Also,\na change in two magnon scattering can be ruled out, since\nit would account for non-linear e\u000bects on the linewidth\nand contribute to \u0001 Binhom [49]. Therefore, the increase\nin Gilbert damping can be attributed to a dynamic cou-\npling, e.g. spin pumping from Py into Co:ZnO. Further-\nmore, the dynamic coupling mechanism is extends over a\nlonger range than the static coupling. With 1 nm spacer\nthe dynamic coupling is only slightly reduced whereas\nthe static coupling is already completely suppressed.7\nTemperature dependent FMR\nIn vicinity to the magnetic phase transition temper-\nature the spin pumping e\u000eciency should be at a max-\nimum [29, 31]. Therefore, the samples are measured\ninside a resonator based FMR setup, as a function of\ntemperature. During the cooldown no magnetic \feld is\napplied and the results shown in Fig. 7 are ZFC mea-\nsurements. For 50 % Co:ZnO/Py the resonance posi-\ntions shifts of Py to lower magnetic \felds as the tem-\nperature decreases as can be seen in Fig. 7(a). Not only\nthe resonance position is shifting, but also the linewidth\nis changing with temperature as shown in Fig. 7(b). The\nlinewidth has a maximum at a temperature of 15 K which\ncorresponds well to T Ndetermined by M(T) SQUID\nmeasurements for a 50 % Co:ZnO layer [32]. This max-\nium of the linewidth in the vicinity of T Nis also ob-\nserved for 60 % Co:ZnO/Py and even 30 % Co:ZnO/Py,\nas shown in Fig. 7(c). The measured maximum of 30 %\nCo:ZnO/Py and 60 % Co:ZnO/Py are at 10.7 K, 19.7 K\nrespectively and are marked with an open symbol in\nFig. 7(c). For comparison the N\u0013 eel temperatures de-\ntermined from M(T) measurements [32] are plotted as\ndashed line. Py on the other hand shows only a slight\nincrease in linewidth with decreasing temperature. The\nobserved e\u000bects at low temperatures vanish for the 60 %\nCo:ZnO/2 nm Al/Py heterostructure.\nFigure 7(d) shows the temperature dependence of the\nresonance \feld for all samples. For Py Bresonly decreases\nslightly whereas for 50 % and 60 % Co:ZnO a strong shift\nofBrescan be observed. This shift evidences a magnetic\ncoupling between the Py and the Co:ZnO layer. Even\nin the heterostructure with 30 % Co:ZnO/Py a clear de-\ncrease in resonance position below 10 K (the previously\ndetermined T N[32]) is visible. This shift of the resonance\nposition is only observed at low temperatures. At room\ntemperature no shift of the resonance position at 9.5 GHz\nhas been observed as shown in Fig. 5(a). From the low-\ntemperature behavior of the single Py layer and eq. 1 it\nis obvious that the gyromagnetic ratio is not changing\nstrongly with temperature, therefore shift of the reso-\nnance position in the heterostructure can be attributed\nto a change in anisotropy. From the SQUID measure-\nments at 2 K, see Fig. 3(b) and Fig. 4(b) EB between the\ntwo layers has been determined, which acts as additional\nanisotropy [20] and therefore causes the shift of the reso-\nnance position. Both the shift of the resonance position\nand the maximum in FMR linewidth vanish if the Py is\nseparated from 60 % Co:ZnO by a 2 nm Al spacer layer.\nSo, also at low temperatures the static EB coupling and\nthe dynamic coupling can be suppressed by an Al spacer\nlayer.\nM(T) measurements indicated a more robust long-\nrange magnetic order in 60 % Co:ZnO by a weak sepa-\nration of the \feld heated and ZFC curves lasting up to\n200 K [32]. Additionally, the coercive \feld measurements\non the 60 % Co:ZnO/Py hetersotructure revealed a weak\ncoupling above T N. However, this has not been observedfor lower Co concentrations. In the heterostructure with\n30 % Co:ZnO the FMR resonance position and linewidth\nreturn quickly to the room temperature value for temper-\natures above the T Nof 10 K. For both 50 % Co:ZnO/Py\nand 60 % Co:ZnO/Py the resonance positions are still de-\ncreased and the linewidths are increased above their re-\nspective N\u0013 eel temperatures and are only slowly approach-\ning the room temperature value. In the 60 % Co:ZnO/Py\nheterostructure measurements between 100 K and 200 K\nrevealed that a reduced EB is still present. It is known for\nthe blocking temperatures of superparamagnetic struc-\ntures that in FMR a higher blocking temperature com-\npared to SQUID is obtained due to much shorter probing\ntimes in FMR of the order of nanoseconds compared to\nseconds in SQUID [50]. Hence, large dopant con\fgura-\ntions in Co:ZnO still appear to be blocked blocked on\ntimescales of the FMR whereas they already appear un-\nblocked on timescales of the SQUID measurements.\nV. Conclusion\nThe static and dynamic magnetic coupling of Co:ZnO,\nwhich is weakly paramagnetic at room temperature and\nan uncompensated AFM at low temperatures, with ferro-\nmagnetic Py was investigated by means of SQUID mag-\nnetometry and FMR. At room temperature no static in-\nteraction is observed in the M(H) curves. After cooling\nto 2 K an EB between the two layers is found resulting\nin an increase of coercive \feld and a horizontal shift.\nAdditionally, a vertical shift is present caused by the un-\ncompensated moments in the Co:ZnO. While this vertical\nshift is nearly una\u000bected by the insertion of an Al spacer\nlayer between Co:ZnO and Py the EB vanishes already\nat a spacer thickness of 1 nm.\nThe FMR measurements at room temperature re-\nveal an increase of the Gilbert damping parameter for\n50 % Co:ZnO/Py and 60 % Co:ZnO/Py, whereas 30 %\nCo:ZnO/Py is in the range of an individual Py \flm. At\nroom temperature the resonance position is not a\u000bected\nfor all the heterostructures. For the 60 % Co doped sam-\nple \u0001\u000b= 3.7\u000210-3, which is equivalent to an increase\nby a factor of 1.64. In contrast to the static magnetic\ncoupling e\u000bects, an increased linewidth is still observed\nin the heterostructure containing a 1 nm Al spacer layer.\nAt lower temperatures the resonance position shifts\nof the heterostructures to lower resonance \felds, due to\nthe additional EB anisotropy. The temperature depen-\ndence of the linewidth shows a maximum at tempera-\ntures, which by comparison with M(T) measurements\ncorrespond well to T Nof single Co:ZnO layers and thus\ncorroborate the increase of the damping parameter and\nthus spin pumping e\u000eciency in vicinity to the magnetic\nphase transition. Furthermore, the shift of the resonance\nposition has been observed at temperatures well above\nTNfor 50 % Co:ZnO/Py and 60 % Co:ZnO/Py. Up to\nnow only indications for a long range AFM order in 60 %\nCo:ZnO/Py had been found by static M(T) measure-8\n/s50/s53 /s53/s48 /s55/s53 /s49/s48/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48 /s51/s53 /s52/s48 /s52/s53 /s53/s48 /s53/s53/s49/s48/s49/s49/s49/s50/s49/s51/s49/s52\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54/s49/s56/s50/s48/s50/s50/s40/s100/s41/s40/s98/s41\n/s40/s99/s41\n/s32/s32/s110/s111/s114/s109/s46/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s66/s32/s40/s109/s84/s41/s32/s84/s32/s61/s32/s32/s32/s52/s46/s48/s75\n/s32/s84/s32/s61/s32/s49/s52/s46/s57/s75\n/s32/s84/s32/s61/s32/s51/s49/s46/s52/s75\n/s32/s84/s32/s61/s32/s53/s48/s46/s50/s75/s40/s97/s41\n/s32/s32/s66\n/s112/s112/s32/s40/s109/s84/s41\n/s116/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s53/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s84\n/s78/s32/s100/s101/s116/s101/s114/s109/s105/s110/s101/s100/s32\n/s102/s114/s111/s109/s32/s77/s40/s84/s41/s32/s83/s81/s85/s73/s68/s32/s91/s51/s50/s93\n/s32/s32/s66\n/s114/s101/s115/s32/s40/s109/s84/s41\n/s116/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s80/s121\n/s32/s51/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s53/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s65/s108/s47/s80/s121/s84\n/s78/s32/s54/s48/s37/s84\n/s78/s32/s53/s48/s37/s84\n/s78/s32/s51/s48/s37\n/s32/s32/s66\n/s112/s112/s32/s40/s109/s84/s41\n/s116/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s32/s80/s121\n/s32/s51/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s53/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s80/s121\n/s32/s54/s48/s37/s67/s111/s58/s90/s110/s79/s47/s65/s108/s47/s80/s121\nFIG. 7: By decreasing the temperature the resonance position of 50 % Co:ZnO/Py shifts to lower resonance \felds (a) and\nthe linewidth increases, showing a maxium at the T N(b). A similar behavior is observed for the heterostructures with 30 %\nand 60 % Co doping while a single Py \flm does not exhibit a maximum when cooling (c). The maximum is marked as open\nsymbol in the temperature dependence, while the T Ndetermined from M(T) [32] are shown as dashed lines. Furthermore, the\nresonance position of the heterostructures with Co:ZnO shifts at low temperatures (d).\nments. The dynamic coupling, however, is sensitive to\nthose interactions due to the higher time resolution in\nFMR resulting in a shift of the resonance position above\nthe T Ndetermined from M(T) SQUID.\nAcknowledgment\nThe authors gratefully acknowledge funding by the\nAustrian Science Fund (FWF) - Project No. P26164-N20 and Project No. ORD49-VO. All the mea-\nsured raw data can be found in the repository at\nhttp://doi.org/10.17616/R3C78N. 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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-FL�� , 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": "2106.11597v1.Choice_of_Damping_Coefficient_in_Langevin_Dynamics.pdf",        "content": "Choice of Damping Coefficient in Langevin Dynamics\nRobert D. Skeel\u0003andCarsten Hartmanny\nAbstract. This article considers the application of Langevin dynamics to sampling and investigates how to\nchoose the damping parameter in Langevin dynamics for the purpose of maximizing thoroughness of\nsampling. Also, it considers the computation of measures of sampling thoroughness.\n1. Introduction. Langevindynamicsisapopulartoolformolecularsimulation. Itrequires\nthe choice of a damping coefficient, which is the reciprocal of a diffusion coefficient. (More\ngenerally this might be a diffusion tensor.) The special case of a constant scalar diffusion\ncoefficient is the topic of this article. The motivation for this study is a suspicion that proposed\nnovel MCMC propagators based on Langevin dynamics (in particular, stochastic gradient\nmethods for machine learning [4, 9]) might be obtaining their advantage at the expense of\nreduced sampling efficiency, as, say, measured by effective sample size.\nFor simulations intended to model the dynamics, the appropriate choice of \ris based on\nphysics. Generally, the dissipation and fluctuation terms are there to account for omitted\ndegrees of freedom. In their common usage as thermostats, they model the effect of forces due\nto atoms just outside the set of explicitly represented atoms. These are essentially boundary\neffects, which disappear in the thermodynamic limit Natoms!1, whereNatomsis the number\nof explicitly represented atoms. Since the ratio of the number of boundary atoms to interior\natoms is of order N\u00001=3\natoms, it might be expected that \ris chosen to be proportional to N\u00001=3\natoms.\nThere is second possible role for the addition of fluctuation-dissipation terms in a dynamics\nsimulation: with a small damping coefficient, these terms can also play a role in stabilizing\na numerical integrator [21], which might be justified if the added terms are small enough to\nhave an effect no greater than that of the discretization error.\nThe bulk of molecular simulations, however, are “simply” for the purpose of drawing ran-\ndom samples from a prescribed distribution and this is the application under consideration\nhere. The appropriate choice of \roptimizes the efficiency of sampling. A measure of this\nis the effective sample size N=\u001cwhereNis the number of samples and \u001cis the integrated\nautocorrelation time. The latter is, however, defined in terms of an observable. An observable\nis an expectation of a specified function of the configuration, which for lack of a better term,\nis referred to here as a preobservable . As an added complication, the accuracy of an estimate\nof an integrated autocorrelation time (IAcT) depends on sampling thoroughness [13, Sec. 3],\nso a conservative approach is indicated. Ref. [13, Sec. 3.1] advocates the use of the maximum\npossible IAcT and shows how it might be a surrogate for sampling thoroughness. The max-\nimum possible IAcT is about the same (except for a factor of 2) as the decorrelation time of\nRef. [30], defined to be “the minimum time that must elapse between configurations for them\nto become fully decorrelated (i.e., with respect to any quantity)”.\n\u0003School of Mathematics and Statistical Sciences, Arizona State University, 900 S Palm Walk, Tempe, AZ 85281,\nUSA, E-mail: rskeel@asu.edu\nyInstitute of Mathematics, Brandenburgische Technische Universität Cottbus-Senftenberg, 03046 Cottbus, Ger-\nmany, E-mail: hartmanc@b-tu.de\n1arXiv:2106.11597v1  [stat.CO]  22 Jun 2021Therefore, for sampling, it is suggested that \rbe chosen to achieve a high level of sampling\nthoroughness, as measured by the maximum possible IAcT. An initial study of this question\nis reported in Ref. [38, Sec. 5], and the purpose of the present article is to clarify and extend\nthese results.\nTo begin with, we analyse an underdamped Langevin equation with a quadratic potential\nenergy function. (See Eq. (12) below.) The main purpose of analyzing this model problem\nis, of course, to obtain insight and heuristics that can be applied to general potential energy\nfunctions. Needed for choosing the optimal gamma is a substitute for the lowest frequency.\nFor the model problem, this can be obtained from the covariance matrix for the position\ncoordinates, which is not difficult to compute for a general potentials. And for estimating\n\u001cq;max, the analysis suggests using the set of all quadratic polynomials, which can be achieved\nusing the algorithm of reference [13, Sec. 3.5].\nFor molecular simulation, the suggestion is that one might choose linear combinations of\nfunctions of the form j~ rj\u0000~ rij2and(~ rj\u0000~ ri)\u0001(~ rk\u0000~ ri)where each ~ riis an atomic position or\ncenter of mass of a group of atoms. Such functions share with the potential energy function\nthe property of being invariant under a rigid body movement.\n1.1. Results and discussion. Section 5 analyzes integrated autocorrelation times for the\nstandard model problem of a quadratic potential energy function. An expression is derived for\nthe IAcT for any preobservable; this is applied in Sec. 5.2 to check the accuracy of a method\nfor estimating the IAcT. In Sec. 5, we also determine the maximum IAcT, denoted by \u001cq;max,\nover all preobservables defined on configurations, as well as the damping coefficient \r\u0003that\nminimizes\u001cq;max. It is shown that it is polynomials of degree \u00142that produce the largest\nvalue of\u001cq;max. And that choosing \requal to the lowest frequency, which is half of the optimal\nvalue of\rfor that frequency, minimizes \u001cq;max. These results extend those of Ref. [38, Sec. 5],\nwhich obtains a (less relevant) result for preobservables defined on phase space rather than\nconfiguration space.\nSections 6 and 7 test the heuristics derived from the quadratic potential energy on some\nsimple potential energy functions giving rise to multimodal distributions. Results suggest that\nthe heuristics for choosing the maximizing preobservable and optimal gamma are effective.\nOne of the test problems is one constructed by Ref. [23] to demonstrate the superiority of\nBAOAB over other Langevin integrators. Experiments for this problem in Sec. 6 are consistent\nwith this claim of superiority.\nIn defining “quasi-reliability” and the notion of thorough sampling, Ref. [13] makes an\nunmotivated leap from maximizing over preobservables that are indicator functions to maxi-\nmizing over arbitrary preobservables. The test problem of Sec. 7 provides a cursory look at\nthis question, though the matter may warrant further study.\nObtaining reliable estimates of the IAcT without generating huge sets of samples very\nmuch hinders this investigation. To this end, Sec. 4.1 explores an intriguing way of calculating\nan estimate for the phase space \u001cmax, which avoids the difficult calculation of IAcTs. For\nthe model problem, it give more accurate results for \u001cmaxthan estimating IAcTs, due to\nthe difficulty of finding a set of functions that play the same role as quadratic polynomials\nwhen maximizing IAcTs. The literature offers interesting suggestions that might help in the\ndevelopment of better schemes for estimating IAcTs, and it may be fruitful to recast some of\n2these ideas using the formalisms employed in this article. In particular, Ref. [30] offers a novel\napproach based on determining whether using every \u001cth sample creates a set of independent\nsamples. Additionally, there are several conditions on covariances [16, Theorem 3.1] that can\nbe checked or enforced.\n1.2. Related work. While the major part of the literature on Markov chain Monte Carlo\n(MCMC) methods with stochastic differential equations focuses on the overdamped Langevin\nequation (e.g. [35, 3] and the references given there), there have been significant advances,\nboth from an algorithmic and a theoretical point of view, in understanding the underdamped\nLangevindynamics[34]. Forexample,inRefs. [39,7]Langevindynamicshasbeenstudiedfrom\ntheperspectiveofthermostattingandenhancmentofspecificvibrationalmodesorcorrelations,\nin Refs. [8, 17, 25] Langevin dynamics has been used to tackle problems in machine learning\nand stochastic optimisation. From a theoretical point of view, the Langevin equation is more\ndifficult to analyse than its overdamped counterpart, since the noise term is degenerate and\nthe associated propagator is non-symmetric; recent work on optimising the friction coefficient\nfor sampling is due to [11, 36, 4], theoretical analyses using both probabilistic and functional\nanalytical methods have been conducted in [10, 5, 12]; see also [27, Secs. 2.3–2.4] and the\nreferences therein.\nRelevant in this regard are Refs. [20, 26, 33], in which non-reversible perturbations of\nthe overdamped Langevin equation are proposed, with the aim of increasing the spectral gap\nof the propagator or reducing the asymptotic variance of the sampler. Related results on\ndecorrelation times for the overdamped Langevin using properties of the dominant spectrum\nof the infinitesimal generator of the associated Markov process have been proved in [22, Sec. 4].\nA key point of this article is that quantities like spectral gaps or asymptotic variances are\nnot easily accessible numerically, therefore computing goal-oriented autocorrelation times (i.e.\nfor specific observables that are of interest) that can be computed from simulation data is a\nsensible approach. With that being said, it would be a serious omission not to mention the\nwork of Ref. [30], which proposes the use of indicator functions for subsets of configuration\nspace in order to estimate asymptotic variance and effective sample size from autocorrelation\ntimes using trajectory data.\nFinally, we should also mention that many stochastic optimisation methods that are nowa-\ndays popular in the machine learning comminity, like ADAM or RMSProp, adaptively control\nthe damping coefficient, though in an ad-hocway, so as to improve the convergence to a local\nminimum. They share many features with adaptive versions of Langevin thermostats that are\nused in moecular dynamics [24], and therefore it comes as no surprise that the Langevin model\nis the basis for the stochastic modified equation approach that can be used to analyse state of\nthe art momentum-based stochastic optimisation algorithms like ADAM [1, 28].\n2. Preliminaries. The computational task is to sample from a probability density \u001aq(q)\nproportional to exp(\u0000\fV(q)), whereV(q)is a potential energy function and \fis inverse\ntemperature. In principle, these samples are used to compute an observable E[u(Q)], where\nQis a random variable from the prescribed distribution and u(q)is a preobservable (possible\n3an indicator function). The standard estimate is\nE[u(Q)]\u0019bUN=1\nNN\u00001X\nn=0u(Qn);\nwhere the samples Qnare from a Markov chain, for which \u001aq(q)(or a close approximation\nthereof) is the stationary density. Assume the chain has been equilibrated, meaning that Q0is\ndrawn from a distribution with density \u001aq(q). An efficient and popular way to generate such\na Markov chain is based on Langevin dynamics, whose equations are\n(1)dQt=M\u00001Ptdt;\ndPt=F(Qt) dt\u0000\rPtdt+q\n2\r\n\fMhdWt;\nwhereF(q) =\u0000rV(q),Mis a matrix chosen to compress the range of vibrational frequencies,\nMhMT\nh=M, and Wtis a vector of independent standard Wiener processes. The invariant\nphase space probability density \u001a(q;p)is given by\n\u001a(q;p) =1\nZexp(\u0000\f(V(q) +1\n2pTM\u00001p));\nwhereZ > 0is a normalisation constant that guarantees that \u001aintegrates to 1. We call \u001aq(q)\nits marginal density for q. We suppose \u001a>0.\nIt is common practice in molecular dynamics to use a numerical integrator, which intro-\nduces a modest bias, that depends on the step size \u0001t. As an illustration, consider the BAOAB\nintegrator [23]. Each step of the integrator consists of the following substeps:\nB:Pn+1=4=Pn+1\n2\u0001tF(Qn),\nA:Qn+1=2=Qn+1\n2\u0001tM\u00001Pn+1=4,\nO:Pn+3=4= exp(\u0000\r\u0001t)Pn+1=4+Rn+1=2,\nA:Qn+1=Qn+1=2+1\n2\u0001tM\u00001Pn+3=4,\nB:Pn+1=Pn+3=4+1\n2\u0001tF(Qn+1=2),\nwhere Rn+1=2is a vector of independent Gaussian random variables with mean 0and covari-\nance matrix (1\u0000exp(\u00002\r\u0001t))\f\u00001M.\nIn the following, we use the shorthand Z= (Q;P)to denote a phase space vector. It is\nknown [16, Sec. 2] that the variance of the estimate bUNforE[u(Z)]is\n(2) Var[bUN]\u0019\u001c\nNVar[u(Z)];\nwhich is exact relative to 1=Nin the limit N!1. Here\u001cis theintegrated autocorrelation\ntime (IAcT)\n(3) \u001c= 1 + 2+1X\nk=1C(k)\nC(0)\nandC(k)is the autocovariance at lag kdefined by\n(4) C(k) =E[(u(Z0)\u0000\u0016)(u(Zk)\u0000\u0016)]\n4with\u0016=E[u(Z0)] =E[u(Zk). Here and in what follows the expectation E[\u0001]is understood\nover all realisations of the (discretized) Langevin dynamics, with initial conditions Z0drawn\nfrom the equilibrium probability density function \u001a.\n2.1. Estimating integrated autocorrelation time. Estimates of the IAcT based on es-\ntimating covariances C(k)suffer from inaccuracy in estimates of C(k)due to a decreasing\nnumber of samples as kincreases. To get reliable estimates, it is necessary to underweight\nor omit estimates of C(k)for larger values of k. Many ways to do this have been proposed.\nMost attractive are those [16, Sec. 3.3] that take advantage of the fact that the time series is\na Markov chain.\nOnethatisusedinthisstudyisashortcomputerprogramcalled acor[18]thatimplements\na method described in Ref. [31]. It recursively reduces the series to one half its length by\nsumming successive pairs of terms until the estimate of \u001cbased on the reduced series is deemed\nreliable. The definition of “reliable” depends on heuristically chosen parameters. A greater\nnumber of reductions, called reducsin this paper, employs greater numbers of covariances, but\nat the risk of introducing more noise.\n2.2. Helpful formalisms for analyzing MCMC convergence. It is helpful to introduce\nthe linear operator Tdefined by\nTu(z) =Z\n\u001a(z0jz)u(z0)dz0\nwhere\u001a(z0jz)is the transition probability density for the Markov chain. Then one can express\nan expectation of the form E[v(Z0)u(Z1)], arising from a covariance, as\nE[v(Z0)u(Z1)] =hv;Tui\nwhere the inner product h\u0001;\u0001iis defined by\n(5) hv;ui=Z\nv(z)u(z)\u001a(z) dz:\nThe adjoint operator\nTyv(z) =1\n\u001a(z)Z\n\u001a(zjz0)v(z0)\u001a(z0)dz0\nis what Ref. [37] calls the forward transfer operator, because it propagates relative probability\ndensities forward in time. On the other hand, Ref. [29] calls Tythe backward operator and\ncallsTitself the forward operator. To avoid confusion, use the term transfer operator forT.\nThe earlier work[13, 38] isin terms ofthe operator Ty. To get an expression for E[v(Z0)u(Zk)],\nwrite\nE[v(Z0)u(Zk)] =ZZ\nv(z)u(z0)\u001ak(z0jz)\u001a(z) dzdz0\nwhere\u001ak(z0jz)is the iterated transition probability density function defined recursively by\n\u001a1(z0jz) =\u001a(zjz0)and\n\u001ak(z0jz) =Z\n\u001a(z0jz00)\u001ak\u00001(z00jz)dz00; k = 2;3;::::\n5By induction on k\nTku(z) =TTk\u00001u(z) =Z\n\u001ak(z0jz)u(z0)dz0;\nwhence,\nE[v(Z0)u(Zk)] =hv;Tkui:\n2.2.1. Properties of the transfer operator and IAcT. It is useful to establish some prop-\nerties ofTand the IAcT that will be used throughout the article. In particular, we shall\nprovide a formula for \u001c(u)in terms of the transfer operator that will be the starting point for\nsystematic improvements and that will later on allow us to estimate \u001cby solving a generalised\neigenvalue problem.\nClearly,T1 = 1, and 1 is an eigenvalue of T. Here, where the context requires a function,\nthe symbol 1 denotes the constant function that is identically 1. Where the context requires\nan operator, it denotes the identity operator. To remove the eigenspace corresponding to the\neigenvalue\u0015= 1fromT, define the orthogonal projection operator\nEu=h1;ui1\nand consider instead the operator\nT0=T\u0000E:\nIt is assumed that the eigenvalues \u0015ofT0satisfyj\u0015j<1, in other words, we assume that\nthe underlying Markov chain is ergodic. Stationarity of the target density \u001a(z)w.r.t.\u001a(zjz0)\nimplies thatTy1 = 1and thatTyT1 = 1. Therefore,TyTis a stochastic kernel. This implies\nthat the spectral radius of TyTis 1, and, since it is a symmetric operator, one has that\n(6) hTu;Tui=hu;TyTui\u0014hu;ui:\nThe IAcT, given by Eq. (3), requires autocovariances, which one can express in terms of\nT0as follows:\n(7)C(k) =h(1\u0000E)u;(1\u0000E)Tkui\n=h(1\u0000E)u;(1\u0000E)Tk\n0ui\n=h(1\u0000E)u;Tk\n0ui;\nwhich follows because Eand1\u0000Eare symmetric. Substituting Equation (7) into Equation (3)\ngives\n(8) \u001c(u) =h(1\u0000E)u;Dui\nh(1\u0000E)u;ui;whereD= 2(1\u0000T0)\u00001\u00001:\nIt can be readily seen that \u001cis indeed nonnegative. With v= (1\u0000T0)\u00001u, the numerator in\nEq. (8) satisfies\nh(1\u0000E)u;Dui=h(1\u0000E)(1\u0000T0)v;(1 +T0)vi\n=hv;vi\u0000hTv;Tvi\n\u00150:\nTherefore,\u001c(u)\u00150if(1\u0000E)u6= 0, where the latter is equivalent to u6=E[u]being not a\nconstant.\n63. Sampling Thoroughness and Efficiency. Less than “thorough” sampling can degrade\nestimates of an IAcT. Ref. [13, Sec. 1] proposes a notion of “quasi-reliability” to mean the\nabsence of evidence in existing samples that would suggest a lack of sampling thoroughness. A\nnotion of sampling thoroughness begins by considering subsets Aof configuration space. The\nprobability that Q2Acan be expressed as the expectation E[1A]where 1Ais the indicator\nfunction for A. A criterion for thoroughness might be that\n(9) jc1A\u0000Pr(Q2A)j\u0014tolwherec1A=1\nNNX\nn=11A(Qn):\nThis is not overly stringent, since it does not require that there are any samples in sets Aof\nprobability\u0014tol.\nThe next step in the development of this notion is to replace the requirement jc1A\u0000Pr(Q2\nA)j\u0014tolby something more forgiving of the random error in c1A. For example, we could\nrequire instead that\n(Var[c1A])1=2\u00140:5tol;\nwhich would satisfy Eq. (9) with 95% confidence, supposing an approximate normal distribu-\ntion for the estimate. (If we are not willing to accept the Gaussian assumption, Chebychev’s\ninequality tells us that we reach 95% confidence level if we replace the right hand side by\n0:05tol.)\nNow let\u001cAbe the integrated autocorrelation time for 1A. Because\nVar[c1A]\u0019\u001cA1\nNVar[1A(Z)]\n=\u001cA1\nNPr(Z2A)(1\u0000Pr(Z2A))\n\u00141\n4N\u001cA;\nit is enough to have (1=4N)\u001cA\u0014(1=4)tol2for all sets of configurations Ato ensure thorough\nsampling (assuming again Gaussianity). The definition of good coverage might then be ex-\npressed in terms of the maximum \u001c(1A)over allA. Note that the sample variance may not be\na good criterion if all the candidate sets Ahave small probability Pr(Z2A), in which case it\nis rather advisable to consider the relativeerror [6].\nRef. [13, Sec 3.1] then makes a leap, for the sake of simplicity, from considering just indi-\ncator functions to arbitrary functions. This leads to defining \u001cq;max= supVar[u(Q)]>0\u001c(u). The\ncondition Var[u(Q)]>0is equivalent to (1\u0000E)u6= 0.\nA few remarks on the efficient choice of preobservables are in order.\nRemark 1. Generally, if there are symmetries present in both the distribution and the pre-\nobservables of interest, this may reduce the amount of sampling needed. Such symmetries can\nbe expressed as bijections  qfor whichu( q(q)) =u(q)and\u001aq( q(q)) =\u001aq(q). Examples in-\nclude translational and rotational invariance, as well as interchangeability of atoms and groups\nof atoms. Let \tqdenote the set of all such symmetries. The definition of good coverage then\n7need only include sets A, which are invariant under all symmetries  q2\tq. The extension\nfrom indicator sets 1Ato general functions leads to considering Wq=fu(q)ju( q(q)) =u(q)\nfor all q2\tqgand defining\n\u001cq;max= sup\nu2W0q\u001c(u)\nwhereW0\nq=fu2WqjVar[u(Q)]>0g.\nRemark 2. Another consideration that might dramatically reduce the set of relevant preob-\nservables is the attractiveness of using collective variables \u0010=\u0018(q)to characterize structure and\ndynamics of molecular systems. This suggests considering only functions defined on collective\nvariable space, hence, functions of the form \u0016u(\u0018(q)).\n4. Computing the Maximum IAcT. The difficulty of getting reliable estimates for \u001c(u)in\norder to compute the maximum IAcT makes it interesting to consider alternative formulation.\n4.1. A transfer operator based formulation. Although, there is little interest in sampling\nfunctionsofauxiliaryvariableslikemomenta, itmaybeusefultoconsiderphasespacesampling\nefficiency. Specifically, a maximum over phase space is an upper bound and it might be easier\nto estimate. Putting aside exploitation of symmetries, the suggestion is to using \u001cmax=\nsupVar[u(Z)]>0\u001c(u). One has, with a change of variables, that\n\u001c((1\u0000T0)v) =\u001c2(v)\nwhere\n\u001c2(v) =h(1\u0000T)v;(1 +T)vi\nh(1\u0000T)v;(1\u0000T)vi:\nThis follows from h(1\u0000E)(1\u0000T0)v;(1\u0006T0)vi=h(1\u0000T)v;(1\u0006T)v\u0007Evi=h(1\u0000T)v;(1\u0006T)vi.\nTherefore,\n\u001cmax= sup\nVar[(1\u0000T0)v(Z)]>0\u001c((1\u0000T0)v)\n= sup\nVar[(1\u0000T0)v(Z)]>0\u001c2(v)\n= sup\nVar[v(Z)]>0\u001c2(v):\nThe last step follows because (1\u0000T0)is nonsingular.\nNeeded for an estimate of \u001c2(v)ishTv;Tvi. To evaluatehTv;Tvi, proceed as follows: Let\nZ0\nn+1be an independent realization of Zn+1fromZn. In particular, repeat the step, but with\nan independent stochastic process having the same distribution. Then\n(10)E[v(Z1)v(Z0\n1)] =Z Z\nv(z)v(z0)Z\n\u001a(zjz00)\u001a(z0jz00)\u001a(z00)dz00dzdz0\n=hTv;Tvi:\nFor certain simple preobservables and propagators having the simple form of BAOAB, the\nsamplesv(Zn)v(Z0\nn)might be obtained at almost no extra cost, and their accuracy improved\nand their cost reduced by computing conditional expectations analytically.\n8This approach has been tested on the model problem of Sec. 5, a Gaussian process, and\nfound to be significantly better than the use of acor. Unfortunately, this observation is not\ngeneralisable: For example, for a double well potential, it is difficult to find preobservables\nv(z), giving a computable estimate of \u001cmaxwhich comes close to an estimate from using acor\nwithu(z) =z1.\nAnother drawback is that the estimates, though computationally inexpensive, require ac-\ncessing intermediate values in the calculation of a time step, which are not normally an output\noption of an MD program. Therefore we will discuss alternatives in the next two paragraphs.\n4.2. A generalised eigenvalue problem. Letu(z)be a row vector of arbitary basis func-\ntionsui(z),i= 1;2;:::; imaxthat span a closed subspace of the Hilbert space associated with\nthe inner product h\u0001;\u0001idefined by (5) and consider the linear combination u(z) =u(z)Tx. One\nhas\n\u001c(u) =h(1\u0000E)u;Dui\nh(1\u0000E)u;ui=xTDx\nxTC0x\nwhere\nD=h(1\u0000E)u;DuTiand C0=h(1\u0000E)u;uTi:\nIf the span of the basis is sufficiently extensive to include preobservables having the greatest\nIAcTs (e.g. polynomials, radial basis functions, spherical harmonics, etc.), the calculation of\n\u001cmaxreduces to that of maximizing xTDx=(xTC0x)over all x, which is equivalent to solving\nthe symmetric generalized eigenvalue problem\n(11)1\n2(D+DT)x=\u0015C0x:\nIt should be noted that the maximum over all linear combinations of the elements of\nu(z)can be arbitrarily greater than use of any of the basis functions individually. Moreover,\nin practice, the coefficients in (11) will be random in that they have to be estimated from\nsimulation data, which warrants special numerical techniques. These techniques, including\nclassical variance reduction methods, Markov State Models or specialised basis functions, are\nnot the main focus of this article and we therefore refer to the articles [19, 32], and the\nreferences given there.\nRemark 3. B records different notions of reversibility of the transfer operator that entail spe-\ncific restrictions on the admissible basis functions that guarantee that the covariance matrices,\nand thus C0, remain symmetric.\n4.3. The use of acor.It is not obvious how to use an IAcT estimator to construct\nmatrix off-diagonal elements Dij=h(1\u0000E)ui;DuT\nji,j6=i, from the time series fu(Zm)g.\nNevertheless, it makes sense to use arcoras a preprocessing or predictor step to generate an\ninitial guess for an IAcT. The acorestimate for a scalar preobservable u(z)has the form\nb\u001c=bD=bC0\nwhere\nbC0=bC0(fu(Zn)\u0000^Ug;fu(Zn)\u0000^Ug)\n9and\nbD=bD(fu(Zn)\u0000^Ug;fu(Zn)\u0000^Ug)\narebilinearfunctionsoftheirargumentsthatdependonthenumberofreductions reducswhere\n^Udenotes the empirical mean of fu(Zm)g.\nThe tests reported in Secs. 5–7 then use the following algorithm. (In what follows we\nassume thatfu(Zm)ghas been centred by subtracting the empirical mean.)\nAlgorithm 1 Computing the IAcT\nFor each basis function, compute b\u001c, and record the number of reductions, set reducsto the\nmaximum of these.\nThen compute D= (Dij)ijfrombD(fui(zm)g;fuj(zn)g)with a number of reductions equal\ntoreducs.\nifD+DThas a non-positive eigenvalue then\nredo the calculation using reducs\u00001reductions.\nend if\nRef. [13, Sec. 3.5] uses a slightly different algorithm that proceeds as follows:\nAlgorithm 2 Computing the IAcT as in [13, Sec. 3.5]\nSetreducsto the value of reducsfor the basis function having the largest estimated IAcT.\nThen run acorwith a number of reductions equal to reducsto determine a revised Dand\na maximizing x.\nForuTx, determine the number of reductions reducs0.\nifreducs0<reducs then,\nredo the calculation with reducs =reducs0and repeat until the value of reducsno longer\ndecreases.\nend if\nIn the experiments reported here, the original algorithm sometimes does one reduction\nfewer than the new algorithm.\nRemark 4. Theoretically, the matrix D+DTis positive definite. If it is not, that suggests\nthat the value of reducsis not sufficiently conservative, in which case reducsneeds to be reduced.\nA negative eigenvalue might also arise if the Markov chain does not converge due to a stepsize\n\u0001tthat is too large. This can be confirmed by seeing whether the negative eigenvalue persists\nfor a larger number of samples.\n5. Analytical Result for the Model Problem. The question of optimal choice for the\ndamping coefficient is addressed in Ref. [38, Sec. 5.] for the standard model problem F(q) =\n\u0000Kq, whereKis symmetric positive definite, for which the Langevin equation is\n(12)dQt=M\u00001Ptdt;\ndPt=\u0000KQtdt\u0000\rPtdt+q\n2\r\n\fMhdWt:\n10Changing variables Q0=MT\nhQandP0=M\u00001\nhPand dropping the primes gives dQt=Ptdt,\ndPt=\u0000M\u00001\nhKM\u0000T\nhQtdt\u0000\rPtdt+p\n2\r=\fdWt:\nWith an orthogonal change of variables, this decouples into scalar equations, each of which\nhas the form\ndQt=Ptdt;dPt=\u0000!2Qtdt\u0000\rPtdt+p\n2\r=\fdWt\nwhere!2is an eigenvalue of M\u00001\nhKM\u0000T\nh, or, equivalently, an eigenvalue of M\u00001K. Changing\nto dimensionless variables t0=!t,\r0=\r=!,Q0= (\fm)1=2!Q,P0= (\f=m)1=2P, and dropping\nthe primes gives\n(13) dQt=Ptdt;dPt=\u0000Qtdt\u0000\rPtdt+p\n2\rdWt:\nForanMCMCpropagator, assumeexactintegrationwithstepsize \u0001t. FromRef.[38, Sec.5.1],\none hasT= (e\u0001tL)y= exp(\u0001tLy)where\nLyf=p@\n@qf\u0000q@\n@pf\u0000\rp@\n@pf+\r@2\n@p2f:\nThe Hilbert space defined by the inner product from Eq. (5) has, in this case, a decomposition\ninto linear subspaces Pk= spanfHem(q)Hen(p)jm+n=kg(denoted by P0\nkin Ref. [38,\nSec. 5.3]). Let\nuT\nk= [Hek(q)He0(p);Hek\u00001(q)He1(p); :::; He0(q)Hek(p)];\nand, in particular,\nuT\n1= [q; p];\nuT\n2= [q2\u00001; qp; p2\u00001];\nuT\n3= [q3\u00003q;(q2\u00001)p; q(p2\u00001); p3\u00003p];\nuT\n4= [q4\u00006q2+ 3;(q3\u00003q)p;(q2\u00001)(p2\u00001);\nq(p3\u00003p); p4\u00006p+ 3]:\nWith a change of notation from Ref. [38, Sec. 5.3], LuT\nk=uT\nkAk, with Akgiven by\n(14) Ak=2\n666640 1\n\u0000k\u0000\r...\n......k\n\u00001\u0000k\r3\n77775:\nOne can show, using arguments similar to those in [38, Sec. 5.3], that Pkclosed under ap-\nplication ofLy. Therefore,LyuT\nk=uT\nkBkfor somek+ 1byk+ 1matrix Bk. Forming\nthe inner product of ukwith each side of this equation gives Bk=C\u00001\nk;0huk;LyuT\nkiwhere\nCk;0=huk;uT\nki. It follows that\nBk=C\u00001\nk;0huk;LyuT\nki=C\u00001\nk;0hLuk;uT\nki\n11and\nLyuT\nk=uT\nkC\u00001\nk;0AT\nkCk;0:\nThe Hermite polynomials ukare orthogonal and\nCk;0= diag(k!0!;(k\u00001)!1!; :::; 0!k!):\nAlso,EuT\nk=0T. Accordingly,\nT0uT\nk=TuT\nk=uT\nkC\u00001\nk;0exp(\u0001tAT\nk)Ck;0\nand\n(15) DuT\nk=uT\nkC\u00001\nk;0Dk\nwhere\nDk=Ck;0\u0010\n2(I\u0000C\u00001\nk;0exp(\u0001tAT\nk)Ck;0)\u00001\u0000I\u0011\n=\u0000coth(\u0001t\n2AT\nk)Ck;0:\nA formula for \u001c(u)is possible if u(q)can be expanded in Hermite polynomials as u=P1\nk=1ckHek. Then, from Eq. (15), DHek2Pk, not to mention Hek2Pk. Using these facts\nand the mutual orthogonality of the subspaces Pk, it can be shown that\n(16) \u001c(u) =P1\nk=1k!c2\nk\u001c(Hek)P1\nk=1k!c2\nk:\nFrom this it follows that maxu\u001c(u) = maxk\u001c(Hek).\nSince Hek=uT\nkxwithx= [1;0;:::; 0]T, one has\n(17) \u001c(Hek) = (Dk)11=(Ck;0)11= (coth(\u0000\u0001t\n2Ak))11:\nAsymptotically \u001c(Hek) =\u0000(2=\u0001t)(A\u00001\nk)11, in the limit as \u0001t!0. In particular,\n(18) A\u00001\n1=\u0014\u0000\r\u00001\n1 0\u0015\nand\n(19) A\u00001\n2=\u00001\n2\r2\n4\r2+ 1\u00002\r1\n\r 0 0\n1 0 13\n5:\nWriting\u001c(Hek)as an expansion in powers of \u0001t,\n\u001c(Hek) =Tk(\r)=\u0001t+O(\u0001t);\n12Figure 1. From top to bottom on the right Tk(\r)vs.\r,k= 1;2;3;4\none hasT1(\r) = 2\randT2(\r) =\r+ 1=\r. Fig. 1 plots Tk(\r),k= 1;2;3;4,1=2\u0014\r\u00144.\nEmpirically, maxkTk=Tmaxdef= maxfT1;T2g.\nRestoring the original variables, one has\n\u001cq;max=Tmax(\r=!)=(!\u0001t) +O(!\u0001t):\nThe leading term increases as !decreases, so \u001cq;maxdepends on the lowest frequency !1. And\n\u001cq;maxisminimizedat \r=!1, whichishalfofthecriticalvalue \r= 2!1. Contrastthiswiththe\nresult [38, Sec. 5.] for the phase space maximum IAcT, which is minimized for \r= (p\n6=2)!1.\nRemark 5. The result is consistent with related results from [4, 12] that consider optimal\ndamping coefficients that maximise the speed of convergence measured in relative entropy.\nSpecifically, calling \u0011t=N(\u0016t;\u0006t)the law of the solution to (13), with initial conditions\n(Qt;Pt) = (q;p); see A for details. Then, using [2, Thm. 4.9], we have\nKL(\u0011t;\u001a)\u0014Mexp(\u00002\u000bt);\nwhereM2(1;1)and\u000bdenotes the spectral abcissa of the matrix Ain A, i.e. the negative\nreal part of the eigenvalue that is closest to the imaginary axis. Here\nKL(f;g) =Z\nlogf(z)\ng(z)f(z)dz\ndenotes the relative entropy (or: Kullback-Leibler divergence) between two phase space proba-\nbility densities fandg, assuming that\nZ\nfg(z)=0gf(z)dz= 0:\n(Otherwise we set KL(f;g) =1.) It is a straightforward calculation to show that the maximum\nvalue for\u000b(that gives the fastest decay of KL(\u0011t;\u001a)) is attained at \r= 2, which is in agreement\n13with the IAcT analysis. For analogous statements on the multidimensional case, we refer to\n[4].\nWe should mention that that there may be cases, in which the optimal damping coefficient\nmay lead to a stiff Langevin equation, depending on the eigenvalue spectrum of the Hessian\nof the potential energy function. As a consequence, optimizing the damping coefficient may\nreduce the maximum stable step size \u0001tthat can be used in numerical simulations.\n5.1. Application to more general distributions. Note that for the model problem, the\nmatrixKcan be extracted from the covariance matrix\nCov[Q] = (1=\f)K\u00001:\nTherefore, as a surrogate for the lowest frequency !1, and as a recommended value for \r,\nconsider using\n\r\u0003= (\u0015min(M\u00001K))1=2= (\f\u0015max(Cov[Q]M))\u00001=2:\n5.2. Sanity check. As a test of the accuracy of acorand the analytical expression (16),\nthe IAcT is calculated by acorfor a time series generated by the exact analytical propagator\n(given in A) for the reduced model problem given by Eq. (12). For the preobservable, we\nchoose\nu(q) =He3(q)=p\n3!\u0000He2(q)=p\n2!\nwhere He2(q) =q2\u00001andHe3(q) =q3\u00003qare Hermite polynomials of degree 2 and 3;\nas damping coefficient, we choose \r= 2, which is the critical value; the time increment is\n\u0001t= 0:5, which is about 1/12th of a period.\nIn this and the other results reported here, equilibrated initial values are obtained by\nrunning for 50000 burn-in steps. As the dependence of the estimate on Nis of interest here,\nwe runM= 103independent realisations for each value of N, from which we can estimate the\nrelative error\n\u000eN(\u001c(u)) =p\nVar[\u001c(u)]\nE[\u001c(u)];\nwhich we expect to decay as N\u00001=2. Fig. 2 shows the relative error in the estimated IAcT \u001c(u)\nforN= 213,214, ..., 222. The least-squares fit of the log relative error as a function of logN\nhas slopem= 0:4908. Thus we observe a nearly perfect N\u00001=2decay of the relative error, in\naccordance with the theoretical prediction.\n6. A simple example. The procedure to determine the optimal damping coefficient in\nthe previous section is based on linear Langevin systems. Even though the considerations of\nSection 5 do not readily generalize to nonlinear systems, it is plausible to use the harmonic\napproximation as a proxy for more general systems, since large IAcT values are often due\nto noise-induced metastability, in which case local harmonic approximations inside metastable\nregions are suitable. For estimating the maximum IAcT, the model problem therefore suggests\nthe use of linear, quadratic and cubic functions of the coordinates, where the latter is suitable\nto capture the possible non-harmonicity of the potential energy wells in the metastable regime.\nThe first test problem, which is from Ref. [23], possesses an asymmetric multimodal dis-\ntribution. It uses U(q) =1\n4q4+ sin(1 + 5q)and\f= 1, and it generates samples using BAOAB\n14104105106\nN10-210-1relative error Nm=-0.4908\nN (M=103)Figure 2. Relative error in estimated IAcT \u001cas a function of sample size N. The relative error \u000eN=p\nVar[\u001c]=E[\u001c]has been computed by averaging over M= 103independent realisations of each simulation.\nwith a step size \u0001t= 0:2, which is representative of step sizes used in Ref. [23]. Fig. 3 plots\nwith dotted lines the unnormalized probability density function.\n6.1. Choice of basis. A first step is to find a preobservable that produces a large IAcT.\nIt would be typical of actual practice to try to select a good value for \r. To this end, choose\n\r=\r\u0003= 1:276, To obtain this value, do a run of sample size N= 2\u0001106using\r= 1, as in\none of the tests in Ref. [23].\nWith a sample size N= 107, the maximum IAcT is calculated for polynomials of increasing\ndegree using the approach described in Secs. 4.2–4.3. Odd degrees produces somewhat greater\nmaxima than even degrees. For cubic, quintic, and septic polynomials, \u001cmaxhas values 59.9,\n63.9, 65.8, respectively As a check that the sample size is adequate, the calculations are redone\nwith half the sample size. Fig. 3 shows how the maximizing polynomial evolves as its degree\nincreases from 3 to 5 to 7.\n6.2. Optimal choice of damping coefficient. The preceding results indicate that septic\npolynomials are a reasonable set of functions for estimating \u001cq;max. For 25 values of \r, ranging\nfrom 0.2 to 5, the value of \u001cq;maxwas thus estimated, each run consisting of N= 107samples.\nThe optimal value is \r= 1:8 = 1:4\r\u0003, which is close the heuristic choice \r\u0003for a damping\ncoefficient. Fig. 4 plots \u001cq;maxvs. the ratio \r=\r\u0003.\nWith respect to this example, Ref. [23, Sec. 5] states, “We were concerned that the im-\nproved accuracy seen in the high \rregime might come at the price of a slower convergence to\nequilibrium”. The foregoing results indicate that the value \r= 1used in one of the tests is\nnear the apparent optimal value \r= 1:8. Hence, the superior accuracy of BAOAB over other\nmethods observed in the low \rregime does not come at the price of slower convergence.\n15Figure 3. In dotted lines is the unnormalized probability density function. From top to bottom on the right\nare the cubic, quintic, and septic polynomials that maximize the IAcT over all polynomials of equal degree.\nFigure 4.\u001cq;maxvs.\r=\r\u0003using septic polynomials as preobservables\n7. Sum of three Gaussians. The next, perhaps more challenging, test problem uses the\nsum of three (equidistant) Gaussians for the distribution, namely.\nexp(\u0000V(x;y)) = exp(\u0000((x\u0000d)2+y2)=2)\n+ exp(\u0000((x+d=2)2+ (y\u0000p\n3d=2)2)=2)\n+ exp(\u0000((x+d=2)2+ (y+p\n3d=2)2)=2))\nwheredis a parameter that measures the distance of the three local minima from the origin.\nIntegrating the Langevin system using BAOAB with a step size \u0001t= 0:5as for the model\nproblem, which is what V(x;y)becomes if d= 0. Shown in Fig. 5 are the first 8\u0001104points\nof a trajectory where d= 4:8.\n7.1. Choice of basis. To compare \u001cmaxfor different sets of preobservables, choose \r=\n\r\u0003= 0:261, and with \rso chosen, run the simulation with d= 4:8forN= 107steps. To\n16Figure 5. A typical time series for a sum of three Gaussians\ncompute\r\u0003, run the simulation for N= 2\u0001106steps with\r= 1(which is optimal for d= 0).\nHere are the different sets of preobservables and the resulting values of \u001cmax:\n1. linear polynomials of xandy, for which \u001cmax= 18774,\n2. quadratic polynomials of xandy, for which \u001cmax= 19408,\n3. linear combinations of indicator functions f1A;1B;1Cgfor the three conformations\nA=f(x;y) :jyj\u0014p\n3xg\nB=f(x;y) :y\u00150andy\u0015p\n3xg\nC=f(x;y) :y\u00140andy\u0014\u0000p\n3xg;\nfor which\u001cmax= 18492,\n4.1Aalone, for which \u001c= 12087,\n5.1Balone, for which \u001c= 5056,\n6.1Calone, for which \u001c= 4521.\nAs consequence of these results, the following section uses quadratic polynomials to estimate\n\u001cq;max.\n7.2. Optimal choice of damping coefficient. Shown in Fig. 6 is a plot of \u001cq;maxvs. the\nratio\r=\r\u0003. To limit the computing time, we set the parameter to d= 4:4rather than 4.8 as\nin Sec. 7.1; for d= 4:4, we have\r?= 0:285, obtained using the same protocol as does Sec. 7.1.\nWe consider 0:05\u0014\r\u00142:2in increments of 0.01 from 0.05 to 0.2, and in increments of 0.1\nfrom 0.2 to 2.2. Each data point is based on a run of N= 2\u0001107time steps. Even though the\nvariance of the estimator is not negligible for our choice of simulation parameters, it is clearly\nvisible that the minimum of \u001cq;maxis attained at \r\u0019\r\u0003.\n8. Conclusions. We have discussed the question of how to choose the damping coefficient\nin (underdamped) Langevin dynamics that leads to efficient sampling of the stationary proba-\nbilitydistributionorexpectationsofcertainobservableswithrespecttothisdistribution. Here,\nefficient sampling is understood as minimizing the maximum possible (worst-case) integrated\n17Figure 6.\u001cq;maxvs. the ratio \r=\r\u0003\nautocorrelation time (IAcT). We propose a numerical method that is based on the concept\nof phase space preobservables that span a function space over which the worst-case IAcT is\ncomputed using trajectory data; the optimal damping coefficient can then chosen on the basis\nof this information.\nBased on heuristics derived from a linear Langevin equation, we derive rules of thumb for\nchoosing good preobservables for more complicated dynamics. The results for the linear model\nproblem are in agreement with recent theoretical results on Ornstein-Uhlenbeck processes with\ndegenerate noise, and they are shown to be a good starting point for a systematic analysis of\nnonlinear Langevin samplers.\nAppendix A. Analytical propagator for reduced model problem.\nThis section derives the analytical propagator for Eq. (13). In vector form, the equation is\ndZt=AZdt+bdWtwhereA=\u00140 1\n\u00001\u0000\r\u0015\nandb= [0;p2\r]T. The variation of parameters solution is\nZt= etAZ0+Rtwhere Rt=Zt\n0e(t\u0000s)Abdt:\nThe stochastic process Rtis Gaussian with mean zero and covariance matrix\n\u0006 =E[RtRT\nt] =Zt\n0e(t\u0000s)AbbTe(t\u0000s)ATdWt:\nTo evaluate this expressions, use A=X\u0003X\u00001where\nX=\u00141 1\n\u0000\r\u0000\u0000\r+\u0015\n; X\u00001=1\n\u000e\u0014\r+ 1\n\u0000\r\u0000\u00001\u0015\n;\n\u0003 = diag(\u0000\r\u0000;\u0000\r+),\n\r\u0006=1\n2(\r\u0006\u000e);and\u000e=p\n\r2\u00004!2:\n18Noting that exp(\u0000\r\u0006t) = exp(\u0000\rt=2)(cosh(\u000et=2)\u0007sinh(\u000et=2)), one has\netA= e\u0000\rt=2cosh\u000et\n2\u00141 0\n0 1\u0015\n+ e\u0000\rt=2t\n2sinhc\u000et\n2\u0014\r 2\n\u00002\u0000\r\u0015\n;\nwhere sinhcs= (sinhs)=s.\nThen\n\u0006 =XZt\n0e(t\u0000s)\u0003X\u00001bbTX\u0000Te(t\u0000s)\u0003dtXT\n=2\r\n\u000e2XZt\n0e(t\u0000s)\u0003\u00141\u00001\n\u00001 1\u0015\ne(t\u0000s)\u0003dtXT\n=2\r\n\u000e2X2\n6641\u0000e\u00002\r\u0000t\n2\r\u0000\u00001\u0000e\u0000\rt\n\r\n\u00001\u0000e\rt\n\r1\u0000e\u00002\r+t\n2\r+3\n775XT:\nNoting that exp(\u00002\r\u0006t) = exp(\u0000\rt)(1 + 2 sinh2(\u000et=2))\u00072 sinh(\u000et=2) cosh(\u000et=2)), one has\n\u0006 = (1\u0000e\u0000\rt)\u00141 0\n0 1\u0015\n\u0000\rt2\n2e\u0000\rt(sinhc\u000et\n2)2\u0014\r\u00002\n\u00002\r\u0015\n+\rte\u0000\rtsinhc\u000et\n2cosh\u000et\n2\u0014\u00001 0\n0 1\u0015\n:\nAppendix B. Different notions of reversibility.\nWe briefly mention earlier work and discuss different reversiblity concepts for transfer\noperators.\nB.1. Quasi-reversibility. Ref. [13, Sec. 3.4] introduces a notion of quasi-reversibility. A\ntransfer operator Tis quasi-reversible if\nTy=RyTR\nwhereRis an operator such that R2= 1. This somewhat generalizes the (suitably modified)\ndefinitions in Refs. [13, 38]. The principal example of such an operator is Ru=u\u000eRwhere\nRis a bijection such that R\u000eR= idandu\u000eR=uforu2W, e.g, momenta flipping.\nThe value of the notion of quasi-reversibility is that it enables the construction of basis\nfunctions that lead to a matrix of covariances that possesses a type of symmetric structure [38,\nSec. 3.1]. This property is possessed by “adjusted” schemes that employ an acceptance test,\nand by the limiting case \u0001t!0of unadjusted methods like BAOAB.\nB.2. Modified detailed balance. A quite different generalization of reversibility, termed\n“modified detailed balance”, is proposed in Ref. [14] as a tool for making it a bit easier to prove\nstationarity.\nModified detailed balance is introduced in Ref. [14] as a concept to make it easier to\nprove stationarity. In terms of the transfer operator, showing stationarity means showing that\nF1 = 1, where 1is the constant function 1.\n19Ref. [14, Eq. (15)] defines modified detailed balance in terms of transition probabilities.\nThe definition is equivalent to F=R\u00001FyR\u00001under the assumption that Rpreserves the\nstationary distribution. This readily generalizes to\n(20) F=R2FyR1\nwhereR1andR2are arbitrary except for the assumption that each of them preserve the\nstationary distribution. Stationarity follows from Eq. (20) because Fy1 = 1for any adjoint\ntransfer operator and R11 =R21 = 1by assumption.\nReference [14] has errors, which are corrected in Ref. [15].\nREFERENCES\n[1] J. An, J. Lu, and L. Ying. 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The result is the so-called\n`destabilization paradox', where the critical force for \rutter instability decreases by an order\nof magnitude when the coe\u000ecient of internal damping becomes in\fnitesimally small. Until\nnow external damping, such as that related to air drag, is believed to provide only a stabi-\nlizing e\u000bect, as one would intuitively expect. Contrary to this belief, it will be shown that\nthe e\u000bect of external damping is qualitatively the same as the e\u000bect of internal damping,\nyielding a pronounced destabilization paradox. Previous results relative to destabilization by\nexternal damping of the Ziegler's and P\r uger's elastic structures are corrected in a de\fnitive\nway leading to a new understanding of the destabilizating role played by viscous terms.\nKeywords: P\r uger column, Beck column, Ziegler destabilization paradox, external\ndamping, follower force, mass distribution\n1. Introduction\n1.1. A premise: the Ziegler destabilization paradox\nIn his pioneering work Ziegler (1952) considered asymptotic stability of a two-linked\npendulum loaded by a tangential follower force P, as a function of the internal damping in\nthe viscoelastic joints connecting the two rigid and weightless bars (both of length l, Fig.\n1(c)). The pendulum carries two point masses: the mass m1at the central joint and the\nEmail addresses: mirko.tommasini@unitn.it (Mirko Tommasini), kirillov@mi.ras.ru (Oleg N.\nKirillov), diego.misseroni@unitn.it (Diego Misseroni), davide.bigoni@unitn.it (Davide Bigoni)\nCorresponding author: Davide Bigoni, davide.bigoni@unitn.it; +39 0461 282507\nPreprint submitted to Elsevier October 19, 2021arXiv:1611.03886v1  [physics.class-ph]  1 Oct 2016Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nmassm2mounted at the loaded end of the pendulum. The follower force Pis always aligned\nwith the second bar of the pendulum, so that its work is non-zero along a closed path, which\nprovides a canonical example of a nonconservative positional force.\nFor two non-equal masses ( m1= 2m2) and null damping, Ziegler found that the pendulum\nis marginally stable and all the eigenvalues of the 2 \u00022 matrix governing the dynamics are\npurely imaginary and simple, if the load falls within the interval 0 \u0014P <P\u0000\nu, where\nP\u0000\nu=\u00127\n2\u0000p\n2\u0013k\nl\u00192:086k\nl; (1)\nandkis the sti\u000bness coe\u000ecient, equal for both joints. When the load Preaches the value\nP\u0000\nu, two imaginary eigenvalues merge into a double one and the matrix governing dynamics\nbecomes a Jordan block. With the further increase of Pthis double eigenvalue splits into\ntwo complex conjugate. The eigenvalue with the positive real part corresponds to a mode\nwith an oscillating and exponentially growing amplitude, which is called \rutter, or oscilla-\ntory, instability. Therefore, P=P\u0000\numarks the onset of \rutter in the undamped Ziegler's\npendulum.\nWhen the internal linear viscous damping in the joints is taken into account, Ziegler\nfound another expression for the onset of \rutter: P=Pi, where\nPi=41\n28k\nl+1\n2c2\ni\nm2l3; (2)\nandciis the damping coe\u000ecient, assumed to be equal for both joints. The peculiarity of\nEq. (2) is that in the limit of vanishing damping, ci\u0000! 0, the \rutter load Pitends to\nthe value 41 =28k=l\u00191:464k=l, considerably lower than that calculated when damping is\nabsent from the beginning, namely, the P\u0000\nugiven by Eq. (1). This is the so-called `Ziegler's\ndestabilization paradox' (Ziegler, 1952; Bolotin, 1963).\nThe reason for the paradox is the existence of the Whitney umbrella singularity on\nthe boundary of the asymptotic stability domain of the dissipative system (Bottema, 1956;\nKrechetnikov and Marsden, 2007; Kirillov and Verhulst, 2010)3.\nIn structural mechanics, two types of viscous dampings are considered: (i.) one, called\n`internal', is related to the viscosity of the structural material, and (ii.) another one, called\n`external', is connected to the presence of external actions, such as air drag resistance during\n3In the vicinity of this singularity, the boundary of the asymptotic stability domain is a ruled surface\nwith a self-intersection, which corresponds to a set of marginally stable undamped systems. For a \fxed\ndamping distribution, the convergence to the vanishing damping case occurs along a ruler that meets the\nset of marginally stable undamped systems at a point located far from the undamped instability threshold,\nyielding the singular \rutter onset limit for almost all damping distributions. Nevertheless, there exist\nparticular damping distributions that, if \fxed, allow for a smooth convergence to the \rutter threshold of\nthe undamped system in case of vanishing dissipation (Bottema, 1956; Bolotin, 1963; Banichuk et al., 1989;\nKirillov and Verhulst, 2010; Kirillov, 2013).\n2Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\noscillations. These two terms enter the equations of motion of an elastic rod as proportional\nrespectively to the fourth spatial derivative of the velocity and to the velocity of the points\nof the elastic line.\nOf the two dissipative terms only the internal viscous damping is believed to yield the\nZiegler destabilization paradox (Bolotin, 1963; Bolotin and Zhinzher, 1969; Andreichikov\nand Yudovich, 1974).\n1.2. A new, destabilizing role for external damping\nDi\u000berently from internal damping, the role of external damping is commonly believed to\nbe a stabilizing factor, in an analogy with the role of stationary damping in rotor dynamics\n(Bolotin, 1963; Crandall, 1995). A full account of this statement together with a review of\nthe existing results is provided in Appendix A.\nSince internal and external damping are inevitably present in any experimental realization\nof the follower force (Saw and Wood, 1975; Sugiyama et al., 1995; Bigoni and Noselli, 2011), it\nbecomes imperative to know how these factors a\u000bect the \rutter boundary of both the P\r uger\ncolumn and of the Ziegler pendulum with arbitrary mass distribution. These structures are\nfully analyzed in the present article, with the purpose of showing: (i.) that external damping\nis a destabilizing factor, which leads to the destabilization paradox for all mass distributions;\n(ii.) that surprisingly, for a \fnite number of particular mass distributions, the \rutter loads\nof the externally damped structures converge to the \rutter load of the undamped case (so\nthat only in these exceptional cases the destabilizing e\u000bect is not present); and (iii.) that\nthe destabilization paradox is more pronounced in the case when the mass of the column or\npendulum is smaller then the end mass.\nTaking into account also the destabilizing role of internal damping, the results presented\nin this article demonstrate a completely new role of external damping as a destabilizing\ne\u000bect and suggest that the Ziegler destabilization paradox has a much better chance of being\nobserved in the experiments with both discrete and continuous nonconservative systems than\nwas previously believed.\n2. Ziegler's paradox due to vanishing external damping\nThe linearized equations of motion for the Ziegler pendulum (Fig. 1(c)), made up of two\nrigid bars of length l, loaded by a follower force P, when both internal and external damping\nare present, have the form (Plaut and Infante, 1970; Plaut, 1971)\nMx+ciDi_x+ceDe_x+Kx= 0; (3)\nwhere a superscript dot denotes time derivative and ciandceare the coe\u000ecients of internal\nand external damping, respectively, in front of the corresponding matrices DiandDe\nDi=\u00122\u00001\n\u00001 1\u0013\n;De=l3\n6\u00128 3\n3 2\u0013\n; (4)\n3Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nandMandKare respectively the mass and the sti\u000bness matrices, de\fned as\nM=\u0012m1l2+m2l2m2l2\nm2l2m2l2\u0013\n;K=\u0012\u0000Pl+ 2k Pl\u0000k\n\u0000k k\u0013\n; (5)\nin whichkis the elastic sti\u000bness of both viscoelastic springs acting at the hinges.\n16 16 16 163/c112 /c112/c112 /c112 3/c112 5/c112 7/c112\n/c970/c112\n88 420.00\n-0.05-0.10\n-0.15\n-0.20\n-0.25\n-0.30/c68F\nDA\nBC\nP m2l\nm1l\nk, c , ciek, c , ciea)\nc)b)\n16 16 16 163/c112 /c112/c112 /c112 3/c112 5/c112 7/c112\n/c970/c112\n88 421.52.02.53.0\nFA\nBCD\nm =2m12\ninternalexternalidealFLUTTER\n1.2502.086\n1.464\nFigure 1: (a) The (dimensionless) tangential force F, shown as a function of the (transformed via cot \u000b=\nm1=m2) mass ratio \u000b, represents the \rutter domain of (dashed/red line) the undamped, or `ideal', Ziegler\npendulum and the \rutter boundary of the dissipative system in the limit of vanishing (dot-dashed/green\nline) internal and (continuous/blue line) external damping. (b) Discrepancy \u0001 Fbetween the critical \rutter\nload for the ideal Ziegler pendulum and for the same structure calculated in the limit of vanishing external\ndamping. The discrepancy quanti\fes the Ziegler's paradox.\nAssuming a time-harmonic solution to the Eq. (3) in the form x=ue\u001btand introducing\nthe non-dimensional parameters\n\u0015=\u001bl\nkp\nkm2; E =cel2\npkm2; B =ci\nlpkm2; F =Pl\nk; \u0016 =m2\nm1; (6)\nan eigenvalue problem is obtained, which eigenvalues \u0015are the roots of the characteristic\npolynomial\np(\u0015) = 36\u00154+ 12(15B\u0016+ 2E\u0016+ 3B+E)\u00153+\n(36B2\u0016+ 108BE\u0016 + 7E2\u0016\u000072F\u0016+ 180\u0016+ 36)\u00152+\n6\u0016(\u00005EF+ 12B+ 18E)\u0015+ 36\u0016: (7)\n4Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nIn the undamped case, when B= 0 andE= 0, the pendulum is stable, if 0 \u0014F <F\u0000\nu,\nunstable by \rutter, if F\u0000\nu\u0014F\u0014F+\nu, and unstable by divergence, if F >F+\nu, where\nF\u0006\nu(\u0016) =5\n2+1\n2\u0016\u00061p\u0016: (8)\nIn order to plot the stability map for all mass distributions 0 \u0014\u0016 <1, a parameter\n\u000b2[0;\u0019=2] is introduced, so that cot \u000b=\u0016\u00001and hence\nF\u0006\nu(\u000b) =5\n2+1\n2cot\u000b\u0006p\ncot\u000b: (9)\nThe curves (9) form the boundary of the \rutter domain of the undamped, or `ideal',\nZiegler's pendulum shown in Fig. 1(a) (red/dashed line) in the load versus mass distribution\nplane (Oran, 1972; Kirillov, 2011). The smallest \rutter load F\u0000\nu= 2 corresponds to m1=m2,\ni.e. to\u000b=\u0019=4. When\u000bequals\u0019=2, the mass at the central joint vanishes ( m1= 0) and\nF\u0000\nu=F+\nu= 5=2. When\u000bequals arctan (0 :5)\u00190:464, the two masses are related as\nm1= 2m2andF\u0000\nu= 7=2\u0000p\n2.\nIn the case when only internal damping is present ( E= 0) the Routh-Hurwitz criterion\nyields the \rutter threshold as (Kirillov, 2011)\nFi(\u0016;B) =25\u00162+ 6\u0016+ 1\n4\u0016(5\u0016+ 1)+1\n2B2: (10)\nFor\u0016= 0:5 Eq. (10) reduces to Ziegler's formula (2). The limit for vanishing internal\ndamping is\nlim\nB!0Fi(\u0016;B) =F0\ni(\u0016) =25\u00162+ 6\u0016+ 1\n4\u0016(5\u0016+ 1): (11)\nThe limitF0\ni(\u0016) of the \rutter boundary at vanishing internal damping is shown in green in\nFig. 1(a). Note that F0\ni(0:5) = 41=28 andF0\ni(1) = 5=4. For 0\u0014\u0016<1the limiting curve\nF0\ni(\u0016) has no common points with the \rutter threshold F\u0000\nu(\u0016) of the ideal system, which\nindicates that the internal damping causes the Ziegler destabilization paradox for every mass\ndistribution.\nIn a route similar to the above, by employing the Routh-Hurwitz criterion, the critical\n\rutter load of the Ziegler pendulum with the external damping Fe(\u0016;E) can be found\nFe(\u0016;E) =122\u00162\u000019\u0016+ 5\n5\u0016(8\u0016\u00001)+7(2\u0016+ 1)\n36(8\u0016\u00001)E2\n\u0000(2\u0016+ 1)p\n35E2\u0016(35E2\u0016\u0000792\u0016+ 360) + 1296(281 \u00162\u0000130\u0016+ 25)\n180\u0016(8\u0016\u00001)\nand its limit calculated when E!0, which provides the result\nF0\ne(\u0016) =122\u00162\u000019\u0016+ 5\u0000(2\u0016+ 1)p\n281\u00162\u0000130\u0016+ 25\n5\u0016(8\u0016\u00001): (12)\n5Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nThe limiting curve (12) is shown in blue in Fig. 1(a). It has a minimum min \u0016F0\ne(\u0016) =\n\u000028 + 8p\n14\u00191:933 at\u0016= (31 + 7p\n14)=75\u00190:763.\nRemarkably, for almost all mass ratios, except two (marked as A and C in Fig. 1(a)),\nthe limit of the \rutter load F0\ne(\u0016) isbelow the critical \rutter load F\u0000\nu(\u0016) of the undamped\nsystem. It is therefore concluded that external damping causes the discontinuous decrease in\nthe critical \rutter load exactly as it happens when internal damping vanishes. Qualitatively ,\nthe e\u000bect of vanishing internal and external damping is the same . The only di\u000berence is\nthe magnitude of the discrepancy: the vanishing internal damping limit is larger than the\nvanishing external damping limit, see Fig. 1(b), where \u0001 F(\u0016) =Fe(\u0016)\u0000F\u0000\nu(\u0016) is plotted.\n0.00B\n0.010.020.030.040.050.060.070.080.090.10\n0.1 0.0 0.3 0.2 0.5 0.4 0.7 0.6 0.8 0.9\nEb) a)\nE0.1 0.0 0.3 0.2 0.5 0.4 0.7 0.6 0.8 0.9 1.02.002.102.202.302.40\n2.052.152.252.35\nFFLUTTER\nFLUTTER\n2.086\nFigure 2: Analysis of the Ziegler pendulum with \fxed mass ratio, \u0016=m2=m1= 1=2: (a) contours of the\n\rutter boundary in the internal/external damping plane, ( B;E), and (b) critical \rutter load as a function\nof the external damping E(continuous/blue curve) along the null internal damping line, B= 0, and (dot-\ndashed/orange curve) along the line B=\u0000\n8=123 + 5p\n2=164\u0001\nE.\nFor example, \u0001 F\u0019\u00000:091 at the local minimum for the discrepancy, occurring at the\npoint B with \u000b\u00190:523. The largest \fnite drop in the \rutter load due to external damping\noccurs at\u000b=\u0019=2, marked as point D in Fig. 1(a,b):\n\u0001F=11\n20\u00001\n20p\n281\u0019\u00000:288: (13)\nFor comparison, at the same value of \u000b, the \rutter load drops due to internal damping of\nexactly 50%, namely, from 2 :5 to 1:25, see Fig. 1(a,b).\nAs a particular case, for the mass ratio \u0016= 1=2, considered by Plaut and Infante (1970)\nand Plaut (1971), the following limit \rutter load is found\nF0\ne(1=2) = 2; (14)\n6Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\n16 16 16 163/c112 /c112 /c112 /c112 3/c112 5/c112 7/c112\n/c970/c112\n88 42 16 16 16 163/c112 /c112 /c112 /c112 3/c112 5/c112 7/c112\n/c970/c112\n88 42b) a)\n-1.0/c98\n-0.8-0.6-0.4-0.20.00.2\n1.52.02.53.0\nFA C0.111\n0.524-2/15FLUTTERideal\nFigure 3: Analysis of the Ziegler pendulum. (a) Stabilizing damping ratios \f(\u0016) according to Eq. (19) with\nthe points A and C corresponding to the tangent points A and C in Fig. 1(a) and to the points A and C of\nvanishing discrepancy \u0001 F= 0 in Fig. 1(b). (b) The limits of the \rutter boundary for di\u000berent damping ratios\n\fhave: two or one or none common points with the \rutter boundary (dashed/red line) of the undamped\nZiegler pendulum, respectively when \f < 0:111 (continuous/blue curves), \f\u00190:111 (continuous/black\ncurve), and \f >0:111 (dot-dashed/green curves).\nonly slightly inferior to the value for the undamped system, F\u0000\nu(1=2) = 7=2\u0000p\n2\u00192:086.\nThis discrepancy passed unnoticed in (Plaut and Infante, 1970; Plaut, 1971) but gives evi-\ndence to the destabilizing e\u000bect of external damping. To appreciate this e\u000bect, the contours\nof the \rutter boundary in the ( B;E) - plane are plotted in Fig. 2(a) for three di\u000berent values\nofF. The contours are typical of a surface with a Whitney umbrella singularity at the origin\n(Kirillov and Verhulst, 2010). At F= 7=2\u0000p\n2 the stability domain assumes the form of a\ncusp with a unique tangent line, B=\fE, at the origin, where\n\f=8\n123+5\n164p\n2\u00190:108: (15)\nFor higher values of Fthe \rutter boundary is displaced from the origin, Fig. 2(a), which\nindicates the possibility of a continuous increase in the \rutter load with damping. Indeed,\nalong the direction in the ( B;E) - plane with the slope (15) the \rutter load increases as\nF(E) =7\n2\u0000p\n2 +\u001247887\n242064+1925\n40344p\n2\u0013\nE2+o(E2); (16)\nsee Fig. 2(b), and monotonously tends to the undamped value as E!0. On the other\nhand, along the direction in the ( B;E) - plane speci\fed by the equation B= 0, the following\n7Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\ncondition is obtained\nF(E) = 2 +14\n99E2+o(E2); (17)\nsee Fig. 2(b), with the convergence to a lower value F= 2 asE!0.\nIn general, the limit of the \rutter load along the line B=\fEwhenE!0 is\nF(\f) =504\f2+ 1467\f+ 104\u0000(4 + 21\f)p\n576\f2+ 1728\f+ 121\n30(1 + 14\f)\u00147\n2\u0000p\n2; (18)\nan equation showing that for almost all directions the limit is lower than the ideal \rutter\nload. The limits only coincide in the sole direction speci\fed by Eq. (15), which is di\u000berent\nfrom theE-axis, characterized by \f= 0. As a conclusion, pure external damping yields the\ndestabilization paradox even at \u0016= 1=2, which was unnoticed in (Plaut and Infante, 1970;\nPlaut, 1971).\nIn the limit of vanishing external ( E) and internal ( B) damping, a ratio of the two\n\f=B=E exists for which the critical load of the undamped system is attained, so that the\nZiegler's paradox does not occur. This ratio can therefore be called `stabilizing', it exists for\nevery mass ratio \u0016=m2=m1, and is given by the expression\n\f(\u0016) =\u00001\n3(10\u0016\u00001)(\u0016\u00001)\n25\u00162+ 6\u0016+ 1+1\n12(13\u0016\u00005)(3\u0016+ 1)\n25\u00162+ 6\u0016+ 1\u0016\u00001=2: (19)\nEq. (19) reduces for \u0016= 1=2 to Eq. (15) and gives \f=\u00002=15 in the limit \u0016!1 . With\nthe damping ratio speci\fed by Eq. (19) the critical \rutter load has the following Taylor\nexpansion near E= 0:\nF(E;\u0016) =F\u0000\nu(\u0016) +\f(\u0016)(5\u0016+ 1)(41\u0016+ 7)\n6(25\u00162+ 6\u0016+ 1)E2\n+636\u00163+ 385\u00162\u0000118\u0016+ 25\n288(25\u00162+ 6\u0016+ 1)\u0016E2+o(E2); (20)\nyielding Eq. (16) when \u0016= 1=2. Eq. (20) shows that the \rutter load reduces to the\nundamped case when E= 0 (called `ideal' in the \fgure).\nWhen the stabilizing damping ratio is null, \f= 0, convergence to the critical \rutter load\nof the undamped system occurs by approaching the origin in the ( B;E) - plane along the E\n- axis. The corresponding mass ratio can be obtained \fnding the roots of the function \f(\u0016)\nde\fned by Eq. (19). This function has only two roots for 0 \u0014\u0016<1, one at\u0016\u00190:273 (or\n\u000b\u00190:267, marked as point A in Fig. 3(a)) and another at \u0016\u00192:559 (or\u000b\u00191:198, marked\nas point C in Fig. 3(a)).\nTherefore, if \f= 0 is kept in the limit when the damping tends to zero, the limit of the\n\rutter boundary in the load versus mass ratio plane will be obtained as a curve showing\ntwo common points with the \rutter boundary of the undamped system, exactly at the mass\n8Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nE Eb) a)\n0.1 0.0 0.3 0.2 0.5 0.4 0.7 0.6 0.8 0.9 1.02.052.072.09\n2.062.08F\n0.1 0.0 0.3 0.2 0.5 0.4 0.7 0.6 0.8 0.9 1.0B\n0.000\n-0.025-0.0500.0250.0500.0750.100\n2.102.112.122.132.142.15\n2.07F=2.10F=2.07 F=2.04FLUTTER FLUTTER\nB=0\nFigure 4: Analysis of the Ziegler pendulum with \fxed mass ratio, \u0016\u00192:559: (a) contours of the \rutter\nboundary in the internal/external damping plane, ( B;E), and (b) critical \rutter load as a function of external\ndampingE(continuous/blue curve) along the null internal damping line, B= 0.\nratios corresponding to the points denoted as A and C in Fig. 1(a), respectively characterized\nbyF\u00192:417 andF\u00192:070.\nIf for instance the mass ratio at the point C is considered and the contour plots are\nanalyzed of the \rutter boundary in the ( B;E) - plane, it can be noted that at the critical\n\rutter load of the undamped system, F\u00192:07, the boundary evidences a cusp with only\none tangent coinciding with the Eaxis, Fig. 4(a). It can be therefore concluded that at the\nmass ratio\u0016\u00192:559 the external damping alone has a stabilizing e\u000bect and the system does\nnot demonstrate the Ziegler paradox due to small external damping, see Fig. 4(b), where\nthe the \rutter load F(E) is shown.\nLooking back at the damping matrices (4) one may ask, what is the property of the\ndamping operator which determines its stabilizing or destabilizing character. The answer to\nthis question (provided by (Kirillov and Seyranian, 2005b; Kirillov, 2013) via perturbation\nof multiple eigenvalues) involves all the three matrices M(mass), D(damping), and K\n(sti\u000bness). In fact, the distributions of mass, sti\u000bness, and damping should be related in a\nspeci\fc manner in order that the three matrices ( M,D,K) have a stabilizing e\u000bect (see\nAppendix B for details).\n3. Ziegler's paradox for the P\r uger column with external damping\nThe Ziegler's pendulum is usually considered as the two-dimensional analog of the Beck\ncolumn, which is a cantilevered (visco)elastic rod loaded by a tangential follower force (Beck,\n9Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\n1952). Strictly speaking, this analogy is not correct because the Beck column has a di\u000berent\nmass distribution (the usual mass distribution of the Ziegler pendulum is m1= 2m2) and this\nmass distribution yields di\u000berent limiting behavior of the stability threshold (Section 2). For\nthis reason, in order to judge the stabilizing or destabilizing in\ruence of external damping in\nthe continuous case and to compare it with the case of the Ziegler pendulum, it is correct to\nconsider the Beck column with the point mass at the loaded end, in other words the so-called\n`P\r uger column' (P\r uger, 1955).\nA viscoelastic column of length l, made up of a Kelvin-Voigt material with Young modulus\nEand viscosity modulus E\u0003, and mass per unit length mis considered, clamped at one end\nand loaded by a tangential follower force Pat the other end (Fig. 5(c)), where a point mass\nMis mounted.\nThe moment of inertia of a cross-section of the column is denoted by Iand a distributed\nexternal damping is assumed, characterized by the coe\u000ecient K.\nSmall lateral vibrations of the viscoelastic P\r uger column near the undeformed equilib-\nrium state is described by the linear partial di\u000berential equation (Detinko, 2003)\nEI@4y\n@x4+E\u0003I@5y\n@t@x4+P@2y\n@x2+K@y\n@t+m@2y\n@t2= 0; (21)\nwherey(x;t) is the amplitude of the vibrations and x2[0;l] is a coordinate along the\ncolumn. At the clamped end ( x= 0) Eq. (21) is equipped with the boundary conditions\ny=@y\n@x= 0; (22)\nwhile at the loaded end ( x=l), the boundary conditions are\nEI@2y\n@x2+E\u0003I@3y\n@t@x2= 0; EI@3y\n@x3+E\u0003I@4y\n@t@x3=M@2y\n@t2: (23)\nIntroducing the dimensionless quantities\n\u0018=x\nl; \u001c =t\nl2q\nEI\nm; p =Pl2\nEI; \u0016 =M\nml;\n\r=E\u0003\nEl2q\nEI\nm; k =Kl2p\nmEI(24)\nand separating the time variable through y(\u0018;\u001c) =lf(\u0018) exp(\u0015\u001c), the dimensionless bound-\nary eigenvalue problem is obtained\n(1 +\r\u0015)@4\n\u0018f+p@2\n\u0018f+ (k\u0015+\u00152)f= 0;\n(1 +\r\u0015)@2\n\u0018f(1) = 0;\n(1 +\r\u0015)@3\n\u0018f(1) =\u0016\u00152f(1);\nf(0) =@\u0018f(0) = 0; (25)\n10Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nde\fned on the interval \u00182[0;1].\nA solution to the boundary eigenvalue problem (25) was found by Pedersen (1977) and\nDetinko (2003) to be\nf(\u0018) =A(cosh(g2\u0018)\u0000cos(g1\u0018)) +B(g1sinh(g2\u0018)\u0000g2sin(g1\u0018)) (26)\nwith\ng2\n1;2=p\np2\u00004\u0015(\u0015+k)(1 +\r\u0015)\u0006p\n2(1 +\r\u0015): (27)\nImposing the boundary conditions (25) on the solution (26) yields the characteristic equation\n\u0001(\u0015) = 0 needed for the determination of the eigenvalues \u0015, where\n\u0001(\u0015) = (1 +\r\u0015)2A1\u0000(1 +\r\u0015)A2\u0016\u00152(28)\nand\nA1=g1g2\u0000\ng4\n1+g4\n2+ 2g2\n1g2\n2coshg2cosg1+g1g2(g2\n1\u0000g2\n2) sinhg2sing1\u0001\n;\nA2= (g2\n1+g2\n2) (g1sinhg2cosg1\u0000g2coshg2sing1): (29)\n16 16 16 163/c112 /c112 /c112 /c112 3/c112 5/c112 7/c1120/c112\n88 42pa)\nexternal\nintenalr8101214161820\nMml\nPb)\n20.0\n19.518.017.517.0\n16/c112/c112\n80external\n/c97FLUTTER\nc)ideal\nidealA\nB\nFigure 5: Analysis of the P\r uger column [scheme reported in (c)]. (a) Stability map for the P\r uger's column\nin the load-mass ratio plane. The dashed/red curve corresponds to the stability boundary in the undamped\ncase, the dot-dashed/green curve to the case of vanishing internal dissipation ( \r= 10\u000010andk= 0 ) and\nthe continuous/blue curve to the case of vanishing external damping ( k= 10\u000010and\r= 0). (b) detail of\nthe curve reported in (a) showing the destabilization e\u000bect of external damping: small, but not null.\n11Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nTransforming the mass ratio parameter in Eq. (28) as \u0016= tan\u000bwith\u000b2[0;\u0019=2] allows\nthe exploration of all possible ratios between the end mass and the mass of the column\ncovering the mass ratios \u0016from zero (\u000b= 0) to in\fnity ( \u000b=\u0019=2). The former case, without\nend mass, corresponds to the Beck column, whereas the latter corresponds to a weightless\nrod with an end mass, which is known as the `Dzhanelidze column' (Bolotin, 1963).\nIt is well-known that the undamped Beck column loses its stability via \rutter at p\u0019\n20:05 (Beck, 1952). In contrast, the undamped Dzhanelidze's column loses its stability via\ndivergence at p\u001920:19, which is the root of the equation tanpp=pp(Bolotin, 1963).\nThese values, corresponding to two extreme situations, are connected by a marginal stability\ncurve in the ( p;\u000b)-plane that was numerically evaluated in (P\r uger, 1955; Bolotin, 1963;\nOran, 1972; Sugiyama et al., 1976; Pedersen, 1977; Ryu and Sugiyama, 2003). The instability\nthreshold of the undamped P\r uger column is shown in Fig. 5 as a dashed/red curve.\n16 16 16 163/c112 /c112 /c112 /c112 3/c112 5/c112 7/c1120/c112\n88 428.0\n7.010.0\n9.011.012.013.014.015.016.017.018.019.020.021.022.023.024.025.0\nSTABILITY\nSTABILITY\n/c103=10 , k=0/UNI207b¹⁰/c103=0.050, k=0\n/c103=0.100, k=0k=5, =0 /c103k=10 , =0 /UNI207b¹⁰/c103\nk=10, =0 /c103\nk1 0/c61/c103=/UNI207b¹⁰\n/c97p\nk 0.010/c61/c49/c44 /c103 =\nFigure 6: Evolution of the marginal stability curve for the P\r uger column in the ( \u000b;p) - plane in the case\nofk= 0 and\rtending to zero (green curves in the lower part of the graph) and in the case of \r= 0 andk\ntending to zero (blue curves in the upper part of the graph). The cases of k=\r= 10\u000010and ofk= 1 and\n\r= 0:01 are reported with continuous/red lines.\n12Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nFor every \fxed value \u000b2[0;\u0019=2), the undamped column loses stability via \rutter when\nan increase in pcauses the imaginary eigenvalues of two di\u000berent modes to approach each\nother and merge into a double eigenvalue with one eigenfunction. When plies above the\ndashed/red curve, the double eigenvalue splits into two complex eigenvalues, one with the\npositive real part, which determines a \rutter unstable mode.\nAt\u000b=\u0019=2 the stability boundary of the undamped P\r uger column has a vertical tangent\nand the type of instability becomes divergence (Bolotin, 1963; Oran, 1972; Sugiyama et al.,\n1976).\nSettingk= 0 in Eq. (28) the location in the ( \u000b;p)-plane of the marginal stability curves\ncan be numerically found for the viscoelastic P\r uger column without external damping, but\nfor di\u000berent values of the coe\u000ecient of internal damping \r, Fig. 6(a). The thresholds tend to\na limit which does not share common points with the stability boundary of the ideal column,\nas shown in Fig. 5(a), where this limit is set by the dot-dashed/green curve.\nThe limiting curve calculated for \r= 10\u000010agrees well with that obtained for \r= 10\u00003\nin (Sugiyama et al., 1995; Ryu and Sugiyama, 2003). At the point \u000b= 0, the limit value of\nthe critical \rutter load when the internal damping is approaching zero equals the well-known\nvalue for the Beck's column, p\u001910:94. At\u000b=\u0019=4 the limiting value becomes p\u00197:91,\nwhile for the case of the Dzhanelidze column ( \u000b=\u0019=2) it becomes p\u00197:49.\nAn interesting question is what is the limit of the stability diagram for the P\r uger column\nin the (\u000b;p)-plane when the coe\u000ecient of internal damping is kept null ( \r= 0), while the\ncoe\u000ecient of external damping ktends to zero.\nThe answer to this question was previously known only for the Beck column ( \u000b= 0), for\nwhich it was established, both numerically (Bolotin and Zhinzher, 1969; Plaut and Infante,\n1970) and analytically (Kirillov and Seyranian, 2005a), that the \rutter threshold of the\nexternally damped Beck's column is higher than that obtained for the undamped Beck's\ncolumn (tending to the ideal value p\u001920:05, when the external damping tends to zero). This\nvery particular example was at the basis of the common and incorrect opinion (maintained\nfor decades until now) that the external damping is only a stabilizing factor, even for non-\nconservative loadings. Perhaps for this reason the e\u000bect of the external damping in the\nP\r uger column has, so far, simply been ignored.\nThe evolution of the \rutter boundary for \r= 0 andktending to zero is illustrated by the\nblue curves in Fig. 6. It can be noted that the marginal stability boundary tends to a limiting\ncurve which has two common tangent points with the stability boundary of the undamped\nP\r uger column, Fig. 5(b). One of the common points, at \u000b= 0 andp\u001920:05, marked as\npoint A, corresponds to the case of the Beck column. The other corresponds to \u000b\u00190:516 and\np\u001916:05, marked as point B. Only for these two `exceptional' mass ratios the critical \rutter\nload of the externally damped P\r uger column coincides with the ideal value when k!0.\nRemarkably, for all other mass ratios the limit of the critical \rutter load for the vanishing\nexternal damping is located below the ideal value, which means that the P\r uger column fully\ndemonstrates the Ziegler destabilization paradox due to vanishing external damping , exactly\nas it does in the case of the vanishing internal damping, see Fig. 5(a), where the two limiting\n13Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\ncurves are compared.\nNote that the discrepancy in case of vanishing external damping is smaller than in case\nof vanishing internal damping, in accordance with the analogous result that was established\nin Section 2 for the Ziegler pendulum with arbitrary mass distribution. As for the discrete\ncase, also for the P\r uger column the \rutter instability threshold calculated in the limit when\nthe external damping tends to zero has only two common points with the ideal marginal\nstability curve. The discrepancy is the most pronounced for the case of Dzhanelidze column\nat\u000b=\u0019=2, where the critical load drops from p\u001920:19 in the ideal case to p\u001916:55 in\nthe case of vanishing external damping.\n4. Conclusions\nSince the \fnding of the Ziegler's paradox for structures loaded by nonconservative follower\nforces, internal damping (due to material viscosity) was considered a destabilizing factor,\nwhile external damping (due for instance to air drag resistance) was believed to merely\nprovide a stabilization. This belief originates from results obtained only for the case of Beck's\ncolumn, which does not carry an end mass. This mass is present in the case of the P\r uger's\ncolumn, which was never analyzed before from the point of view of the Ziegler paradox. A\nrevisitation of the Ziegler's pendulum and the analysis of the P\r uger column has revealed\nthat the Ziegler destabilization paradox occurs as related to the vanishing of the external\ndamping, no matter what is the ratio between the end mass and the mass of the structure.\nResults presented in this article clearly show that the destabilizing role of external damping\nwas until now misunderstood, and that experimental proof of the destabilization paradox\nin a mechanical laboratory is now more plausible than previously thought. Moreover, the\nfact that external damping plays a destabilizing role may have important consequences in\nstructural design and this opens new perspectives for energy harvesting devices.\nAcknowledgements\nThe authors gratefully acknowledge \fnancial support from the ERC Advanced Grant In-\nstabilities and nonlocal multiscale modelling of materials FP7-PEOPLE-IDEAS-ERC-2013-\nAdG (2014-2019).\nReferences\nReferences\nAndreichikov, I. P., Yudovich, V. I., 1974. Stability of viscoelastic bars. Izv. AN SSSR. Mekh.\nTv. Tela 9(2), 78{87.\nBanichuk, N. V., Bratus, A. S., Myshkis, A. D. 1989. Stabilizing and destabilizing e\u000bects in\nnonconservative systems. PMM USSR 53(2), 158{164.\n14Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nBeck, M. 1952. Die Knicklast des einseitig eingespannten, tangential gedr uckten Stabes. Z.\nangew. Math. Phys. 3, 225.\nBigoni, D., Noselli, G., 2011. Experimental evidence of \rutter and divergence instabilities\ninduced by dry friction. J. Mech. Phys. Sol. 59, 2208{2226.\nBolotin, V. V., 1963. Nonconservative Problems of the Theory of Elastic Stability. Pergamon\nPress, Oxford.\nBolotin, V. V., Zhinzher, N. I., 1969. E\u000bects of damping on stability of elastic systems\nsubjected to nonconservative forces. Int. J. Solids Struct. 5, 965{989.\nBottema, O., 1956. The Routh-Hurwitz condition for the biquadratic equation. Indag. Math.\n18, 403{406.\nChen, L. W., Ku, D. M., 1992. Eigenvalue sensitivity in the stability analysis of Beck's\ncolumn with a concentrated mass at the free end. J. Sound Vibr. 153(3), 403{411.\nCrandall, S. H., 1995. The e\u000bect of damping on the stability of gyroscopic pendulums. Z.\nAngew. Math. Phys. 46, S761{S780.\nDetinko, F. M., 2003. Lumped damping and stability of Beck column with a tip mass. Int.\nJ. Solids Struct. 40, 4479{4486.\nDone, G. T. S., 1973. Damping con\fgurations that have a stabilizing in\ruence on non-\nconservative systems. Int. J. Solids Struct. 9, 203{215.\nKirillov, O. N., 2011. Singularities in structural optimization of the Ziegler pendulum. Acta\nPolytechn. 51(4), 32{43.\nKirillov, O. N., 2013. Nonconservative Stability Problems of Modern Physics. De Gruyter\nStudies in Mathematical Physics 14. De Gruyter, Berlin.\nKirillov, O. N., Seyranian, A. P., 2005. The e\u000bect of small internal and external damping on\nthe stability of distributed nonconservative systems. J. Appl. Math. Mech. 69(4), 529{552.\nKirillov, O. N., Seyranian, A. P., 2005. Stabilization and destabilization of a circulatory\nsystem by small velocity-dependent forces, J. Sound Vibr. 283(35), 781{800.\nKirillov, O. N., Seyranian, A. P., 2005. Dissipation induced instabilities in continuous non-\nconservative systems, Proc. Appl. Math. Mech. 5, 97{98.\nKirillov, O. N., Verhulst, F., 2010. Paradoxes of dissipation-induced destabilization or who\nopened Whitney's umbrella? Z. Angew. Math. Mech. 90(6), 462{488.\n15Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nKrechetnikov, R., Marsden, J. E., 2007. Dissipation-induced instabilities in \fnite dimensions.\nRev. Mod. Phys. 79, 519{553.\nOran, C., 1972. On the signi\fcance of a type of divergence. J. Appl. Mech. 39, 263{265.\nPanovko, Ya. G., Sorokin, S. V., 1987. On quasi-stability of viscoelastic systems with the\nfollower forces, Izv. Acad. Nauk SSSR. Mekh. Tverd. Tela. 5, 135{139.\nPedersen, P. 1977. In\ruence of boundary conditions on the stability of a column under\nnon-conservative load. Int. J. Solids Struct. 13, 445{455.\nP\r uger, A., 1955. Zur Stabilit at des tangential gedr uckten Stabes. Z. Angew. Math. Mech.\n35(5), 191.\nPlaut, R. H., 1971. A new destabilization phenomenon in nonconservative systems. Z. Angew.\nMath. Mech. 51(4), 319{321.\nPlaut, R. H., Infante, E. F., 1970. The e\u000bect of external damping on the stability of Beck's\ncolumn. Int. J. Solids Struct. 6(5), 491{496.\nRyu, S., Sugiyama, Y., 2003. Computational dynamics approach to the e\u000bect of damping on\nstability of a cantilevered column subjected to a follower force. Comp. Struct. 81, 265{271.\nSaw, S. S., Wood, W. G., 1975. The stability of a damped elastic system with a follower\nforce. J. Mech. Eng. Sci. 17(3), 163{176.\nSugiyama, Y., Kashima, K., Kawagoe, H., 1976. On an unduly simpli\fed model in the\nnon-conservative problems of elastic stability. J. Sound Vibr. 45(2), 237{247.\nSugiyama, Y., Katayama, K., Kinoi, S. 1995. Flutter of cantilevered column under rocket\nthrust. J. Aerospace Eng. 8(1), 9{15.\nWalker, J. A. 1973. A note on stabilizing damping con\fgurations for linear non-conservative\nsystems. Int. J. Solids Struct. 9, 1543{1545.\nWang, G., Lin, Y. 1993. A new extension of Leverrier's algorithm. Lin. Alg. Appl. 180,\n227{238.\nZhinzher, N. I. 1994. E\u000bect of dissipative forces with incomplete dissipation on the stability\nof elastic systems. Izv. Ross. Akad. Nauk. MTT 1, 149{155.\nZiegler, H. 1952. Die Stabilit atskriterien der Elastomechanik. Archive Appl. Mech. 20, 49{56.\n16Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nAppendix A. - The stabilizing role of external damping and the destabilizing\nrole of internal damping\nA critical review of the relevant literature is given in this Appendix, with the purpose of\nexplaining the historical origin of the misconception that the external damping introduces a\nmere stabilizing e\u000bect for structures subject to \rutter instability.\nPlaut and Infante (1970) considered the Ziegler pendulum with m1= 2m2, without\ninternal damping (in the joints), but subjected to an external damping proportional to the\nvelocity along the rigid rods of the double pendulum4. In this system the critical \rutter load\nincreases with an increase in the external damping, so that they presented a plot showing\nthat the \rutter load converges to a value which is very close to P\u0000\nu. However, they did not\ncalculate the critical value in the limit of vanishing external damping, which would have\nrevealed a value slightly smaller than the value corresponding to the undamped system5.\nIn a subsequent work, Plaut (1971) con\frmed his previous result and demonstrated that\ninternal damping with equal damping coe\u000ecients destabilizes the Ziegler pendulum, whereas\nexternal damping has a stabilizing e\u000bect, so that it does not lead to the destabilization\nparadox. Plaut (1971) reports a stability diagram (in the external versus internal damping\nplane) that implicitly indicates the existence of the Whitney umbrella singularity on the\nboundary of the asymptotic stability domain. These conclusions agreed with other studies\non the viscoelastic cantilevered Beck's column (Beck, 1952), loaded by a follower force which\ndisplays the paradox only for internal Kelvin-Voigt damping (Bolotin and Zhinzher, 1969;\nPlaut and Infante, 1970; Andreichikov and Yudovich, 1974; Kirillov and Seyranian, 2005a)\nand were supported by studies on the abstract settings (Done, 1973; Walker, 1973; Kirillov\nand Seyranian, 2005b), which have proven the stabilizing character of external damping,\nassumed to be proportional to the mass (Bolotin, 1963; Zhinzher, 1994).\nThe P\r uger column [a generalization of the Beck problem in which a concentrated mass\nis added to the loaded end, P\r uger (1955), see also Sugiyama et al. (1976), Pedersen (1977),\nand Chen and Ku (1992)] was analyzed by Sugiyama et al. (1995) and Ryu and Sugiyama\n(2003), who numerically found that the internal damping leads to the destabilization paradox\nfor all ratios of the end mass to the mass of the column. The role of external damping was\ninvestigated only by Detinko (2003) who concludes that large external damping provides a\nstabilizing e\u000bect.\nThe stabilizing role of external damping was questioned only in the work by Panovko\nand Sorokin (1987), in which the Ziegler pendulum and the Beck column were considered\nwith a dash-pot damper attached to the loaded end (a setting in which the external damper\ncan be seen as something di\u000berent than an air drag, but as merely an additional structural\n4Note that di\u000berent mass distributions were never analyzed in view of external damping e\u000bect. In the\nabsence of damping, stability investigations were carried out by Oran (1972) and Kirillov (2011).\n5In fact, the \rutter load of the externally damped Ziegler pendulum with m1= 2m2, considered by Plaut\nand Infante (1970) and Plaut (1971) tends to the value P= 2 which is smaller than P\u0000\nu\u00192:086, therefore\nrevealing the paradox.\n17Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nelement, as suggested by Zhinzher (1994)). In fact the dash-pot was shown to always yield\nthe destabilization paradox, even in the presence of internal damping, no matter what the\nratio is between the coe\u000ecients of internal and external damping (Kirillov and Seyranian,\n2005c; Kirillov, 2013).\nIn summary, there is a well-established opinion that external damping stabilizes struc-\ntures loaded by nonconservative positional forces.\nAppendix B. - A necessary condition for stabilization of a general 2 d.o.f.\nsystem\nKirillov and Seyranian (2005b) considered the stability of the system\nMx+\"D_x+Kx= 0; (A.1)\nwhere\">0 is a small parameter and M=MT,D=DT, and K6=KTare real matrices of\nordern. In the case n= 2, the characteristic polynomial of the system (A.1),\nq(\u001b;\") = det( M\u001b2+\"D\u001b+K);\ncan be written by means of the Leverrier algorithm (adopted for matrix polynomials by\nWang and Lin (1993)) in a compact form:\nq(\u001b;\") = det M\u001b4+\"tr(D\u0003M)\u001b3+ (tr( K\u0003M) +\"2detD)\u001b2+\"tr(K\u0003D)\u001b+ det K;(A.2)\nwhere D\u0003=D\u00001detDandK\u0003=K\u00001detKare adjugate matrices and tr denotes the trace\noperator.\nLet us assume that at \"= 0 the undamped system (A.1) with n= 2 degrees of freedom\nbe on the \rutter boundary, so that its eigenvalues are imaginary and form a double complex-\nconjugate pair \u001b=\u0006i!0of a Jordan block. In these conditions, the real critical frequency\n!0at the onset of \rutter follows from q(\u001b;0) in the closed form (Kirillov, 2013)\n!2\n0=r\ndetK\ndetM: (A.3)\nA dissipative perturbation \"Dcauses splitting of the double eigenvalue i!0, which is\ndescribed by the Newton-Puiseux series \u001b(\") =i!0\u0006ip\nh\"+o(\"), where the coe\u000ecient his\ndetermined in terms of the derivatives of the polynomial q(\u001b;\") as\nh:=dq\nd\"\u00121\n2@2q\n@\u001b2\u0013\u00001\f\f\f\f\f\n\"=0;\u001b=i!0=tr(K\u0003D)\u0000!2\n0tr(D\u0003M)\n4i!0detM: (A.4)\nSince the coe\u000ecient his imaginary, the double eigenvalue i!0splits generically into two com-\nplex eigenvalues, one of them with the positive real part yielding \rutter instability (Kirillov\n18Published in Journal of the Mechanics and Physics of Solids (2016) 91: 204-215\ndoi: http://dx.doi.org/10.1016/j.jmps.2016.03.011\nand Seyranian, 2005b). Consequently, h= 0 represents a necessary condition for\"Dto be\nastabilizing perturbation (Kirillov and Seyranian, 2005b).\nIn the case of the system (3), with matrices (5), it is readily obtained\n!2\n0=k\nl2pm1m2: (A.5)\nAssuming D=Di, eq. (A.4) and the representations (5) and (A.5) yield\nh=hi:=i\nm1l25\u0016\u00002p\u0016+ 1\n4\u0016; (A.6)\nso that the equation hi= 0 has as solution the complex-conjugate pair \u0016= (\u00003\u00064i)=25.\nTherefore, for every real mass distribution \u0016\u00150 the dissipative perturbation with the matrix\nD=Diof internal damping results to be destabilizing.\nSimilarly, eq. (A.4) with D=Deand representations (A.5), (5), and F=F\u0000\nu(\u0016) yield\nh=he:=il\n48m18\u00162\u000011p\n\u00163\u00006\u0016+ 5p\u0016\n\u00162; (A.7)\nso that the constraint he= 0 is satis\fed only by the two following real values of \u0016\n\u0016A\u00190:273; \u0016C\u00192:559: (A.8)\nThe mass distributions (A.8) correspond exactly to the points A and C in Fig. 1, which are\ncommon for the \rutter boundary of the undamped system and for that of the dissipative\nsystem in the limit of vanishing external damping. Consequently, the dissipative perturbation\nwith the matrix D=Deof external damping can have a stabilizing e\u000bect for only two\nparticular mass distributions (A.8). Indeed, as it is shown in the present article, the external\ndamping is destabilizing for every \u0016\u00150, except for \u0016=\u0016Aand\u0016=\u0016C.\nConsequently, the stabilizing or destabilizing e\u000bect of damping with the given matrix D\nis determined not only by its spectral properties, but also by how it `interacts' with the mass\nand sti\u000bness distributions. The condition which selects possibly stabilizing triples ( M,D,\nK) in the general case of n= 2 degrees of freedom is therefore the following\ntr(K\u0003D) =!2\n0tr(D\u0003M): (A.9)\n19"    },    {        "title": "2303.03852v1.Electrically_tunable_Gilbert_damping_in_van_der_Waals_heterostructures_of_two_dimensional_ferromagnetic_metals_and_ferroelectrics.pdf",        "content": "Page 1 of 15 \n Electrically tunable Gilbert damping in van der Waals heterostructures of two-\ndimensional ferromagnetic meta ls and ferroelectrics \nLiang Qiu,1 Zequan Wang,1 Xiao-Sheng Ni,1 Dao-Xin Yao1,2 and Yusheng Hou 1,* \nAFFILIATIONS \n1 Guangdong Provincial Key Laboratory of Magnetoelectric Physics and Devices, State \nKey Laboratory of Optoelectronic Materials and Technologies, Center for Neutron \nScience and Technology, School of Physics, Sun Yat-Sen University, Guangzhou, \n510275, China  \n2 International Quantum Academy, Shenzhen 518048, China \n \nABSTRACT  \nTuning the Gilbert damping of ferromagnetic (FM) metals via a nonvolatile way  is \nof importance to exploit and design next-generation novel spintronic devices. Through \nsystematical first-principles calculations, we study the magnetic properties of the van \nder Waals heterostructure of two-dimensional FM metal  CrTe 2 and ferroelectric (FE) \nIn2Te3 monolayers. The ferromagnetism of CrTe 2 is maintained in CrTe 2/In2Te3 and its \nmagnetic easy axis can be switched from in-plane to out- of-plane by reversing the FE \npolarization of In 2Te3. Excitingly, we find that the Gilbert damping of CrTe 2 is tunable \nwhen the FE polarization of In 2Te3 is reversed from upward to downward. By analyzing \nthe k-dependent contributions to the Gilbert damping, we unravel that such tunability \nresults from the changed intersections between the bands of CrTe 2 and Fermi level on \nthe reversal of the FE polarizations of In 2Te3 in CrTe 2/In2Te3. Our work provides a n \nappealing way to electrically tailor Gilbert dampings of two-dimensional FM metals by \ncontacting them with ferroelectrics.  \n \n*Authors to whom correspondence should be addressed:  \n[Yusheng Hou, houysh@mail.sysu.edu.cn]  \n \n \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 2 of 15 \n Since the atomically thin long-range ferromagnetic ( FM) orders at finite \ntemperatures are discovered in CrI 31 monolayer (ML) and Cr 2Ge2Te62 bilayer, two-\ndimensional (2D) van der Waals (vdW) FM materials have attracted intensive \nattention.3-5 Up to now, many novel vdW ferromagnets such as Fe 3GeTe 2,6 Fe5GeTe 2,7 \nVSe 28,9 and MnSe 210 have been synthesized in experiments. Due to the intrinsic \nferromagnetism in these vdW FM materials, it is highly fertile to engineer emergent \nphenomena through magnetic proximity effect in their heterostructures.11 For instance , \nan unprecedented control of the spin and valley pseudospins in WSe 2 ML is reported in \nCrI 3/WSe 2.12 By contacting the thin films of three-dimensional topological insulators \nand graphene with CrI 3, high-temperature quantum anomalous Hall effect and vdW spin \nvalves are proposed in CrI 3/Bi2Se3/CrI 313 and CrI 3/graphene/CrI 3,14 respectively. On the \nother hand, the magnetic properties of these vdW FM materials can also be controlled \nby means of external perturbations such as gating and moiré patterns.3 In CrI 3 bilayer, \nHuang et al. observed a voltage-controlled switching between antiferromagnetic (AFM) \nand FM states.15 Via an ionic gate, Deng et al. even increased the Curie temperature \n(TC) of the thin flake of vdW FM metal Fe 3GeTe 2 to room temperature, which is much \nhigher than its bulk TC.6 Very recently, Xu et al. demonstrated a coexisting FM and \nAFM state in a twisted bilayer CrI 3.16 These indicate that vdW FM materials are \npromising platforms to design and implement spintronic devices in the 2D limit.4,11  \nRecently, of great interest is the emergent vdW magnetic material CrTe 2 which is \na new platform for realizing room-temperature intrinsic ferromagnetism.17,18 Especially, \nCrTe 2 exhibits greatly tunable magneti sm. In the beginning, its ground state is believed \nto be the nonmagnetic 2 H phase,19 while several later researches suggest that either the \nFM or AFM 1 T phases should be the ground state of CrTe 2.17,18,20- 23 Currently, the \nconsensus is that the structural ground state of CrTe 2 is the 1 T phase. With respect to its \nmagnetic ground state, a first-principles study shows that the FM and AFM ground \nstates in CrTe 2 ML depend on its in-plane lattice constants.24 It is worth noting that the \nTC of FM CrTe 2 down to the few-layer limit can be higher than 300 K,18 making it have \nwide practical application prospects in spintronics.  \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 3 of 15 \n Building heterostructures of FM and ferroelectric (FE) materials offers an effective \nway to control nonvolatile magnetism via an electric field. Experimentally, Eerenstein \net al. presented an electric-field-controlled nonvolatile converse magnetoelectric effect \nin a multiferroic heterostructure La0.67Sr0.33MnO 3/BaTiO 3.25 Later, Zhang et al. reported \nan electric-field-driven control of nonvolatile magnetization in a heterostructure of FM \namorphous alloy Co40Fe40B20 and FE Pb(Mg 1/3Nb2/3)0.7Ti0.3O3.26 Theoretically, Chen et \nal. demonstrated based on first-principles calculations that the interlayer magnetism of \nCrI 3 bilayer in CrI 3/In2Se3 is switchable between FM and AFM couplings by the \nnonvolatile control of the FE polarization direction of In 2Se3.27 In spite of these \ninteresting findings, using FE substrates to electrically tune the Gilbert damping of \nferromagnets, an important factor determining the operation speed of spintronic devices, \nis rarely investigated in 2D FM/FE vdW heterostructures. Therefore, it is of great \nimportance to explore the possibility of tuning the Gilbert damping in such kind of \nheterostructures.  \nIn this work, we first demonstrate that the magnetic ground state of 1 T-phase CrTe 2 \nML will change from the zigzag AFM (denoted as z-AFM) to FM orders with increasing \nits in-plane lattice constants. By building a vdW heterostructure of CrTe 2 and FE In2Te3 \nMLs, we show that the magnetic easy axis of CrTe 2 can be tuned from in-plane to out-\nof-plane by reversing the FE polarization of In 2Te3, although its ferromagnetism is kept . \nImportantly, we find that the Gilbert damping of CrTe 2 is tunable with a wide range on \nreversing the FE polarization of In 2Te3 from upward to downward. Through looking \ninto the k-dependent contributions to the Gilbert damping, we reveal that such tunability \noriginates from the changed intersections between the bands of CrTe 2 and Fermi level \nwhen the FE polarizations of In 2Te3 is reversed in CrTe 2/In2Te3. Our work demonstrates \nthat putting 2D vdW FM metals on FE substrates is an attractive method to electrically \ntune their Gilbert dampings.  \nCrTe 2, a member of the 2D transition metal dichalcogenide family, can potentially \ncrystalize into several different layered structures such as 1 T, 1Td, 1H and 2 H phases.28 \nIt is believed that the 1 T phase is the most stable among all of the se possible phases in \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 4 of 15 \n both bulk and ML. This phase has a hexagonal lattice and belongs to the P3_m1 space \ngroup, with each Cr atom surrounded by the octahedrons of Te atoms (Fig. 1a). In view \nof the hot debates on the magnetic ground state in CrTe 2 ML, we establish a 2× 2√3 \nsupercell and calculate the total energies of several different magnetic structures (Fig. \nS1 in Supplementary Materials) when its lattice constant varies from 3.65 to 4.00 Å. As \nshown in Fig. 1b, our calculations show that z-AFM order is the magnetic ground state \nwhen the lattice constant is from 3.65 to 3.80 Å. By contrast, the FM order is the \nmagnetic ground state when the lattice constant is in the range from 3.80 to 4.00 Å. \nNote that our results are consistent with the experimentally observed z- AFM23 and \nFM29 orders in CrTe 2 with a lattice constant of 3.70 and 3.95 Å, respectively. Since we \nare interested in the Gilbert damping of ferromagnets and the experimentally grow n \nCrTe 2 on ZrTe 2 has a lattice constant of 3.95 Å,29 we will focus on CrTe 2 ML with this \nlattice constant hereinafter.  \n \n \nFIG. 1. (a) Side (the top panel) and top (the bottom panel) views of CrTe 2 ML. The NN \nand second- NN exchange paths are shown by red arrows in the top view. (b) The phase \ndiagram of the magnetic ground state of CrTe 2 ML  with different lattice constants. Insets \nshow the schematic illustrations of the z-AFM and FM orders. The up and down spins \nare indicated by the blue and red balls, respectively. The stars highlight the experimental \nlattice constants of CrTe 2 in Ref.23 and Ref.29.  \n \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 5 of 15 \n To obtain an deeper understanding on the ferromagnetism of CrTe 2 ML, we adopt \na spin Hamiltonian consisting of Heisenberg exchange couplings and single-ion \nmagnetic anisotropy (SIA) as follows:30  \n𝐻 = 𝐽 1∑ 𝑆𝑖⋅ 𝑆𝑗 ⟨𝑖𝑗⟩ + 𝐽 2∑ 𝑆𝑖⋅ 𝑆𝑗 ⟨⟨𝑖𝑗⟩⟩ − 𝐴 ∑ ( 𝑆𝑖𝑧)2\n𝑖  (1) \nIn Eq. (1), J1 and J2 are the nearest neighbor (NN) and second- NN Heisenberg exchange \ncouplings. Note that a negative (positive) J means a FM (AFM) Heisenberg exchange \ncouples. Besides, A parameterizes the SIA term. First of all, our DFT calculations show \nthat the magnetic moment of CrTe 2 ML is 3.35 μB/Cr, consistent with previous DFT \ncalculations.31 As shown in Table I, the calculated J1 and J2 are both FM and J1 is much \nstronger than J2. Both FM  J1 and J2 undoubtedly indicate that CrTe 2 ML has a FM \nmagnetic ground state. Finally, the SIA parameter A is obtained by calculating the \nenergy difference between two FM states with out-of-plane and in-plane magnetizations. \nOur calculations obtain A=1.81 meV/Cr, indicating that CrTe 2 ML has an out-of-plane \nmagnetic easy axis. Hence, our calculations show that CrTe 2 ML exhibits an out-of-\nplane FM order, consistent with experimental observations.29  \n \nTABLE I. Listed are the in-plane lattice constant s a, Heisenberg exchange couplings J \n(in unit of meV) and SIA (in unit of meV/Cr) of CrTe 2 ML and CrTe 2/In2Te3. \nSystem  a (Å) J1 J2  A \nCrTe 2 3.95 -24.56  -0.88  1.81 \nCrTe 2/In2Te3(↑) 7.90 -20.90  -1.80  -1.44  \nCrTe 2/In2Te3(↓) 7.90 -19.33  -0.88   0.16  \n \nTo achieve an electrically tunable Gilbert damping in CrTe 2 ML, we establish its \nvdW heterostructure with F E In2Te3 ML. In building this heterostructure, w e stack a \n2×2 supercell of CrTe 2 and a √3 ×√3  supercell of In2Te3 along the (001) direction. \nBecause the magnetic properties of CrTe 2 ML are the primary topic and the electronic \nproperties of In2Te3 ML are basically not affected by a strain (Fig. S2), we stretch the \nlattice constant of the latter to match that of the former. Fig. 2a shows the most stable \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 6 of 15 \n stacking configuration in CrTe 2/In2Te3 with an upward FE polarization [denoted as \nCrTe 2/In2Te3(↑)]. At the interface in this configuration, one of four Cr atoms and one of \nfour Te atoms at the bottom of CrTe 2 sits on the top of the top-layer Te atom s of In2Te3. \nIn CrTe 2/In2Te3 with a downward FE polarization [denoted as CrTe 2/In2Te3(↓ )], the \nstacking configuration at its interface is same as that in CrTe 2/In2Te3(↑ ). The only \ndifference between CrTe 2/In2Te3(↑ ) and CrTe 2/In2Te3(↓ ) is that the middle-layer Te \natoms of In 2Te3 in the former is farther to CrTe 2 than that in the latter (Fig. 2a and 2c). \nIt is noteworthy that the bottom-layer Te atoms of CrTe 2 do not stay at a plane anymore \nin the relaxed CrTe 2/In2Te3 (see more details in Fig. S3), suggesting non-negligible \ninteractions between CrTe 2 and In 2Te3. \n \n  \nFIG. 2. (a) The schematic stacking configuration and (b) charge density difference 𝛥ρ \nof CrTe 2/In2Te3(↑). (c) and (d) same as (a) and (b) but for CrTe 2/In2Te3(↓). In (b) and \n(d), color bar indicates the weight of negative (blue) and positive (red) charge density \ndifferences. (e) The total DOS of CrTe 2/In2Te3. (f) and (g) show the PDOS of CrTe 2 and \nIn2Te3 in CrTe 2/In2Te3, respectively. In (e)-(g), upward and downward polarizations are \nindicated by black and red lines, respectively.  \n \nTo shed light on the effect of the FE polarization of In 2Te3 on the electronic \nproperty of CrTe 2/In2Te3, we first investigate the spatial distribution of charge density \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 7 of 15 \n difference \n2 2 3 2 2 3 CrTe In Te CrTe In Te     = − −  with different FE polarization directions. \nAs shown in Fig. 2b and 2d, we see that there is an obvious charge transfer at the \ninterfaces of both CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓), which is further confirmed by \nthe planar averaged 𝛥ρ (Fig. S4). Additionally, the charge transfer in CrTe 2/In2Te3(↑) \nis distinctly less than th at in CrTe 2/In2Te3(↓). Fig. 2e shows that the total density of \nstates (DOS) near Fermi level are highly different in CrTe 2/In2Te3(↑ ) and \nCrTe 2/In2Te3(↓). By projecting the DOS onto CrTe 2 and In 2Te3, Fig. 2f shows that the \nprojected DOS (PDOS) of CrTe 2 in CrTe 2/In2Te3(↑) is larger than that in CrTe 2/In2Te3(↓) \nat Fermi level. Interestingly, the PDOS of In 2Te3 in CrTe 2/In2Te3(↑) is larger than that \nin CrTe 2/In2Te3(↓) below Fermi level while the situation is inversed above Fermi level \n(Fig. 2g). By looking into the five Cr- d orbital projected DOS in CrTe 2/In2Te3(↑) and \nCrTe 2/In2Te3(↓) (Fig. S5), we see that there are obviously different occupations for xyd, \n22xyd− and 223zrd− orbitals near Fermi level. All of these imply that the reversal of the \nFE polarization of In 2Te3 may have an unignorable influence on the magnetic properites \nof CrTe 2/In2Te3.   \nDue to the presence of the FE In 2Te3, the inversion symmetry is inevitably broken \nand nonzero Dzyaloshinskii-Moriya interactions (DMIs) may exist in CrTe 2/In2Te3. In \nthis case, we add a DMI term into Eq. (1) to investigate the magnetism of CrTe 2/In2Te3 \nand the corresponding spin Hamiltonian is in the form of32  \n𝐻 = 𝐽 1∑ 𝑆𝑖⋅ 𝑆𝑗 ⟨𝑖𝑗⟩ + 𝐽 2∑ 𝑆𝑖⋅ 𝑆𝑗 ⟨⟨𝑖𝑗⟩⟩ +∑ 𝑫𝑖𝑗⋅ (𝑆 𝑖× 𝑆 𝑗) ⟨𝑖𝑗⟩ − 𝐴 ∑ ( 𝑆𝑖𝑧)2\n𝑖  (2). \nIn Eq. (2), Dij is the DMI vector of the NN Cr-Cr pairs. As the C6-rotational symmetry \nwith respect to Cr atoms in CrTe 2 is reduced to the C3-rotational symmetry, the NN \nDMIs are split into four different DMIs (Fig. S6). For simplicity, the J1 and J2 are still \nregard ed to be six-fold. From Table I, we see that the NN J1 of both CrTe 2/In2Te3(↑) and \nCrTe 2/In2Te3(↓) are still FM but slightly smaller than that of free-standing CrTe 2 ML. \nMoreover, the second- NN FM J2 is obviously enhanced in CrTe 2/In2Te3(↑) compared \nwith CrTe 2/In2Te3(↓) and free-standing CrTe 2 ML. To calculated the NN DMIs, we build \na √3×√3 supercell of CrTe 2/In2Te3 and the four-state method33 is employed here. As \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 8 of 15 \n listed in Table S1, the FE polarization direction of In 2Te3 basically has no qualitative \neffect on the DMIs in CrTe 2/In2Te3 although it affects their magnitudes. More explicitly, \nthe magnitudes of the calculated DMIs range from 1.22 to 2.81 meV, which are about \none order smaller than the NN J1. Finally, we find that the SIA of CrTe 2/In2Te3 is \nstrongly dependent on the FE polarization of In 2Te3. When In 2Te3 has an upward FE \npolarization, the SIA of CrTe 2/In2Te3(↑) is negative, indicating an in-plane magnetic \neasy axis. However, when the FE polarization of In 2Te3 is downward, CrTe 2/In2Te3(↓) \nhas a positive SIA, indicating an out-of-plane magnetic easy axis. It is worth noting that \nCrTe 2/In2Te3(↓) has a much weak SIA than the free-standing CrTe 2 ML, although they \nboth have positive SIAs. The different Heisenberg exchange couplings, DMIs and SIAs \nin CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓) clearly unveil that the magnetic properties of \nCrTe 2 are tuned by the FE polarization of In2Te3.  \nTo obtain the magnetic ground state of CrTe 2/In2Te3, MC simulations are carried \nout. As shown in Fig. S7, CrTe 2/In2Te3(↑) has an in-plane FM magnetic ground state \nwhereas CrTe 2/In2Te3(↓ ) has an out-of-plane one. Such magnetic ground states are \nunderstandable. Firstly, the ratios between DMIs and the NN Heisenberg exchange \ncouplings are small and most of them are out of the typical range of 0.1–0.2 for the \nappearance of magnetic skyrmions.34 Secondly, the SIAs of the CrTe 2/In2Te3(↑) and the \nCrTe 2/In2Te3(↓ ) prefer in-plane and out-of-plane magnetic easy axes, respectively . \nTaking them together, we obtain that the FM Heisenberg exchange couplings dominate \nover the DMIs and thus give rise to a FM magnetic ground state with its magnetization \ndetermined by the SIA,35 consistent with our MC simulated results.  \nFigure 3a shows the Γ-dependent Gilbert dampings of CrTe 2/In2Te3 with upward \nand downward FE polarizations of In2Te3. Similar to previous studies,36,37 the Gilbert \ndampings of CrTe 2/In2Te3 decrease first and then increase as the scattering rate Γ \nincreases. Astonishingly, the Gilbert dampings of CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓) \nare distinctly different at the same scattering rate Γ ranging from 0.001 to 1.0 eV . To \nhave a more intuitive sense on the effect of the FE polarizations of In 2Te3 on the Gilbert \ndampings in CrTe 2/In2Te3, we calculate the ratio   =  at any given Γ, where \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 9 of 15 \n ( ) is the Gilbert damping of CrTe 2/In2Te3(↑) [CrTe 2/In2Te3(↓)]. As shown in Fig. \n3b, the ratio 𝜂 ranges from 6 to around 1.3 with increasing Γ. As the FE polarization \nof In 2Te3 can be switched from upward to downward by an external electric field, the \nGilbert damping of CrTe 2/In2Te3 is electrically tunable in practice.  \n \n \nFIG. 3. (a) The Γ-dependent Gilbert dampings of CrTe 2/In2Te3 with upward (black line) \nand downward (red line) FE polarizations of In 2Te3. (b) The Gilbert damping ratio 𝜂 \nas a function of the scattering rate Γ.  \n \nTo gain a deep insight into how the FE polarization of In 2Te3 tunes the Gilbert \ndamping in CrTe 2/In2Te3, we investigate the k-dependent contributions to the Gilbert \ndampings of CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓). As shown in Fig. 4a and 4b, the bands \naround Fermi level have qiute different intermixing between CrTe 2 and In 2Te3 states \nwhen the FE polarizaiton of In 2Te3 is reversed. Explicitly, there are obvious intermixing \nbelow Fermi leve in CrTe 2/In2Te3(↑) while the intermixing mainly takes place above \nFermi level in CrTe 2/In2Te3(↓). Especially, the bands intersected by Fermi level are at \ndifferent k points in CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓). Through looking into the k-\ndependent contributions to the ir Gilbert dampings (Fig. 4c and 4d), we see that large \ncontributions are from the k points (highlighted by arrows in Fig. 4) at which the bands \nof CrTe 2 cross Fermi level. In addition, these large contributions are different. Such k-\ndipendent contribution to Gilbert dampings is understandable. Based on the scattering \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 10 of 15 \n theory of Gilbert damping 38, Gilbert damping parameter is calculated using the \nfollowing Eq. (3) 36  \n( ) ( ) , , , , , , (3)kk\nk i k j k j k i F k i F k j\nk ij SE E E EM u u\n      = − − −\nHH, \nwhere EF is Fermi level and Ek,i is the enery of band i at a given k point. Due to the delta \n( ) ( ) ,, F k i F k jE E E E −−  , only the valence and conduction bands near Fermi level \nmake dominant contribution to the Gilbert damping. Additionally, their contributions \nalso depend on factor , , , ,kk\nk i k j k j k iuu   \nHH. Overall , through changing the \nintersections between the bands of CrTe 2 and Fermi level, the reversal of the FE \npolarization of In 2Te3 can modulate the contributions to Gilbert damping. Consequently, \nthe total Gilbert dampings are different in CrTe 2/In2Te3(↑) and CrTe 2/In2Te3(↓).  \n \n \nFIG. 4. (a) Band structure calculated with spin-orbit coupling and (c) the k-dependent \ncontributions to the Gilbert damping in CrTe 2/In2Te3(↑). (b) and (d) same as (a) and (c) \nbut for CrTe 2/In2Te3(↓). In (a) and (b), Fermi levels are indicated by horizontal dash \nlines and the states from CrTe 2 and In 2Te3 are shown by red and blue, respectively.  \n \n    From experimental perspectives, the fabrication of CrTe 2/In2Te3 vdW \nheterostructure should be feasible. On the one hand, CrTe 2 with the lattice constant of \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 11 of 15 \n 3.95 Å has been successfully grown on ZrTe 2 substrate by the molecular beam epitaxy.29 \nOn the other hand, In 2Te3 is also synthesized.39 Taking these and the vdW nature of \nCrTe 2 and In 2Te3 together, a practical scheme of growing CrTe 2/In2Te3 is sketched in \nFig. S8 : first grow CrTe 2 ML on ZrTe 2 substrate29 and then put In2Te3 ML on CrTe 2 to \nform the desired CrTe 2/In2Te3 vdW heterostructure.     \nIn summary, by constructing a vdW heterostructure of 2D FM metal CrTe 2 and FE \nIn2Te3 MLs, we find that the magnetic properties of CrTe 2 are engineered by the reversal \nof the FE polariton of In 2Te3. Although the ferromagnetism of CrTe 2 is maintained in \nthe presence of the FE In2Te3, its magnetic easy axis can be tuned from in-plane to out-\nof-plane by reversing the FE polarization of In 2Te3. More importantly, the Gilbert \ndamping of CrTe 2 is tunable with a wide range when reversing the FE polarization of \nIn2Te3 from upward to downward. Such tunability of the Gilbert damping in \nCrTe 2/In2Te3 results from the changed intersections between the bands of CrTe 2 and \nFermi level on reversing the FE polarizations of In 2Te3. Our work introduces a \nremarkably useful method to electrically tune the Gilbert dampings of 2D vdW FM \nmetals by contacting them with ferroelectrics, and should stimulate more experimental \ninvestigations in this realm.  \n \nSee the supplementary material for the details of computational methods31,36,40- 50 \nand other results mentioned in the main text.  \n \nThis project is supported by National Nature Science Foundation of China (No. \n12104518, 92165204, 11974432), NKRDPC-2018YFA0306001, NKRDPC-\n2022YFA1402802, GBABRF-2022A1515012643 and GZABRF-202201011118 . \nDensity functional theory calculations are performed at Tianhe- II. \n \nAUTHOR DECLARATIONS  \nConflict of Interest  \nThe authors have no conflicts to disclose.  \nThis is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401Page 12 of 15 \n Author Contributions  \nLiang Qiu : Investigation (equal); Methodology (equal); Writing –original draft (equal). \nZequan Wang : Methodology (equal). Xiao -sheng Ni : Investigation (equal); \nMethodology (equal).  Dao-Xin Yao : Supervision (equal); Funding acquisition (equal); \nInvestigation (equal); Writing – review &editing (equal). Yusheng Hou : \nConceptualization (equal); Funding acquisition (equal); Investigation (equal); Project \nadministration(equal); Resources (equal); Supervision (equal); Writing – review \n&editing (equal).  \n \nDATA A V AILABILITY  \nThe data that support the findings of this study are available from the \ncorresponding authors upon reasonable request.  \n \n \nREFERENCES  \n1 B. Huang, G. Clark, E. Navarro-Moratalla, D. R. Klein, R. Cheng, K. L. Seyler, \nD. Zhong, E. 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However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.\nPLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0145401"    },    {        "title": "1506.00723v1.Current_Driven_Motion_of_Magnetic_Domain_Wall_with_Many_Bloch_Lines.pdf",        "content": "Journal of the Physical Society of Japan LETTERS\nCurrent-Driven Motion of Magnetic Domain Wall with Many Bloch\nLines\nJunichi Iwasaki1\u0003and Naoto Nagaosa1;2y\n1Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan\n2RIKEN Center for Emergent Matter Science (CEMS),Wako, Saitama 351-0198, Japan\nThe current-driven motion of a domain wall (DW) in a ferromagnet with many Bloch lines (BLs) via the\nspin transfer torque is studied theoretically. It is found that the motion of BLs changes the current-velocity\n(j-v) characteristic dramatically. Especially, the critical current density to overcome the pinning force is\nreduced by the factor of the Gilbert damping coe\u000ecient \u000beven compared with that of a skyrmion. This\nis in sharp contrast to the case of magnetic \feld driven motion, where the existence of BLs reduces the\nmobility of the DW.\nDomain walls (DWs) and bubbles1,2)are the spin tex-\ntures in ferromagnets which have been studied inten-\nsively over decades from the viewpoints of both funda-\nmental physics and applications. The memory functions\nof these objects are one of the main focus during 70's, but\ntheir manipulation in terms of the magnetic \feld faced\nthe di\u000eculty associated with the pinning which hinders\ntheir motion. The new aspect introduced recently is the\ncurrent-driven motion of the spin textures.3,4)The \row\nof the conduction electron spins, which follow the direc-\ntion of the background localized spin moments, moves\nthe spin texture due to the conservation of the angu-\nlar momentum. This e\u000bect, so called the spin transfer\ntorque, is shown to be e\u000bective to manipulate the DWs\nand bubbles compared with the magnetic \feld. Magnetic\nskyrmion5,6)is especially an interesting object, which is\na swirling spin texture acting as an emergent particle\nprotected by the topological invariant, i.e., the skyrmion\nnumberNsk, de\fned by\nNsk=1\n4\u0019Z\nd2rn(r)\u0001\u0012@n(r)\n@x\u0002@n(r)\n@y\u0013\n(1)\nwith n(r) being the unit vector representing the direc-\ntion of the spin as a function of the two-dimensional spa-\ntial coordinates r. This is the integral of the solid angle\nsubtended by n, and counts how many times the unit\nsphere is wrapped. The solid angle and skyrmion number\nNskalso play essential role when one derives the equation\nof motion for the center of mass of the spin texture, i.e.,\nthe gyro-motion is induced by Nskin the Thiele equation,\nwhere the rigid body motion is assumed.7,8)\nBeyond the Thiele equation,7)one can derive the equa-\ntion of motion of a DW in terms of two variables, i.e.,\nthe wall-normal displacement q(t;\u0010;\u0011 ) and the wall-\nmagnetization orientation angle  (t;\u0010;\u0011 ) (see Fig. 1)\n\u0003iwasaki@appi.t.u-tokyo.ac.jp\nynagaosa@ap.t.u-tokyo.ac.jp\nψqFig. 1. Schematic magnetization distribution of DW with many\nBloch lines.\nwhere\u0010and\u0011are general coordinates specifying the\npoint on the DW:9)\n\u000e\u001b\n\u000e = 2M\r\u00001h\n_q\u0000\u000b\u0001_ \u0000vs\n?\u0000\f\u0001vs\nk(@k )i\n;(2)\n\u000e\u001b\n\u000eq=\u00002M\r\u00001h\n_ +\u000b\u0001\u00001_q+vs\nk(@k )\u0000\f\u0001\u00001vs\n?i\n;\n(3)\nHere, _ means the time-derivative. kand?indicate\nthe components parallel and perpendicular to the DW\nrespectively. Mis the magnetization, \ris the gyro-\nmagnetic ratio, and \u001b, \u0001 are the energy per area and\nthickness of the DW. vsis the velocity of the conduction\nelectrons, which produces the spin transfer torque. \u000bis\nthe Gilbert damping constant, and \frepresents the non-\nadiabatic e\u000bect. These equations indicate that qand \nare canonical conjugate to each other. This is understood\nby the fact that the generator of the spin rotation nor-\nmal to the DW, which is proportional to sin  in Fig. 1,\ndrives the shift of q. (Note that  is measured from the\n\fxed direction in the laboratory coordinates.)\nIn order to reduce the magnetostatic energy, the spins\nin the DW tend to align parallel to the DW, i.e., Bloch\nwall. When the DW is straight, this structure is coplanar\nand has no solid angle. From the viewpoint of eqs. (2)\n1arXiv:1506.00723v1  [cond-mat.mes-hall]  2 Jun 2015J. Phys. Soc. Jpn. LETTERS\nand (3), the angle  is \fxed around the minimum, and\nslightly canted when the motion of qoccurs, i.e., _ = 0.\nHowever, it often happens that the Bloch lines (BLs)\nare introduced into the DW as shown schematically in\nFig. 1. The angle  rotates along the DW and the N\u0013 eel\nwall is locally introduced. It is noted here that the solid\nangle becomes \fnite in the presence of the BLs. Also with\nmany BLs in the DW, the translation of BLs activates\nthe motion of the angle  , i.e., _ 6= 0, which leads to the\ndramatic change in the dynamics.\nIn the following, we focus on the straight DW which\nextends along x-direction and is uniform in z-direction.\nThus, the general coordinates here are ( \u0010;\u0011) = (x;z).\nq(t;x;z ) is independent of the coordinates q(t;x;z ) =\nq(t), and the functional derivative \u000e\u001b=\u000eq in eq. (3) be-\ncomes the partial derivative @\u001b=@q . In the absence of\nBLs, we set  (t;x;z ) = (t), and\u000e\u001b=\u000e in eq. (2) also\nbecomes@\u001b=@ . Then the equation of motion in the ab-\nsence of BL is\n@\u001b\n@ = 2M\r\u00001h\n_q\u0000\u000b\u0001_ \u0000vs\n?i\n; (4)\n@\u001b\n@q=\u00002M\r\u00001h\n_ +\u000b\u0001\u00001_q\u0000\f\u0001\u00001vs\n?i\n; (5)\nWith many BLs, the sliding motion of Bloch lines along\nDW, which activates _ , does not change the wall energy,\ni.e.,\u000e\u001b=\u000e in eq. (2) vanishes.2)Here, for simplicity, we\nconsider the periodic BL array with the uniform twist\n (t;x;z ) = (x\u0000p(t))=~\u0001 where ~\u0001 is the distance between\nBLs, which leads to\n0 = 2M\r\u00001h\n_q+\u000b\u0001~\u0001\u00001_p\u0000vs\n?\u0000\f\u0001~\u0001\u00001vs\nki\n;(6)\n@\u001b\n@q=\u00002M\r\u00001h\n\u0000~\u0001\u00001_p+\u000b\u0001\u00001_q+~\u0001\u00001vs\nk\u0000\f\u0001\u00001vs\n?i\n;\n(7)\nFirst, let us discuss the magnetic \feld driven motion\nwithout current. The e\u000bect of the external magnetic \feld\nHextis described by the force @\u001b=@q =\u00002MHextin\neqs. (5) and (7). vs\nkandvs\n?are set to be zero. In the\nabsence of BL, as mentioned above, the phase  is static\n_ = 0 with the slight tilt of the spin from the easy-plane,\nand one obtains from eq. (5)\n_q=\u0001\rHext\n\u000b: (8)\nThis is a natural result, i.e., the mobility is inversely\nproportional to the Gilbert damping \u000b. is determined\nby eq. (4) with this value of the velocity _ q.\nIn the presence of many BLs, eqs. (6) and (7) give the\nvelocities of DW and BL sliding driven by the magnetic\n\feld as\n_q=\u000b\n1 +\u000b2\u0001\rHext; (9)_p=\u00001\n1 +\u000b2~\u0001\rHext: (10)\nComparing eqs. (8) and (9), the mobility of the DW is re-\nduced by the factor of \u000b2since\u000bis usually much smaller\nthan unity. We also note that the velocity of the BL slid-\ning _pis larger than that of the wall _ qby the factor of\n\u000b. Physically, this means that the e\u000bect of the external\nmagnetic \feld Hextmostly contributes to the rapid mo-\ntion of the BLs along the DW rather than the motion of\nthe DW itself. These results have been already reported\nin refs.2,9,10)\nNow let us turn to the motion induced by the current\nvs. In the absence of BL, again we put _ = 0 in eqs. (4)\nand (5). Assuming that there is no pinning force or ex-\nternal magnetic \feld, i.e., @\u001b=@q = 0, one obtains from\neq. (5)\n_q=\f\n\u000bvs\n?; (11)\nand eq. (4) determines the equilibrium value of  . When\nthe pinning force @\u001b=@q =Fpinis \fnite, there appears a\nthreshold current density ( vs\n?)cwhich is determined by\nputting _q= 0 in eq. (5) as\n(vs\n?)c=\r\u0001\n2M\fFpin; (12)\nwhich is inversely proportional to \f.11)Since eq. (11) is\nindependent of vs\nk, the threshold current density\u0010\nvs\nk\u0011\nc\nis\u0010\nvs\nk\u0011\nc=1.\nIn the presence of the many BLs, on the other hand,\neqs. (6) and (7) give\n@\u001b\n@q=\u00002M\r\u00001\u00141 +\u000b2\n\u000b\u0001\u00001_q\n\u00001 +\u000b\f\n\u000b\u0001\u00001vs\n?\u0000\f\u0000\u000b\n\u000b~\u0001\u00001vs\nk\u0015\n;\n(13)\nwhich is the main result of this paper. From eq. (13), the\ncurrent-velocity characteristic in the absence of both the\npinning and the external \feld ( @\u001b=@q =0) is\n_q=1 +\u000b\f\n1 +\u000b2vs\n?\u0000\f\u0000\u000b\n1 +\u000b2\u0001~\u0001\u00001vs\nk\n'vs\n?+ (\f\u0000\u000b)\u0001~\u0001\u00001vs\nk; (14)\nwhere the fact \u000b;\f\u001c1 is used in the last step. If we\nneglect the term coming from vs\nk, the current-velocity\nrelation becomes almost independent of \u000band\fin\nsharp contrast to eq. (11). This is similar to the univer-\nsal current-velocity relation in the case of skyrmion,12)\nwhere the solid angle is \fnite and also the transverse\nmotion to the current occurs. Note that vs\nkslightly con-\ntributes to the motion when \u000b6=\f, while it does\nnot in the absence of BL. Even more dramatic is the\ncritical current density in the presence of the pinning\n2J. Phys. Soc. Jpn. LETTERS\n30\n20\n101525\n520\n1015\n5\n3.0\n2.0\n1.01.52.5\n0.50.6\n0.4\n0.20.30.5\n0.10.4 0.2 1.0 0.8 0.6 0.4 0.2 1.0 0.8 0.6\n4000 2000 10000 8000 6000 4000 2000 10000 8000 6000qq\nqq\nt tt tw/o BL\nw/ BLs0.429\n0.707Pinning\nq(a)\n(c) (d)(b)\nFig. 2. The wall displacement qas a fucntion of tfor the DWs\nwithout BL and with BLs. (a) vs\n?= 22 :0. The inset shows the\npinning force Fpin. (b) vs\n?= 21 :0. (c) vs\n?= 0:0043. (d) vs\n?=\n0:0042.\n(@\u001b=@q =Fpin). When we apply only the current per-\npendicular to the DW, i.e., vs\nk= 0, putting _ q= 0 in\neq. (13) determines the threshold current density as\n(vs\n?)c=\r\u0001\n2M\u000b\n1 +\u000b\fFpin; (15)\nwhich is much reduced compared with eq. (12) by the\nfactor of\u000b\f\n1+\u000b\f\u001c1. Note that ( vs\n?)cin eq. (15) is even\nsmaller than the case of skyrmion12)by the factor of\n\u000b. Similarly, the critical current density of the motion\ndriven byvs\nkis given by\n\u0010\nvs\nk\u0011\nc=\r~\u0001\n2M\u000b\nj\f\u0000\u000bjFpin; (16)\nwhich can also be smaller than eq. (12).\nNext we look at the numerical solutions of q(t) driven\nby the current vs\n?perpendicular to the wall under the\npinning force. We assume the following pinning force:\n(\r\u0001=2M)Fpin(q) =v\u0003(q=\u0001) exp\u0002\n\u0000(q=\u0001)2\u0003\n(see the in-\nset of Fig. 2(a)). We employ the unit of \u0001 = v\u0003=\n1 and the parameters ( \u000b;\f) are \fxed at ( \u000b;\f) =\n(0:01;0:02). Here, we compare two DWs without BL\nand with BLs. The maximum value of the pinning force\n(\r\u0001=2M)Fpin\nmax= 0:429 determines the threshold current\ndensity (vs\n?)cas (vs\n?)c= 21:4 and (vs\n?)c= 0:00429 in the\nabsence of BL and in the presence of many BLs, respec-\ntively. In Fig. 2(a), both DWs overcome the pinning at\nthe current density vs\n?= 22:0, although the velocity of\nthe DW without BL is suppressed in the pinning poten-\ntial. At the current density vs\n?= 21:0 below the threshold\nvalue in the absence of BL, the DW without BL is pinned,\nwhile that with BLs still moves easily (Fig. 2(b)). The\nvelocity suppression in the presence of BLs is observed\nat much smaller current density vs\n?= 0:0043 (Fig. 2(c)),\nand \fnally it stops at vs\n?= 0:0042 (Fig. 2(d)).\nAll the discussion above relies on the assumption thatthe wall is straight and  rotates uniformly. When the\nbending of the DW and non-uniform distribution of BLs\nare taken into account, the average velocity and the\nthreshold current density take the values between two\ncases without BL and with many BLs. The situation\nchanges when the DW forms closed loop, i.e., the do-\nmain forms a bubble. The bubble with many BLs and\nlargejNskjis called hard bubble because the repulsive\ninteraction between the BLs makes it hard to collapse\nthe bubble.2)At the beginning of the motion, the BLs\nmove along the DW, which results in the tiny critical cur-\nrent. In the steady state, however, the BLs accumulate\nin one side of the bubble.13,14)Then, the con\fguration\nof the BLs is static and the Thiele equation is justi\fed\nas long as the force is slowly varying within the size of\nthe bubble. The critical current density ( vs)cis given by\n(vs)c/Fpin=Nsk(Nsk(\u001d1): the skyrmion number of\nthe hard bubble), and is reduced by the factor of Nsk\ncompared with the skyrmion with Nsk=\u00061.\nIn conclusion, we have studied the current-induced\ndynamics of the DW with many BLs. The \fnite _ in\nthe steady motion activated by BLs sliding drastically\nchanges the dynamics, which has already been reported\nin the \feld-driven case. In contrast to the \feld-driven\ncase, where the mobility is suppressed by introducing\nBLs, that in the current-driven motion is not necessarily\nsuppressed. Instead, the current-velocity relation shows\nuniversal behavior independent of the damping strength\n\u000band non-adiabaticity \f. Furthermore, the threshold\ncurrent density in the presence of impurities is tiny even\ncompared with that of skyrmion motion by the factor of\n\u000b. These \fndings will stimulate the development of the\nracetrack memory based on the DW with many BLs.\nAcknowledgments We thank W. Koshibae for useful discus-\nsion. This work is supported by Grant-in-Aids for Scienti\fc Re-\nsearch (S) (No. 24224009) from the Ministry of Education, Cul-\nture, Sports, Science and Technology of Japan. J. I. was supported\nby Grant-in-Aids for JSPS Fellows (No. 2610547).\n1) A. Hubert and R. Sch afer, Magnetic Domains: The Analysis\nof Magnetic Microstructures (Springer-Verlag, Berlin, 1998).\n2) A. P. Malozemo\u000b and J.C. Slonczewski, Magnetic Domain\nWalls in Bubble Materials (Academic Press, New York, 1979).\n3) J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1{L7 (1996).\n4) L. Berger, Phys. Rev. B 54, 9353{9358 (1996).\n5) S. M uhlbauer et al., Science 323, 915 (2009).\n6) X. Z. Yu et al., Nature 465, 901 (2010).\n7) A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973).\n8) K. Everschor et al., Phys. Rev. B 86, 054432 (2012).\n9) J. C. Slonczewski, J. Appl. Phys. 45, 2705 (1974).\n10) A. P. Malozemo\u000b and J. C. Slonczewski, Phys. Rev. Lett. 29,\n952 (1972).\n11) G. Tatara et al., J. Phys. Soc. Japan 75, 64708 (2006).\n12) J. Iwasaki, M. Mochizuki and N. Nagaosa, Nat. Commun. 4,\n1463 (2013).\n13) G. P. Vella-Coleiro, A. Rosencwaig and W. J. Tabor, Phys.\nRev. Lett. 29, 949 (1972)\n14) A. A. Thiele, F. B. Hagedorn and G. P. Vella-Coleiro, Phys.\n3J. Phys. Soc. Jpn. LETTERS\nRev. B 8, 241 (1973).\n4"    },    {        "title": "2308.05955v2.Dynamical_Majorana_Ising_spin_response_in_a_topological_superconductor_magnet_hybrid_by_microwave_irradiation.pdf",        "content": "Dynamical Majorana Ising spin response in a topological superconductor-magnet\nhybrid by microwave irradiation\nYuya Ominato,1, 2Ai Yamakage,3and Mamoru Matsuo1, 4, 5, 6\n1Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China.\n2Waseda Institute for Advanced Study, Waseda University, Shinjuku, Tokyo 169-8050, Japan.\n3Department of Physics, Nagoya University, Nagoya 464-8602, Japan\n4CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing 100190, China\n5Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan\n6RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan\n(Dated: March 20, 2024)\nWe study a dynamical spin response of surface Majorana modes in a topological superconductor-\nmagnet hybrid under microwave irradiation. We find a method to toggle between dissipative and\nnon-dissipative Majorana Ising spin dynamics by adjusting the external magnetic field angle and\nthe microwave frequency. This reflects the topological nature of the Majorana modes, enhancing\nthe Gilbert damping of the magnet, thereby, providing a detection method for the Majorana Ising\nspins. Our findings illuminate a magnetic probe for Majorana modes, paving the path to innovative\nspin devices.\nIntroduction.— The quest for Majoranas within matter\nstands as one of the principal challenges in the study of\ncondensed matter physics, more so in the field of quan-\ntum many-body systems [1]. The self-conjugate nature\nof Majoranas leads to peculiar electrical characteristics\nthat have been the subject of intensive research, both\ntheoretical and experimental [2]. In contrast, the focus of\nthis paper lies on the magnetic properties of Majoranas,\nspecifically the Majorana Ising spin [3–8]. A distinctive\ncharacteristic of Majorana modes, appearing as a surface\nstate in topological superconductors (TSC), is its exceed-\ningly strong anisotropy, which makes it behave as an Ising\nspin. In particular, this paper proposes a method to ex-\nplore the dynamical response of the Majorana Ising spin\nthrough the exchange interaction at the magnetic inter-\nface, achieved by coupling the TSC to a ferromagnet with\nferromagnetic resonance (FMR) (as shown in Fig.1 (a)).\nFMR modulation in a magnetic hybrid system has at-\ntracted much attention as a method to analyze spin ex-\ncitations in thin-film materials attached to magnetic ma-\nterials [9, 10]. Irradiating a magnetic material with mi-\ncrowaves induces dynamics of localized spin in magnetic\nmaterials, which can excite spins in adjacent thin-film\nmaterials via the magnetic proximity effect. This setup\nis called spin pumping, and has been studied intensively\nin the field of spintronics as a method of injecting spins\nthrough interfaces [11, 12]. Recent studies have theoret-\nically proposed that spin excitation can be characterized\nby FMR in hybrid systems of superconducting thin films\nand magnetic materials [13–18]. Therefore, it is expected\nto be possible to analyze the dynamics of surface Majo-\nrana Ising spins using FMR in hybrid systems.\nIn this work, we consider a TSC-ferromagnetic insula-\ntor (FI) hybrid system as shown in Fig. 1 (a). The FMR\nis induced by microwave irradiation on the FI. At the\ninterface between the TSC and the FI, the surface Ma-\n(b)\n(c)(a)\nFI~~\n~~Microwave\nϑS\nY, yX\nxZhdcHex\nTSC\n(d)\nhdchdc+δhα+δα\nHz\nFIG. 1. (a) The TSC-FI hybrid schematic reveals how,\nunder resonance frequency microwave irradiation, localized\nspins commence precessional motion, consequently initiating\nthe dynamical Majorana Ising spin response at the TSC inter-\nface. (b) In the TSC context, the liaison between a spin-up\nelectron and a spin-down hole with the surrounding sea of\nspin-triplet Cooper pairs drastically modulate their proper-\nties; notably, a spin-down hole can engage with a spin-triplet\nCooper pair, thereby inheriting a negative charge. (c) No-\ntably, spin-triplet Cooper pairs amass around holes and scat-\nter around electrons, thereby eroding the rigid distinction be-\ntween the two. (d) The interplay between the Majorana mode\nand the localized spin manipulates the FMR spectrum, trig-\ngering a frequency shift and linewidth broadening.\njorana modes interact with the localized spins in the FI.\nAs a result, the localized spin dynamics leads to the dy-\nnamical Majorana Ising spin response (DMISR), which\nmeans the Majorana Ising spin density is dynamically in-\nduced, and it is possible to toggle between dissipative and\nnon-dissipative Majorana Ising spin dynamics by adjust-\ning the external magnetic field angle and the microwave\nfrequency. Furthermore, the modulation of the localizedarXiv:2308.05955v2  [cond-mat.mes-hall]  19 Mar 20242\nspin dynamics due to the interface interaction leads to a\nfrequency shift and a linewidth broadening, which reflect\nthe properties of the Majorana Ising spin dynamics. This\nwork proposes a setup for detecting Majorana modes and\npaves the way for the development of quantum comput-\ning and spin devices using Majoranas.\nModel.— We introduce a model Hamiltonian Hconsist-\ning of three terms\nH=HM+HFI+Hex. (1)\nThe first, second, and third terms respectively describe\nthe surface Majorana modes on the TSC surface, the bulk\nFI, and the proximity-induced exchange coupling. Our\nfocus is on energy regions significantly smaller than the\nbulk superconducting gap. This focus allows the spin ex-\ncitation in the TSC to be well described using the surface\nMajorana modes. The subsequent paragraphs provide\ndetailed explanations of each of these three terms.\nThe first terms HMdescribes the surface Majorana\nmodes,\nHM=1\n2Z\ndrψT(r)\u0010\nℏvˆkyσx−ℏvˆkxσy\u0011\nψ(r),(2)\nwhere r= (x, y),ˆk= (−i∂x,−i∂y),vis a constant\nvelocity, and σ= (σx, σy, σz) are the Pauli matrices.\nThe two component Majorana field operator is given by\nψ(r) = ( ψ→(r), ψ←(r))T, with the spin quantization\naxis along the xaxis. The Majorana field operators sat-\nisfy the Majorana condition ψσ(r) =ψ†\nσ(r) and the an-\nticommutation relation {ψσ(r), ψσ′(r)}=δσσ′δ(r−r′)\nwhere σ, σ′=→,←. We can derive HMby using surface-\nlocalized solutions of the BdG equation based on the bulk\nTSC Hamiltonian. The details of the derivation of HM\nare provided in the Supplemental Material [19].\nA notable feature of the surface Majorana modes is\nthat the spin density is Ising like, which we call the Majo-\nrana Ising spin [3–8]. The feature follows naturally from\nthe Majorana condition and the anticommutation rela-\ntion. The Majorana Ising spin density operator is given\nbys(r) := ψT(r)(σ/2)ψ(r) = (0 ,0,−iψ→(r)ψ←(r))\n(See the Supplemental Material for details [19]). The\nanisotropy of the Majorana Ising spin is the hallmark of\nthe surface Majorana modes on the TSC surface.\nThe second term HFIdescries the bulk FI and is given\nby the ferromagnetic Heisenberg model,\nHFI=− JX\n⟨n,m⟩Sn·Sm−ℏγhdcX\nnSZ\nn, (3)\nwhere J>0 is the exchange coupling constant, Snis the\nlocalized spin at site n,⟨n, m⟩means summation for near-\nest neighbors, γis the electron gyromagnetic ratio, and\nhdcis the static external magnetic field. We consider the\nspin dynamics of the localized spin under microwave irra-\ndiation, applying the spin-wave approximation. This al-\nlows the spin excitation to be described by a free bosonic\noperator, known as a magnon [20].The third term Hexrepresents the proximity exchange\ncoupling at the interface between the TSC and the FI,\nHex=−Z\ndrX\nnJ(r,rn)s(r)·Sn=HZ+HT,(4)\nHZ=−cosϑZ\ndrX\nnJ(r,rn)sz(r)SZ\nn, (5)\nHT=−sinϑZ\ndrX\nnJ(r,rn)sz(r)SX\nn, (6)\nwhere the angle ϑis shown in Fig. 1 (a). HZis the\ncoupling along the precession axis and HTis the coupling\nperpendicular to the precession axis. In our setup, HZ\nleads to gap opening of the energy spectrum of the surface\nMajorana modes and HTgives the DMISR under the\nmicrowave irradiation.\nDynamical Majorana Ising spin response.— We con-\nsider the microwave irradiation on the FI. The coupling\nbetween the localized spins and the microwave is given\nby\nV(t) =−ℏγhacX\nn\u0000\nSX\nncosωt−SY\nnsinωt\u0001\n,(7)\nwhere hacis the microwave amplitude, and ωis the mi-\ncrowave frequency. The microwave irradiation leads to\nthe precessional motion of the localized spin. When the\nfrequency of the precessional motion and the microwave\ncoincide, the FMR occurs. The FMR leads to the DMISR\nvia the exchange interaction. The DMISR is character-\nized by the dynamic spin susceptibility of the Majorana\nmodes, ˜ χzz(q, ω), defined as\n˜χzz(q, ω) :=Z\ndre−iq·rZ\ndtei(ω+i0)tχzz(r, t),(8)\nwhere χzz(r, t) := −(L2/iℏ)θ(t)⟨[sz(r, t), sz(0,0)]⟩\nwith the interface area L2and the spin den-\nsity operator in the interaction picture, sz(r, t) =\nei(HM+HZ)t/ℏsz(r)e−i(HM+HZ)t/ℏ. For the exchange cou-\npling, we consider configuration average and assume\n⟨P\nnJ(r,rn)⟩ave=J1, which means that HZis treated\nas a uniform Zeeman like interaction and the interface\nis specular [21]. Using eigenstates of Eq. (2) and after a\nstraightforward calculation, the uniform spin susceptibil-\nity is given by\n˜χzz(0, ω)\n=−X\nk,λ|⟨k, λ|σz|k,−λ⟩|2f(Ek,λ)−f(Ek,−λ)\n2Ek,λ+ℏω+i0,\n→ −Z\ndED (E)E2−M2\n2E2f(E)−f(−E)\n2E+ℏω+i0, (9)\nwhere |k, λ⟩is an eigenstate of HMwith eigenenergy\nEk,λ=λp\n(ℏvk)2+M2, (λ=±).M=J1Scosϑis\nthe Majorana gap, f(E) = 1 /(eE/kBT+ 1) is the Fermi3\ndistribution function, and D(E) is the density of states\ngiven by\nD(E) =L2\n2π(ℏv)2|E|θ(|E| − |M|), (10)\nwith the Heaviside step function θ(x). It is important to\nnote that the behavior of the uniform spin susceptibil-\nity is determined by the interband contribution, which is\nproportional to the Fermi distribution function, i.e., the\ncontribution of the occupied states. This mechanism is\nsimilar to the Van Vleck paramagnetism [22]. The con-\ntribution of the occupied states often plays a crucial role\nin topological responses [23].\nReplacing the localized spin operators with their statis-\ntical average values, we find the induced Majorana Ising\nspin density, to the first order of J1S, is given by\nZ\ndr⟨sz(r, t)⟩= ˜χzz\n0(0,0)J1Scosϑ\n+ Re[˜ χzz\n0(0, ω)]hac\nαhdcJ1Ssinϑsinωt, (11)\nwhere ˜ χzz\n0(0,0) is the spin susceptibility for M= 0. The\nfirst term originates from HZand gives a static spin den-\nsity, while the second term originates from HTand gives\na dynamic spin density. Figure 2 shows the induced Ising\nspin density as a function of time at several angles. As\nshown in Eq. (11), the Ising spin density consists of the\nstatic and dynamic components. The dynamic compo-\nnent is induced by the precessional motion of the local-\nized spin, which means one can induce the DMISR using\nthe dynamics of the localized spin.\nThe inset in Fig. 2 shows Im˜ χzz(0, ω) as a function of\nϑat a fixed frequency. When the frequency ℏωis smaller\nthan the Majorana gap, Im˜ χzz(0, ω) is zero. Once the\nfrequency overcomes the Majorana gap, Im˜ χzz(0, ω) be-\ncomes finite. The implications of these behaviors are that\nif the magnon energy is smaller than the Majorana gap,\nthere is no energy dissipation due to the DMISR. How-\never, once the magnon energy exceeds the Majorana gap,\nfinite energy dissipation associated with the DMISR oc-\ncurs at the surface of the TSC. Therefore, one can toggle\nbetween dissipative and non-dissipative Majorana Ising\nspin dynamics by adjusting the precession axis angle and\nthe microwave frequency.\nFMR modulation.— The retarded component of the\nmagnon Green’s function is given by GR(rn, t) =\n−(i/ℏ)θ(t)⟨[S+\nn(t), S−\n0(0)]⟩with the interaction picture\nS±\nn(t) =eiHFIt/ℏS±\nne−iHFIt/ℏ. The FMR signal is char-\nacterized by the spectral function defined as\nA(q, ω) :=−1\nπIm\"X\nne−iq·rnZ\ndtei(ω+i0)tGR(rn, t)#\n.\n(12)\nSSImχzz(0, ω) ˜⟨s   z⟩\n2\n1ωtϑ\nFInon-dissipativenon-dissipativedissipativedissipativeTSC\nFITSC000.00.51.0\nπ/4\nπ/2\n0 π/4 π/20\nϑ2π\nπFIG. 2. The induced Ising spin density, with a unit\n˜χzz\n0(0,0)J1S, is presented as a function of ωtandϑ. The\nfrequency and temperature are set to ℏω/J1S= 1.5 and\nkBT/J 1S= 0.1, respectively. The coefficient, hac/αhdc, is\nset to 0 .3. The static Majorana Ising spin density arises\nfrom HZ. When the precession axis deviates from the di-\nrection perpendicular to the interface, the precessional mo-\ntion of the localized spins results in the dynamical Majorana\nIsing spin response (DMISR). Energy dissipation due to the\nDMISR is zero for small angles ϑas the Majorana gap ex-\nceeds the magnon energy. However, once the magnon energy\novercomes the Majorana gap, the energy dissipation becomes\nfinite. Therefore, one can toggle between dissipative and non-\ndissipative DMISR by adjusting ϑ.\nFor uniform external force, the spectral function is given\nby\nA(0, ω) =2S\nℏ1\nπ(α+δα)ω\n[ω−γ(hdc+δh)]2+ [(α+δα)ω]2.\n(13)\nThe peak position and width of the FMR signal is given\nbyhdc+δhandα+δα, respectively. hdcandαcorre-\nspond to the peak position and the linewidth of the FMR\nsignal of the FI alone. δhandδαare the FMR modu-\nlations due to the exchange interaction HT. We treat\nHM+HFI+HZas an unperturbed Hamiltonian and HT\nas a perturbation. In this work, we assume the specular\ninterface, where the coupling J(r,rn) is approximated\nasDP\nn,n′J(r,rn)J(r′,rn′)E\nave=J2\n1. The dynamics\nof the localized spins in the FI is modulated due to the\ninteraction between the localized spins and the Majo-\nrana Ising spins. In our setup, the peak position and the\nlinewidth of the FMR signal are modulated and the FMR4\nmodulation is given by\nδh= sin2ϑSJ2\n1\n2NγℏRe˜χzz(0, ω), (14)\nδα= sin2ϑSJ2\n1\n2NℏωIm˜χzz(0, ω), (15)\nwhere Nis the total number of sites in the FI. These for-\nmulas were derived in the study of the FMR in magnetic\nmultilayer systems including superconductors. One can\nextract the spin property of the Majorana mode from the\ndata on δhandδα. Because of the Ising spin anisotropy,\nthe FMR modulation exhibits strong anisotropy, where\nthe FMR modulation is proportional to sin2ϑ.\nFigure 3 shows the FMR modulations (a) δαand (b)\nδh. The FMR modulation at a fixed frequency increases\nwith angle ϑand reaches a maximum at π/2, as can be\nread from Eqs. (14) and (15). When the angle ϑis fixed\nand the frequency ωis increased, δαbecomes finite above\na certain frequency at which the energy of the magnon\ncoincides with the Majorana gap. When ϑ < π/ 2 and\nℏω≈2M,δαlinearly increases as a function of ωjust\nabove the Majorana gap. The localized spin damping is\nenhanced when the magnon energy exceeds the Majorana\ngap. At ϑ=π/2 and ω≈0, the Majorana gap vanishes\nandδαis proportional to ω/T. In the high frequency\nregion ℏω/J 1S≫1,δαconverges to its upper threshold.\nThe frequency shift δhis almost independent of ωand\nhas a finite value even in the Majorana gap. This behav-\nior is analogous to the interband contribution to the spin\nsusceptibility in strongly spin-orbit coupled band insula-\ntors, and is due to the fact that the effective Hamiltonian\nof the Majorana modes includes spin operators. It is im-\nportant to emphasize that although the Majorana modes\nhave spin degrees of freedom, only the zcomponent of the\nspin density operator is well defined. This is a hallmark\nof Majorana modes, which differs significantly from elec-\ntrons in ordinary solids. Note that δhis proportional to\nthe energy cutoff, which is introduced to converge energy\nintegral for Re˜ χzz(0, ω). The energy cutoff corresponds\nto the bulk superconducting gap, which is estimated as\n∆∼0.1[meV] ( ∼1[K]). Therefore, our results are ap-\nplicable in the frequency region below ℏω∼0.1[meV]\n(∼30[GHz]). In addition, we assume that Majorana gap\nis estimated to be J1S∼0.01[meV] ( ∼0.1[K]).\nDiscussion.— Comparing the present results with spin\npumping (SP) in a conventional metal-ferromagnet hy-\nbrid, the qualitative behaviors are quite different. In con-\nventional metals, spin accumulation occurs due to FMR.\nIn contrast, in the present system, no corresponding spin\naccumulation occurs due to the Ising anisotropy. Also, in\nthe present calculations, the proximity-induced exchange\ncoupling is assumed to be an isotropic Heisenberg-like\ncoupling. However, in general, the interface interaction\ncan also be anisotropic. Even in such a case, it is no qual-\nitative change in the case of ordinary metals, although a\n0.00.5\n(a) (b)\nϑℏω/J1S 0\nπ/4\nπ/2024\nϑℏω/J1S 0\nπ/4\nπ/2024δ  α δ  h10\n0FIG. 3. The temperature is set to kBT/J 1S= 0.1. (a)\nThe damping modulation δαonly becomes finite when the\nmagnon energy exceeds the Majorana gap; otherwise, it van-\nishes. This behavior corresponds to the energy dissipation of\nthe Majorana Ising spin. (b) The peak shift is finite, except\nforϑ= 0, and is almost independent of ω. This behavior\nresembles the spin response observed in strongly spin-orbit\ncoupled band insulators, where the interband contribution to\nspin susceptibility results in a finite spin response, even within\nthe energy gap.\ncorrection term due to anisotropy is added [24]. There-\nfore, the Ising anisotropy discussed in the present work\nis a property unique to the Majorana modes and can\ncharacterize the Majorana excitations.\nLet us comment on the universal nature of the toggling\nbetween non-dissipative and dissipative dynamical spin\nresponses observed in our study. Indeed, such toggling\nbecomes universally feasible when the microwave fre-\nquency and the energy gap are comparable, and when the\nHamiltonian and spin operators are non-commutative,\nindicating that spin is not a conserved quantity. The\nnon-commutativity can be attributed to the presence of\nspin-orbit couplings [25–27], and spin-triplet pair corre-\nlations [28].\nMicrowave irradiation leads to heating within the FI,\nso that thermally excited magnons due to the heating\ncould influence the DMISR. Phenomena resulting from\nthe heating, which can affect interface spin dynamics, in-\nclude the spin Seebeck effect (SSE) [29], where a spin\ncurrent is generated at the interface due to a tempera-\nture difference. In hybrid systems of normal metal and\nFI, methods to separate the inverse spin Hall voltage due\nto SP from other signals caused by heating have been\nwell studied [30]. Especially, it has been theoretically\nproposed that SP and SSE signals can be separated us-\ning a spin current noise measurement [24]. Moreover, SP\ncoherently excites specific modes, which qualitatively dif-\nfers from SSE induced by thermally excited magnons [14].\nTherefore, even if heating occurs in the FI in our setup,\nthe properties of Majorana Ising spins are expected to\nbe captured. Details of the heating effect on the DMISR\nwill be examined in the near future.\nWe also mention the experimental feasibility of our the-\noretical proposals. As we have already explained, the\nFMR modulation is a very sensitive spin probe. Indeed,\nthe FMR modulation by surface states of 3D topological5\ninsulators [31] and graphene [32–36] has been reported\nexperimentally. Therefore, we expect that the enhanced\nGilbert damping due to Majorana Ising spin can be ob-\nservable in our setup when the thickness of the ferromag-\nnetic insulator is sufficiently thin.\nFinally, it is pertinent to mention the potential candi-\ndate materials where surface Majorana Ising spins could\nbe detectable. Notably, UTe 2[37], Cu xBi2Se3[38, 39],\nSrxBi2Se3and Nb xBi2Se3[40] are reported to be in a p-\nwave superconducting state and theoretically can host\nsurface Majorana Ising spins. Recent NMR measure-\nments indicate that UTe 2could be a bulk p-wave su-\nperconductor in the Balian-Werthamer state [41], which\nhosts the surface Majorana Ising spins with the per-\npendicular Ising anisotropy, as considered in this work.\nAxBi2Se3(A= Cu, Sr, Nb) is considered to possess in-\nplane Ising anisotropy [8], differing from the perpendic-\nular Ising anisotropy explored in this work. Therefore,\nwe expect that it exhibits anisotropy different from that\ndemonstrated in this work.\nConclusion.— We present herein a study of the spin\ndynamics in a topological superconductor (TSC)-magnet\nhybrid. Ferromagnetic resonance under microwave irra-\ndiation leads to the dynamically induced Majorana Ising\nspin density on the TSC surface. One can toggle between\ndissipative and non-dissipative Majorana Ising spin dy-\nnamics by adjusting the external magnetic field angle and\nthe microwave frequency. Therefore, our setup provides\na platform to detect and control Majorana excitations.\nWe expect that our results provide insights toward the\ndevelopment of future quantum computing and spintron-\nics devices using Majorana excitations.\nAcknowledgments.— The authors are grateful to R.\nShindou for valuable discussions. This work is partially\nsupported by the Priority Program of Chinese Academy\nof Sciences, Grant No. XDB28000000. We acknowl-\nedge JSPS KAKENHI for Grants (Nos. JP20K03835,\nJP21H01800, JP21H04565, and JP23H01839).\nSUPPLEMENTAL MATERIAL\nSurface Majorana modes\nIn this section, we describe the procedure for deriv-\ning the effective Hamiltonian of the surface Majorana\nmodes. We start with the bulk Hamiltonian of a three-\ndimensional topological superconductor. Based on the\nbulk Hamiltonian, we solve the BdG equation to demon-\nstrate the existence of a surface-localized solution. Us-\ning this solution, we expand the field operator and show\nthat it satisfies the Majorana condition when the bulk\nexcitations are neglected. As a result, on energy scales\nmuch smaller than the bulk superconducting gap, the\nlow-energy excitations are described by surface-localized\nMajorana modes. The above procedure is explained inmore detail in the following. Note that we use rfor three-\ndimensional coordinates and r∥for two-dimensional ones\nin the Supplemental Material.\nWe start with the mean-field Hamiltonian given by\nHSC=1\n2Z\ndrΨ†\nBdG(r)HBdGΨBdG(r), (16)\nwithr= (x, y, z ). We consider the Balian-Werthamer\n(BW) state, in which the pair potential is given by\n∆ˆk=∆\nkF\u0010\nˆk·σ\u0011\niσywith the bulk superconducting gap\n∆. Here, we do not discuss the microscopic origin of the\npair correlation leading to the BW state. As a result, the\nBdG Hamiltonian HBdGis given by\nHBdG=\nεˆk−EF 0 −∆\nkFˆk−∆\nkFˆkx\n0 εˆk−EF∆\nkFˆkx∆\nkFˆk+\n−∆\nkFˆk+∆\nkFˆkx−εˆk+EF 0\n∆\nkFˆkx∆\nkFˆk− 0 −εˆk+EF\n,\n(17)\nwith ˆk±=ˆky±iˆkz,ˆk=−i∇, and εˆk=ℏ2ˆk2\n2m. The four\ncomponent Nambu spinor ΨBdG(r) is given by\nΨBdG(r) :=\nΨ→(r)\nΨ←(r)\nΨ†\n→(r)\nΨ†\n←(r)\n, (18)\nwith the spin quantization axis along the xaxis. The\nmatrices of the spin operators are represented as\nσx=\u00121 0\n0−1\u0013\n, (19)\nσy=\u0012\n0 1\n1 0\u0013\n, (20)\nσz=\u00120−i\ni0\u0013\n. (21)\nThe fermion field operators satisfy the anticommutation\nrelations\n{Ψσ(r),Ψσ′(r′)}= 0, (22)\n{Ψσ(r),Ψ†\nσ′(r′)}=δσσ′δ(r−r′), (23)\nwith the spin indices σ, σ′=→,←.\nTo diagonalize the BdG Hamiltonian, we solve the BdG\nequation given by\nHBdGΦ(r) =EΦ(r). (24)\nWe assume that a solution is written as\nΦ(r) =eik∥·r∥f(z)\nu→\nu←\nv→\nv←\n, (25)6\nwithk∥= (kx, ky) and r∥= (x, y). If we set the four\ncomponents vector to satisfy the following equation (Ma-\njorana condition)\n\n0 0 1 0\n0 0 0 1\n1 0 0 0\n0 1 0 0\n\nu→\nu←\nv→\nv←\n=±\nu→\nu←\nv→\nv←\n, (26)\nwe can obtain a surface-localized solution. If we take a\npositive (negative) sign, we obtain a solution localized\non the top surface (bottom surface). As we will consider\nsolutions localized on the bottom surface below, we take\na negative sign. Finally, we obtain the normalized eigen-\nvectors of the BdG equation given by\nΦλ,k∥(r) =eik∥·r∥\n√\nL2fk∥(z)uλ,k∥, (27)\nwith\nfk∥(z) =Nk∥sin(k⊥z)e−κz, (28)\nNk∥=s\n4κ(k2\n⊥+κ2)\nk2\n⊥, (29)\nκ=m∆\nℏ2kF, (30)\nk⊥=q\nk2\nF−k2\n∥−κ2, (31)\nand\nu+,k∥=\nu+,→k∥\nu+,←k∥\nv+,→k∥\nv+,←k∥\n=1√\n2\nsinϕk∥+π/2\n2\n−cosϕk∥+π/2\n2\n−sinϕk∥+π/2\n2\ncosϕk∥+π/2\n2\n,(32)\nu−,k∥=\nu−,→k∥\nu−,←k∥\nv−,→k∥\nv−,←k∥\n=1√\n2\n−cosϕk∥+π/2\n2\n−sinϕk∥+π/2\n2\ncosϕk∥+π/2\n2\nsinϕk∥+π/2\n2\n.(33)\nThe eigenenergy is given by Eλ,k∥=λ∆k∥/kF. We can\nshow that the eigenvectors satisfy\nu−,−k∥=u+,k∥. (34)\nConsequently, the field operator is expanded as\nΨBdG(r) =X\nk∥\u0012\nγk∥eik∥·r∥\n√\nL2+γ†\nk∥e−ik∥·r∥\n√\nL2\u0013\n×fk∥(z)u+,k∥+ (bulk modes) ,(35)\nwhere γk∥(γ†\nk∥) is the quasiparticle creation (annihila-\ntion) operator with the eigenenergy E+,k∥. Substitutingthe above expression into Eq. (16) with omission of bulk\nmodes and performing the integration in the z-direction,\nwe obtain the effective Hamiltonian for the surface states\nHM=1\n2Z\ndr∥ψT(r∥)\u0010\nℏvˆkyσx−ℏvˆkxσy\u0011\nψ(r∥),(36)\nwhere v= ∆/ℏkFand we introduced the two component\nMajorana field operator\nψ(r∥) =\u0012ψ→(r∥)\nψ←(r∥)\u0013\n, (37)\nsatisfying the Majorana condition\nψσ(r∥) =ψ†\nσ(r∥), (38)\nand the anticommutation relation\nn\nψσ(r∥), ψσ′(r′\n∥)o\n=δσσ′δ(r∥−r′\n∥). (39)\nThe spin density operator of the Majorana mode is\ngiven by\ns(r∥) =ψ†(r∥)σ\n2ψ(r∥). (40)\nThexcomponent is given by\nsx(r∥) =\u0000\nψ†\n→(r∥), ψ†\n←(r∥)\u0001\u00121/2 0\n0−1/2\u0013\u0012ψ→(r∥)\nψ←(r∥)\u0013\n=1\n2\u0002\nψ†\n→(r∥)ψ→(r∥)−ψ†\n←(r∥)ψ←(r∥)\u0003\n=1\n2\u0002\nψ2\n→(r∥)−ψ2\n←(r∥)\u0003\n= 0. (41)\nIn a similar manner, the yandzcomponents are given\nby\nsy(r∥) =\u0000\nψ†\n→(r∥), ψ†\n←(r∥)\u0001\u00120 1/2\n1/2 0\u0013\u0012ψ→(r∥)\nψ←(r∥)\u0013\n=1\n2\u0002\nψ†\n→(r∥)ψ←(r∥) +ψ†\n←(r∥)ψ→(r∥)\u0003\n=1\n2\b\nψ→(r∥), ψ←(r∥)\t\n= 0, (42)\nand\nsz(r∥) =\u0000\nψ†\n→(r∥), ψ†\n←(r∥)\u0001\u00120−i/2\ni/2 0\u0013\u0012ψ→(r∥)\nψ←(r∥)\u0013\n=−i\n2\u0000\nψ†\n→(r∥)ψ←(r∥)−ψ†\n←(r∥)ψ→(r∥)\u0001\n=−iψ→(r∥)ψ←(r∥), (43)\nrespectively. As a result, the spin density operator is\ngiven by\ns(r∥) =\u0000\n0,0,−iψ→(r∥)ψ←(r∥)\u0001\n. (44)\nOne can see that the spin density of the Majorana mode\nis Ising like.7\nMajorana Ising spin dynamics\nIn this section, we calculate the Ising spin density in-\nduced on the TSC surface by the proximity coupling Hex.\nHexconsists of two terms, HZandHT.HZleads to the\nstatic spin density and HTleads to the dynamic spin\ndensity. First, we calculate the static spin density. Next,\nwe calculate the dynamic spin density.\nThe total spin density operator is given by\nsz\ntot=Z\ndr∥sz(r∥). (45)\nThe statistical average of the static spin density is calcu-\nlated as\n⟨sz\ntot⟩=−X\nk∥M\n2Ek∥\u0002\nf(Ek∥)−f(−Ek∥)\u0003\n→ −\u0012L\n2πℏv\u00132Z∆\nMEdEZ2π\n0dϕM\n2E[f(E)−f(−E)]\n=−Z∆\n0dED (E)f(E)−f(−E)\n2EM. (46)\nAt the zero temperature limit T→0, the static spin\ndensity is given by\n⟨sz\ntot⟩=1\n2L2\n2π(ℏv)2(∆−M)M≈˜χzz\n0(0,0)M, (47)\nwhere ˜ χzz\n0(0,0) = D(∆)/2 and we used ∆ ≫M.\nThe dynamic spin density is given by the perturbative\nforce\nHT(t) =Z\ndr∥sz(r∥)F(r∥, t), (48)\nwhere F(r∥, t) is given by\nF(r∥, t) =−sinϑX\nnJ(r∥,rn)\nSX\nn(t)\u000b\n≈ −sinϑJ1Sγhacp\n(ω−γhdc)2+α2ω2cosωt\n=:Fcosωt. (49)\nThe time dependent statistical average of the Ising spin\ndensity, to the first order of J1S, is given by\nZ\ndr∥\nsz(r∥, t)\u000b\n=Z\ndr∥Z\ndr′\n∥Z\ndt′χzz(r∥−r′\n∥, t′)F(r′\n∥, t−t′)\n= Re\u0002\n˜χzz(0, ω)Fe−iωt\u0003\n≈Re[˜χzz\n0(0, ω)]Fcosωt, (50)\nwhere we used Re˜ χzz\n0(0, ω)≫Im˜χzz\n0(0, ω). The real part\nof ˜χzz(0, ω) is given by\nRe˜χzz(0, ω) =−PZ\ndED (E)E2−M2\n2E2f(E)−f(−E)\n2E+ℏω,\n(51)where Pmeans the principal value. When the integrand\nis expanded with respect to ω, the lowest order correc-\ntion term becomes quadratic in ω. In the frequency range\nconsidered in this work, this correction term is signifi-\ncantly smaller compared to the static spin susceptibility\nRe˜χzz(0,0). Therefore, the spin susceptibility exhibits\nalmost no frequency dependence and remains constant\nas a function of ω. The imaginary part of ˜ χzz(0, ω) is\ngiven by\nIm˜χzz(0, ω)\n=πD(ℏω/2)(ℏω/2)2−M2\n2(ℏω/2)2[f(−ℏω/2)−f(ℏω/2)].\n(52)\nFMR modulation due to the proximity exchange\ncoupling\nIn this section, we provide a brief explanation for the\nderivation of the FMR modulations δhandδα. The FMR\nmodulations can be determined from the retarded com-\nponent of the magnon Green’s function, which is given\nby\n˜GR(k, ω) =2S/ℏ\nω−ωk+iαω−(2S/ℏ)ΣR(k, ω),(53)\nwhere we introduce the Gilbert damping constant αphe-\nnomenologically. In the second-order perturbation calcu-\nlation with respect to HT, the self-energy is given by\nΣR(k, ω) =−\u0012sinϑ\n2\u00132X\nq∥|˜J(q∥,k)|2˜χzz(q∥, ω),(54)\nwhere ˜J(q∥,0) is given by\n˜J(q∥,k) =1\nL2√\nNZ\ndr∥X\nnJ(r∥,rn)ei(q∥·r∥+k·rn)\n(55)\nThe pole of ˜GR(k, ω) signifies the FMR modulations,\nincluding both the frequency shift and the enhanced\nGilbert damping. These are given by\nδh=2S\nγℏReΣR(0, ω), δα =−2S\nℏωImΣR(0, ω).(56)\nFrom the above equations and Eq. (54), it is apparent\nthat FMR modulations provide information regarding\nboth the properties of the interface coupling and the dy-\nnamic spin susceptibility of the Majorana modes.\nThe form of matrix element ˜J(q∥,0) depends on the\ndetails of the interface. In this work, we assume the\nspecular interface. |˜J(q∥,0)|2is given by\n|˜J(q∥,0)|2=J2\n1\nNδq∥,0. (57)8\nUsing Eq. (57), the self-energy for the uniform magnon\nmode is given by\nΣR(0, ω) =−\u0012sinϑ\n2\u00132J2\n1\nN˜χzz(0, ω). (58)\n[1] F. Wilczek, Nat. 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Jpn. 92, 063701 (2023)."    },    {        "title": "1604.07552v1.First_principles_studies_of_the_Gilbert_damping_and_exchange_interactions_for_half_metallic_Heuslers_alloys.pdf",        "content": "arXiv:1604.07552v1  [cond-mat.mtrl-sci]  26 Apr 2016First principles studies of the Gilbert damping and exchang e interactions for\nhalf-metallic Heuslers alloys\nJonathan Chico,1,∗Samara Keshavarz,1Yaroslav Kvashnin,1Manuel Pereiro,1Igor\nDi Marco,1Corina Etz,2Olle Eriksson,1Anders Bergman,1and Lars Bergqvist3,4\n1Department of Physics and Astronomy, Materials Theory Divi sion,\nUppsala University, Box 516, SE-75120 Uppsala, Sweden\n2Department of Engineering Sciences and Mathematics,\nMaterials Science Division, Lule˚ a University of Technolo gy, Lule˚ a, Sweden\n3Department of Materials and Nano Physics, School of Informa tion and Communication Technology,\nKTH Royal Institute of Technology, Electrum 229, SE-16440 K ista, Sweden\n4SeRC (Swedish e-Science Research Center), KTH Royal Instit ute of Technology, SE-10044 Stockholm, Sweden\n(Dated: September 28, 2018)\nHeusler alloys havebeen intensivelystudied dueto thewide varietyof properties thatthey exhibit.\nOne of these properties is of particular interest for techno logical applications, i.e. the fact that some\nHeusler alloys are half-metallic. In the following, a syste matic study of the magnetic properties\nof three different Heusler families Co 2MnZ, Co 2FeZ and Mn 2VZ with Z = (Al, Si, Ga, Ge) is per-\nformed. A key aspect is the determination of the Gilbert damp ing from first principles calculations,\nwith special focus on the role played by different approximat ions, the effect that substitutional\ndisorder and temperature effects. Heisenberg exchange inte ractions and critical temperature for\nthe alloys are also calculated as well as magnon dispersion r elations for representative systems,\nthe ferromagnetic Co 2FeSi and the ferrimagnetic Mn 2VAl. Correlations effects beyond standard\ndensity-functional theory are treated using both the local spin density approximation including the\nHubbard Uand the local spin density approximation plus dynamical mea n field theory approx-\nimation, which allows to determine if dynamical self-energ y corrections can remedy some of the\ninconsistencies which were previously reported for these a lloys.\nI. INTRODUCTION\nThe limitations presented by traditional electronic de-\nvices, such as Joule heating, which leads to higher en-\nergyconsumption, leakagecurrentsandpoorscalingwith\nsize amongothers1, havesparkedprofoundinterest in the\nfields of spintronics and magnonics. Spintronics applica-\ntions rely in the transmission of information in both spin\nand charge degrees of freedom of the electron, whilst in\nmagnonics information is transmitted via magnetic exci-\ntations, spin waves or magnons. Half-metallic materials\nwith a large Curie temperature are of great interest for\nthese applications. Due to the fact that they are con-\nductors in only one of the spin channels makes them\nideal candidates for possible devices2. Half-metals also\nhave certain advantages for magnonic applications, due\nto the fact that they are insulators in a spin channel and\nthus can have a smaller total density of states at the\nFermi energy than metals. This can result into a small\nGilbert damping, which is an instrumental prerequisite\nfor magnonic applications3.\nThe name “full Heusler alloys”refer to a set of com-\npounds with formula X 2YZ with X and Y typically being\ntransition metals4. The interest in them stems from the\nfactthattheirpropertiescanbecompletelydifferentfrom\nthose of their constituents. Heusler compounds can be\nsuperconducting5(Pd2YSn), semiconductors6(TiCoSb),\nhalf-metallic7(Co2MnSi), and can show a wide array of\nmagnetic configurations: ferromagnetic7(Co2FeSi), fer-\nrimagnetic8(Mn2VAl) or antiferromagnetic9(CrMnSb).\nDue to such a wide variety of behaviours, full Heusleralloys have been studied in great detail since their dis-\ncovery in 1903, leading to the discovery of new Heusler\nfamilies such as the half-Heuslers, with formula XYZ,\nand the inverse Heuslers, with formula X 2YZ. The lat-\nter tend to exhibit a different crystal structure and have\nbeen predicted to show quite remarkable properties10.\nMany Heusler alloys have also been predicted to be\nhalf-metallic, in particular Co 2MnSi has been the focus\nofmany theoreticaland experimental works7,11,12, due to\nits large Curie temperature of 985 K13, half-metallicity\nand low damping parameter, which makes it an ideal\ncandidate for possible spintronic applications. Despite\nthe large amount of research devoted to the half-metallic\nHeusleralloys,suchasCo 2MnSi, onlyrecentlytheoretical\npredictions of the Gilbert damping parameter have been\nmade for some Heusler alloys14,15.\nIn the present work first principle calculations of the\nfull Heusler families Co 2MnZ, Co 2FeZ and Mn 2VZ with\nZ = (Al, Si, Ga, Ge) are performed, with special empha-\nsis on the determination of the Gilbert damping and the\ninteratomic exchange interactions. A study treatment of\nthesystemswithdifferentexchangecorrelationpotentials\nis also performed.\nThe paper is organized as follows, in section II the\ncomputational methods used are presented. Then, in\nsection III, magnetic moments and spectral properties\nare discussed. In section IV the results for the exchange\nstiffness parameter, the critical temperature obtained via\nMonteCarlosimulationsandmagnondispersionrelations\nare presented. Finally in section V, the calculated damp-\ning parameter for the different Heusler is presented and2\ndiscussed.\nII. COMPUTATIONAL METHODS\nThe full Heusler alloys(X 2YZ) havea crystalstructure\ngiven by the space group Fm-3m with X occupying the\nWyckoffposition 8c (1\n4,1\n4,1\n4), while Ysits in the 4a(0,0,0)\nand Z in the 4b (1\n2,1\n2,1\n2).\nTo determine the properties of the systems first prin-\nciples electronic structure calculations were performed.\nThey were mainly done by means of the Korringa-Kohn-\nRostocker Green’s function formalism as implemented in\nthe SPR-KKRpackage16. The shape ofthe potential was\nconsidered by using both the Atomic Sphere Approxi-\nmation (ASA) and a full potential (FP) scheme. The\ncalculations of exchange interactions were performed in\nscalar relativistic approximation while the full relativis-\ntic Dirac equation was used in the damping calculations.\nThe exchange correlation functional was treated using\nboth the Local Spin Density Approximation (LSDA), as\nconsidered by Vosko, Wilk, and Nusair (VWN)17, and\nthe Generalized Gradient Approximation (GGA), as de-\nvised by Perdew, Burke and Ernzerhof (PBE)18. For\ncases in which substitutional disorder is considered, the\nCoherent Potential Approximation (CPA) is used19,20.\nStatic correlation effects beyond LSDA or GGA are\ntaken into account by using the LSDA+ Uapproach,\nwherethe Kohn-ShamHamiltonianissupplemented with\nan additional term describing local Hubbard interac-\ntions21, for thed-states of Co, Mn and Fe. The U-matrix\ndescribing this on-site interactions was parametrized\nthrough the Hubbard parameter Uand the Hund ex-\nchangeJ, using values UCo=UMn=UFe= 3 eV and\nJCo=JMn=JFe= 0.8 eV, which are in the range of the\nvalues considered in previous theoretical studies13,22–24.\nThis approach is used for the Heusler alloys families\nCo2MnZ and Co 2FeZ, as previous studies have shown\nthat for systems such as Co 2FeSi it might be necessary to\nreproduce several experimental observations, although,\nthis topic is still up for debate23. Since part of correla-\ntioneffectsofthe3 dorbitalsisalreadyincludedinLSDA,\ntheir contribution has to be subtracted before adding the\n+Uself-energy. This contribution to be removed is usu-\nally called “double-counting”(DC) correction and there\nis no unique way of defining it (see e.g. Ref. 25). We\nhave used two of the most widely used schemes for the\nDC, namely the Atomic Limit (AL), also known as Fully\nLocalized Limit (FLL)26, and the Around Mean Field\n(AMF)27. The dependence of the results on this choice\nwill be extensively discussed in the following sections.\nIn order to shine some light on the importance of\nthe dynamical correlations for the magnetic properties\nof the selected Heusler alloys, a series of calculations\nwere performed in the framework of DFT plus Dynami-\ncal Mean Field Theory (DMFT)28,29, as implemented in\nthe full-potential linear muffin-tin orbital (FP-LMTO)\ncode RSPt30. As for LSDA+ U, the DMFT calculationsare performed for a selected set of metal 3 dorbitals on\ntop of the LSDA solution in a fully charge self-consistent\nmanner.31,32Theeffectiveimpurityproblem, whichisthe\ncore of the DMFT, is solved through the spin-polarized\nT-matrix fluctuation-exchange (SPTF) solver33. This\ntype of solver is perturbative and is appropriate for the\nsystems with moderate correlationeffects, where U/W <\n1 (Wdenotes the bandwidth).34Contrary to the prior\nDMFT studies35,36, we have performed the perturba-\ntion expansion of the Hartree-Fock-renormalizedGreen’s\nfunction ( GHF) and not of the bare one. Concerning the\nDC correction, we here use both the FLL approach, de-\nscribed above, as well as the so-called “Σ(0)”correction.\nIn the latter case, the orbitally-averaged static part of\nthe DMFT self-energy is removed, which is often a good\nchoice for metals29,37. Finally, in order to extract infor-\nmationaboutthemagneticexcitationsin thesystems, we\nhave performed a mapping onto an effective Heisenberg\nHamiltonian\nˆH=−/summationdisplay\ni/negationslash=jJij/vector ei/vector ej, (1)\nwhereJijis anexchangeinteractionbetweenthe spinslo-\ncated at site iandj, while the /vector ei(/vector ej) representsthe unity\nvectoralongthe magnetizationdirectionatsite i (j). The\nexchange parameters then are computed by making use\nof the well established LKAG (Liechtenstein, Katsnel-\nson, Antropov, and Gubanov) formalism, which is based\non the magnetic force theorem38–40. More specific de-\ntails about the implementation of the LKAG formalism\nin RSPt can be found in Ref. 41. We also note that the\nperformance of the RSPt method was recently published\nin Ref.42and it was found that the accuracy was similar\nto that of augmented plane wave methods.\nFrom the exchange interactions between magnetic\natoms, it is possible to obtain the spin wave stiffness,\nD, which, for cubic systems is written as43\nD=2\n3/summationdisplay\ni,jJij√mimj|rij|2exp/parenleftbigg\n−ηrij\nalat/parenrightbigg\n,(2)\nwhere the mi’s are the magnetic moments of a given\natom,rijisthedistancebetweenthetwoconsideredmag-\nneticmoments, alatisthelatticeparameter, ηisaconver-\ngence parameter used to ensure the convergence of Eq. 2,\nthe value of Dis taken under the limit η→0. To ensure\nthe convergence of the summation, it is also important\nto take into consideration long range interactions. Hence\nthe exchange interactions are considered up to 6 lattice\nconstants from the central atom.\nThe obtained exchange interactions were then used to\ncalculate the critical temperature by making use of the\nBindercumulant, obtainedfromMonteCarlosimulations\nas implemented in the UppASD package44. This was\ncalculated for three different number of cell repetitions\n(10x10x10, 15x15x15 and 20x20x20), with the intersec-\ntion point determining the critical temperature of the\nsystem45.3\nThe Gilbert damping, α, is calculated via linear re-\nsponse theory46. Temperature effects in the scattering\nprocess of electrons are taken into account by consider-\ning an alloy analogy model within CPA with respect to\nthe atomic displacements and thermal fluctuations of the\nspin moments47. Vertex corrections are also considered\nhere, because they provide the “scattering in”term of the\nBoltzmann equation and it corrects significant error in\nthe damping, whenever there is an appreciable s-p or s-d\nscattering in the system16,48.\nFrom the calculated exchange interactions, the adia-\nbatic magnon spectra (AMS) can be determined by cal-\nculating the Fourier transform of the interatomic ex-\nchange interactions49. This is determined for selected\ncases and is compared with the magnon dispersion re-\nlation obtained from the dynamical structure factor,\nSk(q,ω), resulting fromspin dynamics calculations. The\nSk(q,ω) is obtained from the Fourier transform of the\ntime and spatially displaced spin-spin correlation func-\ntion,Ck(r−r′,t)50\nSk(q,ω) =1√\n2πN/summationdisplay\nr,r′eiq·(r−r′)/integraldisplay∞\n−∞eiωtCk(r−r′,t)dt.\n(3)\nThe advantage of using the dynamical structure factor\nover the adiabatic magnon spectra is the capability of\nstudying temperature effects as well as the influence of\nthe damping parameter determined from first principles\ncalculations or from experimental measurements.\nIII. ELECTRONIC STRUCTURE\nThe calculated spin magnetic moments for the selected\nsystems are reported in Table I. These values are ob-\ntained from SPR-KKR with various approximations of\nthe exchange correlation potential and for different geo-\nmetrical shapes of the potential itself. For the Co 2MnZ\nfamily, when Z = (Si ,Ge), the obtained spin mag-\nnetic moments do not seem to be heavily influenced by\nthe choice of exchange correlation potential or potential\nshape. However, for Z = (Al ,Ga) a large variation is\nobserved in the spin moment when one includes the Hub-\nbard parameter U.\nFor the Co 2FeZ systems, a pronounced difference can\nbe observed in the magnetic moments between the LSDA\nand the experimental values for Z = (Si ,Ge). Previ-\nous theoretical works13,22,24suggested that the inclusion\nof a +Uterm is necessary to obtain the expected spin\nmagnetic moments, but such a conclusion has been re-\ncently questioned23. To estimate which double counting\nschemewould be most suitableto treatcorrelationeffects\nin this class of systems, an interpolation scheme between\nthe FLL and AMF treatments was tested, as described\nin Ref. 59 and implemented in the FP-LAPW package\nElk60. It was found that both Co 2MnSi and Co 2FeSi\nare better described with the AMF scheme, as indicatedby their small αUparameter of ∼0.1 for both materials\n(αU= 0denotes completeAMF and αU= 1FLL), which\nis in agreement with the recent work by Tsirogiannis and\nGalanakis61.\nTo test whether a more sophisticated way to treat cor-\nrelation effects improves the description of these mate-\nrials, electronic structure calculations for Co 2MnSi and\nCo2FeSi using the DMFT scheme were performed. The\nLSDA+DMFT[Σ(0)] calculations yielded total spin mo-\nments of 5.00 µBand 5.34 µBfor respectively Co 2MnSi\nand Co 2FeSi. These values are almost equal to those ob-\ntained in LSDA, which is also the case in elemental tran-\nsition metals32. As mentioned above for LSDA+ U, the\nchoice of the DC is crucial for these systems. The main\nreason why no significant differences are found between\nDMFT and LSDA values is that the employed “Σ(0)”DC\nalmost entirely preserves the static part of the exchange\nsplitting obtained in LSDA62. For instance, by using\nFLL DC, we obtained a total magnetization of 5.00 µB\nand 5.61 µBin Co2MnSi and Co 2FeSi, respectively. We\nnote that the spin moment of Co 2FeSi still does not reach\nthe value expected from the Slater-Pauling rule, but the\nDMFT modifies it in a right direction, if albeit to a\nsmaller degree that the LSDA+ Uschemes.\nAnother important aspect of the presently studied sys-\ntems is the fact that they are predicted to be half-\nmetallic. In Fig. 1, the density of states (DOS) for\nboth Co 2MnSi and Co 2FeSi is presented using LSDA and\nLSDA+U. For Co 2MnSi, the DOS at the Fermi energy\nis observed to exhibit a very clear gap in one of the spin\nchannels, in agreement with previous theoretical works7.\nFor Co 2FeSi, instead a small pseudo-gap region is ob-\nserved in one of the spin channels, but the Fermi level\nis located just at the edge of the boundary as shown in\nprevious works24. Panels a) and b) of Fig. 1 also show\nthat some small differences arise depending on the ASA\nor FP treatment. In particular, the gap in the minority\nspin channel is slightly reduced in ASA.\nWhen correlation effects are considered within the\nLSDA+Umethod, the observed band gap for Co 2MnSi\nbecomes larger, while the Fermi level is shifted and still\nremainsin the gap. When applyingLSDA+ Uto Co2FeSi\nin the FLL scheme, EFis shifted farther away from the\nedgeofthe gap, whichexplainswhythemoment becomes\nalmostanintegerasexpected fromtheSlater-Paulingbe-\nhaviour7,24,63. Moreover,onecanseethatinASAthegap\nin the spin down channel is much smaller in comparison\nto the results obtained in FP.\nWhen the dynamical correlation effects are considered\nvia DMFT, the overall shape of DOS remains to be quite\nsimilartothatofbareLSDA,especiallyclosetotheFermi\nlevel, as seen in Fig. A.1 in the Appendix A. This is re-\nlated to the fact that we use a perturbative treatment\nof the many-body effects, which favours Fermi-liquid be-\nhaviour. Similarly to LSDA+ U, the LSDA+DMFT cal-\nculations result in the increased spin-down gaps, but the\nproducedshiftofthebandsisnotaslargeasinLSDA+ U.\nThis is quite natural, since the inclusion ofthe dynamical4\nTABLE I. Summary of the spin magnetic moments obtained using different approximations as obtained from SPR-KKR for the\nCo2MnZ and Co 2FeZ families with Z = (Al ,Si,Ga,Ge). Different exchange correlation potential approximati ons and shapes of\nthe potential have been used. The symbol†signifies that the Fermi energy is located at a gap in one of the spin channels.\nQuantity Co2MnAl Co 2MnGa Co 2MnSi Co 2MnGe Co 2FeAl Co 2FeGa Co 2FeSi Co 2FeGe\nalat[˚A] 5.75515.77515.65525.743535.730515.737515.640235.75054\nmASA\nLDA[µB] 4.04†4.09†4.99†4.94†4.86†4.93†5.09 5.29\nmASA\nGGA[µB] 4.09†4.15†4.99†4.96†4.93†5.00†5.37 5.53\nmASA\nLDA+UAMF [µB] 4.02†4.08 4.98†4.98†4.94†4.99†5.19 5.30\nmASA\nLDA+UFLL [µB] 4.77 4.90 5.02†5.11 5.22 5.36 5.86†5.94†\nmFP\nLDA[µB] 4.02†4.08†4.98†4.98†4.91†4.97†5.28 5.42\nmFP\nGGA[µB] 4.03†4.11 4.98†4.99†4.98†5.01†5.55 5.70\nmFP\nLDA+UAMF [µB] 4.59 4.99 4.98†5.13 5.12 5.40 5.98†5.98†\nmFP\nLDA+UFLL [µB] 4.03†4.17 4.99†4.99†4.99†5.09 5.86†5.98†\nmexp[µB] 4.04554.09564.96574.84574.96555.15576.00245.7458\n0369n↑tot[sts./eV]\n0\n3\n6\n9\n-6 -3 0 3n↓tot[sts./eV]\nE-EF[eV]ASA\nFPa)\n0369n↑tot[sts./eV]\n0\n3\n6\n9\n-6 -3 0 3n↓tot[sts./eV]\nE-EF[eV]ASA\nFPb )\n0369n↑tot[sts./eV]\n0\n3\n6\n9\n-6 -3 0 3n↓tot[sts./eV]\nE-EF[eV]FP [FLL]\nFP [AMF]c)\n0369n↑tot[sts./eV]\n0\n3\n6\n9\n-6 -3 0 3n↓tot[sts./eV]\nE-EFASA [AMF]\nFP [FLL]\nFP [AMF]d )\n[eV]\nFIG. 1. (Color online) Total density of states for different e xchange correlation potentials with the dashed line indica ting the\nFermi energy, sub-figures a) and b) when LSDA is used for Co 2MnSi and Co 2FeSi respectively. Sub-figures c) and d) show the\nDOS when the systems (Co 2MnSi and Co 2FeSi respectively) are treated with LSDA+ U. It can be seen that the half metalicity\nof the materials can be affected by the shape of the potential a nd the choice of exchange correlation potential chosen.\ncorrelations usually tends to screen the static contribu-\ntions coming from LSDA+ U.\nAccording to Ref. 35 taking into account dynami-\ncal correlations in Co 2MnSi results in the emergence of\nthe non-quasiparticle states (NQS’s) inside the minority-\nspin gap, which at finite temperature tend to decrease\nthe spin polarisation at the Fermi level. These NQS’s\nwere first predicted theoretically for model systems64and stem from the electron-magnon interactions, which\nare accounted in DMFT (for review, see Ref. 2). Our\nLSDA+DMFT results for Co 2MnSi indeed show the ap-\npearance of the NQS’s, as evident from the pronounced\nimaginary part of the self-energy at the bottom of the\nconduction minority-spin band (see Appendix B). An\nanalysis of the orbital decomposition of the self-energy\nreveals that the largest contribution to the NQS’s comes5\nfrom the Mn- TEgstates. However, in our calculations,\nwhere the temperature was set to 300K, the NQS’s ap-\npeared above Fermi level and did not contribute to the\nsystem’s depolarization, in agreementwith the recent ex-\nperimental study12.\nWe note that a half-metallic state with a magnetic\nmoment of around 6 µBfor Co 2FeSi was reported in a\nprevious LSDA+DMFT[FLL] study by Chadov et al.36.\nIn their calculations, both LSDA+ Uand LSDA+DMFT\ncalculations resulted in practically the same positions of\nthe unoccupied spin-down bands, shifted to the higher\nenergies as compared to LSDA. This is due to techni-\ncal differences in the treatment of the Hartree-Fock con-\ntributions to the SPTF self-energy, which in Ref. 36 is\ndone separately from the dynamical contributions, while\nin this study a unified approach is used. Overall, the\nimprovements in computational accuracy with respect to\npreviousimplementationscouldberesponsiblefortheob-\ntained qualitative disagreement with respect to Refs. 35\nand 36. Moreover, given that the results qualitatively\ndepend on the choice of the DC term, the description of\nthe electronic structure of Co 2FeSi is not conclusive.\nThe discrepancies in the magnetic moments presented\nin Table I with respect to the experimental values can in\npart be traced back to details of the density of the states\naround the Fermi energy. The studied Heusler alloys are\nthought to be half-metallic, which in turn lead to inte-\nger moments following the Slater-Pauling rule7. There-\nfore, any approximation that destroys half-metallicity\nwill have a profound effect on their magnetic properties7.\nFor example, for Co 2FeAl when the potential is treated\nin LSDA+ U[FLL] with ASA the Fermi energy is located\nat a sharp peak close to the edge of the band gap, de-\nstroyingthehalf-metallicstate(Seesupplementarymate-\nrial Fig.1). A similar situation occurs in LSDA+ U[AMF]\nwith a full potential scheme. It is also worth mention-\ning that despite the fact that the Fermi energy for many\nof these alloys is located inside the pseudo-gap in one of\nthe spin channels, this does not ensure a full spin po-\nlarization, which is instead observed in systems as e.g.\nCo2MnSi. Another important factor is the fact that EF\ncan be close to the edge of the gap as in Co 2MnGa when\nthe shape of the potential is considered to be given by\nASA and the exchange correlation potential is dictated\nby LSDA, hence the half-metallicity of these alloys could\nbe destroyed due to temperature effects.\nThe other Heusler family investigated here is the ferri-\nmagnetic Mn 2VZ with Z = (Al ,Si,Ga,Ge). The lattice\nconstants used in the simulations correspond to either\nexperimental or previous theoretical works. These data\nare reported in Table II together with appropriate ref-\nerences. Table II also illustrates the magnetic moments\ncalculated using different exchange correlation potentials\nand shapes of the potential. It can be seen that in gen-\neral there is a good agreement with previous works, re-\nsulting in spin moments which obey the Slater-Pauling\nbehaviour.\nFor these systems, the Mn atoms align themselves inTABLE II. Lattice constants used for the electronic struc-\nture calculations and summary of the magnetic properties fo r\nMn2VZ with Z = (Al ,Si,Ga,Ge). As for the ferromagnetic\nfamilies, different shapes of the potential and exchange cor -\nrelations potential functionals were used. The magnetic mo -\nments follow quite well the Slater-Pauling behavior with al l\nthe studied exchange correlation potentials. The symbol†\nsignifies that the Fermi energy is located at a gap in one of\nthe spin channels.\nQuantity Mn2VAl Mn 2VGa Mn 2VSi Mn 2VGe\nalat[˚A] 5.687655.905666.06656.09567\nmASA\nLDA[µB] 1.87 1.97†1.00†0.99†\nmASA\nGGA[µB] 1.99†2.04†1.01†1.00†\nmFP\nLDA[µB] 1.92 1.95†0.99†0.99\nmFP\nGGA[µB] 1.98†2.02†0.99†0.99†\nmexp[µB] — 1.8666— —\nan anti-parallel orientation with respect to the V mo-\nments, resulting in a ferrimagnetic ground state. As for\nthe ferromagnetic compounds, the DOS shows a pseu-\ndogap in one of the spin channels (see supplementary\nmaterial Fig.8-9) indicating that at T= 0 K these com-\npoundscouldbehalf-metallic. An importantfactoristhe\nfact that the spin polarization for these systems is usu-\nally considered to be in the opposite spin channel than\nfor the ferromagneticalloys presently studied, henceforth\nthe total magnetic moment is usually assigned to a neg-\native sign such that it complies with the Slater-Pauling\nrule7,65.\nIV. EXCHANGE INTERACTIONS AND\nMAGNONS\nIn this section, the effects that different exchange cor-\nrelation potentials and geometrical shapes of the poten-\ntial haveoverthe exchangeinteractionswill be discussed.\nA. Ferromagnetic Co 2MnZ and Co 2FeZ with\nZ= (Al,Si,Ga,Ge)\nIn Table III the calculated spin wave stiffness, D, is\nshown. In general there is a good agreement between\nthe calculated values for the Co 2MnZ family, with the\nobtained values using LSDA or GGA being somewhat\nlarger than the experimental measurements. This is in\nagreement with the observations in the previous section,\nin which the same exchange correlation potentials were\nfound to be able to reproducethe magnetic moments and\nhalf-metallicbehaviourfortheCo 2MnZfamily. Inpartic-\nular, for Co 2MnSi the ASA calculations are in agreement\nwith experiments68,69and previous theoretical calcula-\ntions70. It is important to notice that the experimen-\ntal measurements are performed at room temperature,\nwhich can lead to softening of the magnon spectra, lead-\ning to a reduced spin wave stiffness.6\nHowever, for the Co 2FeZ family neither LSDA or GGA\ncan consistently predict the spin wave stiffness, with\nZ=(Al, Ga) resulting in an overestimated value of D,\nwhile for Co 2FeSi the obtained value is severely underes-\ntimated. However, for some materials in this family, e.g.\nCo2FeGathespinwavestiffnessagreeswith previousthe-\noretical results70. These data reflect the influence that\ncertain approximations have on the location of the Fermi\nlevel, which previously has been shown to have profound\neffects on the magnitude of the exchange interactions71.\nThis can be observed in the half-metallic Co 2MnSi; when\nit is treated with LSDA+ U[FLL] in ASA the Fermi level\nis located at the edge of the gap (see Fig. 1c). Result-\ning in a severely underestimated spin wave stiffness with\nrespect to both the LSDA value and the experimental\nmeasurements (see Table III). The great importance of\nthe location of the Fermi energy on the magnetic proper-\nties can be seen in the cases of Co 2MnAl and Co 2MnGa.\nIn LSDA+ U[FLL], these systems show non integer mo-\nments which are overestimated with respect to the ex-\nperimental measurements (see Table I), but also results\nin the exchange interactions of the system preferring a\nferrimagnetic alignment. Even more the exchange inter-\nactions can be severely suppressed when the Hubbard U\nisused. Forexample, forCo 2MnGe inASAthe dominant\ninteraction is between the Co-Mn moments, in LSDA the\nobtainedvalueis0.79mRy, while inLSDA+ U[FLL]isre-\nduced to 0.34 mRy, also, the nearest neighbour Co 1-Co2\nexchange interaction changes from ferromagnetic to anti-\nferromagnetic when going from LSDA to LSDA+ U[FLL]\nwhich lead the low values obtainedfor the spin wavestiff-\nness. As will be discussed below also for the low Tcfor\nsome of these systems.\nIt is important to notice, that the systems that exhibit\nthe largest deviation from the experimental values, are\nusually those that under a certain exchange correlation\npotential and potential geometry loosetheir half-metallic\ncharacter. Such effect are specially noticeable when one\ncompares LSDA+ U[FLL] results in ASA and FP, where\nhalf-metallicity is more easily lost in ASA due to the\nfact that the pseudogap is much smaller under this ap-\nproximation than under FP (see Fig. 1). In general, it\nis important to notice that under ASA the geometry of\nthe potential is imposed, that is non-spherical contribu-\ntions to the potential are neglected. While this has been\nshown to be very successful to describe many properties,\nit does introduce an additional approximation which can\nlead to anill treatment ofthe properties ofsome systems.\nHence, care must be placed when one is considering an\nASA treatment for the potential geometry, since it can\nlead to large variations of the exchange interactions and\nthus is one of the causes of the large spread on the values\nobserved in Table III for the exchange stiffness and in\nTable IV for the Curie temperature.\nOne of the key factors behind the small values of the\nspin stiffness for Co 2FeSi and Co 2FeGe, in comparison\nwith the rest of the Co 2FeZ family, lies in the fact that\nin LSDA and GGA an antiferromagnetic long-range Fe-Fe interaction is present (see Fig. C.2 in Appendix C).\nAs the magnitude of the Fe-Fe interaction decreases the\nexchange stiffness increases, e.g. as in LSDA+ U[AMF]\nwith afull potential scheme. Theseexchangeinteractions\nare one of the factors behind the reduced value of the\nstiffness, this is evident when comparing with Co 2FeAl,\nwhich while having similar nearest neighbour Co-Fe ex-\nchange interactions, overall displays a much larger spin\nwave stiffness for most of the studied exchange correla-\ntion potentials.\nUsing LSDA+DMFT[Σ(0)] for Co 2MnSi and Co 2FeSi,\nthe obtained stiffness is 580 meV ˚A2and 280 meV ˚A2re-\nspectively, whilst in LSDA+DMFT[FLL] for Co 2MnSi\nthestiffnessis630meV ˚A2andforCo 2FeSiis282meV ˚A2.\nAs can be seen for Co 2MnSi there is a good agree-\nment between the KKR LSDA+ U[FLL], the FP-LMTO\nLSDA+DMFT[FLL] and the experimental values.\nThe agreement with experiments is particularly good\nwhen correlation effects are considered as in the\nLSDA+DMFT[Σ(0)] approach. On the other hand, for\nCo2FeSi the spin wave stiffness is severely underesti-\nmated which is once again consistent with what is shown\nin Table III.\nUsing the calculated exchange interactions, the criti-\ncal temperature, Tc, for each system can be calculated.\nUsing the ASA, the Tcof both the Co 2MnZ and Co 2FeZ\nsystems is consistently underestimated with respect to\nexperimental results, as shown in Table IV. The same\nunderestimation has been observed in previous theo-\nretical studies78, for systems such as Co 2Fe(Al,Si) and\nCo2Mn(Al,Si). However, using a full potential scheme\ninstead leads to Curie temperatures in better agreement\nwith the experimental values, specially when the ex-\nchange correlation potential is considered to be given by\nthe GGA (see Table IV). Such observation is consistent\nwith what was previouslymentioned, regardingthe effect\nofthe ASA treatmentonthe spin wavestiffness andmag-\nnetic moments, where in certain cases, ASA was found to\nnot be the best treatment to reproduce the experimen-\ntal measurements. As mentioned above, this is strongly\nrelated to the fact that in general ASA yields a smaller\npseudogapin the half-metallic materials, leading to mod-\nification of the exchange interactions. Thus, in general, a\nfullpotentialapproachseemstobeabletobetterdescribe\nthe magnetic properties in the present systems, since the\npseudogaparoundthe Fermienergyisbetter describedin\na FP approach for a given choice of exchange correlation\npotential.\nThe inclusionofcorrelationeffects forthe Co 2FeZfam-\nily, lead to an increase of the Curie temperature, as for\nthe spin stiffness. This is related to the enhancement of\nthe interatomic exchange interactions as exemplified in\nthe case of Co 2FeSi. However, the choice of DC once\nmore is shown to greatly influence the magnetic proper-\nties. For the Co 2FeZ family, AMF results in much larger\nTcthan the FLL scheme, whilst for Co 2MnZ the dif-\nferences are smaller, with the exception of Z=Al. All\nthese results showcase how important a proper descrip-7\nTABLE III. Summary of the spin wave stiffness, Dfor Co 2MnZ and Co 2FeZ with Z = (Al ,Si,Ga,Ge). For the Co 2MnZ family\nboth LSDA and GGA exchange correlation potentials yield val ues close to the experimental measurements. However, for th e\nCo2FeZ family a larger data spread is observed. The symbol∗implies that the ground state for these systems was found to b e\nFerri-magnetic from Monte-Carlo techniques and the critic al temperature presented here is calculated from the ferri- magnetic\nground state.\nQuantity Co2MnAl Co 2MnGa Co 2MnSi Co 2MnGe Co 2FeAl Co 2FeGa Co 2FeSi Co 2FeGe\nDASA\nLDA[meV˚A2] 282 291 516 500 644 616 251 206\nDASA\nGGA[meV˚A2] 269 268 538 515 675 415 267 257\nDASA\nLDA+UFLL [meV ˚A2] 29∗487∗205 94 289 289 314 173\nDASA\nLDA+UAMF [meV ˚A2] 259 318 443 417 553 588 235 214\nDFP\nLDA[meV˚A2] 433 405 613 624 692 623 223 275\nDFP\nGGA[meV˚A2] 483 452 691 694 740 730 323 344\nDFP\nLDA+UFLL [meV ˚A2] 447 400 632 577 652 611 461 436\nDFP\nLDA+UAMF [meV ˚A2] 216 348 583 579 771 690 557 563\nDexp[meV˚A2] 190722647357568-5346941374370754967671577—\ntion of the pseudogap region is in determining the mag-\nnetic properties of the system.\nAnother observation, is the fact that even if a given\ncombination of exchange correlation potential and geo-\nmetrical treatment of the potential can yield a value of\nTcin agreementwith experiments, it does not necessarily\nmeans that the spin wave stiffness is correctly predicted\n(see Table III and Table IV).\nWhen considering the LSDA+DMFT[Σ(0)] scheme,\ncritical temperatures of 688 K and 663 K are ob-\ntained for Co 2MnSi and Co 2FeSi, respectively. Thus,\nthe values of the Tcare underestimated in compari-\nson with the LSDA+ Uor LSDA results. The reason\nfor such behaviour becomes clear when one looks di-\nrectly on the Jij’s, computed with the different schemes,\nwhich are shown in Appendix C. These results sug-\ngest that taking into account the dynamical correlations\n(LSDA+DMFT[Σ(0)]) slightly suppresses most of the\nJij’s as compared to the LSDA outcome. This is an\nexpected result, since the employed choice of DC correc-\ntion preserves the exchange splitting obtained in LSDA,\nwhile the dynamical self-energy, entering the Green’s\nfunction, tends to lower its magnitude. Since these two\nquantities are the key ingredients defining the strength\nof the exchange couplings, the Jij’s obtained in DMFT\nare very similar to those of LSDA (see e.g. Refs. 41\nand 81). The situation is a bit different if one employs\nFLL DC, since an additional static correction enhances\nthe local exchange splitting.82For instance, in case of\nCo2MnSi the LSDA+DMFT[FLL] scheme provided a Tc\nof 764 K, which is closer to the experiment. The con-\nsistently better agreement of the LSDA+ U[FLL] and\nLSDA+DMFT[FLL] estimates of the Tcwith experimen-\ntal values might indicate that explicit account for static\nlocal correlations is important for the all considered sys-\ntems.\nUsing the calculated exchange interactions, it is also\npossible to determine the adiabatic magnon spectra\n(AMS). In Fig. 2 is shown the effect that different ex-\nchange correlation potentials have overthe description ofthe magnon dispersion relation of Co 2FeSi is shown. The\nmost noticeable effect between different treatments of\nthe exchange correlation potential is shifting the magnon\nspectra, while its overall shape seems to be conserved.\nThis is a direct result from the enhancement of nearest\nneighbour interactions (see Fig. C.2).\nWhen comparing the AMS treatment with the dy-\nnamical structure factor, S(q,ω), atT= 300 K and\ndamping parameter αLSDA= 0.004, obtained from first\nprinciples calculations (details explained in section V),\na good agreement at the long wavelength limit is found.\nHowever, a slight softening can be observed compared\nto the AMS. Such differences can be explained due to\ntemperature effects included in the spin dynamics sim-\nulations. Due to the fact that the critical temperature\nof the system is much larger than T= 300 K (see Ta-\nble IV), temperature effects are quite small. The high\nenergy optical branches are also softened and in general\nare much less visible. This is expected since the correla-\ntion was studied using only vectors in the first Brillouin\nzone and as has been shown in previous works50, a phase\nshift is sometimes necessary to properly reproduce the\noptical branches, implying the need of vectors outside\nthe first Brillouin zone. Also, Stoner excitations dealing\nwith electron-holeexcitations arenot included in this ap-\nproach,whichresultintheLandaudampingwhichaffects\nthe intensity of the optical branches. Such effects are not\ncaptured by the present approach, but can be studied\nby other methods such as time dependent DFT83. The\nshape of the dispersion relationalong the path Γ −Xalso\ncorresponds quite well with previous theoretical calcula-\ntions performed by K¨ ubler84.\nB. Ferrimagnetic Mn 2VZ with Z = (Al,Si,Ga,Ge)\nAsmentionedabove,theMnbasedMn 2VZfullHeusler\nfamily has a ferrimagnetic ground state, with the Mn\natoms orienting parallel to each other and anti-parallel\nwith respect to the V moments. For all the studied sys-8\nTABLE IV. Summary of the critical temperature for Co 2MnZ and Co 2FeZ with Z = (Al ,Si,Ga,Ge), with different exchange\ncorrelation potentials and shape of the potentials. The sym bol∗implies that the ground state for these systems was found to b e\nFerri-magnetic from Monte-Carlo techniques and the critic al temperature presented here is calculated from the ferri- magnetic\nground state.\nQuantity Co2MnAl Co 2MnGa Co 2MnSi Co 2MnGe Co 2FeAl Co 2FeGa Co 2FeSi Co 2FeGe\nTLDA\ncASA [K] 360 350 750 700 913 917 655 650\nTGGA\ncASA [K] 350 300 763 700 975 973 800 750\nTLDA+U\ncASAFLL[K] 50∗625∗125 225 575 550 994 475\nTLDA+U\ncASAAMF[K] 325 425 650 600 950 950 650 625\nTLDA\ncFP [K] 525 475 875 825 1050 975 750 750\nTGGA\ncFP [K] 600 525 1000 925 1150 1100 900 875\nTLDA+U\ncFPFLL[K] 525 475 950 875 1050 975 1050 1075\nTLDA+U\ncFPAMF[K] 450 450 1000 875 1275 1225 1450 1350\nTexp\nc[K] 69778694 98513905 10007910938011002498158\nTABLE V. Summary of the spin wave stiffness, D, and the\ncritical temperature for Mn 2VZ with Z = (Al ,Si,Ga,Ge) for\ndifferent shapes of the potential and exchange correlation p o-\ntentials.\nQuantity Mn2VAl Mn 2VGa Mn 2VSi Mn 2VGe\nDASA\nLDA[meV˚A2] 314 114 147\nDASA\nGGA[meV˚A2] 324 73 149\nDFP\nLDA[meV˚A2] 421 206 191\nDFP\nGGA[meV˚A2] 415 91 162\nDexp[meV˚A2] 53485— — —\nTLDA\ncASA [K] 275 350 150 147\nTGGA\ncASA [K] 425 425 250 250\nTLDA\ncFP [K] 425 450 200 200\nTGGA\ncFP [K] 600 500 350 350\nTexp\nc[K] 7688578366— —\ntemstheMn-Mnnearestneighbourexchangeinteractions\ndominates. In Table V the obtained spin wave stiffness,\nD, and critical temperature Tcare shown. For Mn 2VAl,\nit can be seen that the spin wave stiffness is trend when\ncompared to the experimental value. The same under-\nestimation can be observed in the critical temperature.\nFor Mn 2VAl, one may notice that the best agreement\nwith experiments is obtained for GGA in FP. An inter-\nesting aspect of the high Tcobserved in these materials\nis the fact that the magnetic order is stabilized due to\nthe anti-ferromagnetic interaction between the Mn and\nV sublattices, since the Mn-Mn interaction is in general\nmuch smaller than the Co-Co, Co-Mn and Co-Fe inter-\nactions present in the previously studied ferromagnetic\nmaterials.\nFor these systems it can be seen that in general the FP\ndescriptionyields Tc’swhichareinbetter agreementwith\nexperiment, albeit if the values are still underestimated.\nAs for the Co based systems the full potential technique\nimproves the description of the pseudogap, it is impor-\ntant to notice that for most systems both in ASA and\nFP the half-metallic characteris preserved. However, the\ndensity of states at the Fermi level changes which could\nlead to changes in the exchange interactions.As for the ferromagnetic systems one can calculate the\nmagnon dispersion relation and it is reported in Fig. 3\nfor Mn 2VAl. A comparison with Fig. 2 illustrates some\nof the differences between the dispersion relation of a fer-\nromagnet and of a ferrimagnetic material. In Fig. 3 some\noverlap between the acoustic and optical branches is ob-\nserved, as well as a quite flat dispersion relation for one\nof the optical branches. Such an effect is not observed in\nthe studied ferromagnetic cases. In general the different\nexchange correlation potentials only tend to shift the en-\nergy of the magnetic excitations, while the overall shape\nof the dispersion does not change noticeably, which is\nconsistent with what was seen in the ferromagnetic case.\nThe observed differences between the LSDA and GGA\nresults in the small qlimit, corresponds quite well with\nwhatisobservedinTableV, wherethe spinwavestiffness\nfor GGA with the potential given by ASA is somewhat\nlargerthan the LSDA case. This is directly related to the\nobservation that the nearest neighbour Mn-Mn and Mn-\nV interactions are large in GGA than in LSDA. Again,\nsuch observation is tied to the DOS at the Fermi level,\nsince Mn 2VAl is not half-metallic in LSDA, on the other\nhand in GGA the half-metallic state is obtained (see Ta-\nble. II.\nV. GILBERT DAMPING\nThe Gilbert damping is calculated for all the previ-\nously studied systems using ASA and a fully relativistic\ntreatment. In Fig. 4, the temperature dependence of the\nGilbert damping for Co 2MnSi is reported for different\nexchange-correlationpotentials. Whencorrelationeffects\nare neglected or included via the LSDA+ U[AMF], the\ndampingincreaseswith temperature. Onthe otherhand,\nin the LSDA+ U[FLL] scheme, the damping decreases as\na function of temperature, and its overall magnitude is\nmuch larger. Such observation can be explained from the\nfact that in this approximation a small amount of states\nexists at the Fermi energyin the pseudogapregion, hence\nresulting in a larger damping than in the half-metallic9\n0100200300400500\nΓ X W L ΓEnergy [meV]FP-LSDA\nDMFT[Σ(0)]a)\nFIG. 2. (Color online) a) Adiabatic magnon spectra for\nCo2FeSi for different exchange correlation potentials. In the\ncase of FP-LSDA and LSDA+DMFT[Σ(0)] the larger devia-\ntionsareobservedinthecase ofhighenergies, withtheDMFT\ncurve having a lower maximum than the LSDA results. In b)\na comparison of the adiabatic magnon spectra (solid lines)\nwith the dynamical structure factor S(q,ω) atT= 300 K,\nwhen the shape of the potential is considered to be given\nby the atomic sphere approximation and the exchange cor-\nrelation potential to be given by LSDA, some softening can\nbe observed due to temperature effects specially observed at\nhigher q-points.\ncases(see Fig. 1c).\nIn general the magnitude of the damping, αLSDA=\n7.4×10−4, is underestimated with respect to older ex-\nperimental measurements at room temperature, which\nyielded values of α= [0.003−0.006]86andα∼0.025\nfor polycrystalline samples87, whilst it agrees with previ-\nously performed theoretical calculations14. Such discrep-\nancy between the experimental and theoretical results\ncould stem from the fact that in the theoretical calcula-\ntions only the intrinsic damping is calculated, while in\nexperimental measurements in addition extrinsic effects\nsuch as eddy currents and magnon-magnon scattering\ncan affect the obtained values. It is also known that sam-FIG. 3. (Color online) Adiabatic magnon dispersion relatio n\nfor Mn 2VAl when different exchange correlation potentials\nare considered. In general only a shift in energy is observed\nwhen considering LSDA or GGA with the overall shape being\nconserved.\n00.511.522.533.54\n50 100 150 200 250 300 350 400 450 500Gilbert damping (10-3)\nTemperature [K]LSDA\nGGA\nLSDA+U [FLL]\nLSDA+U [AMF]\nFIG.4. (Color online)TemperaturedependenceoftheGilber t\ndamping for Co 2MnSi for different exchange correlation po-\ntentials. For LSDA, GGA and LSDA+ U[AMF] exchange cor-\nrelation potentials the damping increases with temperatur e,\nwhilst for LSDA+ U[FLL]thedampingdecreases as afunction\nof temperature.\nple capping or sample termination, can have profound ef-\nfects over the half-metallicity of Co 2MnSi88. Recent ex-\nperiments showed that ultra-low damping, α= 7×10−4,\nfor Co 1.9Mn1.1Si can be measured when the capping\nis chosen such that the half-metallicity is preserved89,\nwhich is in very good agreement with the present theo-\nretical calculations.\nIn Fig. 5, the Gilbert damping at T= 300 K for the\ndifferent Heusler alloys as a function of the density of\nstates at the Fermi level is presented. As expected, the\nincreased density of states at the Fermi energy results in10\nFIG. 5. (Color online) Gilbert damping for different Heusler\nalloys at T= 300 K as a function of density of states at the\nFermi energy for LSDA exchange correlation potential. In\ngeneral the damping increases as the density of states at the\nFermi Energy increases (the dotted line is to guide the eyes) .\nan increased damping. Also it can be seen that in gen-\neral, alloys belonging to a given family have quite similar\ndamping parameter, except for Co 2FeSi and Co 2FeGe.\nTheir anomalous behaviour, stems from the fact that\nin the LSDA approach both Co 2FeSi and Co 2FeGe are\nnot half-metals. Such clear dependence on the density of\nstates is expected, since the spin orbit coupling is small\nfor these materials, meaning that the dominating con-\ntribution to the damping comes from the details of the\ndensity of states around the Fermi energy90,91.\n1. Effects of substitutional disorder\nIn order to investigate the possibility to influ-\nence the damping, we performed calculations for the\nchemically disordered Heusler alloys Co 2Mn1−xFexSi,\nCo2MeAl1−xSixand Co 2MeGa 1−xGexwhere Me =\n(Mn,Fe).\nDue to the small difference between the lattice param-\neters of Co 2MnSi and Co 2FeSi, the lattice constant is\nunchanged when varying the concentration of Fe. This\nis expected to play a minor role on the following results.\nWhen one considers only atomic displacement contribu-\ntions to the damping (see Fig. 6a), the obtained values\nare clearlyunderestimated in comparisonwith the exper-\nimental measurements at room temperature92. Under\nthe LSDA, GGA and LSDA+ U[AMF] treatments, the\ndamping is shown to increase with increasing concentra-\ntion of Fe. On the other hand, in LSDA+ U[FLL] the\ndamping at low concentrations of Fe is much larger than\ninthe othercases, andit decreaseswith Feconcentration,\nuntil a minima is found at Fe concentration of x∼0.8.\nThis increase can be related to the DOS at the Fermi\nenergy, which is reported in Fig. 1c for Co 2MnSi. One00.511.522.533.544.55\n0 0.2 0.4 0.6 0.8 1Gilbert damping (10-3)\nFe concentrationLSDA\nGGA\nLSDA+ U[FLL]\nLSDA+ U[AMF]a)\n00.511.522.533.544.5\n0 0.2 0.4 0.6 0.8 1Gilbert damping (10-3)\nFe concentrationLSDA\nGGA\nLSDA+ U[FLL]\nLSDA+ U[AMF]b)\nFIG. 6. (Color online) Gilbert damping for the random alloy\nCo2Mn1−xFexSi as a function of the Fe concentration at T=\n300 K when a) only atomic deisplacements are considered and\nb) when both atomic displacements and spin fluctuations are\nconsidered.\ncan observe a small amount of states at EF, which could\nlead to increased values of the damping in comparison\nwith the ones obtained in traditional LSDA. As for the\npure alloys, a general trend relating the variation of the\nDOS at the Fermi level and the damping with respect to\nthe variation of Fe concentration can be obtained, anal-\nogous to the results shown in Fig. 5.\nWhen spin fluctuations are considered in addition to\nthe atomic displacements contribution, the magnitude of\nthe damping increases considerably, as shown in Fig. 6b.\nThis is specially noticeable at low concentrations of Fe.\nMn rich alloys have a Tclower than the Fe rich ones,\nthus resulting in larger spin fluctuations at T= 300 K.\nThe overall trend for LSDA and GGA is modified at low\nconcentrations of Fe when spin fluctuations are consid-\nered, whilst for LSDA+ U[FLL] the changes in the trends\noccur mostly at concentrations between x= [0.3−0.8].\nAn important aspect is the overall good agreement of11\nLSDA, GGA and LSDA+ U[AMF]. Instead results ob-\ntained in LSDA+ U[FLL] stand out as different from the\nrest. This is is expected since as was previously men-\ntioned the FLL DC is not the most appropriate scheme\nto treat these systems. An example of such inadequacy\ncan clearly be seen in Fig. 6b for Mn rich concentrations,\nwhere the damping is much larger with respect to the\nother curves. As mentioned above, this could result from\nthe appearance of states at the Fermi level.\nOverall the magnitude of the intrinsic damping pre-\nsented here is smaller than the values reported in experi-\nments92, whichreportvaluesforthedampingofCo 2MnSi\nofα∼0.005 and α∼0.020 for Co 2FeSi, in comparison\nwith the calculated values of αLSDA= 7.4×10−4and\nαLSDA= 4.1×10−3for Co 2MnSi and Co 2FeSi, respec-\ntively. In experiments also a minimum at the concentra-\ntion of Fe of x∼0.4 is present, while such minima is not\nseen in the present calculations. However, similar trends\nas those reported here (for LSDA and GGA) are seen in\nthe work by Oogane and Mizukami15. A possible reason\nbehind the discrepancy between theory and experiment,\ncould stem from the fact that as the Fe concentration\nincreases, correlation effects also increase in relative im-\nportance. Such a situation cannot be easily described\nthrough the computational techniques used in this work,\nandwill affectthe detailsofthe DOSatthe Fermienergy,\nwhich in turn could modify the damping. Another im-\nportant factor influencing the agreement between theory\nand experiments arise form the difficulties in separating\nextrinsic and intrinsic damping in experiments93. This,\ncombined with the large spread in the values reported in\nvarious experimental studies87,94,95, points towards the\nneed of improving both theoretical and experimental ap-\nproaches,ifoneintendstodeterminetheminimumdamp-\ning attainable for these alloys with sufficient accuracy.\nUp until now in the present work, disorder effects\nhave been considered at the Y site of the Heusler struc-\nture. In the following chemical disorder will be consid-\nered on the Z site instead. Hence, the chemical structure\nchanges to the type Co 2MeZA\n1−xZB\nx(Me=Fe,Mn). The\nalloys Co 2MeAlxSi1−xand Co 2MeGa xGe1−xare consid-\nered. The lattice constant for the off stoichiometric com-\npositions is treated using Vegard’s law96, interpolating\nbetween the values given in Table I.\nIn Fig. 7 the dependence of the damping on the con-\ncentration of defects is reported, as obtained in LSDA.\nFor Co 2FeGaxGe1−xas the concentration of defects in-\ncreases the damping decreases. Such a behaviour can\nbe understood by inspecting the density of states at the\nFermi level which follows the same trend, it is important\nto notice that Co 2FeGa is a half-metallic system, while\nCo2FeGe is not (see table I). On the other hand, for\nCo2FeAlxSi1−x, the damping increases slightly with Al\nconcentration, however, for the stoichiometric Co 2FeAl\nis reached the damping decreases suddenly, as in the pre-\nvious case. This is a direct consequence of the fact that\nCo2FeAl is a half metal and Co 2FeSi is not, hence when\nthe half-metallic state is reached a sudden decrease ofthe damping is observed. For the Mn based systems, as\nthe concentration of defects increases the damping in-\ncreases, this stark difference with the Fe based systems.\nFor Co 2MnAlxSi1−xthis is related to the fact that both\nCo2MnAl and Co 2MnSi are half-metals in LSDA, hence,\nthe increase is only related to the fact that the damp-\ning for Co 2MnAl is larger than the one of Co 2MnSi, it\nis also relevant to mention, that the trend obtained here\ncorresponds quite well with what is observed in both ex-\nperimental and theoretical results in Ref.86. A similar\nexplanation can be used for the Co 2MnGa xGe1−xalloys,\nas both are half-metallic in LSDA. As expected, the half\nmetallic Heuslers have a lower Gilbert damping than the\nother ones, as shown in Fig. 7.\n00.511.522.533.544.5\n0 0.2 0.4 0.6 0.8 1Gilbert damping (10-3)\nConcentration of defectsCo2FeAlxSi1-xCo2FeGaxGe1-xCo2MnAlxSi1-xCo2MnGaxGe1-x\nFIG. 7. (color online) Dependence of the Gilbert damping\nfor the alloys Co 2MeAlxSi1−xand Co 2MeGa xGe1−xwith Me\ndenoting Mn or Fe under the LSDA exchange correlation po-\ntential.\nVI. CONCLUSIONS\nThe treatment of several families of half-metallic\nHeusler alloys has been systematically investigated us-\ning several approximations for the exchange correlation\npotential, as well as for the shape of the potential. Spe-\ncial care has been paid to the calculation of their mag-\nnetic properties, such as the Heisenberg exchange inter-\nactions and the Gilbert damping. Profound differences\nhave been found in the description of the systems de-\npending on the choice of exchange correlation potentials,\nspeciallyforsystems in whichcorrelationeffects might be\nnecessarytoproperlydescribethepresumedhalf-metallic\nnature of the studied alloy.\nIn general, no single combination of exchange correla-\ntion potential and potential geometry was found to be\nable to reproduce all the experimentally measured mag-\nnetic properties of a given system simultaneously. Two\nof the key contributing factors are the exchange correla-12\ntion potential and the double counting scheme used to\ntreat correlation effects. The destruction of the half-\nmetallicity of any alloy within the study has profound\neffects on the critical temperature and spin wave stiff-\nness. A clear indication of this fact is that even if the\nFLL double counting scheme may result in a correct de-\nscription of the magnetic moments of the system, the\nexchange interactions may be severely suppressed. For\nthe systems studied with DMFT techniques either mi-\nnor improvement or results similar to the ones obtained\nfrom LSDA is observed. This is consistent with the in-\nclusion of local d−dscreening, which effectively dimin-\nishes the strength of the effective Coulomb interaction\nwith respect to LSDA+ U(for the same Hubbard param-\neterU). In general, as expected, the more sophisticated\ntreatment forthe geometricalshape ofthe potential, that\nis a full potential scheme, yields results closer to experi-\nments, which in these systems, is intrinsically related to\nthe description of the pseudogap region.\nFinally, the Gilbert damping is underestimated with\nrespect to experimental measurements, but in good\nagreement with previous theoretical calculations. One of\nthe possible reasons being the difficulty from the experi-\nmental point of view of separating intrinsic and extrinsic\ncontributions to the damping, as well as the strong de-\npendence of the damping on the crystalline structure.\nA clear correlation between the density of states at the\nFermi level and the damping is also observed, which is\nrelated to the presence of a small spin orbit coupling\nin these systems. This highlights the importance that\nhalf-metallic materials, and their alloys, have in possible\nspintronic and magnonic applications due to their low in-\ntrinsic damping, and tunable magnetodynamic variables.\nThese results could spark interest from the experimental\ncommunity due to the possibility of obtaining ultra-low\ndamping in half-metallic Heusler alloys.\nVII. ACKNOWLEDGEMENTS\nThe authors acknowledge valuable discussions with\nM.I. Katsnelsson and A.I. Lichtenstein. The work was\nfinanced through the VR (Swedish Research Council)\nand GGS (G¨ oran Gustafssons Foundation). O.E. ac-\nknowledges support form the KAW foundation (grants\n2013.0020 and 2012.0031). O.E. and A.B acknowledge\neSSENCE. L.B acknowledge support from the Swedish\ne-Science Research Centre (SeRC). The computer sim-\nulations were performed on resources provided by the\nSwedish National Infrastructure for Computing (SNIC)\nat the National Supercomputer Centre (NSC) and High\nPerformance Computing Center North (HPC2N).\nAppendix A: DOS from LSDA+DMFT\nHere we show the DOS in Co 2MnSi and Co 2FeSi ob-\ntained from LSDA and LSDA+DMFT calculations. The0369n↑tot[sts./eV]\n0\n3\n6\n9-6 -3 0 3n↓tot[sts./eV]\nE-EF[eV]LSDA\n0369n↑tot[sts./eV]\n0\n3\n6\n9-6 -3 0 3n↓tot[sts./eV]\nE-EF[eV]LSDA\nDMFT[Σ(0)]\nDMFT[FLL]\nFIG. A.1. (color online) DOS in Co 2FeSi (top panel) and\nCo2MnSi (bottom panel) obtained in different computational\nsetups.\nresults shown in Fig. A.1 indicate that the DMFT in-\ncreases the spin-down (pseudo-)gap in both Co 2FeSi and\nCo2MnSi. In the latter casethe shift ofthe bands is more\npronounced. InCo 2FeSiitmanifestsitselfinanenhanced\nvalue of the total magnetization. For both studied sys-\ntems, the FLL DC results in relatively larger values of\nthe gaps as compared with the “Σ(0)”estimates. How-\never, for the same choice of the DC this gap appears to\nbe smaller in LSDA+DMFT than in LSDA+ U. Present\nconclusion is valid for both Co 2FeSi and Co 2MnSi (see\nFig. 1 for comparison.)\nAppendix B: NQS in Co 2MnSi\nHere we show the calculated spectral functions in\nCo2MnSi obtained with LSDA+DMFT[Σ(0)] approach.\nAs discussed in the main text, the overall shape of DOS\nis reminiscent of that obtained in LSDA. However, a cer-\ntain amount of the spectral weight appears above the\nminority-spin gap. An inspection of the imaginary part\nof the self-energy in minority-spin channel, shown in the\nbottom panel of Fig. B.1, suggests a strong increase of\nMn spin-down contribution at the corresponding ener-\ngies, thus confirming the non-quasiparticle nature of the\nobtained states. We note that the use of FLL DC formu-\nlationresultsinanenhancedspin-downgapwhichpushes\nthe NQS to appear at even higher energies above EF(see\nAppendix A).13\n-30030PDOS [sts./Ry]\n-0.2-0.10Im [ Σ↑]\nCo Eg\nCo T2g\n-0.2 -0.1 0 0.1 0.2\nE-EF [Ry]-0.2-0.10Im [ Σ↓]\nMn Eg\nMn T2g\nFIG. B.1. (color online) Top panel: DOS in Co 2MnSi pro-\njected onto Mn and Co 3 dstates of different symmetry. Mid-\ndle and bottom panels: Orbital-resolved spin-up and spin-\ndown imaginary parts of the self-energy. The results are\nshown for the “Σ(0)”DC.\nAppendix C: Impact of correlation effects on the\nJij’s in Co 2MnSi and Co 2FeSi\nIn this section we present a comparison of the ex-\nchange parameters calculated in the framework of the\nLSDA+DMFT using different DC terms. The calculated\nJij’s between different magnetic atoms within the first\nfew coordination spheres are shown in Fig. C.1. One can\nsee that the leading interactions which stabilize the fer-\nromagnetism in these systems are the nearest-neighbour\nintra-sublattice couplings between Co and Fe(Mn) atoms\nand, to a lower extend, the interaction between two Co\natoms belonging to the different sublattices. This qual-\nitative behaviour is obtained independently of the em-\nployed method for treating correlation effects and is in\ngood agreement with prior DFT studies. As explained\nin the main text, the LSDA and LSDA+DMFT[Σ(0)] re-\nsults are more similar to each other, whereas most of the\nJij’s extracted from LSDA+DMFT[FLL] are relatively\nenhanced due to inclusion of an additional static contri-\nbution to the exchange splitting. This is also reflected in\nboth values of the spin stiffness and the Tc.\nIn order to have a further insight into the details of\nthe magnetic interactions in the system, we report here\nthe orbital-resolved Jij’s between the nearest-neighbours\nobtained with LSDA. The results, shown in Table. C.1,\nreveal few interesting observations. First of all, all the0.6 1.2 1.8 2.4-0.0500.050.1Jij[mRy]\n0.6 1.2 1.8 2.400.10.2\n0.6 1.2 1.8 2.4\nRij/aalat00.10.20.30.4Jij[mRy]\n0.6 1.2 1.8 2.4\nRij/aalat00.511.5\nLSDA\nLSDA+DMFT [ Σ0]\nLSDA+DMFT [FLL]Co1-Co1Mn-Mn\nCo1-Co2Co-Mn\nFIG. C.1. (color online) The calculated exchange parameter s\nin Co 2MnSi within LSDA and LSDA+DMFT for different\nchoice of DC.\nTABLEC.1. Orbital-resolved Jij’sbetweenthenearestneigh-\nbours in Co 2MnSi in mRy. In the case of Co 1-Co1, the second\nnearest neighbour value is given, due to smallness of the firs t\none. The results were obtained with LSDA.\nTotalEg−EgT2g−T2gEg−T2gT2g−Eg\nCo1-Co10.070 0.077 -0.003 -0.002 -0.002\nCo1-Co20.295 0.357 -0.058 -0.002 -0.002\nCo-Mn 1.237 0.422 -0.079 0.700 0.194\nMn-Mn 0.124 -0.082 0.118 0.044 0.044\nT2g-derived contributions are negligible for all the inter-\nactions involving Co atoms. This has to do with the\nfact that these orbitals are practicallyfilled and therefore\ncan not participate in the exchange interactions. As to\nthe most dominant Co-Mn interaction, the Eg−Egand\nEg−T2gcontributions are both strong and contribute\nto the total ferromagnetic coupling. This is related to\nstrong spin polarisation of the Mn- Egstates.\n00.050.1\n0.6 1.2 1.8 2.4Jij[mRy]\n-0.15-0.1-0.0500.050.1\n0.61.21.82.4\n00.20.40.6\n0.61.21.82.4Jij[mRy]\nRij/alat00.511.522.53\n0.61.21.82.4\nRij/alatLSDA\nLSDA+U[FLL]\nLDA+U[AMF]Co1-Co1 Fe-Fe\nCo1-Co2\nCo-Fe\nFIG. C.2. Exchange interactions for Co 2FeSi within LSDA\nand LSDA+ Uschemes and a full potential approach for dif-\nferent DC choices.14\nCorrelationeffectsalsohaveprofoundeffectsontheex-\nchange interactions of Co 2FeSi. In particular, the Fe-Fe\ninteractions can be dramatically changed when consid-\nering static correlation effects. It is specially noticeable\nhow the anti-ferromagneticexchangeinteractions can de-\ncreasesignificantlywhichcanaffecttheexchangestiffness\nand the critical temperature as described in the maintext. 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Phys. 5, 17 (1921), ISSN 0044-3328."    },    {        "title": "2001.06217v1.Fermi_Level_Controlled_Ultrafast_Demagnetization_Mechanism_in_Half_Metallic_Heusler_Alloy.pdf",        "content": " Fermi Level Controlled Ultrafast Demagnetization Mechanism in Half -Metallic Heusler \nAlloy  \nSantanu Pan1, Takeshi Seki2,3, Koki Takanashi2,3,4, and Anjan Barman1,* \n1Department of Condensed Matter Physics and Material Sciences, S. N. Bose National Centre for \nBasic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700 106, India.  \n2Institute for Materials Research, Tohoku University, Sendai 980 -8577, Japan.  \n3Center for Spintronics Research Network, Tohoku University, Sendai 980 -8577, Japan.  \n4Center f or Science and Innovation in Spintronics, Core Research Cluster, Tohoku University, Sendai \n980-8577, Japan . \n*E-mail: abarman@bose.res.in  \n \n \n \nThe electronic band structure -controlled ultrafast demagnetization mechanism in Co2FexMn 1-\nxSi Heusler alloy is underpinned by systematic variation of composition. We find the spin-flip \nscattering rate controlled by spin density of states at Fermi level  is responsible for non-\nmonotonic variation  of ultrafast demagnetization time (τ M) with x with a maximum  at x = 0.4 . \nFurthermore, Gilbert damping constant exhibits an inverse relationship with  τM due to the \ndominance of inter -band scattering mechanism. This establishes a unified mechanism of \nultrafast spin dynamics based on Fermi level position.  \n \n \n \n \n \n \n \n \n \n \n  The tremendous application potential of spin -polarized Heusler alloys in advanced spintronic s \ndevices  ignites immense interest to investigate the degree and sustainability of their spin-\npolarization under various conditions [1-4]. However, interpreting spin -polarization from the \nconventional methods  such as photoemission, spin transport measurement, point contact \nAndreev reflection and spin-resolved positron annihilation are non -trivial [5-7]. In the quest of \ndeveloping alternative methods, Zhang et al . demonstrated that all -optical ultrafast \ndemagne tization measurement is a reliable technique for probing spin -polarization [8]. They \nobserved a very large ultrafast demagnetization time as a signature of high spin -polarization in \nhalf-metallic CrO 2. However, Co -based half -metallic Heusler alloys exhibit  a comparatively \nsmaller ultrafast demagnetization time (~ 0.3 ps) which raised a serious debate on the \nperception of ultrafast demagnetization mechanism in Heusler alloys [9-11]. A smaller \ndemagnetization time in Heusler alloys than in CrO 2 is explained d ue to the smaller effective \nband gap in the minority spin band  and enhanced spin-flip scattering  (SFS) rate [9]. However, \nfurther experimental evidence shows that the amount of band gap  in minority spin band  cannot \nbe the only deciding factors for SFS medi ated ultrafast demagnetization efficiency [10]. Rather, \none also has to consider the efficiency of optical excitation for majority and minority spin bands \nas well as the optical pump -induced hole dynamics below  Fermi energy  (EF). Consequently, a \nclear interpretation of spin -polarization from ultrafast demagnetization measurement requires \na clear and thorough understanding of its underlying mechanism. Since its inception in 1996 \n[12], several theoretical models and experimental evi dences based on different microscopic \nmechanisms, e.g. spin -flip scattering (SFS) and super -diffusive spin current have been put \nforward to interpret ultrafast demagnetization [13-20]. However, the preceding proposals are \ncomplex and deterring to each othe r. This complexity increases even more in case of special \nclass of material such as the Heusler alloys. The electronic band structure and the associated \nposition of Fermi level can be greatly tuned by tuning the alloy composition of Heusler alloy \n[21,22]. By utilizing this tunability, h ere, we experimentally demonstrate that the ultrafast \ndemagnetization mechanism relies on the spin density of states at Fermi level in case of half -\nmetallic Heusler alloy system. We extracted the value of ultrafast demagnetiz ation time using \nthree temperature modelling [23] and found its non -monotonic dependency on alloy \ncomposition ( x). We have further showed that the Gilbert damping and ultrafast \ndemagnetization time are inversely proportional in CFMS Heusler alloys suggesti ng the inter -\nband scattering as the primary mechanism behind the Gilbert damping in CFMS Heusler alloys . \nOur work has established a unified theory of ultrafast spin dynamics.   A series of Co 2FexMn 1-xSi (CFMS) thin films have been deposited using magnetron co -\nsputtering system for our investigation with  x = 0.00, 0.25, 0.40, 0.50, 0.60, 0.75 and 1.00 . The \nthickness of the CFMS layer was fixed at 30 nm. It is imperative to study the crystalline phase \nwhich is the most crucial parameter that determines other magnetic properties of Heusler alloy. \nPrior to the magnetization dynamics measurement, we invest igate both the crystalline phase as \nwell as growth quality of all the samples. Fig. 1A shows the ex-situ x-ray diffraction (XRD) \npattern for all the samples. The well -defined diffraction peak of CFMS (400) at 2θ = 66.50º \nindicates that the samples are well  crystalline having cubic symmetry. The intense superlattice \npeak at 2θ = 31.90º represents the formation of B2 phase. The presence of other crucial planes \nare investigated by tilting the sample x = 0.4 by 54.5º and 45.2º from the film plane to the \nnormal direction, respectively  and observed the presence of (111) superlattice peak along with \nthe (220) fundamental peak as shown in Fig. 1B and 1C. The presence of (111) superlattice \npeak confirms the best atomic site ordering in the desired L2 1 ordered phase, whereas the (220) \nfundamental peak results from the cubic symmetry.  The intensity ratios of the XRD peaks are \nanalysed to obtain the microscopic atomic site ordering which remain same for the whole range \nof x (given in Supplemental Materials). The epitaxia l growth of the thin films is ensured by \nobserving the in-situ reflection high -energy electron diffraction (RHEED) images. The square \nshaped hysteresis loops obtained using in -plane bias magnetic field shows the samples have in -\nplane magnetization. The nearly increasing trend of saturation magnetization with alloy \ncomposition ( x) follow the Slater -Pauling curve. In -depth details of sample deposition \nprocedure, RHEED pattern and the hysteresis loops are provided in the Supplemental Materials \n[24]. The ultrafast demagnetization dynamics measurements using  time-resolved magneto -\noptical Kerr effect (TRMOKE) magnetometer have been performed  at a fixed probe fluence of \n0.5 mJ/cm2, while the pump fluence have been varied over a large range . Details of the \nTRMOKE technique is provided in Supplemental Materials [24]. The experi mental data of \nvariation of Kerr rotation corresponding to the ultrafast demagnetization  measured for pump \nfluence = 9.5 mJ/cm2 is plotted in Fig. 2A for different values of x. The data points are then \nfitted with a phenomenological expression derived from  the three temperature  model -based \ncoupled rate equations in order to extract the ultrafast demagnetization time (\nMτ) and fast \nrelaxation (\nEτ) time [23], which is given below:  \n  \nME/τ - /τ- 1 2 E 1 M E 1 2\nk3 1/2\n0 E M E MA (A τ -A τ ) τ (A -A )-Δ {[ - e - e ]H( ) A δ( )} G( )( / t 1) ( τ -τ ) (τ -τ )ttθ t t tt= + +       (1)    where A1 represents the magnetization amplitude after equilibrium between electron, spin and \nlattice is restored,  A2 is proportional to the maximum rise in the electron temperature and  A3 \nrepresents the state filling effects during pump -probe temporal overlap described by a Dirac \ndelta function.  H(t) and δ(t) are the Heaviside step and Dirac delta functions , and G(t) is a \nGaussian function which corresponds to the laser pulse.  \nThe \nMτ  extracted from the fit s are plotted as a function of x in Fig. 2B, which shows  a slight \ninitial increment followed by a sharp  decrement with x. In addition, the ultrafast \ndemagnetization rate is found to be slower in the present Heusler alloys than in the 3d metals \n[9]. The theoretical calculation of electronic band structure of CFM S showed no discernible \nchange in the amount of energy gap  in minority spin band  but a change in position of EF with \nx, which lies at the two extreme ends of the gap for x = 0 and x = 1. Thus, the variation of \nMτ  \nwith x clearly indicates  that the composition dependent EF position is somehow  responsible for \nthe variation in  \nMτ . This warrants the investigation of ultrafast demagnetization with \ncontinuously varying x values between 0 and 1. However, a majority of earlier investigations \n[10,11,2 5], being focused on exploring the ultrafast demagnetization only of Co 2MnSi ( x = 0) \nand Co 2FeSi ( x = 1), lack a convincing conclusion about the role of  electronic band structure \non ultrafast demagnetization  mechanism .  \nIn case of 3d transition metal ferromagnets, Elliott -Yafet (EY) -based SFS mechanism is \nbelieved to be responsible for rapid rise in the spin temperature and ultrafast demagnetization \n[15]. In this theory it has been shown that a  scattering event of an excited electron with a \nphonon changes the probability to find that electron in one of the spin states, namely the \nmajority spin -up (\n ) or minority spin -down (\n ) state, thereby delivering angular momentum \nto the lattice from the electronic system. It arises from the band mixing of majority and minority \nspin states with similar energy value near the Fermi surface owing to the spin -orbit coupling \n(SOC). The spin mixing para meter (b2) from the EY theory [26,27] is given by:  \n                             \n2\nk k k k b min ( ψ ψ , ψ ψ )=                                                               (2) \nwhere \nkψ  represent the eigen -state of a single electron and the bar denotes a defined average \nover all electronic states involved in the EY scattering processes. This equation represents that \nthe spin-mixing due to SFS between spin -up and spin -down states depend o n the  number of  \nspin-up (\n ) and spin -down (\n ) states  at the Fermi level, which is already represented by D F.  A compact differential equation regarding rate of ultrafast demagnetization dynamics as \nderived by Koopmans et al. [27], is given below:  \n                                                  \np C\nCeT TR (1 coth( ))TTm dmmdt=−                                                         (3) \nwhere m = M/MS, and Tp, TC, and Te denote the phonon temperature, Curie temperature and \nelectronic temperature, respectively. R is a material specific scaling factor [28], which is \ncalculated to be:  \n                                                               \n2\nsf C ep\n2\nB D S8a T gRk T D=   ,                                                                  (4) \nwhere asf, gep, DS represent the SFS probability, coupling between electron and phonon sub -\nsystem and magnetic moment divided by the  Bohr -magneton (\nB ), whereas TD is the Debye \ntemperature and kB represents the Boltzmann constant. Further, the expression for gep is: \n22\nF P B D ep\nep3πD D k T λg2=\n, where  DP, and λep denote the number of polarization states of spins and \nelectron -phonon coupling constant, respectively , and ℏ is the reduced Planck’s constant. \nMoreover, the ultrafast demagnetization time at low fluence limit can be derived under various \napproximations as:  \n                                                         \n0C\nM 22\nF si B CC F( / T )τπD λ k TT=\n ,                                                               (5)  \nwhere C0 = 1/4, \nsiλ  is a factor scaling with impurity concentration, and F(T/TC) is a function \nsolely dependent on ( T/TC) [29].  \nEarlier, it has been shown that a negligible DF in CrO 2 is responsible  for large ultrafast \ndemagnetization time. The theoretical calculation  for CFMS  by Oogane et al.  shows that DF \ninitially decreases and then increases with x [30] having  a minima at x = 0.4. As DF decreases, \nthe number of effective minority spin states become less, reducing both SOC strength, as shown \nby Mavropoulos et al. [31], and the effective spin -mixing paramet er is given by Eq. (2), and \nvice versa. This will result in a reduced SFS probability  and rate of demagnetization. In \naddition, the decrease in DF makes gep weaker, which, in turn, reduces the value of R as evident \nfrom Eq. (4). As the value of R diminishes, it will slow down the rate of ultrafast \ndemagnetization which is clear from Eq. (3). In essence , a lower value of DF indicates a lower  value of R, i.e. slower demagnetization rate and larger  ultrafast  demagnetization time. Thus, \ndemagnetization time is highest for x = 0.4. O n both sides of x = 0.4, the value of R will increase \nand ultrafast demagnetization time will decline continuously.  Our experimental results, \nsupported by the existing theoretical re sults for  the CFMS samples with varying alloy \ncomposition, clearly show that the position of Fermi level is a crucial decisive factor for the \nrate of ultrafast demagnetization. This happens due to the continuous tunability of DF with x, \nwhich causes an ensuing variation in the number of scattering channels available for SFS.  To \ncapture the effect of pump fluence on the variation of  \nMτ, we have measured the ultrafast \ndemagnetization curves for various applied pump fluences. All the flu ence dependent ultrafast \ndemagnetization curves are fitted with Eq. (1) and the values of corresponding  \nMτ are \nextracted. The change in \nMτ  with fluence is shown in Fig. 2C. A slight change in \nMτ with \nfluence is observed which is negligible in comparison to the change of \nMτ  with x. However, \nthis increment can be explained using the enhanced spin fluctuations at much higher elevated \ntemperature of the spin sy stem [28].  \nAs the primary microscopic channel for spin angular momentum transfer is the same for both \nultrafast demagnetization and magnetic damping, it is expected to find a correlation between \nthem. We have measured the time -resolved Kerr rotation data corresponding to the \nmagnetization precession at an applied in -plane bias magnetic field  (Hb) of 3.5 kOe  as shown \nin Fig. 3A. The macrospin modelling is employed to analyse the time dependent precessional  \ndata obtained by solving the Landau -Lifshitz -Gilbert equation [32] which is given below:  \n                                                 \neffˆˆˆˆγ( ) α( )dm dmm H mdt dt=−  +                                                    (6) \nwhere \nγ  is the gyromagnetic ratio and is related to Lande g factor by  \n/μg=γB . Heff is the \ntotal effective magnetic field consisting of Hb, exchange field ( Hex), dipolar field ( Hdip) and \nanisotropy field (\nKH ). The experimental variation of precession frequency ( f) against Hb is \nfitted with the Kittel formula for uniform precession to extract HK values. The details of the fit \nare discussed in the Supplementa l Materials [24] . \nFor evaluation of  \nα, all the measured data representing single frequency oscillation are fitted \nwith a general damped sine -wave equation superimposed on a bi -exponential decay function, \nwhich is given as:                                  \nfast slow/τ /τ /τ\n12 ( ) A B e B e (0)e sin( ω ζ)tt tM t M t−− −= + + + −  ,                (7) \nwhere \nζ is the initial phase of oscillation and  \nτ is the precessional relaxation time . \nfastτ  and \nslowτ\n are the fast and slow relaxation times, representing the rate of energy transfer in between \ndifferent energy baths (electron, spin and lattice) following the ultrafast demagnetization and \nthe energy transfer rate between the lattice and surrounding, respec tively. A, B1 and B2 are \nconstant coefficients. The value of \nα  is extracted by further analysing \nτ  using  \n                                        \n( )122α[γτ 2 cos( H H ]=− + +bHδφ                                                         (8)  \nwhere \n22\n12\n1S\nS S S2K 2K sin K (2 sin (2 ))4πMM M MφφH⊥ −= + − +  and \n12\n2\nSS2K cos(2 ) 2K cos(4 )\nMMφφH=+ . Here \n\nand \n  represent the angles of Hb and in -plane equilibrium M with respect to the CFMS [110] \naxis [33]. The uniaxial, biaxial and out -of-plane magnetic anisotropies are denoted as K1, K2 \nand \nK⊥, respectively. In our case K2 has a reasonably large value while K1 and \nK⊥  are \nnegligibly small. Plugging in all parameters including the magnetic anisotropy constant K2 in \nEq. (8), we have obtained the values of \nα  to be 0.0041, 0.0035, 0.0046, 0.0055, 0.0061, and \n0.0075 for x = 0.00, 0.40, 0.50, 0.60, 0.75, and 1.00, respectively. Figure 3B shows the variation \nof \nα  with frequency for all the samples. For each sample, \nα  remains constant with frequency, \nwhich rules out the presence of extrinsic mechanisms contributing to the  \nα. Next, we focus on \nthe variation of \nα  with x. Our experimental results show a non -monotonic variation of  \nα with \nx with a minima at x = 0.4 , which  is exactly opposite to the variation of \nMτ  with x. On the basis \nof Kambersky’s SFS model [34], \nα is governed by the spin -orbit interaction and can be \nexpressed as:  \n                                                      \n22\nF\nSγ (δg)αD4ΓM=\n                                                           (9) \nwhere \ngδ and \n1− represent the deviation of g factor from free electron value (~2.0) and \nordinary electron -phonon collision frequency. Eq. (9) suggests that \nα  is directly proportional \nto DF and thus it become s minimum when DF is minimum [3 0]. This leads to the non -monotonic \nvariation of\nα , which agrees well with earlier observation [30]. To eliminate the possible effects  of \nγ  and \nSM , we have plotted the variation of relaxation frequency, \nSMαγ=G  with x which \nalso exhibits similar variation as  \nα (see the supplementary materials [24] ).  \nFinally , to explore the correlation between\nα , \nMτ and alloy composition, we have plotted these \nquantities  against x as shown in Fig. 4A. We observe that \nMτ  and \nα  varies in exactly opposite \nmanner with x, having their respective maxima and minima at x = 0.4. Although \nMτ  and \nα  \nrefer to two different time scales, both of them follow the trend of variation of  DF with x. This \nshows  that the alloy composition -controlled Fermi level tunability and the ensuing SFS is \nresponsible for both ultrafast demagnetization and Gilbert damping . Figure 4B represents the \nvariation of \nMτ  with inverse of  \nα, which establishes an inversely proportional relation between \nthem . Initially under the assumption of two different magnetic fields, i.e. exchange field and \ntotal effective magnetic field, Koopmans et al.  theoretically proposed that Gilbert damping \nparame ter and ultrafast demagnetization time are inversely proportional [29]. However, that \nraised intense debate and in 2010, Fahnle et al.  showed that \nα  can either be proportional or \ninversely proportional to \nMτ  depending upon the dominating microscopic contribution to the \nmagnetic damping [32]. The linear relation sustains when the damping is dominated by \nconductivity -like contribution, whereas the resistivity -like contribution leads to an inverse \nrelation. The basic difference between the conductivity -like and the resistivity -like \ncontribution s lies in the angular momentum transfer mechanism via electron -hole ( e-h) pair \ngeneration. The generation of e-h pair in the same band, i.e. intra -band mechanism leads to t he \nconductivity -like contribution. On the contrary, when e-h pair is generated in different bands \n(inter -band mechanism), the contribution is dominated by resistivity. Our observation of the \ninversely proportional relation between \nα  and \nMτ  clearly indicates that in case of the CFMS \nHeusler alloy  systems, the damping is dominated by resistivity -like contribution arising from \ninter-band e-h pair generation. This is in contrast to the case of Co, Fe and Ni, where the \nconductivity contribution dominates [35]. Typical resistivity (\nρ ) values for Co 2MnSi ( x = 0) \nare 5\ncm− at 5 K and 20 \ncm− at 300 K [36]. The room temperature value of \nρ\ncorresponds to an order of magnitude larger contribution of the inter -band e-h pair generation \nthan the intra -band generation [36]. This is in strong agreement with our experimental results \nand its conclusion. This firmly establishes that unlike convention al transition metal \nferromagnets, damping in CFMS Heusler alloys  is dominated by resistivity -like contribution , \nwhich results in an inversely proportional relation between \nα  and\nMτ .  In summary, we have investigated the ultrafast demagnetization and magnetic Gilbert damping \nin the CFMS Heusler alloy systems with varying alloy composition ( x), ranging from x = 0 \n(CMS) to x = 1 (CFS) and identified a strong correlation between \nMτ  and x, the latter \ncontrolling the position of Fermi level in the electronic band structure of the system. We have \nfound that \nMτ  varies non -monotonically with x, having a maximum value of ~  350 fs for x = \n0.4 corresponding to the lowest DF and highest degree of spin -polarization. In -depth \ninvestigation has revealed that the ultrafast demagnetization process in CFMS is primarily \ngoverned by the composition -controlled variation in spin -flip scattering rate  due to variable DF. \nFurthermore, we have systematically investigated the precessional dynamics with variation in \nx and extracted the value of \nα  from there. Our results have led to a systematic correlation in \nbetween\nMτ ,\nα and x and we have found an inversely proportional relationship between \nMτ  and \nα\n. Our thorough investigation across the alloy composition ranging from CMS to CFS have \nfirmly establishe d the fact that both ultrafast demagnetization and magnetic  Gilbert  damping \nin CFMS are strongly controlled by the spin density of states at Fermi level. Therefore, our \nstudy has enlighten ed a new path for qualitative understanding of spin -polarization from \nultrafast demagnetization time as well as magnetic  Gilbert  dampin g and led a step forward for \nultrafast magnetoelectronic device applications.  \nAcknowledgements  \nThis work was funded by: S. N. Bose National Centre for Basic Sciences under Projects No. \nSNB/AB/12 -13/96 and No. SNB/AB/18 -19/211.  \nReferences  \n[1] T. Kubota,  S. Tsunegi,  M. Oogane,  S. Mizukami, T. Miyazaki,  H. Naganuma, and Y. Ando,   \nAppl. Phys. Lett.  94, 122504 (2009).  \n[2] A. Hirohata , and K. Takanashi, J. Phys. D: Appl. Phys.  47, 193001 (2014).  \n[3] R. J. Soulen  et al., Science  282, 85 (1998).  \n[4] I. I. Mazin, Phys. Rev. Lett.  83, 1427 (1999).  \n[5] K. E. H. M. Hanssen, P. E.  Mijnarends, L. P. L. M. Rabou, and K. H. J. Buschow, Phys. Rev. B  \n42, 1533 (1990).  \n[6] L. Ritchie , Phys. Rev. B  68, 104430 (2003).  \n[7] D. T. Pierce , and F. Meier,  Phys. Rev. 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Lett.  95, 022509 \n(2009).    \n \n \n \n \n \n \n \n \n \n \n    \n \nFig. 1. (A)  X-ray diffraction (XRD) patterns of Co 2FexMn 1-xSi (CFMS) thin films for different alloy \ncomposition ( x) measured in conventional θ-2θ geometry. Both CFMS (200) superlattice and CFMS \n(400) fundamental peaks are marked along with Cr (200) peak. (B) The tilted XRD patterns reveal the \nCFMS (111) superlattice peak for L2 1 structure. (C) CFMS (220) fundamental peak together with Cr \n(110) peak.  \n \n \n \n \n \n \n \n \n  \nFig. 2. (A) Ultrafast demagnetization curves for the samples with different alloy composition ( x) \nmeasured using TRMOKE. Scattered symbols  are the experimental data and solid lines are fit using Eq. \n3. (B) Evolution of \nMτ  with x at pump fluence of 9.5 mJ/cm2. Symbols are experimental results and \ndashed line is guide to eye.  (C) Variation in \nMτ  with pump fluence.  \n \n \n \n \n  \nFig. 3. (A) Time -resolved Kerr rotation data  showing precessional dynamics  for samples with different \nx values . Symbols  are the experimental data and solid lines are fit with damped sine wave equation ( Eq. \n6). The extracted \nα  values are given  below  every curve. (B) Variation of \nα  with precession frequency \n(f) for all samples as shown by symbols, while  solid lines are linear fit.  \n \n \n \n \n \n \n \n \n \n  \n \nFig. 4. (A) Variation of \nMτ  and \nα  with x. Square and circular symbols  denote the experimental results , \nand dashed , dotted  lines are guide to eye.  (B) Variation of \nMτ  with \n1α− . Symbols represent the \nexperimentally obtained values and solid line refers to linear fit.  \n \n \n \n \n \n \n \n \n \n \n \n \n Supplementa l Material s \n \nI. Sample preparation method  \nA series of MgO Substrate /Cr (20 nm)/ Co 2FexMn 1-xSi (30 nm)/Al -O (3 nm) sample stacks \nwere deposited using an ultrahigh vacuum magnetron co -sputtering system. First a 20 -nm-thick \nCr layer was deposited on top of a single crystal MgO (100) substrate at room temperature \n(RT) followed by annealing it at 600 ºC for 1 h. Next, a Co 2FexMn 1-xSi layer of 30 nm thickness \nwas deposited on the Cr layer followed by an in -situ annealing process at 500 ºC for 1 h. \nFinally, each sample stack was capped with a 3 -nm-thick Al -O protective layer. A wide range \nof values of x is chosen, namely, x = 0.00, 0.25, 0.40, 0.50, 0.60, 0.75 and 1.00. To achieve the \ndesired composition of Fe and Mn precisely, the samples were deposited using well controlled \nco-sputtering of Co 2FeSi and Co 2MnSi. Direct deposition of Co 2FexMn 1-xSi on top of MgO \nproduces strain  due to lattice mismatch in the Co 2FexMn 1-xSi layer which alters its intrinsic \nproperties [1S]. Thus, Cr was used as a buffer layer to protect the intrinsic Co 2FexMn 1-xSi layer \nproperties [2S].  \nII. Details of measurement techniques  \nUsing ex-situ x-ray diffraction ( XRD ) measurement we investigated the crystal structure and \ncrystalline phase of the samples. The in-situ reflection high -energy electron diffraction  \n(RHEED ) images were observed after the layer deposition without breaking the vacuum \ncondition in order to investigate the epitaxial relation and surface morphology of Co 2FexMn 1-\nxSi layer. To quantify the values of M S and H C of the samples, we measured the  magnetization \nvs. in -plane magnetic field  (M-H) loops using a vibrating sample magnetometer ( VSM) at room \ntemperature with H directed along the [110] direction of Co 2FexMn 1-xSi. The ultrafast \nmagnetization dynamics for all the samples were measured by using a time-resolved magneto -\noptical Kerr effect ( TRMOKE ) magnetometer [ 3S]. This is a two -colour  pump -probe \nexperiment in non -collinear arrangement. The fundamental output (wavelength, λ = 800 nm, \npulse -width, \ntσ ~ 40 fs) from an amplified laser system (LIBRA, Coherent) acts as probe and \nits second harmonic signal (λ = 400 nm, \ntσ ~ 50 fs) acts as pump beam. For investigating both \nultrafast demagnetization within few hundreds of femtosecond s and p recessional \nmagnetization dynamics in few hundreds of picosecond time scale, we collected the time -\nresolved Kerr signal in two different time regimes. The time resolution during the  measurements was fixed at 50 fs in -0.5 To 3.5 ps and 5 ps in -0.1 ns to 1 .5 ns to trace both the \nphenomena precisely. The pump and probe beams were focused using suitable lenses on the \nsample surface with spot diameters of ~250 µm and ~100 µm, respectively. The reflected signal \nfrom the sample surface was collected and analysed  using a polarized beam splitter and dual \nphoto detector assembly to extract the Kerr rotation and reflectivity signals separately. A fixed \nin-plane external bias magnetic field ( Hb) of 1 kOe was applied to saturate the magnetization \nfor measurement of ult rafast demagnetization dynamics, while it was varied over a wide range \nduring precessional dynamics measurement.  \nIII. Analysis of XRD peaks  \nTo estimate the degree of Co atomic site ordering, one has to calculate the ratio of integrated \nintensity of (200) and (400) peak. Here, we fit the peaks with Lorentzian profile  as shown in \ninset of Fig.  1S and extracted the integrated intensities as a parameter  from the fit . The \ncalculated ratio of I(200) and I(400) with re spect to alloy composition ( x) is sho wn in Fi g. 1S. \nWe note that there is no significant change in the I(200)/I(400) ratio. This result indicates an \noverall good quality atomic site ordering in the broad range of samples used  in our study . \n \nFig. 1S. Variation of integrated intensity ratio I(200)/I(40 0) with x, obtained from XRD patterns. Inset \nshows the fit to the peaks with Lorentzian profile.  \n \nIV. Analysis of RHEED pattern  \n The growth quality of the CFMS thin films was experimentally investigated using in-situ \nRHEED technique. Figure 2S shows the RHEED images captured  along the MgO [100] \ndirection for all the samples. All the images contain main  thick  streak lines in between the thin \nstreak lines , which are marked by the white arrows, suggesting the formation o f ordered phases. \nThe presence of regularly -aligned streak lines confirms the epitaxial growth in all the films.     \n \nFig. 2S. In-situ RHEED images for all the Co 2FexMn 1-xSi films taken along the MgO [100] direction. \nWhite arrows mark the presence of thin streak lines originating from the L2 1 ordered phase.  \n \nV. Analysis of magnetic hysteresis loops  \nFigure 3SA represents the M-H loops measured at room temperature  using VSM for all the \nsamples. All the loops are square in nature, which indicates a very small saturation magnetic \nfield. We have estimated the values of saturation magnetization ( MS) and coercive field ( HC) \nfrom the M-H loops.  Figure 3SB represents MS as a function of x showing a nearly monotonic \nincreasing trend, which is consistent with the Slater -Pauling rule for Heusler alloys [4S], i.e. \nthe increment in MS due to the increase in the number of valence electrons. However, it deviates \nremarkably at x = 1.0. This deviation towards the Fe -rich region is probably due to the slight \ndegradation in the film quality. Figure 3SC shows that HC remains almost constant with \nvariation of x. \n   \nFig. 3S. (A) Variation of M with H for all the samples. (B) Variation of MS as a function of x. \nSymbols are experimentally obtained values and dashed line is a linear fit. (C) Variation of HC \nwith x.    \n \nVI. Anal ysis of frequency ( f) versus bias magnetic field ( Hb) from TRMOKE \nmeasurements  \nWe have experimentally investigated the precessional dynamics of all the samples using \nTRMOKE technique. By varying the external bias magnetic field ( Hb), various precessional \ndynamics have been measured. The post -processing of these data foll owed by fast Fourier \ntransform (FFT) provides the precessional frequency (f) and this is plotted against Hb as shown \nin Fig. 4S .  \nTo determine the value of in-plane magnetic anisotropy  constant , obtained f-Hb curves have \nbeen analysed  with Kittel formula  which is given below:  \n                                  \n2 1 2\nS\nS S S2K 2K 2K γ(4πM )( )2π M M Mbb f H H= + + + +\n                                      (1S) \n  where MS is saturation magnetization and \nγ  denote the gyromagnetic ratio given by\nBgμγ=\nwhile K1 and K2 represent the two -fold uniaxial and four -fold biaxial magnetic anisotropy \nconstant, respectively.  \n \nFig. 4S. Variation of f as a function of Hb. Circular filled symbols represent the experimental data and \nsolid lines are Kittel fit.  \n \nWe have found the values of several parameters from the fit including K1 and K2. K1 has a \nnegligible value while K2 has reasonably large value in our samples. The e xtracted values of \nthe parameters from the fit are tabulated as follows  in Table 1S : \nTable 1S: The extracted values of Lande g factor and the four -fold biaxial magnetic anisotropy \nconstant K 2 for different values of x. \nx g K2 (erg/cm3) \n0.00 2.20 3.1×104 \n0.40 2.20 2.6×104 \n0.50 2.20 3.0×104 \n0.60 2.20 2.5×104 \n0.75 2.20 2.6×104 \n 1.00 2.20 3.4×104 \n \nVII. Variation of r elaxation frequency  with alloy composition   \nWe have estimated the damping coefficient (α) and presented its variation with alloy \ncomposition ( x) in the main manuscript. According to the Slater -Pauling rule, M S increases \nwhen the valence electron number systematically increases. As in our case the valence electron \nnumber changes with x, one may expect a marginal effect of M S on the estimation of damping. \nThus, to rule out any such possibilit ies, we have calculated the  variation of  relaxation \nfrequency ,\nS GαγM= with x, which is represented in Fig. 5S. It can be clearly observed from \nFig. 5S that relaxation frequen cy exactly follows the trend of\nα . This rules out any possible  \nspurious contribution of M S in magnetic damping.   \n \nFig. 5S. Non-monotonic v ariation of G with x for all the samples.  \n \nReferences:  \n[1S] S. Pan, S. Mondal,  T. Seki,  K. Takanashi, , and A. Barman, Influence of the thickness -dependent \nstructural evolution on ultrafast magnetization dynamics in Co 2Fe0.4Mn 0.6Si Heusler alloy thin films.  \nPhys. Rev. B  94, 184417 (2016).  \n[2S] S. Pan, T. Seki, K. Takanashi, and A. Barman, Role of the Cr buffer layer in the thickness -\ndependent ultrafast magnetization dynamics of Co 2Fe0.4Mn 0.6Si Heusler alloy thin films.  Phys. Rev. \nAppl.  7, 064012 (2017).  \n[3S] S. Panda,  S. Mondal, J. Sinha,  S. Choudhury, and A. Barman, All-optical det ection of interfacial \nspin transparency from spin pumping in β -Ta/CoFeB thin films.  Science Adv.  5, eaav7200 (2019).  \n[4S] I. Galanakis,  P. H.  Dederichs, and  N. Papanikolaou, Slater -Pauling behavior and origin of half -\nmetallicity of the full Hesuler alloys.  Phys. Rev. B  66, 174429 (2002).  \n  "    },    {        "title": "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. 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Tatara,  Applied Physics Letters 88 (18) \n(2006).  \n30 T. Tashiro, R. Takahashi, Y. Kajiwara, K. Ando, H. Nakaya ma, T. Yoshino, D. \nKikuchi, and E. Saitoh, presented at the Conference on Spintronics V, San Diego, CA, \n2012 (unpublished).  \n31 K. Ando, S. Takahashi, K. Harii, K. Sasage, J. Ieda, S. Maekawa, and E. Saitoh,  \nPhysical Review Letters 101 (3), 4 (2008).  \n32 V. E. Demidov, S. Urazhdin, H. Ulrichs, V. Tiberkevich, A. Slavin, D. Baither, G. \nSchmitz, and S. O. Demokritov,  Nature Materials 11 (12), 1028 (2012).  \n33 M. Madami, S. Bonetti, G. Consolo, S. Tacchi, G. Carlotti, G. Gubbiotti, F. B. \nMancoff, M. A. Yar, a nd J. Akerman,  Nature Nanotechnology 6 (10), 635 (2011).  9 Figure Captions :  \nFigure 1 : \n(color online). (a) 1 µm x 1µm AFM surface topography of a 20nm YIG film on GGG (111) \n(RMS roughness=0.23  nm). (b) XRD  : \n-2\n scan of the same film using Cu -K\n1 radiation. \nThis spectrum shows that the growth is along the (111) direction with no evidence of parasitic \nphases. The YIG peak are best resolved at high diffraction angle, a zoom around the 888 GGG \npeak is shown in panel (c). \n(c) (d) (e) XRD diffraction s pectrum centered on GGG (888) reflection for different YIG \nthicknesses: 20, 7 and 4 nm. For the 2 thinnest films, the YIG peak is masked by the substrate \npeak . The cubic lattice parameter of the 20 nm thick YIG film (obtained from 888 peak) is \nslightly lar ger than that of the substrate ( aYIG=1.2459 nm , aGGG=1.2383  nm) \n \nFigure 2 : \n(color online) ( a) (b). FMR absorption derivative spectra of 20 and 4 nm thick YIG films at an \nexcitation frequency of 6 GHz.  \n(c) rf excitation frequency dependence of FMR absorption linewidth measured on different \nYIG film thicknesses with an in -plane oriented static field. The black continuous line is a \nlinear fit on the 20  nm thick film from w hich a Gilbert damping coefficient of 2. 3 x 10-4 can \nbe inferred  ( . The damping of the 7 nm and 4 nm films is \nsignificantly larger but must off all the frequency dependence is not linear (see text for \ndiscussion).   \n \nFigure 3 : \n(a) Schematic illustration of the experimental setup: spin waves are excited through the YIG \nwaveguide with a microstrip antenna. The detection is performed by measuring ISHE voltage \ndc signal on a ~200µm wide Pt stripe. (b) External dc magnetic field dependen ce of ISHE \nvoltage measured on Pt electrode showing the polarity inversion when the magnetic field is \nreversed. The peaks occur at the FMR conditions as verified through S 11 spectroscopy on the \nAu antenna (not shown here).  \n(c). (d). (e). ISHE voltage for d ifferent YIG film thicknesses around the resonance magnetic \nfield; microwave frequency is f=3GHz. The peaks can be fitted to a lorentzian shape and the \nextracted linewidth are respectively:  19.2 , 13.2 and 4.6 Oe. The dramatic increase of the \nISHE signal w ith thickness is discussed in the text.   10  \n \nFig 1  \n \n \n  \n11  \nFig 2  \n  \n12  \nFig 3  \n"    },    {        "title": "2202.06154v1.Generalization_of_the_Landau_Lifshitz_Gilbert_equation_by_multi_body_contributions_to_Gilbert_damping_for_non_collinear_magnets.pdf",        "content": "Generalization of the Landau-Lifshitz-Gilbert equation by multi-body contributions to\nGilbert damping for non-collinear magnets\nSascha Brinker,1Manuel dos Santos Dias,2, 1,\u0003and Samir Lounis1, 2,y\n1Peter Gr unberg Institut and Institute for Advanced Simulation,\nForschungszentrum J ulich & JARA, 52425 J ulich, Germany\n2Faculty of Physics, University of Duisburg-Essen and CENIDE, 47053 Duisburg, Germany\n(Dated: February 15, 2022)\nWe propose a systematic and sequential expansion of the Landau-Lifshitz-Gilbert equation utilizing\nthe dependence of the Gilbert damping tensor on the angle between magnetic moments, which arises\nfrom multi-body scattering processes. The tensor consists of a damping-like term and a correction\nto the gyromagnetic ratio. Based on electronic structure theory, both terms are shown to depend\non e.g. the scalar, anisotropic, vector-chiral and scalar-chiral products of magnetic moments: ei\u0001ej,\n(nij\u0001ei)(nij\u0001ej),nij\u0001(ei\u0002ej), (ei\u0001ej)2,ei\u0001(ej\u0002ek)..., where some terms are subjected to the\nspin-orbit \feld nijin \frst and second order. We explore the magnitude of the di\u000berent contributions\nusing both the Alexander-Anderson model and time-dependent density functional theory in magnetic\nadatoms and dimers deposited on Au(111) surface.arXiv:2202.06154v1  [cond-mat.mtrl-sci]  12 Feb 20222\nI. INTRODUCTION\nIn the last decades non-collinear magnetic textures have been at the forefront in the \feld of spintronics due to the\npromising applications and perspectives tied to them1,2. Highly non-collinear particle-like topological swirls, like\nskyrmions3,4and hop\fons5, but also domain walls6can potentially be utilized in data storage and processing devices\nwith superior properties compared to conventional devices. Any manipulation, writing and nucleation of these various\nmagnetic states involve magnetization dynamical processes, which are crucial to understand for the design of future\nspintronic devices.\nIn this context, the Landau-Lifshitz-Gilbert (LLG) model7,8is widely used to describe spin dynamics of materials\nranging from 3-dimensional bulk magnets down to the 0-dimensional case of single atoms, see e.g. Refs.9{12. The\nLLG model has two important ingredients: (i) the Gilbert damping being in general a tensorial quantity13, which can\noriginate from the presence of spin-orbit coupling (SOC)14and/or from spin currents pumped into a reservoir15,16;\n(ii) the e\u000bective magnetic \feld acting on a given magnetic moment and rising from internal and external interactions.\nOften a generalized Heisenberg model, including magnetic anisotropies and magnetic exchange interactions, is utilized\nto explore the ground state and magnetization dynamics characterizing a material of interest. Instead of the con-\nventional bilinear form, the magnetic interactions can eventually be of higher-order type, see e.g.17{23. Similarly to\nmagnetic interactions, the Gilbert damping, as we demonstrate in this paper, can host higher-order non-local contri-\nbutions. Previously, signatures of giant anisotropic damping were found24, while chiral damping and renormalization\nof the gyromagnetic ratio were revealed through measurements executed on chiral domain wall creep motion24{28.\nMost \frst-principles studies of the Gilbert damping were either focusing on collinear systems or were case-by-case\nstudies on speci\fc non-collinear structures lacking a general understanding of the fundamental behaviour of the Gilbert\ndamping as function of the non-collinear state of the system. In this paper, we discuss the Gilbert damping tensor\nand its dependencies on the alignment of spin moments as they occur in arbitrary non-collinear state. Utilizing linear\nresponse theory, we extract the dynamical magnetic susceptibility and identify the Gilbert damping tensor pertaining\nto the generalized LLG equation that we map to that obtained from electronic structure models such as the single\norbital Alexander-Anderson model29or time-dependent density functional theory applied to realistic systems10,30,31.\nApplying systematic perturbative expansions, we \fnd the allowed dependencies of the Gilbert damping tensor on the\ndirection of the magnetic moments. We identify terms that are a\u000bected by SOC in \frst and second order. We generalize\nthe LLG equation by a simple form where the Gilbert damping tensor is amended with terms proportional to scalar,\nanisotropic, vector-chiral and scalar-chiral products of magnetic moments, i.e. terms like ei\u0001ej, (nij\u0001ei)(nij\u0001ej),\nnij\u0001(ei\u0002ej), (ei\u0001ej)2,ei\u0001(ej\u0002ek)..., where we use unit vectors, ei=mi=jmij, to describe the directional dependence\nof the damping parameters and nijrepresents the spin-orbit \feld.\nThe knowledge gained from the Alexander-Anderson model is applied to realistic systems obtained from \frst-principles\ncalculations. As prototypical test system we use 3 dtransition metal adatoms and dimers deposited on the Au(111)\nsurface. Besides the intra-site contribution to the Gilbert damping, we also shed light on the inter-site contribution,\nusually referred to as the non-local contribution.\nII. MAPPING THE GILBERT DAMPING FROM THE DYNAMICAL MAGNETIC SUSCEPTIBILITY\nHere we extract the dynamical transverse magnetic response of a magnetic moment from both the Landau-Lifshitz-\nGilbert model and electronic structure theory in order to identify the Gilbert damping tensor Gij10,11,32,33. In linear\nresponse theory, the response of the magnetization mat siteito a transverse magnetic \feld bapplied at sites jand\noscillating at frequency !reads\nm\u000b\ni(!) =X\nj\f\u001f\u000b\f\nij(!)b\f\nj(!); (1)\nwith the magnetic susceptibility \u001f\u000b\f\nij(!) and\u000b;\fare thex;ycoordinates de\fned in the local spin frame of reference\npertaining to sites iandj.\nIn a general form13the LLG equation is given by\ndmi\ndt=\u0000\rmi\u00020\n@Be\u000b\ni+X\njGij\u0001dmj\ndt1\nA; (2)3\nMagnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)\nMagnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)°2°101234Energy [U]°1.00°0.75°0.50°0.250.000.250.500.751.00DOS [#states/U]m=0.2m=0.3m=0.4m=0.5m=0.6m=0.7m=0.8\n°2°101234Energy [U]°1.0°0.50.00.51.0DOS [#states/U]m=0.2m=0.3m=0.4m=0.5m=0.6m=0.7m=0.8dI/dVVexcitationa)b)\nabc\nFIG. 1. Illustration of the Landau-Lifshitz-Gilbert model and local density of states within the Alexander-Anderson model.\n(a) A magnetic moment (red arrow) precesses in the the presence of an external \feld. The blue arrow indicates the direction\nof a damping term, while the green arrow shows the direction of the precession term. (b) Density of states for di\u000berent\nmagnetizations in the range from 0 :2 to 0:8. Density of states of dimers described within the Alexander-Anderson model for\ndi\u000berent magnetizations in the range from 0 :2 to 0:8. Shown is the ferromagnetic reference state. The magnetizations are\nself-consistently constrained using a longitudinal magnetic \feld, which is shown in the inset. Model parameters: U= 1:0 eV,\nEd= 1:0 eV; t= 0:2 eV;\u0000 = 0:2 eV; 'R= 0 °.\nwhere\r= 2 is the gyromagnetic ratio, Be\u000b\ni=\u0000dHspin=dmiis the e\u000bective magnetic \feld containing the contributions\nfrom an external magnetic \feld Bext\ni, as well as internal magnetic \felds originating from the interaction of the\nmoment with its surrounding. In an atomistic spin model described by e.g. the generalized Heisenberg hamiltonian,\nHspin=P\nimiKimi+1\n2P\nijmiJijmj, containing the on-site magnetic anisotropy Kiand the exchange tensor Jij,\nthe e\u000bective \feld is given by Be\u000b\ni=Bext\ni\u0000Kimi\u0000P\njJijmj(green arrow in Fig. 1a). The Gilbert damping tensor\ncan be separated into two contributions { a damping-like term, which is the symmetric part of the tensor, S, (blue\narrow in Fig. 1a), and a precession-like term A, which is the anti-symmetric part of the tensor. In Appendix A we\nshow how the antisymmetric intra-site part of the tensor contributes to a renormalization of the gyromagnetic ratio.\nTo extract the magnetic susceptibility, we express the magnetic moments in their respective local spin frame of\nreferences and use Rotation matrices that ensure rotation from local to globa spin frame of reference (see Appendix B).\nThe magnetic moment is assumed to be perturbed around its equilibrium value Mi,mloc\ni=Miez\ni+mx\niex\ni+my\niey\ni,\nwhere e\u000b\niis the unit vector in direction \u000bin the local frame of site i. Using the ground-state condition of vanishing\nmagnetic torques, Miez\ni\u0002\u0000\nBext\ni+Bint\ni\u0001\n= 0 and the inverse of the transverse magnetic susceptibility can be identi\fed\nas\n\u001f\u00001\ni\u000bj\f(!) =\u000eij\u0012\n\u000e\u000b\fBe\u000b\niz\nMi+i!\n\rMi\u000f\u000b\f\u0016\u0013\n+1\nMiMj(RiJijRT\nj)\u000b\f+ i!(RiGijRT\nj)\u000b\f; (3)\nfrom which it follows that the Gilbert damping is directly related to the linear in frequency imaginary part of the\ninverse susceptibility\nd\nd!=[\u001f\u00001]\u000b\f\nij=\u000eij\u00121\n\rMi\u000f\u000b\f\u0016\u0013\n+ (RiGijRT\nj)\u000b\f: (4)\nNote thatRiandRjare rotation matrices rotating to the local frames of site iandj, respectively, which de\fne the\ncoordinates \u000b;\f=fx;yg(see Appendix B).\nBased on electronic structure theory, the transverse dynamical susceptibility can be extracted from a Dyson-like\nequation:\u001f\u00001(!) =\u001f\u00001\n0(!)\u0000U, where\u001f0is the susceptibility of non-interacting electron while Uis a many-body\ninteraction Kernel, called exchange-correlation Kernel in the context of time-dependent density functional theory30.\nThe Kernel is generally assumed to be adiabatic, which enables the evaluation of the Gilbert damping directly from4\nthe non-interacting susceptibility. Obviously:d\nd!\u001f\u00001(!) =d\nd!\u001f\u00001\n0(!). For small frequencies !,\u001f0has a simple\n!-dependence11:\n\u001f0(!)\u0019<\u001f0(0) +i!=d\nd!\u001f0j!=0 (5)\nand as shown in Ref.33\nd\nd!\u001f\u00001\n0(!)\u0019[<\u001f0(0)]\u00002=d\nd!\u001f0j!=0: (6)\nStarting from the electronic Hamiltonian Hand the corresponding Green functions G(E\u0006i\u0011) = (E\u0000H\u0006i\u0011)\u00001, one\ncan show that the non-interacting magnetic susceptibility can be de\fned via\n\u001f\u000b\f\n0;ij(!+ i\u0011) =\u00001\n\u0019TrZEF\ndE\u0002\n\u001b\u000bGij(E+!+ i\u0011)\u001b\fImGji(E) +\u001b\u000bImGij(E)\u001b\fGji(E\u0000!\u0000i\u0011)\u0003\n;(7)\nwith\u001bbeing the vector of Pauli matrices. Obviously to identify the Gilbert damping and how it reacts to magnetic\nnon-collinearity, we have to inspect the dependence of the susceptibility, and therefore the Green function, on the\nmisalignment of the magnetic moments.\nIII. MULTI-SITE EXPANSION OF THE GILBERT DAMPING\nAssuming the hamiltonian Hconsisting of an on-site contribution H0and an inter-site term encoded in a hopping\ntermt, which can be spin-dependent, one can proceed with a perturbative expansion of the corresponding Green\nfunction utilizing the Dyson equation\nGij=G0\ni\u000eij+G0\nitijG0\nj+G0\nitikG0\nktkjG0\nj+::: : (8)\nWithin the Alexander-Anderson single-orbital impurity model29,H0\ni=Ed\u0000i \u0000\u0000Uimi\u0001\u001b\u0000Bi\u0001\u001b, whereEd\nis the energy of the localized orbitals, \u0000 is the hybridization in the wide band limit, Uiis the local interaction\nresponsible for the formation of a magnetic moment and Biis an constraining or external magnetic \feld. SOC can be\nincorportated as tsoc\nij=i\u0015ijnij\u0001\u001b, where\u0015ijandnij=\u0000njirepresent respectively the strength and direction of the\nanisotropy \feld. It can be parameterized as a spin-dependent hopping using the Rashba-like spin-momentum locking\ntij=t(cos'R\u001b0\u0000i sin'Rnij\u0001\u001b)34.\nDepending on whether the considered Green function is an on-site Green function Giior an inter-site Green function\nGijdi\u000berent orders in the hopping are relevant. On-site Green functions require an even number of hopping processes,\nwhile inter-site Green functions require at least one hopping process.\nThe on-site Green function G0\nican be separated into a spin-less part Niand a spin dependent part Mi,\nG0\ni=Ni\u001b0+Mi\u0001\u001b ; (9)\nwhere the spin dependent part is parallel to the magnetic moment of site i,Mikmi(note that SOC is added later on\nto the hoppings). Using the perturbative expansion, eq. (8), and the separated Green function, eq. (9), to calculate\nthe magnetic susceptibility, eq. (7), one can systematically classify the allowed dependencies of the susceptibility with\nrespect to the directions of the magnetic moments, e.g. by using diagrammatic techniques as shown in Ref.18for a\nrelated model in the context of higher-order magnetic exchange interactions.\nSince our interest is in the form of the Gilbert damping, and therefore also in the form of the magnetic susceptibility, the\nperturbative expansion can be applied to the magnetic susceptibility. The general form of the magnetic susceptibility\nin terms of the Green function, eq. (7), depends on a combination of two Green functions with di\u000berent energy\narguments, which are labeled as !and 0 in the following. The relevant structure is then identi\fed as33,\n\u001f\u000b\f\nij(!)\u0018Tr\u001b\u000b\niGij(!)\u001b\f\njGji(0): (10)\nThe sake of the perturbative expansion is to gather insights in the possible forms and dependencies on the magnetic\nmoments of the Gilbert damping, and not to calculate explicitly the strength of the Gilbert damping from this5\nexpansion. Therefore, we focus on the structure of eq. (10), even though the susceptibility has more ingredients,\nwhich are of a similar form.\nInstead of writing all the perturbations explicitly, we set up a diagrammatic approach, which has the following\ningredients and rules:\n1. Each diagram contains the operators NandM, which are \u001b\u000band\u001b\ffor the magnetic susceptibility. The\noperators are represented by a white circle with the site and spin index: i\u000b\n2. Hoppings are represented by grey circles indicating the hopping from site itoj:ij. The vertex corresponds\ntotij.\n3. SOC is described as a spin-dependent hopping from site itojand represented by: ij;\u000b. The vertex\ncorresponds to tsoc\nij=i\u0015ij^n\u000b\nij\u001b\u000b.\n4. The bare spin-independent (on-site) Green functions are represented by directional lines with an energy at-\ntributed to it: !. The Green function connects operators and hoppings. The line corresponds to Ni(!).\n5. The spin-dependent part of the bare Green function is represented by: !;\u000b .\u000bindicates the spin direction.\nThe direction ensures the right order within the trace (due to the Pauli matrices, the di\u000berent objects in the\ndiagram do not commute). The line corresponds to Mi(!)m\u000b\ni\u001b\u000b.\nNote that the diagrammatic rules might be counter-intuitive, since local quantities (the Green function) are represented\nby lines, while non-local quantities (the hopping from itoj) are represented by vertices. However, these diagrammatic\nrules allow a much simpli\fed description and identi\fcation of all the possible forms of the Gilbert damping, without\nhaving to write lengthy perturbative expansions.\nSpin-orbit coupling independent contributions.\nTo get a feeling for the diagrammatic approach, we start with the simplest example: the on-site susceptibility without\nany hoppings to a di\u000berent site, which describes both the single atom and the lowest order term for interacting atoms.\nThe possible forms are,\n\u001fii\n\u000b\f(!)/\n!0\ni\u000b i\f+\n!;\r0\ni\u000b i\f\n+\n!0;\r\ni\u000b i\f+\n!;\u000e0;\r\ni\u000b i\f; (11)6\nwhich evaluate to,\n!0\ni\u000b i\f= Tr\u001b\u000b\u001b\fNi!)Ni(0) =\u000e\u000b\fNi(!)Ni(0) (12)\n!;\r0\ni\u000b i\f= Tr\u001b\u000b\u001b\r\u001b\fMi(!)Ni(0)m\r\ni= i\u000f\u000b\r\fMi(!)Ni(0)m\r\ni (13)\n!0;\r\ni\u000b i\f= Tr\u001b\u000b\u001b\f\u001b\rNi(!)Mi(0)m\r\ni= i\u000f\u000b\f\rMi(!)Mi(0)m\r\ni (14)\n!;\u000e0;\r\ni\u000b i\f= Tr\u001b\u000b\u001b\u000e\u001b\f\u001b\rMi(!)Mi(0)m\u000e\nim\r\ni\n= (\u000e\u000b\u000e\u000e\f\r+\u000e\u000b\r\u000e\f\u000e\u0000\u000e\u000b\f\u000e\r\u000e)Mi(!)Mi(0)m\u000e\nim\r\ni: (15)\nThe \frst diagram yields an isotropic contribution, the second and third diagrams yield an anti-symmetric contribution,\nwhich is linear in the magnetic moment, and the last diagram yields a symmetric contribution being quadratic in the\nmagnetic moment. Note that the energy dependence of the Green functions is crucial, since otherwise the sum of\neqs. (13) and (14) vanishes. In particular this means that the static susceptibility has no dependence linear in the\nmagnetic moment, while the the slope of the susceptibility with respect to energy can have a dependence linear in\nthe magnetic moment. The static part of the susceptibility maps to the magnetic exchange interactions, which are\nknown to be even in the magnetic moment due to time reversal symmetry.\nCombining all the functional forms of the diagrams, we \fnd the following possible dependencies of the on-site Gilbert\ndamping on the magnetic moments,\nG\u000b\f\nii(fmg)/f\u000e\u000b\f;\u000f\u000b\f\rm\r\ni;m\u000b\nim\f\nig: (16)\nSince we work in the local frames, mi= (0;0;mz\ni), the last dependence is a purely longitudinal term, which is not\nrelevant for the transversal dynamics discussed in this work.\nIf we still focus on the on-site term, but allow for two hoppings to another atom and back, we \fnd the following new7\ndiagrams,\n!00\n0\ni\u000b i\fij ji\n+\n!;\r00\n0\ni\u000b i\fij ji\n+:::+\n!;\r0;\u000e0\n0\ni\u000b i\fij ji\n+:::\n+\n!;\r0;\u000e0;\u0011\n0\ni\u000b i\fij ji\n+:::+\n!;\r0;\u000e0;\u0011\n0;\u0017\ni\u000b i\fij ji\n: (17)\nThe dashed line in the second diagram can be inserted in any of the four sides of the square, with the other possibilities\nomitted. Likewise for the diagrams with two or three dashed lines, the di\u000berent possible assignments have to be\nconsidered. The additional hopping to the site jyields a dependence of the on-site magnetic susceptibility and\ntherefore also the on-site Gilbert damping tensor on the magnetic moment of site j.\nAnother contribution to the Gilbert damping originates from the inter-site part, thus encoding the dependence of the\nmoment site ion the dynamics of the moment of site jviaGij. This contribution is often neglected in the literature,\nsince for many systems it is believed to have no signi\fcant impact. Using the microscopic model, a di\u000berent class\nof diagrams is responsible for the inter-site damping. In the lowest order in t=Um the diagrams contain already two\nhopping events,\n! !0 0\ni\u000b j\f\nijij\n+\n!;\r !0 0\ni\u000b j\f\nijij\n+:::+\n!;\r !;\u000e0 0\ni\u000b j\f\nijij\n+:::\n+\n!;\r !;\u000e0;\u0011 0\ni\u000b j\f\nijij\n+:::+\n!;\r !;\u000e0;\u0011 0;\u0010\ni\u000b j\f\nijij\n: (18)\nIn total, we \fnd that the spin-orbit independent intra-site and inter-site Gilbert damping tensors can be respectively\nwritten as\nGii=\u0010\nSi+Sij;(1)\ni (ei\u0001ej) +Sij;(2)\ni (ei\u0001ej)2\u0011\nI\n+\u0010\nAi+Aij\ni(ei\u0001ej)\u0011\nE(ei);(19)8\nand\nG\u000b\f\nij=\u0000\nSij+Sdot\nij(ei\u0001ej)\u0001\n\u000e\u000b\f\n+\u0000\nAij+Adot\nij(ei\u0001ej)\u0001\n(E(ei) +E(ej))\u000b\f\n+Scross\nij(ei\u0002ej)\u000b(ei\u0002ej)\f+Sba\nije\f\nie\u000b\nj; (20)\nwhere as mentioned earlier SandArepresent symmetric and asymmetric contributions, Iis the 3\u00023 identity while\nE(ei) =0\n@0ez\ni\u0000ey\ni\n\u0000ez\ni0ex\ni\ney\ni\u0000ex\ni01\nA.\nRemarkably, we \fnd that both the symmetric and anti-symmetric parts of the Gilbert damping tensor have a rich\ndependence with the opening angle of the magnetic moments. We identify, for example, the dot and the square\nof the dot products of the magnetic moments to possibly play a crucial role in modifying the damping, similarly to\nbilinear and biquadratic magnetic interactions. It is worth noting that even though the intra-site Gilbert damping can\nexplicitly depend on other magnetic moments, its meaning remains unchanged. The anti-symmetric precession-like\nterm describes a precession of the moment around its own e\u000bective magnetic \feld, while the diagonal damping-like\nterm describes a damping towards its own e\u000bective magnetic \feld. The dependence on other magnetic moments\nrenormalizes the intensity of those two processes. The inter-site Gilbert damping describes similar processes, but with\nrespect to the e\u000bective \feld of the other involved magnetic moment. On the basis of the LLG equation, eq. (2), it can be\nshown that the term related to Sba\nijwith a functional form of e\f\nie\u000b\njdescribes a precession of the i-th moment around\nthej-th moment with a time- and directional-dependent amplitude, @tmi/(mi\u0002mj) (mi\u0001@tmj). The double\ncross product term yields a time dependence of @tmi/(mi\u0002(mi\u0002mj)) ((mi\u0002mj)\u0001@tmj). Both contributions\nare neither pure precession-like nor pure damping-like, but show complex time- and directional-dependent dynamics.\nSpin-orbit coupling contributions. The spin-orbit interaction gives rise to new possible dependencies of the\ndamping on the magnetic structure. In particular, the so-called chiral damping, which in general is the di\u000berence\nof the damping between a right-handed and a left-handed opening, rises from SOC and broken inversion symmetry.\nUsing our perturbative model, we can identify all possible dependencies up to second order in SOC and third order\nin the magnetic moments.\nIn the diagramms SOC is added by replacing one spin-independent hopping vertex by a spin-dependent one,\n!00\n0\ni\u000b i\fij ji\n!\n!00\n0\ni\u000b i\fij\r ij\n: (21)\nUp to \frst-order in SOC, we \fnd the the following dependencies were found for the on-site Gilbert damping\nGii(fmg)/f\u000f\u000b\f\r^n\r\nij;^n\u000b\nij^n\f\nji;^n\f\nijm\u000b\ni;^n\u000b\nijm\f\ni;\u000e\u000b\f(^nij\u0001mi);\u000e\u000b\f(^nij\u0001mj);\n^n\f\nijm\u000b\nj;^n\u000b\nijm\f\nj;m\u000b\ni(^nij\u0002mi)\f;m\f\ni(^nij\u0002mi)\u000b;\n\u000e\u000b\f^nij\u0001(mi\u0002mj);m\u000b\ni(^nij\u0002mj)\f;m\f\ni(^nij\u0002mj)\u000b;(^nij\u0001mj)\u000f\u000b\f\rm\r\ni;\nm\u000b\nim\f\ni(^nij\u0001mj);(m\u000b\nim\f\nj\u0000m\f\nim\u000b\nj)(^nij\u0001mj);^n\f\nijm\u000b\ni(mi\u0001mj);^n\u000b\nijm\f\ni(mi\u0001mj)g: (22)\nWe identi\fed the following contributions for the on-site and intersite damping to be the most relevant one after the\nnumerical evaluation discussed in the next sections:\nGsoc\nii=Ssoc;ij\ni nij\u0001(ei\u0002ej)I\n+Ssoc;ij;(2)\ni (nij\u0001ei)(nij\u0001ej)I\n+Asoc;ij\ni nij\u0001(ei\u0002ej)E(ei)\n+Asoc;ij;(2)\ni (nij\u0001ej)E(nij); (23)9\nand\nGsoc;\u000b\f\nij =Ssoc\nijnij\u0001(ei\u0002ej)\u000e\u000b\f+Ssoc;ba\nijn\f\nij(ei\u0002ej)\u000b\n+Asoc\nijE\u000b\f(nij): (24)\nThe contributions being \frst-order in SOC are obviously chiral since they depend on the cross product, ei\u0002ej. Thus,\nsimilar to the magnetic Dzyaloshinskii-Moriya interaction, SOC gives rise to a dependence of the Gilbert damping\non the vector chirality, ei\u0002ej. The term chiral damping used in literature refers to the dependence of the Gilbert\ndamping on the chirality, but to our knowledge it was not shown so far how this dependence evolves from a microscopic\nmodel, and how it looks like in an atomistic model.\nExtension to three sites. Including three di\u000berent sites i,j, andkin the expansions allows for a ring exchange\ni!j!k!iinvolving three hopping processes, which gives rise to new dependencies of the Gilbert damping on\nthe directions of the moments.\nAn example of a diagram showing up for the on-site Gilbert damping is given below for the on-site Gilbert damping\nthe diagram,\n!00 0;\r\n0\ni\u000b i\fijjk\nki(25)\nApart from the natural extensions of the previously discussed 2-site quantities, the intra-site Gilbert damping of site i\ncan depend on the angle between the sites jandk,ej\u0001ek, or in higher-order on the product of the angles between site\niandjwithiandk, (ei\u0001ej)(ei\u0001ek). In sixth-order in the magnetic moments the term ( ei\u0001ej)(ej\u0001ek)(ek\u0001ei) yields\nto a dependence on the square of the scalar spin chirality of the three sites, [ ei\u0001(ej\u0002ek)]2. Including SOC, there are\ntwo interesting dependencies on the scalar spin chirality. In \frst-order one \fnds similarly to the recently discovered\nchiral biquadratic interaction18and its 3-site generalization19, e.g. ( nij\u0001ei) (ei\u0001(ej\u0002ek)), while in second order a\ndirect dependence on the scalar spin chirality is allowed, e.g. n\u000b\nijn\f\nki(ei\u0001(ej\u0002ek)). The scalar spin chirality directly\nrelates to the topological orbital moment35{37and therefore the physical origin of those dependencies lies in the\ntopological orbital moment. Even though these terms might not be the most important ones in our model, for speci\fc\nnon-collinear con\fgurations or for some realistic elements with a large topological orbital moment, e.g. MnGe20, they\nmight be important and even dominant yielding interesting new physics.\nIV. APPLICATION TO THE ALEXANDER-ANDERSON MODEL\nMagnetic dimers. Based on a 2-site Alexander-Anderson model, we investigated the dependence of the Gilbert\ndamping on the directions of the magnetic moments using the previously discussed possible terms (see more details\non the method in Appendix C). The spin splitting Ude\fnes the energy scale and all other parameters. The energy of\norbitals is set to Ed= 1:0. The magnetization is self-consistently constrained in a range of m= 0:2 tom= 0:8 using\nmagnetic constraining \felds. The corresponding spin-resolved local density of states is illustrated in Fig. 1b, where\nthe inter-site hopping is set to t= 0:2 and the hybridization to \u0000 = 0 :2. We performed two sets of calculations: one\nwithout spin-dependent hopping, 'R= 0 °, and one with a spin-dependent hopping, 'R= 20 °.\nThe di\u000berent damping parameters are shown in Fig. 2 as function of the magnetization. They are obtained from a\nleast-squares \ft to several non-collinear con\fgurations based on a Lebedev mesh for `= 238. The damping, which is\nindependent of the relative orientation of the two sites, is shown in Fig. 2a. The symmetric damping-like intra-site\ncontributionSidominates the damping tensor for most magnetizations and has a maximum at m= 0:3. The anti-\nsymmetric intra-site contribtuion Ai, which renormalizes the gyromagnetic ratio, approximately changes sign when\nthe Fermi level passes the peak of the minority spin channel at m\u00190:5 and has a signi\fcantly larger amplitude\nfor small magnetizations. Both contributions depend mainly on the broadening \u0000, which mimics the coupling to an10\nMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)\nMagnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)\nMagnetization°1.0°0.50.00.51.0Intra-site damping [U]a\nMagnetizationcGsiGasi\n0.30.50.7Magnetization°0.2°0.10.00.10.2Intra-site damping [U]b\n0.30.50.7MagnetizationdGs,ij,ik,(1)iGs,ij,ik,(2)iGs,ijik,(1)iGas,ij,ik,(1)iGas,jk,(1)iGas,ijik,(1)iGs,soc,crossijGas,soc,crossijGas,soc,(2)ijGs,soc,(2)ijMagnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7MagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)Magnetization−1.0−0.50.00.51.0\nMagnetization\nMagnetization−0.20.00.2\nMagnetization\n0.30.50.7−0.2−0.10.00.10.2\n0.30.50.7\nMagnetizationMagnetizationInter-site damping [eV]Intra-site damping [eV]Intra-site damping [eV]a)\nb)\nc)d)\ne)\nf)abc\nab\nFIG. 2. Damping parameters as function of the magnetization for the dimers described within the Alexander-Anderson\nmodel including spin orbit coupling. A longitudinal magnetic \feld is used to self-consistently constrain the magnetization. The\nparameters are extracted from \ftting to the inverse of the transversal susceptibility for several non-collinear con\fgurations\nbased on a Lebedev mesh. Model parameters in units of U:Ed= 1:0; t= 0:2;\u0000 = 0:2; 'R= 20 °.\nelectron bath and is responsible for the absorption of spin currents, which in turn are responsible for the damping of\nthe magnetization dynamics15,16.\nThe directional dependencies of the intra-site damping are shown in Fig. 2b. With our choice of parameters, the\ncorrection to the damping-like symmetric Gilbert damping can reach half of the direction-independent term. This\nmeans that the damping can vary between \u00190:4\u00001:0 for a ferromagnetic and an antiferromagnetic state at m= 0:4.\nAlso for the renormalization of the gyromagnetic ratio a signi\fcant correction is found, which in the ferromagnetic case\nalways lowers and in the antiferromagnetic case enhances the amplitude. The most dominant contribution induced\nby SOC is the chiral one, which depends on the cross product of the moments iandj, which in terms of amplitude is\ncomparable to the isotropic dot product terms. Interestingly, while the inter-site damping term is in general known\nto be less relevant than the intra-site damping, we \fnd that this does not hold for the directional dependence of the\ndamping. The inter-site damping is shown in Fig. 2c. Even though the directional-independent term, Sij, is nearly\none order of magnitude smaller than the equivalent intra-site contribution, this is not necessarily the case for the\ndirectional-dependent terms, which are comparable to the intra-site equivalents.\nV. APPLICATION TO FIRST-PRINCIPLES SIMULATIONS\nTo investigate the importance of non-collinear e\u000bects for the Gilbert damping in realistic systems, we use DFT and\ntime-dependent DFT to explore the prototypical example of monoatomic 3 dtransition metal adatoms and dimers\ndeposited on a heavy metal surface hosting large SOC (see Fig. 3a for an illustration of the con\fguration). We\nconsider a Cr, Mn, Fe and Co atoms deposited on the fcc-Au(111) surface (details of the simulations are described in\nAppendix D). The parameters and the corresponding functional forms are \ftted to our \frst-principles data using 196\nnon-collinear states based on a Lebedev mesh for `= 238.\nAdatoms on Au(111). To illustrate the di\u000berent e\u000bects on the Gilbert damping, we start by exploring magnetic\nadatoms in the uniaxial symmetry of the Au(111) surface. For the adatoms no non-local e\u000bects can contribute to the\nGilbert damping.\nThe Gilbert damping tensor of a single adatom without SOC has the form shown in relation to eq. (16),\nG0\ni=SiI+AiE(ei): (26)\nNote that SOC can induce additional anisotropies, as shown in eq. (22). The most important ones for the case of a\nsingle adatom are f\u000f\u000b\f\r^n\r\nij;^n\u000b\nij^n\f\njig, which in the C3vsymmetry result in\nGi=G0\ni+Ssoc\ni0\n@0 0 0\n0 0 0\n0 0 11\nA+Asoc\ni0\n@0 1 0\n\u00001 0 0\n0 0 01\nA; (27)11\nDampingCr / Au(111) Mn / Au(111) Fe / Au(111) Co / Au(111)parameters\nSi 0:083 0 :014 0 :242 0 :472\nAi 0:204 0 :100 0 :200 0 :024\nSsoc\ni 0:000 0 :000 0 :116 0 :010\nAsoc\ni 0:000 0 :000\u00000:022 0 :012\n\rrenorm\nx=y 1:42 1 :67 1 :43 1 :91\n\rrenorm\nz 1:42 1 :67 1 :48 1 :87\nTABLE I. Gilbert damping parameters of Cr, Mn, Fe and Co adatoms deposited on the Au(111) surface as parametrized in\neqs. (26) and (27). The SOC \feld points in the z-direction due to the C3vsymmetry. The renormalized gyromagnetic ratio\n\rrenormis calculated according to eqs. (28) for an in-plane magnetic moment and an out-of-plane magnetic moment.\nsince the sum of all SOC vectors points in the out-of-plane direction with ^nij!ez. Thus, the Gilbert damping tensor\nof adatoms deposited on the Au(111) surface can be described by the four parameters shown in eqs. (26) and (27),\nwhich are reported in Table I for Cr, Mn, Fe and Co adatoms. Cr and Mn, being nearly half-\flled, are characterized\nby a small damping-like contribution Si, while Fe and Co having states at the Fermi level show a signi\fcant damping\nof up to 0:47 in the case of Co. The antisymmetric part Aiof the Gilbert damping tensor results in an e\u000bective\nrenormalization of the gyromagnetic ratio \r, as shown in relation to eq. (A5), which using the full LLG equation,\neq. (2), and approximating mi\u0001dmi\ndt= 0 is given by,\n\rrenorm=\r1\n1 +\r(ei\u0001Ai); (28)\nwhere Aidescribes the vector Ai=\u0000\nAi;Ai;Ai+Asoc\ni\u0001\n. For Cr and Fe there is a signi\fcant renormalization of the\ngyromagnetic ratio resulting in approximately 1 :4. In contrast, Co shows only a weak renormalization with 1 :9 being\nclose to the gyromagnetic ratio of 2. The SOC e\u000bects are negigible for most adatoms except for Fe, which shows a\nsmall anisotropy in the renormalized gyromagentic ratio ( \u001910 %) and a large anisotropy in the damping-like term of\nnearly 50 %.\nDimers on Au(111). In contrast to single adatoms, dimers can show non-local contributions and dependencies on\nthe relative orientation of the magnetic moments carried by the atoms. All quantities depending on the SOC vector\nare assumed to lie in the y-z-plane due to the mirror symmetry of the system. A sketch of the dimer and its nearest\nneighboring substrate atoms together with adatoms' local density of states are presented in Fig. 3.\nThe density of states originates mainly from the d-states of the dimer atoms. It can be seen that the dimers exhibit\na much more complicated hybridization pattern than the Alexander-Anderson model. In addition the crystal \feld\nsplits the di\u000berent d-states resulting in a rich and high complexity than assumed in the model. However, the main\nfeatures are comparable: For all dimers there is either a fully occupied majority channel (Mn, Fe, and Co) or a fully\nunoccupied minority channel (Cr). The other spin channel determines the magnetic moments of the dimer atoms\nf4:04;4:48;3:42;2:20g\u0016Bfor respectively Cr, Mn, Fe and Co. Using the maximal spin moment, which is according\nto Hund's rule 5 \u0016B, the \frst-principles results can be converted to the single-orbital Alexander-Anderson model\ncorresponding to approximately m=f0:81;0:90;0:68;0:44g\u0016Bfor the aforementioned sequence of atoms. Thus by\nthis comparison, we expect large non-collinear contributions for Fe and Co, while Cr and Mn should show only weak\nnon-local dependencies.\nThe obtained parametrization is given in Table II. The Cr and Mn dimers show a weak or nearly no directional\ndependence. While the overall damping for both nanostructures is rather small, there is a signi\fcant correction to\nthe gyromagnetic ratio.\nIn contrast, the Fe and Co dimers are characterized by a very strong directional dependence. Originating from the\nisotropic dependencies of the damping-like contributions, the damping of the Fe dimer can vary between 0 :21 in\nthe ferromagnetic state and 0 :99 in the antiferromagnetic state. For the Co dimer the inter-site damping is even\ndominated by the bilinear and biquadratic term, while the constant damping is negligible. In total, there is a very\ngood qualitative agreement between the expectations derived from studying the Alexander-Anderson model and the\n\frst-principles results.12\nDampingCr / Au(111) Mn / Au(111) Fe / Au(111) Co / Au(111)parameters\nSi 0:0911 0 :0210 0 :2307 0 :5235\nSij;(1)\ni 0:0376 0 :0006\u00000:3924\u00000:2662\nSij;(2)\ni 0:0133\u00000:0006 0 :3707 0 :3119\nAi 0:2135 0 :1158 0 :1472 0 :0915\nAij\ni 0:0521 0 :0028\u00000:0710\u00000:0305\nSij\u00000:0356 0 :0028 0 :2932 0 :0929\nSdot\nij\u00000:0344\u00000:0018\u00000:3396\u00000:4056\nSdot;(2)\nij 0:0100 0 :0001 0 :1579 0 :2468\nAij\u00000:0281\u00000:0044 0 :0103 0 :0011\nAdot\nij\u00000:0175 0 :0000\u00000:0234\u00000:0402\nScross\nij 0:0288 0 :0002\u00000:2857\u00000:0895\nSba\nij 0:0331 0 :0036 0 :2181 0 :2651\nSsoc;ij;y\ni 0:0034 0 :0000 0 :0143\u00000:0225\nSsoc;ij;z\ni 0:0011 0 :0000\u00000:0104 0 :0156\nAsoc;ij;y\ni 0:0024\u00000:0001\u00000:0036 0 :0022\nAsoc;ij;z\ni 0:0018\u00000:0005 0 :0039\u00000:0144\nSsoc;y\nij 0:0004 0 :0001 0 :0307 0 :0159\nSsoc;z\nij\u00000:0011 0 :0000\u00000:0233 0 :0206\nSba,soc ;y\nij\u00000:0027 0 :0000\u00000:0184\u00000:0270\nSba,soc ;z\nij 0:0005\u00000:0001 0 :0116\u00000:0411\nTABLE II. Damping parameters of Cr, Mn, Fe and Co dimers deposited on the Au(111) surface. The possible forms of the\ndamping are taken from the analytic model. The SOC \feld is assumed to lie in the y-zplane and inverts under permutation\nof the two dimer atoms.\n\u00003\u00002\u000010123E\u0000EF[eV]\u00006\u0000303DOS [#states/eV]Cr dimerMn dimerFe dimerCo dimersurface\nab\nFIG. 3. aIllustration of a non-collinear magnetic dimer (red spheres) deposited on the (111) facets of Au (grey spheres).\nFrom the initial C3vspatial symmetry of the surface the dimers preserve the mirror plane (indicated grey) in the y-zplane. b\nLocal density of states of the Cr, Mn, Fe and Co dimers deposited on the Au(111) surface. The grey background indicates the\nsurface density of states. The dimers are collinear in the z-direction.\nVI. CONCLUSIONS\nIn this article, we presented a comprehensive analysis of magnetization dynamics in non-collinear system with a special\nfocus on the Gilbert damping tensor and its dependencies on the non-collinearity. Using a perturbative expansion\nof the two-site Alexander-Anderson model, we could identify that both, the intra-site and the inter-site part of the\nGilbert damping, depend isotropically on the environment via the e\u000bective angle between the two magnetic moments,\nei\u0001ej. SOC was identi\fed as the source of a chiral contribution to the Gilbert damping, which similarly to the\nDzyaloshinskii-Moriya and chiral biquadratic interactions depends linearly on the vector spin chirality, ei\u0002ej. We\nunveiled dependencies that are proportional to the three-spin scalar chirality ei\u0001(ej\u0002ek), i.e. to the chiral or\ntopological moment, and to its square. Using the Alexander-Anderson model, we investigated the importance of the13\ndi\u000berent contributions in terms of their magnitude as function of the magnetization. Using the prototypical test\nsystem of Cr, Mn, Fe and Co dimers deposited on the Au(111) surface, we extracted the e\u000bects of the non-collinearity\non the Gilbert damping using time-dependent DFT. Overall, the \frst-principles results agree qualitatively well with\nthe Alexander-Anderson model, showing no dependence for the nearly half-\flled systems Cr and Mn and a strong\ndependence on the non-collinearity for Fe and Co having a half-\flled minority spin-channel. The realistic systems\nindicate an even stronger dependence on the magnetic texture than the model with the used parameters. The Fe and\nthe Co dimer show signi\fcant isotropic terms up to the biquadratic term, while the chiral contributions originating\nfrom SOC have only a weak impact on the total Gilbert damping. However, the chiral contributions can play the\ndeciding role for systems which are degenerate in the isotropic terms, like e.g. spin spirals of opposite chirality.\nWe expect the dependencies of the Gilbert damping on the magnetic texture to have a signi\fcant and non-trivial\nimpact on the spin dynamics of complex magnetic structures. Our \fndings are readily implementable in the LLG\nmodel, which can trivially be amended with the angular dependencies provided in the manuscript. Utilizing multiscale\nmapping approaches, it is rather straightforward to generalize the presented forms for an implementation of the\nmicromagnetic LLG and Thiele equations. The impact of the di\u000berent contributions to the Gilbert damping, e.g. the\nvector (and/or scalar) chiral and the isotropic contributions, can be analyzed on the basis of either free parameters\nor sophisticated parametrizations obtained from \frst principles as discussed in this manuscript. It remains to be\nexplored how the newly found dependencies of the Gilbert damping a\u000bect the excitations and motion of a plethora of\nhighly non-collinear magnetic quasi-particles such as magnetic skyrmions, bobbers, hop\fons, domain walls and spin\nspirals. Future studies using atomistic spin dynamics simulations could shed some light on this aspect and help for\nthe design of future devices based on spintronics.\nACKNOWLEDGMENTS\nThis work was supported by the European Research Council (ERC) under the European Union's Horizon 2020 research\nand innovation program (ERC-consolidator grant 681405 { DYNASORE) and from Deutsche Forschungsgemeinschaft\n(DFG) through SPP 2137 \\Skyrmionics\" (Project LO 1659/8-1). The authors gratefully acknowledge the computing\ntime granted through JARA-HPC on the supercomputer JURECA at the Forschungszentrum J ulich39.\nVII. METHODS\nAppendix A: Analysis of the Gilbert damping tensor\nThe Gilbert damping tensor Gcan be decomposed into a symmetric part Sand an anti-symmetric part A,\nA=G\u0000GT\n2andS=G+GT\n2: (A1)\nWhile the symmetric contribution can be referred to as the damping-like contribution including potential anisotropies,\nthe anti-symmetric Atypically renormalizes the gyromagnetic ratio as can be seen as follows: The three independent\ncomponents of an anti-symmetric tensor can be encoded in a vector Ayielding\nA\u000b\f=\u000f\u000b\f\rA\r; (A2)\nwhere\u000f\u000b\f\ris the Levi-Cevita symbol. Inserting this into the LLG equation yields\ndmi\ndt=\u0000\rmi\u00020\n@Be\u000b\ni+X\njAijdmj\ndt1\nA (A3)\n\u0019\u0000\rmi\u00020\n@Be\u000b\ni\u0000\rX\njAij\u0000\nmj\u0002Be\u000b\nj\u00011\nA: (A4)\nThe last term can be rewritten as\n(Aij\u0001mj)Be\u000b\nj\u0000\u0000\nAij\u0001Be\u000b\nj\u0001\nmj: (A5)14\nFor the local contribution, Aii, the correction is kmiandkBe\u000b\niyielding a renormalization of \rmi\u0002Be\u000b\ni. However,\nthe non-local parts of the anti-symmetric Gilbert damping tensor can be damping-like.\nAppendix B: Relation between the LLG and the magnetic susceptibility\nThe Fourier transform of the LLG equation is given by\n\u0000i!mi=\u0000\rmi\u00020\n@Bext\ni\u0000X\njJijmj\u0000i!X\njGijmj1\nA: (B1)\nTransforming this equation to the local frames of site iandjusing the rotation matrices RiandRjyields\ni!\n\rMimloc\ni=mloc\ni\nMi\u00020\n@RiBext\ni\u0000X\njRiJijRT\njmloc\nj\u0000i!X\njRiGijRT\njmloc\nj1\nA; (B2)\nwhere mloc\ni=Rimiandmloc\nj=Rjmj. The rotation matrices are written as R(#i;'i) = cos(#i=2)\u001b0+\ni sin(#i=2)\u0000\nsin('i)\u001bx\u0000cos('i)\u001by\u0001\n, with (#i;'i) being the polar and azimuthal angle pertaining to the moment\nmi. In the ground state the magnetic torque vanishes. Thus, denoting mloc\ni= (mx\ni; my\ni; Mi), wheremx=y\niare\nperturbations to the ground states, yields for the ground state\n0\n@(RiBext\ni)x\u0000P\nj(RiJijRT\njMjez)x\n(RiBext\ni)y\u0000P\nj(RiJijRT\njMjez)y\n(RiBext\ni)z\u0000P\nj(RiJijRT\njMjez)z1\nA=0\n@0\n0\n(RiBe\u000b\ni)z1\nA: (B3)\nLinearizing the LLG and using the previous result and limiting our expansion to transveral excitations yield\ni!\n\rMimx\ni=my\ni(RiBe\u000b\ni)z\nMi\u0000(RiBext\ni)y+X\nj(RiJijRT\njmloc\nj)y+ i!X\nj(RiGijRT\njmloc\nj)y(B4)\ni!\n\rMimy\ni=\u0000mx\ni(RiBe\u000b\ni)z\nMi+ (RiBext\ni)x\u0000X\nj(RiJijRT\njmloc\nj)x\u0000i!X\nj(RiGijRT\njmloc\nj)x; (B5)\nwhich in a compact form gives\nX\nj\n\f=x;y0\n@\u000eij\u0012\n\u000e\u000b\f(RiBe\u000b\ni)z\nMi+\u000f\u000b\f\u0016i!\n\rMi\u0013\n+X\nj(RiJijRT\nj)\u000b\f+ i!X\nj(RiGijRT\nj)\u000b\f1\nAm\f\nj= (RiBext\ni)\u000b; (B6)\nand can be related to the inverse of the magnetic susceptibility\nX\nj\n\f=x;y\u001f\u00001\ni\u000b;j\f(!)m\f\nj= (RiBext\ni)\u000b: (B7)\nThus, the magnetic susceptibility in the local frames of site iandjis given by\n\u001f\u00001\ni\u000b;j\f(!) =\u000eij\u0012\n\u000e\u000b\f(RiBe\u000b\ni)z\nMi+\u000f\u000b\f\u0016i!\n\rMi\u0013\n+X\nj(RiJijRT\nj)\u000b\f+ i!X\nj(RiGijRT\nj)\u000b\f(B8)\nAppendix C: Alexander-Anderson model{more details\nWe use a single orbital Alexander-Anderson model,\nH=X\nij[\u000eij(Ed\u0000i \u0000\u0000Uimi\u0001\u001b\u0000Bi\u0001\u001b)\u0000(1\u0000\u000eij)tij]; (C1)15\nwhereiandjsum over all n-sites,Edis the energy of the localized orbitals, \u0000 is the hybridization in the wide band\nlimit,Uiis the local interaction responsible for the formation of a magnetic moment, miis the magnetic moment of site\ni,Biis an constraining or external magnetic \feld, \u001bare the Pauli matrices, and tijis the hopping parameter between\nsiteiandj, which can be in general spin-dependent. SOC is added as spin-dependent hopping using a Rashba-like\nspin-momentum locking tij=t(cos'R\u001b0\u0000i sin'Rnij\u0001\u001b), where the spin-dependent hopping is characterized by its\nstrength de\fned by 'Rand its direction nij=\u0000nji34. The eigenenergies and eigenstates of the model are given by,\nHjni= (En\u0000i \u0000)jni: (C2)\nThe single particle Green function can be de\fned using the eigensystem,\nG(E+ i\u0011) =X\nnjnihnj\nE\u0000En+ i\u0011; (C3)\nwhere\u0011is an in\fnitesimal parameter de\fning the retarded ( \u0011!0+) and advanced ( \u0011!0\u0000) Green function. The\nmagnitude of the magnetic moment is determined self-consistently using\nmi=\u00001\n\u0019Im TrZ\ndE\u001bGii(E); (C4)\nwhereGii(E) is the local Green function of site idepending on the magnetic moment. Using the magnetic torque\nexerted on the moment of site i,\ndH\nd^ei=\u0000miBe\u000b\ni; (C5)\nmagnetic constraining \felds can be de\fned ensuring the stability of an arbitrary non-collinear con\fguration,\nBconstr=\u0000Pm\n?mi\njmijBe\u000b\ni) Hconstr=\u0000Bconstr\u0001\u001b ; (C6)\nwherePm\n?is the projection on the plane perpendicular to the moment m. The constraining \felds are added to the\nhamiltonian, eq. (C1), and determined self-consistently.\nAppendix D: Density functional theory{details\nThe density functional theory calculations were performed with the Korringa-Kohn-Rostoker (KKR) Green function\nmethod. We assume the atomic sphere approximation for the the potential and include full charge density in the\nself-consistent scheme40. Exchange and correlation e\u000bects are treated in the local spin density approximation (LSDA)\nas parametrized by Vosko, Wilk and Nusair41, and SOC is added to the scalar-relativistic approximation in a self-\nconsistent fashion42. We model the pristine surfaces utilizing a slab of 40 layers with the experimental lattice constant\nof Au assuming open boundary conditions in the stacking direction, and surrounded by two vacuum regions. No\nrelaxation of the surface layer is considered, as it was shown to be negligible43. We use 450\u0002450k-points in the\ntwo-dimensional Brillouin zone, and the angular momentum expansions for the scattering problem are carried out up\nto`max= 3. Each adatom is placed in the fcc-stacking position on the surface, using the embedding KKR method.\nPreviously reported relaxations towards the surface of 3 dadatoms deposited on the Au(111) surface44indicate a\nweak dependence of the relaxation on the chemical nature of the element. Therefore, we use a relaxation towards the\nsurface of 20 % of the inter-layer distance for all the considered dimers. The embedding region consists of a spherical\ncluster around each magnetic adatom, including the nearest-neighbor surface atoms. 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It is shown that the field-like torque plays an import ant role in finding the balance between\nthe energy supplied by the spin torque and the dissipation du e to the damping, which results in\na steady precession. The validity of the developed theory is confirmed by performing numerical\nsimulations based on the Landau-Lifshitz-Gilbert equatio n.\nSpin torque oscillator (STO) has attracted much at-\ntention as a future nanocommunication device because\nit can produce a large emission power ( >1µW), a high\nquality factor ( >103), a high oscillation frequency ( >1\nGHz), a wide frequency tunability ( >3 GHz), and a nar-\nrowlinewidth ( <102kHz) [1–9]. In particular,STOwith\na perpendicularly magnetized free layer and an in-plane\nmagnetizedpinnedlayerhasbeendevelopedafterthedis-\ncovery of an enhancement of perpendicular anisotropy of\nCoFeB free layer by attaching MgO capping layer [10–\n12]. In the following, we focus on this type of STO. We\nhave investigated the oscillation properties of this STO\nboth experimentally [6, 13] and theoretically [14, 15]. An\nimportant conclusion derived in these studies was that\nfield-like torque is necessary to excite the self-oscillation\nin the absence of an external field, nevertheless the field-\nlike torque is typically one to two orders of magnitude\nsmaller than the spin torque [16–18]. We showed this\nconclusion by performing numerical simulations based on\nthe Landau-Lifshitz-Gilbert (LLG) equation [15].\nThis paper theoretically proves the reason why the\nfield-like torque is necessary to excite the oscillation by\nusing the energy balance equation [19–27]. An effective\nenergy including the effect of the field-like torque is in-\ntroduced. It is shown that introducing field-like torque\nis crucial in finding the energy balance between the spin\ntorque and the damping, and as a result to stabilize a\nsteady precession. A good agreement with the LLG sim-\nulation on the current dependence of the oscillation fre-\nquency shows the validity of the presented theory.\nThesystemunderconsiderationisschematicallyshown\nin Fig. 1 (a). The unit vectorspointing in the magnetiza-\ntion directions of the free and pinned layers are denoted\nasmandp, respectively. The z-axis is normal to the\nfilm-plane, whereas the x-axis is parallel to the pinned\nlayer magnetization. The current Iis positive when elec-\ntrons flow from the free layer to the pinned layer. The\nLLG equation of the free layer magnetization mis\ndm\ndt=−γm×H−γHsm×(p×m)\n−γβHsm×p+αm×dm\ndt,(1)pxz+\n-\nm(a)\n(b)\nmxmy\n1 -1 001\n-1 \nFIG. 1: (a) Schematic view of the system. (b) Schematic\nviews of the contour plot of the effective energy map (dotted) ,\nEq. (2), and precession trajectory in a steady state with I=\n1.6 mA (solid).\nwhereγis the gyromagnetic ratio. Since the external\nfield is assumed to be zero throughout this paper, the\nmagnetic field H= (HK−4πM)mzezconsists of the per-\npendicular anisotropy field only, where HKand 4πMare\nthe crystalline and shape anisotropy fields, respectively.\nSinceweareinterestedintheperpendicularlymagnetized\nfree layer, HKshould be larger than 4 πM. The second\nand third terms on the right-hand-side of Eq. (1) are the\nspin torque and field-like torque, respectively. The spin\ntorque strength, Hs=/planckover2pi1ηI/[2e(1+λm·p)MV], includes\nthe saturation magnetization Mand volume Vof the\nfree layer. The spin polarization of the current and the2\ndependence of the spin torque strength on the relative\nangle of the magnetizations are characterized in respec-\ntive byηandλ[14]. According to Ref. [15], βshould\nbe negative to stabilize the self-oscillation. The values\nof the parameters used in the following calculations are\nM= 1448 emu/c.c., HK= 20.0 kOe,V=π×60×60×2\nnm3,η= 0.54,λ=η2,β=−0.2,γ= 1.732×107\nrad/(Oe·s), andα= 0.005, respectively [6, 15]. The crit-\nical current of the magnetization dynamics for β= 0 is\nIc= [4αeMV/(/planckover2pi1ηλ)](HK−4πM)≃1.2 mA, where Ref.\n[15] shows that the effect of βon the critical current is\nnegligible. Whenthecurrentmagnitudeisbelowthecrit-\nical current, the magnetization is stabilized at mz= 1.\nIn the oscillation state, the energy supplied by the spin\ntorquebalancesthedissipationdue tothedamping. Usu-\nally, the energy is the magnetic energy density defined as\nE=−M/integraltextdm·H[28], which includes the perpendic-\nular anisotropy energy only, −M(HK−4πM)m2\nz/2, in\nthe present model. The first term on the right-hand-side\nof Eq. (1) can be expressed as −γm×[−∂E/∂(Mm)].\nHowever, Eq. (1) indicates that an effective energy den-\nsity,\nEeff=−M(HK−4πM)\n2m2\nz−β/planckover2pi1ηI\n2eλVlog(1+λm·p),\n(2)\nshould be introduced because the first and third terms\non the right-hand-side of Eq. (1) can be summarized as\n−γm×[−∂Eeff/∂(Mm)]. Here, we introduce aneffective\nmagnetic field H=−∂Eeff/∂(Mm) = (β/planckover2pi1ηI/[2e(1 +\nλmx)MV],0,(HK−4πM)mz). Dotted line in Fig. 1 (b)\nschematically shows the contour plot of the effective en-\nergy density Eeffprojected to the xy-plane, where the\nconstant energy curves slightly shift along the x-axis be-\ncause the second term in Eq. (2) breaks the axial sym-\nmetry of E. Solid line in Fig. 1 (b) shows the preces-\nsion trajectory of the magnetization in a steady state\nwithI= 1.6 mA obtained from the LLG equation. As\nshown, the magnetization steadily precesses practically\non a constant energy curve of Eeff. Under a given cur-\nrentI, the effective energy density Eeffdetermining the\nconstant energycurve of the stable precessionis obtained\nby the energy balance equation [27]\nαMα(Eeff)−Ms(Eeff) = 0. (3)\nIn this equation, MαandMs, which are proportional to\nthe dissipation due to the damping and energy supplied\nby the spin torque during a precession on the constant\nenergy curve, are defined as [14, 25–27]\nMα=γ2/contintegraldisplay\ndt/bracketleftBig\nH2−(m·H)2/bracketrightBig\n, (4)\nMs=γ2/contintegraldisplay\ndtHs[p·H−(m·p)(m·H)−αp·(m×H)].\n(5)\nThe oscillation frequency on the constant energy curve(a)\n00.010.02\n-0.01\n-0.020 0.2 0.4 0.6 0.8 1.0\nmzMs, -αM α, Ms-αM αMs\n-αM αMs-αM α\n(b)\n00.010.02\n-0.01\n-0.030 0.2 0.4 0.6 0.8 1.0\nmzMs, -αM α, Ms-αM αMs\n-αM αMs-αM α\n-0.02β=0\nβ=-0.2\nFIG. 2: Dependences of Ms,−αMα, and their difference\nMs−Mαnormalized by γ(HK−4πM) onmz(0≤mz<1)\nfor (a)β= 0, and (b) β=−0.2, where I= 1.6 mA.\ndetermined by Eq. (3) is given by\nf= 1/slashbig/contintegraldisplay\ndt. (6)\nSince we are interested in zero-field oscillation, and from\nthe fact that the cross section of STO in experiment [6]\nis circle, we neglect external field Hextor with in-plane\nanisotropy field Hin−plane\nKmxex. However, the above\nformula can be expanded to system with such effects\nby adding these fields to Hand terms −MHext·m−\nMHin−plane\nKm2\nx/2 to the effective energy.\nIn the absence of the field-like torque ( β= 0), i.e.,\nEeff=E, thereisone-to-onecorrespondencebetween the\nenergy density Eandmz. Because an experimentally\nmeasurable quantity is the magnetoresistance propor-\ntional to ( RAP−RP)max[m·p]∝max[mx] =/radicalbig\n1−m2z,\nit is suitable to calculate Eq. (3) as a function of mz, in-\nstead ofE, whereRP(AP)is the resistance of STO in the\n(anti)parallel alignment of the magnetizations. Figure 2\n(a) shows dependences of Ms,−αMα, and their differ-\nenceMs−αMαonmz(0≤mz<1)forβ= 0, where Ms\nandMαare normalized by γ(HK−4πM). The current is\nset asI= 1.6 mA (> Ic). We also show Ms,−αMα, and\ntheir difference Ms−αMαforβ=−0.2 in Fig. 2 (b),\nwheremxis set as mx=−/radicalbig\n1−m2z. Because −αMα\nis proportional to the dissipation due to the damping,\n−αMαis always −αMα≤0. The implications of Figs.\n2 (a) and (b) are as follows. In Fig. 2 (a), Ms−αMαis\nalways positive. This means that the energy supplied by3\ncurrent (mA)frequency (GHz) \n1.2 1.4 1.6 1.8 2.012\n0345\n: Eq. (6): Eq. (1)\nFIG. 3: Current dependences of peak frequency of |mx(f)|\nobtained from Eq. (1) (red circle), and the oscillation fre-\nquency estimated by using (6) (solid line).\nthe spin torque is always larger than the dissipation due\nto the damping, and thus, the net energy absorbed in\nthe free layer is positive. Then, starting from the initial\nequilibrium state ( mz= 1), the free layer magnetization\nmoves to the in-plane mz= 0, as shown in Ref. [14]. On\nthe other hand, in Fig. 2 (b), Ms−αMαis positive from\nmz= 1to acertain m′\nz, whereasit is negativefrom m′\nzto\nmz= 0 (m′\nz≃0.4 in the case of Fig. 2 (b)). This means\nthat, starting from mz= 1, the magnetization can move\nto a point m′\nzbecause the net energy absorbed by the\nfree layer is positive, which drives the magnetization dy-\nnamics. However, the magnetization cannot move to the\nfilm plane ( mz= 0) because the dissipation overcomes\nthe energy supplied by the spin torque from mz=m′\nztomz= 0. Then, a stable and large amplitude precession\nis realized on a constant energy curve.\nWe confirm the accuracy of the above formula by com-\nparing the oscillation frequency estimated by Eq. (6)\nwiththenumericalsolutionoftheLLGequation, Eq. (1).\nIn Fig. 3, we summarize the peak frequency of |mx(f)|\nforI= 1.2−2.0 mA (solid line), where mx(f) is the\nFourier transformation of mx(t). We also show the oscil-\nlation frequency estimated from Eq. (6) by the dots. A\nquantitatively good agreement is obtained, guaranteeing\nthe validity of Eq. (6).\nIn conclusion, we developed a theoretical formula to\nevaluate the zero-field oscillation frequency of STO in\nthe presence of the field-like torque. Our approach was\nbasedon the energybalance equationbetween the energy\nsuppliedbythe spintorqueandthe dissipationdue tothe\ndamping. An effective energy density was introduced to\ntake into account the effect of the field-like torque. We\ndiscussed that introducing field-like torque is necessary\nto find the energy balance between the spin torque and\nthe damping, which as a result stabilizes a steady preces-\nsion. The validity of the developed theory was confirmed\nby performing the numerical simulation, showing a good\nagreement with the present theory.\nThe authors would like to acknowledge T. Yorozu,\nH. Maehara, H. Tomita, T. Nozaki, K. Yakushiji, A.\nFukushima, K. Ando, and S. Yuasa. This work was sup-\nported by JSPS KAKENHI Number 23226001.\n[1] S. I. Kiselev, J. C. Sankey, I. N. Krivorotov, N. C. Em-\nley, R. J. Schoelkopf, R. A. Buhrman, and D. C. Ralph,\nNature425, 380 (2003).\n[2] W. H. Rippard, M. R. Pufall, S. Kaka, S. E. Russek, and\nT. J. Silva, Phys. Rev. Lett. 92, 027201 (2004).\n[3] D. Houssameddine, U. Ebels, B. Dela¨ et, B. Rodmacq,\nI. Firastrau, F. Ponthenier, M. Brunet, C. Thirion, J.-P.\nMichel, L. Prejbeanu-Buda, et al., Nat. Mater. 6, 447\n(2007).\n[4] S.Bonetti, P.Muduli, F.Mancoff, andJ. Akerman, Appl.\nPhys. Lett. 94, 102507 (2009).\n[5] Z. Zeng, G. Finocchio, B. Zhang, P. K. Amiri, J. A. Ka-\ntine, I. N. Krivorotov, Y. Huai, J. Langer, B. Azzerboni,\nK. L. 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Pitaevskii, Statistical Physics\n(part 2), course of theoretical physics volume 9\n(Butterworth-Heinemann, Oxford, 1980), chap. 7, 1st ed."    },    {        "title": "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": "1309.4897v1.Van_der_Waals_Coefficients_for_the_Alkali_metal_Atoms_in_the_Material_Mediums.pdf",        "content": "arXiv:1309.4897v1  [physics.atom-ph]  19 Sep 2013Van der Waals Coefficients for the Alkali-metal Atoms in the Ma terial Mediums\naBindiya Arora∗andbB. K. Sahoo†\naDepartment of Physics, Guru Nanak Dev University, Amritsar , Punjab-143005, India,\nbTheoretical Physics Division, Physical Research Laborato ry, Navrangpura, Ahmedabad-380009, India\n(Dated: Received date; Accepted date)\nThe damping coefficients for the alkali atoms are determined v ery accurately by taking into\naccount the optical properties of the atoms and three distin ct types of trapping materials such\nas Au (metal), Si (semi-conductor) and vitreous SiO 2(dielectric). Dynamic dipole polarizabilities\nare calculated precisely for the alkali atoms that reproduc e the damping coefficients in the perfect\nconducting medium within 0.2% accuracy. Upon the considera tion of the available optical data of\nthe above wall materials, the damping coefficients are found t o be substantially different than those\nof the ideal conductor. We also evaluated dispersion coeffici ents for the alkali dimers and compared\nthem with the previously reported values. These coefficients are fitted into a ready-to-use functional\nform to aid the experimentalists the interaction potential s only with the knowledge of distances.\nPACS numbers: 34.35.+a, 34.20.Cf, 31.50.Bc, 31.15.ap\nAccurate information on the long-range interactions\nsuch as dispersion (van der Waals) and retarded\n(Casimir-Polder) potentials between two atoms and be-\ntween an atom and surface of the trapping material are\nnecessary for the investigation of the underlying physics\nof atomic collisions especially in the ultracold atomic ex-\nperiments [1–4]. Presence of atom-surface interactions\nlead to a shift in the oscillation frequency of the trap\nwhich alters the trapping frequency as well as magic\nwavelengths for state-insensitive trapping of the trapped\ncondensate. Moreover, this effect has also gained inter-\nest in generating novel atom optical devices known as\nthe “atom chips”. In addition, the knowledge of dis-\npersion coefficients is required in experiments of photo-\nassociation, fluorescence spectroscopy, determination of\nscattering lengths, analysis of feshbach resonances, de-\ntermination of stability of Bose-Einstein condensates\n(BECs), probingextra dimensionsto accommodate New-\ntonian gravity in quantum mechanics etc. [5–10].\nTherehavebeenmanyexperimentalevidencesofanat-\ntractiveforcebetweenneutralatomsandbetweenneutral\natoms with trapping surfaces but their precise determi-\nnations are relatively difficult. In the past two decades,\nseveral groups have evaluated dispersion coefficients C3\ndefining interaction between an atom and a wall using\nvarious approaches [11–13] without rigorous estimate of\nuncertainties. More importantly, they are evaluated for a\nperfect conducting wall which are quite different from an\nactual trapping wall. Since these coefficients depend on\nthedielectricconstantsofthematerialsofthewall, there-\nfore it is worth determining them precisely for trapping\nmaterials with varying dielectric constants (for good con-\nducting, semi conducting, and dielectric mediums) as has\nbeen attempted in [14, 15]. Casimir and Polder [2] had\nestimated that at intermediately largeseparationsthe re-\n∗Email: arorabindiya@gmail.com\n†Email: bijaya@prl.res.intardation effects of the virtual photons passing between\nthe atom and its image weakens the attractive atom-\nwall force and the force scales with a different power law\n(given in details below). In this paper, we carefully ex-\namine these retardation or damping effects which have\nnot been extensively studied earlier. We also parameter-\nized our damping coefficients into a readily usable form\nto be used in experiments.\nThe atom-surface interaction potential resulting from\nthe fluctuating dipole moment of an atom interacting\nwith its image in the surface is formulated by [1, 14]\nUa(R) =−α3\nfs\n2π/integraldisplay∞\n0dωω3α(ιω)/integraldisplay∞\n1dξe−2αfsξωRH(ξ,ǫ(ιω)),\n(1)\nwhereαfsis the fine structure constant, ǫ(ω) is the fre-\nquencydependentdielectricconstantofthesolid, Risthe\ndistance between the atom and the surface and α(ιω) is\nthe ground state dynamic polarizability with imaginary\nargument. The function H(ξ,ǫ(ιω)) is given by\nH(ξ,ǫ) = (1−2ξ2)/radicalbig\nξ2+ǫ−1−ǫξ/radicalbig\nξ2+ǫ−1+ǫξ+/radicalbig\nξ2+ǫ−1−ξ/radicalbig\nξ2+ǫ−1+ξ\nwith the Matsubara frequencies denoted by ξ.\nIn asymptotic regimes, the Matsubara integration is\ndominated by its first term and the potential can be ap-\nproximated to Ua(R) =−C3T\nR3withC3T=α(0)\n4(ǫ(0)−\n1)/(ǫ(0)+1). The potential form can be described more\naccurately at the retardation distances as Ua(R) =−C4\nR4\nand at the non-retarded region as Ua(R) =−C3\nR3[2]. To\nexpress the potential in the intermediate region, these\napproximations are usually modified either to Ua(R) =\n−C4\n(R+λ)R3or toUa(R) =−C3\nR3f3(R) whereλandf3(R)\nare respectively known as the reduced wavelength and\ndamping function. It would be interesting to testify the\nvalidity of both the approximations by evaluating C3,C4\nandf3(R) coefficients togetherfor different atomsin con-\nducting, semi-conducting and dielectric materials. Since\nthe knowledgeofmagneticpermeability ofthe materialis\nrequired to evaluate C4coefficients, hence we determine2\n 0 50 100 150 200 250 300 α(ιω)(a.u.)(a)Li\nNa\nK\nRb\n 0 5 10 15\n0 0.2 0.4 0.6 0.8 1ε(ιω)(a.u.)\nFrequency (a.u.)(b) Au\nSi\nSiO2\nFIG. 1: Dynamic polarizabilities of the Li, Na, K and Rb\natoms and dielectric permittivity of the Au, Si and SiO 2sur-\nfaces along the imaginary axis as functions of frequencies.\nonly the C3andf3(R) coefficients. With the knowledge\nofC3andf3values, the atom-surface interaction poten-\ntialscanbeeasilyreproducedandtheycanbegeneralized\nto other surfaces. In general, the C3coefficient is given\nby\nC3≈1\n4π/integraldisplay∞\n0dωα(ιω)ǫ(ιω)−1\nǫ(ιω)+1. (2)\nForaperfectconductor ǫ→ ∞,ǫ(ιω)−1\nǫ(ιω)+1→1andforother\nmaterialswith their refractiveindices n=√ǫvaryingbe-\ntween 1 and 2,ǫ(ιω)−1\nǫ(ιω)+1≈ǫ(0)−1\nǫ(0)+1is nearly a constant and\ncan be approximated to 0.77. For more preciseness, it is\nnecessary to consider the actual frequency dependencies\nofǫs in the materials. In the present work, three distinct\nmaterials such as Au, Si and SiO 2belonging to conduct-\ning, semi-conducting and dielectric objects respectively,\nare taken into account to find out f3(R) functions and\ncompared against a perfect conducting wall for which\ncase we express [16]\nf3(R) =1\n4πC3/integraldisplay∞\n0dωα(ιω)e−2αfsωRQ(αfsωR),(3)\nwithQ(x) = 2x2+2x+1. To find out f3(R) for the other\nsurfaces, we evaluate Ua(R) by substituting their ǫ(ιω)\nvalues in Eq. (1).\nSimilarly, the leading term in the long-range interac-\ntion between two atoms denoted by aandbis approxi-\nmated by Uab(R) =−Cab\n6\nR6, wherethe Cab\n6is knownasthe\nvan der Waals coefficient and Ris the distance between\ntwo atoms. If retardation effects are included then it is\nmodified to Uab(R) =−Cab\n6\nR6fab\n6(R). The dispersion coef-\nficientCab\n6and the damping coefficient fab\n6(R) betweenTABLE I: Calculated C3coefficients along with their uncer-\ntainties for the alkali-metal atoms and their comparison wi th\nother reported values. Classification of various contribut ions\nare in accordance with [11]a, [12]band [16]c.\nLi Na K Rb\nPerfect Conductor\nCore 0.074 0.332 0.989 1.513\nValence 1.387 1.566 2.115 2.254\nCore-Valence ∼0 ∼0−0.016−0.028\nTail 0.055 0.005 0.003 0.003\nTotal 1.516(2) 1.904(2) 3.090(4) 3.742(5)\nOthers 1.5178a1.8858b2.860b3.362b\n1.889c\nMetal: Au\nCore 0.010 0.051 0.263 0.419\nValence 1.160 1.285 1.804 1.927\nCore-Valence ∼0 ∼0 -0.005 -0.010\nTail 0.029 0.002 0.001 0.002\nTotal 1.199(2) 1.338(1) 2.062(4) 2.338(4)\nOthers [14] 1.210 1.356 2.058 2.79\nSemi-conductor: Si\nCore 0.006 0.033 0.184 0.299\nValence 0.993 1.099 1.543 1.649\nCore-Valence ∼0 ∼0 -0.004 -0.008\nTail 0.023 0.002 0.001 0.001\nTotal 1.022(2) 1.134(1) 1.724(3) 1.942(4)\nDielectric: SiO 2\nCore 0.004 0.022 0.116 0.184\nValence 0.468 0.519 0.726 0.775\nCore-Valence ∼0 ∼0 -0.002 -0.004\nTail 0.012 0.001 0.001 0.001\nTotal 0.4844(8) 0.5424(5) 0.839(1) 0.956(2)\nthe atoms can be estimated using the expressions [16]\nCab\n6=3\nπ/integraldisplay∞\n0dωαa(ιω)αb(ιω),and\nfab\n6=1\nπCab\n6/integraldisplay∞\n0dωαa(ιω)αb(ιω)e−2αfsωRP(αfsωR),\nwhereP(x) =x4+2x3+5x2+6x+3.\nUsing our previously reported E1 matrix elements\n[17, 18] and experimental energies, we plot the dy-\nnamic polarizabilities of the ground states in Fig. 1\nof the considered alkali atoms. The static polarizabil-\nities corresponding to ω= 0 come out to be 164.1(7),\n162.3(2), 289.7(6) and 318.5(8), as given in [17, 18],\nagainst the experimental values 164.2(11) [19], 162.4(2)\n[20], 290.58(1.42) [21] and 318.79(1.42) [21] in atomic\nunit (a.u.) for Li, Na, K and Rb atoms respectively. It\nclearly indicates the preciseness of our estimated results.\nThe main reason for achieving such high accuracies in\nthe estimated static polarizabilities is due to the use of\nE1 matrix elements extracted from the precise lifetime\nmeasurements of few excited states and by fitting our3\n 0 0.2 0.4 0.6 0.8 1f3\n(a) Li (b) NaPerfect conductor\nAu\nSi\nSiO2\n 0 0.2 0.4 0.6 0.8\n02000400060008000f3\nR(a.u.)(c) K\n0200040006000800010000\nR(a.u.)(d) Rb\nFIG. 2: The retardation coefficient f3(R) (dimensionless) for\nLi, Na, K and Rb as a function of atom-wall distance R.\nTABLE II: Fitting parameters aandbforf3coefficients with\na perfectly conducting wall, Au, Si, and SiO 2surfaces.\nLi Na K Rb\nPerfect Conductor\na 0.9843 1.0802 1.1845 1.2598\nb 0.0676 0.0866 0.0808 0.0907\nMetal: Au\na 0.9775 0.9846 1.0248 1.0437\nb 0.0675 0.0614 0.0532 0.0558\nSemi-conductor: Si\na 0.9436 0.9436 0.9749 0.9869\nb 0.0638 0.0718 0.0622 0.0647\nDielectric: SiO 2\na 0.9754 0.9789 1.0238 1.0423\nb 0.0650 0.0746 0.0649 0.0685\nE1 results obtained from the relativistic coupled-cluster\ncalculation at the singles, doubles and partial triples ex-\ncitation level (CCSD(T) method) to the measurements\nof the static polarizabilities of the excited states.\nSubstituting the dynamic polarizabilities in Eq. (2),\nwe evaluatethe C3coefficients fora perfect conductor(to\ncompare with previous studies), for a real metal Au, for\na semi conductor object Si and for a dielectric substance\nof glassy structure SiO 2. These values are given in Table\nI with break down from various individual contributions\nandestimateduncertaintiesarequotedintheparentheses\nafterignoringerrorsfromthe usedexperimentaldata. To\nachieve the claimed accuracy in our results it was neces-\nsaryto use the complete tabulated data for the refraction\nindices of Au, Si, and SiO 2to calculate their dielectric\npermittivities at all the imaginary frequencies [22]. We\nevaluate the imaginary parts of the dielectric constants\nusing the relation Im( ǫ(ω)) = 2n(ω)κ(ω), where n and κ\nare the real and imaginaryparts of the refractiveindex ofa material. The available data for Si and SiO 2are suffi-\nciently extended to lower frequencies. However, they are\nextended to the lower frequencies for Au with the help of\nthe Drude dielectric function [15]\nǫ(ω) = 1−ω2\np\nω(ω+ιγ), (4)\nwith relaxation frequency γ= 0.035 eV and plasma\nfrequency ωp= 9.02 eV. The corresponding real val-\nues at imaginary frequencies are obtained by using the\nKramers-Kronig formula\nRe(ǫ(ιω)) = 1+2\nπ/integraldisplay∞\n0dω′ω′Im(ǫ(ω′))\nω2+ω′2.(5)\nIn bottom part of Fig. 1, the ǫ(ιω) values as a function\nof imaginary frequency are plotted for Au, Si, and SiO 2.\nThe behavior of ǫ(ιω) for various materials is obtained as\nexpected and they match well with the graphical repre-\nsentations given by Caride and co-workers [15].\nAs shown in Table I, C3coefficients increase with the\nincrease in atomic mass. First we present our results for\ntheC3coefficients for the interaction of these atoms with\na perfectly conducting wall. The dominant contribution\nto theC3coefficients is from the valence part of the po-\nlarizability. We also observed that the core contribution\nto theC3coefficients increases with the increasing num-\nber of electrons in the atom which is in agreement with\nthe prediction made in Ref. [12]. Our results are also in\ngood agreement with the results reported by Kharchenko\net al.[16] for Na. Therefore, our results obtained for\nother materials seem to be reliable enough. We noticed\nthat the C3coefficients for a perfect conductor were ap-\nproximately 1.5, 2, and 3.5 times larger than the C3co-\nefficients for Au, Si, and SiO 2respectively. The decrease\nin the coefficient values for the considered mediums can\nbe attributed to the fact that in case of dielectric ma-\nterial the theory is modified for non-unity reflection and\nfor different origin ofthe transmitted wavesfrom the sur-\nface. In addition to this, for Si and SiO 2there are addi-\ntionalinteractionsduetochargedanglingbonds specially\nat shorter separations. The recent estimations with Au\nmedium carried out by Lach et al.[14] are in agreement\nwith our results since the polarizability database they\nhave used is taken from Ref. [12]. These calculations\nseem to be sensitive on the choice of grids used for the\nnumerical integration. An exponential grid yield the re-\nsults more accurately and it is insensitive to choice of the\nsize of the grid in contrast to a linear grid. In fact with\nthe use of a linear grid having a spacing 0 .1, we observed\na 3-5% fall in C3coefficients for the considered atoms.\nThe reason being that most of the contributions to the\nevaluation of these coefficients come from the lower fre-\nquencies which yield inaccuracy in the results for large\ngrid size.\nFig. 2 showsa comparisonof the f3(R) values obtained\nfor Li, Na, K, and Rb atoms as a function of atom-\nwall separationdistance R for the four different materials4\nTABLE III: C6coefficients with fitting parameters for the alkali dimers. Co ntributions from the valence, core and valence-core\npolarizabilities alone are labeled as Cv\n6,Cc\n6andCvc\n6, respectively and Cct\n6corresponds to contributions from the remaining\ncross terms. References:a[23],b[24],c[25],d[26],e[27],f[28],g[29].\nDimer Cv\n6Cc\n6Cvc\n6Cct\n6C6(Total) Others Exp a b\nLi-Li 1351 0.07 ∼0 39 1390(4) 1389(2)a,1388b,1394.6c,1473d0.8592 0.0230\nLi-Na 1428 0.32 ∼0 37 1465(3) 1467(2)a0.8592 0.0245\nLi-K 2201 1.27 ∼0 119 2321(6) 2322(5)a0.8640 0.0217\nLi-Rb 2368 1.94 ∼0 179 2550(6) 2545(7)a0.8666 0.0262\nNa-Na 1515 1.51 ∼0 33 1550(3) 1556(4)a, 1472b,1561c0.8591 0.0262\nNa-K 2316 6.24 ∼0 118 2441(5) 2447(6)a2519e0.8555 0.0231\nNa-Rb 2490 9.60 ∼0 184 2684(6) 2683(7)a0.8686 0.0232\nK-K 3604 29.89 0.01 261 3895(15) 3897(15)a, 3813b,3905c3921f0.8738 0.0207\nK-Rb 3880 46.91 0.02 465 4384(12) 4274(13)a0.8738 0.0207\nRb-Rb 4178 73.96 0.4 465 4717(19) 4691(23)a, 4426b,4635c4698g0.8779 0.0207\nstudied in this work. As seen in the figure, the retarda-\ntion coefficients are the smallest for an ideal metal. At\nvery short separation distances the results for a perfectly\nconductingmaterialdiffersfromtheresultsofAu, Si, and\nSiO2by less than 4%. As the atom-surface distance in-\ncreases, the deviations of f3results for various materials\nfrom the results of an ideal metal are considerable and\nvary as 18%, 15% and 6% for Li; 33%, 14% and 18% for\nNa; 40%, 13% and 26% for K; and 50%, 13% and 33%\nfor Rb in Au, Si, and SiO 2surfaces respectively. The\ndeviation of results between an ideal metal and other di-\nelectric surfaces is smallest for the Li atom and increases\nappreciably for the Rb atom. We use the functional form\ntodescribeaccuratelytheatom-wallinteractionpotential\nat the separate distance R as\nf3(R) =1\na+b(αfsR). (6)\nBy extrapolating data from the above figure, we list the\nextracted aandbvalues for the considered atoms in all\nthe materials in Table II.\nIn Table III, we present our calculated results for the\nC6coefficients for the alkali dimers. In columns II, III\nand IV, we give individual contributions from the va-\nlence, core and valence-core polarizabilities to C6eval-\nuation and column V represents contributions from the\ncross terms which are found to be crucial for obtaining\naccurate results. As can be seen from Table I, the trends\nare almost similar to C3evaluation. A comparison of our\nC6values with other recent calculations and available\nexperimental results is also presented in the same table.\nUsing the similar fitting procedure as for f3, we obtained\nfitting parameters aandbforf6from Fig. 3 which are\nquoted in the last two columns of the above table.\nTo summarize, we haveinvestigatedthe dispersion and\ndamping coefficients for the atom-wall and atom-atominteractions for the Li, Na, K, and Rb atoms and their\ndimers in this work. The interaction potentials of the al-\nkali atomsare studied with Au, Si, and SiO 2surfacesand\n 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1\n 0  2000  4000  6000  8000  10000f6\nR(a.u.)Li-Li\nLi-Na\nLi-K\nLi-Rb\nNa-Na\nNa-K\nNa-Rb\nK-K\nK-Rb\nRb-Rb\nFIG. 3: The retardation coefficient f6(R) (dimensionless) for\nthe alkali dimers as a function of atom-atom distance R.\nfound to be very different than a perfect conductor. It is\nalso shown that the interaction of the atoms in these sur-\nfaces is considerably distinct from each other. A readily\nusable functional form of the retardation coefficients for\nthe interaction between two alkali atoms and alkali atom\nwith the above mediums is provided. Our fit explains\nmore than 99% of total variation in data about average.\nThe results are compared with the other theoretical and\nexperimental values.\nThe work of B.A. is supported by the CSIR, India\n(Grant no. 3649/NS-EMRII). We thank Dr. G. Klim-\nchitskaya and Dr. G. Lach for some useful discussions.\nB.A.alsothanksMr. S.Sokhalforhishelpin somecalcu-\nlations. Computations were carried out using 3TFLOP\nHPC Cluster at Physical Research Laboratory, Ahmed-\nabad.5\n[1] E. M. Lifshitz and L. P Pitaevskii, Statistical Physics ,\nPergamon Press, Oxford, London (1980).\n[2] H. B. G. Casimir and D. Polder, Phys. Rev. 73, 4 (1948).\n[3] F. London, Z. Physik 63, 245 (1930).\n[4] J. Israelachvili, Intermolecular and Surface Forces, A ca-\ndemic Press, San Diego (1992).\n[5] J. L. Roberts et al., Phys. Rev, Lett. 81, 5109 (1998).\n[6] C. Amiot and J. Verges, J. Chem. Phys. 112, 7068\n(2000).\n[7] P. J. Leo, C. J. Williams and P. S. Julienne, Phys. Rev.\nLett.85, 2721 (2000).\n[8] D. M. Harber et al., J. Low Temp. Phys. 133, 229 (2003).\n[9] A. E. Leanhardt et al., Phys. Rev. Lett. 90, 100404\n(2003).\n[10] Y. Lin et al., Phys. Rev. Lett. 92, 050404 (2004).\n[11] A. Derevianko, W. R. Johnson and S. Fritzsche, Phys.\nRev. A57, 2629 (1998).\n[12] A. Derevianko et al., Phys. Rev. Lett. 82, 3589 (1999).\n[13] J. Jiang, Y. Cheng and J. Mitroy, J. Phys. B 46, 125004\n(2013).\n[14] G. Lach, M. Dekieviet and U. D. Jentschura, Int. J. Mod.\nPhys. A 25, 2337 (2010).\n[15] A. O. Caride et al., Phys. Rev. A 71, 042901 (2005).[16] P. Kharchenko, J. F. Babb and A. Dalgarno, Phys. Rev.\nA55, 3566 (1997).\n[17] B. Arora and B. K. Sahoo, Phys. Rev. A 86, 033416\n(2012).\n[18] B. K. Sahoo and B. Arora, Phys. Rev. A 87, 023402\n(2013).\n[19] A. Miffre et al., Eur. Phys. J. D 38, 353 (2006).\n[20] C. R. Ekstrom et al., Phys. Rev. A 51, 3883 (1995).\n[21] W. F. Holmgren et al., Phys. Rev. A 81, 053607 (2010).\n[22] E. D. Palik, Handbook of Optical Constants of Solids ,\nAcademic Press, San Diego (1985).\n[23] A. Derevianko, J. F. Babb and A. Dalgarno, Phys. Rev.\nA83, 052704 (2001).\n[24] M. Marinescu, H. R. Sadeghpour and A. Dalgarno, Phys.\nRev. A49, 982 (1994).\n[25] J. Mitroy and M. W. J. Bromley, Phys. Rev. A 68,\n052714 (2003).\n[26] L. W. Wansbeek et al., Phys. Rev. A 78, 012515 (2008);\nErratum: Phys. Rev. A 82, 029901 (2010)\n[27] I. Russier-Antoine et al., J. Phys. B 33, 2753 (2000).\n[28] A. Pashov et al., Eur. Phys. J. D 46, 241 (2008).\n[29] C. Chin et al., Phys. Rev. A 70, 032701 (2004)."    },    {        "title": "1404.1488v2.Gilbert_damping_in_noncollinear_ferromagnets.pdf",        "content": "arXiv:1404.1488v2  [cond-mat.mtrl-sci]  27 Nov 2014Gilbert damping in noncollinear ferromagnets\nZhe Yuan,1,∗Kjetil M. D. Hals,2,3Yi Liu,1Anton A. Starikov,1Arne Brataas,2and Paul J. Kelly1\n1Faculty of Science and Technology and MESA+Institute for Nanotechnology,\nUniversity of Twente, P.O. Box 217, 7500 AE Enschede, The Net herlands\n2Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\n3Niels Bohr International Academy and the Center for Quantum Devices,\nNiels Bohr Institute, University of Copenhagen, 2100 Copen hagen, Denmark\nThe precession and damping of a collinear magnetization dis placed from its equilibrium are well\ndescribed by the Landau-Lifshitz-Gilbert equation. The th eoretical and experimental complexity\nof noncollinear magnetizations is such that it is not known h ow the damping is modified by the\nnoncollinearity. We use first-principles scattering theor y to investigate transverse domain walls\n(DWs) of the important ferromagnetic alloy Ni 80Fe20and show that the damping depends not only\non the magnetization texture but also on the specific dynamic modes of Bloch and N´ eel DWs in ways\nthat were not theoretically predicted. Even in the highly di sordered Ni 80Fe20alloy, the damping is\nfound to be remarkably nonlocal.\nPACS numbers: 72.25.Rb, 75.60.Ch, 75.78.-n, 75.60.Jk\nIntroduction. —The key common ingredient in various\nproposed nanoscale spintronics devices involving mag-\nnetic droplet solitons [ 1], skyrmions [ 2,3], or magnetic\ndomain walls (DWs) [ 4,5], is a noncollinear magneti-\nzation that can be manipulated using current-induced\ntorques (CITs) [ 6]. Different microscopic mechanisms\nhave been proposed for the CIT including spin trans-\nfer [7,8], spin-orbit interaction with broken inversion\nsymmetry in the bulk or at interfaces [ 9–11], the spin-\nHalleffect[ 12]orproximity-inducedanisotropicmagnetic\nproperties in adjacent normal metals [ 13]. Their contri-\nbutions are hotly debated but can only be disentangled\nif the Gilbert damping torque is accurately known. This\nis not the case [ 14]. Theoretical work [ 15–19] suggest-\ning that noncollinearity can modify the Gilbert damping\ndue to the absorption of the pumped spin current by the\nadjacent precessing magnetization has stimulated exper-\nimental efforts to confirm this quantitatively [ 14,20]. In\nthis Letter, we use first-principles scattering calculations\nto show that the Gilbert damping in a noncollinear alloy\ncan be significantly enhanced depending on the partic-\nular precession modes and surprisingly, that even in a\nhighly disordered alloy like Ni 80Fe20, the nonlocal char-\nacterofthe dampingis verysubstantial. Ourfindingsare\nimportant for understanding field- and/or current-driven\nnoncollinear magnetization dynamics and for designing\nnew spintronics devices.\nGilbert damping in Ni 80Fe20DWs.—Gilbert damping\nis in general described by a symmetric 3 ×3 tensor.\nFor a substitutional, cubic binary alloy like Permalloy,\nNi80Fe20, this tensor is essentially diagonal and isotropic\nand reduces to scalar form when the magnetization is\ncollinear. A value of this dimensionless scalar calculated\nfrom first-principles, αcoll= 0.0046, is in good agree-\nment with values extracted from room temperature ex-\nperiments that range between 0.004 and 0.009 [ 21]. In a\none-dimensional (1D) transverse DW, the Gilbert damp-ing tensor is still diagonal but, as a consequence of the\nlowered symmetry [ 22], it contains two unequal compo-\nnents. The magnetization in static N´ eel or Bloch DWs\n(a) \n(b) \n(c) \nφ\nθ\n x\n y z\nφ\nθ\n0 0.1 0.2 0.3 0.4 \n1/( /h w) (nm -1 )00.01 0.02 0.03 _eff Néel \nBloch \njSO =0 50 20 10 5 3 /h w (nm) \n_oeff _ieff \nFIG. 1. (color online). Sketch of N´ eel (a) and Bloch (b)\nDWs. (c) Calculated effective Gilbert damping parameters\nfor Permalloy DWs (N´ eel, black lines; Bloch, red lines) as a\nfunction of the inverse of the DW width λw. Without spin-\norbit coupling, calculations for the two DW types yield the\nsame results (blue lines). The green dot represents the valu e\nof Gilbert damping calculated for collinear Permalloy. For\neach value of λw, we typically consider 8 different disorder\nconfigurations and the error bars are a measure of the spread\nof the results.2\nliesinsidewelldefinedplanesthatareillustratedinFig. 1.\nAn angle θrepresents the in-plane rotation with respect\nto the magnetizationin the left domainand it variesfrom\n0 toπthrough a 180◦DW. If the plane changes in time,\nas it does when the magnetization precesses, an angle φ\ncan be used to describe its rotation. We define an out-\nof-plane damping component αocorresponding to varia-\ntion inφ, and an in-plane component αicorresponding\nto time-dependent θ. Rigid translation of the DW, i.e.\nmaking the DW center rwvary in time, is a specific ex-\nample of the latter.\nFor Walker-profile DWs [ 23], an effective (dimension-\nless) in-plane ( αeff\ni) and out-of-plane damping ( αeff\no) can\nbe calculated in terms of the scattering matrix Sof the\nsystem using the scattering theory of magnetization dis-\nsipation [ 24,25]. Both calculated values are plotted in\nFig.1(c) as a function of the inverse DW width 1 /λwfor\nN´ eel and Bloch DWs. Results with the spin-orbit cou-\npling (SOC) artificially switched off are shown for com-\nparison; because spin space is then decoupled from real\nspace, the results for the two DW profiles are identical\nand both αeff\niandαeff\novanish in the large λwlimit con-\nfirming that SOC is the origin of intrinsic Gilbert damp-\ning for collinear magnetization. With SOC switched on,\nN´ eel and Bloch DWs have identical values within the\nnumerical accuracy, reflecting the negligibly small mag-\nnetocrystalline anisotropy in Permalloy. Both αeff\niand\nαeff\noapproach the collinear value αcoll[21], shown as a\ngreen dot in the figure, in the wide DW limit. For finite\nwidths, theyexhibit aquadraticandapredominantlylin-\near dependence on 1 /(πλw), respectively, both with and\nwithoutSOC;forlargevaluesof λw, thereisahintofnon-\nlinearity in αeff\no(λw). However, phenomenological theo-\nries [15–17] predict that αeff\nishould be independent of λw\nand equal to αcollwhileαeff\noshould be a quadratic func-\ntion of the magnetization gradient. Neither of these pre-\ndicted behaviours is observed in Fig. 1(c) indicating that\nexisting theoretical models of texture-enhanced Gilbert\ndamping need to be reexamined.\nTheαeffshown in Fig. 1(c) is an effective damping\nconstant because the magnetization gradient dθ/dzof a\nWalker profile DW is inhomogeneous. Our aim in the\nfollowing is to understand the physical mechanisms of\ntexture-enhanced Gilbert damping with a view to deter-\nmining how the local damping depends on the magneti-\nzation gradient, as well as the corresponding parameters\nfor Permalloy, and finally expressing these in a form suit-\nable for use in micromagnetic simulations.\nIn-plane damping αi.—To get a clearer picture of how\nthe in-plane damping depends on the gradient, we calcu-\nlate the energy pumping Er≡Tr/parenleftBig\n∂S\n∂rs∂S†\n∂rs/parenrightBig\nfor a finite\nlengthLof a Bloch-DW-type spin spiral (SS) centered\natrs. In this SS segment (SSS), dθ/dzis constant ex-\ncept at the ends. Figure 2(b) showsthe resultscalculated\nwithout SOC for a single PermalloySSS with dθ/dz= 6◦0 10 20 30 40 \nL (nm) 020 40 Er (nm -2 )Without smearing \nWith smearing 0 4 2 6Winding angle ( /)\n0 1 2 3 4 \nNumber of SSSs z0n//L de/dz L L \n(c) (a) \n(b) \nFIG. 2. (color online). (a) Sketch of the magnetization gra-\ndient for two SSSs separated by collinear magnetization wit h\n(green, dashed) and without (red, solid) a broadening of the\nmagnetization gradient at the ends of the SSSs. The length\nof each segment is L. (b) Calculated energy pumping Eras a\nfunction of Lfor asingle Permalloy Bloch-DW-typeSSSwith-\nout SOC. The upper horizontal axis shows the total winding\nangle of the SSS. (c) Calculated energy pumping Erwithout\nSOC as a function of the number of SSSs that are separated\nby a stretch of collinear magnetization.\nper atomic layer; Fig. 1(c) shows that SOC does not in-\nfluence the quadratic behaviour essentially. Eris seen\nto be independent of Lindicating there is no dissipation\nwhendθ/dzis constant in the absence of SOC. In this\ncase, the only contribution arises from the ends of the\nSSS where dθ/dzchanges abruptly; see Fig. 2(a). If we\nreplace the step function of dθ/dzby a Fermi-like func-\ntion with a smearing width equal to one atomic layer, Er\ndecreasessignificantly(greensquares). Formultiple SSSs\nseparated by collinear magnetization, we find that Eris\nproportional to the number of segments; see Fig. 2(c).\nWhat remains is to understand the physical origin of\nthe damping at the ends of the SSSs. Rigid translation\nof a SSS or of a DW allows for a dissipative spin cur-\nrentj′′\ns∼ −m×∂z∂tmthat breaks time-reversal sym-\nmetry [19]. The divergence of j′′\nsgives rise to a local\ndissipative torque, whose transverse component is the\nenhancement of the in-plane Gilbert damping from the\nmagnetizationtexture. After straightforwardalgebra, we\nobtain the texture-enhanced in-plane damping torque\nα′′/bracketleftbig\n(m·∂z∂tm)m×∂zm−m×∂2\nz∂tm/bracketrightbig\n,(1)\nwhereα′′is a material parameter with dimensions\nof length squared. In 1D SSs or DWs, Eq. ( 1)\nleads to the local energy dissipation rate ˙E(r) =\n(α′′Ms/γ)∂tθ∂t(d2θ/dz2) [25], where Msis the satura-\ntion magnetization and γ=gµB//planckover2pi1is the gyromagnetic\nratio expressed in terms of the Land´ e g-factor and the\nBohr magneton µB. This results shows explicitly that\nthe in-plane damping enhancement is related to finite\nd2θ/dz2. Using the calculated data in Fig. 1(c), we ex-3\ntract a value for the coefficient α′′= 0.016 nm2that is\nindependent of specific textures m(r) [25].\nOut-of-plane damping αo.—We begin our analysis of\nthe out-of-plane damping with a simple two-band free-\nelectron DW model [ 25]. Because the linearity of the\ndamping enhancement does not depend on SOC, we ex-\namine the SOC free case for which there is no differ-\nence between N´ eel and Bloch DW profiles and we use\nN´ eel DWs in the following. Without disorder, we can\nuse the known φ-dependence of the scattering matrix for\nthis model [ 31] to obtain αeff\noanalytically,\nαeff\no=gµB\n4πAMsλw/summationdisplay\nk/bardbl/parenleftbigg/vextendsingle/vextendsingle/vextendsinglerk/bardbl\n↑↓/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsinglerk/bardbl\n↓↑/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingletk/bardbl\n↑↓/vextendsingle/vextendsingle/vextendsingle2\n+/vextendsingle/vextendsingle/vextendsingletk/bardbl\n↓↑/vextendsingle/vextendsingle/vextendsingle2/parenrightbigg\n≈gµB\n4πAMsλwh\ne2GSh,. (2)\nwhereAis the cross sectional area and the convention\nused for the reflection ( r) and transmission ( t) probabil-\nity amplitudes is shown in Fig. 3(a). Note that |tk/bardbl\n↑↓|2and\n|tk/bardbl\n↓↑|2are of the order of unity and much larger than the\nothertwotermsbetweenthebracketsunlesstheexchange\nsplitting is very large and the DW width very small. It\nis then a good approximation to replace the quantities in\nbracketsbythenumberofpropagatingmodesat k/bardbltoob-\ntain the second line of Eq. ( 2), where GShis the Sharvin\nconductance that only depends on the free-electron den-\nsity. Equation ( 2) shows analytically that αeff\nois pro-\nportional to 1 /λwin the ballistic regime. This is repro-\nduced by the results of numerical calculations for ideal\nfree-electron DWs shown as black circles in Fig. 3(b).\nIntroducing site disorder [ 32] into the free-electron\nmodel results in a finite resistivity. The out-of-plane\ndamping calculated for disordered free-electron DWs ex-\nhibits a transition as a function of its width. For narrow\nDWs (ballistic limit), αeff\nois inversely proportional to λw\nand the green, red and blue circles in Fig. 3(b) tend to\nbecomeparalleltothevioletlineforsmallvaluesof λw. If\nλwis sufficiently large, αeff\nobecomes proportional to λ−2\nw\nin agreement with phenomenological predictions [ 15–17]\nwhere the diffusive limit is assumed. This demonstrates\nthe different behaviour of αeff\noin these two regimes.\nWe can construct an expression that describes both\nthe ballistic and diffusive regimes by introducing an ex-\nplicit spatial correlation in the nonlocal form of the out-\nof-plane Gilbert damping tensor that was derived using\nthe fluctuation-dissipation theorem [ 15]\n[αo]ij(r,r′) =αcollδijδ(r−r′)+α′D(r,r′;l0)\n×[m(r)×∂zm(r)]i[m(r′)×∂z′m(r′)]j.(3)\nHereα′isamaterialparameterwithdimensionsoflength\nsquared and Dis a correlation function with an effective\ncorrelation length l0. In practice, we use D(r,r′;l0) =\n1√πAl0e−(z−z′)2/l2\n0, which reduces to δ(r−r′) in the dif-\nfusive limit ( l0≪λw) and reproduces earlier results [ 15–\n17]. In the ballistic limit, both α′andl0are infinite,0.01 0.05 0.1 0.5 \n1/( /h w) (nm -1 )10 -4 10 -3 10 -2 10 -1 _oeff \nBallistic \nl=2.7 !1 cm \nl=25 !1 cm \nl=94 !1 cm 100 50 30 20 10 5 2/h w (nm) \n~1/ hw\n~1/ hw2(a) \n(b) \nand . By definition, for weak splitting 1, but for all commonplace \ns the Fermi wavelength 2 is orders of magnitude smaller than . This \nimplies a wall resistance that is vanishingly small, because of the exponential depen- \ndence. For the example of iron, 2 is only 1 or 2 A , depending on which band \nis in question, whilst the wall thickness is some thousands of A . This leads to a \n10 . The physical reason for this is that waves are only scattered very much \nby potential steps that are abrupt on the scale of the wavelength of that wave, as \nsketched in figure 13. \nFor strong splitting ( it was found to be necessary to restrict the \nculation to a very narrow wall, viz. me 1. In practice this means \nmic abruptness. In this case a variable ¼ ð ÞÞ , trivially \nconnected to the definitions of in equations (2) and (3), determines the DW \nce. The obvious relationship with the Stearns definition of polarisation, \nequation (3), emphasises that the theory is essentially one of tunnelling between \none domain and the next. The DW resistance vanishes as 1, as might be \nd. As !1 uivalent to unity), the material becomes half-metallic \nand the wall resistance also !1 . A multi-domain half-metal, with no opportunity \nfor spin relaxation, is an insulator, no matter how high is. \nCabrera and Falicov satisfied themselves that, once the diamagnetic Lorentz \nforce e that give rise to additional resistance at the wall were properly treated \n[178], their theory could account for the results found in the Fe whiskers. However, \nit does not describe most cases encountered experimentally because the condition Abrupt \nFigure 13. Spin-resolved potential profiles and resulting wavefunctions at abrupt \nand wide (adiabatic) domain walls. The wavefunctions are travelling from left to right. In the \nadiabatic case, the wavelengths of the two wavefunctions are exchanged, but the change in \npotential energy is slow enough that there is no change in the amplitude of the transmitted \nwave. When the wall is abrupt the wavelength change is accompanied by substantial reflection, \nlting in a much lower transmitted amplitude (the reflected part of the wavefunction is not \nshown). This gives rise to domain wall resistance. C. H. Marrows Downloaded By: [University of California, Berkeley] At: 14:36 9 June 2010  V↑ V↓\n↓\n↑  e±ik↑z e±ik↓z\n e±ik↓z e±ik↑z\n. By definition, for weak splitting 1, but for all commonplace \nmi wavelength 2 is orders of magnitude smaller than . This \na wall resistance that is vanishingly small, because of the exponential depen- \ne of iron, 2 is only 1 or 2 A , depending on which band \nis in question, whilst the wall thickness is some thousands of A . This leads to a \n10 . The physical reason for this is that waves are only scattered very much \nby potential steps that are abrupt on the scale of the wavelength of that wave, as \nd in figure 13. \nit was found to be necessary to restrict the \nto a very narrow wall, 1. In practice this means \nabruptness. In this case a variable ¼ ð ÞÞ , trivially \nto the definitions of in equations (2) and (3), determines the DW \n. The obvious relationship with the Stearns definition of polarisation, \non (3), emphasises that the theory is essentially one of tunnelling between \nDW resistance vanishes as 1, as might be \nd. As !1 to , the material becomes half-metallic \n!1 . A multi-domain half-metal, with no opportunity \nis an insulator, no matter how high \nto additional resistance at the wall were properly treated \nld account for the results found in the Fe whiskers. However, \nit does not describe most cases encountered experimentally because the condition at abrupt \nto right. In the \nof the two wavefunctions are exchanged, but the change in \nis slow enough that there is no change in the amplitude of the transmitted \nis abrupt the wavelength change is accompanied by substantial reflection, \nin a much lower transmitted amplitude (the reflected part of the wavefunction is not \nto domain wall resistance. C. H. Marrows Downloaded By: [University of California, Berkeley] At: 14:36 9 June 2010 \n t↑↑ t↓↓\n↓↓\n t↑↓  t↓↑\nFIG. 3. (color online). (a) Cartoon of electronic transport\nin a two-band, free-electron DW. The global quantization\naxis of the system is defined by the majority and minority\nspin states in the left domain. (b) Calculated αeff\nofor two-\nband free-electron DWs as a function of 1 /(πλw) on a log-log\nscale. The black circles show the calculated results for the\nclean DWs, whichare in perfect agreement with theanalytica l\nmodel Eq. ( 2), shown as a dashed violet line. When disorder\n(characterized by the resistivity ρcalculated for the corre-\nsponding collinear magnetization) is introduced, αeff\noshows a\ntransition from a linear dependence on 1 /λwfor narrow DWs\ntoaquadraticbehaviourfor wideDWs. The solid lines arefits\nusing Eq. ( S24). The dashed orange lines illustrate quadratic\nbehaviour.\nbut the product α′D(r,r′;l0) =α′/(√πAl0) is finite and\nrelated to the Sharvin conductance of the system [ 33],\nconsistent with Eq. ( 2). We then fit the calculated val-\nues ofαeff\noshown in Fig. 3(b) using Eq. ( S24) [25]. With\nthe parameters α′andl0listed in Table I, the fit is seen\nto be excellent over the whole range of λw. The out-\nof-plane damping enhancement arises from the pumped\nspin current j′\ns∼∂tm×∂zmin a magnetization tex-\nture [15,17], where the magnitude of j′\nsis related to the\nTABLEI. Fitparameters usedtodescribe thedampingshown\nin Fig.1for Permalloy DWs and in Fig. 3for free-electron\nDWs with Eq. ( S24). The resistivity is determined for the\ncorresponding collinear magnetization.\nSystem ρ(µΩ cm) α′(nm2)l0(nm)\nFree electron 2 .69 45 .0 13 .8\nFree electron 24 .8 1 .96 4 .50\nFree electron 94 .3 0 .324 2 .78\nPy (ξSO= 0) 0 .504 23 .1 28 .3\nPy (ξSO/negationslash= 0) 3 .45 5 .91 13 .14\nconductivity [ 15]. This is the reason why α′is larger in\na system with a lower resistivity in Table I.l0is a mea-\nsure of how far the pumped transverse spin current can\npropagate before being absorbed by the local magnetiza-\ntion. It is worth distinguishing the relevant characteris-\ntic lengths in microscopic spin transport that define the\ndiffusive regimes for different transport processes. The\nmean free path lmis the length scale for diffusive charge\ntransport. The spin-flip diffusion length lsfcharacterizes\nthe length scale for diffusive transport of a longitudinal\nspin current, and l0is the corresponding length scale for\ntransverse spin currents. Only when the system size is\nlarger than the corresponding characteristic length can\ntransport be described in a local approximation.\nWe can use Eq. ( S24) to fit the calculated αeff\noshown\nin Fig.1for Permalloy DWs. The results are shown in\nFig.S4. Since the values of αeff\nowe calculate for N´ eel\nand Bloch DWs are nearly identical, we take their aver-\nage for the SOC case. Intuitively, we would expect the\nout-of-plane damping for a highly disordered alloy like\nPermalloy to be in the diffusive regime corresponding to\na shortl0. But the fitted values of l0are remarkably\nlarge, as long as 28.3 nm without SOC. With SOC, l0\nis reduced to 13.1 nm implying that nonlocal damping\ncan play an important role in nanoscale magnetization\ntextures in Permalloy, whose length scale in experiment\nis usually about 100 nm and can be reduced to be even\nsmaller than l0by manipulating the shape anisotropy of\nexperimental samples [ 34,35].\nAs shown in Table I,l0is positively correlated with\nthe conductivity. The large value of l0and the low re-\nsistivity of Permalloy can be qualitatively understood in\nterms of its electronic structure and spin-dependent scat-\ntering. The Ni and Fe potentials seen by majority-spin\nelectrons around the Fermi level in Permalloy are almost\nidentical [ 25] so that they are only very weakly scattered.\nThe Ni and Fe potentials seen by minority-spin electrons\nare howeverquite different leading to strongscattering in\ntransport. The strong asymmetric spin-dependent scat-\ntering can also be seen in the resistivity of Permalloy\ncalculated without SOC, where ρ↓/ρ↑>200 [21,36]. As\na result, conduction in Permalloy is dominated by the\nweakly scattered majority-spin electrons resulting in a\nlow total resistivity and a large value of l0. This short-\ncircuit effect is only slightly reduced by SOC-induced\nspin-flipscatteringbecausetheSOCin3 dtransitionmet-\nals is in energy terms small compared to the bandwidth\nand exchange splitting. Indeed, αeff\no−αcollcalculated\nwith SOC (the red curve in Fig. S4) shows a greater cur-\nvature at large widths than without SOC, but is still\nquite different from the quadratic function characteristic\nofdiffusive behaviourforthe widest DWs wecould study.\nBothαeff\niandαeff\nooriginate from locally pumped spin\ncurrents proportional to m×∂tm. Because of the spa-\ntially varying magnetization, the spin currents pumped\ntotheleftandrightdonotcancelexactlyandthenetspin0.02 0.05 0.1 0.2 0.5\n1/(πλw) (nm-1)0.0010.010.05αoeff-αcoll\nξSO≠0\nξSO=040 30 20 15 10 5 3 2πλw (nm)\n~1/λw\n~1/λw2\nFIG. 4. (color online). Calculated out-of-plane damping\nαeff\no−αcollfrom Fig. 1plotted as a function of 1 /(πλw) on a\nlog-log scale. The solid lines are fitted using Eq. ( S24). The\ndashed violet and orange lines illustrate linear and quadra tic\nbehaviour, respectively.\ncurrent contains two components, j′′\ns∼ −m×∂z∂tm[19]\nandj′\ns∼∂tm×∂zm[15,17]. For out-of-plane damping,\n∂zmis perpendicular to ∂tmso there is large enhance-\nment due to the lowest order derivative. For the rigid\nmotion of a 1D DW, ∂zmis parallel to ∂tmso thatj′\ns\nvanishes. The enhancement of in-plane damping arising\nfromj′′\nsdue to the higher-orderspatial derivative of mag-\nnetization is then smaller.\nConclusions.— We have discovered an anisotropic\ntexture-enhanced Gilbert damping in Permalloy DWs\nusing first-principles calculations. The findings are ex-\npressed in a form [Eqs. ( 1) and (S24)] suitable for ap-\nplication to micromagnetic simulations of the dynamics\nof magnetization textures. The nonlocal character of the\nmagnetization dissipation suggests that field and/or cur-\nrentdrivenDW motion, whichis alwaysassumedto be in\nthe diffusive limit, needs to be reexamined. The more ac-\ncurate form of the damping that we propose can be used\nto deduce the CITs in magnetization textures where the\nusual way to study them quantitatively is by comparing\nexperimental observations with simulations.\nCurrent-drivenDWs movewith velocities that arepro-\nportional to β/αwhereβis the nonadiabatic spin trans-\nfer torque parameter. The order of magnitude spread in\nvalues of βdeduced for Permalloy from measurements of\nthe velocities of vortex DWs [ 37–40] may be a result of\nassumingthat αis a scalarconstant. Ourpredictions can\nbe tested by reexamining these studies using the expres-\nsions for αgiven in this paper as input to micromagnetic\ncalculations.\nWe would like to thank Geert Brocks and Taher Am-\nlaki for useful discussions. This work was financially\nsupported by the “Nederlandse Organisatie voor Weten-\nschappelijk Onderzoek” (NWO) through the research\nprogramme of “Stichting voor Fundamenteel Onderzoek\nder Materie” (FOM) and the supercomputer facilities5\nof NWO “Exacte Wetenschappen (Physical Sciences)”.\nIt was also partly supported by the Royal Netherlands\nAcademy of Arts and Sciences (KNAW). 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Box 217, 7500 AE Enschede, The Net herlands\n2Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\n3Niels Bohr International Academy and the Center for Quantum Devices,\nNiels Bohr Institute, University of Copenhagen, 2100 Copen hagen, Denmark\nI. COMPUTATIONAL DETAILS.\nTaking the concrete example of Walker profile domain\nwalls (DWs), the effective (dimensionless) in-plane and\nout-of-plane damping parameters can be expressed in\nterms of the scattering matrix Sof the system as, re-\nspectively,\nαeff\ni=gµBλw\n8πAMsTr/parenleftbigg∂S\n∂rw∂S†\n∂rw/parenrightbigg\n, (S1)\nαeff\no=gµB\n8πAMsλwTr/parenleftbigg∂S\n∂φ∂S†\n∂φ/parenrightbigg\n,(S2)\nusing the scattering theory of magnetization dissipation\n[S1,S2]. Heregis the Land´ e g-factor,µBis the Bohr\nmagneton, λwdenotes the DW width, Ais the cross sec-\ntional area, and Msis the saturation magnetization.\nIt is interesting to compare the scheme for calculat-\ning the Gilbert damping of DWs using Eqs. ( S1) and\n(S2) [S1,S2] with that used for collinear magnetiza-\ntion [S3,S4]. Both of them are based upon the energy\npumping theory [ S2,S3]. To calculate the damping αcoll\nfor the collinear case, the magnetization is made to pre-\ncessuniformlyandthelocalenergydissipationishomoge-\nneous throughout the ferromagnet. The total energy loss\ndue to Gilbert damping is then proportional to the vol-\nume of the ferromagnetic material and the homogenous\nlocal damping αcollcan be determined from the damp-\ning per unit volume. When the magnetization of a DW is\nmade to change either by moving its center rwor varying\nits orientation φ, this results in a relatively large preces-\nsion at the center of the DW; the further from the center,\nthe less the magnetization changes. The local contribu-\ntion to the total energydissipationofthe DWis weighted\nby the magnitude of the magnetization precession when\nrworφvaries. For a fixed DW width, the total damping\nis not proportional to the volume of the scattering region\nbut converges to a constant once the scattering region is\nlarge compared to the DW. In practice, αeff\niandαeff\nocal-\nculated using Eqs. ( S1) and (S2) are well converged for\na scattering region 10 times longer than λw. Effectively,\nαeffcan be regardedas a weighted averageof the (dimen-\nsionless) damping constant in the region of a DW. In the\nwide DW limit, αeff\niandαeff\noboth approach αcollwith\nspin-orbit coupling (SOC) and vanish in its absence.\nTo evaluate the effective Gilbert damping of a DWusing Eqs. ( S1) and (S2), we attached semiinfinite (cop-\nper) leads to a finite length of Ni 80Fe20alloy (Permal-\nloy, Py) and rotated the local magnetization to make\na 180◦DW using the Walker profile. Specifically, we\nusedm= (sechz−rw\nλw,0,tanhz−rw\nλw) for N´ eel DWs and\nm= (−tanhz−rw\nλw,−sechz−rw\nλw,0) for Bloch DWs. The\nscatteringpropertiesofthedisorderedregionwereprobed\nby studying how Bloch waves in the Cu leads incident\nfrom the left or right sides weretransmitted and reflected\n[S4,S5]. Thescatteringmatrixwasobtainedusingafirst-\nprinciples “wave-function matching” scheme [ S6] imple-\nmented with tight-binding linearized muffin-tin orbitals\n(TB-LMTOs) [ S7]. SOC was included using a Pauli\nHamiltonian. The calculations were rendered tractable\nby imposing periodic boundary conditions transverse to\nthe transport direction. It turned out that good results\ncould be achieved even when these so-called “lateral su-\npercells” were quite modest in size. In practice, we used\n5×5 lateral supercells and the longest DW we consid-\nered was more than 500 atomic monolayers thick. After\nembedding the DW between collinear Py and Cu leads,\nthe largest scattering region contained 13300 atoms. For\nevery DW width, we averaged over about 8 random dis-\norder configurations.\nA potential profile for the scattering region was con-\nstructed within the framework of the local spin den-\nsity approximation of density functional theory as fol-\nlows. For a slab of collinear Py binary alloy sandwiched\nbetween Cu leads, atomic-sphere-approximation (ASA)\npotentials [ S7] were calculated self-consistently without\nSOC using a surface Green’s function (SGF) method im-\nplemented [ S8] with TB-LMTOs. Chargeand spin densi-\nties for binary alloy AandBsites were calculated using\nthe coherent potential approximation [ S9] generalized to\nlayer structures [ S8]. For the scattering matrix calcu-\nlation, the resulting ASA potentials were assigned ran-\ndomly to sites in the lateral supercells subject to mainte-\nnance of the appropriate concentration of the alloy [ S6]\nand SOC was included. The exchange potentials are ro-\ntated in spin space [ S10] so that the local quantization\naxis for each atomic sphere follows the DW profile. The\nDW width is determined in reality by a competition be-\ntween interatomic exchange interactions and magnetic\nanisotropy. For a nanowire composed of a soft mag-\nnetic material like Py, the latter is dominated by the2\nshape anisotropy that arises from long range magnetic\ndipole-dipole interactions and depends on the nanowire\nprofile. Experimentallyitcanbetailoredbychangingthe\nnanowire dimensions leading to the considerable spread\nof reported DW widths [ S11]. In electronic structure cal-\nculations, that do not contain magnetic dipole-dipole in-\nteractions, we simulate a change of demagnetization en-\nergy by varying the DW width. In this way we can study\nthe dependence of Gilbert damping on the magnetization\ngradient by performing a series of calculations for DWs\nwith different widths.\nFor the self-consistent SGF calculations (without\nSOC), the two-dimensional(2D) Brillouin zone (BZ) cor-\nresponding to the 1 ×1 interface unit cell was sampled\nwith a 120 ×120 grid. The transport calculations includ-\ning SOC were performed with a 32 ×32 2D BZ grid for a\n5×5 lateral supercell, which is equivalent to a 160 ×160\ngrid in the 1 ×1 2D BZ.\nII. EXTRACTING α′′\nWe first briefly derive the form of the in-plane damp-\ning. It has been argued phenomenologically [ S12] that\nfor a noncollinear magnetization texture varying slowly\nin time the lowest order term in an expansion of the\ntransverse component of the spin current in spatial and\ntime derivatives that breaks time-reversal symmetry and\nis therefore dissipative is\nj′′\ns=−ηm×∂z∂tm, (S3)\nwhereηis a coefficient depending on the material and\nmis a unit vector in the direction of the magnetization.\nThe divergence of the spin current,\n∂zj′′\ns=−η/parenleftbig\n∂zm×∂z∂tm+m×∂2\nz∂tm/parenrightbig\n,(S4)\ngives the corresponding dissipative torque exerted on the\nlocal magnetization. While the second term in brackets\nin Eq. (S4) is perpendicular to m, the first term contains\nboth perpendicular and parallel components. Since we\nare only interested in the transverse component of the\ntorque, we subtract the parallel component to find the\ndamping torque\nτ′′=−η/braceleftbig\n(1−mm)·(∂zm×∂z∂tm)+m×∂2\nz∂tm/bracerightbig\n=−η/braceleftbig\n[m×(∂zm×∂z∂tm)]×m+m×∂2\nz∂tm/bracerightbig\n=η/bracketleftbig\n(m·∂z∂tm)m×∂zm−m×∂2\nz∂tm/bracketrightbig\n.(S5)\nThe Landau-Lifshitz-Gilbert equation including the\ndamping torque τ′′reads\n∂tm=−γm×Heff+αcollm×∂tm+γτ′′\nMs\n=−γm×Heff+αcollm×∂tm\n+α′′/bracketleftbig\n(m·∂z∂tm)m×∂zm−m×∂2\nz∂tm/bracketrightbig\n,(S6)where the in-plane damping parameter α′′≡γη/Mshas\nthe dimension of length squared.\nIn the following, we explain how α′′can be extracted\nfrom calculations on Walker DWs and show that it is ap-\nplicable to other profiles. The formulation is essentially\nindependent of the DW type (Bloch or N´ eel) and we use\na Bloch DW in the following derivation for which\nm(z) = [cosθ(z),sinθ(z),0], (S7)\nwhereθ(z) represents the in-plane rotation (see Fig. 1 in\nthe paper). The local energy dissipation associated with\na time-dependent θis given by [ S2]\nγ\nMs˙E(z) =αcoll∂tm·∂tm\n+α′′/bracketleftbig\n(m·∂z∂tm)∂tm·∂zm−∂tm·∂2\nz∂tm/bracketrightbig\n.(S8)\nFor the one-dimensional profile Eq. ( S7), this can be sim-\nplified as\nγ\nMs˙E(z) =αcoll/parenleftbiggdθ\ndt/parenrightbigg2\n−α′′dθ\ndtd\ndt/parenleftbiggd2θ\ndz2/parenrightbigg\n.(S9)\nSubstituting into Eq. ( S9) the Walker profile\nθ(z) =−π\n2−arcsin/parenleftbigg\ntanhz−rw\nλw/parenrightbigg\n,(S10)\nthat we used in the calculations, we obtain for the total\nenergy dissipation associated with the motion of a rigid\nDW for which ˙θ= ˙rwdθ/drw,\n˙E=/integraldisplay\nd3r˙E(z) =2MsA\nγλw/parenleftbigg\nαcoll+α′′\n3λ2w/parenrightbigg\n˙r2\nw.(S11)\nComparing this to the energy dissipation expressed in\nterms of the effective in-plane damping αeff\ni[S2]\n˙E=2MsA\nγλwαeff\ni˙r2\nw, (S12)\nwe arrive at\nαeff\ni(λw) =αcoll+α′′\n3λ2w. (S13)\nUsing Eq. ( S13), we perform a least squares linear fitting\nofαeff\nias a function of λ−2\nwto obtain αcollandα′′. The\nfitting is shown in Fig. S1and the parameters are listed\nin Table SI. Note that αcollis in perfect agreement with\nindependent calculations for collinear Py [ S4].\nTo confirm that α′′is independent of texture, we con-\nsider another analytical DW profile in which the in-plane\nrotation is described by a Fermi-like function,\nθ(z) =−π+π\n1+ez−rF\nλF. (S14)\nHererFandλFdenote the DW center and width, re-\nspectively. Substituting Eq. ( S14) into Eq. ( S9), we find\nthe energy dissipation for “Fermi” DWs to be\n˙E=π2MsA\n6γλF/parenleftbigg\nαcoll+α′′\n5λ2\nF/parenrightbigg\n˙r2\nF,(S15)3\n0 0.5 1.0 1.5\n1/λw2 (nm-2)00.0050.0100.015αieff\nBloch\nNéel\nξSO=0Walker\nFIG. S1. Calculated αeff\nifor Walker-profile Permalloy DWs.\nN´ eel DWs: black circles, Bloch DWs: red circles. Without\nSOC, calculations for the twoDWtypesyield thesame results\n(blue circles). The dashed lines are linear fits using Eq. ( S13).\nwhich suggests the effective in-plane damping\nαeff\ni(λF) =αcoll+α′′\n5λ2\nF. (S16)\nEq. (S16) is plotted as solid lines in Fig. S2with the\nvalues of αcollandα′′taken from Table SI.\nSince the energy pumping can be expressed in terms\nof the scattering matrix Sas\n˙E=/planckover2pi1\n4πTr/parenleftbigg∂S\n∂t∂S†\n∂t/parenrightbigg\n=/planckover2pi1\n4πTr/parenleftbigg∂S\n∂rF∂S†\n∂rF/parenrightbigg\n˙r2\nF,(S17)\nwe can calculate the effective in-plane damping for a\nFermi DW from the Smatrix to be\nαeff\ni=3/planckover2pi1γλF\n2π3MsATr/parenleftbigg∂S\n∂rF∂S†\n∂rF/parenrightbigg\n.(S18)\nWe plot the values of αeff\nicalculated using the derivative\nof the scattering matrix Eq. ( S18) as circles in Fig. S2.\nThe good agreement between the circles and the solid\nlines demonstratesthe validity ofthe form ofthe in-plane\ndamping torque in Eq. ( S6) and that the parameter α′′\ndoes not depend on a specific magnetization texture.\nTABLE SI. Fit parameters to describe the in-plane Gilbert\ndamping in Permalloy DWs.\nDW type αcoll α′′(nm2)\nBloch (4.6 ±0.1)×10−30.016±0.001\nN´ eel (4.5 ±0.1)×10−30.016±0.001\nξSO=0 (2.0 ±1.0)×10−60.017±0.0010 0.5 1.0 1.5 2.0 2.5 3.0\n1/λF2 (nm-2)00.0050.0100.015αieff\nBloch\nξSO=0Fermi\nFIG. S2. Calculated αeff\nifor Permalloy Bloch DWs (red cir-\ncles) with the Fermi profile Eq. ( S14). The blue circles are\nresults calculated without SOC. The solid lines are the an-\nalytical expression Eq. ( S16) using the parameters listed in\nTableSI.\nIII. THE FREE-ELECTRON MODEL USING\nMUFFIN-TIN ORBITALS\nWe take constant potentials, V↑=−0.2 Ry,V↓=\n−0.1Ry inside atomic sphereswith an exchangesplitting\n∆V= 0.1 Ry between majority and minority spins and a\nFermi level EF= 0. The atomic spheres are placed on a\nface-centered cubic (fcc) lattice with the lattice constant\nof nickel, 3.52 ˚A. The magnetic moment on each atom is\nthen 0.072µB. The transport direction is along the fcc\n[111]. In the scattering calculation, we use a 300 ×300\n01020 30 4050 60\nL (nm)306090102030AR (fΩ m2)456(a)\n(b)\n(c)ρ=2.69±0.06 µΩ cm\nρ=24.8±0.5 µΩ cm\nρ=94.3±4.4 µΩ cm\nFIG.S3. Resistancecalculatedforthedisorderedfree-ele ctron\nmodel as a function of the length of the scattering region for\nthree values of V0, the disorder strength: 0.05 Ry (a), 0.15\nRy (b) and 0.25 Ry (c). The lines are the linear fitting used\nto determine the resistivity.4\nk-point mesh in the 2D BZ. The calculated Sharvin con-\nductances for majority and minority channels are 0.306\nand 0.153 e2/hper unit cell, respectively, compared with\nanalytical values of 0.305 and 0.153.\nTo mimic disordered free-electron systems, we intro-\nduce a 5 ×5 lateral supercell and distribute constant\npotentials uniformly in the energy range [ −V0/2,V0/2]\nwhereV0is some given strength [ S13] and spatially at\nrandom on every atomic sphere in the scattering re-gion. The calculated total resistance as a function of the\nlengthLof the (disordered) scattering region is shown in\nFig.S3withV0= 0.05 Ry (a), 0.15 Ry (b) and 0.25 Ry\n(c). The resistivity increases with the impurity strength\nas expected and can be extracted with a linear fitting\nAR(L) =AR0+ρL. For each system, we calculate about\n10randomconfigurationsand takethe averageofthe cal-\nculatedresults. Wellconvergedresultsareobtainedusing\na 32×32k-point mesh for the 5 ×5 supercell.\nIV. FITTING α′ANDl0\nWith a nonlocal Gilbert damping, α(r,r′), the energy dissipation rate is given by [ S2]\n˙E=Ms\nγ/integraldisplay\nd3r˙m(r)·/integraldisplay\nd3r′α(r,r′)·˙m(r′). (S19)\nIf we consider the out-of-plane damping of a N´ eel DW, i.e. for which the angle φvaries in time (see Fig. 1 in the\npaper), we have\n˙m(r) =˙φsechz−rw\nλwˆy. (S20)\nConsidering again a Walker profile, we find the explicit form of the out- of-plane damping matrix element\nαo(z,z′) =αcollδ(z−z′)+α′\nλ2wsechz−rw\nλwsechz′−rw\nλw1√πAl0e−(z−z′\nl0)2. (S21)\nSubstituting Eq. ( S21) and Eq. ( S20) into Eq. ( S19), we obtain explicitly the energy dissipation rate\n˙E=2MsAλw\nγαcoll˙φ2+MsAα′˙φ2\n√πγl0λ2w/integraldisplay\ndzsech2z−rw\nλw/integraldisplay\ndz′sech2z′−rw\nλwe−(z−z′\nl0)2\n. (S22)\nThe calculated effective out-of-plane Gilbert damping for a DW with th e Walker profile is related to the energy\ndissipation rate as [ S2]\n˙E=2MsAλw\nγαeff\no˙φ2. (S23)\nComparing Eqs. ( S22) and (S23), we arrive at\nαeff\no=αcoll+α′\n2√πλ3wl0/integraldisplay\ndzsech2z−rw\nλw/integraldisplay\ndz′sech2z′−rw\nλwe−(z−z′\nl0)2\n. (S24)\nThe last equation is used to fit α′andl0toαeff\nocalculated for different λw. For Bloch DWs, it is straightforward to\nrepeat the above derivation and find the same result, Eq. ( S24).\nV. BAND STRUCTURES OF NI AND FE IN\nPERMALLOY\nIn the coherent potential approximation (CPA) [ S8,\nS9], the single-site approximation involves calculating\nauxiliary (spin-dependent) potentials for Ni and Fe self-\nconsistently. In our transport calculations, these auxil-\niary potentials are distributed randomly in the scattering\nregion. It is instructive to place the Ni potentials (for\nmajority- and minority-spin electrons) on an fcc latticeand to calculate the band structure non-self-consistently.\nThen we do the same using the Fe potentials. The cor-\nresponding band structures are plotted in Fig. S4. At\nthe Fermi level, where electron transport takes place,\nthe majority-spin bands for Ni and Fe are almost identi-\ncal, including their angular momentum character. This\nmeans that majority-spin electrons in a disordered al-\nloy see essentially the same potentials on all lattice sites\nand are only very weakly scattered in transport by the\nrandomly distributed Ni and Fe potentials. In contrast,5\n-9-6-303E-EF (eV)Majority Spin Minority Spin\nX Γ L-9-6-303E-EF (eV)\nX Γ LNi Ni\nFe Fe\nFIG. S4. Band structures calculated with the auxiliary Ni\nand Fe atomic sphere potentials and Fermi energy that were\ncalculated self-consistently forNi 80Fe20usingthecoherentpo-\ntential approximation. The red bars indicates the amount of\nscharacter in each band.\nthe minority-spin bands are quite different for Ni and Fe.\nThiscanbeunderstoodintermsofthe differentexchange\nsplitting between majority- and minority-spin bands; the\ncalculated magnetic moments of Ni and Fe in Permalloy\nin the CPA are 0.63 and 2.61 µB, respectively. The ran-\ndom distribution of Ni and Fe potentials in Permalloy\nthen leads to strong scattering of minority-spin electrons\nin transport.∗Present address: Institut f¨ ur Physik, Johannes\nGutenberg–Universit¨ at Mainz, Staudingerweg 7, 55128\nMainz, Germany; zyuan@uni-mainz.de\n[S1] K. M. D. Hals, A. K. Nguyen, and A. Brataas,\nPhys. Rev. Lett. 102, 256601 (2009) .\n[S2] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer,\nPhys. Rev. B 84, 054416 (2011) .\n[S3] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer,\nPhys. Rev. Lett. 101, 037207 (2008) .\n[S4] A. A. Starikov, P. J. Kelly, A. Brataas,\nY. Tserkovnyak, and G. E. W. Bauer,\nPhys. Rev. Lett. 105, 236601 (2010) .\n[S5] Z. Yuan, Y. Liu, A. A. Starikov, P. J. Kelly, and\nA. Brataas, Phys. Rev. Lett. 109, 267201 (2012) .\n[S6] K. Xia, M. Zwierzycki, M. Talanana, P. J. Kelly, and\nG. E. W. Bauer, Phys. Rev. B 73, 064420 (2006) .\n[S7] O. K. Andersen, Z. Pawlowska, and O. Jepsen,\nPhys. Rev. B 34, 5253 (1986) .\n[S8] I. Turek, V. Drchal, J. Kudrnovsk´ y, M. ˇSob, and\nP. Weinberger, Electronic Structure of Disordered Al-\nloys, Surfaces and Interfaces (Kluwer, Boston-London-\nDordrecht, 1997).\n[S9] P. Soven, Phys. Rev. 156, 809 (1967) .\n[S10] S. Wang, Y. Xu, and K. Xia,\nPhys. Rev. B 77, 184430 (2008) .\n[S11] O. Boulle, G. Malinowski, and M. Kl¨ aui,\nMat. Science and Eng. R 72, 159 (2011) .\n[S12] Y. Tserkovnyak, E. M. Hankiewicz, and G. Vignale,\nPhys. Rev. B 79, 094415 (2009) .\n[S13] A. K. Nguyen and A. Brataas,\nPhys. Rev. Lett. 101, 016801 (2008) ."    },    {        "title": "2403.17732v1.On_a_class_of_nonautonomous_quasilinear_systems_with_general_time_gradually_degenerate_damping.pdf",        "content": "arXiv:2403.17732v1  [math.AP]  26 Mar 2024On a class of nonautonomous quasilinear systems\nwith general time-gradually-degenerate damping\nRichard De la cruz∗and Wladimir Neves†\nMarch 28, 2024\nAbstract\nInthispaper, westudytwosystemswithatime-variable coeffi cientandgeneral time-gradually-degenerate\ndamping. More explicitly, we construct the Riemann solutio ns to the time-variable coefficient Zeldovich\napproximation and time-variable coefficient pressureless g as systems both with general time-gradually-\ndegenerate damping. Applying the method of similar variabl es and nonlinear viscosity, we obtain classical\nRiemann solutions and delta shock wave solutions.\nKeywords: Pressurelessgasdynamicssystem, Zeldovichtypeapproximatesy stem, time-gradually-degenerate\ndamping, Riemann problem, delta shock solution.\n1 Introduction\nOne can find many problems from Continuum Physics that are mathem atically modeled by balance laws, that\nis to say, systems of partial differential equations in the following div ergence form\n∂u\n∂t+d/summationdisplay\nj=1∂Fj(u)\n∂xj=G(u), (1)\nwhere (t,x)∈Rd+1\n+≡(0,∞)×Rdis the set of independent variables, u∈Rndenotes the unknown vector field,\nFj∈Rnis called the flux function and G∈Rnis the vector production, absorption, or damping term. The\nfirst component t >0 is the time variable and x∈Rdis the space variable. Moreover, when G≡0 equation\n(1) is called a system of conservation laws. In fact, denoting Aj(·) =DFj(·), that is the Jacobian matrices of\nthe fluxes, the system ( 1) falls in the general class of nonhomogeneous quasilinear first-ord er systems of partial\ndifferential equations\n∂u\n∂t+d/summationdisplay\nj=1Aj(u)∂u\n∂xj=G(u). (2)\nAlbeit, there are important applications that require to consider sy stems where the coefficients Ajand\nGin (2) may depend also on the independent variables ( t,x), for instance to take into account material\ninhomogeneities, or some special geometries, also external action s, etc., see Francesco Oliveri [ 22] and references\ntherein. Therefore, one has to study the general nonautonomo us quasilinear system of partial differential\nequations\n∂ui\n∂t+d/summationdisplay\nj=1Aj\ni(t,x,u)∂u\n∂xj=Gi(t,x,u),(i= 1,...,n).\n∗School of Mathematics and Statistics, Universidad Pedag´ o gica y Tecnol´ ogica de Colombia, 150003, Tunja, Colombia. E -mail:\nrichard.delacruz@uptc.edu.co\n†Instituto de Matem´ atica, Universidade Federal do Rio de Ja neiro, Cidade Universit´ aria 21945-970, Rio de Janeiro, Br azil.\nE-mail: wladimir@im.ufrj.br\n1We are interested in studying these types of systems, more precis ely, a particular class of such systems which\nis the 2×2 systems, ( n= 2,d= 1), when Ai≡Ai(t,u), and thus the companion function Gi=Gi(t,u),\n(i= 1,2). Moreover, in this case, we recover in a simple way the divergence form. Indeed, taking especially,\nAi(t,u) =αi(t)Ai(u) andGi(t,u) =σi(t)Gi(u), we may write the above system as\n\n\n∂u1\n∂t+α1(t)∂F1(u1,u2)\n∂x=σ1(t)G1(u1,u2),\n∂u2\n∂t+α2(t)∂F2(u1,u2)\n∂x=σ2(t)G2(u1.u2).(3)\nRelated to system ( 3), let us start our study by considering the following class of nonaut onomous quasilin-\near systems with time-variable coefficients and time-dependent (line ar) damping represented by the following\nsystems:\n\nρt+α(t)(ρu)x= 0,\nut+α(t)(u2\n2)x=−σ(t)u,(4)\nand also /braceleftBigg\nρt+α(t)(ρu)x= 0,\n(ρu)t+α(t)(ρu2)x=−σ(t)ρu,(5)\nwhere 0 ≤α∈L1([0,∞)), 0≤σ∈L1\nloc([0,∞)), the unknown ρcan be interpreted as some density, and uis\nthe velocity vector field which carries the density ρ. Companion to ( 4) and (5) the initial data is given by\n(ρ(x,0),u(x,0)) = (ρ0(x),u0(x)) =/braceleftBigg\n(ρ−,u−),ifx <0,\n(ρ+,u+),ifx >0,(6)\nfor arbitrary constant states u±andρ±>0. Therefore, we are considering in fact the Riemann problem, which\nis the building block of the Cauchy problem.\nAt this point, we would like to address the reader to [ 21], where it is studied the following generalized\nBoussinesq system with variable-coefficients, (compare it with the s ystem (4)),\n\n\nut+α1(t)(u2\n2)x+β1(t)ux+γ1(t)ρx= 0,\nρt+α2(t)(ρu)x+β2(t)ρx+γ2(t)uxxx= 0,\nwhereαi,βi,γi, (i= 1,2), are time-dependent coefficients relevant to density, dispersio n and viscosity of the\nfluid. The above system can model the propagation of weakly disper sive and long weakly nonlinear surface\nwaves in shallow water. The authors, under a selection of the spect ral parameters, showed the existence of\nsoliton solutions applying the Darboux transformation and symbolic c omputation.\nOneobservesthatthesecondequationofthesystem( 4)istheBurgersequationwithtimevariablecoefficients\n[8]. In particular, the time variable coefficients can provide more usefu l models in many complicated physical\nsituations [ 8,12,28]. The homogeneous case of the system ( 4), that is to say σ(t) = 0 for all t≥0, is the\nfollowing time variable coefficient system\n\n\nρt+α(t)(ρu)t= 0,\nut+α(t)(u2\n2)x= 0,(7)\nwhich can be interpreted as an extension of Zeldovich approximation system [26,30]. In particular, the system\n(7) withα(·)≡1 is used to model the evolution of density inhomogeneities of matter in the universe [ 24, B.\nLate nonlinear stage, 3. Sticky dust]. Further, let us recall that, the system ( 4) belongs to the class of triangular\nsystems of conservation laws, that arises in a wide variety of models in physics and engineering, see for example\n2[15,23] and references therein. For this reason, the triangular system s have been studied by many authors\nand several rigorous results have been obtained for them. In [ 4,6], the Riemann problem was solved to the\nsystem ( 4) withα(·)≡1 andσ(·) equals to a positive constant, where Delta shocks have to be cons idered.\nRecently, based on the method of similar variables proposed in [ 6], Li [19] studied the Riemann problem to the\nsystem ( 4) withα(·)≡1 andσ(t) =µ\n1+twith physical parameter µ >0. In the literature the external term\nσ(t) =µ\n(1+t)θuwith physical parameters µ >0 andθ≥0 is called a time-gradually-degenerate damping [11,20],\nand it represents the time-gradually-vanishing friction effect.\nOn the other hand, the homogeneous case of the system ( 5) is the following time variable coefficient system\n/braceleftBigg\nρt+α(t)(ρu)t= 0,\n(ρu)t+α(t)(ρu2)x= 0,(8)\nwhich can be seen as an extension of pressureless gas dynamics system [26,30]. We recall that gas dynamics\nwith zero pressure is a simplified scenario where the pressure of the gas is assumed to be negligible, accounting\nfor high-speed flows or rarefied gases. The first study for the us ual pressureless gas dynamics system, that is\n(8) withα(·)≡1, is due to Bouchut [ 1] in 1994. In that paper it was studied the existence of solutions to t he\nRiemann problem for the pressureless gas dynamics system, introd ucing a notion of measure solution and delta\nshock waves were obtained. However, uniqueness was not studied .\nMoreover, the existence of a weak solution to the Cauchy problem w as first obtained independently by E,\nRikov, Sinai [ 9] in 1996, and Brenier, Grenier [ 3] in 1998. In particular, the authors in [ 9] show that, the\nstandard entropy condition ( ρΦ(ρ))t+(ρuΦ(ρ))x≤0 in the sense of distributions, where Φ is a convex function,\nis not enough to express a uniqueness criterion for weak solutions t o the Cauchy problem. Conversely, Wang\nand Ding [ 27] proved that the pressureless gas dynamics system has a unique w eak solution using the Oleinik\nentropy condition when the initial data ρ0,u0are both bounded measurable functions. However, the solution\nfor the Cauchy problem for the pressureless gas dynamics system is in general a Radon measure [ 9].\nIn 2001, Huang and Wang [ 14] studied the Cauchy problem for the system ( 8), when initial data ρ0,u0are\nrespectively a Radon measure and a bounded measurable function. Then, they showed the uniqueness of weak\nsolutions under the Oleinik entropy condition together with an energ y condition in the sense that, ρu2weakly\nconverges to ρ0u2\n0ast→0. We recall that, a particular case of Radon measure solution is the delta shock\nwave solution. A delta shock wave solution is a type of nonclassical wa ve solution in which at least one state\nvariable may develop a Dirac measure. Actually, on physical grounds , delta shock solutions typically display\nconcentration occurrence in a complex system [ 2,18]. On the other hand, it is well known that the solution for\nthe Riemann problem to the pressureless gas dynamics system involv es vacuum and delta shock wave solution\nand the classical Riemann solutions satisfy the Lax entropy conditio n while delta shock wave solution is unique\nunder an over-compressive entropy condition [ 25,29]. In a similar way, Keita and Bourgault [ 17] solved the\nRiemann problem for the pressureless system with linear damping, th at is, the system ( 5) withα(·)≡1 and\nσ(·)≡const., showing vacuum states and delta shock solution and uniqueness un der the Lax entropy condition\nand over-compressive entropy condition, respectively. Finally, De la cruz and Juajibioy [ 5] obtained delta shock\nsolutions for a generalized pressureless system with linear damping.\n1.1 Equivalent ×non-equivaqlent systems\nSince we considered α1=α2=αin both systems ( 4), (5), one may ask when these systems are equivalent or\nnot. Indeed, we observe first that for smooth solutions an elemen tary manipulation of the second equation of\n(5) reads\nρ(ut+α(t)(1\n2u2)x)+u(ρt+α(t)(ρu)x) =−σ(t)ρu.\nTherefore, due to the first equation of ( 5) and for ρ/ne}ationslash= 0, the above equation reduces to the equation ( 4)2, and\nthus for smooth solutions the system ( 5) is equivalent to the system ( 4). Albeit, the question remains open\nwhen the solutions are non-regular.\n3Once placed the above question, we observe that Keita and Bourga ult [17], recently in 2019, studied the\nRiemann problem for the Zeldovich approximation and pressureless g as dynamics systems with linear damping\nwithσ=const. > 0. Moreprecisely, they analyzedin that paper the Riemann problemt o the followingsystems:\n\n\nρt+(ρu)x= 0,\nut+(u2\n2)x=−σu,(9)\n/braceleftBigg\nρt+(ρu)x= 0,\n(ρu)t+(ρu2)x=−σρu,(10)\nwith initial data given by ( 6), and it was proved that\n1.u−< u+. The solution of the Riemann problem ( 9)-(6) and (10)-(6) is given by\n(ρ,u)(x,t) =\n\n(ρ−,u−e−σt), x < u −1−e−σt\nσ,\n(0,σx\neσt−1), u −1−e−σt\nσ≤x≤u+1−e−σt\nσ,\n(ρ+,u+e−σt), x > u +1−e−σt\nσ.\n2.u−> u+. The solution of the Riemann problem ( 9)-(6) is given by\n(ρ,u)(x,t) =\n\n(ρ−,u−e−σt), x <u−+u+\n2σ(1−e−σt),\n(w(t)δ(x−u−+u+\n2σ(1−e−σt)),uδ(t)), x=u−+u+\n2σ(1−e−σt),\n(ρ+,u+e−σt), x >u−+u+\n2σ(1−e−σt),(11)\nwhere\nw(t) =(ρ++ρ−)(u−−u+)\n2σ(1−e−σt) and uδ(t) =u−+u+\n2e−σt.\nHowever, the solution of the Riemann problem ( 10)-(6) is given by\n(ρ,u)(x,t) =\n\n(ρ−,u−e−σt), x <√ρ+u++√ρ−u−√ρ++√ρ−(1−eσt),\n(w(t)δ(x−/integraldisplayt\n0uδ(s)ds),uδ(t)), x=√ρ+u++√ρ−u−√ρ++√ρ−(1−eσt),\n(ρ+,u+e−σt), x >√ρ+u++√ρ−u−√ρ++√ρ−(1−eσt),\nwhere\nw(t) =√ρ−ρ+(u−−u+)\nσ(1−e−σt) and uδ(t) =√ρ+u++√ρ−u−√ρ++√ρ−e−σt.\nConsequently, Keita, Bourgault showed that the systems ( 9) and (10) are equivalent for smooth and also for\ntwo contact-discontinuity solutions, but they differ for delta shoc k solutions. Therefore, it should be expected\nthat a similar scenario of delta shocks are presented here as well, an d the systems ( 4) and (5) are not equivalent\nfor these types of solutions. We remark that the problems here be come much more complicated since σ(·)\nbesides non-constant is just a locally summable function.\n42 The Zeldovich Type Approximate System\nIn this section, we study the Riemann problem to the time-variable co efficient Zeldovich’s approximate system\nand time-variable linear damping, that is to say ( 4)-(6). We extended some ideas from [ 4] to construct the\nviscous solutions to the system ( 4), see (12) below. After we show that the family of viscous solutions {(ρε,uε)}\nconverges to a solution of the Riemann problem ( 4)-(6). Foru−< u+, classical Riemann solutions are obtained.\nWhenu−> u+, we show that a delta shock solution is a solution to the Riemann proble m (4)-(6).\n2.1 Parabolic regularization\nGivenε >0, we consider the following parabolic regularization for the system ( 4),\n\n\nρε\nt+α(t)(ρεuε)x=εβ(t)ρε\nxx,\nuε\nt+1\n2α(t)((uε)2)x+σ(t)uε=εβ(t)uε\nxx,(12)\nwhere conveniently we define β(t) :=α(t)exp(−/integraltextt\n0σ(s)ds). We search for ( ρε,uε) be an approximate solution\nof problem ( 4)-(6), which is defined by the parabolic approximation ( 12) with initial data given by\n(ρε(x,0),uε(x,0)) = (ρ0(x),u0(x)), (13)\nwhere (ρ0,u0) is given by ( 6).\nThen, the main issue of this section is to solve problem ( 12) with initial data ( 13). To this end, we use the\nauxiliary function u(x,t) =/hatwideux(x,t)e−/integraltextt\n0σ(τ)dτand a version of Hopf-Cole transformation which enable us to\nobtain an explicit solution of the viscous system ( 12)-(13). The function /hatwideuwill be explained during the proof\nof the following\nProposition 2.1. Under the assumptions on the functions α,β,σ, the explicit solution of the problem (12)-(13)\nis given by\nρε(x,t) =∂xWε(x,t)anduε(x,t) =u+bε\n+(x,t)+u−bε\n−(x,t)\nbε\n+(x,t)+bε\n−(x,t)exp(−/integraldisplayt\n0σ(s)ds),\nwhere\nWε(x,t) =ρ−/parenleftBig\nx−u−/integraltextt\n0β(s)ds/parenrightBig\nbε\n−(x,t)+ρ+/parenleftBig\nx−u+/integraltextt\n0β(s)ds/parenrightBig\nbε\n+(x,t)\nbε\n−(x,t)+bε\n+(x,t)\n+(ρ+−ρ−)(ε/integraltextt\n0β(s)ds)1/2exp/parenleftBig\n−x2\n4ε/integraltextt\n0β(s)ds/parenrightBig\nπ1/2(bε\n−(x,t)+bε\n+(x,t))\nand\nbε\n±(x,t) :=±1\n(4πε/integraltextt\n0β(s)ds)1/2/integraldisplay±∞\n0exp/parenleftBigg\n−(x−y)2\n4ε/integraltextt\n0β(s)ds−u±y\n2ε/parenrightBigg\ndy.\nProof.1. Firstly we observe that, if ( /hatwideρ,/hatwideu) solves\n/braceleftBigg\n/hatwideρt+α(t)e−/integraltextt\n0σ(τ)dτ/hatwideρx/hatwideux=εβ(t)/hatwideρxx,\n/hatwideut+1\n2α(t)e−/integraltextt\n0σ(τ)dτ(/hatwideux)2=εβ(t)/hatwideuxx,(14)\nwith the initial condition given by\n(ρ(x,0),/hatwideu(x,0)) =/braceleftBigg\n(ρ−x,u−x),ifx <0,\n(ρ+x,u+x),ifx >0,\n5then (ρε,uε) defined by ( /hatwideρε\nx,/hatwideuxe−/integraltextt\n0σ(τ)dτ) solves the problem ( 12)-(13). Indeed, let us recall the generalized\nHopf-Cole transformation, see [ 13,4,16], that is\n/braceleftBigg\n/hatwideρε=Cεe/hatwideu\n2ε,\n/hatwideuε=−2εln(Sε).(15)\nThen, from system ( 14) and the generalized Hopf-Cole transformation ( 15), we have\n/braceleftBigg\nCε\nt=εβ(t)Cε\nxx,\nSε\nt=εβ(t)Sε\nxx,(16)\nwith initial data given by\n(Cε(x,0),Sε(x,0)) =/braceleftBigg\n(ρ−xe−u−x\n2ε,e−u−x\n2ε),ifx <0,\n(ρ+xe−u+x\n2ε,e−u+x\n2ε),ifx >0.(17)\n2. Now, the solution to the problem ( 16)-(17) in terms of the heat kernel is\n/braceleftBigg\nCε(x,t) =aε\n−(x,t)+aε\n+(x,t),\nSε(x,t) =bε\n−(x,t)+bε\n+(x,t),(18)\nwhere\naε\n±(x,t) :=±ρ±\n(4πε/integraltextt\n0β(s)ds)1/2/integraldisplay±∞\n0yexp/parenleftBigg\n−(x−y)2\n4ε/integraltextt\n0β(s)ds−u±y\n2ε/parenrightBigg\ndy\nand\nbε\n±(x,t) :=±1\n(4πε/integraltextt\n0β(s)ds)1/2/integraldisplay±∞\n0exp/parenleftBigg\n−(x−y)2\n4ε/integraltextt\n0β(s)ds−u±y\n2ε/parenrightBigg\ndy.\nMoreover, we have\n/integraldisplay±∞\n0∂y/parenleftBigg\nexp/parenleftBigg\n−(x−y)2\n4ε/integraltextt\n0β(s)ds/parenrightBigg/parenrightBigg\nexp/parenleftBig\n−u±y\n2ε/parenrightBig\ndy=−exp/parenleftBigg\n−x2\n4ε/integraltextt\n0β(s)ds/parenrightBigg\n+u±\n2ε/integraldisplay±∞\n0exp/parenleftBigg\n−(x−y)2\n4ε/integraltextt\n0β(s)ds−u±y\n2ε/parenrightBigg\ndy.(19)\nOn the other hand, it follows that\n/integraldisplay±∞\n0∂y/parenleftBigg\nexp/parenleftBigg\n−(x−y)2\n4ε/integraltextt\n0β(s)ds/parenrightBigg/parenrightBigg\nexp/parenleftBig\n−u±y\n2ε/parenrightBig\ndy=/integraldisplay±∞\n0(x−y)\n2ε/integraltextt\n0β(s)dsexp/parenleftBigg\n−(x−y)2\n4ε/integraltextt\n0β(s)ds−u±y\n2ε/parenrightBigg\ndy\n=x\n2ε/integraltextt\n0β(s)ds/integraldisplay±∞\n0exp/parenleftBigg\n−(x−y)2\n4ε/integraltextt\n0β(s)ds−u±y\n2ε/parenrightBigg\ndy\n−/integraldisplay±∞\n0y\n2ε/integraltextt\n0β(s)dsexp/parenleftBigg\n−(x−y)2\n4ε/integraltextt\n0β(s)ds−u±y\n2ε/parenrightBigg\ndy.\n(20)\nTherefore, from ( 19) and (20) we obtain\n/integraldisplay±∞\n0yexp/parenleftBigg\n−(x−y)2\n4ε/integraltextt\n0β(s)ds−u±y\n2ε/parenrightBigg\ndy=2ε/integraldisplayt\n0β(s)ds·exp/parenleftBigg\n−x2\n4ε/integraltextt\n0β(s)ds/parenrightBigg\n+/parenleftbigg\nx−u±/integraldisplayt\n0β(s)ds/parenrightbigg/integraldisplay±∞\n0exp/parenleftBigg\n−(x−y)2\n4ε/integraltextt\n0β(s)ds−u±y\n2ε/parenrightBigg\ndy.(21)\n63. Finally, we observe that\n∂x/parenleftBigg/integraldisplay±∞\n0exp/parenleftBigg\n−(x−y)2\n4ε/integraltextt\n0β(s)ds−u±y\n2ε/parenrightBigg\ndy/parenrightBigg\n=−/integraldisplay±∞\n0∂y/parenleftBigg\nexp/parenleftBigg\n−(x−y)2\n4ε/integraltextt\n0β(s)ds/parenrightBigg/parenrightBigg\nexp/parenleftBig\n−u±y\n2ε/parenrightBig\ndy\nand from ( 19) we have\n∂x/parenleftBigg/integraldisplay±∞\n0exp/parenleftBigg\n−(x−y)2\n4ε/integraltextt\n0β(s)ds−u±y\n2ε/parenrightBigg\ndy/parenrightBigg\n= exp/parenleftBigg\n−x2\n4ε/integraltextt\n0β(s)ds/parenrightBigg\n−u±\n2ε/integraldisplay±∞\n0exp/parenleftBigg\n−(x−y)2\n4ε/integraltextt\n0β(s)ds−u±y\n2ε/parenrightBigg\ndy.(22)\nTherefore, we may write from ( 18) and (21) that\nCε(x,t) =ρ−/bracketleftBigg\n−(ε/integraltextt\n0β(s)ds)1/2\nπ1/2exp/parenleftBigg\n−x2\n4ε/integraltextt\n0β(s)ds/parenrightBigg\n+/parenleftbigg\nx−u−/integraldisplayt\n0β(s)ds/parenrightbigg\nbε\n−(x,t;1)/bracketrightBigg\n+ρ+/bracketleftBigg\n(ε/integraltextt\n0β(s)ds)1/2\nπ1/2exp/parenleftBigg\n−x2\n4ε/integraltextt\n0β(s)ds/parenrightBigg\n+/parenleftbigg\nx−u+/integraldisplayt\n0β(s)ds/parenrightbigg\nbε\n+(x,t;1)/bracketrightBigg\n.\nMoreover, from ( 18) and (22) we have\nSε\nx(x,t) =−1\n2ε(u−bε\n−(x,t)+u+bε\n+(x,t)).\nApplying the generalized Hopf-Cole transformation ( 15), it follows that\nρε(x,t) =/hatwideρε\nx(x,t) = (Cε(x,t)/Sε(x,t))x,\nuε(x,t) =−2εSε\nx\nSεexp(−/integraldisplayt\n0σ(s)ds),\nand hence the proof is complete.\nRemark 1. One observes that, the solution ( ρε,uε) of the problem ( 12)-(13) is absolutely continuous with\nrespect to time t >0, and smooth in x∈R.\n2.2 The Riemann problem\nIn this section, we study the Riemann problem to the system ( 4) withσ(t)≥0 for allt≥0, which means that\nthe damping can degenerate in some open interval contained in (0 ,∞).\nTo obtain the Riemann solution to the problem ( 4) with initial data ( 6) we use the viscosity system with\ntime-dependent damping ( 12) with initial data ( 13) and analyze the limit behavior as ε→0+of the solutions\n(ρε,uε) obtained in the previous section. To follow, we write bε\n±(x,t) as\nbε\n±(x,t) =±1\n(4πε/integraltextt\n0β(s)ds)1/2/integraldisplay±∞\n0exp/parenleftBigg\n−(x−y)2\n4ε/integraltextt\n0β(s)ds−u±y\n2ε/parenrightBigg\ndy\n=±1\n(πBε(t))1/2exp/parenleftbigg−x2+(x−x±(t))2\nBε(t)/parenrightbigg/integraldisplay±∞\n0exp/parenleftbigg\n−(y+x±(t)−x)2\nBε(t)/parenrightbigg\ndy\n=1\nπ1/2exp/parenleftbigg−x2+(x−x±(t))2\nBε(t)/parenrightbigg/integraldisplay∞\n±(Bε(t))1/2(x±(t)−x)exp(−y2)dy\n=1\nπ1/2exp/parenleftbigg−x2+(x−x±(t))2\nBε(t)/parenrightbigg\nIε,t\n±,\n7wherex±(t) =u±/integraltextt\n0β(s)ds,Bε(t) = 4ε/integraltextt\n0β(s)ds, and\nIε,t\n±=/integraldisplay∞\n±(Bε(t))1/2(x±(t)−x)exp(−y2)dy.\nAsε→0+, due to the asymptotic expansion of the (complementary) error f unction (see [ 10]), we have\nIε,t\n±=\n\n∞/summationdisplay\nn=0(−1)n(2n)!\nn!/parenleftbigg(Bε(t))1/2\n±2(x±(t)−x)/parenrightbigg2n+1\nexp/parenleftbigg\n−(x±(t)−x)2\nBε(t)/parenrightbigg\n,if±(x±(t)−x)>(Bε(t))1/2,\n1\n2π1/2,ifx±(t) =x,\nπ1/2−∞/summationdisplay\nn=0(−1)n(2n)!\nn!/parenleftbigg(Bε(t))1/2\n∓2(x±(t)−x)/parenrightbigg2n+1\nexp/parenleftbigg\n−(x±(t)−x)2\nBε(t)/parenrightbigg\n,if±(x±(t)−x)<−(Bε(t))1/2,\nand therefore we obtain\nbε\n±(x,t) =\n\n±Q±\nπ1/2exp/parenleftBigx2\nBε(t)/parenrightBig\n,if±(x±(t)−x)>(Bε(t))1/2,\n1\n2exp/parenleftBig\n−x2\nBε(t)/parenrightBig\n,ifx±(t) =x,\nexp/parenleftBig−x2+(x±(t)−x)2\nBε(t)/parenrightBig\n±Q±\nπ1/2exp/parenleftBig\n−x2\nBε(t)/parenrightBig\n,if±(x±(t)−x)<−(Bε(t))1/2,(23)\nwhere\nQ±=∞/summationdisplay\nn=0(−1)n(2n)!\nn!/parenleftBig(Bε(t))1/2\n2(x±(t)−x)/parenrightBig2n+1\n=ε1/2/parenleftBigg/parenleftbig/integraltextt\n0β(s)ds/parenrightbig1/2\nx±(t)−x−2ε/parenleftBig/parenleftbig/integraltextt\n0β(s)ds/parenrightbig1/2\nx±(t)−x/parenrightBig3\n+12ε2/parenleftBig/parenleftbig/integraltextt\n0β(s)ds/parenrightbig1/2\nx±(t)−x/parenrightBig5\n−···/parenrightBigg\n.\n2.2.1 Classical Riemann solutions: u−≤u+.\nIn this case, we have the following\nTheorem 2.1. Suppose that u−≤u+. Let(ρε,uε)be the solution of the viscosity problem (12)-(13). Then,\nthe limit\nlim\nε→0+(ρε(x,t),uε(x,t)) = (ρ(x,t),u(x,t))\nexists in the sense of distributions, and the pair (ρ(x,t),u(x,t))solves the time-variable coefficient Zeldovich\napproximate system and time-dependent damping (4)with initial data (6). In addition, if u−< u+, then\n(ρ(x,t),u(x,t)) =\n\n(ρ−,u−exp(−/integraltextt\n0σ(s)ds)),ifx < x−(t),\n(0,x/integraltextt\n0β(s)dsexp(−/integraltextt\n0σ(s)ds)),ifx−(t)< x < x +(t),\n(ρ+,u+exp(−/integraltextt\n0σ(s)ds)),ifx > x+(t),\nand when u−=u+, then\n(ρ(x,t),u(x,t)) =/braceleftBigg\n(ρ−,u−exp(−/integraltextt\n0σ(s)ds)),ifx < x−(t),\n(ρ+,u−exp(−/integraltextt\n0σ(s)ds)),ifx > x−(t).\n8Proof.1. First, let us consider the case x−x−(t)<−(Bε(t))1/2. Forε >0 sufficiently small, due to approxi-\nmations given by ( 23), we may write\nWε(x,t)≈ρ−(x−x−(t))cε\n−−ρ+(Bε(t))1/2\n2π1/2exp/parenleftBig\n−x2\nBε(t)/parenrightBig\n+(ρ+−ρ−)(Bε(t))1/2\n2π1/2exp/parenleftBig\n−x2\nBε(t)/parenrightBig\ncε\n−+(Bε(t))1/2\n2π1/2(x+(t)−x),\nwherecε\n−= exp/parenleftBig\n−x2+(x−(t)−x)2\nBε(t)/parenrightBig\n−(Bε(t))1/2\n2π1/2(x−(t)−x)exp/parenleftBig\n−x2\nBε(t)/parenrightBig\n. Therefore, we obtain\nWε(x,t)≈ρ−(x−x−(t))exp/parenleftBig\n(x−(t)−x)2\nBε(t)/parenrightBig\nexp/parenleftBig\n(x−(t)−x)2\nBε(t)/parenrightBig\n+(Bε(t))1/2\n2π1/2/parenleftBig\n1\nx+(t)−x−1\nx−(t)−x/parenrightBig (24)\nand\nuε(x,t)≈(Bε(t))1/2\n2π1/2/parenleftBig\nu+\nx+(t)−x−u−\nx−(t)−x/parenrightBig\n+u−exp/parenleftBig\n(x−(t)−x)2\nBε(t)/parenrightBig\nexp/parenleftBig\n(x−(t)−x)2\nBε(t)/parenrightBig\n+(Bε(t))1/2\n2π1/2/parenleftBig\n1\nx+(t)−x−1\nx−(t)−x/parenrightBigexp(−/integraldisplayt\n0σ(s)ds). (25)\n2. Similarly, if x−(t)+(Bε(t))1/2< x < x +(t)−(Bε(t))1/2, then we approximate Wεas\nWε(x,t)≈ρ−(x−x−(t))/hatwidecε\n−+ρ+(x−x+(t))/hatwidecε\n++(ρ+−ρ−)(Bε(t))1/2\n2π1/2exp/parenleftBig\n−x2\nBε(t)/parenrightBig\n/hatwidecε\n−+/hatwidecε\n+\nwhere/hatwidecε\n±=±1\nπ1/2/parenleftBig\n(Bε(t))1/2\n2(x±(t)−x)−(Bε(t))3/2\n4(x±(t)−x)3/parenrightBig\nexp/parenleftBig\n−x2\nBε(t)/parenrightBig\n. Therefore,\nWε(x,t)≈Bε(t)/parenleftBig\nρ+\n(x+(t)−x)2−ρ−\n(x−(t)−x)2/parenrightBig\n2/parenleftBig\n1\nx+(t)−x−1\nx−(t)−x/parenrightBig\n+Bε(t)/parenleftBig\n1\n(x−(t)−x)3−1\n(x+(t)−x)3/parenrightBig (26)\nand\nuε(x,t) =u+\nx+(t)−x+u−\nx−x−(t)+∞/summationtext\nn=1(−1)n(2n)!(Bε(t))n\nn!4n/parenleftBig\nu+\n(x+(t)−x)2n+1+u−\n(x−x−(t))2n+1/parenrightBig\n1\nx+(t)−x+1\nx−x−(t)+∞/summationtext\nn=1(−1)n(2n)!(Bε(t))n\nn!4n/parenleftBig\n1\n(x+(t)−x)2n+1+1\n(x−x−(t))2n+1/parenrightBigexp(−/integraldisplayt\n0σ(s)ds).(27)\nMoreover, if x+(t)−x <−(Bε(t))1/2, then we have\nWε(x,t)≈ρ−(Bε(t))1/2\n2π1/2exp/parenleftBig\n−x2\nBε(t)/parenrightBig\n+ρ+(x−x+(t))cε\n++(ρ+−ρ−)(Bε(t))1/2\n2π1/2exp/parenleftBig\n−x2\nBε(t)/parenrightBig\n(Bε(t))1/2\n2π1/2(x−x−(t))exp/parenleftBig\n−x2\nBε(t)/parenrightBig\n+cε\n+\nwherecε\n+= exp/parenleftBig\n−x2+(x+(t)−x)2\nBε(t)/parenrightBig\n−(Bε(t))2\n2π1/2(x−x+(t))exp/parenleftBig\n−x2\nBε(t)/parenrightBig\n, and therefore we get\nWε(x,t)≈ρ+(x−x+(t))exp/parenleftBig\n(x+(t)−x)2\nBε(t)/parenrightBig\nexp/parenleftBig\n(x+(t)−x)2\nBε(t)/parenrightBig\n+(Bε(t))1/2\n2π1/2/parenleftBig\n1\nx+(t)−x−1\nx−(t)−x/parenrightBig (28)\nand\nuε(x,t)≈u+exp/parenleftBig\n(x+(t)−x)2\nBε(t)/parenrightBig\n+(Bε(t))1/2\n2π1/2/parenleftBig\nu+\nx+(t)−x−u−\nx−(t)−x/parenrightBig\nexp/parenleftBig\n(x+(t)−x)2\nBε(t)/parenrightBig\n+(Bε(t))1/2\n2π1/2/parenleftBig\n1\nx+(t)−x−1\nx−(t)−x/parenrightBigexp(−/integraldisplayt\n0σ(s)ds). (29)\n93. Now, for the case u−< u+, from (24), (26), and (28) we have\nlim\nε→0+Wε(x,t) =W(x,t) =\n\nρ−(x−x−(t)),ifx < x−(t),\n0, ifx−(t)< x < x +(t),\nρ+(x−x+(t)),ifx > x+(t)\nand from ( 25), (27), and (29) we have\nlim\nε→0+uε(x,t) =u(x,t) =\n\nu−exp(−/integraltextt\n0σ(s)ds),ifx < x−(t),\nx/integraltextt\n0β(s)dsexp(−/integraltextt\n0σ(s)ds),ifx−(t)< x < x +(t),\nu+exp(−/integraltextt\n0σ(s)ds),ifx > x+(t).\nSinceuε(x,t) is bounded on compact subsets of R2\n+={(x,t) :x∈R,t >0}anduε(x,t)→u(x,t) pointwise as\nε→0+, then uε(x,t)→u(x,t) in the sense of distribution. Also, Wε(x,t) is bounded on compact subsets of\nR2\n+andWε(x,t)→W(x,t) pointwise as ε→0+, then Wε(x,t)→W(x,t) in the sense of distributions and so\nWε\nx(x,t) converges in the distributional sense to Wx(x,t). From Proposition 2.1, we have that lim\nε→0+ρε(x,t) =\nρ(x,t) exists in the sense of distribution and\nρ(x,t) =Wx(x,t) =\n\nρ−,ifx < x−(t),\n0,ifx−(t)< x < x +(t),\nρ+,ifx > x+(t).\nFor the case u−=u+, we have\nlim\nε→0+(ρε(x,t),uε(x,t)) = (ρ(x,t),u(x,t)) =/braceleftBigg\n(ρ−,u−exp(−/integraltextt\n0σ(s)ds)),ifx < x−(t),\n(ρ+,u−exp(−/integraltextt\n0σ(s)ds)),ifx > x−(t).\nFinally, it is not difficult to show that ( ρ(x,t),u(x,t)) solves ( 4), and thus we omit the details.\n2.2.2 Delta shock wave solutions: u−> u+.\nIn this section, we study the Riemann problem to the system ( 4) with initial data ( 6) whenu−> u+. Let us\nrecall that, in particular when α(·)≡1 andσ(·)≡σ=const. > 0, the solution is not bounded and contains a\nweighted delta measure supported on a smooth curve (see [ 17]), which is a delta shock solution given by ( 11).\nHere, we have a more general context with similar results. Therefo re, we first define the meaning of a\ntwo-dimensional weighted delta function.\nDefinition 2.1. Givenw∈L1((a,b)), with−∞< a < b < ∞, and a smooth curve\nL≡ {(x(s),t(s)) :a < s < b },\nwe say that w(·)δLis a two-dimensional weighted delta function supported on L, when for each test function\nϕ∈C∞\n0(R×[0,∞)),\n/an}bracketle{tw(·)δL,ϕ(·,·)/an}bracketri}ht=/integraldisplayb\naw(s)ϕ(x(s),t(s))ds.\nNow, the following definition tells us when a pair ( ρ,u) is a delta shock wave solution to the Riemann\nproblem ( 4)-(6).\nDefinition 2.2. A distribution pair ( ρ,u) is called a delta shock wave solution of the problem ( 4) and (6)\nin the sense of distributions, when there exists a smooth curve Land a function w(·), such that ρanduare\nrepresented respectively by\nρ=/hatwideρ(x,t)+wδL, u=u(x,t)\n10with/hatwideρ,u∈L∞(R×(0,∞)), and satisfy for each the test function ϕ∈C∞\n0(R×(0,∞)),\n\n\n< ρ,ϕ t>+< αρu,ϕ x>= 0,\n/integraldisplay/integraldisplay\nR2\n+/parenleftBig\nuϕt+α(t)\n2u2ϕx−σ(t)uϕ/parenrightBig\ndxdt= 0,\nwhere\n< ρ,ϕ >=/integraldisplay/integraldisplay\nR2\n+/hatwideρϕdxdt+/an}bracketle{twδL,ϕ/an}bracketri}ht,\nand\n< αρu,ϕ > =/integraldisplay/integraldisplay\nR2\n+α(t)/hatwideρuϕdxdt +/an}bracketle{tα(·)wuδδL,ϕ/an}bracketri}ht.\nMoreover, u|L=uδ(·).\nPlaced the previous definitions, we are going to show a solution with a d iscontinuity on x=x(t) for the\nsystem (4) of the form\n(ρ(x,t),u(x,t)) =\n\n(ρ−(x,t),u−(x,t)),ifx < γ(t),\n(w(t)δL,uδ(t)),ifx=γ(t),\n(ρ+(x,t),u+(x,t)),ifx > γ(t),\nwhereρ±(x,t),u±(x,t) are piecewise smooth solutions of system ( 4),δLis the Dirac measure supported on the\ncurveγ∈C1, andγ,w, anduδare to be determined. Then, we have the following\nTheorem 2.2. Suppose u−> u+. Let(ρε,uε)be the solution of the problem (12)-(13). Then the limit\nlim\nε→0+(ρε(x,t),uε(x,t)) = (ρ(x,t),u(x,t))\nexists in the sense of distributions and (ρ(x,t),u(x,t))solves the problem (4)-(6). In addition,\n(ρ(x,t),u(x,t)) =\n\n(ρ−,u−exp(−/integraldisplayt\n0σ(s)ds)),ifx < x(t),\n(w(t)δ(x−x(t)),u−+u+\n2exp(−/integraldisplayt\n0σ(s)ds)),ifx=x(t),\n(ρ+,u+exp(−/integraldisplayt\n0σ(s)ds)),ifx > x(t),\nwhere\nw(t) =1\n2(ρ−+ρ+)(u−−u+)/integraldisplayt\n0α(s)exp(−/integraldisplays\n0σ(τ)dτ)ds,\nx(t) =u−+u+\n2/integraldisplayt\n0α(s)exp(−/integraldisplays\n0σ(τ)dτ)ds.\nProof.1. First, since u−> u+, it follows that x−(t)> x+(t). Forε >0 sufficiently small, if x−x−(t)>\n(Bε(t))1/2, then we may write from ( 23),\nWε(x,t)≈ρ+(x−x+(t))exp/parenleftBig\n(x+(t)−x)2\nBε(t)/parenrightBig\nexp/parenleftBig\n(x+(t)−x)2\nBε(t)/parenrightBig\n+(Bε(t))1/2\n2π1/2/parenleftBig\n1\nx+(t)−x−1\nx−(t)−x/parenrightBig\nand\nuε(x,t)≈u+exp/parenleftBig\n(x+(t)−x)2\nBε(t)/parenrightBig\n+(Bε(t))1/2\n2π1/2/parenleftBig\nu+\nx+(t)−x−u−\nx−(t)−x/parenrightBig\nexp/parenleftBig\n(x+(t)−x)2\nBε(t)/parenrightBig\n+(Bε(t))1/2\n2π1/2/parenleftBig\n1\nx+(t)−x−1\nx−(t)−x/parenrightBigexp(−/integraldisplayt\n0σ(s)ds).\n11Ifx+(t)−x <−(Bε(t))1/2andx=x−(t), then\nWε(x,t)≈−ρ−(Bε(t))1/2\n2π1/2exp/parenleftBig\n−x2\nBε(t)/parenrightBig\n+ρ+(x−x+(t))exp/parenleftBig\n−x2+(x+(t)−x)2\nBε(t)/parenrightBig\n1\n2exp/parenleftBig\n−x2\nBε(t)/parenrightBig\n+exp/parenleftBig\n−x2+(x+(t)−x)2\nBε(t)/parenrightBig\n+(Bε(t))1/2\n2π1/2(x+(t)−x)exp/parenleftBig\n−x2\nBε(t)/parenrightBig\nand\nuε(x,t)≈u−\n2exp/parenleftBig\n−x2\nBε(t)/parenrightBig\n+u+exp/parenleftBig\n−x2+(x+(t)−x)2\nBε(t)/parenrightBig\n+u+(Bε(t))1/2\n2π1/2(x+(t)−x)exp/parenleftBig\n−x2\nBε(t)/parenrightBig\n1\n2exp/parenleftBig\n−x2\nBε(t)/parenrightBig\n+exp/parenleftBig\n−x2+(x+(t)−x)2\nBε(t)/parenrightBig\n+(Bε(t))1/2\n2π1/2(x+(t)−x)exp/parenleftBig\n−x2\nBε(t)/parenrightBigexp(−/integraldisplayt\n0σ(s)ds).\nIfx+(t)+(Bε(t))1/2≤x≤x−(t)−(Bε(t))1/2, then\nWε(x,t)≈ρ−(x−x−(t))exp/parenleftBig\n(x−(t)−x)2\nBε(t)/parenrightBig\n+ρ+(x−x+(t))exp/parenleftBig\n(x+(t)−x)2\nBε(t)/parenrightBig\nexp/parenleftBig\n(x−(t)−x)2\nBε(t)/parenrightBig\n+exp/parenleftBig\n(x+(t)−x)2\nBε(t)/parenrightBig\n+(Bε(t))1/2\n2π1/2/parenleftBig\n1\nx+(t)−x−1\nx−(t)−x/parenrightBig\nand\nuε(x,t)≈u−exp/parenleftBig\n(x−(t)−x)2\nBε(t)/parenrightBig\n+u+exp/parenleftBig\n(x+(t)−x)2\nBε(t)/parenrightBig\n+(Bε(t))1/2\n2π1/2/parenleftBig\nu+\nx+(t)−x−u−\nx−(t)−x/parenrightBig\nexp/parenleftBig\n(x−(t)−x)2\nBε(t)/parenrightBig\n+exp/parenleftBig\n(x+(t)−x)2\nBε(t)/parenrightBig\n+(Bε(t))1/2\n2π1/2/parenleftBig\n1\nx+(t)−x−1\nx−(t)−x/parenrightBigexp(−/integraldisplayt\n0σ(s)ds).\nIfx+(t)−x >(Bε(t))1/2, then\nWε(x,t)≈ρ−(x−x−(t))exp/parenleftBig\n(x−(t)−x)2\nBε(t)/parenrightBig\nexp/parenleftBig\n(x−(t)−x)2\nBε(t)/parenrightBig\n+(Bε(t))1/2\n2π1/2/parenleftBig\n1\nx+(t)−x−1\nx−(t)−x/parenrightBig\nand\nuε(x,t)≈u−exp/parenleftBig\n(x−(t)−x)2\nBε(t)/parenrightBig\n+(Bε(t))1/2\n2π1/2/parenleftBig\nu+\nx+(t)−x−u−\nx−(t)−x/parenrightBig\nexp/parenleftBig\n(x−(t)−x)2\nBε(t)/parenrightBig\n+(Bε(t))1/2\n2π1/2/parenleftBig\n1\nx+(t)−x−1\nx−(t)−x/parenrightBigexp(−/integraldisplayt\n0σ(s)ds).\nIfx−x−(t)<−(Bε(t))1/2andx=x+(t), then\nWε(x,t)≈ρ−(x−x−(t))exp/parenleftBig\n−x2+(x−(t)−x)2\nBε(t)/parenrightBig\n+ρ+(Bε(t))1/2\n2π1/2exp/parenleftBig\n−x2\nBε(t)/parenrightBig\nexp/parenleftBig\n−x2+(x−(t)−x)2\nBε(t)/parenrightBig\n−(Bε(t))1/2\n2π1/2(x−(t)−x)exp/parenleftBig\n−x2\nBε(t)/parenrightBig\n+1\n2exp/parenleftBig\n−x2\nBε(t)/parenrightBig\nand\nuε(x,t)≈u−exp/parenleftBig\n−x2+(x−(t)−x)2\nBε(t)/parenrightBig\n−u−(Bε(t))1/2\n2π1/2(x−(t)−x)exp/parenleftBig\n−x2\nBε(t)/parenrightBig\n+u+\n2exp/parenleftBig\n−x2\nBε(t)/parenrightBig\nexp/parenleftBig\n−x2+(x−(t)−x)2\nBε(t)/parenrightBig\n−(Bε(t))1/2\n2π1/2(x−(t)−x)exp/parenleftBig\n−x2\nBε(t)/parenrightBig\n+1\n2exp/parenleftBig\n−x2\nBε(t)/parenrightBigexp(−/integraldisplayt\n0σ(s)ds).\nTherefore, we have that\nlim\nε→0+Wε(x,t) =/braceleftBigg\nρ−(x−x−(t)),if (x−x+(t))2−(x−x−(t))2<0,\nρ+(x−x+(t)),if (x−x+(t))2−(x−x−(t))2>0.\nObserve that ( x−x+(t))2−(x−x−(t))2= 2(x−(t)−x+(t))(x−x−(t)+x+(t)\n2), and defining\nx−(t)+x+(t)\n2=:x(t),\n12we get\nlim\nε→0+Wε(x,t) =/braceleftBigg\nρ−(x−x−(t)),ifx < x(t),\nρ+(x−x+(t)),ifx > x(t).\nSinceWε(x,t) is bounded on compact subsets of R2\n+andWε(x,t)→W(x,t) pointwise as ε→0+, then\nWε(x,t)→W(x,t) in the sense of distribution and so Wε\nx(x,t) convergesin the distributional sense to Wx(x,t).\nFrom Proposition 2.1we have that lim\nε→0+ρε(x,t) =ρ(x,t) exists in the sense of distribution and\nρ(x,t) =Wx(x,t) =\n\nρ−,ifx < x(t),\n(x−(t)−x+(t))ρ−+ρ+\n2δ(x−x(t)),ifx=x(t)\nρ+,ifx > x(t).(30)\nAnalogously, we obtain\nu(x,t) =\n\nu−exp(−/integraldisplayt\n0σ(s)ds),ifx < x(t),\nu−+u+\n2exp(−/integraldisplayt\n0σ(s)ds),ifx=x(t),\nu+exp(−/integraldisplayt\n0σ(s)ds),ifx > x(t).(31)\n2. Now, we show that ρandu, defined respectively by ( 30), (31) solve the Riemann problem ( 4)-(6) in the\nsense of Definition 2.2. Indeed, for any test function ϕ∈C∞\n0(R×R+) we have\n< ρ,ϕ t>+< αρu,ϕ x>=/integraldisplay∞\n0/integraldisplay\nR(ρϕt+α(t)ρuϕx)dxdt\n+/integraldisplay∞\n0ρ−+ρ+\n2(x−(t)−x+(t))/parenleftbigg\nϕt+α(t)u−+u+\n2exp(−/integraldisplayt\n0σ(s)ds)ϕx/parenrightbigg\ndt\n=/integraldisplay∞\n0/integraldisplayx(t)\n−∞(ρ−ϕt+α(t)ρ−u−exp(−/integraldisplayt\n0σ(s)ds)ϕx)dxdt+/integraldisplay∞\n0/integraldisplay∞\nx(t)(ρ+ϕt+α(t)ρ+u+exp(−/integraldisplayt\n0σ(s)ds)ϕx)dxdt\n+/integraldisplay∞\n0ρ−+ρ+\n2(x−(t)−x+(t))/parenleftbigg\nϕt+α(t)u−+u+\n2exp(−/integraldisplayt\n0σ(s)ds)ϕx/parenrightbigg\ndt\n=−/contintegraldisplay\n−(α(t)ρ−u−exp(−/integraldisplayt\n0σ(s)ds)ϕ)dt+(ρ−ϕ)dx+/contintegraldisplay\n−(α(t)ρ+u+exp(−/integraldisplayt\n0σ(s)ds)ϕ)dt+(ρ+ϕ)dx\n+/integraldisplay∞\n0ρ−+ρ+\n2(x−(t)−x+(t))/parenleftbigg\nϕt+α(t)u−+u+\n2exp(−/integraldisplayt\n0σ(s)ds)ϕx/parenrightbigg\ndt\n=/integraldisplayt\n0/parenleftbigg\nα(t)(ρ−u−−ρ+u+)exp(−/integraldisplayt\n0σ(s)ds)−(ρ−−ρ+)dx(t)\ndt/parenrightbigg\nϕdt\n+/integraldisplay∞\n0ρ−+ρ+\n2(x−(t)−x+(t))dϕ\ndtdt\n=/integraldisplayt\n0/parenleftbigg\nα(t)(ρ−u−−ρ+u+)exp(−/integraldisplayt\n0σ(s)ds)−(ρ−−ρ+)dx(t)\ndt/parenrightbigg\nϕdt\n−/integraldisplay∞\n0d\ndt/parenleftbiggρ−+ρ+\n2(x−(t)−x+(t))/parenrightbigg\nϕdt= 0,\n13and\n/integraldisplay∞\n0/integraldisplay\nR/parenleftbigg\nuϕt+α(t)\n2u2ϕx−σ(t)uϕ/parenrightbigg\ndxdt=/integraldisplay∞\n0/integraldisplay\nR/parenleftbigg\nuϕt+α(t)\n2u2ϕx/parenrightbigg\ndxdt−/integraldisplay∞\n0/integraldisplay\nRσ(t)uϕdxdt\n=/integraldisplay∞\n0/integraldisplayx(t)\n−∞u−exp(−/integraldisplayt\n0σ(s)ds)/parenleftbigg\nϕt+α(t)\n2u−exp(−/integraldisplayt\n0σ(s)ds)ϕx/parenrightbigg\ndxdt\n+/integraldisplay∞\n0/integraldisplay∞\nx(t)u+exp(−/integraldisplayt\n0σ(s)ds)/parenleftbigg\nϕt+α(t)\n2u+exp(−/integraldisplayt\n0σ(s)ds)ϕx/parenrightbigg\ndxdt−/integraldisplay∞\n0/integraldisplay\nRσ(t)uϕdxdt\n=−/contintegraldisplay\n−/parenleftbiggα(t)\n2u2\n−exp(−2/integraldisplayt\n0σ(s)ds)ϕ/parenrightbigg\ndt+/parenleftbigg\nu−exp(−/integraldisplayt\n0σ(s)ds)ϕ/parenrightbigg\ndx\n+/integraldisplay∞\n0/integraldisplayx(t)\n−∞σ(t)u−exp(−/integraldisplayt\n0σ(s)ds)ϕdxdt\n+/contintegraldisplay\n−/parenleftbiggα(t)\n2u2\n+exp(−2/integraldisplayt\n0σ(s)ds)ϕ/parenrightbigg\ndt+/parenleftbigg\nu+exp(−/integraldisplayt\n0σ(s)ds)ϕ/parenrightbigg\ndx\n+/integraldisplay∞\n0/integraldisplay∞\nx(t)σ(t)u+exp(−/integraldisplayt\n0σ(s)ds)ϕdxdt−/integraldisplay∞\n0/integraldisplay\nRσ(t)uϕdxdt\n=/integraldisplay∞\n0/parenleftbiggα(t)\n2(u2\n−−u2\n+)exp(−/integraldisplayt\n0σ(s)ds)−(u−−u+)dx(t)\ndt/parenrightbigg\nϕexp(−/integraldisplayt\n0σ(s)ds)dt= 0.\n3. Finally, we observe that, for each t≥0,\nu+α(t)exp(−/integraldisplayt\n0σ(τ)dτ)<dx(t)\ndt< u+α(t)exp(−/integraldisplayt\n0σ(τ)dτ),\nwhich is an entropy condition to the system ( 4).\n3 Pressureless Type Gas Dynamics System\nThe main issue of this section is to study the Riemann problem of the pr essureless gas system with variable\ncoefficient and time-variable linear damping ( 5). We introduce a similar variable to reduce the system ( 5) to\nhyperbolic conservation laws with variable coefficient to solve the Riem ann problem with u−< u+. To the\ncaseu−> u+, similar to [ 5], we use a nonlinear viscous system and using a similar variable we obtain viscous\nsolutions that converge to a delta shock solution of the Riemann pro blem (5)-(6).\n3.1 Classical Riemann solutions.\nWe observe that under transformation /hatwideu(x,t) =u(x,t)e/integraltextt\n0σ(r)drthe system ( 5) is equivalent to\n/braceleftBigg\nρt+α(t)e−/integraltextt\n0σ(r)dr(ρ/hatwideu)x= 0,\n(ρ/hatwideu)t+α(t)e−/integraltextt\n0σ(r)dr(ρ/hatwideu2)x= 0,(32)\nwith the initial data ( 6). Using the similar variable\nξ=x/integraltextt\n0α(s)e−/integraltexts\n0σ(r)drds, (33)\nthe system ( 32) can be written as/braceleftBigg\n−ξρξ+(ρ/hatwideu)ξ= 0,\n−ξ(ρ/hatwideu)ξ+(ρ/hatwideu2)ξ= 0,(34)\n14and the initial condition ( 6) changes to the boundary condition\n(ρ(±∞),/hatwideu(±∞)) = (ρ±,u±).\nNow, we note that any smooth solution of the system ( 34) satisfies\n/parenleftbigg/hatwideu−ξ ρ\n/hatwideu(/hatwideu−ξ)ρ(2/hatwideu−ξ)/parenrightbigg/parenleftbiggρξ\n/hatwideuξ/parenrightbigg\n=/parenleftbigg0\n0/parenrightbigg\nand it provides either the general solution (constant state) ρ(ξ) =constant and /hatwideu(ξ) =constant, ρ/ne}ationslash= 0, or the\nsingular solution ρ(ξ) = 0 for all ξand/hatwideu(ξ) =ξ, called the vacuum state . Thus the smooth solutions of system\n(34) only contain constants and vacuum solutions. For a bounded disco ntinuity at ξ=η, the Rankine-Hugoniot\ncondition holds, that is to say,\n/braceleftBigg\n−η(ρ−−ρ+)+(ρ−/hatwideu−−ρ+/hatwideu+) = 0,\n−η(ρ−/hatwideu−−ρ+/hatwideu+)+(ρ−/hatwideu2\n−−ρ+/hatwideu2\n+) = 0,\nwhich holds when η=u−=u+. Therefore, two states ( ρ−,u−) and (ρ+,u+) can be connected by a contact\ndiscontinuity if and only if u−=u+. Thus, the contact discontinuity is characterized by ξ=u−=u+.\nSummarizing, we obtainthe solutionwhich consistsoftwocontactdis continuitiesand avacuum statebesides\ntwo constant states. Therefore, the solution can be expressed as\n(ρ(ξ),/hatwideu(ξ)) =\n\n(ρ−,u−),ifξ < u−,\n(0,ξ),ifu−≤ξ≤u+,\n(ρ+,u+),ifξ > u+.\nSinceu(x,t) =/hatwideu(x,t)e−/integraltextt\n0σ(r)drandξ=x/integraltextt\n0α(s)e−/integraltexts\n0σ(r)drds, then for u−< u+the Riemann solution to the\nsystem (5) is\n(ρ(x,t),u(x,t)) =\n\n(ρ−,u−e−/integraltextt\n0σ(r)dr),ifx < u−/integraldisplayt\n0α(s)e−/integraltexts\n0σ(r)drds,\n(0,xe−/integraltextt\n0σ(r)dr\n/integraltextt\n0α(s)e−/integraltexts\n0σ(r)drds),ifu−/integraldisplayt\n0α(s)e−/integraltexts\n0σ(r)drds≤x≤u+/integraldisplayt\n0α(s)e−/integraltexts\n0σ(r)drds,\n(ρ+,u+e−/integraltextt\n0σ(r)dr),ifx > u+/integraldisplayt\n0α(s)e−/integraltexts\n0σ(r)drds.\n3.2 Delta shock wave solutions.\nGivenε >0, we consider the following parabolic regularization to the system ( 5),\n/braceleftBiggρε\nt+α(t)(ρεuε)x= 0,\n(ρεuε)t+α(t)(ρε(uε)2)x=εβ∗(t)uε\nxx−σ(t)ρεuε,(35)\nwhereβ∗(t) =α(t)exp(−/integraltextt\n0σ(s)ds)/integraltextt\n0α(s)exp(−/integraltexts\n0σ(r)dr)ds, with initial condition\n(ρε(x,0),uε(x,0)) = (ρ0(x),u0(x)), (36)\nwhere (ρ0,u0) is given by ( 6).\nUnder the transformation /hatwideuε(x,t) =uε(x,t)e/integraltextt\n0σ(r)drthe system ( 35) becomes\n\n\nρε\nt+α(t)e−/integraltextt\n0σ(r)dr(ρε/hatwideuε)x= 0,\n(ρε/hatwideuε)t+α(t)e−/integraltextt\n0σ(r)dr(ρε(/hatwideuε)2)x=εβ∗(t)/hatwideuε\nxx,(37)\n15and the initial condition ( 36) becomes\n(ρε(x,0),/hatwideuε(x,0)) = (ρε\n0(x),/hatwideuε\n0(x)) =/braceleftBigg\n(ρ−,u−),ifx <0,\n(ρ+,u+),ifx >0(38)\nfor arbitrary constant states u±andρ±>0 as well. By using the similar variable ( 33) the system ( 37) can be\nwritten as \n\n−ξρε\nξ+(ρε/hatwideuε)ξ= 0,\n−ξ(ρε/hatwideuε)ξ+(ρε(/hatwideuε)2)ξ=ε/hatwideuε\nξξ(39)\nand the initial data ( 38) changes to the boundary condition\n(ρ(±∞),/hatwideu(±∞)) = (ρ±,u±) (40)\nfor arbitrary constant states u−> u+andρ±>0. The existence of solutions to the system ( 39) with boundary\ncondition ( 40) was shown in Theorem 3 of [ 5]. More explicitly, in [ 5], the following result was obtained:\nProposition 3.1. There exists a weak solution (ρε,/hatwideuε)∈L1\nloc((−∞,+∞))×C2((−∞,+∞))to the boundary\nproblem (39)-(40).\nFrom Theorem 2 in [ 5], we have that for each ε >0, the function /hatwideuεsatisfies\n/braceleftBigg\nε(/hatwideuε)′′(ξ) = (ρε(ξ)(/hatwideu−ξ))(/hatwideuε)′(ξ),\n/hatwideuε(±∞) =u±,\nwith′=d\ndξand\nρε(ξ) =/braceleftBigg\nρε\n1(ξ),if−∞< ξ < ξε\nς,\nρε\n2(ξ),ifξε\nς< ξ <+∞,\nwhereξε\nςsatisfies/hatwideuε(ξε\nς) =ξε\nς,\nρ1(ξ) =ρ−exp/parenleftBigg\n−/integraldisplayξ\n−∞(/hatwideuε(s))′\n/hatwideuε(s)−sds/parenrightBigg\nandρ2(ξ) =ρ+exp/parenleftbigg/integraldisplay∞\nξ(/hatwideuε(s))′\n/hatwideuε(s)−sds/parenrightbigg\n.\nDefinition 3.1. A distribution pair ( ρ,u) is called a delta shock wave solution of the problem ( 5) and (6) in the\nsense of distributions, when there exist a smooth curve Land a function w(·), such that ρanduare represented\nrespectively by\nρ=/hatwideρ(x,t)+wδLandu=u(x,t),\nwith/hatwideρ,u∈L∞(R×(0,∞)), and satisfy for each the test function ϕ∈C∞\n0(R×(0,∞)),\n/braceleftBigg\n< ρ,ϕ t>+< αρu,ϕ x>= 0,\n< ρu,ϕ t>+< αρu2,ϕx>=< σρu,ϕ >,(41)\nwhere\n< ρ,ϕ >:=/integraldisplay/integraldisplay\nR2\n+/hatwideρϕdxdt+/an}bracketle{twδL,ϕ/an}bracketri}ht\nand for some smooth function G,\n< αρG(·),ϕ >:=/integraldisplay/integraldisplay\nR2\n+α(t)/hatwideρG(u)ϕdxdt+/an}bracketle{tα(·)wG(uδ)δL,ϕ/an}bracketri}ht.\nMoreover, u|L=uδ(t).\n16Now, we denote ς= lim\nε→0+ξε\nς= lim\nε→0+/hatwideuε(ξε\nς) =/hatwideu(ς). Then, according to Theorem 4 in [ 5], we have\nlim\nε→0+(ρε(ξ),/hatwideuε(ξ)) =\n\n(ρ−,u−), ifξ < ς,\n(w0δ(ξ−ς),uδ),ifξ=ς,\n(ρ+,u+), ifξ > ς,\nwhereρεconverges in the sense of distributions to the sum of a step functio n and a Dirac measure δwith weight\nw0=−ς(ρ−−ρ+)+(ρ−u−−ρ+u+) anduδ=/hatwideu(ς). Moreover, ( ς,w0,uδ) satisfies\n\n\nς=uδ,\nw0=−ς(ρ−−ρ+)+(ρ−u−−ρ+u+),\nw0uδ=−ς(ρ−u−−ρ+u+)+(ρ−u2\n−−ρ+u2\n+),(42)\nand the over-compressive entropy condition\nu+< uδ< u−. (43)\nObserve that from the system ( 42) we have\n(ρ−−ρ+)u2\nδ−2(ρ−u−−ρ+u+)uδ+(ρ−u2\n−−ρ+u2\n+) = 0,\nwhich implies\nuδ=√ρ−u−−√ρ+u+√ρ−−√ρ+oruδ=√ρ−u−+√ρ+u+√ρ−+√ρ+.\nOne remarks that, when uδ=√ρ−u−+√ρ+u+√ρ−+√ρ+the entropy condition is valid while uδ=√ρ−u−−√ρ+u+√ρ−−√ρ+\ndoes not satisfy the entropy condition. Moreover, using the seco nd equation of the system ( 42) anduδ=√ρ−u−+√ρ+u+√ρ−+√ρ+, we obtain w0=√ρ−ρ+(u−−u+). Therefore, when ρ−=ρ+, from (42) we obtain\n2(u−−u+)uδ−(u2\n−−u2\n+) = 0\nand hence we have uδ=1\n2(u−+u+) andw0=ρ−(u−−u+). Finally, using the similar variable ( 33), we have\nobtained the following result\nProposition 3.2. Suppose u−> u+. Let(ρε(x,t),/hatwideuε(x,t))be the solution of the problem (37)-(38). Then\nthe limit lim\nε→0+(ρε(x,t),/hatwideuε(x,t)) = (ρ(x,t),/hatwideu(x,t))exists in the distribution sense. Moreover, (ρ(x,t),/hatwideu(x,t))is\ngiven by\n\n\n(ρ−,u−),ifx < uδ/integraldisplayt\n0α(s)e−/integraltexts\n0σ(r)drds,\n(w0/integraldisplayt\n0α(s)e−/integraltexts\n0σ(r)drds·δ(x−uδ/integraldisplayt\n0α(s)e−/integraltexts\n0σ(r)drds),uδ),ifx=uδ/integraldisplayt\n0α(s)e−/integraltexts\n0σ(r)drds,\n(ρ+,u+),ifx > uδ/integraldisplayt\n0α(s)e−/integraltexts\n0σ(r)drds,\nwherew0=√ρ−ρ+(u−−u+)anduδ=√ρ−u−+√ρ+u+√ρ−+√ρ+, whenρ−/ne}ationslash=ρ+. For the case ρ−=ρ+, it follows that,\nw0=ρ−(u−−u+)anduδ=1\n2(u−+u+). In addition, the solution is unique under the over-compres sive entropy\ncondition (43).\nRemark 2. The condition ( 42) is necessary and sufficient to guarantee the existence of delta sh ock solutions\nto the problem ( 37)-(38) withε= 0. In fact, there are two delta shock solutions. Now, the over-c ompressive\nentropy condition ( 43), (see the above proposition), was sufficient to obtain the uniquen ess of the delta shock\nsolution.\n17From the above proposition and since u(x,t) =/hatwideu(x,t)e−/integraltextt\n0σ(r)dr, we can establish a solution to the system\n(5)withinitialdata( 6). Moreover,multiplyingtheentropycondition( 43)byα(t)wegetα(t)u+< uδα(t)< α(t)\nfor allt≥0 and again using that u(x,t) =/hatwideu(x,t)e−/integraltextt\n0σ(r)dr, we have extended the entropy condition ( 43) to\nthe following entropy condition to the system ( 5),\nλ(ρ+,u+)e−/integraltextt\n0σ(r)dr<dx(t)\ndt< λ(ρ−,u−)e−/integraltextt\n0σ(r)dr,for allt≥0, (44)\nwhereλ(ρ,u) =αuis the eigenvalue associated to system ( 5). Then, we have the following\nTheorem 3.1. Suppose u−> u+. Then the Riemann problem (5)-(6)admits under the entropy condition (44)\na unique delta shock solution of the form\n(ρ(x,t),u(x,t)) =\n\n(ρ−,u−e−/integraltextt\n0σ(r)dr),ifx < x(t),\n(w(t)δ(x−x(t)),uδ(t)),ifx=x(t),\n(ρ+,u+e−/integraltextt\n0σ(r)dr),ifx > x(t),(45)\nwhere for ρ−/ne}ationslash=ρ+,\nw(t) =√ρ−ρ+(u−−u+)/integraldisplayt\n0α(s)e−/integraltexts\n0σ(r)drds, u δ(t) =√ρ−u−+√ρ+u+√ρ−+√ρ+e−/integraltextt\n0σ(r)dr,and\nx(t) =√ρ−u−+√ρ+u+√ρ−+√ρ+/integraldisplayt\n0α(s)e−/integraltexts\n0σ(r)drds.\nFor the case ρ−=ρ+, it follws that\nw(t) =ρ−(u−−u+)/integraldisplayt\n0α(s)e−/integraltexts\n0σ(r)drds, u δ(t) =1\n2(u−+u+)e−/integraltexts\n0σ(r)dr,and\nx(t) =1\n2(u−+u+)/integraldisplayt\n0α(s)e−/integraltexts\n0σ(r)drds.\nProof.Suppose that ρ−/ne}ationslash=ρ+. Therefore, in orderto show that ( ρ,u), given by ( 45), is a solution to the problem\n(5)-(6), we consider any test function ϕ∈C∞\n0(R×(0,∞)) and compute\n< ρu,ϕ t>+< ρu2,ϕx>=/integraldisplay∞\n0/integraldisplay\nR(ρuϕt+α(t)ρu2ϕx)dxdt+/integraldisplay∞\n0w(t)(uδ(t)ϕt+α(t)u2\nδ(t)ϕx)dt\n=/integraldisplay∞\n0/integraldisplayx(t)\n−∞(ρ−u−e−/integraltextt\n0σ(r)drϕt+α(t)ρ−u2\n−e−2/integraltextt\n0σ(r)drϕx)dxdt\n+/integraldisplay∞\n0/integraldisplay∞\nx(t)(ρ+u+e−/integraltextt\n0σ(r)drϕt+α(t)ρ+u2\n+e−2/integraltextt\n0σ(r)drϕx)dxdt\n+/integraldisplay∞\n0w(t)√ρ−u−+√ρ+u+√ρ−+√ρ+e−/integraltextt\n0σ(r)dr/parenleftbigg\nϕt+α(t)√ρ−u−+√ρ+u+√ρ−+√ρ+e−/integraltextt\n0σ(r)drϕx/parenrightbigg\ndt\n=−/contintegraldisplay\n−/parenleftBig\nα(t)ρ−u2\n−e−2/integraltextt\n0σ(r)drϕ/parenrightBig\ndt+/parenleftBig\nρ−u−e−/integraltextt\n0σ(r)drϕ/parenrightBig\ndx\n+/contintegraldisplay\n−/parenleftBig\nα(t)ρ+u2\n+e−2/integraltextt\n0σ(r)drϕ/parenrightBig\ndt+/parenleftBig\nρ+u+e−/integraltextt\n0σ(r)drϕ/parenrightBig\ndx\n+/integraldisplay∞\n0/integraldisplay\nRσ(t)ρuϕdxdt +/integraldisplay∞\n0w(t)√ρ−u−+√ρ+u+√ρ−+√ρ+e−/integraltextt\n0σ(r)dr/parenleftbigg\nϕt+dx(t)\ndtϕx/parenrightbigg\ndt\n18=/integraldisplay∞\n0α(t)(ρ−u2\n−−ρ+u2\n+)e−2/integraltextt\n0σ(r)drϕdt−/integraldisplay∞\n0dx(t)\ndt(ρ−u−−ρ+u+)e−/integraltextt\n0σ(r)drϕdt\n+/integraldisplay∞\n0/integraldisplay\nRσ(t)ρuϕdxdt +/integraldisplay∞\n0w(t)√ρ−u−+√ρ+u+√ρ−+√ρ+e−/integraltextt\n0σ(r)drdϕ(t)\ndtdt\n=/integraldisplay∞\n0α(t)(ρ−u2\n−−ρ+u2\n+)e−2/integraltextt\n0σ(r)drϕdt−/integraldisplay∞\n0dx(t)\ndt(ρ−u−−ρ+u+)e−/integraltextt\n0σ(r)drϕdt\n+/integraldisplay∞\n0/integraldisplay\nRσ(t)ρuϕdxdt −/integraldisplay∞\n0d\ndt/parenleftbigg\nw(t)√ρ−u−+√ρ+u+√ρ−+√ρ+e−/integraltextt\n0σ(r)dr/parenrightbigg\nϕdt\n=/integraldisplay∞\n0/integraldisplay\nRσ(t)ρuϕdxdt +/integraldisplay∞\n0σ(t)w(t)√ρ−u−+√ρ+u+√ρ−+√ρ+e−/integraltextt\n0σ(r)drdt=< σρu,ϕ >,\nwhich implies the second equation of ( 41). With a similar argument, it is possible to obtain the first equation of\n(41) and the case when ρ−=ρ+. The uniqueness of the solution will be obtained under the entropy c ondition\n(44).\n4 Riemann problem to the systems (4) and (5) with σ(·)≡0\nIn this section, we consider σ(t) =µν(t) whereµ >0 is a parameter, ν(t)≥0 for allt≥0, andν∈L1\nloc([0,∞)).\nAccording to the Sections 2.2.1and3.1, ifu−< u+, the systems ( 4) and (5) with initial data ( 6) have the\nsolution\n(ρ(x,t),u(x,t)) =\n\n(ρ−,u−exp(−µ/integraltextt\n0ν(s)ds)), ifx < x−(t),\n(0,x/integraltextt\n0α(s)exp(−µ/integraltexts\n0ν(r)dr)dsexp(−µ/integraltextt\n0ν(s)ds)),ifx−(t)< x < x +(t),\n(ρ+,u+exp(−µ/integraltextt\n0ν(s)ds)), ifx > x+(t),\nwherex±(t) =u±/integraltextt\n0α(s)exp(−µ/integraltexts\n0ν(r)dr)ds. Ifu−> u+, the the solution for the problem ( 4)-(6) is\n(ρ(x,t),u(x,t)) =\n\n(ρ−,u−exp(−µ/integraltextt\n0ν(s)ds)), ifx < x(t),\n(w(t)δ(x−x(t)),u−+u+\n2exp(−µ/integraltextt\n0ν(s)ds)),ifx=x(t),\n(ρ+,u+exp(−µ/integraltextt\n0ν(s)ds)), ifx > x(t),\nwherew(t) =1\n2(ρ−+ρ+)(u−−u+)/integraltextt\n0α(s)exp(−µ/integraltexts\n0ν(τ)dτ)dsandx(t) =u−+u+\n2/integraltextt\n0α(s)exp(−µ/integraltexts\n0ν(τ)dτ)ds\nwhile the solution to the problem ( 5)-(6) is\n(ρ(x,t),u(x,t)) =\n\n(ρ−,u−e−µ/integraltextt\n0ν(r)dr,ifx < x(t),\n(w(t)δ(x−x(t)),uδ(t)),ifx=x(t),\n(ρ+,u+e−µ/integraltextt\n0ν(r)dr,ifx > x(t),\nwherew(t) =√ρ+ρ−(u−−u+)/integraltextt\n0α(s)e−µ/integraltexts\n0ν(r)drds,uδ(t) =√ρ−u−+√ρ+u+√ρ−+√ρ+e−µ/integraltextt\n0ν(r)dr,\nandx(t) =√ρ−u−+√ρ+u+√ρ−+√ρ+/integraltextt\n0α(s)e−µ/integraltexts\n0ν(r)drdsifρ−/ne}ationslash=ρ+andw(t) =ρ−(u−−u+)/integraltextt\n0α(s)e−µ/integraltexts\n0ν(r)drds,\nuδ(t) =1\n2(u−+u+)e−µ/integraltextt\n0ν(r)dr, andx(t) =1\n2(u−−u+)/integraltextt\n0α(s)e−µ/integraltexts\n0ν(r)drdsifρ−=ρ+.\nOne observes that the solutions given above are explicit with respec t to the parameter µ >0, and also we\nhave\nlim\nµ→0+exp(−µ/integraldisplayt\n0ν(s)ds) = 1 and lim\nµ→0+/integraldisplayt\n0α(s)exp(−µ/integraldisplays\n0ν(r)dr)ds=/integraldisplayt\n0α(s)ds.\nTherefore, the Riemann solution to the problems ( 4) and (5) withσ(t) = 0 for all t≥0 and initial data ( 6) is\ngiven by\n(ρ(x,t),u(x,t)) =\n\n(ρ−,u−),ifx < u−/integraltextt\n0α(s)ds,\n(0,x/integraltextt\n0α(s)ds),ifu−/integraltextt\n0α(s)ds < x < u +/integraltextt\n0α(s)ds,\n(ρ+,u+),ifx > u+/integraltextt\n0α(s)ds.\n19ifu−< u+. Ifu−> u+, then the Riemann solution to the problem ( 4) withσ(t) = 0 for all t≥0 and initial\ndata (6) is\n(ρ(x,t),u(x,t)) =\n\n(ρ−,u−), ifx < x(t),\n(w(t)δ(x−x(t)),u−+u+\n2),ifx=x(t),\n(ρ+,u+), ifx > x(t),\nwherew(t) =1\n2(ρ−+ρ+)(u−−u+)/integraltextt\n0α(s)dsandx(t) =u−+u+\n2/integraltextt\n0α(s)dsand the Riemann solution to the\nproblem ( 5)-(6) withσ(t) = 0 for all t≥0 is given by\n(ρ(x,t),u(x,t)) =\n\n(ρ−,u−), ifx < x(t),\n(w(t)δ(x−x(t)),uδ(t)),ifx=x(t),\n(ρ+,u+), ifx > x(t),\nwherew(t) =√ρ+ρ−(u−−u+)/integraltextt\n0α(s)ds,uδ(t) =√ρ−u−+√ρ+u+√ρ−+√ρ+,andx(t) =√ρ−u−+√ρ+u+√ρ−+√ρ+/integraltextt\n0α(s)dsifρ−/ne}ationslash=ρ+\nandw(t) =ρ−(u−−u+)/integraltextt\n0α(s)ds,uδ(t) =1\n2(u−+u+), andx(t) =1\n2(u−−u+)/integraltextt\n0α(s)dsifρ−=ρ+.\n5 Comments and Extensions\nThe main goal of this section is to present comments and extensions of ongoing work on the topic developed in\nthis paper.\nWe studied in this paper, the Riemann problems to the time-variable co efficient Zeldovich approximate\nsystem (4) and time-variablecoefficientpressurelessgassystem ( 5) both with generaltime-gradually-degenerate\ndamping. Similar to the results obtained by Keita and Bourgault in [ 17] to the Riemann problems ( 4)-(6) and\n(5)-(6) both with α(·)≡1 andσ(·)≡σ=const. > 0, we have that the systems ( 4) and (5), where αandσ\nare non-negative functions that dependents of time t, are equivalent for smooth and two-contact-discontinuity\nsolutions but they differ for delta shock solutions. Moreover, we sh ow that the uniqueness is obtained under an\nover-compressive entropy condition.\nItisinterestingtoremarkthat, whywehavetofixthesignof α(·) solvingthe Riemannproblem. Indeed, they\nonly need to have one sign (positive or negative) to maintain the Lax e ntropy (in shocks) and over-compressive\nentropy condition in delta shocks (as we need the characteristics n ot to be inverted). Clearly, the sign of σ(·)\njustifies the physical meaning of damping.\nNow, we would like to mention another direction of the work developed here, see [ 7]. Also related to system\n(3), we consider the following nonautonomous quasilinear systems:\n\n\nρt+α1(t)(ρu)x= 0,\nut+α2(t)(u2\n2)x=−σ(t)u,\nand also /braceleftBigg\nρt+α1(t)(ρu)x= 0,\n(ρu)t+α2(t)(ρu2)x=−σ(t)ρu,\nwhereαi∈L1([0,∞)), (i= 1,2), and 0 ≤σ∈L1\nloc([0,∞)). It is not absolutely clear that, all the strategies\nappliedinthispaperworkwiththesesystems,infact, thisisnotthec ase. Indeed, when α1/ne}ationslash=α2theconstruction\nof shocks, rarefactions, contact discontinuities, and delta shoc k solutions is not easy due to the behavior of the\nunder- or over-compressibility of the eigenvalues and left or right s tates. This stands as the focal point of our\nongoing research efforts.\nData availability statement\nData sharing does not apply to this article as no data sets were gene rated or analyzed during the current study.\n20Conflict of Interest\nThe authorRichardDe lacruzacknowledgesthe supportreceivedf rom UniversidadPedag´ ogicay Tecnol´ ogicade\nColombia. TheauthorWladimirNeveshasreceivedresearchgrantsf romCNPqthroughthegrants313005/2023-\n0, 406460/2023-0, and also by FAPERJ (Cientista do Nosso Estado ) through the grant E-26/201.139/2021.\nReferences\n[1] F. Bouchut, On zero pressure gas dynamics. In: Advances in Kinetic Theory and Computing . Series on\nAdvances in Mathematics for Applied Sciences, vol. 22, pp. 171–190 . World Scientific, Singapore (1994).\n[2] Y. Brenier, Solutions with concentration to the Riemann problem f or one-dimensional Chaplygin gas dy-\nnamics, J. Math. Fluid Mech. 7, S326–S331 (2005).\n[3] Y. Brenier and E. Grenier, Sticky particles and scalar conservat ion laws, SIAM Journal on Numerical\nAnalysis, 35(6), 2317-2328 (1998).\n[4] R. De la cruz, Riemann problem for a 2 ×2 hyperbolic system with linear damping, Acta Appl. Math. 170,\n631-647 (2020).\n[5] R. De la cruz and J.C. Juajibioy, Delta shock solution for generalize d zero-pressure gas dynamics system\nwith linear damping, Acta Appl. 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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": "2206.03218v2.Decay_property_of_solutions_to_the_wave_equation_with_space_dependent_damping__absorbing_nonlinearity__and_polynomially_decaying_data.pdf",        "content": "arXiv:2206.03218v2  [math.AP]  11 Aug 2022DECAY PROPERTY OF SOLUTIONS TO THE WAVE\nEQUATION WITH SPACE-DEPENDENT DAMPING,\nABSORBING NONLINEARITY, AND POLYNOMIALLY\nDECAYING DATA\nYUTA WAKASUGI\nAbstract. We study the large time behavior of solutions to the semiline ar\nwave equation with space-dependent damping and absorbing n onlinearity in\nthe whole space or exterior domains. Our result shows how the amplitude of\nthe damping coefficient, the power of the nonlinearity, and th e decay rate of\nthe initial data at the spatial infinity determine the decay r ates of the energy\nand the L2-norm of the solution. In Appendix, we also give a survey of ba sic\nresults on the local and global existence of solutions and th e properties of\nweight functions used in the energy method.\n1.Introduction\nWe study the initial-boundary value problem of the wave equation with space-\ndependent damping and absorbing nonlinearity\n\n\n∂2\ntu−∆u+a(x)∂tu+|u|p−1u= 0, t >0,x∈Ω,\nu(t,x) = 0, t > 0,x∈∂Ω,\nu(0,x) =u0(x), ∂tu(0,x) =u1(x), x∈Ω.(1.1)\nHere, Ω = Rnwithn≥1, or Ω⊂Rnwithn≥2 is an exterior domain, that\nis,Rn\\Ω is compact. We also assume that the boundary ∂Ω of Ω is of class\nC2. When Ω = Rn, the boundary condition is omitted and we consider the initial\nvalue problem. The unknown function u=u(t,x) is assumed to be real-valued.\nThe function a(x) denotes the coefficient of the damping term. Throughout this\npaper, we assume that a∈C(Rn) is nonnegative and bounded. The semilinear\nterm|u|p−1u, wherep >1, is the so-called absorbing nonlinearity, which assists the\ndecay of the solution.\nThe aim of this paper is to obtain the decay estimates of the energy\nE[u](t) :=1\n2/integraldisplay\nΩ(|∂tu(t,x)|2+|∇u(t,x)|2)dx+1\np+1/integraldisplay\nΩ|u(t,x)|p+1dx(1.2)\nand the weighted L2-norm\n/integraldisplay\nΩa(x)|u(t,x)|2dx\nof the solution.\nDate: August 12, 2022.\n2020Mathematics Subject Classification. 35L71, 35L20, 35B40.\nKey words and phrases. wave equation, space-dependent damping, absorbing nonlin earity.\n12 Y. WAKASUGI\nFirst, for the energy E[u](t), we observe from the equation (1.1) that\nd\ndtE[u](t) =−/integraldisplay\nΩa(x)|∂tu(t,x)|2dx,\nwhich gives the energy identity\nE[u](t)+/integraldisplayt\n0/integraldisplay\nΩa(x)|∂tu(s,x)|2dxds=E[u](0).\nSincea(x) is nonnegative, the energy is monotone decreasing in time. Theref ore, a\nnaturalquestionarisesastowhethertheenergytendstozeroa stimegoestoinfinity\nand, if that is true, what the actual decay rate is. Moreover, we c an expect that\nthe amplitude of the damping coefficient a(x), the power pof the nonlinearity, and\nthe spatial decay of the initial data ( u0,u1) will play crucial roles for this problem.\nOur goal is to clarify how these three factors determine the decay property of the\nsolution.\nBefore going to the main result, we shall review previous studies on t he asymp-\ntotic behavior of solutions to linear and nonlinear damped wave equat ions.\nThe study of the asymptotic behavior of solutions to the damped wa ve equation\ngoes back to the pioneering work by Matsumura [52]. He studied the initial value\nproblem of the linear wave equation with the classical damping\n/braceleftbigg∂2\ntu−∆u+∂tu= 0, t > 0,x∈Rn,\nu(0,x) =u0(x), ∂tu(0,x) =u1(x), x∈Rn.(1.3)\nIn this case the energy of the solution uis defined by\nEL(t) :=1\n2/integraldisplay\nRn(|∂tu(t,x)|2+|∇u(t,x)|2)dx. (1.4)\nBy using the Fourier transform, he proved the so-called Matsumur a estimates\n/ba∇dbl∂k\nt∂γ\nxu(t)/ba∇dblL∞≤C(1+t)−n\n2m−k−|γ|\n2/parenleftbig\n/ba∇dblu0/ba∇dblLm+/ba∇dblu1/ba∇dblLm+/ba∇dblu0/ba∇dblH[n\n2]+k+|γ|+1+/ba∇dblu1/ba∇dblH[n\n2]+k+|γ|/parenrightbig\n,\n/ba∇dbl∂k\nt∂γ\nxu(t)/ba∇dblL2≤C(1+t)−n\n2(1\nm−1\n2)−k−|γ|\n2(/ba∇dblu0/ba∇dblLm+/ba∇dblu1/ba∇dblLm+/ba∇dblu0/ba∇dblHk+|γ|+/ba∇dblu1/ba∇dblHk+|γ|−1)\n(1.5)\nfor 1≤m≤2,k∈Z≥0, andγ∈Zn\n≥0, and applied them to semilinear problems.\nIn particular, the above estimate implies\n(1+t)EL(t)+/ba∇dblu(t)/ba∇dbl2\nL2\n≤C(1+t)−n(1\nm−1\n2)(/ba∇dblu0/ba∇dblLm+/ba∇dblu1/ba∇dblLm+/ba∇dblu0/ba∇dblH1+/ba∇dblu1/ba∇dblL2)2.(1.6)\nThis indicates that the spatial decay of the initial data improves the time decay of\nthe solution.\nMoreover, the decay rate in the estimates (1.5) suggeststhat th e solution of (1.3)\nis approximated by a solution of the corresponding heat equation\n∂tv−∆v= 0, t >0,x∈Rn.\nThis is the so-called diffusion phenomenon and firstly proved by Hsiao a nd Liu [18]\nfor the hyperbolic conservation law with damping.\nThere are many improvements and generalizations of the Matsumur a estimates\nand the diffusion phenomenon for (1.3). We refer the reader to [7, 1 7, 20, 21, 28,\n33, 41, 44, 51, 55, 59, 61, 76, 78, 86, 99] and the references th erein.SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 3\nNext, we consider the initial boundary value problem of the linear wav e equation\nwith space-dependent damping\n\n\n∂2\ntu−∆u+a(x)∂tu= 0, t > 0,x∈Ω,\nu(t,x) = 0, t > 0,x∈∂Ω,\nu(0,x) =u0(x), ∂tu(0,x) =u1(x), x∈Ω.(1.7)\nMochizuki [56] firstly studied the case Ω = Rn(n/\\e}atio\\slash= 2) and showed that if a(x)≤\nC/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−αwithα >1, then the wave operator exists and is not identically vanishing.\nNamely, the energy EL(t) defined by (1.4) of the solution does not decay to zero\nin general, and the solution behaves like a solution of the wave equatio n without\ndamping. This means that if the damping is sufficiently small at the spat ial infinity,\nthen the energy ofthe solution does not decay to zero in general. H is result actually\nincludesthetimeandspacedependentdamping, andgeneralizations inthedamping\ncoefficientsand domainscan be found in Mochizuki and Nakazawa[57], Matsuyama\n[54], and Ueda [90].\nOn the other hand, for (1.7) with Ω = Rn, from the result by Matsumura [53],\nwe see that if u0,u1∈C∞\n0(Rn) anda(x)≥C/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−1, thenEL(t) decays to zero as\nt→ ∞(seealsoUesaka[91]). Theseresultsindicatethatforthedampingc oefficient\na(x) =/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−α, the value α= 1 is critical for the energy decay or non-decay.\nRegardingthe precise decayrate ofthe solution to (1.7), Todorov aand Yordanov\n[89] proved that if Ω = Rn,a(x) is positive, radial and satisfies a(x) =a0|x|−α+\no(|x|−α) (|x| → ∞) with some α∈[0,1), and the initial data has compact support,\nthen the solution satisfies\n(1+t)EL(t)+/integraldisplay\nRna(x)|u(t,x)|2dx≤C(1+t)−n−α\n2−α+δ(/ba∇dblu0/ba∇dblH1+/ba∇dblu1/ba∇dblL2)2,\nwhereδ >0 is arbitrary constant and Cdepends on δand the support of the data.\nWe note that if we formally take α= 0 and δ= 0, then the decay rate coincides\nwith that of (1.6). The proof of [89] is based on the weighted energy method with\nthe weight function\nt−n−α\n2−α+2δexp/parenleftbigg\n−/parenleftbiggn−α\n2−α−δ/parenrightbiggA(x)\nt/parenrightbigg\n,\nwhereA(x) is a solution of the Poisson equation ∆ A(x) =a(x). Such weight\nfunctions were firstly introduced by Ikehata and Tanizawa [36] and Ikehata [32]\nfor damped wave equations. Some generalizations of the principal p art to variable\ncoefficients were made by Radu, Todorova, and Yordanov [71, 72]. The assumption\nof the radial symmetry of a(x) was relaxed by Sobajima and the author [81]. More-\nover, in [83, 84], the compactness assumption on the support of th e initial data was\nremoved and polynomially decaying data were treated. The point is th e use of a\nsuitable supersolution of the corresponding heat equation\na(x)∂tv−∆v= 0\nhaving polynomial order in the far field. This approach is also a main too l in this\npaper. For the diffusion phenomenon, we refer the reader to [40, 6 8, 73, 74, 80, 82,\n92].\nWhen the damping coefficient is critical for the energy decay, the sit uation be-\ncomes more delicate. Ikehata, Todorova, and Yordanov [38] stud ied (1.7) in the\ncase where Ω = Rn(n≥3),a(x) satisfies a0/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−1≤a(x)≤a1/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−1with some4 Y. WAKASUGI\na0,a1>0, and the initial data has compact support. They obtained the dec ay\nestimates\nEL(t) =/braceleftBigg\nO(t−a0) (1< a0< n),\nO(t−n+δ) (a0≥n)\nast→ ∞with arbitrarysmall δ >0. This indicates that the decay rate depends on\nthe constant a0. Similar results in the lower dimensional cases and the optimality\nof the above estimates under additional assumptions were also obt ained in [38].\nWe also mention that a(x) is not necessarily positive everywhere. It is known\nthat the so-called geometric control condition (GCC) introduced b y Rauch and\nTaylor [75] and Bardos, Lebeau, and Rauch [2] is sufficient for the en ergy decay of\nsolutions with initial data in the energy space. For the problem (1.7) w ith Ω =Rn,\n(GCC) is read as follows: There exist constants T >0 andc >0 such that for any\n(x0,ξ0)∈Rn×Sn−1, we have\n1\nT/integraldisplayT\n0a(x0+sξ0)ds≥c.\nFor this and related topics, we refer the reader to [1, 5, 9, 29, 45 , 58, 67, 68, 101].\nWe note that for a(x) =/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−αwithα >0, (GCC) is not fulfilled.\nWe note that for the linear wave equation with time-dependent damp ing\n∂2\ntu−∆u+b(t)∂tu= 0,\nthe asymptotic behavior of the solution can be classified depending o n the behavior\nofb(t). See [93, 94, 95, 96, 97, 98].\nThirdly, we consider the semilinear problem\n\n\n∂2\ntu−∆u+∂tu=f(u), t > 0,x∈Ω,\nu(t,x) = 0, t > 0,x∈∂Ω,\nu(0,x) =u0(x), ∂tu(0,x) =u1(x), x∈Ω.(1.8)\nWhenf(u) =|u|p−1uor±|u|pwithp >1, the nonlinearity works as a sourcing\nterm and it may cause the singularity of the solution in a finite time. In t his case, it\nis known that there exists the critical exponent pF(n) = 1+2\nn, that is, if p > pF(n),\nthen (1.8) admits the global solution for small initial data; if p < pF(n), then the\nsolution may blow up in finite time even for the small initial data. The num ber\npF(n) is the so-called Fujita critical exponent named after the pioneerin g work by\nFujita [10] for the semilinear heat equation.\nWhen Ω = Rnandf(u) =±|u|p, Todorova and Yordanov [87] determined the\ncritical exponent for compactly supported initial data. Later on, Zhang [100] and\nKirane and Qafsaoui [46] proved that the critical case p=pF(n) belongs to the\nblow-up case.\nThere are many improvements and related studies to the results ab ove. The\ncompactness assumption of the support of the initial data were re moved by [13,\n20, 21, 36, 60]. The diffusion phenomenon for the global solution was proved by\n[11, 13, 42, 43]. The case where Ω is the half space or the exterior do main was\nstudied by [24, 26, 30, 31, 69, 70, 77] Also, estimates of lifespan fo r blowing-up\nsolutions were obtained by [48, 49, 62, 27, 22, 24, 23].\nWhenf(u) =|u|p−1u, the global existence part can be proved completely the\nsame way as in the case f(u) =±|u|p. However, regardingthe blow-up of solutions,\nthe same proof as before works only for n≤3, since the fundamental solution ofSEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 5\nthe linear damped wave equation is not positive for n≥4, which follows from the\nexplicit formula of the linear wave equation (see e.g., [76, p.1011]). Ike hata and\nOhta [35] obtained the blow-up of solutions for the subcritical case p < pF(n). The\ncritical case p=pF(n) withn≥4 seems to remain open.\nWhenf(u) =−|u|p−1uwithp >1, the nonlinearity works as an absorbing\nterm. In this case with Ω = Rn, Kawashima, Nakao, and Ono [44] proved the large\ndata global existence. Moreover, decay estimates of solutions we re obtained for\np >1 +4\nn. Later on, Nishihara and Zhao [65] and Ikahata, Nishihara, and Zha o\n[34] studied the case 1 < p≤1+4\nn. From their results, we have the energy estimate\n(1+t)E[u](t)+/ba∇dblu(t)/ba∇dbl2\nL2≤C(I0)(1+t)−2(1\np−1−n\n4), (1.9)\nwhere\nI0:=/integraldisplay\nRn/parenleftbig\n|u1(x)|2+|∇u0(x)|2+|u0(x)|p+1+|u0(x)|2/parenrightbig\n/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht2mdx, m > 2/parenleftbigg1\np−1−n\n4/parenrightbigg\nand we recall that E[u](t) is defined by (1.2). Also, the asymptotic behavior was\ndiscussed by [41, 12, 15, 16, 34, 63]. There seems no result for ext erior domain\ncases.\nFinally, weconsiderthesemilinearproblemwithspace-dependentdam pingwhich\nis slightly more general than (1.1):\n\n\n∂2\ntu−∆u+a(x)∂tu=f(u), t > 0,x∈Ω,\nu(t,x) = 0, t > 0,x∈∂Ω,\nu(0,x) =u0(x), ∂tu(0,x) =u1(x), x∈Ω.\nWhen the nonlinearity works as a sourcing term, we expect that the re is the critical\nexponent as in the case a(x)≡1. Indeed, in the case where Ω = Rn,f(u) =±|u|p,\nthe initial data has compact support, and a(x) is positive, radial, and satisfies\na(x) =a0|x|−α+o(|x|−α) (|x| → ∞) withα∈[0,1), Ikehata, Todorova, and\nYordanov [37] determined the critical exponent as pF(n−α) = 1 +2\nn−α. The\nestimate of lifespan for blowing-up solutions was obtained in [24, 27]. T he blow-up\nof solutions for the case f(u) =|u|p−1useems to be an open problem.\nRecently, Sobajima [79] studied the critical damping case a(x) =a0|x|−1in\nan exterior domain Ω with n≥3, and proved the small data global existence of\nsolutions under the conditions a0> n−2 andp >1+4\nn−2+min{n,a0}. The blow-up\npart was investigated by [25, 50, 79]. In particular, when Ω is the ou tside a ball\nwithn≥3,a0≥n, andf(u) =±|u|p, the critical exponent is determined as\np=pF(n−1). Moreover, in Ikeda and Sobajima [25], the blow-up of solutions wa s\nobtained for Ω = Rn(n≥3), 0≤a0<(n−1)2\nn+1,f(u) =±|u|pwithn\nn−1< p≤\npS(n+a0), where pS(n) is the positive root of the quadratic equation\n2+(n+1)p−(n−1)p2= 0\nand is the so-called Strauss exponent. We remark that pS(n+a0)> pF(n−1)\nholds ifa0<(n−1)2\nn+1. From this, we can expect that the critical exponent changes\ndepending on the value a0.\nFor the absorbing nonlinear term f(u) =−|u|p−1uin the whole space case\nΩ =Rnwas studied by Todorova and Yordanov [88] and Nishihara [64]. In [64],\nfor compactly supported initial data, the following two results were proved:6 Y. WAKASUGI\n(i) Ifa(x) =a0/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−αwith some a0>0 andα∈[0,1), then we have\n(1+t)E[u](t)+/integraldisplay\nRna(x)|u(t,x)|2dx≤C(1+t)−n−α\n2−α+δ\nwith arbitrary small δ >0;\n(ii) Ifa0/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−α≤a(x)≤a1/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−αwith some a0,a1>0 andα∈[0,1), then we\nhave\n(1+t)E[u](t)+/integraldisplay\nRna(x)|u(t,x)|2dx≤C\n\n(1+t)−4\n2−α(1\np−1−n−α\n4)(p > psubc(n,α)),\n(1+t)−2\np−1log(2+t) (p=psubc(n,α)),\n(1+t)−2\np−1 (p < psubc(n,α)),\nwhere\npsubc(n,α) := 1+2α\nn−α. (1.10)\nWe note that the decay rate in (i) is the same as that of the linear pro blem (1.7)\nand it is better than that of (ii) if p > pF(n−α). This means pF(n−α) is critical in\nthe sense of the effect of the nonlinearity to the decay rate of the energy. Moreover,\n(ii) shows that the second critical exponent psubc(n,α) appears and it divides the\ndecay rate of the energy. We also note that the estimate for the c asep > psubc(n,α)\ncorresponds to the estimate (1.9). Thus, we may interpret the sit uation in the\nfollowing way: When the damping is weak in the sense of a(x)∼ /a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−αwith\nα∈(0,1), we cannot obtain the same type energy estimate as in (1.9) for a ll\np >1, and the decay rate becomes worse under or on the second critic al exponent\npsubc(n,α). Our main goal in this paper is to give a generalization of the results ( i)\nand (ii) above.\nIn recent years, semilinear wave equations with time-dependent da mping have\nbeen intensively studied. For the progress of this problem, we refe r the reader to\nSections 1 and 2 in Lai, Schiavone, and Takamura [47]. We also refer to [66] and\nthe references therein for a recent study of semilinear wave equa tions with time and\nspace dependent damping.\nTo state our results, we define the solution.\nDefinition 1.1 (Mild and strong solutions) .LetAbe the operator\nA=/parenleftbigg0 1\n∆−a(x)/parenrightbigg\ndefined on H:=H1\n0(Ω)×L2(Ω)with the domain D(A) = (H2(Ω)∩H1\n0(Ω))×H1\n0(Ω).\nLetU(t)denote the C0-semigroup generated by A. Let(u0,u1)∈ HandT∈(0,∞].\nA function\nu∈C([0,T);H1\n0(Ω))∩C1([0,T);L2(Ω))\nis called a mild solution of (1.1)on[0,T)ifU=t(u,∂tu)satisfies the integral\nequation\nU(t) =U(t)/parenleftbiggu0\nu1/parenrightbigg\n+/integraldisplayt\n0U(t−s)/parenleftbigg0\n−|u|p−1u/parenrightbigg\ndsSEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 7\ninC([0,T);H). Moreover, when (u0,u1)∈D(A), a function\nu∈C([0,T);H2(Ω))∩C1([0,T);H1\n0(Ω))∩C2([0,T);L2(Ω))\nis said to be a strong solution of (1.1)on[0,T)ifusatisfies the equation of (1.1)\ninC([0,T);L2(Ω)). IfT=∞, we call ua global (mild or strong) solution.\nFirst, we prepare the existence and regularity of the global solutio n.\nProposition 1.2. LetΩ =Rnwithn≥1, orΩ⊂Rnwithn≥2be an exterior\ndomain with C2-boundary. Let a(x)∈C(Rn)be nonnegative and bounded. Let\n1< p <∞(n= 1,2),1< p≤n\nn−2(n≥3), (1.11)\nand let(u0,u1)∈H1\n0(Ω)×L2(Ω). Then, there exists a unique global mild solution u\nto(1.1). If we further assume (u0,u1)∈(H2(Ω)∩H1\n0(Ω))×H1\n0(Ω), thenubecomes\na strong solution to (1.1).\nRemark 1.3. The assumption ∂Ω∈C2is used to ensure D(A) = (H2(Ω)∩\nH1\n0(Ω))×H1\n0(Ω)(see Cazenave and Haraux [6, Remark 2.6.3] and Brezis [4, Theo-\nrem9.25] ). The restriction of the range of pin(1.11)is due to the use of Gagliardo–\nNirenberg inequality (see Section A.2).\nThe proof of Proposition 1.2 is standard. However, for reader’s co nvenience, we\nwill give an outline of the proof in the appendix.\nTo state our result, we recall that E[u](t) andpsubc(n,α) are defined by (1.2)\nand (1.10), respectively. The main result of this paper reads as follo ws.\nTheorem 1.4. LetΩ =Rnwithn≥1orΩ⊂Rnwithn≥2be an exterior\ndomain with C2-boundary. Let psatisfy(1.11)and(u0,u1)∈H1\n0(Ω)×L2(Ω), and\nletube the corresponding global mild solution of (1.1). Then, the followings hold.\n(i)Assume that a∈C(Rn)is positive and satisfies\nlim\n|x|→∞|x|αa(x) =a0 (1.12)\nwith some constants α∈[0,1)anda0>0. Moreover, we assume that the\ninitial data satisfy\nI0[u0,u1]\n:=/integraldisplay\nΩ/bracketleftbig\n(|u1(x)|2+|∇u0(x)|2+|u0(x)|p+1)/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+|u0(x)|2/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−α/bracketrightbig\n/a\\}b∇acketle{tx/a\\}b∇acket∇i}htλ(2−α)dx\n<∞ (1.13)\nwith some λ∈[0,n−α\n2−α). Then, we have\n(1+t)E[u](t)+/integraldisplay\nΩa(x)|u(t,x)|2dx≤CI0[u0,u1](1+t)−λ\nfort≥0with some constant C=C(n,a,p,λ)>0.\n(ii)Assume that a∈C(Rn)is positive and satisfies\na0/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−α≤a(x)≤a1/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−α8 Y. WAKASUGI\nwith some constants α∈[0,1),a0,a1>0. Moreover, we assume that the\ninitial data satisfy the condition I0[u0,u1]<∞with some λ∈[0,∞), where\nI0[u0,u1]is defined by (1.13). Then, we have\n(1+t)E[u](t)+/integraldisplay\nΩa(x)|u(t,x)|2dx\n≤C(I0[u0,u1]+1)\n×\n\n(1+t)−λ(λ <min{4\n2−α(1\np−1−n−α\n4),2\np−1}),\n(1+t)−λlog(2+t) (λ= min{4\n2−α(1\np−1−n−α\n4),2\np−1}, p/\\e}atio\\slash=psubc(n,α)),\n(1+t)−λ(log(2+ t))2(λ=4\n2−α(1\np−1−n−α\n4) =2\np−1,i.e., p=psubc(n,α)),\n(1+t)−4\n2−α(1\np−1−n−α\n4)(λ >4\n2−α(1\np−1−n−α\n4), p > p subc(n,α)),\n(1+t)−2\np−1log(2+t) (λ >2\np−1, p=psubc(n,α)),\n(1+t)−2\np−1 (λ >2\np−1, p < p subc(n,α))\nfort≥0with some constant C=C(n,a,p,λ)>0.\nRemark 1.5. Under the assumptions of (i), the both conclusions of (i) and (ii)\nare true. In Figure 1, the decay rates of/integraldisplay\nΩa(x)|u(t,x)|2dxis classified in the case\n(n,α) = (3,0.5)(for ease of viewing, the figure is multiplied by 7and0.75in the\nhorizontal and vertical axis, respectively).\nRemark 1.6. From the proof of the above theorem, we also have the followin g\nestimates for the L2-norm of uwithout the weight a(x): Under the assumptions on\n(i) withλ∈[α\n2−α,n−α\n2−α), we have\n/integraldisplay\nΩ|u(t,x)|2dx≤C(1+t)−λ+α\n2−α\nfort >0; Under the assumptions on (ii) with λ∈[α\n2−α,∞), we have\n/integraldisplay\nΩ|u(t,x)|2dx\n≤C\n\n(1+t)−λ+α\n2−α (λ <min{4\n2−α(1\np−1−n−α\n4),2\np−1}),\n(1+t)−λ+α\n2−αlog(2+t) (λ= min{4\n2−α(1\np−1−n−α\n4),2\np−1}, p/\\e}atio\\slash=psubc(n,α)),\n(1+t)−λ+α\n2−α(log(2+ t))2(λ=4\n2−α(1\np−1−n−α\n4) =2\np−1,i.e., p=psubc(n,α)),\n(1+t)−4\n2−α(1\np−1−n−α\n4)+α\n2−α(λ >4\n2−α(1\np−1−n−α\n4), p > p subc(n,α)),\n(1+t)−2\np−1+α\n2−αlog(2+t) (λ >2\np−1, p=psubc(n,α)),\n(1+t)−2\np−1+α\n2−α (λ >2\np−1, p < p subc(n,α))\nfort >0.\nRemark 1.7. (i) Theorem 1.4 generalizes the result of Nishihara [64]to the exte-\nrior domain, general damping coefficient a(x)satisfying (1.12), and polynomially\ndecaying initial data satisfying (1.13).\n(ii) For the simplest case Ω =Rnanda(x)≡1, the result of Theorem 1.4 (ii)\nextends that of Ikehata, Nishihara, and Zhao [34], in the sense that our estimate\nin the region λ >2/parenleftBig\n1\np−1−n\n4/parenrightBig\ncoincides with their estimate (1.9). Moreover, theSEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 9\nn−α\n2−αn−α\nα\n1 psubc(n,α) pF(n−α)pλ\n(1+t)−λ(1+t)−4\n2−α(1\np−1−n−α\n4)(1+t)−2\np−1 (1+t)−λlog(2+t)\n(1+t)−λlog(2+t)(1+t)−2\np−1log(2+t)\n(1+t)−λ(log(2+ t))2\n(1+t)−n−α\n2−α+δλ=4\n2−α(1\np−1−n−α\n4)\nλ=2\np−1\nFigure 1. Classification of decay rates in p-λplane when ( n,α) = (3,1\n2)\nresult of Theorem 1.4 (i) in the case p > pF(n)is better than the estimate obtained\nin[34]. Hence, our result still has a novelty.\nRemark 1.8. The optimality of the decay rates in Theorem 1.4 is an open pro blem.\nWe expect that the estimate in the case (i) is optimal if p > pF(n−α) = 1+2\nn−α,\nsince the decay rate is the same as that of the linear problem (1.7)obtained by [84].\nOn the other hand, in the critical case p=pF(n−α), the estimates in Theorem 1.4\nwill be improved in view of the known results [15, 16]for the classical damping (1.8)\nin the whole space. Moreover, the optimality in the subcriti cal casep < pF(n−α)\nis a difficult problem even when a(x)≡1andΩ =Rn, and we have no idea so far.\nThe strategy of the proof of Theorem 1.4 is as follows. For the both parts (i)\nand (ii), we apply the weighted energy method. The difficulty is how to e stimate\nthe weighted L2-norm of the solution. To overcome it, we take different approache s\nfor (i) and (ii). First, for the part (i), we apply the weighted energy method\ndeveloped by [83, 84]. We shall use a suitable supersolution of the cor responding\nheat equation a(x)∂tv−∆v= 0 as the weight function. Next, for the part (ii), we\nshall use the same type weight function as in Ikehata, Nishihara, an d Zhao [34] with\na modification to fit the space-dependent damping case. In this cas e the absorbing\nsemilinear term helps to estimate the weighted L2-norm of the solution.10 Y. WAKASUGI\nThe rest of the paper is organized in the following way. In the next se ction, we\nprepare the definitions and properties of the weight functions use d in the proof.\nSections 3 and 4 are devoted to the proof of Theorem 1.4 (i) and (ii), respectively.\nIn Appendix A, we give a proof of Proposition 1.2. Finally, in Appendix B, we\nprove the properties of weight functions stated in Section 2.\nWe end up this section with introducing notations used throughout t his paper.\nThe letter Cindicates a generic positive constant, which may change from line to\nline. In particular, C(∗,···,∗) denotes a constant depending only on the quantities\nin the parentheses. For x= (x1,...,x n)∈Rn, we define /a\\}b∇acketle{tx/a\\}b∇acket∇i}ht=/radicalbig\n1+|x|2. We\nsometimes use BR(x0) ={x∈Rn;|x−x0|< R}forR >0 andx0∈Rn.\nLetLp(Ω) be the usual Lebesgue space equipped with the norm\n/ba∇dblf/ba∇dblLp=\n\n/parenleftbigg/integraldisplay\nΩ|f(x)|pdx/parenrightbigg1/p\n(1< p <∞),\nesssup\nx∈Ω|f(x)| (p=∞).\nIn particular, L2(Ω) is a Hilbert space with the innerproduct\n(f,g)L2:=/integraldisplay\nΩf(x)g(x)dx.\nLetHk(Ω) with a nonnegative integer kbe the Sobolev space equipped with the\ninnerproduct and the norm\n(f,g)Hk=/summationdisplay\n|α|≤k(∂αf,∂αg)L2,/ba∇dblf/ba∇dblHk=/radicalbig\n(f,f)Hk,\nrespectively. C∞\n0(Ω) denotes the space of smooth functions on Ω with compact\nsupport. Hk\n0(Ω) is the completion of C∞\n0(Ω) with respect to the norm /ba∇dbl·/ba∇dblHk. For\nan interval I⊂R, a Banach space X, and a nonnegative integer k,Ck(I;X) stands\nfor the space of k-times continuously differentiable functions from ItoX.\n2.Preliminaries\nIn this section, we prepare weight functions for the weighted ener gy method used\nin the proof of Theorem 1.4.\nThese lemmas were shown in [77, 81, 83, 84], however, for the conve nience, we\ngive a proof of them in the appendix.\nFollowing [81], we first take a suitable approximate solution of the Poiss on equa-\ntion ∆A(x) =a(x), which will be used for the construction of the weight function.\nLemma 2.1 ([81, 84]) .Assume that a(x)∈C(Rn)is positive and satisfies the\ncondition lim|x|→∞|x|αa(x) =a0with some constants α∈(−∞,min{2,n})and\na0>0. Letε∈(0,1). Then, there exist a function Aε∈C2(Rn)and positive\nconstants c=c(n,a,ε)andC=C(n,a,ε)such that for x∈Rn, we have\n(1−ε)a(x)≤∆Aε(x)≤(1+ε)a(x), (2.1)\nc/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht2−α≤Aε(x)≤C/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht2−α, (2.2)\n|∇Aε(x)|2\na(x)Aε(x)≤2−α\nn−α+ε. (2.3)\nForthe constructionofourweightfunction, wealsoneed the follow ingKummer’s\nconfluent hypergeometric function.SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 11\nDefinition 2.2 (Kummer’s confluent hypergeometric functions) .Forb,c∈Rwith\n−c /∈N∪{0}, Kummer’s confluent hypergeometric function of first kind is defined\nby\nM(b,c;s) =∞/summationdisplay\nn=0(b)n\n(c)nsn\nn!, s∈[0,∞),\nwhere(d)nis the Pochhammer symbol defined by (d)0= 1and(d)n=/producttextn\nk=1(d+\nk−1)forn∈N; note that when b=c,M(b,b;s)coincides with es.\nForε∈(0,1/2), we define\n/tildewideγε=/parenleftbigg2−α\nn−α+2ε/parenrightbigg−1\n, γε= (1−2ε)/tildewideγε. (2.4)\nDefinition 2.3. Forβ∈R, define\nϕβ,ε(s) =e−sM(γε−β,γε;s), s≥0.\nSinceM(γε,γε,s) =es, we remark that ϕ0,ε(s)≡1. Roughly speaking, if\nwe formally take ε= 0, then {ϕβ,0}β∈Rgives a family of self-similar profiles of\nthe equation |x|−α∂tv= ∆vwith the parameter β. See [83] for more detailed\nexplanation. The next lemma states basic properties of ϕβ,ε.\nLemma 2.4. The function ϕβ,εdefined in Definition 2.3 satisfies the following\nproperties.\n(i)ϕβ,ε(s)satisfies the equation\nsϕ′′(s)+(γε+s)ϕ′(s)+βϕ(s) = 0. (2.5)\n(ii)If0≤β < γε, thenϕβ,ε(s)satisfies the estimates\nkβ,ε(1+s)−β≤ϕβ,ε(s)≤Kβ,ε(1+s)−β\nwith some constants kβ,ε,Kβ,ε>0.\n(iii)For every β≥0,ϕβ,ε(s)satisfies\n|ϕβ,ε(s)| ≤Kβ,ε(1+s)−β\nwith some constant Kβ,ε>0.\n(iv)For every β∈R,ϕβ,ε(s)andϕβ+1,ε(s)satisfy the recurrence relation\nβϕβ,ε(s)+sϕ′\nβ,ε(s) =βϕβ+1,ε(s).\n(v)For every β∈R, we have\nϕ′\nβ,ε(s) =−β\nγεe−sM(γε−β,γε+1;s),\nϕ′′\nβ,ε(s) =β(β+1)\nγε(γε+1)e−sM(γε−β,γε+2;s).\nIn particular, if 0< β < γ ε, thenϕ′\nβ,ε(s)andϕ′′\nβ,ε(s)satisfy\n−Kβ,ε(1+s)−β−1≤ϕ′\nβ,ε(s)≤ −kβ,ε(1+s)−β−1,\nkβ,ε(1+s)−β−2≤ϕ′′\nβ,ε(s)≤Kβ,ε(1+s)−β−2\nwith some constants kβ,ε,Kβ,ε>0.\nFinally, we define the weight function which will be used for our energy method.12 Y. WAKASUGI\nDefinition 2.5. Forβ∈Rand(x,t)∈Rn×[0,∞), we define\nΦβ,ε(x,t;t0) = (t0+t)−βϕβ,ε(z), z=/tildewideγεAε(x)\nt0+t,\nwhereε∈(0,1/2),/tildewideγεis the constant given in (2.4),t0≥1,ϕβ,εis the function\ndefined by Definition 2.3, and Aε(x)is the function constructed in Lemma 2.1.\nSinceϕ0,ε(s)≡1, we again remark that Φ 0,ε(x,t;t0)≡1.\nFort0≥1,t >0, andx∈Rn, we also define\nΨ(x,t;t0) :=t0+t+Aε(x). (2.6)\nProposition 2.6. The function Φβ,ε(x,t;t0)satisfies the following properties:\n(i)For every β≥0, we have\n∂tΦβ,ε(x,t;t0) =−βΦβ+1,ε(x,t;t0).\n(ii)Ifβ≥0, then there exists a constant C=C(n,α,β,ε)>0such that\n|Φβ,ε(x,t;t0)| ≤CΨ(x,t;t0)−β\nfor any(x,t)∈Rn×[0,∞).\n(iii)If0≤β < γε, then there exists a constant c=c(n,α,β,ε)>0such that\nΦβ,ε(x,t;t0)≥cΨ(x,t;t0)−β\nfor any(x,t)∈Rn×[0,∞).\n(iv)Forβ >0, there exists a constant c=c(n,α,β,ε)>0such that\na(x)∂tΦβ,ε(x,t;t0)−∆Φβ,ε(x,t;t0)≥ca(x)Ψ(x,t;t0)−β−1\nfor any(x,t)∈Rn×[0,∞).\nFinally, we prepare a useful lemma for our weighted energy method. The proof\ncan be found in [83, Lemma 3.6] or [77, Lemma 2.5]. However, for the c onvenience,\nwe give its proof in the appendix.\nLemma 2.7. LetΩ =Rnwithn≥1orΩ⊂Rnwithn≥2be an exterior domain\nwithC2-boundary. Let Φ∈C2(Ω)be a positive function and let δ∈(0,1/2). Then,\nfor anyu∈H2(Ω)∩H1\n0(Ω)satisfying suppu∈BR(0) ={x∈Rn;|x|< R}with\nsomeR >0, we have\n/integraldisplay\nΩ(u∆u)Φ−1+2δdx≤ −δ\n1−δ/integraldisplay\nΩ|∇u|2Φ−1+2δdx+1−2δ\n2/integraldisplay\nΩu2(∆Φ)Φ−2+2δdx.\n3.Proof of Theorem 1.4: first part\nIn this section, we prove Theorem 1.4 (i). First, we note that Propo sition 1.2\nimplies the existence of the global mild solution u.\nFollowing the argument in Sobajima [79], we first prove Theorem 1.4 (i) in the\ncase of compactly supported initial data, and after that, we will tr eat the general\ncase by an approximation argument.SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 13\n3.1.Proof for the compactly supported initial data. We first consider the\ncase where the initial data are compactly supported, that is, we as sume that\nsuppu0∪suppu1⊂BR0(0) ={x∈Rn;|x|< R0}. Then, by the finite prop-\nagation property (see Section A.2.7), the corresponding mild solutio nusatisfies\nsuppu(t,·)⊂BR0+t(0).\nLetT0>0 be arbitrary fixed and let T∈(0,T0). Then, we have supp u(t,·)⊂\nBR0+T0(0) for all t∈[0,T]. LetD= Ω∩BR0+T0(0). Then, for t∈[0,T], we can\nconvert the problem (1.1) to the problem in the bounded domain\n\n\n∂2\ntu−∆u+a(x)∂tu+|u|p−1u= 0, t∈(0,T],x∈D,\nu(t,x) = 0, t ∈(0,T],x∈∂D,\nu(0,x) =u0(x), ∂tu(0,x) =u1(x), x∈D\nwith (u0,u1)∈ HD:=H1\n0(D)×L2(D).\nLetADbe the operator\nAD=/parenleftbigg0 1\n∆−a(x)/parenrightbigg\ndefined on HDwith the domain D(AD) = (H2(D)∩H1\n0(D))×H1\n0(D). Then, from\nthe argument in Section A.1, there exists λ∗>0 such that for any λ > λ ∗, the\nresolvent Jλ= (I−λ−1AD)−1is defined as a bounded operator on HD. Take a\nsequence {λj}∞\nj=1such that λj> λ∗forj≥1 and lim j→∞λj=∞, and define\n/parenleftBigg\nu(j)\n0\nu(j)\n1/parenrightBigg\n:=Jλj/parenleftbiggu0\nu1/parenrightbigg\n.\nThen, we have\n(u(j)\n0,u(j)\n1)∈D(AD),lim\nj→∞(u(j)\n0,u(j)\n1) = (u0,u1) inHD (3.1)\n(see e.g. the proof of [19, Theorem 2.18]). Therefore, Proposition 1.2 shows that\nthe mild solution u(j)corresponding to the initial data ( u(j)\n0,u(j)\n1) becomes a strong\nsolution. Moreover, the continuous dependence on the initial data (see Section\nA.2.4) implies\nlim\nj→∞sup\nt∈[0,T]/ba∇dbl(u(j)(t),∂tu(j)(t))−(u(t),∂tu(t))/ba∇dblHD= 0.\nThis means that, if we prove the conclusion of Theorem 1.4 (i) for u(j), that is,\n(1+t)E[u(j)](t)+/integraldisplay\nΩa(x)|u(j)(t,x)|2dx≤CI0[u(j)\n0,u(j)\n1](1+t)−λ\nfort∈[0,T], where the constant Cis independent of j,T,T0,R0, then letting\nj→ ∞and also using the Sobolev embedding /ba∇dblu/ba∇dblLp+1(D)≤C/ba∇dblu/ba∇dblH1(D), we\nhave the same estimate for the original mild solution u. Note that (3.1) implies\nlimj→∞I0[u(j)\n0,u(j)\n1] =I0[u0,u1], sincethe integralis takenoverthe bounded region\nD. Finally, since TandT0are arbitrary and Cis independent of them, we obtain\nthe desired energy estimate for any t≥0.\nTherefore, in the following argument, we may further assume ( u0,u1)∈D(AD)\nanduis the strong solution. This enables us to justify all the computation s in this\nsection.14 Y. WAKASUGI\nIn what follows, we shall use the weight functions Φ β,ε(x,t;t0) and Ψ( x,t;t0)\ndefined by Definition 2.5 and (2.6), respectively. We also recall that t he constant\nγεis given by (2.4). Then, we define the following energies.\nDefinition 3.1. For a function u=u(t,x),α∈[0,1),δ∈(0,1/2),ε∈(0,1/2),\nλ∈[0,(1−2δ)γε),β=λ/(1−2δ),ν >0, andt0≥1, we define\nE1(t;t0,λ) =/integraldisplay\nΩ/bracketleftbigg1\n2/parenleftbig\n|∂tu(t,x)|2+|∇u(t,x)|2/parenrightbig\n+1\np+1|u(t,x)|p+1/bracketrightbigg\nΨ(t,x;t0)λ+α\n2−αdx,\nE0(t;t0,λ) =/integraldisplay\nΩ/parenleftbig\n2u(t,x)∂tu(t,x)+a(x)|u(t,x)|2/parenrightbig\nΦβ,ε(t,x;t0)−1+2δdx,\nE∗(t;t0,λ,ν) =E1(t;t0,λ)+νE0(t;t0,λ),\n˜E(t;t0,λ) = (t0+t)/integraldisplay\nΩ/bracketleftbigg1\n2/parenleftbig\n|∂tu(t,x)|2+|∇u(t,x)|2/parenrightbig\n+1\np+1|u(t,x)|p+1/bracketrightbigg\nΨ(t,x;t0)λdx\nfort≥0.\nSince\n2u∂tu≤a(x)\n2|u|2+2\na(x)|∂tu|2≤a(x)\n2|u|2+CΨα\n2−α|∂tu|2(3.2)\nand Φ−1+2δ\nβ,ε≤CΨλ(see (2.2) and Proposition 2.6 (iii)), we see that there exists a\nsmall constant ν0=ν0(n,a,δ,ε,λ )>0 such that for any ν∈(0,ν0),\nE∗(t;t0,λ,ν)≥1\n2E1(t;t0,λ)+ν\n2/integraldisplay\nΩa(x)|u(t,x)|2Ψ(t,x;t0)λdx(3.3)\nholds.\nWe first prepare the following energy estimates for E1(t;t0,λ) andE0(t;t0,λ).\nLemma3.2. Under the assumptions on Theorem 1.4 (i), there exists t1=t1(n,a,λ,ε)≥\n1such that for t0≥t1andt >0, we have\nd\ndtE1(t;t0,λ)≤ −1\n2/integraldisplay\nΩa(x)|∂tu(t,x)|2Ψ(t,x;t0)λ+α\n2−αdx\n+C/integraldisplay\nΩ/parenleftbig\n|∇u(t,x)|2+|u(t,x)|p+1/parenrightbig\nΨ(t,x;t0)λ+α\n2−α−1dx\nwith some constant C=C(n,α,p,λ)>0.\nProof.Differentiating E1(t;t0,λ), one has\nd\ndtE1(t;t0,λ) =/integraldisplay\nΩ/bracketleftbig\n∂tu∂2\ntu+∇u·∇∂tu+|u|p−1u∂tu/bracketrightbig\nΨλ+α\n2−αdx\n+/parenleftbigg\nλ+α\n2−α/parenrightbigg/integraldisplay\nΩ/bracketleftbigg1\n2/parenleftbig\n|∇u|2+|∂tu|2/parenrightbig\n+1\np+1|u|p+1/bracketrightbigg\nΨλ+α\n2−α−1dx.SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 15\nThe integration by parts and the equation (1.1) imply\nd\ndtE1(t;t0,λ) =−/integraldisplay\nΩa(x)|∂tu|2Ψλ+α\n2−αdx\n−/parenleftbigg\nλ+α\n2−α/parenrightbigg/integraldisplay\nΩ∂tu(∇u·∇Ψ)Ψλ+α\n2−α−1dx\n+/parenleftbigg\nλ+α\n2−α/parenrightbigg/integraldisplay\nΩ/bracketleftbigg1\n2/parenleftbig\n|∇u|2+|∂tu|2/parenrightbig\n+1\np+1|u|p+1/bracketrightbigg\nΨλ+α\n2−α−1dx.\n(3.4)\nLet us estimate the right-hand side. First, the Schwarz inequality g ives\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle−/parenleftbigg\nλ+α\n2−α/parenrightbigg\n∂tu(∇u·∇Ψ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤a(x)\n4|∂tu|2Ψ+C|∇u|2|∇Ψ|2\na(x)Ψ.\nMoreover, by (2.3), we have\n|∇Ψ|2\na(x)Ψ≤|∇Aε(x)|2\na(x)Aε(x)≤2−α\nn−α+ε. (3.5)\nAlso, from the definition of Ψ, (2.2), and a(x)∼ /a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−α, one obtains\nΨ(t,x;t0)−1≤t−1+α\n2−α\n0Aε(x)−α\n2−α≤Ct−2(1−α)\n2−α\n0a(x). (3.6)\nTherefore, taking t1≥1 sufficiently large, we have, for t0≥t1,\n/parenleftbigg\nλ+α\n2−α/parenrightbigg/integraldisplay\nΩ|∂tu|2Ψλ+α\n2−α−1dx≤1\n4/integraldisplay\nΩa(x)|∂tu|2Ψλ+α\n2−αdx.\nUsing the above estimates to (3.4), we deduce\nd\ndtE1(t;t0,λ)≤ −1\n2/integraldisplay\nΩa(x)|∂tu|2Ψλ+α\n2−αdx\n+C/integraldisplay\nΩ/parenleftbig\n|∇u|2+|u|p+1/parenrightbig\nΨλ+α\n2−α−1dx,\nwhich completes the proof. /square\nLemma 3.3. Under the assumptions on Theorem 1.4 (i), for t0≥1andt >0, we\nhave\nd\ndtE0(t;t0,λ)≤ −η/integraldisplay\nΩ/parenleftbig\n|∇u(t,x)|2+|u(t,x)|p+1/parenrightbig\nΨ(t,x;t0)λdx\n+C/integraldisplay\nΩ|∂tu(t,x)|2Ψ(t,x;t0)λdx\nwith some positive constants η=η(n,α,δ,ε,λ )andC=C(n,α,δ,ε,λ ).\nProof.Differentiating E0(t;t0,λ) and using the equation (1.1) yield\nd\ndtE0(t;t0,λ) =/integraldisplay\nΩ/parenleftbig\n2|∂tu|2+2u∂2\ntu+2a(x)u∂tu/parenrightbig\nΦ−1+2δ\nβ,εdx\n−(1−2δ)/integraldisplay\nΩ/parenleftbig\n2u∂tu+a(x)|u|2/parenrightbig\nΦ−2+2δ\nβ,ε∂tΦβ,εdx.16 Y. WAKASUGI\nUsing the equation (1.1), we have\nd\ndtE0(t;t0,λ) = 2/integraldisplay\nΩ|∂tu|2Φ−1+2δ\nβ,εdx+2/integraldisplay\nΩu∆uΦ−1+2δ\nβ,εdx\n−2/integraldisplay\nΩ|u|p+1Φ−1+2δ\nβ,εdx\n−(1−2δ)/integraldisplay\nΩ/parenleftbig\n2u∂tu+a(x)|u|2/parenrightbig\nΦ−2+2δ\nβ,ε∂tΦβ,εdx.\nApplying Lemma 2.7 with Φ = Φ β,εto the second term of the right-hand side, one\nobtains\nd\ndtE0(t;t0,λ)≤2/integraldisplay\nΩ|∂tu|2Φ−1+2δ\nβ,εdx−2δ\n1−δ/integraldisplay\nΩ|∇u|2Φ−1+2δ\nβ,εdx\n−2/integraldisplay\nΩ|u|p+1Φ−1+2δ\nβ,εdx\n−2(1−2δ)/integraldisplay\nΩu∂tuΦ−2+2δ\nβ,ε∂tΦβ,εdx\n−(1−2δ)/integraldisplay\nΩ|u|2Φ−2+2δ\nβ,ε(a(x)∂tΦβ,ε−∆Φβ,ε)dx.(3.7)\nNext, we estimate the terms in the right-hand side. First, we remar k that if λ= 0\n(i.e.,β= 0), then the last two terms in (3.7) vanish, since Φ β,ε≡1. For the case\nβ >0, by Proposition 2.6 (ii) and (iv), we have\n/integraldisplay\nΩ|u|2Φ−2+2δ\nβ,ε(a(x)∂tΦβ,ε−∆Φβ,ε)dx≥η1/integraldisplay\nΩa(x)|u|2Ψλ−1dx\nwith some constant η1=η1(n,α,δ,ε,λ )>0. Moreover, Proposition 2.6 (i), (ii),\nand (iii) imply\n|u∂tuΦ−2+2δ\nβ,ε∂tΦβ,ε| ≤C|u||∂tu||Φ−2+2δ\nβ,ε||Φβ+1,ε| ≤C|u||∂tu|Ψλ−1.\nThis and the Schwarz inequality lead to\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle2(1−2δ)/integraldisplay\nΩu∂tuΦ−2+2δ\nβ,ε∂tΦβ,εdx/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤C/integraldisplay\nΩ|u||∂tu|Ψλ−1dx\n≤C/parenleftbigg/integraldisplay\nΩa(x)|u|2Ψλ−1dx/parenrightbigg1/2/parenleftbigg/integraldisplay\nΩa(x)−1|∂tu|2Ψλ−1dx/parenrightbigg1/2\n≤η1\n2/integraldisplay\nΩa(x)|u|2Ψλ−1dx+C/integraldisplay\nΩ|∂tu|2Ψλdx\nwith some C=C(n,a,δ,ε,λ )>0. Summarizing the above computations, we see\nthat for both cases λ= 0 and λ >0, the last two terms of (3.7) can be estimatedSEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 17\nas\n−2(1−2δ)/integraldisplay\nΩu∂tuΦ−2+2δ\nβ,ε∂tΦβ,εdx\n−(1−2δ)/integraldisplay\nΩ|u|2Φ−2+2δ\nβ,ε(a(x)∂tΦβ,ε−∆Φβ,ε)dx\n≤C/integraldisplay\nΩ|∂tw|2Ψλdx.\nFinally, from Proposition 2.6 (ii) and (iii), one obtains\n2/integraldisplay\nΩ|∂tu|2Φ−1+2δ\nβ,εdx≤C/integraldisplay\nΩ|∂tu|2Ψλdx\nand\n2δ\n1−δ/integraldisplay\nΩ|∇u|2Φ−1+2δ\nβ,εdx+2/integraldisplay\nΩ|u|p+1Φ−1+2δ\nβ,εdx≥η/integraldisplay\nΩ/parenleftbig\n|∇u|2+|u|p+1/parenrightbig\nΨλdx\nwith some positive constants C=C(n,α,δ,ε,λ ) andη=η(n,α,δ,ε,λ ). Putting\nthis all together, we deduce from (3.7) that\nd\ndtE0(t;t0,λ)≤ −η/integraldisplay\nΩ/parenleftbig\n|∇u|2+|u|p+1/parenrightbig\nΨλdx\n+C/integraldisplay\nΩ|∂tu|2Ψλdx,\nand the proof is complete. /square\nCombining Lemmas3.2 and3.3, we havethe followingestimate for E∗(t;t0,λ,ν).\nLemma 3.4. Under the assumptions on Theorem 1.4 (i), there exist consta nts\nν∗=ν∗(n,a,δ,ε,λ )∈(0,ν0)andt2=t2(n,a,p,δ,ε,λ,ν ∗)≥1such that for t0≥t2\nandt >0, we have\nE∗(t;t0,λ,ν∗)+/integraldisplayt\n0/integraldisplay\nΩa(x)|∂tu(s,x)|2Ψ(s,x;t0)λ+α\n2−αdxds\n+/integraldisplayt\n0/integraldisplay\nΩ(|∇u(s,x)|2+|u(s,x)|p+1)Ψ(s,x;t0)λdxds\n≤CE∗(0;t0,λ,ν∗)\nwith some constant C=C(n,a,δ,ε,λ,ν ∗)>0.\nProof.Letν∈(0,ν0), where ν0is taken so that (3.2) holds. From the definition of\nE∗(t;t0,λ,ν) and Lemmas 3.2 and 3.3, one has\nd\ndtE∗(t;t0,λ,ν) =d\ndtE1(t;t0,λ)+νd\ndtE0(t;t0,λ)\n≤ −1\n2/integraldisplay\nΩa(x)|∂tu|2Ψλ+α\n2−αdx\n+C/integraldisplay\nΩ/parenleftbig\n|∇u|2+|u|p+1/parenrightbig\nΨλ+α\n2−α−1dx\n−νη/integraldisplay\nΩ/parenleftbig\n|∇u|2+|u|p+1/parenrightbig\nΨλdx\n+Cν/integraldisplay\nΩ|∂tu|2Ψλdx (3.8)18 Y. WAKASUGI\nfort0≥t1andt >0, where t1≥1 is determined in Lemma 3.2. Noting that (1.12)\nand (2.2) imply\n|∂tu|2Ψλ≤C/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−αAε(x)α\n2−α|∂tu|2Ψλ≤Ca(x)|∂tu|2Ψλ+α\n2−α\nwith some constant C=C(n,a,α,ε)>0, and taking ν=ν∗with sufficiently small\nν∗∈(0,ν0), we deduce\n−1\n2/integraldisplay\nΩa(x)|∂tu|2Ψλ+α\n2−αdx+Cν∗/integraldisplay\nΩ|∂tu|2Ψλdx≤ −1\n4/integraldisplay\nΩa(x)|∂tu|2Ψλ+α\n2−αdx.\nNext, by Ψα\n2−α−1≤(t0+t)α\n2−α−1and taking t2≥t1sufficiently large depending\nonν∗, one obtains\nC/integraldisplay\nΩ/parenleftbig\n|∇u|2+|u|p+1/parenrightbig\nΨλ+α\n2−α−1dx−ν∗η/integraldisplay\nΩ/parenleftbig\n|∇u|2+|u|p+1/parenrightbig\nΨλdx\n≤ −ν∗η\n2/integraldisplay\nΩ/parenleftbig\n|∇u|2+|u|p+1/parenrightbig\nΨλdx\nfort0≥t2. Finally, plugging the above estimates into (3.8) with ν=ν∗, we\nconclude\nd\ndtE∗(t;t0,λ,ν∗)≤ −1\n4/integraldisplay\nΩa(x)|∂tu|2Ψλ+α\n2−αdx\n−ν∗η\n2/integraldisplay\nΩ/parenleftbig\n|∇u|2+|u|p+1/parenrightbig\nΨλdx\nfort0≥t2andt >0. Integrating it over [0 ,t], we have the desired estimate. /square\nLemma 3.5. Under the assumptions on Theorem 1.4 (i), there exists a cons tant\nt2=t2(n,a,p,δ,ε,λ )≥1such that for t0≥t2andt >0, we have\n˜E(t;t0,λ)+/integraldisplay\nΩa(x)|u(t,x)|2Ψ(t,x;t0)λdx≤CI0[u0,u1]\nwith some constant C=C(n,a,p,δ,ε,λ,ν ∗,t0)>0.\nProof.Take the same constants ν∗andt2as in Lemma 3.4. The integration by\nparts and the equation (1.1) imply\nd\ndt˜E(t;t0,λ) =/integraldisplay\nΩ/bracketleftbigg1\n2/parenleftbig\n|∂tu|2+|∇u|2/parenrightbig\n+1\np+1|u|p+1/bracketrightbigg\n(Ψ+λ(t0+t))Ψλ−1dx\n+(t0+t)/integraldisplay\nΩ/parenleftbig\n∂tu∂2\ntu+∇u·∇∂tu+|u|p−1u∂tu/parenrightbig\nΨλdx\n=/integraldisplay\nΩ/bracketleftbigg1\n2/parenleftbig\n|∂tu|2+|∇u|2/parenrightbig\n+1\np+1|u|p+1/bracketrightbigg\n(Ψ+λ(t0+t))Ψλ−1dx\n−(t0+t)/integraldisplay\nΩa(x)|∂tu|2Ψλdx−λ(t0+t)/integraldisplay\nΩ∂tu(∇u·∇Ψ)Ψλ−1dx.\nThe last term of the right-hand side is estimated as\n−λ(t0+t)/integraldisplay\nΩ∂tu(∇u·∇Ψ)Ψλ−1dx≤η(t0+t)/integraldisplay\nΩa(x)|∂tu|2|∇Ψ|2\na(x)Ψλ−1dx\n+C(t0+t)/integraldisplay\nΩ|∇u|2Ψλ−1dxSEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 19\nfor anyη >0. Using (3.5) and taking η=η(n,α,ε) sufficiently small, we have\nd\ndt˜E(t;t0,λ)≤C/integraldisplay\nΩ/parenleftbig\n|∂tu|2+|∇u|2+|u|p+1/parenrightbig\n(Ψ+(t0+t))Ψλ−1dx\n−1\n2(t0+t)/integraldisplay\nΩa(x)|∂tu|2Ψλdx.\nNotingt0+t≤Ψ anda(x)−1≤CΨα\n2−α, we estimate\n/integraldisplay\nΩ|∂tu|2(Ψ+λ(t0+t))Ψλ−1dx≤C/integraldisplay\nΩa(x)|∂tu|2Ψλ+α\n2−αdx.\nTherefore, integrating over [0 ,t] yield\n˜E(t;t0,λ)+1\n2/integraldisplayt\n0(t0+s)/integraldisplay\nΩa(x)|∂tu|2Ψλdxds\n≤˜E(0;t0,λ)+C/integraldisplayt\n0/integraldisplay\nΩa(x)|∂tu|2Ψλ+α\n2−αdxds+C/integraldisplayt\n0/integraldisplay\nΩ/parenleftbig\n|∇u|2+|u|p+1/parenrightbig\nΨλdxds.\nNow, we multiply the both sides of above inequality by a sufficiently small constant\nµ >0, and add it and the conclusion of Lemma 3.4. Then, we obtain\nµ˜E(t;t0,λ)+E∗(t;t0,λ,ν∗)\n+/integraldisplayt\n0/integraldisplay\nΩa(x)|∂tu|2/bracketleftBigµ\n2(t0+s)+(1−Cµ)Ψα\n2−α/bracketrightBig\nΨλdxds\n+(1−Cµ)/integraldisplayt\n0/integraldisplay\nΩ/parenleftbig\n|∇u|2+|u|p+1/parenrightbig\nΨλdxds\n≤µ˜E(0;t0,λ)+CE∗(0;t0,λ,ν∗) (3.9)\nfort0≥t2andt >0. Let us take µsufficiently small so that 1 −Cµ >0. Then, the\nlast three terms in the left-hand side can be dropped. Finally, from t he definitions\nofE∗(t;t0,λ) and˜E(t;t0,λ), we can easily verify\nµ˜E(0;t0,λ)+E∗(0;t0,λ,ν∗)≤CI0[u0,u1]\nwith some constant C=C(a,p,λ,t 0)>0. Thus, we conclude\n˜E(t;t0,λ)+E∗(t;t0,λ,ν∗)≤CI0[u0,u1]\nfort0≥t2andt >0. This and the lower bound (3.3) of E∗(t;t0,λ,ν∗) give the\ndesired estimate. /square\nProof of Theorem 1.4 (i) for compactly supported initial dat a.Takeλ∈[0,n−α\n2−α)\nasin theassumption(1.13), andthen choose δ,ε∈(0,1/2)sothat λ∈[0,(1−2δ)γε)\nholds. Moreover, take the same constants ν∗andt2as in Lemmas 3.4 and 3.5. By\n(3.3), Lemmas 3.4 and 3.5, Definition 3.1, and ( t0+t)λ≤Ψλ, we have\n(t0+t)λ+1E[u](t)+(t0+t)λ/integraldisplay\nΩa(x)|u(t,x)|2dx≤CI0[u0,u1] (3.10)\nfort0≥t2andt >0 with some constant C=C(n,a,p,δ,ε,λ,ν ∗,t0)>0. This\ncompletes the proof. /square20 Y. WAKASUGI\nRemark 3.6. From(3.9), we have a slightly more general estimate\n/integraldisplay\nΩ/parenleftbig\n|∂tu|2+|∇u|2+|u|p+1/parenrightbig/bracketleftbig\n(t0+t)+Ψα\n2−α/bracketrightbig\nΨλ+/integraldisplay\nΩa(x)|u|2Ψλdx\n+/integraldisplayt\n0/integraldisplay\nΩa(x)|∂tu|2/bracketleftbig\n(t0+s)+Ψα\n2−α/bracketrightbig\nΨλdxds\n+/integraldisplayt\n0/integraldisplay\nΩ/parenleftbig\n|∇u|2+|u|p+1/parenrightbig\nΨλdxds\n≤CI0[u0,u1]\nfort0≥t2andt >0. Moreover, from the proof of Lemma 3.3, we can add the term/integraltextt\n0/integraltext\nΩa(x)|u|2Ψλ−1dxdsto the left-hand side when λ >0.\n3.2.Proof for the general case. Here, we give a proof of Theorem 1.4 (i) for\nnon-compactly supported initial data.\nLet (u0,u1)∈H1\n0(Ω)×L2(Ω) satisfy I0[u0,u1]<∞and letube the corre-\nsponding mild solution to (1.1). We take a cut-off function χ∈C∞\n0(Rn) such\nthat\n0≤χ(x)≤1 (x∈Rn), χ(x) =/braceleftBigg\n1 (|x| ≤1),\n0 (|x| ≥2).\nFor each j∈N, we define χj(x) =χ(x/j). Then, we have\n0≤χj(x)≤1 (x∈Rn), χj(x) =/braceleftBigg\n1 (|x| ≤j),\n0 (|x| ≥2j),\n|∇χj(x)| ≤C\nj(x∈Rn),supp∇χj⊂B2j(0)\\Bj(0),\nwhere the constant Cis independent of j.\nLet (u(j)\n0,u(j)\n1) = (χju0,χju1) and let u(j)be the corresponding mild solution to\n(1.1). First, by definition, it is easily seen that\nlim\nj→∞(u(j)\n0,u(j)\n1) = (u0,u1) inH1\n0(Ω)×L2(Ω).\nTherefore, the continuous dependence on the initial data (see Se ction A.2.4) yields\nlim\nj→∞(u(j)(t),∂tu(j)(t)) = (u(t),∂tu(t)) inC([0,T];H1\n0(Ω))∩C1([0,T];L2(Ω))\nfor any fixed T >0. From this and the Sobolev embedding, we deduce\nlim\nj→∞E[u(j)](t) =E[u](t) (3.11)\nfor anyt≥0.\nWe next show\nlim\nj→∞I0[u(j)\n0,u(j)\n1] =I0[u0,u1]. (3.12)\nTo prove this, we use the notation\nI0[u0,u1;D]\n:=/integraldisplay\nD/bracketleftbig\n(|u1(x)|2+|∇u0(x)|2+|u0(x)|p+1)/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+|u0(x)|2/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−α/bracketrightbig\n/a\\}b∇acketle{tx/a\\}b∇acket∇i}htλ(2−α)dxSEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 21\nfor a region D⊂Ω. Using the properties of χjdescribed above and\n|∇(χju0)|2=χ2\nj|∇u0|2+2(∇χj·∇u0)χju0+|∇χj|2|u0|2,\nwe calculate\n|I0[u0,u1]−I0[u(j)\n0,u(j)\n1]| ≤I0[u0,u1;Ω\\Bj(0)]\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nB2j(0)\\Bj(0)2(∇χj·∇u0)χju0/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+λ(2−α)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n+/integraldisplay\nB2j(0)\\Bj(0)|∇χj|2|u0|2/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+λ(2−α)dx.(3.13)\nThe Schwarz inequality gives\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nB2j(0)\\Bj(0)2(∇χj·∇u0)χju0/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+λ(2−α)dx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤I0[u0,u1;Ω\\Bj(0)]+/integraldisplay\nB2j(0)\\Bj(0)|∇χj|2|u0|2/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+λ(2−α)dx.\nFurthermore, using the estimate of ∇χj, one sees that\n/integraldisplay\nB2j(0)\\Bj(0)|∇χj|2|u0|2/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+λ(2−α)dx\n≤Cj−2(1+|2j|2)α/integraldisplay\nB2j(0)\\Bj(0)|u0|2/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−α+λ(2−α)dx\n≤CI0[u0,u1;Ω\\Bj(0)],\nwhere the constant Cis independent of j. Putting this all together into (3.13), we\nhave\n|I0[u0,u1]−I0[u(j)\n0,u(j)\n1]| ≤CI0[u0,u1;Ω\\Bj(0)].\nSinceI0[u0,u1]<∞, the right-hand side tends to zero as j→ ∞. This proves\n(3.12).\nNow we are at the position to proof Theorem 1.4 (i).\nProof of Theorem 1.4 (i) for the general case. Takethesameconstant t2asinLem-\nmas 3.4 and 3.5. Let {(u(j)\n0,u(j)\n1)}∞\nj=1be the sequence defined above and let u(j)\nbe the corresponding mild solution to (1.1) with the initial data ( u(j)\n0,u(j)\n1). Since\neach (u(j)\n0,u(j)\n1) has the compact support, one can apply the result (3.10) in the\nprevious subsection to obtain\n(t0+t)λ+1E[u(j)](t)+(t0+t)λ/integraldisplay\nΩa(x)|u(j)(t,x)|2dx≤CI0[u(j)\n0,u(j)\n1]\nfort0≥t2andt >0. Finally, using (3.11) and (3.12), we have\n(t0+t)λ+1E[u](t)+(t0+t)λ/integraldisplay\nΩa(x)|u(t,x)|2dx≤CI0[u0,u1]\nfort0≥t2andt >0, which completes the proof. /square22 Y. WAKASUGI\n4.Proof of Theorem 1.4: second part\nIn this section, we prove Theorem 1.4 (ii). By the same approximation argument\ndescribed in Section 3, we may assume ( u0,u1)∈D(AD) and consider the strong\nsolution u.\nFirst, we note that, since the larger λis, the stronger the assumption on the\ninitial data is. Thus, without loss of generality, we may assume that λalways\nsatisfies\nλ <min/braceleftbigg2\np−1,4\n2−α/parenleftbigg1\np−1−n−α\n4/parenrightbigg/bracerightbigg\n+ε, (4.1)\nwhereε >0 is a sufficiently small constant specified later. This will be used for th e\nestimate of the remainder term.\nIn contrast to the previous section, in the following, we shall use on ly\nΘ(x,t;t0) :=t0+t+/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht2−α(4.2)\nas a weight function, and we define the following energies.\nDefinition 4.1. For a function u=u(t,x),α∈[0,1),λ∈[0,∞),ν >0, and\nt0≥1, we define\nE1(t;t0,λ) =/integraldisplay\nΩ/bracketleftbigg1\n2/parenleftbig\n|∂tu(t,x)|2+|∇u(t,x)|2/parenrightbig\n+1\np+1|u(t,x)|p+1/bracketrightbigg\nΘ(t,x;t0)λ+α\n2−αdx,\nE0(t;t0,λ) =/integraldisplay\nΩ/parenleftbig\n2u(t,x)∂tu(t,x)+a(x)|u(t,x)|2/parenrightbig\nΘ(t,x;t0)λdx,\nE∗(t;t0,λ,ν) =E1(t;t0,λ)+νE0(t;t0,λ),\n˜E(t;t0,λ) = (t0+t)/integraldisplay\nΩ/bracketleftbigg1\n2/parenleftbig\n|∂tu(t,x)|2+|∇u(t,x)|2/parenrightbig\n+1\np+1|u(t,x)|p+1/bracketrightbigg\nΘ(t,x;t0)λdx\nfort≥0.\nSimilarly to (3.2) and (3.3), we can prove the lower bound\nE∗(t;t0,λ,ν)≥1\n2E1(t;t0,λ)+ν\n2/integraldisplay\nΩa(x)|u(t,x)|2Θ(t,x;t0)λdx,(4.3)\nprovided that ν∈(0,ν0) with some constant ν0>0.\nWe start with the following simple estimates for E1(t;t0,λ) andE0(t;t0,λ).\nLemma4.2. Under the assumptions on Theorem 1.4 (ii), there exists t1=t1(n,α,a 0,λ,ε)≥\n1such that for t0≥t1andt >0, we have\nd\ndtE1(t;t0,λ)≤ −1\n2/integraldisplay\nΩa(x)|∂tu(t,x)|2Θ(t,x;t0)λ+α\n2−αdx\n+C/integraldisplay\nΩ/parenleftbig\n|∇u(t,x)|2+|u(t,x)|p+1/parenrightbig\nΘ(t,x;t0)λ+α\n2−α−1dx\nwith some constant C=C(n,α,a 0,p,λ)>0.\nProof.The proof is almost the same as that of Lemma 3.2. The only difference s\nare the use of\n|∇Θ|2\na(x)Θ= (2−α)2/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−2α|x|2\na(x)(t0+t+/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht2−α)≤(2−α)2\na0(4.4)SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 23\nand\nΘ(t,x;t0)−1≤t−1+α\n2−α\n0/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−α≤1\na0t−1+α\n2−α\n0a(x)\ninstead of (3.5) and (3.6), respectively. Thus, we omit the detail. /square\nLemma 4.3. Under the assumptions on Theorem 1.4 (ii), for t0≥1andt >0,\nwe have\nd\ndtE0(t;t0,λ)≤ −/integraldisplay\nΩ|∇u(t,x)|2Θ(t,x;t0)λdx−2/integraldisplay\nΩ|u(t,x)|p+1Θ(t,x;t0)λdx\n+C/integraldisplay\nΩa(x)|∂tu(t,x)|2Θ(t,x;t0)λ+α\n2−αdx+C/integraldisplay\nΩa(x)|u(t,x)|2Θ(t,x;t0)λ−1dx\nwith some constant C=C(n,α,a 0,λ)>0.\nProof.The equation (1.1) and the integration by parts imply\nd\ndtE0(t;t0,λ) = 2/integraldisplay\nΩ|∂tu|2Θλdx+2/integraldisplay\nΩ/parenleftbig\n∂2\ntu+a(x)∂tu/parenrightbig\nΘλdx\n+λ/integraldisplay\nΩ/parenleftbig\n2u∂tu+a(x)|u|2/parenrightbig\nΘλ−1dx\n= 2/integraldisplay\nΩ|∂tu|2Θλdx+2/integraldisplay\nΩ/parenleftbig\n∆u−|u|p−1u/parenrightbig\nuΘλdx\n+λ/integraldisplay\nΩ/parenleftbig\n2u∂tu+a(x)|u|2/parenrightbig\nΘλ−1dx\n=−2/integraldisplay\nΩ|∇u|2Θλdx−2/integraldisplay\nΩ|u|p+1Θλdx\n+2/integraldisplay\nΩ|∂tu|2Θλdx−2λ/integraldisplay\nΩ(∇u·∇Θ)uΘλ−1dx\n+λ/integraldisplay\nΩ/parenleftbig\n2u∂tu+a(x)|u|2/parenrightbig\nΘλ−1dx. (4.5)\nLet us estimates the right-hand side. Applying the Schwarz inequalit y and (4.4),\nwe obtain\n−2λ/integraldisplay\nΩ(∇u·∇Ψ)uΘλ−1dx≤1\n2/integraldisplay\nΩ|∇u|2Θλdx+C/integraldisplay\nΩ|u|2|∇Θ|2Θλ−2dx\n≤1\n2/integraldisplay\nΩ|∇u|2Θλdx+C/integraldisplay\nΩa(x)|u|2Θλ−1dx.\nMoreover, the Schwarz inequality and Θ−1≤1\na0a(x) imply\nλ/integraldisplay\nΩ2u(t,x)∂tu(t,x)Θλ−1dx≤1\n2/integraldisplay\nΩ|∇u|2Θλdx+C/integraldisplay\nΩ|u|2Θλ−2dx\n≤1\n2/integraldisplay\nΩ|∇u|2Θλdx+C/integraldisplay\nΩa(x)|u|2Θλ−1dx.\nFrom 1≤1\na0a(x)Θα\n2−α, we also obtain\n2/integraldisplay\nΩ|∂tu|2Θλdx≤C/integraldisplay\nΩa(x)|∂tu|2Θλ+α\n2−αdx.24 Y. WAKASUGI\nPutting them all together into (4.5), we conclude\nd\ndtE0(t;t0,λ)≤ −/integraldisplay\nΩ|∇u|2Θλdx−2/integraldisplay\nΩ|u|p+1Θλdx\n+C/integraldisplay\nΩa(x)|∂tu|2Θλ+α\n2−αdx+C/integraldisplay\nΩa(x)|u|2Θλ−1dx.\nThis completes the proof. /square\nCombining Lemmas 4.2 and 4.3, we have the following.\nLemma 4.4. Under the assumptions on Theorem 1.4 (ii), there exist const ants\nν∗=ν∗(n,α,a 0,λ)∈(0,ν0)andt2=t2(n,α,a 0,p,λ,ν ∗)≥1such that for t0≥t2,\nandt >0, we have\nE∗(t;t0,λ,ν∗)+/integraldisplayt\n0/integraldisplay\nΩa(x)|∂tu(s,x)|2Θ(s,x;t0)λ+α\n2−αdxds\n+/integraldisplayt\n0/integraldisplay\nΩ/parenleftbig\n|∇u(s,x)|2+|u(s,x)|p+1/parenrightbig\nΘ(s,x;t0)λdxds\n≤CE∗(0;t0,λ,ν)+C/integraldisplayt\n0/integraldisplay\nΩa(x)|u(s,x)|2Θ(s,x;t0)λ−1dxds\nwith some constant C=C(n,α,a 0,p,λ,ν ∗)>0.\nProof.Letν∈(0,ν0), where ν0is taken so that (4.3) holds. Let t1be the constant\ndetermined by Lemma 4.2. Then, by Lemmas 4.2 and 4.3, we obtain for t0≥t1\nandt >0,\nd\ndtE∗(t;t0,λ,ν) =d\ndtE1(t;t0,λ)+νd\ndtE0(t;t0,λ)\n≤ −1\n2/integraldisplay\nΩa(x)|∂tu|2Θλ+α\n2−αdx\n+C/integraldisplay\nΩ|∇u|2Θλ+α\n2−α−1dx+C/integraldisplay\nΩ|u|p+1Θλ+α\n2−α−1dx\n−ν/integraldisplay\nΩ|∇u|2Θλdx−2ν/integraldisplay\nΩ|u|p+1Θλdx\n+Cν/integraldisplay\nΩa(x)|∂tu|2Θλ+α\n2−αdx+Cν/integraldisplay\nΩa(x)|u|2Θλ−1dx.\nWe take ν=ν∗with sufficiently small ν���∈(0,ν0) such that the constants in front\nof the last two terms satisfy Cν∗<1\n2. Moreover, taking t2>0 sufficiently large\ndepending on ν∗so thatCΘα\n2−α−1< ν∗fort0≥t2, we conclude\nd\ndtE∗(t;t0,λ,ν)≤ −η/integraldisplay\nΩa(x)|∂tu|2Θλ+α\n2−αdx−η/integraldisplay\nΩ|∇u|2Θλdx\n−η/integraldisplay\nΩ|u|p+1Θλdx+Cν/integraldisplay\nΩa(x)|u|2Θλ−1dx\nwith some constant η=η(n,α,a 0,p,λ,ν ∗)>0. Finally, integrating the above\ninequality over [0 ,t] gives the desired estimate. /square\nBesed on Lemma 4.4, we show the following estimate for ˜E(t;t0,λ).SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 25\nLemma 4.5. Under the assumptions on Theorem 1.4 (ii), there exists a con stant\nt2=t2(n,α,a 0,p,λ)≥1such that for t0≥t2andt >0, we have\n˜E(t;t0,λ)+/integraldisplay\nΩa(x)|u(t,x)|2Θ(t,x;t0)λdx\n+/integraldisplayt\n0/integraldisplay\nΩa(x)|∂tu(s,x)|2/bracketleftbig\n(t0+s)+Θ(s,x;t0)α\n2−α/bracketrightbig\nΘ(s,x;t0)λdxds\n+/integraldisplayt\n0/integraldisplay\nΩ/parenleftbig\n|∇u(s,x)|2+|u(s,x)|p+1/parenrightbig\nΘ(s,x;t0)λdxds\n≤CI0[u0,u1]+C/integraldisplayt\n0/integraldisplay\nΩa(x)p+1\np−1Θ(s,x;t0)λ−p+1\np−1dxds\nwith some constant C=C(n,α,a 0,a1,p,λ,t 0)>0.\nProof.Take the same constants ν∗andt2as in Lemma 4.4. By the same compu-\ntation as in Lemma 3.5, we can obtain\n˜E(t;t0,λ)+1\n2/integraldisplayt\n0(t0+s)/integraldisplay\nΩa(x)|∂tu|2Θλdxds\n≤˜E(0;t0,λ)+C/integraldisplayt\n0/integraldisplay\nΩa(x)|∂tu|2Θλ+α\n2−αdxds+C/integraldisplayt\n0/integraldisplay\nΩ/parenleftbig\n|∇u|2+|u|p+1/parenrightbig\nΘλdxds.\nWe multiply the both sides by a sufficiently small constant µ >0, and add it and\nthe conclusion of Lemma 4.4. Then, we obtain\nµ˜E(t;t0,λ)+E∗(t;t0,λ,ν∗)\n+/integraldisplayt\n0/integraldisplay\nΩa(x)|∂tu|2/bracketleftBigµ\n2(t0+s)+(1−Cµ)Θα\n2−α/bracketrightBig\nΘλdxds\n+(1−Cµ)/integraldisplayt\n0/integraldisplay\nΩ/parenleftbig\n|∇u|2+|u|p+1/parenrightbig\nΘλdxds\n≤µ˜E(0;t0,λ)+CE∗(0;t0,λ,ν∗)\nfort0≥t2andt >0. By taking µsufficiently small so that 1 −Cµ >0 holds,\nthe terms including |∂tu|2and|∇u|2in the left-hand side can be dropped. Since\nboth˜E(0;t0,λ) andE∗(0;t0,λ,ν∗) are bounded by CI0[u0,u1] with some constant\nC=C(a1,p,λ,t 0)>0, one obtains\n˜E(t;t0,λ)+/integraldisplay\nΩa(x)|u(t,x)|2Θ(t,x;t0)λdx+/integraldisplayt\n0/integraldisplay\nΩ|u|p+1Θλdxds\n≤CI0[u0,u1]+C/integraldisplayt\n0/integraldisplay\nΩa(x)|u|2Θλ−1dxds (4.6)\nwith some C=C(n,α,a 0,a1,p,λ,t 0)>0. Finally, applying the Young inequality\nto the last term of the right-hand side, we deduce\nC/integraldisplayt\n0/integraldisplay\nΩa(x)|u|2Θλ−1dxds=C/integraldisplayt\n0/integraldisplay\nΩ|u|2Θ2\np+1λ·a(x)Θλ(1−2\np+1)−1dxds\n≤1\n2/integraldisplayt\n0/integraldisplay\nΩ|u|p+1Θλdxds+C/integraldisplayt\n0/integraldisplay\nΩa(x)p+1\np−1Θλ−p+1\np−1dxds.26 Y. WAKASUGI\nThis and (4.6) give the conclusion. /square\nBy virtue of Lemma 4.5, it suffices to estimate the term\nC/integraldisplayt\n0/integraldisplay\nΩa(x)p+1\np−1Θ(s,x;t0)λ−p+1\np−1dxds.\nFor this, we have the following lemma.\nLemma 4.6. Under the assumptions on Theorem 1.4 (ii) and (4.1), we have for\nanyt0>0andt≥0,\n/integraldisplayt\n0/integraldisplay\nΩa(x)p+1\np−1Θ(s,x;t0)λ−p+1\np−1dxds\n≤C\n\n1 ( λ <min{4\n2−α(1\np−1−n−α\n4),2\np−1}),\nlog(t0+t) ( λ= min{4\n2−α(1\np−1−n−α\n4),2\np−1}, p/\\e}atio\\slash=psubc(n,α)),\n(log(t0+t))2(λ=4\n2−α(1\np−1−n−α\n4) =2\np−1,i.e., p=psubc(n,α)),\n(1+t)λ−4\n2−α(1\np−1−n−α\n4)(λ >4\n2−α(1\np−1−n−α\n4), p > p subc(n,α)),\n(1+t)λ−2\np−1log(t0+t) (λ >2\np−1, p=psubc(n,α)),\n(1+t)λ−2\np−1 (λ >2\np−1, p < p subc(n,α))\nwith some constant C=C(n,α,a 1,p,λ)>0.\nProof.Lets∈(0,t). First, we divide Ω into Ω = Ω 1(s)∪Ω2(s), where\nΩ1(s) =/braceleftbig\nx∈Ω;/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht2−α≤t0+s/bracerightbig\n,\nΩ2(s) = Ω\\Ω1(s) =/braceleftbig\nx∈Ω;/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht2−α> t0+s/bracerightbig\n.\nThe corresponding integral is also decomposed into\n/integraldisplay\nΩa(x)p+1\np−1Θ(s,x;t0)λ−p+1\np−1dx=/integraldisplay\nΩ1(s)a(x)p+1\np−1Θ(s,x;t0)λ−p+1\np−1dx\n+/integraldisplay\nΩ2(s)a(x)p+1\np−1Θ(s,x;t0)λ−p+1\np−1dx\n=:I(s)+II(s).\nNote that, in Ω 1(s), the function Θ( s,x;t0) =t0+s+/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht2−αis bounded from both\nabove and below by t0+s. Therefore, we estimate\nI(s)≤C(t0+s)λ−p+1\np−1/integraldisplay\nΩ1(s)a(x)p+1\np−1dx\n≤C(t0+s)λ−p+1\np−1/integraldisplay\nΩ1(s)/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−αp+1\np−1dx\n≤C(t0+s)λ−p+1\np−1h(s), (4.7)\nwhere\nh(s) =\n\n1 ( p < psubc(n,α)),\nlog(t0+s) ( p=psubc(n,α)),\n(t0+s)1\n2−α(n−αp+1\np−1)(p > psubc(n,α)).(4.8)SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 27\nOn the other hand, in Ω 2(s), the function Θ is bounded from both above and below\nby/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht2−α. Thus, we have\nII(s)≤C/integraldisplay\nΩ2(s)/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−αp+1\np−1+(2−α)(λ−p+1\np−1)dx.\nHere, we remarkthat the condition(4.1) ensuresthe finiteness of the aboveintegral,\nprovided that εis taken sufficiently small depending on nandα. A straightforward\ncomputation shows\nII(s)≤C(t0+s)λ−p+1\np−1+1\n2−α(n−αp+1\np−1).\nSince the above estimate is better than (4.7) if p≤psubc(n,α) and is the same if\np > psubc(n,α), we conclude\n/integraldisplay\nΩa(x)p+1\np−1Θ(s,x;t0)λ−p+1\np−1dx≤C(t0+s)λ−p+1\np−1h(s).\nNext, we compute the integral of the function ( t0+s)λ−p+1\np−1h(s) over [0,t]. From\nthe definition (4.8) of h(s), one has the following: If p < psubc(n,α), then\n/integraldisplayt\n0(t0+s)λ−p+1\np−1h(s)ds≤C\n\n1/parenleftbigg\nλ <2\np−1/parenrightbigg\n,\nlog(t0+t)/parenleftbigg\nλ=2\np−1/parenrightbigg\n,\n(t0+t)λ−2\np−1/parenleftbigg\nλ >2\np−1/parenrightbigg\n;\nIfp=psubc(n,α), then\n/integraldisplayt\n0(t0+s)λ−p+1\np−1h(s)ds≤C\n\n1/parenleftbigg\nλ <2\np−1/parenrightbigg\n,\n(log(t0+t))2/parenleftbigg\nλ=2\np−1/parenrightbigg\n,\n(t0+t)λ−2\np−1log(t0+t)/parenleftbigg\nλ >2\np−1/parenrightbigg\n;\nIfp > psubc(n,α), then\n/integraldisplayt\n0(t0+s)λ−p+1\np−1h(s)ds≤C\n\n1/parenleftbigg\nλ <4\n2−α/parenleftbigg1\np−1−n−α\n4/parenrightbigg/parenrightbigg\n,\nlog(t0+t)/parenleftbigg\nλ=4\n2−α/parenleftbigg1\np−1−n−α\n4/parenrightbigg/parenrightbigg\n,\n(t0+t)λ−4\n2−α(1\np−1−n−α\n4)/parenleftbigg\nλ >4\n2−α/parenleftbigg1\np−1−n−α\n4/parenrightbigg/parenrightbigg\n.\nThis completes the proof. /square\nWe are now at the position to prove Theorem 1.4 (ii):28 Y. WAKASUGI\nProof of Theorem 1.4 (ii). By Lemmas 4.5 and 4.6 with the constant t2≥1 deter-\nmined in Lemma 4.5, we have\n˜E(t;t0,λ)+/integraldisplay\nΩa(x)|u(t,x)|2Θ(t,x;t0)λdx\n≤CI0[u0,u1]+C\n\n1 ( λ <min{4\n2−α(1\np−1−n−α\n4),2\np−1}),\nlog(t0+t) ( λ= min{4\n2−α(1\np−1−n−α\n4),2\np−1}, p/\\e}atio\\slash=psubc(n,α)),\n(log(t0+t))2(λ=4\n2−α(1\np−1−n−α\n4) =2\np−1,i.e., p=psubc(n,α)),\n(1+t)λ−4\n2−α(1\np−1−n−α\n4)(λ >4\n2−α(1\np−1−n−α\n4), p > p subc(n,α)),\n(1+t)λ−2\np−1log(t0+t) (λ >2\np−1, p=psubc(n,α)),\n(1+t)λ−2\np−1 (λ >2\np−1, p < p subc(n,α))\nfort0≥t2andt≥0. On the other hand, the definition (4.2) of Θ immediately\ngives the lower bound\n˜E(t;t0,λ)+/integraldisplay\nΩa(x)|u(t,x)|2Θ(t,x;t0)λdx\n≥(t0+t)λ+1E[u](t)+(t0+t)λ/integraldisplay\nΩa(x)|u(t,x)|2dx,\nwhereE(t) is defined by (1.2). Combining them, we have the desired estimate. /square\nAppendix A.Outline of the proof of Proposition 1.2\nIn this section, we give a proof of Proposition 1.2. The solvability and b asic\nproperties of the solution of the linear problem (A.1) below can be fou nd in, for\nexample, [8, 19, 25, 68]. Here, we give an outline of the argument alon g with\n[19]. The existence of the unique mild solution of the semilinear problem ( 1.1) is\nproved by the contraction mapping principle. This argument can be f ound in, e.g.,\n[6, 25, 36, 85]. Here, we will give a proof based on [6].\nA.1.Linear problem. Letn∈N, and let Ω be an open set in Rnwith a compact\nC2-boundary ∂Ω or Ω = Rn. We discuss the linear problem\n\n\n∂2\ntu−∆u+a(x)∂tu= 0, t > 0,x∈Ω,\nu(x,t) = 0, t > 0,x∈∂Ω,\nu(0,x) =u0(x), ∂tu(0,x) =u1(x), x∈Ω.(A.1)\nThe function a(x) is nonnegative, bounded, and continuous in Rn. LetH:=\nH1\n0(Ω)×L2(Ω) be the real Hilbert space equipped with the inner product\n/parenleftbigg/parenleftbigg\nu\nv/parenrightbigg\n,/parenleftbigg\nw\nz/parenrightbigg/parenrightbigg\nH= (u,w)H1+(v,z)L2.\nLetAbe the operator\nA=/parenleftbigg0 1\n∆−a(x)/parenrightbigg\ndefined on Hwith the domain D(A) = (H2(Ω)∩H1\n0(Ω))×H1\n0(Ω), which is dense\ninH.SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 29\nWe first show the estimate/parenleftbigg\nA/parenleftbigg\nu\nv/parenrightbigg\n,/parenleftbigg\nu\nv/parenrightbigg/parenrightbigg\nH≤ /ba∇dbl(u,v)/ba∇dbl2\nH\nfor (u,v)∈D(A). Indeed, we calculate\n/parenleftbigg\nA/parenleftbiggu\nv/parenrightbigg\n,/parenleftbiggu\nv/parenrightbigg/parenrightbigg\nH=/parenleftbigg/parenleftbiggv\n∆u−a(x)v/parenrightbigg\n,/parenleftbiggu\nv/parenrightbigg/parenrightbigg\nH\n= (v,u)H1+(∆u−a(x)v,v)L2\n= (∇v,∇u)L2+(v,u)L2−(∇v,∇u)L2−(a(x)v,v)L2\n≤(v,u)L2≤ /ba∇dbl(u,v)/ba∇dbl2\nH.\nNext, we prove that there exists λ0∈Rsuch that for any λ≥λ0, the operator\nλ−Ais invertible, that is, for any ( f,g)∈ H, we can find a unique ( u,v)∈D(A)\nsatisfying\n(λ−A)/parenleftbigg\nu\nv/parenrightbigg\n=/parenleftbigg\nf\ng/parenrightbigg\n. (A.2)\nIndeed, the above equation is equivalent with\n/braceleftBigg\nλu−v=f,\nλv−∆u+a(x)v=g.\nWe remark that the first equation implies v=λu−f. Substituting this into the\nsecond equation, one has\n(λ2+λa(x))u−∆u=h, (A.3)\nwhereh=g+ (λ+a(x))f∈L2(Ω). Take an arbitrary constant λ0>0 and\nletλ≥λ0be fixed. Associated with the above equation, we define the bilinear\nfunctional\na(z,w) = ((λ2+λa(x))z,w)L2+(∇z,∇w)L2\nforz,w∈H1\n0(Ω). Since λ >0 anda(x) is nonnegative and bounded, ais bounded:\na(z,w)≤C/ba∇dblz/ba∇dblH1/ba∇dblw/ba∇dblH1, and coercive: a(z,z)≥C/ba∇dblz/ba∇dbl2\nH1. Therefore, by the Lax–\nMilgram theorem (see, e.g., [6, Theorem 1.1.4]), there exists a unique u∈H1\n0(Ω)\nsatisfying a(u,ϕ) = (h,ϕ)H1for anyϕ∈H1\n0(Ω). In particular, usatisfies the\nequation (A.3) in the distribution sense. This shows ∆ u∈L2(Ω), and hence, a\nstandard elliptic estimate implies u∈H2(Ω) (see, for example, Brezis [4, Theorem\n9.25]). Defining vbyv=λu−f∈H1\n0(Ω), we find the solution ( u,v)∈D(A) to\nthe equation (A.2).\nThe above properties enable us to apply the Hille–Yosida theorem (se e, e.g., [19,\nTheorem 2.18]), and there exists a C0-semigroup U(t) onHsatisfying the estimate\n/vextenddouble/vextenddouble/vextenddouble/vextenddoubleU(t)/parenleftbigg\nu0\nu1/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH≤eCt/ba∇dbl(u0,u1)/ba∇dblH (A.4)\nwith some constant C >0. Moreover, if ( u0,u1)∈D(A), thenU(t) :=U(t)/parenleftbiggu0\nu1/parenrightbigg\nsatisfies\nd\ndtU(t) =AU(t), t >0. (A.5)30 Y. WAKASUGI\nTherefore, the first component u(t) ofU(t) satisfies\nu∈C([0,∞);H2(Ω))∩C1([0,∞);H1\n0(Ω))∩C2([0,∞);L2(Ω))\nand the equation (A.1) in C([0,∞);L2(Ω)).\nFor (u0,u1)∈ H, letU(t) =/parenleftbiggu(t)\nv(t)/parenrightbigg\n:=U(t)/parenleftbiggu0\nu1/parenrightbigg\n. We next show that usatisfies\nu∈C([0,∞);H1\n0(Ω))∩C1([0,∞);L2(Ω)). (A.6)\nThe property u∈C([0,∞);H1\n0(Ω)) is obvious from U ∈C([0,∞);H). In or-\nder to prove u∈C1([0,∞);L2(Ω)), we employ an approximation argument. Let\n{(u(j)\n0,u(j)\n1)}∞\nj=1be a sequence in D(A) such that lim j→∞(u(j)\n0,u(j)\n1) = (u0,u1) in\nH, and let U(j)(t) =/parenleftbiggu(j)\nv(j)/parenrightbigg\n:=U(t)/parenleftBigg\nu(j)\n0\nu(j)\n1/parenrightBigg\n. From ( u(j)\n0,u(j)\n1)∈D(A),U(j)satis-\nfies the equation (A.5), and hence, one obtains v(j)=∂tu(j). For any fixed T >0,\nthe estimate (A.4) implies\nsup\nt∈[0,T]/ba∇dblu(j)(t)−u(t)/ba∇dblL2≤eCT/ba∇dbl(u(j)\n0−u0,u(j)\n1−u1)/ba∇dblH→0,\nsup\nt∈[0,T]/ba∇dbl∂tu(j)(t)−v(t)/ba∇dblL2≤eCT/ba∇dbl(u(j)\n0−u0,u(j)\n1−u1)/ba∇dblH→0\nasj→ ∞. This shows u∈C1([0,T];L2(Ω)) and ∂tu=v. SinceT >0 is arbitrary,\nwe obtain (A.6).\nA.2.Semilinear problem. Let us turn to study the semilinear problem (1.1).\nA.2.1.Uniqueness of the mild solution. We first show the uniqueness of the mild\nsolution of the integral equation\nU(t) =/parenleftbigg\nu(t)\nv(t)/parenrightbigg\n=U(t)/parenleftbigg\nu0\nu1/parenrightbigg\n+/integraldisplayt\n0U(t−s)/parenleftbigg\n0\n−|u(s)|p−1u(s)/parenrightbigg\nds(A.7)\ninC([0,T0);H) for arbitrary fixed T0>0. Hereafter, as long as there is no risk\nof confusion, we call both Uand the first component uofUmild solutions. Let\nT0>0 andC0=eCT0, whereCis the constant in (A.4). Let U(t) =/parenleftbiggu\nv/parenrightbigg\nand\nW(t) =/parenleftbigg\nw\nz/parenrightbigg\nbe two solutions to (A.7) in C([0,T0);H). TakeT∈(0,T0) arbitrary\nand put K:= supt∈[0,T](/ba∇dblU(t)/ba∇dblH+/ba∇dblW(t)/ba∇dblH. Then, the estimate (A.4) implies\n/ba∇dblU(t)−W(t)/ba∇dblH≤C0/integraldisplayt\n0/ba∇dbl|w(s)|p−1w(s)−|u(s)|p−1u(s)/ba∇dblL2ds.\nSince the nonlinearity satisfies\n||w|p−1w−|u|p−1u| ≤C(|w|+|u|)p−1|u−w|SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 31\nandpfulfills the condition(1.11), weapply the H¨ olderandthe Gagliardo–Nir enberg\ninequality /ba∇dblu/ba∇dblL2p≤C/ba∇dblu/ba∇dblH1to obtain\n/ba∇dblU(t)−W(t)/ba∇dblH≤C0/integraldisplayt\n0/ba∇dbl|u(s)|p−1u(s)−|w(s)|p−1w(s)/ba∇dblL2ds\n≤C0C/integraldisplayt\n0(/ba∇dblu(s)/ba∇dblL2p+/ba∇dblw(s)/ba∇dblL2p)p−1/ba∇dblu(s)−w(s)/ba∇dblL2pds\n≤C0C/integraldisplayt\n0(/ba∇dblu(s)/ba∇dblH1+/ba∇dblw(s)/ba∇dblH1)p−1/ba∇dblu(s)−w(s)/ba∇dblH1ds\n≤C0CKp−1/integraldisplayt\n0/ba∇dblU(s)−W(s)/ba∇dblHds (A.8)\nfort∈[0,T]. Therefore, by the Gronwall inequality, we have /ba∇dblU(t)−W(t)/ba∇dblH= 0\nfort∈[0,T]. Since T∈(0,T0) is arbitrary, we conclude U(t) =W(t) for all\nt∈[0,T0).\nA.2.2.Existence of the mild solution. Here, we show the existence of the mild\nsolution.\nLetT0>0 be arbitrarily fixed. For T∈(0,T0) andU=/parenleftbigg\nu\nv/parenrightbigg\n∈C([0,T];H), we\ndefine the mapping\nΦ(U)(t) =U(t)/parenleftbigg\nu0\nu1/parenrightbigg\n+/integraldisplayt\n0U(t−s)/parenleftbigg\n0\n−|u(s)|p−1u(s)/parenrightbigg\nds.\nLetC0=eCT0, whereCis the constant in (A.4). Then, we have/vextenddouble/vextenddouble/vextenddouble/vextenddoubleU(t)/parenleftbiggu0\nu1/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nH≤C0/ba∇dbl(u0,u1)/ba∇dblH\nfort∈(0,T0). LetK= 2C0/ba∇dbl(u0,u1)/ba∇dblHand define\nMT,K:=/braceleftBigg\nU=/parenleftbigg\nu\nv/parenrightbigg\n∈C([0,T];H); sup\nt∈[0,T]/ba∇dbl(u(t),v(t))/ba∇dblH≤K/bracerightBigg\n.\nMT,Kis a complete metric space with respect to the metric\nd(U,W) = sup\nt∈[0,T]/ba∇dbl(u(t)−w(t),v(t)−z(t))/ba∇dblH\nforU=/parenleftbiggu\nv/parenrightbigg\nandW=/parenleftbiggw\nz/parenrightbigg\n. We shall prove that Φ is the contraction mapping on\nMT,R, provided that Tis sufficiently small.\nFirst, we show that Φ(U)∈MT,KforU ∈MT,K. By the estimate (A.4) and the\nGagliardo–Nirenberg inequality, we obtain for t∈[0,T],\n/ba∇dblΦ(U)(t)/ba∇dblH≤K\n2+C0/integraldisplayt\n0/ba∇dbl|u(s)|p−1u(s)/ba∇dblL2ds\n≤K\n2+C0/integraldisplayt\n0/ba∇dblu(s)/ba∇dblp\nL2pds\n≤K\n2+C0C/integraldisplayt\n0/ba∇dblu(s)/ba∇dblp\nH1ds\n≤K\n2+C0CTKp. (A.9)32 Y. WAKASUGI\nTherefore, taking Tsufficiently small so that\nK\n2+C0CTKp≤K\nholds, we see that Φ(U)∈MT,K. Moreover, for U=/parenleftbigg\nu\nv/parenrightbigg\n,W=/parenleftbigg\nw\nz/parenrightbigg\n∈MT,R, the\nsame computation as in (A.8) yields for t∈[0,T],\nd(Φ(U),Φ(W))≤C0CTKp−1d(U,W).\nThus, retaking Tsmaller if needed so that\nC0CTKp−1≤1\n2,\nwe have the contractivity of Φ. Thus, by the contraction mapping principle, we see\nthat there exists a fixed point U=/parenleftbiggu\nv/parenrightbigg\n∈MT,K, that is, Usatisfies the integral\nequation (A.7). We postpone to verify u∈C1([0,T];L2(Ω)) and ∂tu=vafter\nproving the approximation property below.\nA.2.3.Blow-up alternative. LetTmax=Tmax(u0,u1)bethemaximalexistencetime\nof the mild solution defined by\nTmax= sup/braceleftbigg\nT∈(0,∞];∃U=/parenleftbigg\nu\nv/parenrightbigg\n∈C([0,T);H) satisfies (A.7)/bracerightbigg\n.\nWe show that if Tmax<∞, the corresponding unique mild solution U=/parenleftbiggu\nv/parenrightbigg\nmust\nsatisfy\nlim\nt→Tmax−0/ba∇dblU(t)/ba∇dblH=∞. (A.10)\nIndeed, if m:= liminf t→Tmax−0/ba∇dblU(t)/ba∇dblH<∞, thenthereexistsamonotoneincreas-\ningsequence {tj}∞\nj=1in(0,Tmax)suchthatlim j→∞tj=Tmaxandlim j→∞/ba∇dblU(tj)/ba∇dblH=\nm. LetT0> Tmaxbe arbitrary fixed and let C0=eCT0as in Section A.2.2. Ap-\nplying the same argument as in Section A.2.2 with replacement ( u0,u1) byU(tj),\none can find there exists Tdepending only on p,m, andC0such that there exists\na mild solution on the interval [ tj,tj+T]. However, this contradicts the definition\nofTmaxwhenjis large. Thus, we have (A.10).\nA.2.4.Continuous dependence on the initial data. Let (u0,u1)∈ HandT < T 0<\nTmax(u0,u1). We take C0=eCT0as in Section A.2.2. Let {(u(j)\n0,u(j)\n1)}∞\nj=1be a\nsequence in Hsuch that ( u(j)\n0,u(j)\n1)→(u0,u1) inHasj→ ∞. Then, we will prove\nthat, for sufficiently large j,Tmax(u(j)\n0,u(j)\n1)> Tand the corresponding solution\nU(j)with the initial data ( u(j)\n0,u(j)\n1) satisfies\nlim\nj→∞sup\nt∈[0,T]/ba∇dblU(j)(t)−U(t)/ba∇dblH= 0. (A.11)\nLetC1= 2supt∈[0,T]/ba∇dblU(t)/ba∇dblHand let\nτj:= sup/braceleftBigg\nt∈[0,Tmax(u(j)\n0,u(j)\n1)); sup\nt∈[0,T]/ba∇dblU(j)(t)/ba∇dblH≤2C1/bracerightBigg\n.SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 33\nSince (u(j)\n0,u(j)\n1)→(u0,u1) inHasj→ ∞, we have /ba∇dbl(u(j)\n0,u(j)\n1)/ba∇dblH≤C1for large\nj, which ensures τj>0 for such j. Moreover, the same computation as in (A.8)\nand the Gronwall inequality imply, for t∈[0,min{τj,T}],\n/ba∇dblU(j)(t)−U(t)/ba∇dblH≤C0/ba∇dblU(j)(0)−U(0)/ba∇dblHexp/parenleftBig\nCCp−1\n1T/parenrightBig\n.(A.12)\nNote that the right-hand side tends to zero as j→ ∞. From this and the definition\nofC1, we obtain\n/ba∇dblU(j)(t)/ba∇dblH≤C1(t∈[0,min{τj,T}])\nfor large j. By the definition of τj, the above estimate implies τj> T, and hence,\nTmax(u(j)\n0,u(j)\n1)> T. From this, the estimate (A.12) holds for t∈[0,T]. Letting\nj→ ∞in (A.12) gives (A.11).\nA.2.5.Regularity of solution. Next, we discuss the regularity of the solution. Let\n(u0,u1)∈D(A) andTmax=Tmax(u0,u1). Then, we will show that the correspond-\ning mild solution Usatisfies\nU ∈C([0,Tmax);D(A))∩C1([0,Tmax);H).\nTakeT∈(0,Tmax) arbitrary. First, from Section A.1, the linear part ofthe mild so-\nlution satisfies UL(t) =U(t)/parenleftbiggu0\nu1/parenrightbigg\n∈C([0,∞);D(A))∩C1([0,∞);H). This implies,\nforh >0 andt∈[0,T−h],\n/ba∇dblUL(t+h)−UL(t)/ba∇dblH≤Ch. (A.13)\nThus, it suffices to show\nUNL(t) :=/integraldisplayt\n0U(t−s)/parenleftbigg0\n−|u(s)|p−1u(s)/parenrightbigg\nds\n∈C([0,T];D(A))∩C1([0,T];H). (A.14)\nBy the changing variable t+h−s/mapsto→s, we calculate\nUNL(t+h)−UNL(t) =/integraldisplayt+h\n0U(t−s)/parenleftbigg0\n−|u(s)|p−1u(s)/parenrightbigg\nds\n−/integraldisplayt\n0U(t−s)/parenleftbigg\n0\n−|u(s)|p−1u(s)/parenrightbigg\nds\n=/integraldisplayt\n0U(s)/parenleftbigg0\n−|u|p−1u(t+h−s)+|u|p−1u(t−s)/parenrightbigg\nds\n+/integraldisplayt+h\ntU(s)/parenleftbigg0\n−|u|p−1u(t+h��s)/parenrightbigg\nds.\nTherefore, the same computation as in (A.8) and (A.9) implies\n/ba∇dblUNL(t+h)−UNL(t)/ba∇dblH≤C/integraldisplayt\n0/ba∇dblu(s+h)−u(s)/ba∇dblH1ds+Ch.\nCombining this with (A.13), one obtains\n/ba∇dblU(t+h)−U(t)/ba∇dblH≤Ch+/integraldisplayt\n0/ba∇dblU(s+h)−U(s)/ba∇dblHds.34 Y. WAKASUGI\nThe Gronwall inequality implies\n/ba∇dblU(t+h)−U(t)/ba∇dblH≤Ch.\nThis further yields\n/ba∇dbl−|u|p−1u(t+h)+|u|p−1u(t)/ba∇dblH1≤Ch,\nthat is, the nonlinearity is Lipschitz continuous in H1\n0(Ω). From this, we can\nsee−|u|p−1u∈W1,∞(0,T;H1\n0(Ω)) (see e.g. [6, Corollary 1.4.41]). Thus, we can\ndifferentiate the expression\n/integraldisplayt\n0U(t−s)/parenleftbigg\n0\n−|u|p−1u(s)/parenrightbigg\nds=/integraldisplayt\n0U(s)/parenleftbigg\n0\n−|u|p−1u(t−s)/parenrightbigg\nds\nwith respect to tinH, and it implies UNL∈C1([0,T];H). Finally, for h >0 and\nt∈[0,T−h], we have\n1\nh(U(t)−I)UNL(t) =1\nh/integraldisplayt\n0U(t+h−s)/parenleftbigg0\n−|u|p−1u(s)/parenrightbigg\nds−1\nh/integraldisplayt\n0U(t−s)/parenleftbigg0\n−|u|p−1u(s)/parenrightbigg\nds\n=1\nh(UNL(t+h)−UNL(t))−1\nh/integraldisplayt+h\ntU(t+h−s)/parenleftbigg\n0\n−|u|p−1u(s)/parenrightbigg\nds.\nThis implies U(t)∈D(A) and\nd\ndtUNL(t) =AUNL(t)+/parenleftbigg\n0\n−|u|p−1u(t)/parenrightbigg\n.\nMoreover, the above equation and U ∈C1([0,T];H) lead to U ∈C([0,T];D(A)).\nThis proves the property (A.14). We also remark that the first com ponentuofU\nis a strong solution to (1.1).\nA.2.6.Approximation of the mild solution by strong solutions. Let (u0,u1)∈ H\nandTmax=Tmax(u0,u1). Let{(u(j)\n0,u(j)\n1)}∞\nj=1be a sequence in D(A) satisfying\nlimj→∞(u(j)\n0,u(j)\n1) = (u0,u1) inH. TakeT∈(0,Tmax) arbitrary. Then, the results\nof Sections A.2.4 and A.2.5 imply that Tmax(u(j)\n0,u(j)\n1)> Tfor large j, and the\ncorresponding mild solution U(j)=/parenleftbiggu(j)\nv(j)/parenrightbigg\nwith the initial data ( u(j)\n0,u(j)\n1) satisfies\nU(j)∈C([0,T];D(A))∩C1([0,T];H). Moreover, ∂tu(j)=v(j)holds and u(j)is a\nstrong solution to (1.1). By the result of Section A.2.4, we see that\nlim\nj→∞sup\nt∈[0,T]/ba∇dblu(j)(t)−u(t)/ba∇dblH1= 0,\nlim\nj→∞sup\nt∈[0,T]/ba∇dbl∂tu(j)(t)−v(t)/ba∇dblL2= 0,\nwhich yields u∈C1([0,T];L2(Ω)) and ∂tu=v. Namely, we have the property\nstated at the end of Section A.2.2.SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 35\nA.2.7.Finite propagation property. Here, we show the finite propagation property\nfor the mild solution. In what follows, we use the notations BR(x0) :={x∈\nRn;|x−x0|< R}forx0∈RnandR >0. LetT∈(0,Tmax(u0,u1)) andR >\n0. Assume that ( u0,u1)∈ Hsatisfies supp u0∪suppu1⊂BR(0)∩Ω. Letu∈\nC([0,T];H1\n0(Ω))∩C1([0,T];L2(Ω)) be the mild solution of (1.1). Then, we have\nsuppu(t,·)⊂Bt+R(0)∩Ω (t∈[0,T]). (A.15)\nTo prove this, we modify the argument of [39] in which the classical so lution is\ntreated. Let ( t0,x0)∈[0,T]×Ω be a point such that |x0|> t0+Rand define\nΛ(t0,x0) ={(t,x)∈(0,T)×Ω; 0< t < t 0,|x−x0|< t0−t}\n=/uniondisplay\nt∈(0,t0)({t}×(Bt0−t(x0)∩Ω))).\nIt suffices to show u= 0 in Λ( t0,x0). We also put St0−t:=∂Bt0−t(x0)∩Ω and\nSb,t0−t:=Bt0−t(x0)∩∂Ω. Note that ∂(Bt0−t(x0)∩Ω) =St0−t∪Sb,t0−tholds.\nFirst, we further assume ( u0,u1)∈D(A). Then, by the result of Section A.2.5,\nubecomes the strong solution. This ensures that the following compu tations make\nsense.\nDefine\nE(t;t0,x0) :=1\n2/integraldisplay\nBt0−t(x0)∩Ω(|∂tu(t,x)|2+|∇u(t,x)|2+|u(t,x)|2)dx\nfort∈[0,t0]. By differentiating in tand applying the integration by parts, we have\nd\ndtE(t;t0,x0) =/integraldisplay\nBt0−t(x0)∩Ω/parenleftbig\n∂2\ntu−∆u+u/parenrightbig\n∂tudx\n−1\n2/integraldisplay\nSt0−t∪Sb,t0−t(|∂tu|2+|∇u|2+|u|2−2(n·∇u)∂tu)dS,\nwherenis the unit outward normal vector of St0−t∪Sb,t0−tanddSdenotes the\nsurface measure. The Schwarz inequality implies the second term of the right-\nhand side is nonpositive, and hence, we can omit it. Using the equation (1.1)\nto the first term and the Gagliardo–Nirenberg inequality /ba∇dblu(t)/ba∇dblL2p(Bt0−t(x0)∩Ω)≤\nC/ba∇dblu(t)/ba∇dblH1(Bt0−t(x0)∩Ω), we can see that\nd\ndtE(t;t0,x0)≤C/parenleftBig\n/ba∇dblu(t)/ba∇dbl2p\nH1(Bt0−t(x0)∩Ω)+/ba∇dbl∂tu(t)/ba∇dbl2\nL2(Bt0−t(x0)∩Ω)+/ba∇dblu(t)/ba∇dbl2\nL2(Bt0−t(x0)∩Ω)/parenrightBig\n≤CE(t;t0,x0),\nwhere we have also used /ba∇dblu(t)/ba∇dblH1(Bt0−t(x0)∩Ω)is bounded for t∈(0,t0). Noting\nthat the support condition of the initial data implies E(0;t0,x0) = 0, we obtain\nfrom the above inequality that E(t;t0,x0) = 0 for t∈[0,t0]. This yields u= 0 in\nΛ(t0,x0).\nFinally, for the general case ( u0,u1)∈ H, we take an arbitrary small ε >0 and\na sequence {(u(j)\n0,u(j)\n1)}∞\nj=1inD(A) such that supp u(j)\n0∪suppu(j)\n1⊂BR+ε(0)∩Ω\nand lim j→∞(u(j)\n0,u(j)\n1) = (u0,u1) inH. Here, we remark that such a sequence can\nbe constructed by the form ( u(j)\n0,u(j)\n1) = (φε˜u(j)\n0,φε˜u(j)\n1), where {(˜u(j)\n0,˜u(j)\n1)}is a\nsequencein D(A) which convergesto( u0,u1) inHasj→ ∞, andφε∈C∞\n0(Rn) is a\ncut-off function satisfy 0 ≤φε≤1,φε= 1 onBR(0), and φε= 0 onRn\\BR+ε(0).36 Y. WAKASUGI\nThen, the result of Section A.2.5 shows that the corresponding str ong solution\nu(j)to (u(j)\n0,u(j)\n1) satisfies supp u(j)(t,·)⊂BR+ε+t(0). Moreover, the result of\nSection A.2.6 leads to lim j→∞u(j)=uinC([0,T];H1\n0(Ω)). Hence, we conclude\nsuppu(t,·)⊂BR+ε+t(0). Since εis arbitrary, we have (A.15).\nA.2.8.Existence of the global solution. Finally, we show the existence of the global\nsolution to (1.1). Let ( u0,u1)∈ Hand suppose that Tmax(u0,u1) is finite. Then,\nby the blow-up alternative (Section A.2.3), the corresponding mild so lutionumust\nsatisfy\nlim\nt→Tmax−0/ba∇dbl(u(t),∂tu(t))/ba∇dblH=∞. (A.16)\nLet{(u(j)\n0,u(j)\n1)}∞\nj=1be a sequence in D(A) such that lim j→∞(u(j)\n0,u(j)\n1) = (u0,u1)\ninH, andletu(j)bethecorrespondingstrongsolutionwiththeinitialdata( u(j)\n0,u(j)\n1).\nUsing the integration by parts and the equation (1.1), we calculate\nd\ndt/bracketleftbigg1\n2/parenleftBig\n/ba∇dbl∂tu(j)(t)/ba∇dbl2\nL2+/ba∇dbl∇u(j)(t)/ba∇dbl2\nL2/parenrightBig\n+1\np+1/ba∇dblu(j)(t)/ba∇dblp+1\nLp+1/bracketrightbigg\n=−/ba∇dbl∂tu(j)(t)/ba∇dbl2\nL2.\nThis and the Gagliardo–Nirenberg inequality imply\n/ba∇dbl∂tu(j)(t)/ba∇dbl2\nL2+/ba∇dbl∇u(j)(t)/ba∇dbl2\nL2≤C/parenleftBig\n/ba∇dblu(j)\n1/ba∇dbl2\nL2+/ba∇dbl∇u(j)\n0/ba∇dbl2\nL2+/ba∇dblu(j)\n0/ba∇dblp+1\nH1/parenrightBig\n.\nMoreover, by\nu(t) =u0+/integraldisplayt\n0∂tu(s)ds,\none obtains the bound\n/ba∇dbl(u(j)(t),∂tu(j)(t))/ba∇dbl2\nH≤C(1+T)2/parenleftBig\n/ba∇dblu(j)\n1/ba∇dbl2\nL2+/ba∇dbl∇u(j)\n0/ba∇dbl2\nL2+/ba∇dblu(j)\n0/ba∇dblp+1\nH1/parenrightBig\n(A.17)\nfort∈[0,T]. Thisandtheblow-upalternative(SectionA.2.3)show Tmax(u(j)\n0,u(j)\n1) =\n∞for allj. The bound (A.17) with T=Tmax(u0,u1) also yields that\nsup\nj∈Nsup\nt∈[0,Tmax(u0,u1)]/ba∇dbl(u(j)(t),∂tu(j)(t))/ba∇dbl2\nH<∞. (A.18)\nOn the other hand, from the result of Section A.2.6, we have\nlim\nj→∞sup\nt∈[0,T]/ba∇dbl(u(j)(t)−u(t),∂tu(j)(t)−∂tu(t))/ba∇dblH= 0 (A.19)\nfor anyT∈(0,Tmax(u0,u1)). However, (A.18) and (A.19) contradict (A.16). Thus,\nwe conclude Tmax(u0,u1) =∞.\nAppendix B.Proof of Preliminary lemmas\nB.1.Proof of Lemma 2.1.\nProof of Lemma 2.1. We define\nb1(x) = ∆/parenleftbigga0\n(n−α)(2−α)/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht2−α/parenrightbigg\n=a0/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−α+a0α\nn−α/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−α−2\nandb2(x) =a(x)−b1(x). By\nb2(x)\na(x)=1\n/a\\}b∇acketle{tx/a\\}b∇acket∇i}htαa(x)/parenleftbigg\n/a\\}b∇acketle{tx/a\\}b∇acket∇i}htαa(x)−a0−a0α\nn−α/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−2/parenrightbiggSEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 37\nand the assumption (1.12), there exists a constant Rε>0 such that |b2(x)| ≤εa(x)\nholds for |x|> Rε. Letηε∈C∞\n0(Rn) satisfy 0 ≤ηε(x)≤1 forx∈Rnand\nηε(x) = 1 for |x|< Rε. LetN(x) denote the Newton potential, that is,\nN(x) =\n\n|x|\n2(n= 1),\n1\n2πlog1\n|x|(n= 2),\nΓ(n/2+1)\nn(n−2)πn/2|x|2−n(n≥3).\nWe define\nAε(x) =A0+a0\n(n−α)(2−α)/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht2−α−N∗(ηεb2),\nwhereA0>0 is a sufficiently large constant determined later. We show that the\naboveAε(x) has the desired properties. First, we compute\n∆Aε(x) =b1(x)+ηε(x)b2(x) =a(x)−(1−ηε)b2(x),\nwhich implies (2.1). Next, since ηεb2has the compact support, N∗(ηεb2) satisfies\n|N∗(ηεb2)(x)| ≤C/braceleftBigg\n1+log/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht(n= 2)\n/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht2−n(n= 1,n≥3),|∇N∗(ηεb2)(x)| ≤C/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht1−n\nwith some constant C=C(n,Rε,/ba∇dbla/ba∇dblL∞,α,a0,ε)>0, and the former estimate\nleads to (2.2), provided that A0is sufficiently large. Moreover, the latter estimate\nshows\nlim\n|x|→∞|∇Aε(x)|2\na(x)Aε(x)= lim\n|x|→∞1\n/a\\}b∇acketle{tx/a\\}b∇acket∇i}htαa(x)·1\n/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα−2Aε(x)/vextendsingle/vextendsingle/vextendsingle/vextendsinglea0\nn−α/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−1x−/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα−1∇N∗(ηεb2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n=2−α\nn−α,\nwhich implies the inequality (2.3) for sufficiently large x. Finally, taking A0suffi-\nciently large, we have (2.3) for any x∈Rn. /square\nB.2.Properties of Kummer’s function. To proveLemma 2.4, we preparesome\nproperties of Kummer’s function.\nLemma B.1. Kummer’s confluent hypergeometric function M(b,c;s)satisfies the\nproperties listed as follows.\n(i)M(b,c;s)satisfies Kummer’s equation\nsu′′(s)+(c−s)u′(s)−bu(s) = 0.\n(ii)Ifc≥b >0, thenM(b,c;s)>0fors≥0and\nlim\ns→∞M(b,c;s)\nsb−ces=Γ(c)\nΓ(b). (B.1)\nIn particular, M(b,c;s)satisfies\nC(1+s)b−ces≤M(b,c;s)≤C′(1+s)b−ces(B.2)\nwith some positive constants C=C(b,c)andC′=C(b,c)′.38 Y. WAKASUGI\n(iii)More generally, if −c /∈N∪{0}andc≥b, then, while the sign of M(b,c;s)\nis indefinite, it still has the asymptotic behavior\nlim\ns→∞M(b,c;s)\nsb−ces=Γ(c)\nΓ(b), (B.3)\nwhere we interpret that the right-hand side is zero if −b∈N∪ {0}. In\nparticular, M(b,c;s)has a bound\n|M(b,c;s)| ≤C(1+s)b−ces(B.4)\nwith some positive constant C=C(b,c).\n(iv)M(b,c;s)satisfies the relations\nsM(b,c;s) =sM′(b,c;s)+(c−b)M(b,c;s)−(c−b)M(b−1,c;s),\ncM′(b,c;s) =cM(b,c;s)−(c−b)M(b,c+1;s).\nProof.The property (i) is directly obtained from the definition of M(b,c;s). When\nc=b >0, (ii) is obviousfrom M(b,b;s) =es. Whenc > b >0, we havethe integral\nrepresentation (see [3, (6.1.3)])\nM(b,c;s) =Γ(c)\nΓ(b)Γ(c−b)/integraldisplay1\n0tb−1(1−t)c−b−1etsdt,\nwhich implies M(b,c;s)>0. Moreover, [3, (6.1.8)] shows the asymptotic behavior\n(B.1). The estimate (B.2) is obvious, since the right-hand side of (B.1 ) is positive\nandM(b,c;s)>0 fors≥0. Next, the property (iii) clearly holds if c=bor\n−b∈N∪{0}, sinceM(b,c;s) is a polynomial of order −bif−b∈N∪{0}. For the\ncasesc > band−b /∈N∪{0}, note that for any m∈N∪{0}we have\ndm\ndsmM(b,c;s) =(b)m\n(c)mM(b+m,c+m;s),\nwhich implies |dm\ndsmM(b,c;s)| → ∞ass→ ∞. By taking m∈N∪ {0}so that\nb+m >0 and applying l’Hˆ opital theorem we deduce\nlim\ns→∞M(b,c;s)\nsb−ces= lim\ns→∞dm\ndsmM(b,c;s)\ndm\ndsm(sb−ces)=(b)m\n(c)mlim\ns→∞M(b+m,c+m;s)\nsb−ces+o(sb−ces)\n=(b)mΓ(c+m)\n(c)mΓ(b+m)=Γ(c)\nΓ(b).\nThe estimate (B.4) is easily follows from the asymptotic behavior (B.3) and we\nhave (iii). Finally, the property (iv) can be found in [3, p.200]. /square\nB.3.Proof of Lemma 2.4.\nProof of Lemma 2.4. The property (i) is directly follows from Lemma B.1 (i). For\n(ii), noting that 0 ≤β < γ εand applying Lemma B.1 (ii) with b=γε−βand\nc=γε, we have ϕβ(s)>0 fors≥0 and\nlim\ns→∞sβϕβ,ε(s) =Γ(γε)\nΓ(γε−β).\nThis proves the property (ii). Next, by Lemma B.1 (iii) with b=γε−βandc=γε,\none still obtains lim s→∞sβϕε(s) = Γ(γε)/Γ(γε−β), where the right-hand side isSEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 39\ninterpreted as zero if β−γε∈N∪{0}. In particular, this (or the estimate (B.4))\ngives\n|ϕβ,ε(s)| ≤Kβ,ε(1+s)−β\nwith some constant Kβ,ε>0. Thus, we have (iii). Noting that\nϕ′\nβ,ε(s) =e−s[−M(γε−β,γε;s)+M′(γε−β,γε;s)] (B.5)\nand applying the first assertion of Lemma B.1 (iv), we have the prope rty (iv).\nFinally, from (B.5) and the second assertion of Lemma B.1 (iv), we obt ain\nγεϕ′\nβ,ε(s) =−βe−sM(γε−β,γε+1;s).\nDifferentiating again the above identity gives\nγεϕ′′\nβ,ε(s) =−βe−s[−M(γε−β,γε+1;s)+M′(γε−β,γε+1;s)].\nTherefore, the second assertion of Lemma B.1 (iv) implies\nγε(γε+1)ϕ′′\nβ,ε(s) =β(β+1)e−sM(γε−β,γε+2;s).\nIn particular, if 0 < β < γ ε, then Lemma B.1 (ii) shows that M(γε−β,γε+1;s)\n(resp.M(γε−β,γε+ 2;s) ) is bounded from above and below by (1 + s)−β−1es\n(resp. (1+ s)−β−2es), and hence, we have the assertions of (v). /square\nB.4.Proof of Proposition 2.6. We are now in a position to prove Proposition\n2.6.\nProof of Proposition 2.6. Letz=/tildewideγεAε(x)/(t0+t). FromDefinition2.5andLemma\n2.4 (iv), one obtains\n∂tΦβ,ε(t,x;t0) =−(t0+t)−β−1/bracketleftbig\nβϕβ,ε(z)+zϕ′\nβ,ε(z)/bracketrightbig\n=−(t0+t)−β−1βϕβ+1,ε(z)\n=−βΦβ+1,ε(t,x;t0),\nwhich proves (i). Applying Lemma 2.4 (iii), we have\n|Φβ,ε(t,x;t0)| ≤Kβ,ε(t0+t)−β/parenleftbigg\n1+/tildewideγεAε(x)\nt0+t/parenrightbigg−β\n≤C(t0+t+Aε(x))−β\n=CΨ(t,x;t0)−β\nwith some constant C=C(n,α,β,ε)>0. This implies (ii). Next, by Lemma 2.4\n(ii), Φ β,ε(t,x;t0) satisfies\nΦβ,ε(t,x;t0)≥kβ,ε(t0+t)−β/parenleftbigg\n1+/tildewideγεAε(x)\nt0+t/parenrightbigg−β\n≥c(t0+t+Aε(x))−β\n=cΨ(t,x;t0)−β40 Y. WAKASUGI\nwith some constant c=c(n,α,β,ε)>0, and (iii) is verified. For (iv), we again put\nz= ˜γεAε(x)/(t0+t) and compute\na(x)∂tΦβ,ε(x,t;t0)−∆Φβ,ε(x,t;t0)\n=−a(x)(t0+t)−β−1\n×/parenleftbigg\nβϕβ,ε(z)+zϕ′\nβ,ε(z)+ ˜γε∆Aε(x)\na(x)ϕ′\nβ,ε(z)+ ˜γε|∇Aε(x)|2\na(x)Aε(x)zϕ′′\nβ,ε(z)/parenrightbigg\n.\nUsing the equation (2.5) and the definition (2.4), we rewrite the right -hand side as\n˜γεa(x)(t0+t)−β−1/parenleftbigg\n1−2ε−∆Aε(x)\na(x)/parenrightbigg\nϕ′\nβ,ε(z)\n+a(x)(t0+t)−β−1/parenleftbigg\n1−˜γε|∇Aε(x)|2\na(x)Aε(x)/parenrightbigg\nϕ′′\nβ,ε(z).\nBy (2.1) and (2.3) in Lemma 2.1, we have\n1−2ε−∆Aε(x)\na(x)≤ −ε,\n1−˜γε|∇Aε(x)|2\na(x)Aε(x)≥ε/parenleftbigg2−α\nn−α+2ε/parenrightbigg−1\n>0.\nFrom them and the property (v) of Lemma 2.4, we conclude\na(x)∂tΦβ,ε(x,t;t0)−∆Φβ,ε(x,t;t0)≥ −ε˜γεa(x)(t0+t)−β−1ϕ′\nβ,ε/parenleftbigg˜γεAε(x)\nt0+t/parenrightbigg\n≥εkβ,εa(x)(t0+t)−β−1/parenleftbigg\n1+˜γεAε(x)\nt0+t/parenrightbigg−β−1\n≥ca(x)(t0+t+Aε(x))−β−1\n=ca(x)Ψ(x,t;t0)−β−1\nwith some constant c=c(n,α,β,ε)>0, which completes the proof. /square\nB.5.Proof of Lemma 2.7.\nProof of Lemma 2.7. Putting v= Φ−1+δu, noting ∇u= (1−δ)Φ−δ(∇Φ)v+\nΦ1−δ∇v, and applying integration by parts imply\n/integraldisplay\nΩ|∇u|2Φ−1+2δdx\n=/integraldisplay\nΩ|∇v|2Φdx+2(1−δ)/integraldisplay\nΩv(∇v·∇Φ)dx+(1−δ)2/integraldisplay\nΩ|v|2|∇Φ|2\nΦdx\n=/integraldisplay\nΩ|∇v|2Φdx−(1−δ)/integraldisplay\nΩ|v|2∆Φdx+(1−δ)2/integraldisplay\nΩ|v|2|∇Φ|2\nΦdx\n≥ −(1−δ)/integraldisplay\nΩ|u|2(∆Φ)Φ−2+2δdx+(1−δ)2/integraldisplay\nΩ|u|2|∇Φ|2Φ−3+2δdx.SEMILINEAR SPACE-DEPENDENT DAMPED WAVE EQUATION 41\nByu∆u=−|∇u|2+∆(u2\n2), integration by parts, and applying the above estimate,\nwe have/integraldisplay\nΩu∆uΦ−1+2δdx\n=−/integraldisplay\nΩ|∇u|2Φ−1+2δdx+1\n2/integraldisplay\nΩ|u|2∆(Φ−1+2δ)dx\n=−/integraldisplay\nΩ|∇u|2Φ−1+2δdx−1−2δ\n2/integraldisplay\nΩ|u|2(∆Φ)Φ−2+2δdx\n+(1−δ)(1−2δ)/integraldisplay\nΩ|u|2|∇Φ|2Φ−3+2δdx\n≤ −δ\n1−δ/integraldisplay\nΩ|∇u|2Φ−1+2δdx+1−2δ\n2/integraldisplay\nΩ|u|2(∆Φ)Φ−2+2δdx.\nThis completes the proof. /square\nAcknowledgements\nThis work was supported by JSPS KAKENHI Grant Numbers JP18H01 132 and\nJP20K14346. 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Partial Differential Equations 15(1990), 205–235.\nLaboratory of Mathematics, Graduate School of Engineering , Hiroshima University,\nHigashi-Hiroshima, 739-8527, Japan\nEmail address :wakasugi@hiroshima-u.ac.jp"    },    {        "title": "1209.3120v1.Skyrmion_Dynamics_in_Multiferroic_Insulator.pdf",        "content": "arXiv:1209.3120v1  [cond-mat.str-el]  14 Sep 2012Skyrmion Dynamics in Multiferroic Insulator\nYe-Hua Liu,1You-Quan Li,1and Jung Hoon Han2,3,∗\n1Zhejiang Institute of Modern Physics and Department of Physi cs,\nZhejiang University, Hangzhou 310027, People’s Republic of China\n2Department of Physics and BK21 Physics Research Division,\nSungkyunkwan University, Suwon 440-746, Korea\n3Asia Pacific Center for Theoretical Physics, Pohang, Gyeong buk 790-784, Korea\n(Dated: October 31, 2018)\nRecent discovery of Skyrmion crystal phase in insulating mu ltiferroic compound Cu 2OSeO 3calls\nfor new ways and ideas to manipulate the Skyrmions in the abse nce of spin transfer torque from\nthe conduction electrons. It is shown here that the position -dependent electric field, pointed along\nthe direction of the average induced dipole moment of the Sky rmion, can induce the Hall motion\nof Skyrmion with its velocity orthogonal to the field gradien t. Finite Gilbert damping produces\nlongitudinal motion. We find a rich variety of resonance mode s excited by a.c. electric field.\nPACS numbers: 75.85.+t, 75.70.Kw, 76.50.+g\nSkyrmions are increasingly becoming commonplace\nsightings among spiral magnets including the metallic\nB20 compounds[1–5] and most recently, in a multiferroic\ninsulator Cu 2OSeO3[6]. Both species of compounds dis-\nplaysimilarthickness-dependentphasediagrams[5,6] de-\nspitetheircompletelydifferentelectricalproperties,high-\nlighting the generality of the Skyrmion phase in spiral\nmagnets. Along with the ubiquity of Skyrmion matter\ncomes the challenge of finding means to control and ma-\nnipulatethem, inadevice-orientedmannerakintoefforts\nin spintronics community to control the domain wall and\nvortex motion by electrical current. Spin transfer torque\n(STT) is a powerful means to induce fast domain wall\nmotion in metallic magnets[7, 8]. Indeed, current-driven\nSkyrmion rotation[9] and collective drift[10], originating\nfrom STT, have been demonstrated in the case of spiral\nmagnets. Theory of current-induced Skyrmion dynam-\nics has been worked out in Refs. [11, 12]. In insulating\ncompounds such as Cu 2OSeO3, however, the STT-driven\nmechanism does not work due to the lack of conduction\nelectrons.\nAs with other magnetically driven multiferroic\ncompounds[13], spiral magnetic order in Cu 2OSeO3is\naccompanied by finite electric dipole moment. Recent\nwork by Seki et al.[14] further confirmed the mecha-\nnism of electric dipole moment induction in Cu 2OSeO3\nto be the so-called pd-hybridization[15–17]. In short, the\npd-hybridization mechanism claims the dipole moment\nPijfor every oxygen-TM(transition metal) bond propor-\ntional to ( Si·ˆeij)2ˆeijwhereiandjstand for TM and\noxygen sites, respectively, and ˆ eijis the unit vector con-\nnecting them. Carefully summing up the contributions\nof such terms over a unit cell consisting of many TM-\nO bonds, Seki et al.were able to deduce the dipole\nmoment distribution associated with a given Skyrmionic\nspin configuration[14]. It is interesting to note that the\nnumerical procedure performed by Seki et al.is pre-\ncisely the coarse-graining procedure which, in the text-book sense of statistical mechanics, is tantamount to the\nGinzburg-Landau theory of order parameters. Indeed we\ncan show that Seki et al.’s result for the dipole moment\ndistribution is faithfully reproduced by the assumption\nthat the local dipole moment Piis related to the local\nmagnetization Siby\nPi=λ(Sy\niSz\ni,Sz\niSx\ni,Sx\niSy\ni) (1)\nwith some coupling λ. A similar expression was pro-\nposed earlier in Refs. [18, 19] as the GL theory\nof Ba 2CoGe2O7[20], another known pd-hybridization-\noriginated multiferroic material with cubic crystal struc-\nture. Each site icorresponds to one cubic unit cell of\nCu2OSeO3with linear dimension a∼8.9˚A, and we have\nnormalized Sito have unit magnitude. The dimension of\nthe coupling constant is therefore [ λ] = C·m.\nHaving obtained the proper coupling between dipole\nmoment and the magnetizaiton vector in Cu 2OSeO3one\ncan readily proceed to study the spin dynamics by solv-\ning Landau-Lifshitz-Gilbert (LLG) equation. Very small\nvalues of Gilbert damping parameter are assumed in the\nsimulation as we are dealing with an insulating magnet.\nA new, critical element in the simulation is the term aris-\ning from the dipolar coupling\nHME=−/summationdisplay\niPi·Ei=−λ\n2/summationdisplay\niSi\n0Ez\niEy\ni\nEz\ni0Ex\ni\nEy\niEx\ni0\nSi,(2)\nwhere we have used the magneto-electric coupling ex-\npression in Eq. (1). In essence this is a field-dependent\n(voltage-dependent) magnetic anisotropy term. The to-\ntal Hamiltonian for spin is given by H=HHDM+\nHME, whereHHDMconsists of the Heisenberg and the\nDzyaloshinskii-Moriya (DM) exchange and a Zeeman\nfield term. Earlier theoretical studies showed HHDMto\nstabilize the Skyrmion phase[1, 21–24].2\nTwo field orientations can be chosen independently in\nexperiments performed on insulating magnets. First, the\ndirection of magnetic field Bdetermines the plane, or-\nthogonal to B, in which Skyrmions form. Second, the\nelectric field Ecan be applied to couple to the induced\ndipole moment of the Skyrmion and used as a “knob” to\nmove it around. Three field directions used in Ref. [14]\nand the induced dipole moment in eachcase areclassified\nas (I)B/bardbl[001],P= 0, (II) B/bardbl[110],P/bardbl[001], and\n(III)B/bardbl[111],P/bardbl[111]. One can rotate the spin axis\nappearing in Eq. (1) accordingly so that the z-direction\ncoincides with the magnetic field orientation in a given\nsetup and the x-direction with the crystallographic[ 110].\nIn each of the cases listed above we obtain the magneto-\nelectric coupling, after the rotation,\nH(I)\nME=−λ\n2/summationdisplay\niEi([Sy\ni]2−[Sx\ni]2),\nH(II)\nME=−λ\n2/summationdisplay\niEi([Sz\ni]2−[Sx\ni]2),\nH(III)\nME=−λ\n2√\n3/summationdisplay\niEi(3[Sz\ni]2−1).(3)\nIn cases (II) and (III) the E-field is chosen parallel to the\ninduced dipole moment P,Ei=EiˆP, to maximize the\neffect of dipolar coupling. In case (I) where there is no\nnet dipole moment for Skyrmions we chose E/bardbl[001] to\narrive at a simple magneto-electric coupling form shown\nabove.\nSuppose now that the E-field variation is sufficiently\nslow on the scale of the lattice constant ato allow the\nwriting of the continuum energy,\nHME=−λd\na/integraldisplay\nd2rE(r)ρD(r). (4)\nIt is assumed that all variables behave identically along\nthe thickness direction, oflength d. The “dipolarcharge”\ndensityρD(r)couplestotheelectricfield E(r)inthesame\nwayasthe conventionalelectric chargedoes tothe poten-\ntial field in electromagnetism. The analogy is also useful\nin thinking about the Skyrmion dynamics under the spa-\ntially varying E-field as we will show. The continuum\nform of dipolar charge density in Eq. 4 is\nρ(I)\nD(ri) =1\n2a2([Sy\ni]2−[Sx\ni]2),\nρ(II)\nD(ri) =1\n2a2([Sz\ni]2−[Sx\ni]2−1),\nρ(III)\nD(ri) =√\n3\n2a2([Sz\ni]2−1). (5)\nDivision by the unit cell area a2ensures that ρD(r) has\nthe dimension of areal density. Values for the ferromag-\nnetic case, Sz\ni= 1, is subtracted in writing down the def-\ninition (5) in order to isolate the motion of the Skyrmion\nFIG. 1: (color online) (a) Skyrmion configuration and (b)-(d )\nthe corresponding distribution of dipolar charge density f or\nthree magnetic field orientations as in Ref. 14. (b) B/bardbl[001]\n(c)B/bardbl[110] (d) B/bardbl[111]. For each case, electric field is\nchosen as E/bardbl[001],E/bardbl[001] and E/bardbl[111], respectively. See\ntext for the definition of dipolar charge density. As schemat i-\ncally depicted in (a), the Skyrmion executes a Hall motion in\nresponse to electric field gradient.\nrelative to the ferromagnetic background. Due to the\nsubtraction, the dipolar charge is no longer equivalent to\nthe dipole moment of the Skyrmion. The distribution of\ndipolar charge density for the Skyrmion spin configura-\ntion in the three cases are plotted in Fig. 1. In case (I)\nthe total dipolar charge is zero. In cases (II) and (III)\nthe net dipolar charges are both negative with the re-\nlation,Q(II)\nD/Q(III)\nD=√\n3/2, where QD, of order unity,\nis obtained by integrating ρD(r) over the space of one\nSkyrmion and divide the result by the number of spins\nNSkinside the Skyrmion. If the field variation is slow on\nthe scale of the Skyrmion, then the point-particle limit is\nreached by writing ρD(r) =QDNSk/summationtext\njδ(r−rj) whererj\nspans the Skyrmion positions, and identical charge QDis\nassumed for all the Skyrmions. We arrive at the “poten-\ntial energy” of the collection of Skyrmion particles,\nHME=−λQDNSkd\na/summationdisplay\njE(rj). (6)\nA force acting on the Skyrmion will be given as the gra-\ndientFi=−∇iHME. Inter-Skyrmion interaction is ig-\nnored.\nThe response of Skyrmions to a given force, on the\nother hand, is that of an electric charge in strong\nmagnetic field, embodied in the Berry phase action3\n(−2πS¯hQSkd/a3)/summationtext\nj/integraltext\ndt(rj×˙rj)·ˆz, whereQSkis the\nquantized Skyrmion charge[12, 25], and Sis the size of\nspin. Equation of motion follows from the combination\nof the Berry phase action and Eq. (6),\nvj=λ\n4πS¯ha2NSkQD\nQSkˆz×∇jE(rj), (7)\nwherevjis thej-th Skyrmion velocity. Typical Hall ve-\nlocity can be estimated by replacing |∇E|with ∆E/lSk,\nwhere ∆Eis the difference in the field strength between\nthe left and the right edge of the Skyrmion and lSkis its\ndiameter. Taking a2NSk∼l2\nSkwe find the velocity\nλlSk\n4πS¯h∆E∼10−6∆E[m2/V·s], (8)\nwhich gives the estimated drift velocity of 1 mm/s for the\nfield strength difference of 103V/m across the Skyrmion.\nExperimental input parameters of lSk= 10−7m, and\nλ= 10−32C·m were taken from Ref. [14] in arriving at\nthe estimation, as well as the dipolar and the Skyrmion\nchargesQD≈ −1 andQSk=−1. We may estimate the\nmaximum allowed drift velocity by equating the dipolar\nenergy difference λ∆Eacross the Skyrmion to the ex-\nchange energy J, also corresponding to the formation en-\nergy of one Skyrmion[24]. The maximum expected veloc-\nity thus obtained is enormous, ∼104m/s forJ∼1meV,\nimplying that with the right engineering one can achieve\nrather high Hall velocity of the Skyrmion. In an encour-\naging step forward, electric field control of the Skyrmion\nlattice orientation in the Cu 2OSeO3crystal was recently\ndemonstrated[26].\nResults of LLG simulation is discussed next. To start,\na sinusoidal field configuration Ei=E0sin(2πxi/Lx) is\nimposed on a rectangular Lx×Lysimulation lattice with\nLxmuch larger than the Skyrmion size. In the absence\nof Gilbert damping, a single Skyrmion placed in such\nan environment moved along the “equi-potential line” in\nthey-direction as expected from the guiding-center dy-\nnamics of Eq. (7). In cases (II) and (III) where the\ndipolar charges are nonzero the velocity of the Skyrmion\ndrift is found to be proportional to their respective dipo-\nlar charges QDas shown in Fig. 2. The drift velocity\ndecreased continuously as we reduced the field gradient,\nobeying the relation (7) down to the zero velocity limit.\nThe dipolar charge is zero in case (I), and indeed the\nSkyrmion remains stationary for sufficiently smooth E-\nfield gradient. Even for this case, Skyrmions can move\nfor field gradient modulations taking place on the length\nscale comparable to the Skyrmion radius, for the reason\nthat the forces acting on the positive dipolar charge den-\nsity blobs (red in Fig. 1(a)) are not completely canceled\nby those on the negative dipolar charge density blobs\n(blue in Fig. 1(a)) for sufficiently rapid variations of thefield strength gradient. A small but non-zero drift veloc-\nityensues, asshowninFig. 2. Longitudinalmotionalong\nthe field gradient begins to develop with finite Gilbert\ndamping, driving the Skyrmion center to the position of\nlowest “potential energy” E(r). For the Skyrmion lat-\ntice, imposing a uniform field gradient across the whole\nlattice may be too demanding experimentally, unless the\nmagnetic crystal is cut in the form of a narrow strip the\nwidth of which is comparable to a few Skyrmion radii.\nIn this case we indeed observe the constant drift of the\nSkyrmions along the length of the strip in response to\nthe field gradient across it. The drift speed is still pro-\nportional to the field gradient, but about an order of\nmagnitude less than that of an isolated Skyrmion under\nthe same field gradient. We observed the excitation of\nbreathing modes of Skyrmions when subject to a field\ngradient, and speculate that such breathing mode may\ninterfere with the drift motion as the Skyrmions become\nclosed-packed.\n0 500 1000 1500 2000−30−25−20−15−10−505\nty(ab. unit)\n  \nv(I)\nHall=−7×10−4\nv(II)\nHall=−1.32×10−2\nv(III)\nHall=−1.49×10−2\nv(II)\nHall\nv(III)\nHall≈Q(II)\nD\nQ(III)\nD=√\n3\n2case (I)  : B||[001]\ncase (II) : B||[110]\ncase (III): B||[111]\nFIG. 2: (color online) Skyrmion position versus time for cas es\n(I) through (III) for sinusoidal electric field modulation ( see\ntext) with the Skyrmion center placed at the maximum field\ngradient position. The average Hall velocities (in arbitra ry\nunits) in cases (II) and (III) indicated in the figure are ap-\nproximately proportional to the respective dipolar charge s, in\nagreement with Eq. (7). A small velocity remains in case\n(I) due to imperfect cancelation of forces across the dipola r\ncharge profile.\nSeveral movie files are included in the supplementary\ninformation. II.gif and III.gif give Skyrmion motion for\nEi=E0sin(2πxi/Lx) onLx×Ly= 66×66 lattice for\nmagneto-electric couplings (II) and (III) in Eq. (3). III-\nGilbert.gif gives the same E-field as III.gif, with finite\nGilbert damping α= 0.2. I.gif describes the case (I)\nwhere the average dipolar charge is zero, with a rapidly\nvarying electric field Ei=E0sin(2πxi/λx) andλxcom-\nparable to the Skyrmion radius. The case of a narrow\nstrip with the field gradient across is shown in strip.gif.\nMochizuki’s recent simulation[27] revealed that inter-4\nnal motion of Skyrmions can be excited with the uniform\na.c. magnetic field. Some of his predictions were con-\nfirmed by the recent microwave measurement[29]. Here\nwe show that uniform a.c. electric field can also ex-\ncite several internal modes due to the magneto-electric\ncoupling. Time-localized electric field pulse was applied\nin the LLG simulation and the temporal response χ(t)\nwas Fourier analyzed, where the response function χ(t)\nrefersto(1 /2)/summationtext\ni([Sy\ni(t)]2−[Sx\ni(t)]2), (1/2)/summationtext\ni([Sz\ni(t)]2−\n[Sx\ni(t)]2), and (√\n3/2)/summationtext\ni[Sz\ni(t)]2for cases (I) through\n(III), respectively. (In Mochizuki’s work, the response\nfunction was the component of total spin along the a.c.\nmagnetic field direction.)\nIn case (I), the uniform electric field perturbs the ini-\ntial cylindrical symmetry ofSkyrmion spin profileso that/summationtext\ni([Sx\ni(t)]2−[Sy\ni(t)]2) becomes non-zero and the over-\nall shape becomes elliptical. The axes of the ellipse then\nrotates counter-clockwise about the Skyrmion center of\nmass as illustrated in supplementary figure, E-mode.gif.\nThere are two additional modes of higher energies with\nbroken cylindrical symmetry in case (I), labeled X1 and\nX2 in Fig. 3 and included as X1-mode.gif and X2-\nmode.gif in the supplementary. The rotational direction\nof the X1-mode is the same as in E-mode, while it is the\nopposite for X2-mode.\nAs in Ref. [27], we find sharply defined breathing\nmodesin cases(II) and(III) atthe appropriateresonance\nfrequency ω, in fact the same frequency at which the a.c.\nmagnetic field excites the breathing mode. The verti-\ncal dashed line in Fig. 3 indicates the common breath-\ning mode frequency. Movie file B1-mode.gif shows the\nbreathing mode in case (III). Additional, higher energy\nB2-mode (B2-mode.gif) was found in cases (II) and (III),\nwhich is the radial mode with one node, whereas the B1\nmode is nodeless.\nIn addition to the two breathing modes, E-mode and\nthe two X-modes are excited in case (II) as well due\nto the partly in-plane nature of the spin perturbation,\n−(λE(t)/2)/summationtext\ni([Sz\ni(t)]2−[Sx\ni(t)]2). In contrast, case\n(III), where the perturbation −(√\n3λE(t)/2)/summationtext\ni[Sz\ni(t)]2\nis purely out-of-plane, one only finds the B-modes. As\na result, case (I) and (III) have no common resonance\nmodes or peaks, while case (II) has all the peaks (though\nX1 and X2 peaks are small). Compared to the magnetic\nfield-induced resonances, a richer variety of modes are\nexcited by a.c. electric field. In particular, the E-mode\nhas lower excitation energy than the B-mode and has a\nsharp resonance feature, which should make its detection\na relatively straightforward task. Full analytic solution\nof the excited modes[28] will be given later.\nIn summary, motivated by the recent discov-\nery of magneto-electric material Cu 2OSeO3exhibiting\nSkyrmion lattice phase, we have outlined the theory of\nSkyrmion dynamics in such materials. Electric field gra-\ndient is identified as the source of Skyrmion Hall motion.00.20.40.60.81Imχ(ω) (ab. unit)\n  \nB1\nR1\nR2(a)B||z,Bω||x,y\nB||z,Bω||z\n00.05 0.10.15 0.20.25 0.30.3500.20.40.60.81\nωImχ(ω) (ab. unit)\n  \nE\nX1X2B1\nB2(b)case (I)  : B||[001],Eω||[001]\ncase (II) : B||[110],Eω||[001]\ncase (III): B||[111],Eω||[111]\nFIG. 3: (color online) (a) Absorption spectra for a.c. unifo rm\nmagnetic field as in Mochizuki’s work (reproduced here for\ncomparison). (b) Absorption spectra for a.c. uniform elect ric\nfield in cases (I) through (III). In case (I) where there is no\nnet dipolar charge we find three low-energy modes E, X1, and\nX2. For case (III) where the dipolar charge is finite we find\nB1 and B2 radial modes excited. Case (II) exhibits all five\nmodes. Detailed description of each mode is given in the text .\nSeveralresonantexcitationsbya.c. electricfieldareiden-\ntified.\nJ. H. H. is supported by NRF grant (No. 2010-\n0008529, 2011-0015631). Y. Q. L. is supported by NSFC\n(Grant No. 11074216). J. H. H. acknowledges earlier\ncollaboration with N. Nagaosa, Youngbin Tchoe, and J.\nZang on a related model and informative discussion with\nY. Tokura.\n∗Electronic address: hanjh@skku.edu\n[1] S. M¨ uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A.\nRosch, A. Neubauer, R. Georgii, and P. B¨ oni, Science\n323, 915 (2009).\n[2] W. M¨ unzer, A. Neubauer, T. Adams, S. M¨ uhlbauer, C.\nFranz, F. Jonietz, R. Georgii, P. B¨ oni, B. Pedersen, M.\nSchmidt, A. Rosch, and C. Pfleiderer, Phys. Rev. B 81,\n041203(R) (2010).\n[3] X. Z. Yu, Y. Onose, N. Kanazawa, J. H. Park, J. H. Han,\nY. Matsui, N. Nagaosa, and Y. Tokura, Nature 465, 901\n(2010).\n[4] X. Z. Yu, N. Kanazawa, Y. Onose, K. Kimoto, W. Z.\nZhang, S. Ishiwata, Y. Matsui, and Y. Tokura, Nature\nMat.10, 106 (2011).\n[5] N. Kanazawa, Y. Onose, T. Arima, D. Okuyama, K.\nOhoyama, S. Wakimoto, K. Kakurai, S. Ishiwata, and\nY. Tokura, Phys. Rev. Lett. 106, 156603 (2011).\n[6] S. Seki, X. Z. Yu, S. Ishiwata, and Y. Tokura, Science5\n336, 198 (2012).\n[7] G. Tatara, H. Kohno, and J. Shibata, Phys. Rep. 468,\n213 (2008).\n[8] D. C. Ralph and M. Stiles, J. Magn. Mag. Mat. 320,\n1190 (2008).\n[9] F. Jonietz, S. M¨ uhlbauer, C. Pfleiderer, A. Neubauer, W.\nM¨ unzer, A. Bauer, T. Adams, R. Georgii, P. B¨ oni, R. A.\nDuine, K. Everschor, M. Garst, and A. Rosch, Science\n330, 1648 (2010).\n[10] T. Schulz, R. Ritz, A. Bauer, M. Halder, M.Wagner, C.\nFranz, C. Pfleiderer, K. Everschor, M. Garst, and A.\nRosch, Nat. Phys. 8, 301 (2012).\n[11] K. Everschor, M. Garst, R. A. Duine, and A. Rosch,\nPhys. Rev. B 84, 064401 (2011).\n[12] J. Zang, M. Mostovoy, J. H. Han, and N. Nagaosa, Phys.\nRev. Lett. 107, 136804 (2011).\n[13] Y. Tokura and S. Seki, Adv. Mat. 21, 1 (2009).\n[14] S. Seki, S. Ishiwata, and Y. Tokura, arXiv:1206.4404v1\n(2012).\n[15] Chenglong Jia, Shigeki Onoda, Naoto Nagaosa, and Jung\nHoon Han, Phys. Rev. B 74, 224444 (2006).\n[16] Chenglong Jia, Shigeki Onoda, Naoto Nagaosa, and Jung\nHoon Han, Phys. Rev. B 76, 144424 (2007).\n[17] Taka-hisa Arima, J. Phys. Soc. Jpn. 76, 073702 (2008).\n[18] Judit Romh´ anyi, Mikl´ os Lajk´ o, and Karlo Penc, Phys.Rev. B84, 224419 (2011).\n[19] J. H. Han, unpublished note.\n[20] H. Murakawa, Y. Onose, S. Miyahara, N. Furukawa, and\nY. Tokura, Phys. Rev. Lett. 105, 137202 (2010).\n[21] A. N. Bogdanov and D. A. Yablonskii, Sov. Phys. JETP\n68, 101 (1989); A. Bogdanov and A. Hubert, J. Magn.\nMagn. Mater. 138, 255 (1994).\n[22] U. K. Roßler, A. N. Bogdanov, and C. Pfleiderer, Nature\n442, 797 (2006).\n[23] Su Do Yi, Shigeki Onoda, Naoto Nagaosa, and Jung\nHoon Han, Phys. Rev. B 80, 054416 (2009).\n[24] Jung Hoon Han, Jiadong Zang, Zhihua Yang, Jin-Hong\nPark, and Naoto Nagaosa, Phys. Rev. B 82, 094429\n(2010).\n[25] Michael Stone, Phys. Rev. B 53, 16573 (1996).\n[26] J. S. White, et al. arXiv:1208.1146 (2012).\n[27] Masahito Mochizuki, Phys. Rev. Lett. 108, 017601\n(2012).\n[28] Olga Petrova and Oleg Tchernyshyov, Phys. Rev. B 84,\n214433 (2011); Imam Makhfudz, Benjamin Kr¨ uger, and\nOleg Tchernyshyov, arXiv:1208.3123 (2012).\n[29] Y. Onose, Y. Okamura, S. Seki, S. Ishiwata, and Y.\nTokura, Phys. Rev. Lett. 109, 037603 (2012)."    },    {        "title": "1810.07384v2.Perpendicularly_magnetized_YIG_films_with_small_Gilbert_damping_constant_and_anomalous_spin_transport_properties.pdf",        "content": "Perpendicularly magnetized  YIG films with small  Gilbert \ndamping constant and anomalous spin transport properties  \nQianbiao  Liu1, 2, Kangkang  Meng1*, Zedong  Xu3, Tao Zhu4, Xiao guang Xu1, Jun \nMiao1 & Yong Jiang1* \n \n1. Beijing Advanced Innovation Center for Materials Genome Engineering, University of Science and \nTechnology Beijing, Beijing 100083, China  \n2. Applied and Engineering physics, Cornell University, Ithaca, NY 14853, USA  \n3. Department of Physics, South University of Science and Technology of China , Shenzhen 518055, \nChina  \n4. Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China  \nEmail: kkmeng@ustb.edu.cn ; yjiang@ustb.edu.cn  \n \nAbstract: The Y 3Fe5O12 (YIG) films with perpendicular magnetic anisotropy (PMA) \nhave recently attracted a great deal of attention for spintronics  applications. Here, w e \nreport the induced PMA in the ultrathin YIG films grown on \n(Gd 2.6Ca0.4)(Ga 4.1Mg 0.25Zr0.65)O12 (SGGG) substrate s by epitaxial strain without \npreprocessing. Reciprocal space mapping shows  that the film s are  lattice -matched to \nthe substrate s without strain relaxation. Through ferromagnetic resonance and \npolarized neutron reflectometry measurements, we find that these YIG films have \nultra-low Gilbert damping constant  (α < 1×10-5) with a magnetic dead layer as thin as \nabout 0.3 nm  at the YIG/SGGG interfaces. Moreover, the transport behavior of the \nPt/YIG/SGGG films reveals an enhancement of spin mixing conductance and a large \nnon-monotonic magnetic field dependence of anomalous Hall effect as compared with \nthe Pt/YIG/Gd 3Ga5O12 (GGG) films. The non- monotonic anomalous Hall signal is \nextracted in the temperature range from 150 to 350 K, which has been ascribed to the possible non -collinear magnetic order at the Pt/YIG interface induced by uniaxial \nstrain.  \n \nThe spin transport in ferrim agnetic insulator (FMI)  based devices has received \nconsiderable interest due to its free of current -induced Joule heating and beneficial for \nlow-power spintronic s applications  [1, 2]. Especially, the high-quality Y3Fe5O12 (YIG)  \nfilm as a widely  studied FMI  has low damping  constant,  low magnetostriction and \nsmall magnetocrystalline anisotropy,  making it a key material for magnonics  and spin \ncaloritronics . Though the magnon s can carr y information over distances as long as \nmillimeters  in YIG film , there remain s a challenge to control its magnetic anisotropy \nwhile maintaining the low damping constant [3] , especially for  the thin film with \nperpendicular  magnetic anisotropy (PMA) , which  is very useful for spin polarizers,  \nspin-torque oscillators, magneto -optical d evices  and m agnon valve s [4-7]. In addition, \nthe spin- orbit torque (SOT) induced magnetization switching with low current \ndensities has been realized in non -magnetic heavy metal  (HM)/FMI heterostructures , \npaving the road towards  ultralow -dissipation SOT de vices based on FMI s [8-10]. \nFurthermore, p revious theoretical studies have pointed that  the current density will \nbecome much smaller if the domain structures were topologically protected (chiral) [11].  However,  most FMI  films  favor in-plane easy axis dominated by shape \nanisotropy , and the investigation is eclipsed as compared with ferromagnetic materials \nwhich show abundant and interesting domain structures such as chiral domain walls and magnetic skyrmions et al.  [12-17]. Recently, the interface- induc ed chiral domain walls  have been observed in centrosymmetric oxides Tm 3Fe5O12 (TmIG) thin films, \nand the  domain walls  can be propelled by spin current from an adjacent platinum \nlayer  [18]. Similar with the TmIG films, the possible chiral  magnetic structures are \nalso expected in the YIG films with lower damping  constan t, which  would further \nimprove the chiral domain walls’ motion speed.  \nRecently, several ways  have been reported to attain the perpendicular ly \nmagnetized YIG films , one of which is utiliz ing the lattice distortion and \nmagnetoelastic effect induced by  epitaxial strain [1 9-22]. It is noted that  the strain \ncontrol can not only enable the field -free magnetization switching but also assist the \nstabilization of the non- collinear  magnetic textures  in a broad range of magnetic field \nand temperature. Therefore, abundant and interesting physical phenomena  would \nemerge in epitaxial grown YIG films with PMA. However, either varying the buffer \nlayer or doping would increase the Gilbert damping constant of YIG, which will  \naffect the efficiency of the SOT induced magnetization switching  [20, 21]. On the \nother hand, these preprocessing would lead to a more complicate magnetic structures  \nand impede  the further discussion of spin transport properties such as possible \ntopological Hall effect  (THE).  \nIn this work, we realized the PMA of ultrathin YIG films deposited on SGGG \nsubstrates  due to epitaxial strain . Through ferromagnetic resonance (FMR) and \npolarized neutron reflectometry (PNR) measurements, we have found that the YIG \nfilms had small Gilbert damping constant with a magnetic dead layer as thin as about \n0.3 nm  at the YIG/SGGG interfaces. Moreover, we have carried out the transport measurements of the Pt/YIG/SGGG films and observed a large non -monotonic \nmagnet ic field dependence of the anomalous Hall resistivity, which did not exis t in \nthe compared Pt/YIG/GGG films. The non -monotonic anomalous Hall signal was \nextracted in the temperature range from 150 to 350 K, and we ascribed it  to the \npossible non -collinear magnetic order at the Pt/YIG interfaces induced by uniaxial \nstrain.  \n \nResults  \nStructural and magnetic characterization.  The epitaxial YIG films with varying \nthickness from 3 to 90 nm  were grown on the [111] -oriented GGG substrate s (lattice \nparameter a = 1.237 nm) and SGGG substrates  (lattice parameter a = 1.248 nm)  \nrespectively by pulsed laser deposition technique  (see methods). After the deposition, \nwe have investigated the surface morphology of the two kinds of films using atomic \nforce microscopy (AFM) as shown in Fig. 1 ( a), and the two films have a  similar and \nsmall surface roughness ~0.1 nm. Fig. 1 ( b) shows the  enlarged  XRD ω-2θ scan \nspectra of the YIG (40 nm) thin film s grow n on the two different substrates (more \ndetails are shown in the Supplementary Note 1 ), and they  all show predominant (444) \ndiffraction peaks without any other diffraction peaks, excluding impurity phases or other crystallographic orientation s and indicat ing the single -phase  nature. According \nto the (444) diffraction pe ak position and the reciprocal space map of the (642) \nreflection of a 40 -nm-thick YIG film grown on SGGG as shown in Fig. 1(c), we have \nfound that the lattice constant of SGGG  (~1.248 nm) substrate was larger than the YIG layer  (~1.236 nm). We quantify thi s biaxial strain as ξ  = (aOP - aIP)/aIP, where a OP \nand aIP represent the pseudo  cubic lattice constant calculated from the ou t-of-plane  \nlattice constant d(4 4 4) OP and in-plane lattice constant d(1 1 0) IP, respectively, \nfollowing the equation of  \n2 2 2lkhad\n++= , with h, k, and l  standing for the Miller \nindices of the crystal planes . It indicates that the SGGG substrate provides a tensile \nstress ( ξ ~ 0.84%) [21]. At the same time, the magnetic properties of the YIG films \ngrown on the two different substrates were measured via VSM magnetometry at room \ntemperature. According to the magnetic field ( H) dependence of the magnetization  (M) \nas shown in Fig. 1 (d), the magnetic anisotropy of the YIG film grown on SGGG \nsubstrate has been modulated by strain, while the two films have similar in -plane \nM-H curves.  \nTo further investigate the quality of the YIG films  grown on SGGG substrates  \nand exclude  the possibility of the strain induced large stoichiometry  and lattice \nmismatch, compositional analyse s were carried out using x -ray photoelectron \nspectroscopy (XPS) and PNR. As shown in Fig. 2 (a), the difference of binding \nenergy between the 2p 3/2 peak and the satellite peak is about 8.0 eV, and the Fe ions \nare determined to be in the 3+ valence state.  It is found that there is  no obvious \ndifference for Fe elements in the YIG films grown on GGG and SGGG substrates. \nThe Y 3 d spectrums show a small energy shift as shown in Fig. 2 (b) and the binding \nenergy shift may be related to the lattice strain and the variation of bond length  [21]. \nTherefore, the stoichiometry of the YIG  surface has not been dramatically modified \nwith the strain control.  Furthermore, we have performed the PNR meas urement  to probe the depth dependent struc ture and magnetic information of YIG films grown on \nSGGG substrates. The PNR  signals and scattering length density (SLD) profiles  for \nYIG (12.8 nm)/SGGG films by applying an in- plane magnetic field of 900 mT at \nroom temperature are shown in Fig. 2 ( c) and ( d), respectively.  In Fig. 2(c), R++ and \nR-- are the nonspin -flip reflectivities, where the spin polarizations are the same for the \nincoming and reflected neutrons. The inset of Fig. 2(c) shows the experimental and  \nsimulated spin -asymmetry (SA), defined as SA = ( R++ – R--)/(R++ + R--), as a function \nof scattering vector Q. A reasonable fitting was obtained with a three- layer model for \nthe single YIG film, containing the interface layer , main YIG layer and surface layer. \nThe nuclear SLD and  magnetic SLD are directly proportional to the nuclear scattering \npotential and the magnetization , respectively . Then, the depth- resolved structural and \nmagnetic SLD profiles delivered by fitting are s hown in Fig. 2(d) . The Z -axis \nrepresents the distance for the vertical direction of the film, where Z  = 0 indicates the \nposition at the YIG/SGGG interface. It is obvious that  there is few Gd diffusion into \nthe YIG film, and the dead layer  (0.3 nm ) is much thinner than the reported values \n(5-10 nm) between YIG  (or T mIG) and substrates [23 -25]. The net magnetization of \nYIG is 3.36 μB (～140  emu/cm3), which is similar with that of bulk YIG [2 6]. The \nPNR results also showed that besides the YIG/ SGGG interface region, there is also  \n1.51- nm-thick  nonmagnetic surface layer, which may be  Y2O3 and is likely to be \nextremely important in magnetic proximity effect [ 23]. \n Dynamical characterization and spin transport properties.  To quantitatively \ndetermine  the magnetic anisotropy and dynamic properties of the YIG films, the FMR \nspectra were measured at room temperature using an electron paramagnetic resonance \nspectrometer with rotating the films. Fig. 3(a) shows the geometric configuration of the angle reso lved FMR measurements. We use the FMR absorption line shape to \nextract the resonance field  (H\nres) and peak -to-peak linewidth ( ΔHpp) at different θ  for \nthe 40 -nm-thick YIG fil ms grown on GGG and SGGG substrates, respectively.  The \ndetails for 3 -nm-thick  YIG film are show n in the Supp lementary  Note 2 . According to \nthe angle  dependence of H res as shown in Fig. 3(b), one can find that as compared \nwith the YIG films grown on GGG substrate s, the minimum Hres of the 40- nm-thick \nYIG film grown on SGGG substrate increases with varying θ from 0° to 90° .On the \nother hand,  according to the frequency dependence of Hres for the YIG (40 nm)  films \nwith applying H  in the XY  plane as shown in Fig. 3(c), in contrast to the YIG/GGG \nfilms, the H res in YIG/SGGG films could not be fitted by the in-plane magnetic \nanisotropy Kittel formula  21)] 4 ( )[2(/\neff res res πM H Hπγ/ f + = . All these results \nindicate that the easy axis of YIG  (40 nm) /SGGG films  lies out -of-plane. The angle  \ndependent  ΔHpp for the  two films are  also compared as shown in Fig. 3(d) , the \n40-nm-thick YIG film grown on SGGG substrate has  an optimal value of Δ Hpp as low \nas 0.4 mT at θ =64°, and the corresponding FMR absorption line and Lorentz fitting \ncurve are shown in Fig. 3(e).  Generally , the ΔHpp is expected to be minimum \n(maximum) along  magnetic easy (hard) axis, which is basically coincident with the \nangle dependent  ΔHpp for the YIG films grown on GGG substrates.  However, as shown in Fig. 3(d), the  ΔHpp for the YIG/SGGG films  shows an anomalous variation. \nThe lowest  ΔHpp at θ=64° could be ascribed to the high YIG film quality and  ultrathin  \nmagnetic dead layer at the YIG/SGGG interface.  It should be noted that , as compared \nwith YIG/GGG films , the Δ Hpp is independent on the frequency from 5 GHz to 14 \nGHz as shown in Fig. 3(f). Then, w e have calculate d the Gilbert damping constant α  \nof the  YIG (40 nm)/SGGG films by extracting the Δ Hpp at each frequency as shown in \nFig. 3(f). The obtained α  is smaller tha n 1 × 10−5, which  is one order of magnitude \nlower than t he report in  Ref. [20]  and would  open new perspectives for the \nmagnetization dynamics.  According to the theor etical theme, the ΔHpp consists of \nthree parts: Gilbert damping, two magnons scattering relaxation process and \ninhomogeneities, in which both the Gilbert damping and the two magnons scattering \nrelaxation process depend on frequency. Therefore, the large Δ Hpp in the YIG/SGGG \nfilms  mainly  stems from the inhomogeneities, w hich will be discussed next with the \nhelp of the transport measurements. All of the above results have proven that the  \nultrathin YIG films grown on SGGG substrate s have not only evident  PMA  but also \nultra-low Gilbert damping constant.  \nFurthermore, we have also investigated the spin transport properties for  the high \nquality YIG film s grown on SGGG substrate s, which  are basically sensitive to  the \nmagnetic details of YIG. The magnetoresistance (MR) has been proved as a powerful \ntool to effectively explore magnetic information originating from the interfaces [ 27]. \nThe temperature dependent spin Hall magnetoresistance (SMR) of the Pt (5 nm)/YIG \n(3 nm) films grown on the two different substrates were measured using a small and non-perturbative current densit y (~ 106 A/cm2), and the s ketches of the measurement  \nis shown in Fig. 4 (a). The β scan of the longitudinal  MR, which is defined as \nMR=ΔρXX/ρXX(0)=[ρXX(β) -ρXX(0)]/ρXX(0) in the YZ  plane  for the two films  under a 3 T \nfield (enough to saturate the magnetization  of YIG ), shows cos2β behavior s with \nvarying temperature  for the Pt/YIG/GGG and Pt/YIG/SGGG films as shown in Fig. 4 \n(b) and (c), respectively. T he SMR of  the Pt/YIG /SGGG films  is larger than that of \nthe Pt/YIG /GGG films with the same thickness of YIG  at room temperature,  \nindica ting an enhanced  spin mixing conductance ( G↑↓) in the Pt/YIG /SGGG films. \nHere, it should be noted that the spin transport properties for the Pt layers ar e \nexpected to be the same because of the similar resistivity and film s quality . Therefore, \nthe SGGG substrate not only induces the PMA but also enhances G ↑↓ at the Pt/YIG \ninterface. Then, we have also investigated the field dependent Hall resistivities  in the \nPt/YIG/SGGG films at the temperature range from 260 to 350 K  as shown in Fig. 4(d).  \nThough the conduction electrons cannot penetrate into the FMI layer, the possible \nanomalous Hall effect (AHE) at the HM/FMI interface is proposed to emerge, and the \ntotal Hall resistivity can usually be expressed as the sum of various contributions  [28, \n29]: \nS-A S H ρ  ρ H R ρ + + =0 ,         (1)  \nwhere R0 is the normal Hall coefficient, ρ S the transverse manifestation of SMR, and \nρS-A the spin Hall anomalous Hall effect (SAHE) resistivity. Notably, the external field \nis applied out -of-plane, and  ρs (~Δρ1mxmy) can be neglected [ 29]. Interestingly, the \nfilm grown on SGGG substrate shows a bump and dip feature during the hysteretic measurements in the temperature range from 260 to 350 K. In the following \ndiscussion, we term the part of extra anomalous signals as the anomalous SAHE resistivity ( ρ\nA-S-A). The ρ A-S-A signals clearly coexist with the large background of \nnormal Hall effect. Notably, the broken (space) inversion symmetry with strong \nspin-orbit coupling (SOC)  will induce the Dzyaloshinskii -Moriya interaction (DMI) . \nIf the DMI could be compared with the Heisenberg exchange interaction and the \nmagnetic anisotropy  that were controlled by st rain, it c ould stabilize non-collinear \nmagnetic textures  such as skyrmions, producing  a fictitious magnetic field and the \nTHE . The ρA-S-A signals  indicate that a chiral spin texture may exist, which  is similar \nwith B20-type compounds Mn 3Si and Mn 3Ge [ 30,31]. To more clearly demonstrate \nthe origin of the anomalous  signals, we have subtracted the normal Hall term , and the \ntemperature dependence of ( ρS-A + ρ A-S-A) has been shown in Fig. 4 (e). Then, we can \nfurther discern the peak and hump structure s in the temperature range from 260 to 350 \nK. The SAHE  contribution ρS-A can be expressed  as 𝜌𝑆−𝐴=𝛥𝜌2𝑚𝑍 [32, 33],\n where  \n𝛥𝜌2 is the coefficient depending on the imaginary part of G ↑↓, and mz is the unit \nvector of the magnetization  orientation along  the Z direction . The extracted Hall \nresist ivity ρA-S-A has been shown in Fig. 4 (f), and the temperature dependence of the \nlargest ρA-S-A (𝜌𝐴−𝑆−𝐴Max) in all the films have been shown in Fig. 4 (g). Finite values of \n𝜌𝐴−𝑆−𝐴Max exist  in the temperature range from 150 to 350 K , which is much d ifferen t \nfrom that in B20 -type bulk chiral magnets which are subjected to low temperature and \nlarge magnetic field [34]. The large non -monotonic magnetic field dependence of anomalous Hall resistivity  could not stem from the We yl points, and the more detailed \ndiscussion was shown in the Supplementary Note 3.  \nTo further discuss the origin of the anomalous transport signals, we have \ninvestigated the small field dependence of the Hall resistances for Pt  (5 nm) /YIG (40 \nnm)/SGGG films as shown in Fig. 5(a). The out-of-plane hysteresis loop  of \nPt/YIG/SGGG is not central symmetry, which indicates the existence of an  internal \nfield leading to opposite  velocities  of up to down and down to domain walls in the  \npresence of current along the +X  direction. The large field dependences of the Hall \nresistances are shown in Fig. 5(b), which could not be described by Equation (1). \nThere are large variations for the Hall signals when the external magnetic field  is \nlower than the saturation field ( Bs) of YIG film (~50 mT at 300 K and ~150 mT at 50 \nK). More interestingly, we have firstly applied a large out -of-plane external magnetic \nfield of +0.8 T ( -0.8 T) above Bs to saturate the out -of-plane magnetization \ncomp onent MZ > 0 ( MZ < 0), then decreased the field to zero, finally the Hall \nresistances were measured in the small field range ( ± 400 Oe), from which  we could \nfind that the shape was reversed as shown in Fig. 5(c). Here, we infer that the magnetic structures  at the Pt/YIG interface grown on SGGG substrate could not be a \nsimple linear magnetic order.  Theoretically , an additional chirality -driven Hall effect \nmight be present in the ferromagnetic regime due to spin canting [3 5-38]. It has been \nfound that  the str ain from an insulating substrate could produce a tetragonal distortion, \nwhich would drive an orbital selection, modifying  the electronic properties and the \nmagnetic ordering of manganites. For A\n1-xBxMnO 3 perovskites, a compressive strain makes the ferromagnetic configuration relatively more stable than the \nantiferromagnetic state [3 9]. On the other hand, the strain would induce the spin \ncanting [ 40]. A variety of experiments and theories have reported that the ion \nsubstitute, defect and magnetoelast ic interaction would cant the magnetization of YIG \n[41-43]. Therefore, if we could modify the magnetic order by epitaxial strain, the \nnon-collinear magnetic structure is expected to emerge in the YIG film.  For YIG \ncrystalline structure, the two Fe sites ar e located on the octahedrally coordinated 16(a) \nsite and the tetrahedrally coordinated 24(d) site, align ing antiparallel with each other  \n[44]. According to the XRD and RSM results, the tensile strain due to SGGG \nsubstrate would result in the distortion ang le of the facets of the YIG unit cell smaller \nthan 90 ° [45]. Therefore, the magneti zations  of Fe at two sublattice s should  be \ndiscussed separately rather than as a whole. Then, t he anomalous  signals  of \nPt/YIG/SGGG films  could be  ascribed to the emergence o f four different Fe3+ \nmagnetic orientation s in strained Pt/YIG films, which are shown in Fig. 5(d).  For \nbetter to understand our results, w e assume that, in analogy with ρ S, the ρA-S-A is larger \nthan  ρA-S and scales linearly with m ymz and mxmz. With applying a large external field \nH along  Z axis, the uncompensated magnetic moment at the tetrahedrally coordinated \n24(d) is along with the external fields H  direction for |H | > Bs, and the magnetic \nmoment tends to be along A (-A) axis  when the external fields is swept  from 0.8 T \n(-0.8 T) to 0 T. Then,  if the Hall resistance  was measured at small out -of-plane field , \nthe uncompensated magnetic moment  would switch from A (-A) axis to B (-B) axis. In \nthis case, the ρ A-S-A that scales with Δ ρ3(mymz+mxmz) would change the sign because the mz is switched from the Z  axis to - Z axis as shown in Fig. 5(c). However, there is \nstill some problem that needs to be further clarified. There are no anomalous  signals  \nin Pt/YIG/GGG films that could be ascribed to the weak strength of Δρ3 or the strong \nmagnetic anisotropy . It is still valued for further discussion of the origin of Δ ρ3 that \nwhether it could stem from the skrymions et al ., but until now we have not observed \nany chiral domain structures in Pt/YIG/SGGG films through the Lorentz transmission \nelectron microscopy. Therefore, we hope that future work would  involve more \ndetailed magnetic microscopy imaging and microstructure analysis, which can further elucidate the real microscopic origin of the large non -monotonic magnetic field \ndependence of anomalous Hall resistivity.   \n \nConclusion  \nIn conclusion, the YIG film with PMA could be realized using both epitaxial strain \nand growth -induced anisotropies. These YIG films grown on SGGG substrates had \nlow G ilbert damping constants (<1 ×10\n-5) with a magnetic dead layer as thin as about \n0.3 nm  at the YIG/SGGG interface. Moreover, we observe d a large non -monotonic \nmagnetic field dependence of anomalous Hall resistivity in Pt/YIG/SGGG films, \nwhich did not exist in Pt/YIG/GGG films. The non -monotonic anomalous portion of \nthe Hall signal was extracted in the temperature range from 150 to 350 K and w e \nascribed it to the possible non -collinear magnetic order at the Pt/YIG interface \ninduced by uniaxial strain. The present work not only demonstrate that the strain \ncontrol can effectively tune the electromagnetic properties of FMI but also open up the exp loration of non -collinear spin texture for fundamental physics and magnetic \nstorage technologies based on FMI.  \n \nMethods  \nSample preparation.  The epitaxial YIG films with varying thickness from 3 to 90 \nnm were grown on the [111] -oriented GGG substrate s (lattice parameter a =1.237 nm) \nand SGGG substrates  (lattice parameter a  =1.248 nm)  respectively by pulsed laser \ndeposition technique . The growth  temperature was TS =780 ℃ and the oxyg  \npressure  was varied from  10 to 50 Pa . Then,  the films were annealed at 780℃ for 30 \nmin at the oxygen pressure of  200 Pa . The Pt (5nm) layer was deposited on  the top of  \nYIG films  at room temperature by magnetron sputtering. After the deposition, the \nelectron beam lithography and Ar ion milling were used to pattern Hall bars, and a lift-off process was used to form contact electrodes . The size of all the Hall bars is 20 \nμm×120 μm. \nStructural and magnetic characterization. The s urface morphology was measured \nby AFM (Bruke Dimension Icon).  Magnetization measurements were carried out \nusing a Physical Property Measurement System (PPMS) VSM.  A detailed \ninvestigation of the magnetic information  of Y IG was investigated by  PNR at the \nSpallation Neutron Source of China.   \nFerromagnetic resonance measurements. The measurement setup is depicted in Fig. \n3(a). For FMR measurements, the DC magnetic field was modulated with an AC field. \nThe transmitted signal was detected by a lock -in amplifier. We observed the FMR spectrum of the sample by sweeping the external magnetic field. The data obtained \nwere then fitted to a sum of symmetric and antisymmetric Lorentzian functions to \nextract the linewidth.  \nSpin transport measurements . The measurements  were carried out  using PPMS \nDynaCool.  \n \nAcknowledgments  \nThe authors thanks Prof. L. Q. Yan and Y. Sun for the technical assistant in \nferromagnetic resonance  measurement . This work was partially supported by the \nNational Science Foundation of China (Grant Nos. 51971027, 51927802, 51971023 , \n51731003, 51671019, 51602022, 61674013, 51602025), and the Fundamental Research Funds for the Central Universities (FRF- TP-19-001A3).  \n References  \n[1] Wu, M.-Z. & Hoffmann , A. Recent advances in magnetic insulators from \nspintronics to microwave applications. Academic Press , New York, 64 , 408 \n(2013) . \n[2] Maekawa,  S. Concepts in spin electronics. Oxford Univ., ( 2006) . \n[3] Neusser, S. & Grundler, D. Magnonics: spin waves on the nanoscale. Adv. 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N. & Saslow W. M. Defect interactions and canting in ferromagnets. Phys. Rev. B 38, 11718 (1988).  \n[42] Rosencwaig  A. Localized canting model for substituted ferrimagnets. I. Singly \nsubstituted YIG systems. Can. J. Phys. 48, 2857- 2867(1970).  \n[43] AULD B. A. Nonlinear magnetoelastic interactions. Proceedings of the IEEE, 53,  \n1517- 1533 (1965).  \n[44] Ching W. Y., Gu Z. & Xu Y N. Th eoretical calculation of the optical properties \nof Y\n3Fe5O12. J. Appl. Phys. 89,  6883- 6885 (2001).  [45] Baena A., Brey L. & Calder ón M. J. Effect of strain on the orbital and magnetic \nordering of manganite thin films and their interface with an insulator. Phys. Rev. \nB 83, 064424 (2011).  \n   \nFigure Captions  \n \nFig. 1 Structural  and magnetic  properties of YIG films.  (a) AFM images of the  \nYIG films  grown on  the two  substrates (scale bar, 1 μ m). (b) XRD ω-2θ scans of the \ntwo different YIG films  grown on  the two substrates . (c) High -resolution XRD \nreciprocal space map of t he YIG film deposited on the SGGG substrate. (d) Field \ndependence of the normalized magnetization of the YIG films  grown  on the two \ndifferent substrates . \n \n \nFig. 2 Structural  and magnetic  properties of YIG films.  Room temperature XPS \nspectra of (a) Fe 2p and (b) Y 3d for YIG films grown  on the two substrates . (c) P NR \nsignals (with a 900 mT in -plane field) for the spin -polarized R++ and R-- channels. \nInset: The experimental and simulated SA  as a function of scattering vector Q.  (d) \nSLD profiles of the  YIG/SGGG films.  The nuclear SLD and  magnetic SLD is directly \nproportional to the nuclear scattering potential and the magnetization , respectively.  \n \n \n \n \nFig. 3 Dynamical properties of YIG films . (a) The geometric configuration of the \nangle dependent FMR measurement. (b) The angle  dependence of the H res for the YIG \nfilms on GGG and SGGG substrates. (c) The frequency dependence of the H res for \nYIG films grown on GGG and S GGG substrates. (d) The ang le dependence of Δ Hpp \nfor the YIG films on GGG and SGGG substrates. (e) FMR spectrum of the \n40-nm-thick YIG film grown on SGGG substrate with 9.46 GHz at θ =64°. (f) The \nfrequency dependence of Δ Hpp for the 40 -nm-thick YIG films grown on GGG and \nSGGG substr ates.  \n \nFig. 4 Spin transport properties  of Pt/YIG (3nm) films . (a) The definition of the \nangle, the axes and the measurement configurations. ( b) and ( c) Longitudinal MR at \ndifferent temperatures in  Pt/YIG/GGG and Pt/YIG/SGGG films respectively (The \napplied magnetic field is 3 T). (d) Total Hall resistivities vs  H for Pt/YIG/SGGG films \nin the temperature range from 260 to 300 K. (e)  (ρS-A+ρA-S-A) vs H for two films in the \ntemperature range from 260 to 300 K. (f) ρ A-S-A vs H for Pt/YIG/SGGG films at 300K. \nInset: ρS-A and ρS-A + ρ A-S-A vs H for Pt/YIG/SGGG films at 300K. (g) Temperature \ndependence of the 𝜌𝐴−𝑆−𝐴𝑀𝑎𝑥. \n \n \n \nFigure 5 S pin transport properties  of Pt/YIG ( 40 nm) films . (a) and (b) The Hall \nresistances vs H for the Pt/YIG/SGGG films in the temperature range from 50 to 300 \nK in small  and large magnetic field range, respectively. (c) The Hall resistances vs  H \nat small magnetic field range after sweeping a large out -of-plane  magnetic field +0.8 \nT (black line) and - 0.8 T (red line) to zero.  (d) An illustration of the orientations of the \nmagnetizations Fe ( a) and Fe ( d) in YIG films with the normal in -plane magnetic \nanisotropy (IMA), the ideal  strain induced PMA and the actual magnetic anisotropy \ngrown on SGGG in our work.  \n"    },    {        "title": "1909.05881v2.Spin_Transport_in_Thick_Insulating_Antiferromagnetic_Films.pdf",        "content": "Spin Transport in Thick Insulating Antiferromagnetic Films\nRoberto E. Troncoso1, Scott A. Bender2, Arne Brataas1, and Rembert A. Duine1;2;3\n1Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Trondheim, Norway\n2Utrecht University, Leuvenlaan 4, 3584 CE Utrecht, The Netherlands and\n3Department of Applied Physics, Eindhoven University of Technology,\nP.O. Box 513, 5600 MB Eindhoven, The Netherlands\nSpin transport of magnonic excitations in uniaxial insulating antiferromagnets (AFs) is investi-\ngated. In linear response to spin biasing and a temperature gradient, the spin transport properties\nof normal-metal{insulating antiferromagnet{normal-metal heterostructures are calculated. We fo-\ncus on the thick-\flm regime, where the AF is thicker than the magnon equilibration length. This\nregime allows the use of a drift-di\u000busion approach, which is opposed to the thin-\flm limit considered\nby Bender et al. 2017, where a stochastic approach is justi\fed. We obtain the temperature- and\nthickness-dependence of the structural spin Seebeck coe\u000ecient Sand magnon conductance G. In\ntheir evaluation we incorporate e\u000bects from \feld- and temperature-dependent spin conserving inter-\nmagnon scattering processes. Furthermore, the interfacial spin transport is studied by evaluating\nthe contact magnon conductances in a microscopic model that accounts for the sub-lattice sym-\nmetry breaking at the interface. We \fnd that while inter-magnon scattering does slightly suppress\nthe spin Seebeck e\u000bect, transport is generally una\u000bected, with the relevant spin decay length being\ndetermined by non-magnon-conserving processes such as Gilbert damping. In addition, we \fnd that\nwhile the structural spin conductance may be enhanced near the spin \rip transition, it does not\ndiverge due to spin impedance at the normal metal|magnet interfaces.\nI. INTRODUCTION\nSpin-wave excitations in magnetic materials are a cor-\nnerstone in spintronics for the transport of spin-angular\nmomentum1,2. The usage of antiferromagnetic materials\nhas gained a renewed interest due to their high potential\nfor practical applications. The most attractive proper-\nties of antiferromagnets (AFs) are the lack of stray \felds\nand the fast dynamics that can operate in the THz fre-\nquency range3. Those attributes have the potential to\ntackle current technological bottlenecks, like the absence\nof practical solutions to generate and detect electromag-\nnetic waves in the spectrum ranging from 0.3 THz to 30\nTHz (the terahertz gap)2. Nevertheless, the control and\naccess to the high-frequency response of AFs is challeng-\ning. New proposals circumvent one of these obstacles by\nmanipulating metallic AFs with charge currents, through\nthe so-called spin-orbit torques4{6. Antiferromagnetic in-\nsulators , however, o\u000ber a compelling alternative since the\nJoule heating caused by moving electrons is absent. In\nsuch systems, the study of transport instead focuses on\ntheir magnetic excitations.\nIn insulating AFs the spin-angular momentum is trans-\nferred by their quantized low-energy excitations, i.e.,\nmagnons. Since the AF in its groundstate is composed of\ntwo collinear magnetic sublattices, magnons carry oppo-\nsite spin angular momentum. The transport of magnons\nhas been experimentally achieved through the longitu-\ndinal spin Seebeck e\u000bect in AF jNM7{13(NM, normal\nmetal) and FMjAFjNM14{18(FM, ferromagnets) het-\nerostructures, in which magnons were driven by a ther-\nmal gradient across the AF. Alternatively, thermal injec-\ntion of magnons in AFs has been studied19,20by a spin\naccumulation at the contact with adjacent metals. Inaddition, it was shown that thermal magnon transport\ntakes place at zero spin bias when the sublattice symme-\ntry is broken at the interfaces, e.g., induced by interfacial\nmagnetically uncompensated AF order21. Complemen-\ntarily, coherent spin transport induced by spin accumu-\nlation has been earlier considered and predicted to result\nin spin super\ruidity22,23or Bose-Einstein condensates of\nmagnons24. Recently, it has been shown via non-local\nspin transport measurements that magnons remarkably\npropagate at long distances in insulating AFs; \u000b-Fe2O325,\nCr2O326, MnPS 327and also in NiO via spin-pumping ex-\nperiments with YIG14. Their exceptional transport prop-\nerties, as well as those reported in Refs.7{11, are governed\nby the spin conductance and spin Seebeck coe\u000ecients.\nRezendeetal:28discussed theoretically the spin Seebeck\ne\u000bect in AFs in contact with a normal metal. They ob-\ntain the Seebeck coe\u000ecient in terms of temperature and\nmagnetic \feld, \fnding a good qualitative agreement with\nmeasurements in MnF 2/Pt7. In addition, it was found\nthat magnon scattering processes a\u000bect signi\fcantly the\nspin Seebeck coe\u000ecient. Hitherto there has been no com-\nplete studies on the underlying mechanism for spin trans-\nport coe\u000ecients, e.g., their thickness-, temperature- and\n\feld-dependence, e\u000bects derived from magnon-magnon\ninteractions or when the sublattice symmetry is broken\nat the interfaces.\nIn this work, we describe spin transport though a left\nnormal-metal{insulating antiferromagnet{right normal-\nmetal (LNMjAFjRNM) heterostructure. As depicted in\nFig. 1, magnon transport is driven by either a temper-\nature gradient or spin biasing. We focus on the thick-\n\flm limit, where the thickness dof the AF is greater\nthan the internal equilibration length lfor the magnon\ngas. This limit implies a di\u000busive regime where magnonsarXiv:1909.05881v2  [cond-mat.mes-hall]  4 Feb 20202\nare in a local equilibrium described by a local tempera-\nture and chemical potential. This is in contrast to our\nearlier, stochastic treatment of thin-\flms ( d\u001cl) where\nspin waves do not establish a local equilibrium19. Specif-\nically, we study the spin transport by evaluating, via a\nphenomenological theory, the structural spin Seebeck co-\ne\u000ecientSand magnon conductance G. Furthermore,\nwe investigate their temperature- and magnetic-\feld de-\npendence by computing the interfacial conductance coef-\n\fcients in a microscopic model for the NM jAF interface\nand evaluating the various coe\u000ecients using a Boltzmann\napproach.\nAF LNM RNM µ\nnT(x)d\nˆxˆzjH\nFIG. 1. A normal-metal{insulating antiferromagnet{normal-\nmetal heterostructure. An external \feld His applied along\nthezdirection. A spatially dependent temperature T(x) and\na spin bias\u0016is considered. As a result, a magnon spin current\nj\rows through the AF of thickness d.\nThe paper is outlined as follows. In Sec. II, we intro-\nduce the microscopic Hamiltonian for the bulk AF and\nits interaction with the metallic contacts. In Sec. III,\nwe formulate the phenomenological spin di\u000busion model,\nincluding scattering between magnon branches, and ob-\ntain expressions for the structural Seebeck coe\u000ecient and\nmagnon conductance. In Sec. IV, we compute the coef-\n\fcients for interfacial magnon transport from the micro-\nscopic model for the contacts. Based on this result, we\nestimate bulk transport coe\u000ecients assuming the interac-\ntion parameters are \feld and temperature independent.\nWe conclude in Sec. VI with a discussion of our results.\nIn the appendices, we detail various technical aspects of\nthe calculations.\nII. MODEL\nWe begin by de\fning the microscopic model for the\nLNMjAFjRNM heterostructure. The total Hamiltonian\nis^H=^HAF+^HI+^He, where ^HAFdescribes the AF spin\nsystem while ^HIrepresents their interfacial contact with\nthe normal metals. The Hamiltonian ^Hedescribes the\nelectronic states at the left- and right-lead. The coupling\nwith LNM and RNM is modeled by a simple interfacialexchange Hamiltonian,\n^HI=\u0000Z\ndxX\niJi\u001ai(x)^si\u0001^S(x); (1)\nwhereJiis the exchange coupling between the electronic\nspin density ^S(x) and the localized spin operator ^siat\nsiteithat labels the lattice along the interface. Here\n\u001ai(x) is the density of the localized AF electron orbital\nrepresenting e\u000bective spin densities at the interface. We\nwill return to the study of ^HIin Sec. IV to determine\nthe contact spin conductance.\nThe AF spin Hamiltonian is introduced by labelling\neach square sublattice site by the position i. The nearest-\nneighbour Hamiltonian is\n^HAF=JX\nhiji^si\u0001^sj\u0000HX\ni^siz+\u0014\n2sX\ni\u0000^s2\nix+^s2\niy\u0001\n;(2)\nwith ^sithe spin operator at site i,J > 0 the antiferro-\nmagnetic exchange biasing, Hthe magnetic \feld, and \u0014\nthe uniaxial easy-axis anisotropy. We are interested in\nsmall spin \ructuations (magnons) around the collinear\nbipartite ground state. The latter is the relevant ground\nstate to expand around for magnetic \feds below the spin-\n\rop \feldHsf. Magnons are introduced by the Holstein-\nPrimako\u000b transformation29,\n^siz=s\u0000^ay\ni^ai;^si\u0000= ^ay\niq\n2s\u0000^ay\ni^ai; (3a)\n^siz=\u0000s+^by\ni^bi;^si\u0000=q\n2s\u0000^by\ni^bi^bi; (3b)\nand^si+=^sy\ni\u0000, when ibelongs to sublattice aandb, re-\nspectively. We expand the spin Hamiltonian, Eq. (2),\nin powers of magnon operators that includes magnon-\nmagnon interactions, up to the fourth order, ^HAF=\n^H(2)\nAF+^H(4)\nAF. To lowest order in s, excitations of ^H(2)\nAF\nare diagonalized through the Bogoliubov transformation\n(see Appendix A for de\fnition), by the operators ^ \u000bqand\n^\fqthat carry spin angular momentum + ~^zand\u0000~^zre-\nspectively,\n^H(2)\nAF=X\nqh\n\u000f\u000b(q)^\u000by\nq^\u000bq+\u000f\f(q)^\fy\nq^\fqi\n: (4)\nWe refer to the magnons described by the operator \u000b\n(\f) as\u000b-(\f)magnons, respectively. The dispersion re-\nlation is\u000f\u000b;\f(q) =\u0006H+q\n(6Js)2\u0000\n1\u0000\r2q\u0001\n+H2cin a\n3-dimensional lattice, where \u0006stands for the \u000b- and\f-\nmagnon branch, respectively. Here, H2\nc\u0011\u00142+ 2\u00146Jsis\nthe critical \feld corresponding to the spin-\rop transition,\nwhile\rq= (1=3)P3\nn=1cos(qna), where ais the lattice\nspacing. Magnon-magnon interactions are represented\nby the interacting Hamiltonian ^H(4)\nAF. In the diagonal\nbasis, the interacting Hamiltonian becomes a lengthy ex-\npression that is detailed in Eq. (A7) (Appendix A). It\nconsists of nine di\u000berent scattering processes among \u000b-\nand\f-magnons. Some of these processes allow for the\nexchange of population of \u000b- and\f-magnons, see Fig. 6.3\nIII. SPIN TRANSPORT:\nPHENOMENOLOGICAL THEORY\nWe now outline the phenomenological spin trans-\nport theory for magnons across the LNM jAFjRNM het-\nerostructure. In the subsections that follow, we esti-\nmate the structural spin Seebeck coe\u000ecient and struc-\ntural magnon conductances. The basic assumption is\nthat the equilibration length for magnon-magnon inter-\nactions is much shorter than the system length d, so that\nthe two magnon gases are parametrized by local chemical\npotentials\u0016\u000band\u0016\fand temperatures T\u000bandT\f. In\nkeeping with our treatment of ferromagnets30, we assume\nthat strong, inelastic spin-preserving processes \fx the lo-\ncal magnon temperatures to that of the local phonon\ntemperature. This assumption is reasonable since the\nrate at which magnon temperature equilibrates with the\nphonon bath is dominated by both magnon-conserving\nand magnon-nonconserving scattering processes31. Thus,\nthe magnon temperature reaches its equilibrium faster\nthan the magnon chemical potential. The local phonon\ntemperature, in turn, is assumed to be linear across the\nAF, and to be equal to the electronic temperatures in\neach of the metallic leads. Only the magnon chemical\npotentials\u0016\u000band\u0016\fare then left to be determined.\nWe then express phenomenologically the spin conser-\nvation laws in terms of the chemical potentials. Its mi-\ncroscopic derivation can be established from the Boltz-\nmann equation as is explained in Appendix B. De\fning\nthe magnon densities n\u000bandn\f, these read\n_n\u000b+r\u0001j\u000b=\u0000r\u000b\u0016\u000b\u0000g\u000b\u000b\u0016\u000b\u0000g\u000b\f\u0016\f; (5a)\n_n\f+r\u0001j\f=\u0000r\f\u0016\f\u0000g\f\u000b\u0016\u000b\u0000g\f\f\u0016\f: (5b)\nHere,ridescribes relaxation of spin into the lattice re-\nsulting from inelastic magnon-phonon interactions that\ndo not conserve magnon number. In addition, gijde-\nscribes inelastic spin-conserving processes that accounts\nfor, e.g., magnon-magnon and magnon-phonon scatter-\ning, where the total number of magnons n\u000b+n\fmay\nchange but the spin \u0018n\u000b\u0000n\fis constant. In what fol-\nlows, the coe\u000ecients gij, by assumption, have their ori-\ngin in the coupling between magnons. The currents of \u000b-\nand\f-magnons, denoted as j\u000bandj\f, are given by j\u000b=\n\u0000\u001b\u000br\u0016\u000b\u0000&\u000brTandj\f=\u0000\u001b\fr\u0016\f\u0000&\frT, where\u001b\u000b;\f\nand&\u000b;\fare the bulk magnon spin conductivities and\nSeebeck coe\u000ecients, respectively. In writing the particle\ncurrents in the form above, we have neglected magnon-\nmagnon drag, which stems from magnon-magnon inter-\nactions that transfer linear momentum from one magnon\nband to another in such a way that the total spin cur-\nrent is conserved. Such drag gives rise to cross-terms like\nj\u000b/r\u0016\f. We shall simply limit the discussion to the\nregime in which such momentum scattering in subdomi-\nnant to e.g. elastic disorder scattering. The bulk conti-\nnuity equations, Eqs. (5a) and (5b), are complemented\nby the boundary conditions at the NM jAF interfaces onthe spin currents j(s)\n\u000b=~j\u000bandj(s)\n\f=\u0000~j\f,\nx\u0001j(s)\n\u000b(x=\u0000d=2) =G\u000b[\u0016L\u0000\u0016\u000b(\u0000d=2)]; (6a)\nx\u0001j(s)\n\f(x=\u0000d=2) =G\f[\u0016L+\u0016\f(\u0000d=2)]; (6b)\nx\u0001j(s)\n\u000b(x=d=2) =\u0000G\u000b[\u0016R\u0000\u0016\u000b(d=2)]; (6c)\nx\u0001j(s)\n\f(x=d=2) =\u0000G\f[\u0016R+\u0016\f(d=2)]; (6d)\nwithxthe unit vector along x-axis and where we have\nchosen the left and right interfaces to correspond to the\nplanesx=\u0000d=2 andx=d=2. Inside the left and right\nnormal metals the respective spin accumulations, the dif-\nference between spin-up and spin-down chemical poten-\ntial, are\u0016Land\u0016R. HereG\u000b;\fare the contact magnon\nspin conductances of each interface. The contact Seebeck\ncoe\u000ecient does not appear, as we are assuming a contin-\nuous temperature pro\fle across the structure, i.e., there\nis no temperature di\u000berence between magnons at the in-\nterface and normal metal leads. For \fxed spin accumu-\nlations\u0016L=R, Eqs. (5a-6d) form a closed set of equations\nwith the parameters g,r\u000b;\f,&\u000b;\f,\u001b\u000b;\fandG\u000b;\fto be\nestimated from microscopic calculations (see Sec. IV).\nThe inelastic spin-conserving terms gijcan be signif-\nicantly simpli\fed by additional considerations. Impos-\ning spin conservation one \fnds that g\u000b\u000b=g\f\u000band\ng\u000b\f=g\f\f. This result is obtained from Eqs. (5a-6d) by\nequating _n\u000b\u0000_n\f= 0 in the absence of magnon currents\nand disregarding the relaxation term ri. In addition, we\ncan estimate the \feld- and temperature-dependence of\nthe coe\u000ecients g\u000b\u000bandg\f\f, in particular near the spin-\n\rop transition. For this purpose, we use Fermi's golden\nrule to calculate the transition rate of \u000b-magnons ( \f-\nmagnons), de\fned as \u0000 \u000b\f(\u0000\f\u000b), that represents the in-\nstantaneous leakage of magnons due to the conversion\nbetween\u000b- and\f-magnons. Among the di\u000berent scatter-\ning processes displayed in Fig. 6, few of them conserve\nthe number of \u000b- or\f-magnons and thus do not con-\ntribute to the transition rate. As detailed in Appendix\nB, we sum over all the scattering rates and \fnd that\n\u0000\u000b\f= \u0000\f\u000b, which derives as a consequence of conser-\nvation of spin-angular momentum. Moreover, and more\nimportantly, up to linear order in the chemical potential\n\u0000\u000b\f=\u0000g(\u0016\u000b+\u0016\f). Therefore, g\u000b\u000b=g\u000b\f\u0011g, mean-\ning that a single scattering rate describes the inelastic\nspin-conserving process. The coe\u000ecient gis expressed in\nterms of a complex integral, given in Eq. (B7), that can\nbe estimated in certain limits. In the high temperature\nregime, where the thermal energy is much higher than the\nmagnon gap, we obtain g=\u0000\n2\u0019N\n=~s2\u0001\n(kBT=Jsz )3\nwith \n a dimensionless integral de\fned in Appendix B.\nIn the steady state limit the magnon chemical poten-\ntials are described by Eqs. (5a) and (5b), and are of the\ngeneral form \u0016\u000b\u0018\u0016\f\u0018e\u0006x=\u0015. The collective spin decay\nlength\u0015admits two solutions,\n2\u0015\u00002\n1= \u0003\u00002\n\u000b+ \u0003\u00002\n\f\u0000q\n4\u0015\u00002\n\f\u0015\u00002\u000b+ (\u0003\u00002\u000b\u0000\u0003\u00002\n\f)2(7)\n2\u0015\u00002\n2= \u0003\u00002\n\u000b+ \u0003\u00002\n\f+q\n4\u0015\u00002\u000b\u0015\u00002\n\f+ (\u0003\u00002\u000b\u0000\u0003\u00002\n\f)2(8)4\nwhere\u0015\u00002\n\f=g=\u001b\f,\u0015\u00002\n\u000b=g=\u001b\u000b, \u0003\u00002\n\u000b= (g+r\u000b)=\u001b\u000band\n\u0003\u00002\n\f= (g+r\f)=\u001b\f. In the absence of magnetic \feld,\nthe magnon-bands are degenerate and therefore \u000band\f\nhave equal properties. Thus the collective spin di\u000busion\nlengths become \u0015\u00002\n1=r=\u001band\u0015\u00002\n2= (2g+r)=\u001bthat dif-\nfer due to the inelastic spin-conserving scattering ( \u0018g).\nIn the following sections we evaluate the structural spin\nSeebeck coe\u000ecient and structural magnon conductance.\nWe will consider separately two scenarios, a tempera-\nture gradient and spin bias across the LNM jAFjRNM\nheterostructure in Sec. III A and III B, respectively.\nA. Spin Seebeck E\u000bect\nLet us assume a linear temperature gradient, with no\nspin accumulation in the normal metals. We solve for\nthe spin current at the right interface, js=x\u0001j(s)\n\u000b(d=2)+\nx\u0001j(s)\n\f(d=2) in presence of the temperature gradient \u0001 T.\nThen, the spin current \rowing through the right interface\nis related to the thermal gradient by js=S\u0001T, where\nSis the structural spin Seebeck coe\u000ecient. The general\nsolution forSis found in Appendix C (Eq. C4). In what\nfollows we examine several regimes of interest.\nFirst, we consider the zero applied magnetic \feld case,\nbut allow for sublattice symmetry breaking at the nor-\nmal metal|magnet interfaces. Here, we have that the\ndispersion relations for the \u000b- and\f-magnons are identi-\ncal. Furthermore, the bulk transport properties becomes\nindependent of the magnon band, i.e., \u001b=\u001b\u000b=\u001b\fand\n&=&\u000b=&\f. In this limit one \fnds\nS=2\u001b&(G\f\u0000G\u000b)\n(G\u000b1G\f2+G\u000b2G\f1)d\u00151Coth\u0014d\n2\u00151\u0015\n; (9)\nwithGinthe e\u000bective conductances de\fned by Gin\u0011\nGi+ (\u001bi=\u0015n) Coth [d=2\u0015n] fori=\u000b;\fandn= 1;2. We\nsee thatSis proportional to the bulk spin Seebeck con-\nductivity&. In the absence of symmetry breaking at the\ninterfaces,G\u000b=G\f, the spin Seebeck e\u000bect vanishes as\nexpected. When there is no magnetic \feld, it is thus es-\nsential to have systems with uncompensated interfaces to\nget a \fnite Seebeck e\u000bect.\nIn order to understand the dependence of Eq. (9) on\nthe \flm thickness d, it is useful to distinguish two thick-\nness regimes. Let us \frst introduce a \\thin\" \flm regime,\nd\u001cdin\u0011\u0015nCoth\u00001(Gi\u0015n=\u001b) forn= 1;2 andi=\u000b;\f.\nIn this limitGni\u0019(\u001b=\u0015n) Coth [d=2\u0015n]. The spin See-\nbeck coe\u000ecient becomes,\nS\u0019\u00152(G\f\u0000G\u000b)\ndCoth [d=2\u00152]&\n\u001b; (10)\nwhich in the extreme thin \flm limit ( d\u001c\u00152), be-\ncomes independent of d,S!(G\f\u0000G\u000b)2&=\u001b. This\ncan be understood as the sum of two independent par-\nallel channels, each with e\u000bective conductances renor-\nmalized by the bulk transport parameters. When Gi>\u001b=\u0015n, we may also de\fne a \\thick\" regime ( d\u001ddin\u0011\n\u0015nCoth\u00001(Gi\u0015n=\u001b) for alli;n) in which the contact re-\nsistance dominates, i.e., Gin\u0019Gi(thick \flm). In this\ncase, one obtains,\nS\u0019Coth\u0014d\n2\u00151\u0015\u001b&\nd\u00151\u0010\nG\u00001\n\u000b\u0000G\u00001\n\f\u0011\n; (11)\nandS \u0018 (\u001b&=d\u0015 1)G\u00001\nTat long distances d\u001d\u00151. In\nthis case, the net interfacial conductance behaves as the\nsum of a series spin-channels, each with conductance G\u000b\nandG\f. Note that as the Seebeck coe\u000ecient is de\fned\nthrough the relation js=S\u0001T, the\u00181=d-dependence\nmeans that js/@xTis independent of d; a Seebeck\ne\u000bect can thus originate for a thick AF due to a di\u000berence\nbetween the impedances of the magnon-bands just at the\ninterface where the signal is measured.\nSecond, we consider e\u000bects of a \fnite applied magnetic\n\feld. In addition, we assume no symmetry-breaking at\ninterfaces,G\u000b=G\f=G. In the \\thick\" \flm regime, we\nobtainsS\u0019\u0000 (&\u000b\u0000&\f)=d, which is simply the bulk value\nof the Seebeck coe\u000ecient. However, allowing symmetry\nbreaking at the interfaces we can obtain Sin the weak\ncoupling regime, i.e., g\u001cr\u000b;r\f, corresponding to slow\nscattering between the magnon branches (compared to\nGilbert damping). Expanding the collective spin decay\nlength, Eqs. (7) and (8), to linear order in g=r\u000b;\f, we get\n\u00151\u0019p\n\u001b\f=r\f(1\u0000g=r\f) and\u00152\u0019p\n\u001b\u000b=r\u000b(1\u0000g=r\u000b).\nThis expansion lead to corrections in the structural See-\nbeck coe\u000ecient,S\u0019S(0)+O(g=r), where\nS(0)=S(0)\n\f+S(0)\n\u000b=G\f&\f\ndG(0)\n\f1\u0000G\u000b&\u000b\ndG(0)\n\u000b2; (12)\nwithG(0)\nnithe lowest order of the e\u000bective conductances.\nIt is interesting to note that Eq. (12) consists of two\ncompletely decoupled parallel channels. In the partic-\nular thick \flm regime ( d\u001ddin), it reduces to S(0)\u0019\n\u0000(&\u000b\u0000&\f)=d, which is consistent with the result obtained\nat \fnite \feld in the \\thick\" \flm regime and G\u000b=G\f.\nAlthough we allow for symmetry-breaking at the inter-\nface here, all of the interfacial properties are washed out\nin the thick \flm regime.\nLast, consider the regime in which interactions are\nstrong:g\u001dr\u000b;r\fandd\u001d\u00152. Naively, one might\nexpect interactions to greatly reduce the spin Seebeck ef-\nfect. In fact, one \fnds that all dependence on gdrops\nout:\nS=G\u000b+G\f\nd\"(\u001b\u000b\n\u001b\f)2&\f\u0000&\u000b\n(\u001b\u000b\n\u001b\f)2&\f+&\u000b#\n; (13)\nThus, even with strong interactions between magnon\nbands, the spin Seebeck e\u000bect becomes independent of\ngand nonzero. The e\u000bects of interband interactions are\nshown in Fig. (4a); while there is a slight suppression\nof the signal, the spin Seebeck e\u000bect is qualitatively un-\nchanged by large scattering.5\nB. Spin biasing\nAside from a temperature gradient, a spin current may\nbe generated by means of an electrically driven spin bi-\nasing across the spin (usually via the spin Hall e\u000bect in\na normal metal contact)25. To model this, we consider\nthe temperature constant throughout the structure, but\na spin accumulation \u0016=\u0016^zis applied at the left inter-\nface, giving rise to a spin current j=G\u0016\rowing out\nof the opposite interface, parametrized by the structural\nconductance coe\u000ecient G. The full steady-state solution\nto Eqs. (5a) and (5b) is given by Eq. (C7) in Appendix C.\nIn order to \fnd simple relations for the spin conductance,\nwe focus on three regimes.\nFirst, we consider the case of sublattice symmetry and\nzero magnetic \feld At the interfaces, this entails G\u000b=\nG\f=G. In the bulk, this implies that bulk magnon\nspin conductivities and Seebeck coe\u000ecients are identical\nfor each magnon branch. Here we \fnd that only one\ncollective spin decay length, \u0015r=p\n\u001b=r, plays a role in\ntransport. One obtains,\nG=2G2\u001b=\u0015r\n[\u001b2=\u00152r+G2] sinh\u0010\nd\n\u0015r\u0011\n+ 2(\u001b=\u0015r)Gcosh\u0010\nd\n\u0015r\u0011:\n(14)\n(Note that as the \feld - or symmetry-breaking at the in-\nterfaces - is turned on, the magnon-magnon interactions\nstart to play a role). In the thin \flm regime ( d\u001c\u0015r),\nG\u0019G, which is just the series conductance of two paral-\nlel channels, each with interfacial conductance G=2 (due\nto the two interfaces through which the spin current must\npass). In the opposite limit, d\u001d\u0015r, we \fnd\nG\u00194(\u001b=\u0015r)G2\n(\u001b=\u0015r+G)2e\u0000d=\u0015r; (15)\nexhibiting an exponential decay over the distance \u0015r.\nSecond, we consider the strongly interacting case where\ng\u001dr\u000b;r\fandd\u001d\u00152. Here, one \fnds that while the\nconductance generally depends on g, in this regime the\nconductance is \fnite and independent of g:\nG=GS+GB (16)\nwhere\nGS=\u0010\nG\u000b\u001b2\n\f+\u001b2\n\u000bG\f\u00112\n(\u001b\u000b+\u001b\f)=\u0015r\nsinh\u0010\nd\n\u0015r\u0011Q\n\u0011=\u0006\u0010\nG(\u0011)\n\u000br\u001b2\n\f+\u001b2\u000bG(\u0011)\n\fr\u0011 (17)\nreduces to Eq. (14) at zero \feld, while\nGB=1\n2\u0000\n\u001b2\n\f\u0000\u001b2\n\u000b\u0001X\n\u0011=\u0006\u0011G(\u0011)\n\frG\u000b\u0000G(\u0011)\n\u000brG\f\nG(\u0011)\n\u000br\u001b2\n\f+\u001b2\u000bG(\u0011)\n\fr(18)\nis nonzero only when the magnetic \feld is applied; here\nG(\u0000)\nir\u0011Gi+ (\u001bi=\u0015r) Tanh [d=2\u0015r] whileG(+)\nir\u0011Gi+(\u001bi=\u0015r) Coth [d=2\u0015r]; the decay length \u0015ris given by\nthe limit of \u00151in the large glimit, yielding \u0015\u00002\nr=\n(r\u000b=\u001b\u000b+r\f=\u001b\f)=2. Thus, we \fnd that strong interactions\ndo not radically alter the structural spin conductance in\nthe sense that the spin conductance neither vanishes or\ndiverges in this regime. When d\u001d\u0015r, Eq. (16) simpli\fes\nto:\nG= 2(\u001b\u000b+\u001b\f)\n\u0015r\u0010\nG\u000b\u001b2\n\f+\u001b2\n\u000bG\f\u00112\n\u0010\nG\u000br\u001b2\n\f+\u001b2\u000bG\fr\u00112e\u0000d=\u0015r: (19)\nThus we \fnd that for large inter-band scattering, the\nnonlocal signal does not depend on gbut only on the\ndecay processes (e.g. Gilbert damping) via ri.\nThird, we consider a \fnite magnetic \feld and the limit\nwhen magnons are non-interacting. In the zero coupling\nregime,g= 0, one \fnds that the structural conductance\nis the sum of the parallel channels, G=G\u000b+G\f. Here,\nGi=(\u001bi=\u0015ir)G2\ni\n[\u001b2\ni=\u00152\nir+G2\ni] sinh\u0010\nd\n\u0015ir\u0011\n+ 2(\u001bi=\u0015ir)Gicosh\u0010\nd\n\u0015ir\u0011:\n(20)\nwhere\u0015\u00002\nir=ri=\u001biis determined by decay processes. For\nd\u001d\u0015ir, we \fnd that\nGi=2(\u001bi=\u0015ir)G2\ni\n((\u001bi=\u0015ir) +Gi)2e\u0000d=\u0015ir(21)\nwhich shows an exponential decay over distance. When\n\u000b- and\f-magnons are identical at the bulk and inter-\nfaces, both Eqs. (21) and (19) reduce to Eq. (14).\nIn the following sections we calculate and estimate the\nvarious parameters that enter into the phenomenological\ntheory above.\nIV. TRANSPORT COEFFICIENTS:\nMICROSCOPIC THEORY\nIn this section, we compute the interfacial spin con-\nductances from a microscopic model for the interface. In\naddition, the bulk magnon conductance, as well as the\nbulk Seebeck coe\u000ecient, are obtained in linear response\nfrom transport kinetic theory. Based on these results the\nstructural Seebeck coe\u000ecient is evaluated and plotted in\nFigs. 4.\nA. Contact magnon spin conductance\nIn this section, we compute interface transport co-\ne\u000ecients appropriate to our bulk drift-di\u000busion theory\nabove, allowing for the boundary conditions, Eqs (6a) to\nbe computed.\nLet us suppose that the spin degrees of freedom of the\nAF are coupled to those of the normal metals by the\nexchange Hamiltonian (1). Here it is understood that i6\nlabels the lattice along the interface (see Fig. 2). Speci\f-\ncally, the lattice is the set of vectors R2=fna^z+ma^yg.\nThe integers nandmare such that icorresponds to\na- andb-atoms when n+mare even and odd, respec-\ntively. In this model, we assume that aandbatoms\nare evenly spaced, which is not essential in what fol-\nlows. Besides, the itinerant electronic density, corre-\nsponding to evanescent modes in the x-direction, de-\ncays over an atomistic distance inside the AF. The spin\ndensity of itinerant electrons in the normal metal is\n^S(x) = ( ~=2)P\n\u001b\u001b0^\ty\n\u001b(x)\u001c\u001b\u001b0^\t\u001b0(x), where ^\t\u001b(x) is\nthe electron operator and \u001cthe Pauli matrix vector. The\nexchange coupling Ji=Ja, ifi2a, andJi=Jb, ifi2b,\nwhile the local spin density at each lattice site iis mod-\nulated by the function \u001ai(x) =j\u001ei(x)j2, with\u001eithe lo-\ncalized orbital at position i. Note that in general the\norbitals for the aandbsublattices may be di\u000berent.\nPrinted by Mathematica for Students\n\t\t\nPrinted by Mathematica for Students\n\t\taJa⇢a(x?)Jb⇢b(x?)(uncompensated) (compensated) \nFIG. 2. E\u000bective spin densities of AF jNM interface as experi-\nenced by normal metal electrons scattering o\u000b of the interface,\nfor the compensated and uncompensated cases.\nBased on the model represented by the contact Hamil-\ntonian (1), we wish to obtain the magnonic spin current\nacross the interface using Fermi's Golden Rule. We ex-\npand ^HIin terms of magnon operators up to order ni=s,\nobtaining ^HI=^H(k)\nI+^H(sf)\nI. The \frst term is the coher-\nent Hamiltonian ^H(k)\nI=P\nkk0Ukk0\u0010\n^cy\nk\"^ck0\"\u0000^cy\nk#^ck0#\u0011\n,\nwith ^ck\u001b\u0010\n^cy\nk\u001b\u0011\nthe fermionic operator that annihilate\n(create) and electron with spin- \u001band momentum k.\nThe term ^H(k)\nIgives rise to coherent spin torques, and\nmagnonic corrections to it, \u0018n\u0002\u0016. Since we are assum-\ning a \fxed order parameter nand focus only on thermal\nmagnon spin currents, we need not consider this term.\nThe second contribution, ^H(sf)\nI, is the spin-\rip Hamilto-\nnian that describes processes in which both branches of\nmagnons are annihilated and created at the interface by\nspin-\rip scattering of electrons. This term reads,\n^H(sf)\nI=X\nqkk0\u0010\nV\u000b\nqkk0^\u000by\nq^cy\nk#^ck0\"+V\f\nqkk0^\fy\nq^cy\nk\"^ck0#\u0011\n+h:c:;\n(22)where the matrix elements are\nV\u000b\nqkk0\u0011\u0000r\n8S\nNZ\ndx\t\u0003\nk(x) \tk0(x)\n\u0002(\u001a\u0003\na(q;x)Jacosh\u0012q\u0000\u001a\u0003\nb(q;x)Jbsinh\u0012q) (23)\nand\nV\f\nqkk0\u0011\u0000r\n8S\nNZ\ndx\t\u0003\nk(x) \tk0(x)\n\u0002(\u001ab(q;x)Jbcosh\u0012q\u0000\u001aa(q;x)Jasinh\u0012q):(24)\nHere, the function \t k(x) represent the eigenstates of the\nnonmagnetic Hamiltonian. Speci\fcally, in the yz direc-\ntions, the wavefunction is a delocalized Bloch state of\nthe interfacial lattice, which we assume here for simplic-\nity to be common to the both the metal and insulators\n(as is common in such heterostuctures); in the x direc-\ntion, the state is an evanescent mode on the insulator\nside of the interface, and a Bloch-like state of the metal-\nlic lattice on the other, which reduces to the usual 3D\nmetallic Bloch state far inside the metal. The quantities\n\u001aaand\u001abare de\fned by \u001aa(q;x) =P\ni2a\u001ai(x)eiq\u0001iand\n\u001ab(q;x) =P\ni2b\u001ai(x)eiq\u0001i, with\u001ai(x) =j\u001ei(x)j2as the\ndensity of the localized AF electron orbital at site i. The\nquantities cosh \u0012qand sinh\u0012qoriginate from the Bogoli-\nubov transformation that diagonalizes the noninteracting\nmagnon Hamiltonian32.\nIt is instructive to consider the simplest case of in-\nterfacial spin transport. This occurs when the inter-\nface is fully compensated, i.e., \u001ai2a(x) =\u001ai2b(x+i))\nandJa=Jb, see right side of Fig. 2. Because the\nnormal metal electronic states \t k(x) are Bloch states\nof the interfacial nonmagnetic Hamiltonian, then trans-\nlation by the lattice spacing ain they(orz) direc-\ntion transform, \t k!eikya\tk. Using\u001aa(b)(q;x) =\neiqya\u001ab(a)(q;x+a^y) =ei2qya\u001aa(b)(q;x+ 2a^y), it follows\nthatV\u000b\nqkk0=ei2a(q+k\u0000k0)yV\u000b\nqkk0. Applying translational\ninvariance on the full Hamiltonian, under x!x+a^y,\none has that this is independent of q+k\u0000k0, and it\nfollows that V\u000b\nqkk0=ei\u001e\u0010\nV\f\nqk0k\u0011\u0003\n, with the phase factor\nde\fned by \u001e=a\u0000\nqy+ky\u0000k0\ny\u0001\n. Since all of the inter-\nfacial transport coe\u000ecients are proportional to jVj2, we\nestablish that they become identical for both magnonic\nbranches, at zero \feld, for the case of fully compensated\ninterface.\nIt is also interesting to note the role played by Umk-\nlapp scattering processes at the interface. Suppose again\na fully compensated interface. Then, in the small q\nlimit, one \fnds cosh \u0012q\u0019sinh\u0012q, and the matrix ele-\nmentsV\u000b\nqkk0andV\f\nqkk0vanish when ( q\u0000k+k0)?= 0\n(specular scattering of electrons), where the subscript\n\\?\" designates the in-plane components. However, when\n(q\u0000k+k0)?=Gmn, where Gmn=n(\u0019=a)^y+m(\u0019=a)^zis\nthe reciprocal lattice vector, the matrix elements do not\nvanish for odd values of m+n, and transport for each7\nspecies becomes possible. We therefore expect Umklapp\nscattering processes to play a crucial role in the low tem-\nperature behavior of the magnon conductance, as well as\nother interfacial linear transport coe\u000ecients. This is con-\nsistent with Takei etal:21,22, where Umklapp scattering is\nfound to be responsible for a \fnite spin-mixing conduc-\ntance, describing coherent spin torques, at an AF jNM\ninterface. Umklapp processes, however, may only hap-\npen when part of the yzcross section of the normal metal\njkj= 2kFsurface lies outside the magnetic Brillioun zone\nof the lattice interface, for instance in a spherical Fermi\nsurface this conditions becomes 2 kF>\u0019= a.\nWe return to the general case in order to obtain the\ncontact magnon conductances G\u000b;\f. This can be done\nby a straightforward application of Fermi's Golden rule\nto calculate the magnonic spin current \rowing across the\ninterface. The magnon current is expressed as30\nji= 2D2\nFZ\nd\u000fg(i)\n\u000fjVi(\u000f)j2(\u000f\u0000\u00160\ni) [fim(\u000f)\u0000fie(\u000f)];(25)\nand therefore, the magnon spin current through the in-\nterface becomes js=~(j\u000b\u0000j\f). Here we have de\fned\nfori=\u000b;\f,\njVi(\u000f)j2=\u0019dAF\nD2\nFX\nqkk01\ng(i)\n\u000f\f\fVi\nqkk0\f\f2\u000e(\u000fk\u0000\u000fF)\n\u0002\u000e(\u000fk0\u0000\u000fF)\u000e(\u000f\u0000\u000fq);(26)\nwithDFas the normal metal density of states and g(i)\n\u000f\nthei-magnon density of states. In Eq. (25) \u00160\n\u000b;\f=\u0007\u0016,\nwhere we recall that \u0016is the spin accumulation. The\nBose-Einstein distribution for the i-magnons is fim(\u000f) =\n1=[e(\u000f\u0000\u0016i)=kBTi\u00001], andfie(\u000f) = 1=[e(\u000f\u0000\u00160\ni)=kBTe\u00001] cor-\nresponds to the e\u000bective electron-hole-excitation density\nexperienced by the i-magnons.\nIn a simple model, we may take the atomic\ndensities for both sublattices as \u001ai(x) =\u000e(x\u0000\nri) and the normal metal wavefunctions \t k(x) =\neik?\u0001xFk(x)=pVNM. Here the function Fk(x) de-\nscribes the decay within the AF and VNM the nor-\nmal metal volume. Then, de\fning the spin-mixing con-\nductanceg\"#\ni\u001116\u0019NdAFVAF(s\u0011DFJi=VNM)2, where\n\u0011=R\ndxFk0(x)F\u0003\nk(x), which we take to be momentum\nindependent for simplicity, one may write the i-magnon\nspin current ~jiin Eq. (25) as\n~ji=1\n16\u0019\u0012\ng\"#\na\u0018(i)\naa+g\"#\nb\u0018(i)\nbb+ 2q\ng\"#\nag\"#\nb\u0018(i)\nab\u0013\n;(27)\nwhere the functions \u0018(i)\nll0, which carry units of energy, are\ngiven by\n\u0018(i)\nll0=1\nD2\nF(sVAF)X\nq;k;k0X\nm;nF(ll0)\nmnq(\u000fq\u0000\u0016i) (nim\u0000nie)\n\u0002\u000e(\u000fF\u0000\u000fk)\u000e(\u000fF\u0000\u000fk0)\u000ek0\u0000k\u0000q;Gmn;(28)\nwithF(aa)\nmnq= cosh2\u0012q,F(bb)\nmnq= sinh2\u0012qandF(ab)\nmnq=\nF(ba)\nmnq= (\u00001)m+n+1cosh\u0012qsinh\u0012q. The motivation for\n1246810J_bJ_a51015GIl2MG_bêG_a\nPrinted by Mathematica for StudentsG↵/G\u000010\t15\t5\t1\t2\t4\t6\t8\t10\t24681002004006008001000\nPrinted by Mathematica for StudentsTHc1\t\nwith Umklappwithout UmklappJa/JbG↵/g\"#↵FIG. 3. Ratio of interfacial conductances of the two magnon\nbranches\u000band\fat zero \feld for di\u000berent ratios of interfacial\nsublattice exchange constants. Sublattice symmetry breaking\n(Ja6=Jb) is necessary to obtain a structural spin Seebeck\ne\u000bect in the absence of magnetic \felds (see Eq. (9)). Inset:\ntemperature dependence of G\u000bincluding and excluding Umk-\nlapp scattering ( m6= 0 and/or n6= 0 in Eq. (28)). All curves\nare obtained for kF= 4=a, and 6Js2= 2Hc.\nexpressing the spin current in the form of Eq. (27) is\nthat in the case of particle-hole symmetry at the inter-\nface, ~ji= (g\"#=4\u0019)\u0018\u0018(g\"#=4\u0019)(~!\u0000\u0016i)(ni=s). In\nparticular, the m=n= 0 term in Eq. (28) gives spec-\ntral scattering processes, while all others ( m6=n6= 0)\ncorrespond to Umklapp scattering.\nThe contact spin conductances G\u000bandG\fare ob-\ntained by the linearization of the i-magnon current given\nby Eq. (27), i.e., Gi= (@ji=@\u0016i)j\u0016i=0. The ratio of in-\nterfacial conductances of the two magnon branches \u000band\n\fis shown in Fig. 3 at zero \feld and as a function of\nratios of interfacial sublattice exchange constants Ja=Jb.\nThe ratioG\u000b=G\freaches a maximum value to later sat-\nurates when Ja=Jbis increased. In particular, we note\nthatG\u000b=G\fwhen the interfacial exchange constants\nare equal. Thus, the breaking of sublattice symmetry\n(Ja6=Jb) is necessary in realizing a structural spin See-\nbeck e\u000bect in the absence of a \feld, as is seen from Eq.\n(9). In the inset of Fig. 3 we display the temperature\ndependence of G\u000b. In this plot we have included (solid\nline) and excluded (dashed line) Umklapp scattering.\nB. Bulk magnon conductances and spin Seebeck\ncoe\u000ecients\nIn this section, we evaluate the bulk magnon con-\nductances\u001b\u000b;\fand bulk spin Seebeck coe\u000ecients &\u000b;\f.\nThese are obtained from standard kinetic theory of trans-\nport. Unlike previous works28,33,34, here we consider the\nmagnonic transport driven, in addition to thermal gra-\ndients, by spin biasing. The generic expressions for the8\nFIG. 4. (a) Structural Seebeck coe\u000ecient Sand (b) struc-\ntural spin conductance Gas functions of temperature T=Hc\nfor a \feldH= 0:2Hc. The temperature dependence of the\ninter-magnon scattering is given by g= \n(T=Tc)3(see Ap-\npendix B). Shown for both plots are \n = 0 ;1;10;103;105,\ncorresponding to a shift from blue to red coloring. While\nincreased scattering slightly diminishes the SSE, it has no\ndiscernible e\u000bect on the spin conductance for these particular\nparameters. For these plots, the parameters g\"#\na= 1=100a2\n(which for a= 1\u0017A corresponds to g\"#\u00181=nm2),kF= 1=a,\n6Js2= 2Hc,\u000b= 10\u00003andd= 100 awere used.\n0.20.40.60.81.00.00.51.01.52.0\nPrinted by Mathematica for StudentsH/HcH/HcG\n0.20.40.60.81.02468101214\nPrinted by Mathematica for Students\u0000r\nH/Hc\nFIG. 5. Main \fgure: behavior of conductance Gnear spin \rop.\nWhile the spin di\u000busion length \u0015rdiverges asjHj!Hc, the\nconductanceG, though sharply increasing, does not actually\ndiverge because of bottlenecking by the interface impedances;\nfor noninteracting magnons it has a maximium value of\nmax(G\u000b=2;G\f=2) (see Eq. (34)). The colors and parameters\nare identical to those shown in Fig. 4.\nmagnon current in the bulk are,\nji=\u0000Zdq\n(2\u0019)3\u001civ2\ni@fi\n@x; (29)\nwhere the integration is over the Brillouin zone and i=\n\u000b;\f. The magnon relaxation time is \u001ciand the magnon\ngroup velocity along the xdirection is vi=@\u000fi=~@kx.\nThe number of i-magnons with momentum qis denoted\nbyfiand given by the Bose-Einstein distribution func-tion. This yields the transport coe\u000ecients,\n\u001bi=4~J4H2\nc\n9~2Zdq\n(2\u0019)3\u001cisin2(aqx)\r2\nq\n1 +~J2\u0000\n1\u0000\r2q\u0001\fe\f\u000fi\n(e\f\u000fi\u00001)2;(30)\n&i=4~J4H2\nc\n9~2Zdq\n(2\u0019)3\u001cisin2(aqx)\r2\nq\n1 +~J2\u0000\n1\u0000\r2q\u0001\u000fi\f2e\f\u000fi\n(e\f\u000fi\u00001)2;(31)\nwhere ~J\u00116Js2=Hcis roughly the N\u0013 eel temperature in\nunits ofHcand\f= 1=kBT. Similarly, we may ob-\ntain expressions for the damping rates rifrom _nij\u000b=\n(2\u0019)\u00003R\ndqni=\u001cigwith\u001cigthe Gilbert damping lifetime.\nFrom the above relation riis extracted and obeys\nri=2\u000b\n~Zdq\n(2\u0019)3\f\u000f2\nie\f\u000fi\n(e\f\u000fi\u00001)2; (32)\nand\u000bis being the Gilbert damping constant.\nThe momentum relaxation rate, entering in the trans-\nport coe\u000ecients obtained in Eqs. (30) and (31), has\ncontributions from di\u000berent sources; Gilbert damping,\ndisorder scattering, magnon-phonon scattering and \u000b-\nmagnon{\f-magnon scattering. For simplicity, we con-\nsider the regime in which Gilbert damping dominates\ntransport:\n~\u001c\u00001\ni\u0019~\u001c\u00001\nig; (33)\nwhere ~\u001c\u00001\nig= 2\u000b\u000fi=Hc. Note that tilde represents units\nofHc.\nIn Figs. 4 (a) and (b) we show the temperature-\ndependence of the spin Seebeck coe\u000ecient and struc-\ntural spin condutance, respectively using the interface\nand bulk transport coe\u000ecients above. The interaction\nparameter ggrows with temperature (see Appendix B\nand C). As shown in Fig. 4 (a), however, the e\u000bects of g\nare minimal, suppressing the spin Seebeck signal slightly\nand negligibly a\u000becting the structural conductance.\nV. SPIN-FLOP TRANSITION\nThe spin-\rop transition occurs as jHj!Hcfrom be-\nlow. Here, the magnon spectrum becomes gapless and\nquadratic at low energies for one of the two magnon\nbands (say, the \f-band, for purposes of discussion).\nWhen Gilbert damping dominates the transport time\n(Eq. (33)), the bulk conductance \u001b\fin Eq. (30) demon-\nstrates an infrared divergence, while &\f,r\fandG\fare\n\fnite. It is straightforward to show that the Seebeck coef-\n\fcient, Eq. (C4), does not diverge in this case, consistent\nwith19.\nIn contrast to19, however, the structural conductance\nGdoes not diverge in the di\u000busive regime. Here, it is\ninstructive to consider the noninteracting case, Eq. (20),\nwhich reduces to\nG=G2\n\f\n(\u001b\f=\u0015\fr)(d=\u0015\fr) + 2G\f; (34)9\nwhich shows an algebraic , rather than exponential, de-\ncay in \flm thickness, due to a diverging decay length \u0015\fr\n(\u0018p\u001b\f). For a thin \flm, this becomes G=G\f=2; while\nthe AF bulk shows zero spin resistivity ( \u001b\u00001\n\f= 0) due\nto the Bose-Einstein divergence at low energies, struc-\ntural transport is bottlenecked by the interface resistance\nG\u00001\n\f, which is only well de\fned in the di\u000busive regime.\n(The e\u000bect is similar to a superconducting circuit, which\nwith perfectly conducting components, shows a \fnite re-\nsistance due to normal metal contacts.) While the signal\ndoes not diverge, there is a clear enhancement due to\nthe diminished spin resistivity, as well as long-distance\ntransport (algebraic in d), manifesting as a peak in the\nsignal near the spin-\rop transition25(see Fig. 5). A full\ncalculation for nonlocal spin injection - including spin\nHall/inverse spin Hall e\u000bects absent here - would show\nadditional impedances to spin \row due to spin resistance\nin the normal metal injector and detector.\nVI. CONCLUSION AND DISCUSSION\nIn summary, we have presented a study of spin trans-\nport of magnons in insulating AFs in contact with nor-\nmal metals. We focus on the thick-\flm limit, wherein\na di\u000busive regime can be assumed and magnons are in\na local thermodynamical equilibrium. The excitation\nof magnon currents is considered in linear response and\ndriven by either a temperature gradient and/or spin bi-\nasing. The spin transport is studied by evaluating the\nstructural magnon conductance and spin Seebeck e\u000bect\nwithin a phenomenological spin-di\u000busion transport the-\nory. These parameters were calculated in terms of bulk\ntransport coe\u000ecients as well as contact magnon conduc-\ntances. While the former were computed through kinetic\ntheory of transport, the latter are obtained from a mi-\ncroscopic model of the NM jAF interface. Furthermore,\nwe allowed for the breaking of sublattice symmetry at\nthe interface assuming an uncompensated magnetic or-\nder. In addition, the \feld- and temperature-dependence\nof the inter-magnon scattering rates, which redistribute\nangular momentum between the magnon branches, were\nestimated. We \fnd that the e\u000bects of inter-magnon scat-\ntering, which lock the two magnon bands together, isnegligible. Furthermore, we show that even as the bulk\nspin resistivity vanishes near the spin \rop transition, nor-\nmal metal|magnet interface spin impedance ultimately\nbottleneck transport, irrespective of interactions, in con-\ntrast to the stochastic theory19for thin \flms.\nThe phenomenological approach above ultimately\nbreaks down for strong interactions (which occur near\nthe spin-\rop transition), where the individual \u000band\f\nclouds are no longer internally equilibrated with well-\nde\fned chemical potentials an interactions. Instead, a\ntreatment of the interacting clouds (e.g. a kinetic theory\napproach) beyond the quasiequilibrium approach that is\nadopted here is needed. In addition, more sophisticated\ntreatments of the transport time \u001cihave been shown to\nmore realistically reproduce experimental results33; such\nquasi-empirical transport times could be incorporated di-\nrectly into our phenomenology. Most importantly, our\nsomewhat arti\fcial assumption that the magnetic \feld is\napplied along the easy-axis in not necessarily realized in\nexperiment. Instead, even simple bipartite AFs such as\nthose modeled by our phenomenology above show com-\nplex paramagnetic behavior in response to a \feld applied\nalong di\u000berent axes. In these scenarios, heterostuctures\nmay manifest both antiferromagnetic and ferromagnetic\ntransport behaviors25. Future work, such as the drift-\ndi\u000busion approach discussed above, is needed to fully\nunderstand such scenarios at a more fundamental level.\nACKNOWLEDGMENTS\nThis work was supported by the European Union's\nHorizon 2020 Research and Innovation Programme under\nGrant DLV-737038 \"TRANSPIRE\", the Research Coun-\ncil of Norway through is Centres of Excellence funding\nscheme, Project No. 262633, \"QuSpin\" and European\nResearch Council via Advanced Grant No. 669442 \"Insu-\nlatronics\". Also we acknowledge funding from the Sticht-\ning voor Fundamenteel Onderzoek der Materie (FOM)\nand the European Research Council via Consolidator\nGrant number 725509 SPINBEYOND.\nNote added{ During the submission of our work, we\nbecame aware of another related article35that considers\nmagnon transport in AFs.\nAppendix A: Magnon-magnon interactions\nWe start out by de\fning the AF Hamiltonian. Introducing a square lattice, labelling the sites in the lattice by i,\non sub-latticesAandB, the nearest-neighbor Hamiltonian reads\n^HAF=JX\nhi2A;j2Bi^si\u0001^sj\u0000HX\ni2A;B^siz\u0000\u0014\n2sX\ni2A;B^s2\niz; (A1)\nwhereJ >0 is the exchange coupling, Hthe magnetic \feld and \u0014>0 the uniaxial easy-axis anisotropy. Introducing\nthe Holstein-Primako\u000b transformation, assuming a bipartite ground state, the spin operators in the limit of small spin10\n\ructuations reads\n^sA\niz=s\u0000ay\niai; ^sB\niz=\u0000s+by\nibi; (A2a)\n^sA\ni+=p\n2sai\u00001p\n2say\niaiai; ^sB\ni+=p\n2sby\ni\u00001p\n2sby\niby\nibi; (A2b)\n^sA\ni\u0000=p\n2say\ni\u00001p\n2say\niay\niai ^sB\ni\u0000=p\n2sbi\u00001p\n2sby\nibibi: (A2c)\nThe AF Hamiltonian Eq. (A1) is expanded up to fourth order in the magnon operators, Fourier transformed\nthrough the relations ai=1p\nNP\nieik\u0001iakandbi=1p\nNP\nieik\u0001jbkand expressed as HAF=E0+H(2)\nAF+H(4)\nAFwhere\n^H(2)\nAF= (Jsz+\u0014)X\nqh\n(1 +h)ay\nqaq+ (1\u0000h)by\nqbq+\u0018\rq(aqb\u0000q+ay\nqby\n\u0000q)i\n(A3)\n^H(4)\nAF=\u0000Jz\n2NX\nq1q2q3q4\u000eq1+q2\u0000q3\u0000q4h\n2\rq2\u0000q4ay\nq1by\n\u0000q4aq3b\u0000q2+\u0014\n2s\u0000\nay\nq1ay\nq2aq3aq4+by\nq1by\nq2bq3bq4\u0001\n+\rq4\u0010\nby\nq1b\u0000q2bq3aq4+by\nq3by\n\u0000q2bq1ay\nq4+ay\nq1a\u0000q2aq3bq4+ay\nq3ay\n\u0000q2aq1by\nq4\u0011i\n(A4)\nwithh=H=(Jsz+\u0014),\u0018=Jsz= (Jsz+\u0014) and\rq=2\nzP\n\u000ecos [q\u0001\u000e] wherezis the coordination number. The quadratic\npart of the Hamiltonian, Eq. (A3), is diagonalized by the Bogoliubov transformation\n^aq=lq^\u000bq+mq^\fy\n\u0000q (A5)\n^by\n\u0000q=mq^\u000bq+lq^\fy\n\u0000q (A6)\nwith the coe\u000ecients lq=\u0010\n(Jsz+\u0014)+\u000fq\n2\u000fq\u00111=2\n,mq=\u0000\u0010\n(Jsz+\u0014)\u0000\u000fq\n2\u000fq\u00111=2\n\u0011\u0000\u001fqlqand\u000fq= (Jsz+\u0014)q\n1\u0000\u00182\r2q, resulting\nin Eq. (4). In the diagonal basis, the interacting Hamiltonian Eq. (A4) \fnally becomes\n^H(4)\nAF=X\nq1q2q3q4\u000eq1+q2\u0000q3\u0000q4h\nV(1)\nq1q2q3q4\u000by\nq1\u000by\nq2\u000bq3\u000bq4+V(2)\nq1q2q3q4\u000by\nq1\f\u0000q2\u000bq3\u000bq4+V(3)\nq1q2q3q4\u000by\nq1\u000by\nq2\u000bq3\fy\n\u0000q4\n+V(4)\nq1q2q3q4\u000by\nq1\f\u0000q2\u000bq3\fy\n\u0000q4+V(5)\nq1q2q3q4\f\u0000q1\f\u0000q2\u000bq3\fy\n\u0000q4+V(6)\nq1q2q3q4\u000by\nq1\f\u0000q2\fy\n\u0000q3\fy\n\u0000q4\n+V(7)\nq1q2q3q4\u000by\nq1\u000by\nq2\fy\n\u0000q3\fy\n\u0000q4+V(8)\nq1q2q3q4\f\u0000q1\f\u0000q2\u000bq3\u000bq4+V(9)\nq1q2q3q4\f\u0000q1\f\u0000q2\fy\n\u0000q3\fy\n\u0000q4i\n(A7)\nwhere the scattering amplitudes are V(a)\nq1q2q3q4=\u0000\u0000Jz\nN\u0001\nlq1lq2lq3lq4\b(a)\n1234. The functions \b(a)are the following\nexpressions11\n\b(1)\n1234 =\rq2\u0000q4\u001fq2\u001fq4\u00001\n2(\rq2\u001fq2+\rq4\u001fq4+\rq2\u001fq1\u001fq3\u001fq4+\rq4\u001fq1\u001fq2\u001fq3) +\u0014\n2Jzs(1 +\u001fq1\u001fq2\u001fq3\u001fq4) (A8)\n\b(2)\n1234 =\u0000\rq2\u0000q4\u001fq4\u0000\rq1\u0000q4\u001fq1\u001fq2\u001fq4+\rq4\u001fq1\u001fq3+\rq4\u001fq2\u001fq4+1\n2(\u001fq3\u001fq4(\rq1+\rq2\u001fq1\u001fq2) + (\rq2+\rq1\u001fq1\u001fq2))\n\u0000\u0014\nJzs(\u001fq2+\u001fq1\u001fq3\u001fq4) (A9)\n\b(3)\n1234 =\u0000\rq2\u0000q4\u001fq2\u0000\rq2\u0000q3\u001fq2\u001fq3\u001fq4+\rq2\u001fq1\u001fq3+\rq2\u001fq2\u001fq4+1\n2(\u001fq1\u001fq2(\rq3+\rq4\u001fq3\u001fq4) + (\rq4+\rq3\u001fq3\u001fq4))\n\u0000\u0014\nJzs(\u001fq4+\u001fq1\u001fq2\u001fq3) (A10)\n\b(4)\n1234 =\rq2\u0000q4+\rq1\u0000q4\u001fq1\u001fq2+\rq2\u0000q3\u001fq3\u001fq4+\rq1\u0000q3\u001fq1\u001fq2\u001fq3\u001fq4+2\u0014\nJzs(\u001fq2\u001fq4+\u001fq1\u001fq3)\n\u0000(\u001fq1(\rq3+\rq4\u001fq3\u001fq4) +\u001fq3(\rq1+\rq2\u001fq1\u001fq4) +\u001fq4(\rq2+\rq1\u001fq1\u001fq2) +\u001fq2(\rq4+\rq3\u001fq3\u001fq4)) (A11)\n\b(5)\n1234 =\u0000\rq2\u0000q4\u001fq1\u0000\rq2\u0000q3\u001fq1\u001fq3\u001fq4+\rq2\u001fq2\u001fq3+\rq2\u001fq1\u001fq4+1\n2((\rq3+\rq4\u001fq3\u001fq4) +\u001fq1\u001fq2(\rq4+\rq3\u001fq3\u001fq4))\n\u0000\u0014\nJzs(\u001fq3+\u001fq1\u001fq2\u001fq4) (A12)\n\b(6)\n1234 =\u0000\rq2\u0000q4\u001fq3\u0000\rq1\u0000q4\u001fq1\u001fq2\u001fq3+\rq4\u001fq1\u001fq4+\rq4\u001fq2\u001fq3+1\n2((\rq1+\rq2\u001fq1\u001fq2) +\u001fq3\u001fq4(\rq2+\rq1\u001fq1\u001fq2))\n\u0000\u0014\nJzs(\u001fq1+\u001fq2\u001fq3\u001fq4) (A13)\n\b(7)\n1234 =\rq2\u0000q4\u001fq2\u001fq3\u00001\n2(\rq2\u001fq1+\rq4\u001fq3+\rq4\u001fq1\u001fq2\u001fq4+\rq2\u001fq2\u001fq3\u001fq4) +\u0014\n2Jzs(\u001fq3\u001fq4+\u001fq1\u001fq2) (A14)\n\b(8)\n1234 =\rq2\u0000q4\u001fq1\u001fq4\u00001\n2(\rq4\u001fq3+\rq2\u001fq1+\rq2\u001fq2\u001fq3\u001fq4+\rq4\u001fq1\u001fq2\u001fq4) +\u0014\n2Jzs(\u001fq1\u001fq2+\u001fq3\u001fq4) (A15)\n\b(9)\n1234 =\rq2\u0000q4\u001fq1\u001fq3\u00001\n2(\rq4\u001fq4+\rq2\u001fq2+\rq2\u001fq1\u001fq3\u001fq4+\rq4\u001fq1\u001fq2\u001fq3) +\u0014\n2Jzs(1 +\u001fq1\u001fq2\u001fq3\u001fq4) (A16)\nwhere\u001fq=\u0000\u0010\n1\u0000\u000fq\n1+\u000fq\u00111=2\n. Note the symmetry relations among these coe\u000ecients \b(3)\n1234 = \b(2)\n3412, \b(6)\n1234 = \b(5)\n3412\nand \b(8)\n1234 = \b(7)\n3412. The form of these expressions di\u000ber from Ref.36, where a Dyson-Maleev transformation was\nconsidered.\nAppendix B: Scattering Lengths\nIn this section, we compute the \feld and temperature dependences of g\u000b\u000bandg\f\fthrough the Fermi's golden\nrule. To start with, we introduce the Boltzmann equation for the distribution of \u000b- and\f-magnons,f\u000b(x;q;t) and\nf\f(x;q;t) respectively,\n@f\u000b\n@t+@f\u000b\n@x\u0001@!\u000b\nq\n@q= \u0000\u000b[q] + \u0000\u000b\f[q]; (B1)\n@f\f\n@t+@f\f\n@x\u0001@!\f\nq\n@q= \u0000\f[q] + \u0000\f\u000b[q]; (B2)\nwhere\u000f\u000b\nq=~!\u000b\nq. The right-hand side are the total net rates of scattering into and out of a magnon state with\nwave vector q. The magnon spin di\u000busion equations [Eqs. (5a) and (5b)] are obtained by linearizing the Boltzmann\nequations in terms of the small perturbations, e.g. the chemical potential. This is achieved, in addition, by integrating\nEqs. (B1) and (B2) over all possible wave vectors q.\nThe terms \u0000\u000band \u0000\foriginate from multiple e\u000bects such as, magnon-phonon collisions, elastic magnon scattering\nwith defects, and magnon number and energy-conserving intraband magnon-magnon interaction. It is worth to\nmention that the estimation of each of those contribution, as was done in Ref.31for ferromagnets, is out of the\nscope of our work. However, we implement the basic assumption that the equilibration length for magnon-magnon\ninteractions is much shorter than the system size, so that the two magnon gases are parametrized by local chemical\npotentials\u0016\u000band\u0016\fand temperatures T\u000bandT\f. Moreover, as was pointed out in Sec. IV, the magnon relaxation\ninto the phonon bath is parametrized by the Gilbert damping.12\nNow we focus on the magnon-magnon collisions described by \u0000\u000b\fand \u0000\f\u000b. These terms describe interband\ninteraction between magnons that exchange the population of di\u000berent magnon species. To calculate \u0000\u000b\fand \u0000\f\u000b\nwe consider the interacting Hamiltonian given by Eq. (A7) that represents all scattering processes among \u000b- and\n\f-magnons (depicted in Fig. 6). We emphasize that those processes represented in Fig. 6(b), (d) and (e), do not\nconserve the number of \u000b-magnons or \f-magnons, even though the di\u000berence n\u000b\u0000n\fis constant due to conservation\nof spin-angular momentum. This inelastic spin-conserving processes contribute to the transfer of one magnon mode\ninto the other, and thus determining the coe\u000ecients gij. We quantify this e\u000bect evaluating perturbatively the rate of\nchange of magnons using Fermi's golden rule.\nFIG. 6. Diagrammatic representation for the scattering processes experienced by \u000b- and\f-magnons. In (a), (c) and (f) are\nrepresented the processes with scattering amplitude V(1),V(4)andV(9), respectively. In (b), (d) and (e) is shown those inelastic\nprocesses that do not conserve the number of magnons. These are scattered by the interacting potential with amplitude V(3),\nV(6)andV(7), respectively. Those processes with amplitude V(2),V(5)andV(8)are the hermitian conjugate of the above and\nthus are omitted.\nBased on time-dependent perturbation theory, the transition rate between an initial jiiand a \fnal statejfiis given\nby Fermi's Golden Rule, which reads \u0000 = (2 \u0019=~)P\ni;fWi\f\f\fhfj^Vjii\f\f\f2\n\u000e(\u000ff\u0000\u000fi). The sum runs over all possible initial\nand \fnal states, Wiis the Boltzmann weight that gives the probability of being in some initial state, ^Vis the matrix\nelement of the Hamiltonian corresponding to the interactions and the delta function ensures energy conservation.\nWe recognize that a \fnal state can be either any of those described in Eq. (A7). However, those processes that\nconserve the number of particles, i.e., \u000b-magnons and \f-magnons, have a null transition rate. Only the states described\nin Fig. 6(b), (d) and (e), will contribute to a \fnite transition rate. The net transition rate of scattering into and out\nof a magnon state with wave vector q, reads\n\u0000\u000b\f[q] =2\u0019\n~X\nq1q2q3\u001a\f\f\fV(3)\nqq1q2q3\f\f\f2h\u0000\n1 +f\u000b\nq\u0001\u0010\n1 +f\f\n\u0000q1\u0011\u0000\n1 +f\u000b\nq2\u0001\nf\u000b\nq3\u0000f\u000b\nqf\f\n\u0000q1f\u000b\nq2\u0000\n1 +f\u000b\nq3\u0001i\n\u0002\u000eq\u0000q1+q2\u0000q3\u000e\u0010\n\u000f\u000b\nq+\u000f\f\n\u0000q1+\u000f\u000b\nq2\u0000\u000f\u000b\nq3\u0011\n+\f\f\fV(6)\nqq1q2q3\f\f\f2h\u0000\n1 +f\u000b\nq\u0001\u0010\n1 +f\f\n\u0000q1\u0011\u0010\n1 +f\f\n\u0000q2\u0011\nf\f\n\u0000q3\u0000f\u000b\nqf\f\n\u0000q1f\f\n\u0000q2\u0010\n1 +f\f\n\u0000q3\u0011i\n\u0002\u000eq\u0000q1\u0000q2+q3\u000e\u0010\n\u000f\u000b\nq+\u000f\f\n\u0000q1+\u000f\f\n\u0000q2\u0000\u000f\f\n\u0000q3\u0011o\n(B3)\nand\n\u0000\f\u000b[q] =2\u0019\n~X\nq1q2q3\u001a\f\f\fV(3)\nqq1q2q3\f\f\f2h\u0010\n1 +f\f\n\u0000q\u0011\u0000\n1 +f\u000b\nq1\u0001\u0000\n1 +f\u000b\nq2\u0001\nf\u000b\nq3\u0000f\f\n\u0000qf\u000b\nq1f\u000b\nq2\u0000\n1 +f\u000b\nq3\u0001i\n\u0002\u000e\u0000q+q1+q2\u0000q3\u000e\u0010\n\u000f\f\n\u0000q+\u000f\u000b\nq1+\u000f\u000b\nq2\u0000\u000f\u000b\nq3\u0011\n+\f\f\fV(6)\nqq1q2q3\f\f\f2h\u0010\n1 +f\f\n\u0000q\u0011\u0000\n1 +f\u000b\nq1\u0001\u0010\n1 +f\f\n\u0000q2\u0011\nf\f\n\u0000q3\u0000f\f\n\u0000qf\u000b\nq1f\f\n\u0000q2\u0010\n1 +f\f\n\u0000q3\u0011i\n\u0002\u000e\u0000q+q1\u0000q2+q3\u000e\u0010\n\u000f\f\n\u0000q+\u000f\u000b\nq1+\u000f\f\n\u0000q2\u0000\u000f\f\n\u0000q3\u0011o\n: (B4)13\nWe note that processes described by Fig. 6(e) do not contribute to the change of magnon density by invoking\nconservation of energy. The total rates \u0000 \u000b\fand \u0000\f\u000b, obtained by summing up over all wave vectors q, are de\fned by\n\u0000\u000b\f=~X\nq\u0000\u000b\f[q]; (B5)\n\u0000\f\u000b=~X\nq\u0000\f\u000b[q]; (B6)\nwhich describes the net imbalance of the magnon densities n\u000bandn\fby the successive scatterings events between\nboth magnon modes. Next, we will show that Eqs. (B5) and (B6) scale linearly with the magnon chemical potentials\nwhen the magnon distribution are close to the equilibrium.\nIn order to calculate \u0000 \u000b\fand \u0000\f\u000bwe consider that magnons are near thermodynamic equilibrium. Thus,\ntheir distributions are parameterized by the Bose-Einstein distribution as f\u000b\nq=\u0010\ne(\u000f\u000b\nq\u0000\u0016\u000b)=kBT\u00001\u0011\u00001\nandf\f\nq=\n\u0010\ne(\u000f\f\nq\u0000\u0016\f)=kBT\u00001\u0011\u00001\n, whereTis the temperature of the phonon bath. At equilibrium the rates obey \u0000 \u000b\f= \u0000\f\u000b= 0,\nwhich is established when the chemical potentials satisfy \u0016\u000b+\u0016\f= 0. This can be clearly seen when the distribution\nis expanded up to linear order on \u0016\u000band\u0016\f. Using this expansion on Eqs. (B3) and (B4), is found that the total\ntransition rates \u0000 \u000b\fand \u0000\f\u000bbecome equals and proportional to the sum of the chemical potentials. Precisely, we\nobtain \u0000\u000b\f= \u0000\f\u000b=\u0000g(\u0016\u000b+\u0016\f) with the coe\u000ecient ggiven by\ng=2\u0019\n~kBTX\nqq1q2q3\u000eq\u0000q1+q2\u0000q3\u0014\f\f\fV(3)\nqq1q2q3\f\f\f2\nf\u000b;0\nqf\f;0\nq1f\u000b;0\nq2\u0000\n1 +f\u000b;0\nq3\u0001\n\u000e\u0000\n\u000f\u000b\nq+\u000f\f\nq1+\u000f\u000b\nq2\u0000\u000f\u000b\nq3\u0001\n+\f\f\fV(6)\nqq1q2q3\f\f\f2\nf\u000b;0\nqf\f;0\nq1\u0000\n1 +f\f;0\nq2\u0001\nf\f;0\nq3\u000e\u0000\n\u000f\u000b\nq+\u000f\f\nq1+\u000f\f\nq3\u0000\u000f\f\nq2\u0001\u0015\n;(B7)\nwheref\u000b;0andf\f;0denote the equilibrium distribution evaluated at the chemical potential \u0016e\n\u000b=\u0000\u0016e\n\f. Comparing\nwith the phenomenological Eqs. (5a) and (5b), we obtain g\u000b\u000b=g\u000b\f=g\f\f=g\f\u000b=g.\nDespite the complex expression for the factor g, it can be estimated in certain temperature regimes. For instance, at\nhigh temperatures the thermal energy is much higher than the magnon gap, therefore \u000f\u000b;\f(q)=kBT\u0019(Jsz=kBT)ajqj,\ni.e., the exchange energy is the only magnetic coupling that becomes relevant. Thus, at large temperatures we obtain\ng=2\u0019\n~N\ns2\u0012kBT\nJsz\u00133\n\n (B8)\nwhere \n is a dimensionless integral de\fned as\n\n =Zdp1\n(2\u0019)3dp2\n(2\u0019)3dp3\n(2\u0019)3dp4\n(2\u0019)3\u000e(p1+p2\u0000p3\u0000p4)\u0014\f\f\fv(3)\np1p2p3p4\f\f\f2\nf\u000b;0\npf\f;0\np1f\u000b;0\np2\u0010\n1 +f\u000b;0\np3\u0011\n\u000e(p + p1+ p2\u0000p3)\n+\f\f\fv(6)\np1p2p3p4\f\f\f2\nf\u000b;0\npf\f;0\np1\u0010\n1 +f\f;0\np2\u0011\nf\f;0\np3\u000e(p + p1+ p3\u0000p2)\u0015\n: (B9)\nTo obtain Eq. (B8) the continuum limit was taken by the replacementP\nq!V((Jsz=kBT)a)\u00003R\ndp=(2\u0019)3on\nEq. (B7), where the dimensionless wavevector p = ( Jsz=kBT)ajqjwas introduced. We notice that in the limit of\nvery large temperatures the Bose factors approach the Raleigh-Jeans distribution, i.e., f\u000b\nq\u0018f\f\nq\u0018kBT=(Jsz)ajqj,\nand \n becomes independent of temperature. The dimensionless scattering amplitudes v(i)\nq1q2q3q4=V(i)\nq1q2q3q4=v0, with\nv0= (Jsz)3=Ns(kBT)2, are evaluated and their asymptotic behaviour obeys v(3)\np1p2p3p4=v(6)\np1p2p3p4=\u00002v(7)\np1p2p3p4\nwith\nv(3)\np1p2p3p4\u0019\u00002\u00121\np1p2p3p4\u00131=2\n: (B10)\nAppendix C: Seebeck coe\u000ecient and spin conductance\nTo \fnd the structural spin Seebeck coe\u000ecient and spin conductance we \frst express the general solution for the\nmagnon chemical potential,\n\u0016\u000b(x) = (Asinh [x=\u0015 1] +Bcosh [x=\u0015 1]) + (Dsinh [x=\u0015 2] +Ecosh [x=\u0015 2]) (C1)\n\u0016\f(x) =C(Asinh [x=\u0015 1] +Bcosh [x=\u0015 1]) + (Dsinh [x=\u0015 2] +Ecosh [x=\u0015 2]) (C2)14\nwhereCis a constant that is obtained from the eigenvalue problem that determines \u00151and\u00152. From the boundary\nconditions, Eqs. (6a-6d), we \fnd the unknown coe\u000ecients A,B,DandE. The net spin current crossing the right\nlead is,\njs=j(s)\n\u000b(d=2) +j(s)\n\f(d=2); (C3)\nwherej(s)\n\u000b=~j\u000bandj(s)\n\f=\u0000~j\f. In the absence of a spin accumulation, \u0016L=\u0016R= 0, we calculate the magnon\ncurrentsj\u000bandj\fto obtainS=js=d. Thus, the structural spin Seebeck coe\u000ecient is:\nS=\u00172(G\u000b2G\f&\f\u0000G\f1G\u000b&\u000b) +\u00171(G\fG\u000b1&\f\u0000G\f2G\u000b&\u000b) + (G\f1\u0000G\f2)G\f&\u000b+\u00171\u00172(G\u000b2\u0000G\u000b1)G\u000b&\f\n(G\u000b1G\f2\u00171+G\u000b2G\f1\u00172)d(C4)\nwhich is written in terms of the e\u000bective conductances\nGin\u0011Gi+\u0012\u001bi\n\u0015n\u0013\nCoth\u0014d\n2\u0015n\u0015\n(C5)\nforn= 1;2 andi=\u000b;\f, and\n\u00171= (~g\f\u0000~g\u000b+ ~r\f\u0000~r\u000b+\u000e)=2~g\f\n\u00172= (~g\u000b\u0000~g\f+ ~r\u000b\u0000~r\f+\u000e)=2~g\f (C6)\nwhere\u000e=p\n4~g\f~g\u000b+ (~g\u000b\u0000~g\f+ ~r\u000b\u0000~r\f)2, and ~r\u000b=r\u000b=\u001b\u000b, and ~g\u000b=g=\u001b\u000b, with similar expressions for \f-\nparameters.\nThe structural conductance is obtained by following the same procedure as before. However, in this case, we assume\nthat\u0016L6=\u0016RandrT= 0. In the general case, Gis given by:\nG=\u00001\n2\u00151(\u001b\u000b\u00171+\u001b\f)G(+)\n\f2G\u000b+\u00172G(+)\n\u000b2G\f\n\u00171G(+)\n\u000b1G(+)\n\f2+\u00172G(+)\n\u000b2G(+)\n\f1Tanh\u0014d\n2\u00151\u0015\n+1\n2\u00152(\u001b\f\u0000\u001b\u000b\u00172)G(+)\n\f1G\u000b\u0000\u00171G(+)\n\u000b1G\f\n\u00171G(+)\n\u000b1G(+)\n\f2+\u00172G(+)\n\u000b2G(+)\n\f1Tanh\u0014d\n2\u00152\u0015\n+1\n2\u00151(\u001b\u000b\u00171+\u001b\f)G(\u0000)\n\f2G\u000b+\u00172G(\u0000)\n\u000b2G\f\n\u00171G(\u0000)\n\u000b1G(\u0000)\n\f2+\u00172G(\u0000)\n\u000b2G(\u0000)\n\f1Coth\u0014d\n2\u00151\u0015\n+1\n2\u00152(\u001b\u000b\u00172\u0000\u001b\f)G(\u0000)\n\f1G\u000b\u0000\u00171G(\u0000)\n\u000b1G\f\n\u00171G(\u0000)\n\u000b1G(\u0000)\n\f2+\u00172G(\u0000)\n\u000b2G(\u0000)\n\f1Coth\u0014d\n2\u00152\u0015\n(C7)\nwhereG(\u0000)\nin\u0011Gi+ (\u001bi=\u0015n) Tanh [d=2\u0015n] whileG(+)\nin\u0011Gi+ (\u001bi=\u0015n) Coth [d=2\u0015n].\n1X. W. P. W. J. Jungwirth, T. 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B 3, 961 (1971)."    },    {        "title": "1709.10365v1.Non_local_Gilbert_damping_tensor_within_the_torque_torque_correlation_model.pdf",        "content": "Non-local Gilbert damping tensor within the torque-torque correlation model\nDanny Thonig,1,\u0003Yaroslav Kvashnin,1Olle Eriksson,1, 2and Manuel Pereiro1\n1Department of Physics and Astronomy, Material Theory, Uppsala University, SE-75120 Uppsala, Sweden\n2School of Science and Technology, Orebro University, SE-701 82 Orebro, Sweden\n(Dated: July 19, 2018)\nAn essential property of magnetic devices is the relaxation rate in magnetic switching which\ndepends strongly on the damping in the magnetisation dynamics. It was recently measured that\ndamping depends on the magnetic texture and, consequently, is a non-local quantity. The damping\nenters the Landau-Lifshitz-Gilbert equation as the phenomenological Gilbert damping parameter\n\u000b, that does not, in a straight forward formulation, account for non-locality. E\u000borts were spent\nrecently to obtain Gilbert damping from \frst principles for magnons of wave vector q. However,\nto the best of our knowledge, there is no report about real space non-local Gilbert damping \u000bij.\nHere, a torque-torque correlation model based on a tight binding approach is applied to the bulk\nelemental itinerant magnets and it predicts signi\fcant o\u000b-site Gilbert damping contributions, that\ncould be also negative. Supported by atomistic magnetisation dynamics simulations we reveal the\nimportance of the non-local Gilbert damping in atomistic magnetisation dynamics. This study gives\na deeper understanding of the dynamics of the magnetic moments and dissipation processes in real\nmagnetic materials. Ways of manipulating non-local damping are explored, either by temperature,\nmaterials doping or strain.\nPACS numbers: 75.10.Hk,75.40.Mg,75.78.-n\nE\u000ecient spintronics applications call for magnetic ma-\nterials with low energy dissipation when moving magnetic\ntextures, e.g. in race track memories1, skyrmion logics2,3,\nspin logics4, spin-torque nano-oscillator for neural net-\nwork applications5or, more recently, soliton devices6. In\nparticular, the dynamics of such magnetic textures |\nmagnetic domain walls, magnetic Skyrmions, or magnetic\nsolitons | is well described in terms of precession and\ndamping of the magnetic moment mias it is formulated\nin the atomistic Landau-Lifshitz-Gilbert (LLG) equation\nfor sitei\n@mi\n@t=mi\u0002\u0012\n\u0000\rBeff\ni+\u000b\nms@mi\n@t\u0013\n; (1)\nwhere\randmsare the gyromagnetic ratio and the\nmagnetic moment length, respectively. The precession\n\feldBeff\niis of quantum mechanical origin and is ob-\ntained either from e\u000bective spin-Hamilton models7or\nfrom \frst-principles8. In turn, energy dissipation is\ndominated by the ad-hoc motivated viscous damping in\nthe equation of motion scaled by the Gilbert damping\ntensor\u000b. Commonly, the Gilbert damping is used as\na scalar parameter in magnetization dynamics simula-\ntions based on the LLG equation. Strong e\u000borts were\nspend in the last decade to put the Gilbert damping\nto a \frst-principles ground derived for collinear mag-\nnetization con\fgurations. Di\u000berent methods were pro-\nposed: e.g. the breathing Fermi surface9{11, the torque-\ntorque correlation12, spin-pumping13or a linear response\nmodel14,15. Within a certain accuracy, the theoretical\nmodels allow to interpret16and reproduce experimental\ntrends17{20.\nDepending on the model, deep insight into the fun-\ndamental electronic-structure mechanism of the Gilbertdamping\u000bis provided: Damping is a Fermi-surface ef-\nfect and depending on e.g. scattering rate, damping\noccurs due to spin-\rip but also spin-conservative tran-\nsition within a degenerated (intraband, but also inter-\nband transitions) and between non-degenerated (inter-\nband transitions) electron bands. As a consequence of\nthese considerations, the Gilbert damping is proportional\nto the density of states, but it also scales with spin-orbit\ncoupling21,22. The scattering rate \u0000 for the spin-\rip tran-\nsitions is allocated to thermal, but also correlation ef-\nfects, making the Gilbert damping strongly temperature\ndependent which must be a consideration when applying\na three-temperature model for the thermal baths, say\nphonon14, electron, and spin temperature23. In particu-\nlar, damping is often related to the dynamics of a collec-\ntive precession mode (macrospin approach) driven from\nan external perturbation \feld, as it is used in ferromag-\nnetic resonance experiments (FMR)24. It is also estab-\nlished that the Gilbert damping depends on the orien-\ntation of the macrospin25and is, in addition, frequency\ndependent26.\nMore recently, the role of non-collective modes to the\nGilbert damping has been debated. F ahnle et al.27\nsuggested to consider damping in a tensorial and non-\nisotropic form via \u000bithat di\u000bers for di\u000berent sites i\nand depends on the whole magnetic con\fguration of the\nsystem. As a result, the experimentally and theoret-\nically assumed local Gilbert equation is replaced by a\nnon-local equation via non-local Gilbert damping \u000bijac-\ncounting for the most general form of Rayleigh's dissi-\npation function28. The proof of principles was given for\nmagnetic domain walls29,30, linking explicitly the Gilbert\ndamping to the gradients in the magnetic spin texture\nrm. Such spatial non-locality, in particular, for discrete\natomistic models, allows further to motivate energy dis-arXiv:1709.10365v1  [cond-mat.mtrl-sci]  29 Sep 20172\nij\nαij\nq\nFIG. 1: Schematic illustration of non-local energy dissipation\n\u000bijbetween site iandj(red balls) represented by a power\ncord in a system with spin wave (gray arrows) propagation q.\nsipation between two magnetic moments at sites iand\nj, and is represented by \u000bij, as schematically illustrated\nin Fig. 1. An analytical expression for \u000bijwas already\nproposed by various authors14,31,32, however, not much\nwork has been done on a material speci\fc, \frst-principle\ndescription of the atomistic non-local Gilbert damping\n\u000bij. An exception is the work by Gilmore et al.32who\nstudied\u000b(q) in the reciprocal space as a function of the\nmagnon wave vector qand concluded that the non-local\ndamping is negligible. Yan et al.29and Hals et al.33, on\nthe other hand, applied scattering theory according to\nBrataas et al.34to simulate non-collinearity in Gilbert\ndamping, only in reciprocal space or continuous meso-\nscopic scale. Here we come up with a technical descrip-\ntion of non-locality of the damping parameter \u000bij, in\nreal space, and provide numerical examples for elemental,\nitinerant magnets, which might be of high importance in\nthe context of ultrafast demagnetization35.\nThe paper is organized as follows: In Section I, we\nintroduce our \frst-principles model formalism based on\nthe torque-torque correlation model to study non-local\ndamping. This is applied to bulk itinerant magnets bcc\nFe, fcc Co, and fcc Ni in both reciprocal and real space\nand it is analysed in details in Section II. Here, we will\nalso apply atomistic magnetisation dynamics to outline\nthe importance in the evolution of magnetic systems. Fi-\nnally, in the last section, we conclude the paper by giving\nan outlook of our work.\nI. METHODS\nWe consider the torque-torque correlation model in-\ntroduced by Kambersk\u0013 y10and further elaborated on by\nGilmore et al.12. Here, \fnite magnetic moment rotations\ncouple to the Bloch eigenenergies \"n;kand eigenstates\njnki, characterised by the band index nat wave vec-tork, due to spin-orbit coupling. This generates a non-\nequilibrium population state (a particle-hole pair), where\nthe excited states relax towards the equilibrium distribu-\ntion (Fermi-Dirac statistics) within the time \u001cn;k=1=\u0000,\nwhich we assume is independent of nandk. In the adi-\nabatic limit, this perturbation is described by the Kubo-\nGreenwood perturbation theory and reads12,36in a non-\nlocal formulation\n\u000b\u0016\u0017(q) =g\u0019\nmsZ\n\nX\nnmT\u0016\nnk;mk+q\u0000\nT\u0017\nnk;mk+q\u0001\u0003Wnk;mk+qdk:\n(2)\nHere the integral runs over the whole Brillouin zone\nvolume \n. A frozen magnon of wave vector qis consid-\nered that is ascribed to the non-locality of \u000b. The scat-\ntering events depend on the spectral overlap Wnk;mk+q=R\n\u0011(\")Ank(\";\u0000)Amk+q(\";\u0000) d\"between two bands \"n;k\nand\"m;k+q, where the spectral width of the electronic\nbandsAnkis approximated by a Lorentzian of width \u0000.\nNote that \u0000 is a parameter in our model and can be spin-\ndependent as proposed in Ref. [37]. In other studies, this\nparameter is allocated to the self-energy of the system\nand is obtained by introducing disorder, e.g., in an al-\nloy or alloy analogy model using the coherent potential\napproximation14(CPA) or via the inclusion of electron\ncorrelation38. Thus, a principle study of the non-local\ndamping versus \u0000 can be also seen as e.g. a temperature\ndependent study of the non-local damping. \u0011=@f=@\"is\nthe derivative of the Fermi-Dirac distribution fwith re-\nspect to the energy. T\u0016\nnk;mk+q=hnkj^T\u0016jmk+qi, where\n\u0016=x;y;z , are the matrix elements of the torque oper-\nator ^T= [\u001b;Hso] obtained from variation of the mag-\nnetic moment around certain rotation axis e.\u001band\nHsoare the Pauli matrices and the spin-orbit hamilto-\nnian, respectively. In the collinear ferromagnetic limit,\ne=ezand variations occur in xandy, only, which al-\nlows to consider just one component of the torque, i.e.\n^T\u0000=^Tx\u0000i^Ty. Using Lehmann representation39, we\nrewrite the Bloch eigenstates by Green's function G, and\nde\fne the spectral function ^A= i\u0000\nGR\u0000GA\u0001\nwith the\nretarded (R) and advanced (A) Green's function,\n\u000b\u0016\u0017(q) =g\nm\u0019Z Z\n\n\u0011(\")^T\u0016^Ak\u0010\n^T\u0017\u0011y^Ak+qdkd\":(3)\nThe Fourier transformation of the Green's function G\n\fnally is used to obtain the non-local Gilbert damping\ntensor23between site iat positionriand sitejat position\nrj,\n\u000b\u0016\u0017\nij=g\nm\u0019Z\n\u0011(\")^T\u0016\ni^Aij\u0010\n^T\u0017\nj\u0011y^Ajid\": (4)\nNote that ^Aij= i\u0000\nGR\nij\u0000GA\nji\u0001\n. This result is consis-\ntent with the formulation given in Ref. [31] and Ref. [14].\nHence, the de\fnition of non-local damping in real space3\nand reciprocal space translate into each other by a\nFourier transformation,\n\u000bij=Z\n\u000b(q) e\u0000i(rj\u0000ri)\u0001qdq: (5)\nNote the obvious advantage of using Eq. (4), since it\nallows for a direct calculation of \u000bij, as opposed to tak-\ning the inverse Fourier transform of Eq. (5). For \frst-\nprinciples studies, the Green's function is obtained from\na tight binding (TB) model based on the Slater-Koster\nparameterization40. The Hamiltonian consists of on-site\npotentials, hopping terms, Zeeman energy, and spin-orbit\ncoupling (See Appendix A). The TB parameters, includ-\ning the spin-orbit coupling strength, are obtained by \ft-\nting the TB band structures to ab initio band structures\nas reported elsewhere23.\nBeyond our model study, we simulate material spe-\nci\fc non-local damping with the help of the full-potential\nlinear mu\u000en-tin orbitals (FP-LMTO) code \\RSPt\"41,42.\nFurther numerical details are provided in Appendix A.\nWith the aim to emphasize the importance of non-\nlocal Gilbert damping in the evolution of atomistic\nmagnetic moments, we performed atomistic magnetiza-\ntion dynamics by numerical solving the Landau-Lifshitz\nGilbert (LLG) equation, explicitly incorporating non-\nlocal damping23,34,43\n@mi\n@t=mi\u00020\n@\u0000\rBeff\ni+X\nj\u000bij\nmj\ns@mj\n@t1\nA:(6)\nHere, the e\u000bective \feld Beff\ni =\u0000@^H=@miis allo-\ncated to the spin Hamiltonian entails Heisenberg-like ex-\nchange coupling\u0000P\nijJijmi\u0001mjand uniaxial magneto-\ncrystalline anisotropyP\niKi(mi\u0001ei)2with the easy axis\nalongei.JijandKiare the Heisenberg exchange cou-\npling and the magneto-crystalline anisotropy constant,\nrespectively, and were obtained from \frst principles44,45.\nFurther details are provided in Appendix A.\nII. RESULTS AND DISCUSSION\nThis section is divided in three parts. In the \frst part,\nwe discuss non-local damping in reciprocal space q. The\nsecond part deals with the real space de\fnition of the\nGilbert damping \u000bij. Atomistic magnetization dynam-\nics including non-local Gilbert damping is studied in the\nthird part.\nA. Non-local damping in reciprocal space\nThe formalism derived by Kambersk\u0013 y10and Gilmore12\nin Eq. (2) represents the non-local contributions to the\nenergy dissipation in the LLG equation by the magnonwave vector q. In particular, Gilmore et al.32con-\ncluded that for transition metals at room temperature\nthe single-mode damping rate is essentially independent\nof the magnon wave vector for qbetween 0 and 1% of\nthe Brillouin zone edge. However, for very small scat-\ntering rates \u0000, Gilmore and Stiles12observed for bcc Fe,\nhcp Co and fcc Ni a strong decay of \u000bwithq, caused by\nthe weighting function Wnm(k;k+q) without any sig-\nni\fcant changes of the torque matrix elements. Within\nour model systems, we observed the same trend for bcc\nFe, fcc Co and fcc Ni. To understand the decay of the\nGilbert damping with magnon-wave vector qin more de-\ntail, we study selected paths of both the magnon qand\nelectron momentum kin the Brillouin zone at the Fermi\nenergy\"Ffor bcc Fe (q;k2\u0000!Handq;k2H!N),\nfcc Co and fcc Ni ( q;k2\u0000!Xandq;k2X!L) (see\nFig. 2, where the integrand of Eq. (2) is plotted). For\nexample, in Fe, a usually two-fold degenerated dband\n(approximately in the middle of \u0000H, marked by ( i)) gives\na signi\fcant contribution to the intraband damping for\nsmall scattering rates. There are two other contributions\nto the damping (marked by ( ii)), that are caused purely\nby interband transitions. With increasing, but small q\nthe intensities of the peaks decrease and interband tran-\nsitions become more likely. With larger q, however, more\nand more interband transitions appear which leads to an\nincrease of the peak intensity, signi\fcantly in the peaks\nmarked with ( ii). This increase could be the same or-\nder of magnitude as the pure intraband transition peak.\nSimilar trends also occur in Co as well as Ni and are\nalso observed for Fe along the path HN. Larger spectral\nwidth \u0000 increases the interband spin-\rip transitions even\nfurther (data not shown). Note that the torque-torque\ncorrelation model might fail for large values of q, since\nthe magnetic moments change so rapidly in space that\nthe adiababtic limit is violated46and electrons are not\nstationary equilibrated. The electrons do not align ac-\ncording the magnetic moment and the non-equilibrium\nelectron distribution in Eq. (2) will not fully relax. In\nparticular, the magnetic force theorem used to derive\nEq. (3) may not be valid.\nThe integration of the contributions in electron mo-\nmentum space kover the whole Brillouin zone is pre-\nsented in Fig. 3, where both `Loretzian' method given\nby Eq. (2) and Green's function method represented\nby Eq. (3) are applied. Both methods give the same\ntrend, however, di\u000ber slightly in the intraband region,\nwhich was already observed previously by the authors\nof Ref. [23]. In the `Lorentzian' approach, Eq. (2), the\nelectronic structure itself is una\u000bected by the scattering\nrate \u0000, only the width of the Lorentian used to approx-\nimateAnkis a\u000bected. In the Green function approach,\nhowever, \u0000 enters as the imaginary part of the energy\nat which the Green functions is evaluated and, conse-\nquently, broadens and shifts maxima in the spectral func-\ntion. This o\u000bset from the real energy axis provides a more\naccurate description with respect to the ab initio results\nthan the Lorentzian approach.4\nΓHq(a−1\n0)\nΓ H\nk(a−1\n0)\nFe\nΓX\nΓ X\nk(a−1\n0)\n Co\nΓX\nΓ X\nk(a−1\n0)\n Ni\n(i) (ii) (ii)\nFIG. 2: Electronic state resolved non-local Gilbert damping obtained from the integrand of Eq. (3) along selected paths in the\nBrillouin zone for bcc Fe, fcc Co and fcc Ni. The scattering rate used is \u0000 = 0 :01 eV. The abscissa (both top and bottom in\neach panels) shows the momentum path of the electron k, where the ordinate (left and right in each panel) shows the magnon\npropagation vector q. The two `triangle' in each panel should be viewed separately where the magnon momentum changes\naccordingly (along the same path) to the electron momentum.\nWithin the limits of our simpli\fed electronic structure\ntight binding method, we obtained qualitatively similar\ntrends as observed by Gilmore et al.32: a dramatic de-\ncrease in the damping at low scattering rates \u0000 (intra-\nband region). This trend is common for all here ob-\nserved itinerant magnets typically in a narrow region\n0<jqj<0:02a\u00001\n0, but also for di\u000berent magnon propa-\ngation directions. For larger jqj>0:02a\u00001\n0the damping\ncould again increase (not shown here). The decay of \u000b\nis only observable below a certain threshold scattering\nrate \u0000, typically where intra- and interband contribu-\ntion equally contributing to the Gilbert damping. As\nalready found by Gilmore et al.32and Thonig et al.23,\nthis point is materials speci\fc. In the interband regime,\nhowever, damping is independent of the magnon propa-\ngator, caused by already allowed transition between the\nelectron bands due to band broadening. Marginal vari-\nations in the decay with respect to the direction of q\n(Inset of Fig. 3) are revealed, which was not reported be-\nfore. Such behaviour is caused by the break of the space\ngroup symmetry due to spin-orbit coupling and a selected\nglobal spin-quantization axis along z-direction, but also\ndue to the non-cubic symmetry of Gkfork6= 0. As a re-\nsult, e.g., in Ni the non-local damping decays faster along\n\u0000Kthan in \u0000X. This will be discussed more in detail in\nthe next section.\nWe also investigated the scaling of the non-local\nGilbert damping with respect to the spin-orbit coupling\nstrength\u0018dof the d-states (see Appendix B). We observe\nan e\u000bect that previously has not been discussed, namely\nthat the non-local damping has a di\u000berent exponential\nscaling with respect to the spin-orbit coupling constant\nfor di\u000berentjqj. In the case where qis close to the Bril-\nlouin zone center (in particular q= 0),\u000b/\u00183\ndwhereas\nfor wave vectors jqj>0:02a\u00001\n0,\u000b/\u00182\nd. For largeq,\ntypically interband transitions dominate the scatteringmechanism, as we show above and which is known to\nscale proportional to \u00182. Here in particular, the \u00182will\nbe caused only by the torque operator in Eq. (2). On the\nother hand, this indicates that spin-mixing transitions\nbecome less important because there is not contribution\nin\u0018from the spectral function entering to the damping\n\u000b(q).\nThe validity of the Kambserk\u0013 y model becomes ar-\nguable for\u00183scaling, as it was already proved by Costa\net al.47and Edwards48, since it causes the unphysical\nand strong diverging intraband contribution at very low\ntemperature (small \u0000). Note that there is no experi-\nmental evidence of such a trend, most likely due to that\nsample impurities also in\ruence \u0000. Furthermore, various\nother methods postulate that the Gilbert damping for\nq= 0 scales like \u00182 9,15,22. Hence, the current applied\ntheory, Eq. (3), seems to be valid only in the long-wave\nlimit, where we found \u00182-scaling. On the other hand,\nEdwards48proved that the long-wave length limit ( \u00182-\nscaling) hold also in the short-range limit if one account\nonly for transition that conserve the spin (`pure' spin\nstates), as we show for Co in Fig. 11 of Appendix C. The\ntrends\u000bversusjqjas described above changes drastically\nfor the `corrected' Kambersk\u0013 y formula: the interband re-\ngion is not a\u000bected by these corrections. In the intraband\nregion, however, the divergent behaviour of \u000bdisappears\nand the Gilbert damping monotonically increases with\nlarger magnon wave vector and over the whole Brillouin\nzone. This trend is in good agreement with Ref. [29].\nFor the case, where q= 0, we even reproduced the re-\nsults reported in Ref. [21]; in the limit of small scattering\nrates the damping is constant, which was also reported\nbefore in experiment49,50. Furthermore, the anisotropy\nof\u000b(q) with respect to the direction of q(as discussed\nfor the insets of Fig. 3) increases by accounting only for\npure-spin states (not shown here). Both agreement with5\n510−22Fe\n0.000\n0.025\n0.050\n0.075\n0.100\nq: Γ→H\n2510−2α(q)Co\nq: Γ→X\n510−225\n10−310−210−110+0\nΓ (eV)Ni\nq: Γ→X\nFIG. 3: (Color online) Non-local Gilbert damping as a func-\ntion of the spectral width \u0000 for di\u000berent reciprocal wave vector\nq(indicated by di\u000berent colors and in units a\u00001\n0). Note that q\nprovided here are in direct coordinates and only the direction\ndi\u000bers between the di\u000berent elementals, itinerant magnets.\nThe non-local damping is shown for bcc Fe (top panel) along\n\u0000!H, for fcc Co (middle panel) along \u0000 !X, and for fcc Ni\n(bottom panel) along \u0000 !X. It is obtained from `Lorentzian'\n(Eq. (2), circles) and Green's function (Eq. (3), triangles)\nmethod. The directional dependence of \u000bfor \u0000 = 0:01 eV is\nshown in the inset.\nexperiment and previous theory motivate to consider \u00182-\nscaling for all \u0000.\nB. Non-local damping in real space\nAtomistic spin-dynamics, as stated in Section I (see\nEq. (6)), that includes non-local damping requires\nGilbert damping in real-space, e.g. in the form \u000bij. This\npoint is addressed in this section. Such non-local con-\ntributions are not excluded in the Rayleigh dissipation\nfunctional, applied by Gilbert to derive the dissipation\ncontribution in the equation of motion51(see Fig. 4).\nDissipation is dominated by the on-site contribution\n-101 Fe\nαii= 3.552·10−3\n˜αii= 3.559·10−3\n-101αij·10−4Co\nαii= 3.593·10−3\n˜αii= 3.662·10−3\n-10\n1 2 3 4 5 6\nrij/a0Ni\nαii= 2.164·10−2\n˜αii= 2.319·10−2FIG. 4: (Color online) Real-space Gilbert damping \u000bijas\na function of the distance rijbetween two sites iandjfor\nbcc Fe, fcc Co, and fcc Ni. Both the `corrected' Kambersk\u0013 y\n(red circles) and the Kambersk\u0013 y (blue squares) approach is\nconsidered. The distance is normalised to the lattice constant\na0. The on-site damping \u000biiis shown in the \fgure label. The\ngrey dotted line indicates the zero line. The spectral width is\n\u0000 = 0:005 eV.\n\u000biiin the itinerant magnets investigated here. For both\nFe (\u000bii= 3:55\u000110\u00003) and Co ( \u000bii= 3:59\u000110\u00003) the\non-site damping contribution is similar, whereas for Ni\n\u000biiis one order of magnitude higher. O\u000b-site contri-\nbutionsi6=jare one-order of magnitude smaller than\nthe on-site part and can be even negative. Such neg-\native damping is discernible also in Ref. [52], however,\nit was not further addressed by the authors. Due to\nthe presence of the spin-orbit coupling and a preferred\nglobal spin-quantization axis (in z-direction), the cubic\nsymmetry of the considered itinerant magnets is broken\nand, thus, the Gilbert damping is anisotropic with re-\nspect to the sites j(see also Fig. 5 left panel). For ex-\nample, in Co, four of the in-plane nearest neighbours\n(NN) are\u000bNN\u0019\u00004:3\u000110\u00005, while the other eight are\n\u000bNN\u0019\u00002:5\u000110\u00005. However, in Ni the trend is opposite:\nthe out-of-plane damping ( \u000bNN\u0019\u00001:6\u000110\u00003) is smaller\nthan the in-plane damping ( \u000bNN\u0019 \u00001:2\u000110\u00003). In-\nvolving more neighbours, the magnitude of the non-local6\ndamping is found to decay as 1=r2and, consequently, it\nis di\u000berent than the Heisenberg exchange parameter that\nasymptotically decays in RKKY-fashion as Jij/1=r353.\nFor the Heisenberg exchange, the two Green's functions\nas well as the energy integration in the Lichtenstein-\nKatsnelson-Antropov-Gubanov formula54scales liker\u00001\nij,\nG\u001b\nij/ei(k\u001b\u0001rij+\b\u001b)\njrijj(7)\nwhereas for simplicity we consider here a single-band\nmodel but the results can be generalized also to the multi-\nband case and where \b\u001bdenotes a phase factor for spin\n\u001b=\";#. For the non-local damping the energy integra-\ntion is omitted due to the properties of \u0011in Eq. (4) and,\nthus,\n\u000bij/sin\u0002\nk\"\u0001rij+ \b\"\u0003\nsin\u0002\nk#\u0001rij+ \b#\u0003\njrijj2:(8)\nThis spatial dependency of \u000bijsuperimposed with\nRuderman-Kittel-Kasuya-Yosida (RKKY) oscillations\nwas also found in Ref. [52] for a model system.\nFor Ni, dissipation is very much short range, whereas in\nFe and Co `damping peaks' also occur at larger distances\n(e.g. for Fe at rij= 5:1a0and for Co at rij= 3:4a0).\nThe `long-rangeness' depends strongly on the parameter\n\u0000 (not shown here). As it was already observed for the\nHeisenberg exchange interaction Jij44, stronger thermal\ne\u000bects represented by \u0000 will reduce the correlation length\nbetween two magnetic moments at site iandj. The same\ntrend is observed for damping: larger \u0000 causes smaller\ndissipation correlation length and, thus, a faster decay\nof non-local damping in space rij. Di\u000berent from the\nHeisenberg exchange, the absolute value of the non-local\ndamping typically decreases with \u0000 as it is demonstrated\nin Fig. 5.\nNote that the change of the magnetic moment length\nis not considered in the results discussed so far. The\nanisotropy with respect to the sites iandjof the non-\nlocal Gilbert damping continues in the whole range of the\nscattering rate \u0000 and is controlled by it. For instance, the\nsecond nearest neighbours damping in Co and Ni become\ndegenerated at \u0000 = 0 :5 eV, where the anisotropy between\n\frst-nearest neighbour sites increase. Our results show\nalso that the sign of \u000bijis a\u000bected by \u0000 (as shown in\nFig. 5 left panel). Controlling the broadening of Bloch\nspectral functions \u0000 is in principal possible to evaluate\nfrom theory, but more importantly it is accessible from\nexperimental probes such as angular resolved photoelec-\ntron spectroscopy and two-photon electron spectroscopy.\nThe importance of non-locality in the Gilbert damping\ndepend strongly on the material (as shown in Fig. 5 right\npanel). It is important to note that the total | de\fned as\n\u000btot=P\nj\u000bijfor arbitrary i|, but also the local ( i=j)\nand the non-local ( i6=j) part of the Gilbert damping do\nnot violate the thermodynamic principles by gaining an-\ngular momentum (negative total damping). For Fe, the\n-101\n1. NN.\n2. NN.Fe\n34567αii\nαtot=/summationtext\njαijαq=0.1a−1\n0αq=0\n-10αij·10−4Co\n123456\nαij·10−3\n-15-10-50\n10−210−1\nΓ (eV)Ni\n5101520\n10−210−1\nΓ (eV)FIG. 5: (Color online) First (circles) and second nearest\nneighbour (triangles) Gilbert damping (left panel) as well as\non-site (circles) and total Gilbert (right panel) as a function of\nthe spectral width \u0000 for the itinerant magnets Fe, Co, and Ni.\nIn particular for Co, the results obtained from tight binding\nare compared with \frst-principles density functional theory\nresults (gray open circles). Solid lines (right panel) shows the\nGilbert damping obtained for the magnon wave vectors q= 0\n(blue line) and q= 0:1a\u00001\n0(red line). Dotted lines are added\nto guide the eye. Note that since cubic symmetry is broken\n(see text), there are two sets of nearest neighbor parameters\nand two sets of next nearest neighbor parameters (left panel)\nfor any choice of \u0000.\nlocal and total damping are of the same order for all\n\u0000, where in Co and Ni the local and non-local damp-\ning are equally important. The trends coming from our\ntight binding electron structure were also reproduced by\nour all-electron \frst-principles simulation, for both de-\npendency on the spectral broadening \u0000 (Fig. 5 gray open\ncircles) but also site resolved non-local damping in the\nintraband region (see Appendix A), in particular for fcc\nCo.\nWe compare also the non-local damping obtain from\nthe real and reciprocal space. For this, we used Eq. (3)\nby simulating Nq= 15\u000215\u000215 points in the \frst magnon\nBrillouin zone qand Fourier-transformed it (Fig. 6). For7\n-1.0-0.50.00.51.0αij·10−4\n5 10 15 20 25 30\nrij/a0FFT(α(q));αii= 0.003481\nFFT(G(k));αii= 0.003855\nFIG. 6: (Color online) Comparing non-local Gilbert damping\nobtained by Eq. (5) (red symbols) and Eq. (4) (blue symbols)\nin fcc Co for \u0000 = 0 :005 eV. The dotted line indicates zero\nvalue.\nboth approaches, we obtain good agreement, corroborat-\ning our methodology and possible applications in both\nspaces. The non-local damping for the \frst three nearest\nneighbour shells turn out to converge rapidly with Nq,\nwhile it does not converge so quickly for larger distances\nrij. The critical region around the \u0000-point in the Bril-\nlouin zone is suppressed in the integration over q. On\nthe other hand, the relation \u000btot=P\nj\u000bij=\u000b(q= 0)\nfor arbitrary ishould be valid, which is however violated\nin the intraband region as shown in Fig. 5 (compare tri-\nangles and blue line in Fig. 5): The real space damping\nis constant for small \u0000 and follows the long-wavelength\nlimit (compare triangles and red line in Fig. 5) rather\nthan the divergent ferromagnetic mode ( q= 0). Two\nexplanations are possible: i)convergence with respect to\nthe real space summation and ii)a di\u000berent scaling in\nboth models with respect to the spin-orbit coupling. For\ni), we carefully checked the convergence with the summa-\ntion cut-o\u000b (see Appendix D) and found even a lowering\nof the total damping for larger cut-o\u000b. However, the non-\nlocal damping is very long-range and, consequently, con-\nvergence will be achieved only at a cut-o\u000b radius >>9a0.\nForii), we checked the scaling of the real space Gilbert\ndamping with the spin-orbit coupling of the d-states\n(see Appendix B). Opposite to the `non-corrected' Kam-\nbersk\u0013 y formula in reciprocal space, which scales like\n\u00183\nd, we \fnd\u00182\ndfor the real space damping. This indi-\ncates that the spin-\rip scattering hosted in the real-space\nGreen's function is suppressed. To corroborate this state-\nment further, we applied the corrections proposed by\nEdwards48to our real space formula Eq. (4), which by\ndefault assumes \u00182(Fig. 4, red dots). Both methods, cor-\nrected and non-corrected Eq. (4), agree quite well. The\nsmall discrepancies are due to increased hybridisations\nand band inversion between p and d- states due to spin-\norbit coupling in the `non-corrected' case.\nFinally, we address other ways than temperature (here\nrepresented by \u0000), to manipulate the non-local damping.\nIt is well established in literature already for Heisenberg\nexchange and the magneto crystalline anisotropy that\n-0.40.00.40.81.2αij·10−4\n1 2 3 4 5 6 7\nrij/a0αii= 3.49·10−3αii= 3.43·10−3FIG. 7: (Color online) Non-local Gilbert damping as a func-\ntion of the normalized distancerij=a0for a tetragonal dis-\ntorted bcc Fe crystal structure. Here,c=a= 1:025 (red circles)\nandc=a= 1:05 (blue circles) is considered. \u0000 is put to 0 :01 eV.\nThe zero value is indicated by dotted lines.\ncompressive or tensial strain can be used to tune the mag-\nnetic phase stability and to design multiferroic materials.\nIn an analogous way, also non-local damping depends on\ndistortions in the crystal (see Fig. 7).\nHere, we applied non-volume conserved tetragonal\nstrain along the caxis. The local damping \u000biiis marginal\nbiased. Relative to the values of the undistorted case,\na stronger e\u000bect is observed for the non-local part, in\nparticular for the \frst few neighbours. Since we do a\nnon-volume conserved distortion, the in-plane second NN\ncomponent of the non-local damping is constant. The\ndamping is in general decreasing with increasing distor-\ntion, however, a change in the sign of the damping can\nalso occur (e.g. for the third NN). The rate of change\nin damping is not linear. In particular, the nearest-\nneighbour rate is about \u000e\u000b\u00190:4\u000110\u00005for 2:5% dis-\ntortion, and 2 :9\u000110\u00005for 5% from the undistorted case.\nFor the second nearest neighbour, the rate is even big-\nger (3:0\u000110\u00005for 2:5%, 6:9\u000110\u00005for 5%). For neigh-\nbours larger than rij= 3a0, the change is less signi\fcant\n(\u00000:6\u000110\u00005for 2:5%,\u00000:7\u000110\u00005for 5%). The strongly\nstrain dependent damping motivates even higher-order\ncoupled damping contributions obtained from Taylor ex-\npanding the damping contribution around the equilib-\nrium position \u000b0\nij:\u000bij=\u000b0\nij+@\u000bij=@uk\u0001uk+:::. Note that\nthis is in analogy to the magnetic exchange interaction55\n(exchange striction) and a natural name for it would\nbe `dissipation striction'. This opens new ways to dis-\nsipatively couple spin and lattice reservoir in combined\ndynamics55, to the best of our knowledge not considered\nin todays ab-initio modelling of atomistic magnetisation\ndynamics.\nC. Atomistic magnetisation dynamics\nThe question about the importance of non-local damp-\ning in atomistic magnetization dynamics (ASD) remains.8\n0.40.50.60.70.80.91.0M\n0.0 0.5 1.0 1.5 2.0 2.5 3.0\nt(ps)0.5\n0.1\n0.05\n0.01αtot\nαij\n0.5 1.0 1.5 2.0 2.5 3.0\nt(ps)Fe\nCo\nFIG. 8: (Color online) Evolution of the average magnetic mo-\nmentMduring remagnetization in bcc Fe (left panel) and\nfcc Co (right panel) for di\u000berent damping strength according\nto the spectral width \u0000 (di\u000berent colors) and both, full non-\nlocal\u000bij(solid line) and total, purely local \u000btot(dashed line)\nGilbert damping.\nFor this purpose, we performed zero-temperature ASD\nfor bcc Fe and fcc Co bulk and analysed changes in the\naverage magnetization during relaxation from a totally\nrandom magnetic con\fguration, for which the total mo-\nment was zero (Fig. 8)\nRelated to the spectral width, the velocity for remag-\nnetisation changes and is higher, the bigger the e\u000bective\nGilbert damping is. For comparison, we performed also\nASD simulations based on Eq. (2) with a scalar, purely\nlocal damping \u000btot(dotted lines). For Fe, it turned out\nthat accounting for the non-local damping causes a slight\ndecrease in the remagnetization time, however, is overall\nnot important for relaxation processes. This is under-\nstandable by comparing the particular damping values\nin Fig. 5, right panel, in which the non-local part ap-\npear negligible. On the other hand, for Co the e\u000bect\non the relaxation process is much more signi\fcant, since\nthe non-local Gilbert damping reduces the local contribu-\ntion drastically (see Fig. 5, right panel). This `negative'\nnon-local part ( i6=j) in\u000bijdecelerates the relaxation\nprocess and the relaxation time is drastically increased\nby a factor of 10. Note that a `positive' non-local part\nwill accelerate the relaxation, which is of high interest for\nultrafast switching processes.\nIII. CONCLUDING REMARKS\nIn conclusion, we have evaluated the non-locality of\nthe Gilbert damping parameter in both reciprocal and\nreal space for elemental, itinerant magnets bcc Fe, fcc\nCo and fcc Ni. In particular in the reciprocal space,\nour results are in good agreement with values given in\nthe literature32. The here studied real space damping\nwas considered on an atomistic level and it motivates\nto account for the full, non-local Gilbert damping in\nmagnetization dynamic, e.g. at surfaces56or for nano-\nstructures57. We revealed that non-local damping canbe negative, has a spatial anisotropy, quadratically scales\nwith spin-orbit coupling, and decays in space as r\u00002\nij.\nDetailed comparison between real and reciprocal states\nidenti\fed the importance of the corrections proposed by\nEdwards48and, consequently, overcome the limits of the\nKambersk\u0013 y formula showing an unphysical and experi-\nmental not proved divergent behaviour at low tempera-\nture. We further promote ways of manipulating non-local\nGilbert damping, either by temperature, materials dop-\ning or strain, and motivating `dissipation striction' terms,\nthat opens a fundamental new root in the coupling be-\ntween spin and lattice reservoirs.\nOur studies are the starting point for even further in-\nvestigations: Although we mimic temperature by the\nspectral broadening \u0000, a precise mapping of \u0000 to spin\nand phonon temperature is still missing, according to\nRefs. [14,23]. Even at zero temperature, we revealed a\nsigni\fcant e\u000bect of the non-local Gilbert damping to the\nmagnetization dynamics, but the in\ruence of non-local\ndamping to \fnite temperature analysis or even to low-\ndimensional structures has to be demonstrated.\nIV. ACKNOWLEDGEMENTS\nThe authors thank Lars Bergqvist, Lars Nordstr om,\nJustin Shaw, and Jonas Fransson for fruitful discus-\nsions. O.E. acknowledges the support from Swedish Re-\nsearch Council (VR), eSSENCE, and the KAW Founda-\ntion (Grants No. 2012.0031 and No. 2013.0020).\nAppendix A: Numerical details\nWe performkintegration with up to 1 :25\u0001106mesh\npoints (500\u0002500\u0002500) in the \frst Brillouin zone for bulk.\nThe energy integration is evaluated at the Fermi level\nonly. For our principles studies, we performed a Slater-\nKoster parameterised40tight binding (TB) calculations58\nof the torque-torque correlation model as well as for the\nGreen's function model. Here, the TB parameters have\nbeen obtained by \ftting the electronic structures to those\nof a \frst-principles fully relativistic multiple scattering\nKorringa-Kohn-Rostoker (KKR) method using a genetic\nalgorithm. The details of the \ftting and the tight binding\nparameters are listed elsewhere23,59. This puts our model\non a \frm, \frst-principles ground.\nThe tight binding Hamiltonian60H=H0+Hmag+\nHsoccontains on-site energies and hopping elements H0,\nthe spin-orbit coupling Hsoc=\u0010S\u0001Land the Zeeman\ntermHmag=1=2B\u0001\u001b. The Green's function is obtained\nbyG= (\"+ i\u0000\u0000H)\u00001, allows in principle to consider\ndisorder in terms of spin and phonon as well as alloys23.\nThe bulk Greenian Gijin real space between site iandj\nis obtained by Fourier transformation. Despite the fact\nthat the tight binding approach is limited in accuracy, it\nproduces good agreement with \frst principle band struc-\nture calculations for energies smaller than \"F+ 5 eV.9\n-1.5-1.0-0.50.00.51.01.5\n5 10 15 20 25 30\nrij(Bohr radii)Γ≈0.01eVTB\nTBe\nDFT\nαDFT\nii= 3.9846·10−3\nαTB\nii= 3.6018·10−3-1.5-1.0-0.50.00.51.01.5\nΓ≈0.005eV\nαDFT\nii= 3.965·10−3\nαTB\nii= 3.5469·10−3αij·10−4\nFIG. 9: (Colour online) Comparison of non-local damping ob-\ntained from the Tight Binding method (TB) (red \flled sym-\nbols), Tight Binding with Edwards correction (TBe) (blue\n\flled symbols) and the linear mu\u000en tin orbital method (DFT)\n(open symbols) for fcc Co. Two di\u000berent spectral broadenings\nare chosen.\nEquation (4) was also evaluated within the DFT and\nlinear mu\u000en-tin orbital method (LMTO) based code\nRSPt. The calculations were done for a k-point mesh\nof 1283k-points. We used three types of basis func-\ntions, characterised by di\u000berent kinetic energies with\n\u00142= 0:1;\u00000:8;\u00001:7 Ry to describe 4 s, 4pand 3dstates.\nThe damping constants were calculated between the 3 d\norbitals, obtained using using mu\u000en-tin head projection\nscheme61. Both the \frst principles and tight binding im-\nplementation of the non-local Gilbert damping agree well\n(see Fig. 9).\nNote that due to numerical reasons, the values of\n\u0000 used for the comparisons are slightly di\u000berent in\nboth electronic structure methods. Furthermore, in the\nLMTO method the orbitals are projected to d-orbitals\nonly, which lead to small discrepancies in the damping.\nThe atomistic magnetization dynamics is also per-\nformed within the Cahmd simulation package58. To\nreproduce bulk properties, periodic boundary condi-\ntions and a su\u000eciently large cluster (10 \u000210\u000210)\nare employed. The numerical time step is \u0001 t=\n0:1 fs. The exchange coupling constants Jijare\nobtained from the Liechtenstein-Kastnelson-Antropov-\nGubanovski (LKAG) formula implemented in the \frst-\nprinciples fully relativistic multiple scattering Korringa-\nKohn-Rostoker (KKR) method39. On the other hand,\nthe magneto-crystalline anisotropy is used as a \fxed pa-\nrameter with K= 50\u0016eV.\n012345678α·10−3\n0.0 0.02 0.04 0.06 0.08 0.1\nξd(eV)2.02.22.42.62.83.03.2γ\n0.0 0.1 0.2 0.3 0.4\nq(a−1\n0)-12-10-8-6-4-20αnn·10−5\n01234567\nαos·10−3 1.945\n1.797\n1.848\n1.950\n1.848\n1.797\n1.950FIG. 10: (Color online) Gilbert damping \u000bas a function of\nthe spin-orbit coupling for the d-states in fcc Co. Lower panel\nshows the Gilbert damping in reciprocal space for di\u000berent\nq=jqjvalues (di\u000berent gray colours) along the \u0000 !Xpath.\nThe upper panel exhibits the on-site \u000bos(red dotes and lines)\nand nearest-neighbour \u000bnn(gray dots and lines) damping.\nThe solid line is the exponential \ft of the data point. The\ninset shows the \ftted exponents \rwith respect wave vector\nq. The colour of the dots is adjusted to the particular branch\nin the main \fgure. The spectral width is \u0000 = 0 :005 eV.\nAppendix B: Spin-orbit coupling scaling in real and\nreciprocal space\nKambersk\u0013 y's formula is valid only for quadratic spin-\norbit coupling scaling21,47, which implies only scattering\nbetween states that preserve the spin. This mechanism\nwas explicitly accounted by Edwards48by neglecting the\nspin-orbit coupling contribution in the `host' Green's\nfunction. It is predicted for the coherent mode ( q= 0)21\nthat this overcomes the unphysical and not experimen-\ntally veri\fed divergent Gilbert damping for low tem-\nperature. Thus, the methodology requires to prove the\nfunctional dependency of the (non-local) Gilbert damp-\ning with respect to the spin-orbit coupling constant \u0018\n(Fig. 10). Since damping is a Fermi-surface e\u000bects, it\nis su\u000ecient to consider only the spin-orbit coupling of\nthe d-states. The real space Gilbert damping \u000bij/\u0018\r\nscales for both on-site and nearest-neighbour sites with\n\r\u00192. For the reciprocal space, however, the scaling is\nmore complex and \rdepends on the magnon wave vec-\ntorq(inset in Fig. 10). In the long-wavelength limit,\nthe Kambersk\u0013 y formula is valid, where for the ferromag-\nnetic magnon mode with \r\u00193 the Kambersk\u0013 y formula\nis inde\fnite according to Edwards48.10\n10−32510−2α(q)\n10−310−210−110+0\nΓ (eV)0.000\n0.025\n0.050\n0.075\n0.100\nq: Γ→XCo\nFIG. 11: (Colour online) Comparison of reciprocal non-local\ndamping with (squares) or without (circles) corrections pro-\nposed by Costa et al.47and Edwards48for Co and di\u000berent\nspectral broadening \u0000. Di\u000berent colours represent di\u000berent\nmagnon propagation vectors q.\nAppendix C: Intraband corrections\nFrom the same reason as discussed in Section B, the\nrole of the correction proposed by Edwards48for magnon\npropagations di\u000berent than zero is unclear and need to\nbe studied. Hence, we included the correction of Ed-\nward also to Eq. (3) (Fig. 11). The exclusion of the spin-\norbit coupling (SOC) in the `host' clearly makes a major\nqualitative and quantitative change: Although the in-\nterband transitions are una\u000bected, interband transitions\nare mainly suppressed, as it was already discussed by\nBarati et al.21. However, the intraband contributions are\nnot totally removed for small \u0000. For very small scat-\ntering rates, the damping is constant. Opposite to the\n`non-corrected' Kambersk\u0013 y formula, the increase of the\nmagnon wave number qgives an increase in the non-\nlocal damping which is in agreement to the observation\nmade by Yuan et al.29, but also with the analytical modelproposed in Ref. [52] for small q. This behaviour was ob-\nserved for all itinerant magnets studied here.\nAppendix D: Comparison real and reciprocal\nGilbert damping\nThe non-local damping scales like r\u00002\nijwith the dis-\ntance between the sites iandj, and is, thus, very long\nrange. In order to compare \u000btot=P\nj2Rcut\u000bijfor arbi-\ntraryiwith\u000b(q= 0), we have to specify the cut-o\u000b ra-\ndius of the summation in real space (Fig. 12). The inter-\nband transitions (\u0000 >0:05 eV) are already converged for\nsmall cut-o\u000b radii Rcut= 3a0. Intraband transitions, on\nthe other hand, converge weakly with Rcutto the recipro-\ncal space value \u000b(q= 0). Note that \u000b(q= 0) is obtained\nfrom the corrected formalism. Even with Rcut= 9a0\nwhich is proportional to \u001930000 atoms, we have not\n0.81.21.62.0αtot·10−3\n4 5 6 7 8 9\nRcut/a00.005\n0.1\nFIG. 12: Total Gilbert damping \u000btotfor fcc Co as a function\nof summation cut-o\u000b radius for two spectral width \u0000, one in\nintraband (\u0000 = 0 :005 eV, red dottes and lines) and one in the\ninterband (\u0000 = 0 :1 eV, blue dottes and lines) region. The\ndotted and solid lines indicates the reciprocal value \u000b(q= 0)\nwith and without SOC corrections, respectively.\nobtain convergence.\n\u0003Electronic address: danny.thonig@physics.uu.se\n1S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320,\n190 (2008), URL http://www.sciencemag.org/cgi/doi/\n10.1126/science.1145799 .\n2J. Iwasaki, M. Mochizuki, and N. Nagaosa, Nature\nNanotech 8, 742 (2013), URL http://www.nature.com/\ndoifinder/10.1038/nnano.2013.176 .\n3A. Fert, V. Cros, and J. 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B\n76, 035107 (2007), URL https://link.aps.org/doi/10.\n1103/PhysRevB.76.035107 ."    },    {        "title": "2212.09164v1.Exponential_decay_of_solutions_of_damped_wave_equations_in_one_dimensional_space_in_the__L_p__framework_for_various_boundary_conditions.pdf",        "content": "arXiv:2212.09164v1  [math.OC]  18 Dec 2022EXPONENTIAL DECAY OF SOLUTIONS OF DAMPED WAVE EQUATIONS\nIN ONE DIMENSIONAL SPACE IN THE LpFRAMEWORK FOR VARIOUS\nBOUNDARY CONDITIONS\nYACINE CHITOUR AND HOAI-MINH NGUYEN\nAbstract. We establish the decay of the solutions of the damped wave equ ations in one dimen-\nsional space for the Dirichlet, Neumann, and dynamic bounda ry conditions where the damping\ncoefficient is a function of space and time. The analysis is bas ed on the study of the corresponding\nhyperbolic systems associated with the Riemann invariants . The key ingredient in the study of\nthese systems is the use of the internal dissipation energy t o estimate the difference of solutions\nwith their mean values in an average sense.\nContents\n1. Introduction 1\n2. The well-posedness in Lp-setting 5\n2.1. Proof of Proposition 2.1 7\n2.2. Proof of Proposition 2.2 9\n3. Some useful lemmas 9\n4. Exponential decay in Lp-framework for the Dirichlet boundary condition 13\n4.1. Proof of Theorem 1.2 14\n4.2. Proof of Theorem 1.1 16\n4.3. On the case anot being non-negative 16\n5. Exponential decay in Lp-framework for the Neuman boundary condition 19\n5.1. Proof of Theorem 5.2 19\n5.2. Proof of Theorem 5.1 21\n6. Exponential decay in Lp-framework for the dynamic boundary condition 21\n6.1. Proof of Theorem 6.2 22\n6.2. Proof of Theorem 6.1 23\nReferences 23\n1.Introduction\nThis paper is devoted to the decay of solution of the damped wa ve equations in one dimensional\nspace in the Lp-framework for 1 < p <+∞for various boundary conditions where the damping\ndepends on space and time. More precisely, we consider the da mped wave equation\n(1.1)/braceleftBigg\n∂ttu−∂xxu+a∂tu= 0 in R+×(0,1),\nu(0,·) =u0, ∂tu(0,·) =u1on (0,1),\nequipped with one of the following boundary conditions:\n(1.2) Dirichlet boundary condition: u(t,0) =u(t,1) = 0,fort≥0,\n12 Y. CHITOUR AND H.-M. NGUYEN\n(1.3) Neumann boundary condition: ∂xu(t,0) =∂xu(t,1) = 0,fort≥0,\nand, for κ >0,\n(1.4) dynamic boundary condition: ∂xu(t,0)−κ∂tu(t,0) =∂xu(t,1)+κ∂tu(t,1) = 0,fort≥0.\nHereu0∈W1,p(0,1) (with u0(0) =u0(1) = 0, i.e., u0∈W1,p\n0(0,1), in the case where the\nDirichlet boundarycondition is considered), and u1∈Lp(0,1) aretheinitial conditions. Moreover,\na∈L∞/parenleftbig\nR+×(0,1)/parenrightbig\nis assumed to verify the following hypothesis:\n(1.5)a≥0,and∃λ,ε0>0,(x0−ε0,x0+ε0)⊂(0,1) such that a≥λonR+×(x0−ε0,x0+ε0),\ni.e,ais non-negative and a(t,x)≥λ >0 fort≥0 and for xin some open subset of (0 ,1). The\nregion where a >0 represents the region in which the damping term is active.\nThe decay of the solutions of ( 1.1) equipped with either ( 1.2), or (1.3), or (1.4) has been\nextensively investigated in the case where ais independent of t, i.e.,a(t,x) =a(x) and mainly\nin theL2-framework, i.e. within an Hilbertian setting. In this case , concerning the Dirichlet\nboundary condition, under the additional geometric multip lier condition on a, by the multiplier\nmethod, see e.g., [ 20,24], one can prove that the solution decays exponentially, i.e ., there exist\npositive constants Candγindependent of usuch that\n(1.6)/bardbl∂tu(t,·)/bardblL2(0,1)+/bardbl∂xu(t,·)/bardblL2(0,1)≤Ce−γt/parenleftBig\n/bardbl∂tu(0,·)/bardblL2(0,1)+/bardbl∂xu(0,·)/bardblL2(0,1)/parenrightBig\n, t≥0.\nTheassumption that asatisfies the geometric multiplier condition is equivalent to the requirement\nthata(x)≥λ >0 on some neighbourhood of 0 or 1. Based on more sophisticate a rguments in\nthe seminal work of Bardos, Lebeau, and Rauch on the controll ability of the wave equation [ 3],\nLebeau [23] showed that ( 1.6) also holds withoutthe geometric multiplier condition on a, see also\nthe work of Rauch andTaylor [ 31]. When the dampingcoefficient ais also time-dependent, similar\nresults have been obtained recently by Le Rousseau et al. in [ 22]. It is worth noticing that strong\nstabilization, i.e., the energy decay to zero for each traje ctory, has been established previously\nusing LaSalle’s invariance argument [ 14,15]. The analysis of the nonlinear setting associated\nwith (1.1) can be found in [ 6,17,26,27,34] and the references therein. Similar results holds for\nthe Neumann boundary condition [ 3,22,26,34]. Concerning the dynamic boundary condition\nwithout interior damping effect, i.e., a≡0, the analysis for L2-framework was previously initiated\nby Quinn and Russell [ 30]. They proved that the energy exponentially decays in L2-framework\nin one dimensional space. The exponential decay for higher d imensional space was proved by\nLagnese [ 21] using the multiplier technique (see also [ 30]). The decay hence was established for\nthe geometric multiplier condition and this technique was l ater extended in [ 25], see also [ 1] for a\nnice account on these issues.\nMuch less is known about the asymptotic stability of ( 1.1) equipped with either ( 1.2), or (1.3),\nor(1.4)inLp-framework. Thisisprobablyduetothefactthat forlinearw ave equationsconsidered\nin domains of Rdwithd≥2 is not a well defined bounded operator in general in Lpframework\nwithp/ne}ationslash= 2, a result due to Peral [ 29]. As far as we know, the only work concerning exponential\ndecay in the Lp-framework is due to Kafnemer et al. [ 19], where the Dirichlet boundary condition\nwas considered. For the damping coefficient abeing time-independent, they showed that the\ndecay holds under the additional geometric multiplier cond ition on afor 1< p <+∞. Their\nanalysis is via the multiplier technique involving various non-linear test functions. In the case\nof zero damping and with a dynamic boundary condition, previ ous results have been obtained\nin [7] where the problem has been reduced to the study of a discrete time dynamical system over\nappropriate functional spaces.3\nThe goal of this paper is to give a unified approach to deal with all the boundary considered in\n(1.2), (1.3), and (1.4) in theLp-framework for 1 < p <+∞under the condition ( 1.5). Our results\nthus hold even in the case where ais a function of time and space. The analysis is based on the\nstudy of the corresponding hyperbolic systems associated w ith the Riemann invariants for which\nnew insights are required.\nConcerning the Dirichlet boundary condition, we obtain the following result.\nTheorem 1.1. Let1< p <+∞,ε0>0,λ >0, and let a∈L∞/parenleftbig\nR+×(0,1)/parenrightbig\nbe such that a≥0\nanda≥λ >0inR+×(x0−ε0,x0+ε0)⊂R+×(0,1)for some x0∈(0,1). Then there exist positive\nconstants Candγdepending only on p,/bardbla/bardblL∞/parenleftbig\nR+×(0,1)/parenrightbig,ε0, andλsuch that for all u0∈W1,p\n0(0,1)\nandu1∈Lp(0,1), the unique weak solution u∈C([0,+∞);W1,p\n0(0,1))∩C1([0,+∞);Lp(0,1))of\n(1.1)and(1.2)satisfies\n(1.7) /bardbl∂tu(t,·)/bardblLp(0,1)+/bardbl∂xu(t,·)/bardblLp(0,1)≤Ce−γt/parenleftBig\n/bardblu1/bardblLp(0,1)+/bardbl∂xu0/bardblLp(0,1)/parenrightBig\n, t≥0.\nThe meaning of the (weak) solutions given Theorem 1.1is given in Section 2(see Definition 2.1)\nand their well-posedness is also established there (see Pro position 2.1). Our analysis is via the\nstudy of the decay of solutions of hyperbolic systems which a re associated with ( 1.1) via the\nRiemann invariants. Such a decay for the hyperbolic system, even in the case p= 2, is new to our\nknowledge. The analysis of these systems has its own interes t and is motivated by recent analysis\non the controllability of hyperbolic systems in one dimensi onal space [ 9–12].\nAs in [16,19], we set\n(1.8)ρ(t,x) =ux(t,x)+ut(t,x) and ξ(t,x) =ux(t,x)−ut(t,x) for (t,x)∈R+×(0,1).\nOne can check that for a smooth solution uof (1.1) and (1.2), the pair of functions ( ρ,ξ) defined\nin (1.8) satisfies the system\n(1.9)\n\nρt−ρx=−1\n2a(ρ−ξ) in R+×(0,1),\nξt+ξx=1\n2a(ρ−ξ) in R+×(0,1),\nρ(t,0)−ξ(t,0) =ρ(t,1)−ξ(t,1) = 0 in R+.\nOnecannothope the decay of a general solutions of ( 1.9) since any pair ( c,c) wherec∈Ris a\nconstant is a solution of ( 1.9). Nevertheless, for ( ρ,ξ) being defined by ( 1.9) for a solution uof\n(1.1), one also has the following additional information\n(1.10)ˆ1\n0ρ(t,x)+ξ(t,x)dx= 0 fort≥0.\nConcerningSystem( 1.9) itself (i.e., withoutnecessarily assuming( 1.10)), weprovethefollowing\nresult, which takes into account ( 1.10).\nTheorem 1.2. Let1< p <+∞,ε0>0,λ >0, anda∈L∞/parenleftbig\nR+×(0,1)/parenrightbig\nbe such that a≥0\nanda≥λ >0inR+×(x0−ε0,x0+ε0)⊂R+×(0,1)for some x0∈(0,1). There exist a\npositive constant Cand a positive constant γdepending only on on p,/bardbla/bardblL∞/parenleftbig\nR+×(0,1)/parenrightbig,ε0, andλ\nsuch that the unique solution (ρ,ξ)of(1.9)with the initial condition ρ(0,·) =ρ0andξ(0,·) =ξ0\nsatisfies\n(1.11) /bardbl(ρ−c0,ξ−c0)(t,·)/bardblLp(0,1)≤Ce−γt/bardbl(ρ(0,·)−c0,ξ(0,·)−c0)/bardblLp(0,1), t≥0,4 Y. CHITOUR AND H.-M. NGUYEN\nwhere\n(1.12) c0:=1\n2ˆ1\n0/parenleftbig\nρ(0,x)+ξ(0,x)/parenrightbig\ndx,\nIn Theorem 1.2, we consider the broad solutions. It is understood through t he broad solution\nin finite time: for T >0 and 1≤p <+∞, a broad solution uof the system\n(1.13)\n\nρt−ρx=−1\n2a(ρ−ξ) in (0 ,T)×(0,1),\nξt+ξx=1\n2a(ρ−ξ) in (0 ,T)×(0,1),\nρ(t,0)−ξ(t,0) =ρ(t,1)−ξ(t,1) = 0 in (0 ,T),\nρ(0,·) =ρ0, ξ(0,·) =ξ0 in (0,1),\nis a pair of functions ( ρ,ξ)∈C([0,T];/bracketleftbig\nLp(0,1)/bracketrightbig2/parenrightbig\n∩C([0,1];/bracketleftbig\nLp(0,T)/bracketrightbig2/parenrightbig\nwhich obey the charac-\nteristic rules, see e.g., [ 10]. Thewell-posedness of ( 1.13) can befoundin [ 10] (see also theappendix\nof [13]). The analysis there is mainly for the case p= 2 but the arguments extend naturally for\nthe case 1 ≤p <+∞.\nIn theLp-framework, the Neumann boundary condition and its corresp onding hyperbolic sys-\ntems are discussed in Section 5and the dynamic boundary condition and its corresponding hy -\nperbolic systems are discussed in Section 6. Concerning the dynamic boundary condition, the\ndecay holds even under the assumption a≥0. The analysis for the Neumann case shares a large\npart in common with the one of the Dirichlet boundary conditi on. The difference in their analysis\ncomes from taking into account differently the boundary condi tion. The analysis of the dynamic\ncondition is similar but much simpler.\nThe study of the wave equation in one dimensional space via th e corresponding hyperbolic\nsystem is known. Thecontrollability and stability of hyper bolicsystems has been also investigated\nextensively. This goes back to the work of Russel [ 32,33] and Rauch and Taylor [ 31]. Many\nimportant progress has been obtained recently, see, e.g., [ 4] and the references therein. It is worth\nnoting that many works have been devoted to the L2-framework. Less is studied in the Lp-scale.\nIn this direction, we want to mention [ 9] where the exponential stability is studied for dissipativ e\nboundary condition.\nConcerning the wave equation in one dimensional space, the e xponential decay in L2-setting for\nthe dynamic boundary condition is also established via its c orresponding hyperbolic systems [ 30].\nHowever, to our knowledge, the exponential decay for the Dir ichlet and Neumann boundary\nconditions has not been established even in L2-framework via this approach. Our work is new\nand quite distinct from the one in [ 30] and has its own interest. First, the analysis in [ 30] uses\nessentially the fact that the boundary is strictly dissipat ive, i.e.,κ >0 in (1.4). Thus the analysis\ncannot be used for the Dirichlet and Neumann boundary condit ions. Moreover, it is not clear\nhow to extend it to the Lp-framework. Concerning our analysis, the key observation i s that the\ninformation of the internal energy allows one to control the difference of the solutions and its\nmean value in the interval of time (0 ,T) in an average sense. This observation is implemented in\ntwo lemmas (Lemma 3.2and Lemma 3.3) after using a standard result (Lemma 3.1) presented\nin Section 3. These two lemmas are the main ingredients of our analysis fo r the Dirichlet and\nNeumann boundary conditions. The proof of the first lemma is m ainly based on the characteristic\nmethod while as the proof of the second lemma is inspired from the theory of functions with\nbounded mean oscillations due to John and Nirenberg [ 18]. As seen later that, the analysis for\nthe dynamic boundary condition is much simpler for which the use of Lemma 3.1is sufficient.5\nAninterestingpointofouranalysisisthefactthatthesele mmasdonotdependontheboundary\nconditions used. In fact, one can apply it in a setting where a bound of the internal energy is\naccessible. This allows us to deal with all the boundary cond itions considered in this paper by the\nsame way. Another point of our analysis which is helpful to be mentioned is that the asymptotic\nstability for hyperbolic systems in one dimensional space h as been mainly studied for general\nsolutions. This is not the case in the setting of Theorem 1.2where the asymptotic stability holds\nunder condition ( 1.10). It is also worth noting that the time-dependent coefficient s generally make\nthe phenomena more complex, see [ 13] for a discussion on the optimal null-controllable time.\nThe analysis in this paper cannot handle the cases p= 1 and p= +∞. Partial results in this\ndirection for the Dirichlet boundary condition can be found in [19] whereais constant and in\nsome range. These cases will be considered elsewhere by differ ent approaches.\nThe paper is organized as follows. The well-posedness of ( 1.1) equipped with one of the bound-\nary conditions ( 1.2) and (1.3) is discussed in Section 2, where a slightly more general context is\nconsidered (the boundary condition ( 1.4) is considered directly in Section 6; comments on this\npoint is given in Remark 6.3). Section 4is devoted to the proof of Theorem 1.1and Theorem 1.2.\nWe also relaxed slightly the non-negative assumption on ain Theorem 1.1and Theorem 1.2there\n(see Theorem 4.1and Theorem 4.2) using a standard perturbative argument. The Neumann\nboundary condition is studied in Section 5and the Dynamic boundary condition is considered in\nSection6.\n2.The well-posedness in Lp-setting\nInthissection, wegivethemeaningof thesolutions of theda mpedwave equation ( 1.1)equipped\nwith either the Dirichlet boundary condition ( 1.2) or the Neumann boundary condition ( 1.3) and\nestablish their well-posedness in the Lp-framework with 1 ≤p≤+∞. We will consider a slightly\nmore general context. More precisely, we consider the syste m\n(2.1)/braceleftBigg\n∂ttu−∂xxu+a∂tu+b∂xu+cu=fin (0,T)×(0,1),\nu(0,·) =u0, ∂tu(0,·) =u1in (0,1),\nequipped with either\n(2.2) Dirichlet boundary condition: u(t,0) =u(t,1) = 0 for t∈(0,T),\nor\n(2.3) Neumann boundary condition: ∂xu(t,0) =∂xu(t,1) = 0 for t∈(0,T).\nHerea,b,c∈L∞((0,T)×(0,1)) andf∈Lp((0,T)×(0,1)).\nWe begin with the Dirichlet boundary condition.\nDefinition 2.1. LetT >0,1≤p <+∞,a,b,c∈L∞((0,T)×(0,1)),f∈Lp((0,T)×(0,1)),\nu0∈W1,p\n0(0,1), andu1∈Lp(0,1). A function u∈C([0,T];W1,p\n0(0,1))∩C1([0,T];Lp(0,1))is\ncalled a (weak) solution of (2.1)and(2.2)(up to time T) if\n(2.4) u(0,·) =u0, ∂tu(0,·) =u1in(0,1),6 Y. CHITOUR AND H.-M. NGUYEN\nand\n(2.5)d2\ndt2ˆ1\n0u(t,x)v(x)dx+ˆ1\n0ux(t,x)vx(x)dx+ˆ1\n0a(t,x)ut(t,x)v(x)dx\n+ˆ1\n0b(t,x)ux(t,x)v(x)dx+ˆ1\n0c(t,x)u(t,x)v(x)dx=ˆ1\n0f(t,x)v(x)dx\nin the distributional sense in (0,T)for allv∈C1\nc(0,1).\nDefinition 2.1can be modified to deal with the case p= +∞as follows.\nDefinition 2.2. LetT >0,a,b,c∈L∞((0,T)×(0,1)),f∈L∞((0,T)×(0,1)),u0∈W1,∞\n0(0,1),\nandu1∈L∞(0,1). A function u∈L∞([0,T];W1,∞\n0(0,1))∩W1,∞([0,T];L∞(0,1))is called a\n(weak) solution of (2.1)and(2.2)(up to time T) ifu∈C([0,T];W1,2\n0(0,1))∩C1([0,T];L2(0,1))\n1and satisfies (2.4)and(2.5).\nConcerning the Neumann boundary condition, we have the foll owing definition.\nDefinition 2.3. LetT >0,1≤p <+∞,a,b,c∈L∞((0,T)×(0,1)),f∈Lp((0,T)×(0,1)),\nu0∈W1,p(0,1), andu1∈Lp(0,1). A function u∈C([0,T];W1,p(0,1))∩C1([0,T];Lp(0,1))is\ncalled a (weak) solution of (2.1)and(2.3)(up to time T) if(2.4)is valid and\n(2.6)d2\ndt2ˆ1\n0u(t,x)v(x)dx+ˆ1\n0ux(t,x)vx(x)dx\n+ˆ1\n0b(t,x)ux(t,x)v(x)dx+ˆ1\n0c(t,x)u(t,x)v(x)dx+ˆ1\n0a(t,x)ut(t,x)v(x)dx=ˆ1\n0f(t,x)v(x)dx\nholds in the distributional sense in (0,T)for allv∈C1([0,1]).\nDefinition 2.1can be modified to deal with the case p= +∞as follows.\nDefinition 2.4. LetT >0,a,b,c∈L∞((0,T)×(0,1)),f∈L∞((0,T)×(0,1)),u0∈W1,∞(0,1),\nandu1∈L∞(0,1). A function u∈L∞([0,T];W1,∞(0,1))∩W1,∞([0,T];L∞(0,1))is called a\n(weak) solution of (2.1)and(2.3)(up to time T) ifu∈C([0,T];W1,2(0,1))∩C1([0,T];L2(0,1))\n2,(2.4)is valid, and (2.5)holds in the distributional sense in (0,T)for allv∈C1([0,1]).\nConcerningthe well-posedness of the Dirichlet system ( 2.1) and (2.2), we establish the following\nresult.\nProposition 2.1. LetT >0,1≤p≤+∞, anda,b,c∈L∞((0,T)×(0,1)), and let u0∈\nW1,p\n0(0,1),u1∈Lp(0,1), andf∈Lp/parenleftbig\n(0,T)×(0,1)/parenrightbig\n. Then there exists a unique (weak) solution\nuof(2.1)and(2.2). Moreover, it holds\n(2.7)\n/bardbl∂tu(t,·)/bardblLp(0,1)+/bardbl∂xu(t,·)/bardblLp(0,1)≤C/parenleftBig\n/bardblu1/bardblLp(0,1)+/bardbl∂xu0/bardblLp(0,1)+/bardblf/bardblLp/parenleftbig\n(0,T)×(0,1)/parenrightbig/parenrightBig\n, t≥0\nfor some positive constant C=C(p,T,/bardbla/bardblL∞,/bardblb/bardblL∞,/bardblc/bardblL∞)which is independent of u0,u1, and\nf.\n1By interpolation, one can use C([0,T];W1,2\n0(0,1))∩C1([0,T];L2(0,1)) instead of C([0,T];W1,2\n0(0,1))∩\nC1([0,T];L2(0,1)) for any 1 ≤q <+∞. This condition is used to give the meaning of the initial con ditions.\n2By interpolation, one can use C([0,T];W1,2\n0(0,1))∩C1([0,T];L2(0,1)) instead of C([0,T];W1,2(0,1))∩\nC1([0,T];L2(0,1)) for any 1 ≤q <+∞. This condition is used to give the meaning of the initial con ditions.7\nConcerning the well-posedness of the Neumann system ( 2.1) and (2.3), we prove the following\nresult.\nProposition 2.2. LetT >0,1≤p≤+∞, anda,b,c∈L∞((0,T)×(0,1)), and let u0∈\nW1,p(0,1),u1∈Lp(0,1), andf∈Lp/parenleftbig\n(0,T)×(0,1)/parenrightbig\n. Then there exists a unique (weak) solution\nuof(2.1)and(2.3)and\n(2.8)\n/bardbl∂tu(t,·)/bardblLp(0,1)+/bardbl∂xu(t,·)/bardblLp(0,1)≤C/parenleftBig\n/bardblu1/bardblLp(0,1)+/bardbl∂xu0/bardblLp(0,1)+/bardblf/bardblLp/parenleftbig\n(0,T)×(0,1)/parenrightbig/parenrightBig\n, t≥0\nfor some positive constant C=C(p,T,/bardbla/bardblL∞,/bardblb/bardblL∞,/bardblc/bardblL∞)which is independent of u0,u1, and\nf.\nRemark 2.1. The definition of weak solutions and the well-posedness are s tated for p= 1 and\np= +∞as well. The existence and the well-posedness is well-known in the case p= 2. The\nstandard analysis in the case p= 2 is via the Galerkin method.\nThe rest of this section is devoted to the proof of Propositio n2.1and Proposition 2.2in\nSection2.1and Section 2.2, respectively.\n2.1.Proof of Proposition 2.1.The proof is divided into two steps in which we prove the\nuniqueness and the existence.\n•Step 1: Proof of the uniqueness. Assume that uis a (weak) solution of ( 2.1) withf= 0 in\n(0,T)×(0,1) andu0=u1= 0 in (0 ,1). We will show that u= 0 in (0 ,T)×(0,1). Set\n(2.9) g(t,x) =−a(t,x)∂tu(t,x)−b(t,x)∂xu(t,x)−c(t,x)u(t,x).\nThenuis a weak solution of the system\n(2.10)\n\n∂ttu−∂xxu=g in (0,T)×(0,1),\nu(t,0) =u(t,1) = 0 for t∈(0,T),\nu(0,·) = 0, ∂tu(0,·) = 0 in (0 ,1).\nExtenduandgin (0,T)×Rby appropriate reflection in xfirst by odd extension in ( −1,0), i.e.,\nu(t,x) =−u(t,−x) andg(t,x) =−g(t,−x) in (0,T)×(−1,0) and so on, and still denote the\nextension by uandg. Thenu∈C([0,T];W1,p(−k,k))∩C1([0,t];Lp(−k,k)) andg∈Lp/parenleftbig\n(0,T)×\n(−k,k)/parenrightbig\nfork≥1 and for 1 ≤p <+∞, and similar facts holds for p= +∞. We also obtain that\nu(0,·) = 0 and ∂tu(0,·) = 0 inR, and\n(2.11) ∂ttu−∂xxu=gin (0,T)×Rin the distributional sense .\nThe d’Alembert formula gives, for t≥0, that\n(2.12) u(t,x) =1\n2ˆt\n0ˆx+t−τ\nx−t+τg(τ,y)dydτ.\nWe then obtain for t≥0\n(2.13) ∂tu(t,x) =1\n2ˆt\n0g(τ,x+t−τ)+g(τ,x−t+τ)dτ\nand\n(2.14) ∂xu(t,x) =1\n2ˆt\n0g(τ,x+t−τ)−g(τ,x−t+τ)dτ.8 Y. CHITOUR AND H.-M. NGUYEN\nUsing (2.9), we derive from ( 2.12), (2.13) and (2.14) that, for 1 ≤p <+∞and fort≥0,\n(2.15)ˆ1\n0|∂tu(t,x)|p+|∂tu(t,x)|p+|∂xu(t,x)|pdx\n≤Cˆt\n0ˆ1\n0/parenleftBig\n|∂tu(s,y)|p+|∂xu(s,y)|p+|u(s,y)|p/parenrightBig\ndyds,\nand, for p= +∞,\n(2.16)/bardblu(t,·)/bardblL∞(0,1)+/bardbl∂tu(t,·)/bardblL∞(0,1)+/bardbl∂xu(t,·)/bardblL∞(0,1)\n≤Ct/parenleftBig\n/bardbl∂tu(t,·)/bardblL∞/parenleftbig\n(0,t)×(0,1)/parenrightbig+/bardbl∂xu(t,·)/bardblL∞/parenleftbig\n(0,t)×(0,1)/parenrightbig+/bardblu(t,·)/bardblL∞/parenleftbig\n(0,t)×(0,1)/parenrightbig/parenrightBig\n,\nfor positive constant Conly depending only on p,T,/bardbla/bardblL∞,/bardblb/bardblL∞,/bardblc/bardblL∞. In the sequel, such\nconstants will again be denoted by C.\nIt is immediate to deduce from the above equations that u= 0 on [0 ,1/2C]×(0,1) and then\nu= 0 in (0 ,T)×(0,1). The proof of the uniqueness is complete.\n•Step 2: Proof of the existence. Let ( an), (bn), and (cn) be smooth functions in [0 ,T]×[0,1]\nsuch that supp an,suppbn,suppcn∩0×[0,1] =∅,\n(an,bn,cn)⇀(a,b,c) weakly star in/parenleftBig\nL∞/parenleftbig\n(0,T)×(0,1)/parenrightbig/parenrightBig3\n,\nand\n(an,bn,cn)→(a,b,c) in/parenleftBig\nLq/parenleftbig\n(0,T)×(0,1)/parenrightbig/parenrightBig3\nfor 1≤q <+∞.\nLetu0,n∈C∞\nc(0,1) andu1,n∈C∞\nc(0,1) be such that, if 1 ≤p <+∞,\nu0,n→u0inW1,p\n0(0,1) and u1,n→u1inLp(0,1),\nand, ifp= +∞then the following two facts hold\nu0,n⇀ u0weakly star in W1,∞\n0(0,1) and u1,n⇀ u1weakly star in L∞(0,1),\nand, for 1 ≤q <+∞,\nu0,n→u0inW1,q\n0(0,1) and u1,n→u1inLq(0,1).\nThe existence of ( an,bn,cn) and the existence of u0,nandu1,nfollows from the standard theory\nof Sobolev spaces, see, e.g., [ 5].\nLetunbe the weak solution corresponding to ( an,bn,cn) with initial data ( u0,n,u1,n). Thenun\nis smooth in [0 ,T]×[0,1]. Set\ngn(t,x) =−an(t,x)∂tun(t,x)−bn(t,x)∂xun(t,x)−cn(t,x)un(t,x) in (0,T)×(0,1).\nExtendun,gn, andfin (0,T)×Rby first odd refection in ( −1,0) and so on, and still denote the\nextension by unandgn, andf. We then have\n(2.17) ∂ttun−∂xxun=gn+fin (0,T)×R,9\nThe d’Alembert formula gives\nu(t,x) =1\n2ˆt\n0ˆx+t−τ\nx−t+τgn(τ,y)+f(τ,y)dydτ\n+1\n2/parenleftBig\nun(0,x−t)+un(0,x+t)/parenrightBig\n+1\n2ˆx+t\nx−t∂tun(0,y)dy.\nAs in the proof of the uniqueness, we then have, for 1 ≤p <+∞and 0< t < T,\n(2.18)ˆ1\n0|un(t,x)|p+|∂tun(t,x)|p+|∂xun(t,x)|pdx\n≤Cˆt\n0ˆ1\n0/parenleftBig\n|∂tu(s,y)|p+|∂xu(s,y)|p/parenrightBig\ndyds\n+C/parenleftbigg\n/bardblun(0,·)/bardblp\nW1,p+/bardbl∂tun(0,·)/bardblp\nLp+ˆt\n0ˆ1\n0|f(s,y)|pdyds/parenrightbigg\n,\nand, for p= +∞,\n(2.19)/bardblu(t,·)/bardblL∞(0,1)+/bardbl∂tu(t,·)/bardblL∞(0,1)+/bardbl∂xu(t,·)/bardblL∞(0,1)\n≤Ct/parenleftbigg\n/bardbl∂tu(t,·)/bardblL∞/parenleftbig\n(0,t)×(0,1)/parenrightbig+/bardbl∂xu(t,·)/bardblL∞/parenleftbig\n(0,t)×(0,1)/parenrightbig/parenrightbigg\n+C/parenleftbigg\n/bardblun(0,·)/bardblW1,∞+/bardbl∂tun(0,·)/bardblL∞+/bardblf/bardblL∞/parenleftbig\n(0,t)×(0,1)/parenrightbig/parenrightbigg\n.\nLettingn→+∞, we derive ( 2.8) from (2.18) and (2.19).\nTo derive that u∈C([0,T];W1,p\n0(0,1))∩C1([0,T];Lp(0,1)) in the case 1 ≤p <+∞and\nu∈C([0,T];W1,2\n0(0,1))∩C1([0,T];L2(0,1)) otherwise, one just notes that ( un) is a Cauchy\nsequence in these spaces correspondingly.\nThe proof is complete. /square\nRemark 2.2. Our proof on the well-posedness is quite standard and is base d on the d’Alembert\nformula. This formula was also used previously in [ 19].\nRemark 2.3. There are several ways to give the notion of weak solution eve n in the case p= 2,\nsee, e.g., [ 2,8]. The definitions given here is a nature modification of the ca sep= 2 given in [ 2].\n2.2.Proof of Proposition 2.2.The proof of Proposition 2.2is similar to the one of Propo-\nsition2.1. To apply the d’Alembert formula, one just needs to extend va rious function appro-\npriately and differently. For example, in the proof of the uniq ueness, one extend uandgin\n(0,T)×Rby appropriate reflection in xfirst by even extension in ( −1,0), i.e.,u(t,x) =u(t,−x)\nandg(t,x) =g(t,−x) in (0,T)×(−1,0) and so on. The details are left to the reader. /square\n3.Some useful lemmas\nIn this section, we prove three lemmas which will be used thro ugh out the rest of the paper.\nThe first one is quite standard and the last two ones are the mai n ingredients of our analysis for\nthe Dirichlet and Neumann boundary condition. We begin with the following lemma.10 Y. CHITOUR AND H.-M. NGUYEN\nLemma 3.1. Let1< p <+∞,0< T <ˆT0, anda∈L∞((0,T)×(0,1))be such that a≥0in\n(0,T)×(0,1). There exists a positive constant Cdepending only on p,ˆT0, and/bardbla/bardblL∞such that,\nfor(ρ,ξ)∈/bracketleftbig\nLp/parenleftbig\n(0,T)×(0,1)/parenrightbig/bracketrightbig2,\n(3.1)ˆT\n0ˆ1\n0a|ρ−ξ|p(t,x)dxdt≤/braceleftBiggCmpifp≥2,\nC(mp+m2/p\np)if1< p <2,\nwhere\n(3.2) mp=ˆ1\n0ˆT\n0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dtdx.\nProof.The proof of Lemma 3.1is quite standard. For the convenience of the reader, we pres ent\nits proof. There exists a positive constant Cpdepending only on psuch that\n•for 2≤p <+∞, it holds, for α,β∈R,\n(α−β)(α|α|p−2−β|β|p−2)≥Cp|α−β|p;\n•for 1< p <2, it holds, for α,β∈R3\n(α−β)(α|α|p−2−β|β|p−2)≥Cpmin/braceleftbig\n|α−β|p,|α−β|2/bracerightbig\n.\nUsing this, we derive that\nˆT\n0ˆ1\n0\n|ρ−ξ|≥1a|ρ−ξ|pdxdt+ˆT\n0ˆ1\n0\n|ρ−ξ|<1a|ρ−ξ|max{p,2}dxdt≤mp.\nThis yields\n(3.3)ˆT\n0ˆ1\n0a|ρ−ξ|p(t,x)dxdt≤Cmpifp≥2,\nand, using H¨ older’s inequality, one gets\n(3.4)ˆT\n0ˆ1\n0a|ρ−ξ|p(t,x)dxdt≤C(mp+m2/p\np) if 1< p≤2,\nThe conclusion follows from ( 3.3) and (3.4). /square\nThefollowing lemmais oneof themain ingredients in theanal ysis of theDirichlet andNeumann\nboundary conditions.\nLemma 3.2. Let1< p <+∞,0< T0< T <ˆT0,ε0>0,λ >0, anda∈L∞((0,T)×(0,1))be\nsuch that T > T 0+4ε0,a≥0anda≥λ >0in(0,T)×(x0−ε0,x0+ε0)⊂(0,T)×(0,1)for\nsomex0∈(0,1). Let(ρ,ξ)be a broad solution of the system\n(3.5)/braceleftBigg\nρt−ρx=−1\n2a(ρ−ξ)in(0,T)×(0,1),\nξt+ξx=1\n2a(ρ−ξ)in(0,T)×(0,1).\nSet\n(3.6) mp=ˆ1\n0ˆT\n0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dtdx.\n3Using the symmetry between αandβ, one can assume |α| ≥ |β|and by considering β/|α|, it is enough to prove\nthese inequalities for α= 1 and β∈(−1,1). One finally reduces the analysis for β∈(0,1) and even βclose to one.\nThe conclusion follows by performing a Taylor expansion wit h respect to 1 −β.11\nThen there exists z∈(x0−ε0/2,x0+ε0/2)such that\n(3.7)ˆε0/2\n0ˆT\n0|ρ(t+s,z)−ρ(t,z)|pdtds+ˆε0/2\n0ˆT\n0|ξ(t+s,z)−ξ(t,z)|pdtds\n+ˆT\n0|ρ(t,z)−ξ(t,z)|pdt+ˆT\n0ˆ1\n0a|ρ−ξ|p(t,x)dxdt\n≤/braceleftBiggCmp ifp≥2,\nC(mp+m2/p\np)if1≤p <2.\nfor some positive constant Cdepending only on ε0,λ,p,T0,ˆT0, and/bardbla/bardblL∞.\nProof.Set\nT1=T−4ε0andT2=T−2ε0.\nThenT > T2> T1> T0.\nWe have, for s∈(−ε0/2,ε0/2) andy∈(x0−ε0/2,x0+ε0/2),\n(3.8)ρ(t,y+2s)−ρ(t,y) =/parenleftBig\nρ(t+2s,y)−ρ(t+s,y+s)/parenrightBig\n+/parenleftBig\nρ(t+s,y+s)−ξ(t+s,y+s)/parenrightBig\n+/parenleftBig\nξ(t+s,y+s)−ξ(t,y)/parenrightBig\n+/parenleftBig\nξ(t,y)−ρ(t,y)/parenrightBig\n.\nBy the characteristics method, we obtain\n(3.9) ξ(t+s,y+s)−ξ(t,y) =1\n2ˆs\n0a(t+τ,y+τ)/parenleftBig\nρ(t+τ,y+τ)−ξ(t+τ,y+τ)/parenrightBig\ndτ\nand\n(3.10)ρ(t+2s,y)−ρ(t+s,y+s)\n=1\n2ˆ2s\nsa(t+τ,y+2s−τ)/parenleftBig\nρ(t+τ,y+2s−τ)−ξ(t+τ,y+2s−τ)/parenrightBig\ndτ.\nCombining ( 3.8), (3.9), and (3.10), after integrating with respect to tfrom 0 to T1, we obtain, for\n0≤s≤ε0/2,\nˆT1\n0|ρ(t+2s,y)−ρ(t,y)|pdt≤4p−1/parenleftbiggˆT2\n0|ρ(t,y+s)−ξ(t,y+s)|pdt\n+2ˆT2\n0ˆ1\n0ap|ρ−ξ|p(t,x)dtdx+ˆT\n0|ρ(t,y)−ξ(t,y)|pdt/parenrightbigg\n.12 Y. CHITOUR AND H.-M. NGUYEN\nIntegrating the above inequality with respect to sfrom 0 to ε0/2, we obtain\n(3.11)ˆε0/2\n0ˆT1\n0|ρ(t+2s,y)−ρ(t,y)|pdtds\n≤4p/parenleftbiggˆx0+ε0\nx0−ε0ˆT\n0|ρ(t,x)−ξ(t,x)|pdtdx\n+ε0ˆT\n0|ρ(t,y)−ξ(t,y)|pdt+ε0ˆ1\n0ˆT\n0ap|ρ−ξ|p(t,x)dtdx/parenrightbigg\n.\nSimilarly, we have\n(3.12)ˆε0/2\n0ˆT1\n0|ξ(t+2s,y)−ξ(t,y)|pdtds\n≤4p/parenleftbiggˆx0+ε0\nx0−ε0ˆT\n0|ρ(t,x)−ξ(t,x)|pdtdx\n+ε0ˆT\n0|ρ(t,y)−ξ(t,y)|pdt+ε0ˆ1\n0ˆT\n0ap|ρ−ξ|p(t,x)dtdx/parenrightbigg\n.\nTakey=z∈(x0−ε0/2,x0+ε0/2) such that\n(3.13)ˆT\n0|ρ(t,z)−ξ(t,z)|pdt≤1\nε0ˆx0+ε0\nx0−ε0ˆT\n0|ρ−ξ|p(t,x)dxdt.\nBy choosing y=zin (3.11) and (3.12), then by using ( 3.13) and the fact that (itself consequence\nof (1.5))\nˆx0+ε0\nx0−ε0ˆT\n0|ρ(t,x)−ξ(t,x)|pdtdx≤C(a,p)ˆ1\n0ˆT\n0ap|ρ−ξ|p(t,x)dtdx,\nfor some positive constant C(a,p) only depending on a,p, one gets the conclusion. /square\nThe next lemma is also a main ingredient of our analysis for th e Dirichlet and Neumann\nboundary conditions.\nLemma 3.3. Let1≤p <+∞andL > l > 0, and let u∈Lp(0,L+l). Then there exists a\npositive constant Cdepending only on p,L, andlsuch that\n(3.14)ˆL\n0|u(x)− L\n0u(y)dy|pdx≤Cˆl\n0ˆL\n0|u(x+s)−u(x)|pdxds.\nHere and in what follows,fflb\nameans1\nb−a´b\naforb > a.\nProof.By scaling, one can assume that L= 1. Fix n≥2 such that 2 /n≤l≤2/(n−1).\nOne first notes that, for x∈[0,1],\n(3.15) x+1/n\nx/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleu(x)− x+1/n\nxu(y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\ndxJensen\n≤ x+1/n\nx x+1/n\nx|u(x)−u(y)|pdxdy\n≤n2ˆ2/n\n0ˆ1\n0|u(x+s)−u(x)|pdxds13\nand\n(3.16)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle x+1/n\nxu(s)ds− x+2/n\nx+1/nu(t)dt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\nJensen\n≤ x+1/n\nx x+2/n\nx+1/n|u(s)−u(t)|pdtds\n≤n2ˆ2/n\n0ˆ1\n0|u(x+s)−u(x)|pdxds.\nFor 0≤k≤n−1, set\nak= k/n+1/n\nk/nu(s)ds.\nWe then derive from ( 3.16) that, for 0 ≤i < j≤n−1,\n|aj−ai|p≤(|ai+1−ai|+···+|aj−aj−1|)p\n≤np−1(|ai+1−ai|p+···+|aj−aj−1|p)\n≤np+1ˆ2/n\n0ˆ1\n0|u(x+s)−u(x)|pdxds.\nThis implies, for 0 ≤k≤n−1,\n(3.17)/vextendsingle/vextendsingle/vextendsingle/vextendsingleak−ˆ1\n0u(t)dt/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\n≤/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nnn−1/summationdisplay\ni=0|ak−ai|/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\n≤1\nnn−1/summationdisplay\ni=0|ak−ai|p≤np+1ˆ2/n\n0ˆ1\n0|u(x+s)−u(x)|pdxds.\nWe have\n(3.18)ˆ1\n0/vextendsingle/vextendsingle/vextendsingle/vextendsingleu(x)−ˆ1\n0u(y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\ndx=n−1/summationdisplay\nk=0ˆk/n+1/n\nk/n/vextendsingle/vextendsingle/vextendsingle/vextendsingleu(x)−ˆ1\n0u(y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\ndx\n≤2p−1n−1/summationdisplay\nk=0ˆk/n+1/n\nk/n|u(x)−ak|pdx+2p−1n−1/summationdisplay\nk=0/vextendsingle/vextendsingle/vextendsingle/vextendsingleak−ˆ1\n0u(y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\nThe conclusion with C= 2pnp+1now follows from ( 3.15), (3.17), and (3.18) after noting that\nL= 1 and 2 /n≤l. /square\nRemark 3.1. Related ideas used in the proof of Lemma 3.3was implemented in the proof of\nCaffarelli-Kohn-Nirenberg inequality for fractional Sobol ev spaces [ 28].\n4.Exponential decay in Lp-framework for the Dirichlet boundary condition\nIn this section, we prove Theorem 1.1and Theorem 1.2. We begin with the proof Theorem 1.2\nin the first section, and then use it to prove Theorem 1.1in the second section. We finally\nextend these results for awhich might be negative in some regions using a standard pert urbation\nargument in the third section.14 Y. CHITOUR AND H.-M. NGUYEN\n4.1.Proof of Theorem 1.2.We will only consider smooth solutions ( ρ,ξ)4. The general case\nwill follow by regularizing arguments. Moreover, replacin g (ρ,ξ) by (ρ−c0,ξ−c0), where the\nconstant c0is defined in ( 1.12), we can assume that\nˆ1\n0(ρ0+ξ0)dx= 0.\nMultiplying the equation of ρwithρ|ρ|p−2, the equation of ξwithξ|ξ|p−2, and integrating the\nexpressions with respect to x, after using the boundary conditions, we obtain, for t >0,\n(4.1)1\npd\ndtˆ1\n0(|ρ(t,x)|p+|ξ(t,x)|p)dx+1\n2ˆ1\n0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dx= 0.\nThis implies\n(4.2)1\np/bardbl(ρ,ξ)(t,·)/bardblp\nLp(0,1)+1\n2ˆt\n0ˆ1\n0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dxdt=1\np/bardbl(ρ0,ξ0)/bardblp\nLp(0,1).\nIntegrating the equations of ρandξ, summing them up and using the boundary conditions, we\nobtain\nd\ndtˆ1\n0/parenleftBig\nρ(t,x)+ξ(t,x)/parenrightBig\ndx= 0 fort >0.\nIt follows that\n(4.3)ˆ1\n0/parenleftBig\nρ(t,x)+ξ(t,x)/parenrightBig\ndx=ˆ1\n0/parenleftBig\nρ(0,x)+ξ(0,x)/parenrightBig\ndx= 0 fort≥0.\nBy (4.2) and (4.3), toderive ( 1.11), it sufficesto prove that thereexists aconstant c >0depending\nonly on/bardbla/bardblL∞(R+×(0,1)),ε0,γ, andpsuch that for any T >2, there exists cT>0 only depending\nonp,T,aso that\n(4.4)ˆT\n0ˆ1\n0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dxdt≥cT/bardbl(ρ0,ξ0)/bardblp\nLp(0,1).\nBy scaling, without loss of generality, one might assume tha t\n(4.5) /bardbl(ρ0,ξ0)/bardblLp(0,1)= 1\nSet\nmp:=ˆT\n0ˆ1\n0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dxdt.\nApplying Lemma 3.1, we have\n(4.6)ˆT\n0ˆ1\n0a|ρ−ξ|p(t,x)dxdt≤C(mp+m2/p\np).\nBy Lemma 3.2there exists z∈(x0−ε0/2,x0+ε0/2) such that\n(4.7)ˆε0/2\n0ˆT\n0|ρ(t+s,z)−ρ(s,z)|pdtds+ˆε0/2\n0ˆT\n0|ξ(t+s,z)−ξ(s,z)|pdtds\n+ˆT\n0|ρ(t,z)−ξ(t,z)|pdt≤C(mp+m2/p\np).\n4We thus assume that ais smooth. Nevertheless, the constants in the estimates whi ch will be derived in the\nproof depend only on p,/bardbla/bardblL∞,λ, andε0.15\nBy Lemma 3.3, we have\n(4.8)ˆT\n0|ρ(t,z)−Aρ|pdt≤Cˆε0/2\n0ˆT\n0|ρ(t+s,z)−ρ(s,z)|pdtds\nand\n(4.9)ˆT\n0|ξ(t,z)−Aξ|pdt≤Cˆε0/2\n0ˆT\n0|ξ(t+s,z)−ξ(s,z)|pdtds.\nwhere we have set\n(4.10) Aρ:= T\n0ρ(s,z)ds, A ξ:= T\n0ξ(s,z)ds.\nCombining ( 4.7), (4.8), and (4.9) yields\n(4.11)ˆT\n0|ρ(t,z)−Aρ|pdt+ˆT\n0|ξ(t,z)−Aξ|pdt\n+ˆT\n0|ρ(t,z)−ξ(t,z)|pdt≤C(mp+m2/p\np).\nWe next prove the following estimates\n(4.12)ˆ1\n0|ρ(0,x)−Aξ|pdx≤C(mp+m2/p\np)\nand\n(4.13)ˆ1\n0|ξ(0,x)−Aρ|pdx≤C(mp+m2/p\np).\nThe arguments being similar, we only provide that of ( 4.12). Forx∈(0,1), one has, by using the\nboundary condition at x= 0, i.e., ρ(·,0) =ξ(·,0),\nρ(0,x) =/parenleftBig\nρ(0,x)−ρ(x,0)/parenrightBig\n+ρ(x,0)\n=/parenleftBig\nρ(0,x)−ρ(x,0)/parenrightBig\n+ξ(x,0)\n=/parenleftBig\nρ(0,x)−ρ(x,0)/parenrightBig\n+/parenleftBig\nξ(x,0)−ξ(x+z,z)/parenrightBig\n+ξ(x+z,z),\nwhich yields, after substracting Aξto both sides of the above equality,\n(4.14)ˆ1\n0|ρ(0,x)−Aξ|pdx≤3p−1/parenleftbiggˆ1\n0|ρ(0,x)−ρ(x,0)|pdx+ˆ1\n0|ξ(x,0)−ξ(x+z,z)|pdx\n+ˆ1\n0|ξ(x+z,z)−Aξ|pdx/parenrightbigg\n.\nWe use the characteristics method and ( 3.9),(3.10) to upper bound the first two integrals in the\nright-hand side of ( 4.14) byC(mp+m2/p\np). As for the third integral in the right-hand side of16 Y. CHITOUR AND H.-M. NGUYEN\n(4.14), we perform the change of variables t=x+zto obtain\nˆ1\n0|ξ(x+z,z)−Aξ|pdx=ˆz+1\nz|ξ(t,z)−Aξ|pdt\n≤ˆT\n0|ξ(t,z)−Aξ|pdt,\nwhich is upper bounded by C(mp+m2/p\np) according to ( 4.11). The proof of ( 4.12) is complete.\nWe now resume the argument for ( 4.4). We start by noticing that, for every t∈(0,T)\n|Aρ−Aξ| ≤ |Aρ−ρ(t,z)|+|Aξ−ρ(t,z)|+|ρ(t,z)−ξ(t,z)|.\nTaking the p-th power, integrating over t∈(0,T) and using ( 4.11), one gets that\n(4.15) |Aρ−Aξ|p≤C(mp+m2/p\np).\nSimilarly, for every x∈(0,1),\nAρ+Aξ=/parenleftbig\nAρ−ξ(0,x)/parenrightbig\n+/parenleftbig\nAξ−ρ(0,x)/parenrightbig\n+/parenleftbig\nρ(0,x)+ξ(0,x)/parenrightbig\n.\nIntegrating over x∈(0,1) and using ( 4.3), then taking the p-th power and using ( 4.12) and (4.13)\nyield\n(4.16) |Aρ+Aξ|p≤C(mp+m2/p\np).\nStill, for x∈(0,1), it holds\n|ρ(0,x)|p+|ξ(0,x)|p≤2p−1/parenleftBig\n|Aρ−ξ(0,Aξ|p+|Aρ−ξ(0,x)|p/parenrightBig\n+|Aρ|p+|Aξ|p.\nIntegrating over x∈(0,1) and using ( 4.5), one gets\n(4.17) 1 ≤ |Aρ|p+|Aξ|p+C(mp+m2/p\np).\nSince it holds |a|p+|b|p≤ |a+b|p+|a−b|pfor every real numbers a,b, one deduces from ( 4.15),\n(4.16) and (4.17) that\n1≤C(mp+m2/p\np)\nand hence mp≥c3for some positive constant depending only on /bardbla/bardblL∞(R+×(0,1)),ε0,γ, andp\n(after fixing for instance T= 3). The proof of the theorem is complete. /square\n4.2.Proof of Theorem 1.1.Using Theorem 1.2, we obtain the conclusion of Theorem 1.1\nfor smooth solutions. The proof in the general case follows f rom the smooth case by density\narguments. /square\n4.3.On the case anot being non-negative. In this section, we first consider the following\nperturbed system of ( 1.9):\n(4.18)\n\nρt−ρx=−1\n2a(ρ−ξ)−b(ρ−ξ) in R+×(0,1),\nξt+ξx=1\n2a(ρ−ξ)+b(ρ−ξ) in R+×(0,1),\nρ(t,0)−ξ(t,0) =ρ(t,1)−ξ(t,1) = 0 in R+.\nWe establish the following result.17\nTheorem 4.1. Let1< p <+∞,ε0>0,λ >0, anda,b∈L∞/parenleftbig\nR+×(0,1)/parenrightbig\nbe such that a≥0\nanda≥λ >0inR+×(x0−ε0,x0+ε0)⊂R+×(0,1)for some x0∈(0,1). There exists a positive\nconstant αdepending only on p,/bardbla/bardblL∞,ε0, andλsuch that if\n(4.19) /bardblb/bardblL∞≤α,\nthen there exist constants C,γ >0depending only on p,/bardbla/bardblL∞,ε0, andλsuch that, if´1\n0ρ0+\nξ0dx= 0, then the solution (ρ,ξ)of(4.18)satisfies\n(4.20) /bardbl(ρ,ξ)(t,·)/bardblLp(0,1)≤Ce−γt/bardbl(ρ0,ξ0)/bardblLp(0,1), t≥0.\nProof.Multiplying the equation of ρwithρ|ρ|p−2, the equation of ξwithξ|ξ|p−2, and integrating\nthe expressions with respect to x, after using the boundary conditions, we obtain\n1\npd\ndtˆ1\n0(|ρ(t,x)|p+|ξ(t,x)|p)dx+1\n2ˆ1\n0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dx\n+ˆ1\n0b(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dx= 0.\nThis implies\n(4.21)1\np/bardbl(ρ,ξ)(t,·)/bardblp\nLp(0,1)+1\n2ˆt\n0ˆ1\n0a(ρ−ξ)(ρ|ρ|p−1−ξ|ξ|p−1)(t,x)dxdt\n+ˆt\n0ˆ1\n0b(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dx=1\np/bardbl(ρ0,ξ0)/bardblp\nLp(0,1).\nIntegrating the equation of ρandξand using the boundary condition, we obtain\nd\ndtˆ1\n0/parenleftBig\nρ(t,x)+ξ(t,x)/parenrightBig\ndx= 0,fort >0.\nIt follows that\n(4.22)ˆ1\n0/parenleftBig\nρ(t,x)+ξ(t,x)/parenrightBig\ndx=ˆ1\n0/parenleftBig\nρ(0,x)+ξ(0,x)/parenrightBig\ndx= 0,fort >0.\nBy (4.21) and (4.22), to derive ( 4.20), it suffices to prove that there exists a constant c >0\ndepending only on /bardbla/bardblL∞(R+×(0,1)),ε0,γ, andpsuch that for T= 35, it holds\n(4.23)ˆT\n0ˆ1\n0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dxdt≥c/bardbl(ρ0,ξ0)/bardblp\nLp(0,1).\nUsing the facts that a≥0 andbis bounded, a simple application of Gronwall’s lemma to ( 4.21)\nyields the existence of α >0 depending only on /bardblb/bardblL∞(R+×(0,1))so that\n(4.24) /bardbl(ρ,ξ)(t,·)/bardblp\nLp(0,1)≤epαt/bardbl(ρ,ξ)(0,·)/bardblp\nLp(0,1)fort∈[0,T].\n5It holds for T >2 withc=cT.18 Y. CHITOUR AND H.-M. NGUYEN\nLet (ρ1,ξ1) be the unique solution of the system\n(4.25)\n\nρ1,t−ρ1,x=−1\n2a(ρ1−ξ1)−b(ρ−ξ) inR+×(0,1),\nξ1,t+ξ1,x=1\n2a(ρ1−ξ1)+b(ρ−ξ) in R+×(0,1),\nρ1(t,0)��ξ1(t,0) =ρ1(t,1)−ξ1(t,1) = 0 in R+,\nρ1(0,·) =ξ1(0,·) = 0 in (0 ,1).\nThus−b(ρ−ξ) andb(ρ−ξ) can be considered as source terms for the system of ( ρ1,ξ1). We then\nderive from ( 4.24) that\n(4.26) /bardbl(ρ1,ξ1)/bardblLp(T,·)≤Cα/bardbl(ρ,ξ)(0,·)/bardblp\nLp(0,1).\nSet\n/tildewideρ=ρ−ρ1and/tildewideξ=ξ−ξ1.\nThen\n(4.27)\n\n/tildewideρt−/tildewideρx=−1\n2a(t,x)(/tildewideρ−/tildewideξ) in R+×(0,1),\n/tildewideξt+/tildewideξx=1\n2a(t,x)(/tildewideρ−/tildewideξ) in R+×(0,1),\n/tildewideρ(t,0)−/tildewideξ(t,0) =/tildewideρ(t,1)−/tildewideξ(t,1) = 0 in R+,\n/tildewideρ(0,·) =ρ0,/tildewideξ(0,·) =ξ0 in (0,1).\nApplying Theorem 1.2, we have\n(4.28) /bardbl(/tildewideρ,/tildewideξ)(T,·)/bardblLp≤c/bardbl(/tildewideρ,/tildewideξ)(0,·)/bardblLp\nfor some positive constant cdepending only on /bardbla/bardblL∞,ε0, andλ. The conclusion now follows\nfrom (4.26) and (4.27). /square\nRegarding the wave equation, we have\nTheorem 4.2. Let1< p <+∞,ε0>0,λ >0, anda,b∈L∞/parenleftbig\nR+×(0,1)/parenrightbig\nbe such that a≥0\nanda≥λ >0inR+×(x0−ε0,x0+ε0)⊂R+×(0,1). There exists a positive constant α\ndepending only on p,/bardbla/bardblL∞,ε0, andλsuch that if\n(4.29) /bardblb/bardblL∞≤α,\nthen there exist positive constants Candγdepending on p,/bardbla/bardblL∞/parenleftbig\nR+×(0,1)/parenrightbig,ε0, andλsuch that\nfor allu0∈W1,p\n0(0,1)andu1∈Lp(0,1), the unique weak solution u∈C([0,+∞);W1,p\n0(0,1))∩\nC1([0,+∞);Lp(0,1))of\n(4.30)\n\n∂ttu−∂xxu+/parenleftBig\na(t,x)+b(t,x)/parenrightBig\n∂tu= 0inR+×(0,1),\nu(t,0) =u(t,1) = 0 inR+,\nu(0,·) =u0, ∂tu(0,·) =u1 in(0,1),\nsatisfies\n(4.31) /bardbl∂tu(t,·)/bardblp\nLp(0,1)+/bardbl∂xu(t,·)/bardblp\nLp(0,1)≤Ce−γt/parenleftBig\n/bardblu1/bardblp\nLp(0,1)+/bardbl∂xu0/bardblp\nLp(0,1)/parenrightBig\n, t≥0.\nProof.The proof of Theorem 4.2is similar to that of Theorem 1.1however instead of using\nTheorem 1.2one apply Theorem 4.1. The details are left to the reader. /square19\n5.Exponential decay in Lp-framework for the Neuman boundary condition\nIn this section, we study the decay of the solutions of the dam ped wave equation equipped the\nNeumann boundary condition and the solutions of the corresp onding hyperbolic systems. Here is\nthe first main result of this section concerning the wave equa tion.\nTheorem 5.1. Let1< p <+∞,ε0>0,λ >0, and let a∈L∞/parenleftbig\nR+×(0,1)/parenrightbig\nbe such that a≥0\nanda≥λ >0inR+×(x0−ε0,x0+ε0)⊂R+×(0,1)for some x0∈(0,1). There exist positive\nconstants Candγdepending only on p,/bardbla/bardblL∞/parenleftbig\nR+×(0,1)/parenrightbig,ε0, andλsuch that for all u0∈W1,p(0,1)\nandu1∈Lp(0,1), the unique weak solution u∈C([0,+∞);W1,p(0,1))∩C1([0,+∞);Lp(0,1))of\n(1.1)and(1.3)satisfies\n(5.1) /bardbl∂tu(t,·)/bardblLp(0,1)+/bardbl∂xu(t,·)/bardblLp(0,1)≤Ce−γt/parenleftBig\n/bardblu1/bardblLp(0,1)+/bardbl∂xu0/bardblLp(0,1)/parenrightBig\n, t≥0.\nAs in the case where the Dirichlet condition is considered, w e use the Riemann invariants to\ntransform ( 1.1) with Neumann boundary condition into a hyperbolic system. Set\n(5.2)ρ(t,x) =ux(t,x)+ut(t,x) and ξ(t,x) =ux(t,x)−ut(t,x),for (t,x)∈R+×(0,1).\nOne can check that for smooth solutions uof (1.1), the pair of functions ( ρ,ξ) defined in ( 1.8)\nsatisfies the system\n(5.3)\n\nρt−ρx=−1\n2a(ρ−ξ) in R+×(0,1),\nξt+ξx=1\n2a(ρ−ξ) in R+×(0,1),\nρ(t,0)+ξ(t,0) =ρ(t,1)+ξ(t,1) = 0 in R+.\nConcerning ( 5.3), we prove the following result.\nTheorem 5.2. Let1< p <+∞,ε0>0,λ >0, anda∈L∞/parenleftbig\nR+×(0,1)/parenrightbig\nbe such that a≥0and\na≥λ >0inR+×(x0−ε0,x0+ε0)⊂R+×(0,1)for some x0∈(0,1). Then there exist positive\nconstants C,γdepending only on on p,/bardbla/bardblL∞/parenleftbig\nR+×(0,1)/parenrightbig,ε0, andλsuch that the unique solution\nuof(5.3)with the initial condition ρ(0,·) =ρ0andξ(0,·) =ξ0satisfies\n(5.4) /bardbl(ρ,ξ)(t,·)/bardblLp(0,1)≤Ce−γt/bardbl(ρ0,ξ0)/bardblLp(0,1).\nThe rest of this section is organized as follows. The first sub section is devoted to the proof of\nTheorem 5.2and the second subsection is devoted to the proof of Theorem 5.1.\n5.1.Proof of Theorem 5.2.The argument is in the spirit of that of Theorem 1.2. As in there,\nwe will only consider smooth solutions ( ρ,ξ). Multiplying the equation of ρwithρ|ρ|p−2, the\nequation of ξwithξ|ξ|p−2, and integrating the expressions with respect to x, after using the\nboundary conditions, we obtain, for t >0,\n(5.5)1\npd\ndtˆ1\n0(|ρ(t,x)|p+|ξ(t,x)|p)dx+1\n2ˆ1\n0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dx= 0.\nThis implies\n(5.6)1\np/bardbl(ρ,ξ)(t,·)/bardblp\nLp(0,1)+1\n2ˆt\n0ˆ1\n0a(ρ−ξ)(ρ|ρ|p−1−ξ|ξ|p−1)(t,x)dxdt=1\np/bardbl(ρ0,ξ0)/bardblp\nLp(0,1).\nBy (5.6), to derive ( 5.4), it suffices to prove that there exists a constant c >0 depending only\non/bardbla/bardblL∞(R+×(0,1)),ε0,γ, andpsuch that for any T >2, there exists cT>0 only depending on20 Y. CHITOUR AND H.-M. NGUYEN\np,T,aso that\n(5.7)ˆT\n0ˆ1\n0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dxdt≥cT/bardbl(ρ0,ξ0)/bardblp\nLp(0,1).\nBy scaling, without loss of generality, one might assume tha t\n(5.8) /bardbl(ρ0,ξ0)/bardblLp(0,1)= 1\nSet\nmp:=ˆT\n0ˆ1\n0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dxdt.\nApplying Lemma 3.1, we have\n(5.9)ˆT\n0ˆ1\n0a|ρ−ξ|p(t,x)dxdt≤C(mp+m2/p\np).\nBy Lemma 3.2there exists z∈(x0−ε0/2,x0+ε0/2) such that\n(5.10)ˆε0/2\n0ˆT\n0|ρ(t+s,z)−ρ(t,z)|pdtds+ˆε0/2\n0ˆT\n0|ξ(t+s,z)−ξ(t,z)|pdtds\n+ˆT\n0|ρ(t,z)−ξ(t,z)|pdt≤C(mp+m2/p\np).\nApplying Lemma 3.3, we obtain\n(5.11)ˆT\n0|ρ(t,z)− T\n0ρ(s,z)ds|pdt≤Cˆε0/2\n0ˆT\n0|ρ(t+s,z)−ρ(t,z)|pdtds\nand\n(5.12)ˆT\n0|ξ(t,z)− T\n0ξ(s,z)ds|pdt≤Cˆε0/2\n0ˆT\n0|ξ(t+s,z)−ξ(t,z)|pdtds.\nCombining ( 5.10), (5.11), and (5.12) yields\n(5.13)ˆT\n0|ρ(t,z)− T\n0ρ(τ,z)dτ|pdt+ˆT\n0|ξ(t,z)− T\n0ξ(τ,z)dτ|pdt\n+ˆT\n0|ρ(t,z)−ξ(t,z)|pdt≤C(mp+m2/p\np).\nUsing the characteristics method to estimate ρ(τ,0) byρ(τ−z,z) andξ(τ,0) byξ(τ+z,z)\nafter using the boundary condition at 0 and choosing appropr iatelyτ, we derive from ( 5.9) that\n(5.13) that\n(5.14)/vextendsingle/vextendsingle/vextendsingle/vextendsingle T\n0ρ(t,z)dt+ T\n0ξ(t,z)dt/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\n≤C(mp+m2/p\np).\nAs done to obtain ( 4.12) and (4.13), we use the characteristic methods to estimate ρ(0,·) via\nξ(t,z) andξ(0,·) viaρ(t,z) after taking into account the boundary conditions (at x= 0 forρ(0,·)\nand atx= 1 forξ(0,·)), we derive from ( 5.9) and (5.13) that\n(5.15)/vextendsingle/vextendsingle/vextendsingle/vextendsingle T\n0ρ(t,z)dt/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle T\n0ξ(t,z)dt/vextendsingle/vextendsingle/vextendsingle/vextendsinglep\n≥1−C(mp+m2/p\np).21\nCombining ( 5.14) and (5.15), we derive (after choosing T= 3) that there exists a postive constant\nc3only depending on /bardbla/bardblL∞(R+×(0,1)),ε0,γ, andpsuch that mp≥c. The proof of the theorem\nis complete. /square\n5.2.Proof of Theorem 5.1.The proof of Theorem 5.1is in the same spirit of Theorem 1.1.\nHowever, instead of using Theorem 1.2, we apply Theorem 5.2. In fact, as in the proof of Theo-\nrem1.1, we have\nˆ1\n0|∂tu(t,x)−∂xu(t,x)|p+|∂tu(t,x)+∂xu(t,x)|pdx\n≤Ce−γtˆ1\n0|∂tu(0,x)−∂xu(0,x)|p+|∂tu(0,x)+∂xu(0,x)|pdx.\nAssertion ( 5.1) follows with two different appropriate positive constants Candγ. /square\nRemark 5.1. We can also consider the setting similar to the one in Section 4.3and establish\nsimilar results. This allows one to deal with a class of afor which ais not necessary to be\nnon-negative. The analysis for this is almost the same lines as in Section 4.3and is not pursued\nhere.\n6.Exponential decay in Lp-framework for the dynamic boundary condition\nIn this section, we study the decay of the solution of the damp ed wave equation equipped the\ndynamic boundary condition and of the solutions of the corre sponding hyperbolic systems. Here\nis the first main result of this section concerning the wave eq uation.\nTheorem 6.1. Let1< p <+∞,κ >0, anda∈L∞/parenleftbig\nR+×(0,1)/parenrightbig\nnon negative. Then there\nexist positive constants C,γdepending only on p,κ, and/bardbla/bardblL∞/parenleftbig\nR+×(0,1)/parenrightbigsuch that for all u0∈\nW1,p(0,1)andu1∈Lp(0,1), there exists a unique weak solution u∈C([0,+∞);W1,p(0,1))∩\nC1([0,+∞);Lp(0,1))such that ∂tu,∂xu∈C([0,1];Lp(0,T))for allT >0of\n(6.1)\n\n∂ttu−∂xxu+a∂tu= 0 inR+×(0,1),\n∂xu(t,0)−κ∂tu(t,0) =∂xu(t,1)+κ∂tu(t,1) = 0 inR+,\nu(0,·) =u0, ∂tu(0,·) =u1 in(0,1),\nsatisfies\n(6.2) /bardbl∂tu(t,·)/bardblLp(0,1)+/bardbl∂xu(t,·)/bardblLp(0,1)≤Ce−γt/parenleftBig\n/bardblu1/bardblLp(0,1)+/bardbl∂xu0/bardblLp(0,1)/parenrightBig\n, t≥0.\nRemark 6.1. In Theorem 6.1, a weak considered solution of ( 6.1) means that ∂ttu(t,x)−\n∂xxu(t,x) +a(t,x)∂tu= 0 holds in the distributional sense, and the boundary and th e initial\nconditions are understood as usual thanks to the regularity imposing condition on the solutions.\nAs previously, we use the Riemann invariants to transform th e wave equation into a hyperbolic\nsystem. Set\n(6.3)ρ(t,x) =ux(t,x)+ut(t,x) and ξ(t,x) =ux(t,x)−ut(t,x) for (t,x)∈R+×(0,1).22 Y. CHITOUR AND H.-M. NGUYEN\nOne can check that for smooth solutions uof (1.1), the pair of functions ( ρ,ξ) defined in ( 1.8)\nsatisfies the system\n(6.4)\n\nρt−ρx=−1\n2a(t,x)(ρ−ξ) in R+×(0,1),\nξt+ξx=1\n2a(t,x)(ρ−ξ) in R+×(0,1),\nξ(t,0) =c0ρ(t,0), ρ(t,1) =c1ξ(t,1) inR+,\nwherec0=c1= (κ−1)/(κ+1).\nRegarding System ( 6.4) withc0,c1not necessarily equal, we prove the following result.\nTheorem 6.2. Let1< p <+∞,c0,c1∈(−1,1), anda∈L∞/parenleftbig\nR+×(0,1)/parenrightbig\nnon negative. Then\nthere exist positive constants C,γdepending only on c0,c1, and/bardbla/bardblL∞/parenleftbig\nR+×(0,1)/parenrightbigsuch that the\nunique solution uof(6.4)with the initial condition ρ(0,·) =ρ0andξ(0,·) =ξ0satisfies\n(6.5) /bardbl(ρ,ξ)(t,·)/bardblLp(0,1)≤Ce−γt/bardbl(ρ0,ξ0)/bardblLp(0,1), t≥0.\nThe rest of this section is organized as follows. The proof of Theorem 6.2is given in the first\nsection and the proof of Theorem 6.1is given in the second section.\n6.1.Proof of Theorem 6.2.We will only consider smooth solutions ( ρ,ξ). Multiplying the\nequation of ρwithρ, the equation of ξwithξ, and integrating the expressions with respect to x,\nafter using the boundary conditions, we obtain, for t >0,\n(6.6)1\npd\ndtˆ1\n0(|ρ(t,x)|p+|ξ(t,x)|p)dx+1\n2ˆ1\n0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dx\n1\np/parenleftBig\n(1−|c1|p)|ξ(t,1)|p+(1−|c0|p)|ρ(t,0)|p/parenrightBig\n= 0.\nThis implies\n(6.7)1\np/bardbl(ρ,ξ)(t,·)/bardblp\nLp(0,1)+1\n2ˆT\n0ˆ1\n0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dxdt\n+1\npˆT\n0/parenleftBig\n(1−|c1|p)|ξ(t,1)|p+(1−|c0|p)|ρ(t,0)|p/parenrightBig\ndt=1\n2/bardbl(ρ0,ξ0)/bardbl2\nL2(0,1).\nTo derive ( 6.5) from (6.7), it suffices to prove that there exists a constant c >0 depending only\non/bardbla/bardblL∞(R+×(0,1)),c0,c1,ε0,γ, andpsuch that for for T= 36, it holds\n(6.8)ˆT\n0ˆ1\n0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dxdt\n+ˆT\n0/parenleftBig\n|ξ(t,1)|p+|ρ(t,0)|p/parenrightBig\ndt≥c/bardbl(ρ0,ξ0)/bardblp\nLp(0,1).\nAfter scaling, one might assume without loss of generality t hat\n(6.9) /bardbl(ρ0,ξ0)/bardblLp(0,1)= 1\n6It holds for T >2 withc=cT.23\nApplying Lemma 3.1, we have\n(6.10)ˆT\n0ˆ1\n0a|ρ−ξ|p(t,x)dxdt≤C(mp+m2/p\np),\nwhere\nmp:=ˆT\n0ˆ1\n0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dxdt.\nUsing the characteristics method (in particular equations (3.9), (3.10)), we derive that\n(6.11) /bardbl(ρ,ξ)(T,·)/bardblp\nLp(0,1)≤CˆT\n0/parenleftBig\n|ξ(t,1)|p+|ρ(t,0)|p/parenrightBig\ndt+CˆT\n0ˆ1\n0ap|ρ−ξ|p(t,x)dxdt.\nAs a consequence of ( 6.7), (6.9), (6.10), and (6.11), we have\nˆT\n0ˆ1\n0a(ρ−ξ)(ρ|ρ|p−2−ξ|ξ|p−2)(t,x)dxdt+ˆT\n0/parenleftBig\n|ξ(t,1)|p+|ρ(t,0)|p/parenrightBig\ndt≥c.\nThe proof of the theorem is complete. /square\nRemark 6.2. In the case a≡0, one can show that the exponential stability for 1 ≤p≤+∞by\nnoting that\n/bardbl/parenleftbig\nρ(t+1,0),ρ(t+1,1)/parenrightbig\n/bardbl ≤max{|c0|,|c1|}/bardbl/parenleftbig\nρ(t,0),ρ(t,1)/parenrightbig\n/bardbl.\nThe conclusion then follows using the characteristics meth od.\n6.2.Proof of Theorem 6.1.Wefirstdealwiththewell-posednessofthesystem. Theuniqu eness\nfollows as in the proof of Proposition 2.1via the d’Alembert formula. The existence can be proved\nbyapproximation arguments. Firstdeal withsmoothsolutio ns (withsmooth a) usingTheorem 6.2\nand then pass to the limit. The details are omitted.\nThe proof of ( 6.5) is in the same spirit of ( 1.7). However, instead of using Theorem 1.2, we\napply Theorem 6.2. The details are left to the reader. /square\nRemark 6.3. One can prove the well-posedness of ( 1.1) and (1.4) directly in Lp-framework.\nNevertheless, tomake thesensefortheboundarycondition, oneneedstoconsider regular solutions\nand then ais required to be more regular than just L∞. We here take advantage of the fact that\nsuch a system has a hyperbolic structure as given in ( 6.4). This give us the way to give sense for\nthe solution by imposing the fact ∂tu,∂xu∈C([0,1];Lp(0,T)) for all T >0.\nReferences\n[1] F. Alabau-Boussouira, On some recent advances on stabilization for hyperbolic equ ationsIn Cannarsa, Pier-\nmarco, Coron, and Jean-Michel, editors, Control of partial differential equations, volume 2048 of Lecture Notes\nin Mathematics, pages 1–100. 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Zuazua, C. Bardos, G. Lebeau and J.\nRauch. MR 953547\n[25] K. Liu., Locally distributed control and damping for the conservati ve systems , SIAM journal on control and\noptimization, 35, 09 1997.\n[26] Patrick Martinez, Stabilization for the wave equation with Neumann boundary c ondition by a locally distributed\ndamping , Contrˆ ole des syst` emes gouvern´ es par des ´ equations aux d´ eriv´ ees partielles (Nancy, 1999), ESAIM\nProc., vol. 8, Soc. Math. Appl. Indust., Paris, 2000, pp. 119 –136. MR 1809565\n[27] Patrick Martinez andJudith Vancostenoble, Exponential stability for the wave equation with weak nonmo notone\ndamping , Portugal. Math. 57(2000), no. 3, 285–310. MR 1781446\n[28] Hoai-Minh Nguyen and Marco Squassina, Fractional Caffarelli-Kohn-Nirenberg inequalities , J. Funct. Anal.\n274(2018), no. 9, 2661–2672. MR 3771839\n[29] Juan C. Peral, Lpestimates for the wave equation , J. Functional Analysis 36(1980), no. 1, 114–145. MR 56897925\n[30] John P. Quinn and David L. Russell, Asymptotic stability and energy decay rates for solutions o f hyperbolic\nequations with boundary damping , Proc. Roy. Soc. Edinburgh Sect. A 77(1977), no. 1-2, 97–127. MR 473539\n[31] Jeffrey Rauch and Michael Taylor, Exponential decay of solutions to hyperbolic equations in b ounded domains ,\nIndiana Univ. Math. J. 24(1974), 79–86. MR 361461\n[32] David L. Russell, A unified boundary controllability theory for hyperbolic an d parabolic partial differential equa-\ntions., 1973.\n[33] DavidL.Russell, Controllability and stabilizability theory for linear par tial differential equations: recent progress\nand open questions , SIAM Rev. 20(1978), no. 4, 639–739. MR 508380\n[34] Enrique Zuazua, Exponential decay for the semilinear wave equation with loc ally distributed damping , Comm.\nPartial Differential Equations 15(1990), no. 2, 205–235. MR 1032629\n(Y. Chitour) Laboratoire des signaux et syst `emes,\nUniversit ´e Paris Saclay,\nFrance\nEmail address :yacine.chitour@l2s.centralesupelc.fr\n(H.-M. Nguyen) Laboratoire Jacques Louis Lions,\nSorbonne Universit ´e\nParis, France\nEmail address :hoai-minh.nguyen@sorbonne-universite.fr"    },    {        "title": "1804.07080v2.Damping_of_magnetization_dynamics_by_phonon_pumping.pdf",        "content": "Damping of magnetization dynamics by phonon pumping\nSimon Streib,1Hedyeh Keshtgar,2and Gerrit E. W. Bauer1, 3\n1Kavli Institute of NanoScience, Delft University of Technology, Lorentzweg 1, 2628 CJ Delft, The Netherlands\n2Institute for Advanced Studies in Basic Science, 45195 Zanjan, Iran\n3Institute for Materials Research & WPI-AIMR & CSRN, Tohoku University, Sendai 980-8577, Japan\n(Dated: July 11, 2018)\nWe theoretically investigate pumping of phonons by the dynamics of a magnetic film into a non-\nmagnetic contact. The enhanced damping due to the loss of energy and angular momentum shows\ninterferencepatternsasafunctionofresonancefrequencyandmagneticfilmthicknessthatcannotbe\ndescribed by viscous (“Gilbert”) damping. The phonon pumping depends on magnetization direction\nas well as geometrical and material parameters and is observable, e.g., in thin films of yttrium iron\ngarnet on a thick dielectric substrate.\nThe dynamics of ferromagnetic heterostructures is at\nthe root of devices for information and communication\ntechnologies [1–5]. When a normal metal contact is at-\ntached to a ferromagnet, the magnetization dynamics\ndrives a spin current through the interface. This effect\nis known as spin pumping and can strongly enhance the\n(Gilbert) viscous damping in ultra-thin magnetic films\n[6–8]. Spin pumping and its (Onsager) reciprocal, the\nspin transfer torque [9, 10], are crucial in spintronics, as\nthey allow electric control and detection of magnetiza-\ntion dynamics. When a magnet is connected to a non-\nmagnetic insulator instead of a metal, angular momen-\ntum cannot leave the magnet in the form of electronic or\nmagnonic spin currents, but they can do so in the form\nof phonons. Half a century ago it was reported [11, 12]\nand explained [13–16] that magnetization dynamics can\ngenerate phonons by magnetostriction. More recently,\nthe inverse effect of magnetization dynamics excited by\nsurface acoustic waves (SAWs) has been studied [17–20]\nand found to generate spin currents in proximity normal\nmetals [21, 22]. The emission and detection of SAWs was\ncombined in one and the same device [23, 24], and adia-\nbatic transformation between magnons and phonons was\nobserved in inhomogeneous magnetic fields [25]. The an-\ngular momentum of phonons [26, 27] has recently come\ninto focus again in the context of the Einstein-de Haas\neffect [28] and spin-phonon interactions in general [29].\nThe interpretation of the phonon angular momentum in\ntermsoforbitalandspincontributions[29]hasbeenchal-\nlenged [30], a discussion that bears similarities with the\ninterpretation of the photon angular momentum [31]. In\nour opinion this distinction is rather semantic since not\nrequired to arrive at concrete results. A recent quantum\ntheory of the dynamics of a magnetic impurity [32] pre-\ndicts a broadening of the electron spin resonance and a\nrenormalized g-factor by coupling to an elastic contin-\nuum via the spin-orbit interaction, which appears to be\nrelated to the enhanced damping and effective gyromag-\nnetic ratio discussed here.\nA phonon current generated by magnetization dynam-\nics generates damping by carrying away angular momen-\ntum and energy from the ferromagnet. While the phonon\nphonon sinkzmagnet\nnon-magnet0\nphononsmHFigure 1. Magnetic film (shaded) with magnetization mat-\ntached to a semi-infinite elastic material, which serves as an\nideal phonon sink.\ncontribution to the bulk Gilbert damping has been stud-\nied theoretically [33–38], the damping enhancement by\ninterfaces to non-magnetic substrates or overlayers has\nto our knowledge not been addressed before. Here we\npresent a theory of the coupled lattice and magnetiza-\ntion dynamics of a ferromagnetic film attached to a half-\ninfinite non-magnet, which serves as an ideal phonon\nsink. We predict, for instance, significantly enhanced\ndamping when an yttrium iron garnet (YIG) film is\ngrown on a thick gadolinium gallium garnet (GGG) sub-\nstrate.\nWe consider an easy-axis magnetic film with static ex-\nternal magnetic field and equilibrium magnetization ei-\nther normal (see Fig. 1) or parallel to the plane. The\nmagnet is connected to a semi-infinite elastic material.\nMagnetization and lattice are coupled by the magne-\ntocrystalline anisotropy and the magnetoelastic interac-\ntion, giving rise to coupled field equations of motion in\nthe magnet [39–42]. By matching these with the lattice\ndynamics in the non-magnet by proper boundary con-\nditions, we predict the dynamics of the heterostructure\nas a function of geometrical and constitutive parameters.\nWe find that magnetization dynamics induced, e.g., by\nferromagnetic resonance (FMR) excites the lattice in the\nattachednon-magnet. Inanalogywiththeelectroniccase\nwecallthiseffect“phononpumping” thataffectsthemag-\nnetization dynamics. We consider only equilibrium mag-\nnetizations that are normal or parallel to the interface,\nin which the pumped phonons are pure shear waves that\ncarry angular momentum. We note that for general mag-arXiv:1804.07080v2  [cond-mat.mes-hall]  16 Jul 20182\nnetization directions both shear and pressure waves are\nemitted, however.\nWe consider a magnetic film (metallic or insulating)\nthat extends from z=\u0000dtoz= 0. It is subject to suffi-\nciently high magnetic fields H0such that magnetization\nis uniform, i.e. M(r) =M:For in-plane magnetizations,\nH0> Ms, where the magnetization Msgoverns the de-\nmagnetizing field [43]. The energy of the magnet|non-\nmagnet bilayer can be written\nE=ET+Eel+EZ+ED+E0\nK+Eme;(1)\nwhich are integrals over the energy densities \"X(r). The\ndifferent contributions are explained in the following.\nThe kinetic energy density of the elastic motion reads\n\"T(r) =(\n1\n2\u001a_u2(r); z> 0\n1\n2~\u001a_u2(r);\u0000d<z< 0; (2)\nand the elastic energy density [44]\n\"el=(\n1\n2\u0015(P\n\u000bX\u000b\u000b(r))2+\u0016P\n\u000b\fX2\n\u000b\f(r); z> 0\n1\n2~\u0015(P\n\u000bX\u000b\u000b(r))2+ ~\u0016P\n\u000b\fX2\n\u000b\f(r);\u0000d<z< 0;\n(3)\nwhere\u000b;\f2fx;y;zg,\u0015and\u0016are the Lamé parameters\nand\u001athe mass density of the non-magnet. The tilded\nparameters are those of the magnet. The strain tensor\nX\u000b\fis defined in terms of the displacement fields u\u000b(r),\nX\u000b\f(r) =1\n2\u0012@u\u000b(r)\n@r\f+@u\f(r)\n@r\u000b\u0013\n: (4)\nEZ=\u0000\u00160VM\u0001Hextis the Zeeman energy for Hext=\nH0+h(t), where h(t)is time-dependent. ED=\n1\n2\u00160VMTDMis the magnetostatic energy with shape-\ndependent demagnetization tensor DandVthe volume\nof the magnet. For a thin film with zaxis along the sur-\nface normal n0,Dzz= 1while the other components van-\nish.E0\nK=K1V(m\u0002n0)2is the uniaxial magnetocrys-\ntalline anisotropy in the absence of lattice deformations,\nwhere m=M=MsandK1is the anisotropy constant.\nThe magnetoelastic energy Emecouples the magnetiza-\ntion to the lattice, as discussed in the following.\nThe magnetoelastic energy density can be expanded as\n\"me(r) =1\nM2sX\n\u000b;\fM\u000b(r)M\f(r)\n\u0002[B\u000b\fX\u000b\f(r) +C\u000b\f\n\u000b\f(r)]:(5)\nFor an isotropic medium the magnetoelastic constants\nB\u000b\fread [45]\nB\u000b\f=\u000e\u000b\fBk+ (1\u0000\u000e\u000b\f)B?: (6)\nRotational deformations as expressed by the tensor\n\n\u000b\f(r) =1\n2\u0012@u\u000b(r)\n@r\f\u0000@u\f(r)\n@r\u000b\u0013\n(7)are often disregarded [39–42, 46], but lead to a position\ndependence of the easy axis n(r)from the equilibrium\nvalue n0=ezand an anisotropy energy density [29, 47,\n48]\n\"K(r) =K1\nM2s[M\u0002n(r)]2: (8)\nTo first order in the small deformation\n\u000en(r) =n(r)\u0000n0=0\n@\nxz(r)\n\nyz(r)\n01\nA; (9)\n\"K(r) =\"0\nK+ 2K1(n0\u0000mzm)\u0001\u000en(r):(10)\nFrom \n\u000b\f=\u0000\n\f\u000bit follows that (for non-chiral crystal\nstructures) C\u000b\f=\u0000C\f\u000b. For the uniaxial anisotropy\nconsidered here Cxz=Cyz=\u0000K1. The magnetoelastic\ncoupling due to the magnetocrystalline anisotropy thus\ncontributes [47]\n\"K\nme(r) =\u00002K1\nM2sMz(r) [Mx(r)\nxz(r) +My(r)\nyz(r)]:\n(11)\nPureYIGismagneticallyverysoft, sothemagnetoelastic\nconstants are much larger than the anisotropy constant\n[49, 50]\nBk= 3:48\u0002105Jm\u00003; B?= 6:96\u0002105Jm\u00003;\nK1=\u00006:10\u0002102J m\u00003; (12)\nbut this ratio can be very different for other magnets.\nWe find below that for the Kittel mode dynamics both\ncoupling processes cannot be distinguished, even though\nthey can characteristically affect the magnon-phonon\ncoupling for finite wave numbers.\nThe magnetization dynamics within the magnetic film\nis described by the Landau-Lifshitz-Gilbert (LLG) equa-\ntion [51, 52]\n_m=\u0000\r\u00160m\u0002He\u000b+\u001c(\u000b)\nm; (13)\nwhere\u0000\ris the gyromagnetic ratio, the effective mag-\nnetic field which includes the magnetoelastic coupling\nHe\u000b=\u0000rmE=(\u00160VMs); (14)\nand the Gilbert damping torque [52]\n\u001c(\u000b)\nm=\u000bm\u0002_m: (15)\nThe equation of motion of the elastic continuum reads\n[44]\nu(r;t) =c2\nt4u(r;t) + (c2\nl\u0000c2\nt)r[r\u0001u(r;t)];(16)\nwith longitudinal and transverse sound velocities\ncl=s\n\u0015+ 2\u0016\n\u001a; ct=r\u0016\n\u001a; (17)3\nwhere elastic constants and mass density of non-magnet\nand magnet can differ.\nA uniform precession of the magnetization interacts\nwith the lattice deformation at the surfaces of the mag-\nnetic film [13, 14] and at defects in the bulk. The present\ntheorythenholdswhenthethicknessofthemagneticfilm\nd\u001cp\nA, whereAis the cross section area. The Kittel\nmode induces lattice distortions that are uniform in the\nfilm planeu\u000b(r) =u\u000b(z)[14]. The elastic energy density\nis then affected by shear waves only:\n\"el(z) =(\n\u0016\n2\u0000\nu02\nx(z) +u02\ny(z)\u0001\n; z> 0\n~\u0016\n2\u0000\nu02\nx(z) +u02\ny(z)\u0001\n;\u0000d<z< 0;(18)\nwhereu0\n\u000b(z) =@u\u000b(z)=@z. The magnetic field\nHext=\u0000hx(t); hy(t); H 0\u0001Twith monochromatic drive\nhx;y(t) = Re\u0000\nhx;ye\u0000i!t\u0001\nand static component H0along\nthezaxis. At the FMR frequency !?=!H+!Awith\n!H=\r\u00160H0and!A=\r(2K1=Ms\u0000\r\u0016Ms). The equi-\nlibrium magnetization is perpendicular for !?>0. The\nmagnetoelastic energy derived above then simplifies to\nEz\nme=(B?\u0000K1)A\nMsX\n\u000b=x;yM\u000b[u\u000b(0)\u0000u\u000b(\u0000d)];(19)\nwhichresultsinsurfaceshearforces F\u0006(0) =\u0000F\u0006(\u0000d) =\n\u0000(B?\u0000K1)Am\u0006, withF\u0006=Fx\u0006iFy. These forces\ngenerate a stress or transverse momentum current in the\nzdirection (see Supplemental Material)\nj\u0006(z) =\u0000\u0016(z)u0\n\u0006(z); (20)\nwith\u0016(z) =\u0016forz >0and\u0016(z) = ~\u0016for\u0000d < z < 0,\nandu\u0006=ux\u0006iuy, which is related to the transverse mo-\nmentump\u0006(z) =\u001a( _ux(z)\u0006i_uy(z))by Newton’s equa-\ntion:\n_p\u0006(z) =\u0000@\n@zj\u0006(z): (21)\nThe boundary conditions require momentum conserva-\ntion and elastic continuity at the interfaces,\nj\u0006(\u0000d) = (B?\u0000K1)m\u0006;(22)\nj\u0006(0+)\u0000j\u0006(0\u0000) =\u0000(B?\u0000K1)m\u0006;(23)\nu\u0006(0+) =u\u0006(0\u0000): (24)\nWe treat the magnetoelastic coupling as a small pertur-\nbation and therefore we approximate the magnetization\nm\u0006entering the above boundary conditions as indepen-\ndent of the lattice displacement u\u0006. The loss of angular\nmomentum (see Supplemental Material) affects the mag-\nnetization dynamics in the LLG equation in the form of a\ntorque, which we derive from the magnetoelastic energy\n(19),\n_m\u0006jme=\u0006i!c\nd[u\u0006(0)\u0000u\u0006(\u0000d)]\n=\u0006i!cRe(v)m\u0006\u0007!cIm(v)m\u0006;(25)where!c=\r(B?\u0000K1)=Ms(for YIG:!c= 8:76\u0002\n1011s\u00001) andv= [u\u0006(0)\u0000u\u0006(\u0000d)]=(dm\u0006). We can\ndistinguish an effective field\nHme=!c\n\r\u00160Re(v)ez; (26)\nand a damping coefficient\n\u000b(?)\nme=\u0000!c\n!Imv: (27)\nThe latter can be compared with the Gilbert damping\nconstant\u000bthat enters the linearized equation of motion\nas\n_m\u0006j\u000b=\u0006i\u000b_m\u0006=\u0006\u000b!m\u0006: (28)\nWith the ansatz\nu\u0006(z;t) =(\nC\u0006eikz\u0000i!t; z> 0\nD\u0006ei~kz\u0000i!t+E\u0006e\u0000i~kz\u0000i!t;\u0000d<z< 0;\n(29)\nwe obtain\nv=Ms!c\n!\rd~\u001a~ct2h\ncos(~kd)\u00001i\n\u0000i\u001act\n~\u001a~ctsin(~kd)\nsin(~kd) +i\u001act\n~\u001a~ctcos(~kd);(30)\nand the damping coefficient for perpendicular magneti-\nzation\n\u000b(?)\nme=\u0010!c\n!\u00112Ms\n\rd~\u001a~ct\u001act\n~\u001a~ct4 sin4\u0010~kd\n2\u0011\nsin2(~kd) +\u0010\n\u001act\n~\u001a~ct\u00112\ncos2(~kd);\n(31)\nwhere!=ctk= ~ct~k. The oscillatory behavior of the\ndamping\u000b(?)\nmecomes from the interference of the elastic\nwaves that are generated at the top and bottom surfaces\nof the magnetic film. When they constructively (destruc-\ntively) interfere at the FMR frequency, the damping is\nenhanced(suppressed), becausethemagnon-phononcou-\npling and phonon emission are large (small).\nWhen\u001act\u001c~\u001a~ct(soft substrate) or when acoustic\nimpedances are matched ( \u001act= ~\u001a~ct), damping at the\nresonance ~kd= (2n+ 1)\u0019withn2N0[14] simplifies to\n\u000b(?)\nme!\u0010!c\n!\u001124Ms\n\rd\u001act: (32)\nWhen\u001act\u001d~\u001a~ct(hard substrate), the magnet is acousti-\ncally pinned at the interface and the acoustic resonances\nare at ~kd= (2n+ 1)\u0019=2[14] with\n\u000b(?)\nme!\u0010!c\n!\u00112Ms\n\rd~\u001a~ct\u001act\n~\u001a~ct: (33)\nIn contrast to Gilbert damping, \u000b(?)\nmedepends on the\nfrequency and vanishes in the limits !!0and!!1.\nTherefore, it does not obey the LLG phenomenology4\nand in the non-linear regime does not simply enhance\n\u000bin Eq. (15). The magnetization damping \u000b0in bulk\nmagnetic insulators, on the other hand, is usually of\nthe Gilbert type. It is caused by phonons as well, but\nnot necessarily the magnetoelastic coupling. A theory\nof Gilbert damping [38] assumes a bottleneck process\nby sound wave attenuation, which appears realistic for\nmagnets with high acoustic quality such as YIG. In the\npresent phonon pumping model, energy and angular mo-\nmentum is lost by the emission of sound waves into an\nattached perfect phononwaveguide, so thepumpingpro-\ncess dominates. Such a scenario could also dominate the\ndamping in magnets in which the magnetic quality is rel-\natively higher than the acoustic one.\nWhen the field is rotated to Hext =\u0000\nhx(t); H 0; hz(t)\u0001T, the equilibrium magnetization\nis in the in-plane ydirection and the magnetoelastic\nenergy couples only to the strain uy,\nEy\nme=(B?\u0000K1)A\nMsMz[uy(0)\u0000uy(\u0000d)]:(34)\nThe FMR frequency for in-plane magnetization !k=\n!Hp\n1\u0000!A=!Hwith!A< !H. The magnetoelastic\ncoupling generates again only transverse sound waves.\nThe linearized LLG equation including the phononic\ntorques reads now\n_mx= (!H+!me)mz\u0000\r\u00160hz\u0000!Amz\n+ (\u000b+\u000bme) _mz; (35)\n_mz=\u0000!Hmx+\r\u00160hx\u0000\u000b_mx; (36)\nwhere\u000bmeis given by Eq. (27) and !me=\r\u00160Hmewith\neffective field Hme=Hme\u0001ezgiven by Eq. (26). Both\nHmeand\u000bmecontribute only to _mx. The phonon pump-\ning is always less efficient for the in-plane configuration:\n\u000b(k)\nme=1\n1 + (!k=!H)2\u000b(?)\nme: (37)\nAs an example, we insert parameters for a thin YIG\nfilm on a semi-infinite gadolinium gallium garnet (GGG)\nsubstrate at room temperature. We have chosen YIG\nbecause of its low intrinsic damping and high quality in-\nterface to the GGG substrate. Substantially larger mag-\nnetoelastic coupling in other materials should be offset\nagainst generally larger bulk damping. For GGG, \u001a=\n7:07\u0002103kg m\u00003,cl= 6411 m s\u00001, andct= 3568 m s\u00001\n[53]. For YIG, Ms= 1:4\u0002105A m\u00001,\r= 1:76\u0002\n1011s\u00001T\u00001,~\u001a= 5170 kg m\u00003,~cl= 7209 m s\u00001,~ct=\n3843 m s\u00001, and!c= 8:76\u00021011s\u00001[49, 50]. The ratio\nof the acoustic impedances ~\u001a~ct=\u001act= 0:79. The damp-\ning enhancement \u000b(?)\nmeis shown in Fig. 2 over a range\nof FMR frequencies and film thicknesses. The FMR fre-\nquencies!?=!H+!Aand!k=!Hp\n1\u0000!A=!Hfor\nthe normal and in-plane configurations are tunable by\nthe static magnetic field component H0via!H=\r\u00160H0.\n0 200 400 600 800 1000\nmagnetic film thickness d[nm]0246810FMR frequency ν[GHz]\n<10−510−410−3\nα(⊥)\nmeFigure 2. Damping enhancement \u000b(?)\nmeby phonon pumping\nin a YIG film on a semi-infinite GGG substrate, as given by\nEq. (31).\nThe damping enhancement peaks at acoustic resonance\nfrequencies \u0017\u0019n~ct=(2d). The counter-intuitive result\nthat the damping increases for thicker films can be un-\nderstood by the competition between the magnetoelastic\neffect that increases with thickness at the resonances and\nwins against the increase in total magnetization. How-\never, with increasing thickness the resonance frequencies\ndecrease below a minimum value at which FMR can be\nexcited. For a fixed FMR frequency \u000bme!0ford!1:\nFor comparison, the Gilbert damping in nanometer thin\nYIG films is of the order \u000b\u001810\u00004[54] which is larger\nthan corresponding values for single crystals. We con-\ncludethattheenhanceddampingisatleastpartlycaused\nby interaction with the substrate and not by a reduced\ncrystal quality.\nThe resonances in the figures are very broad because\nthe\u001act\u0019~\u001a~ctimplies very strong coupling of the discrete\nphonons in the thin magnetic layer with the phonon con-\ntinuum in the substrate. When an acoustic mismatch\nis introduced, the broad peaks increasingly sharpen, re-\nflecting the increased lifetime of the magnon polaron res-\nonances in the magnet.\nThe frequency dependent effective magnetic field H(?)\nme\nis shown in Fig. 3. The frequency dependence of H(?)\nme\nimplies a weak frequency dependence of the effective gy-\nromagnetic ratio\n\r(?)\ne\u000b=\r \n1 +\r\u00160H(?)\nme\n!!\n: (38)\nIn the limit of vanishing film thickness, \u00160H(?)\nme!\n\u0000(B?\u0000K1)2=(Ms~\u0016).\nWe assumed that the non-magnet is an ideal phonon\nsink, which means that injected sound waves do not\nreturn. In the opposite limit in which the phonons\ncannot escape, i.e. when the substrate is a thin film\nwith high acoustic quality, the additional damping van-\nishes. This can be interpreted in terms of a phonon5\n0 200 400 600 800 1000\nmagnetic film thickness d[nm]0246810FMR frequency ν[GHz]\n−60−45−30−15015304560\nµ0H(⊥)\nme[µT]\nFigure3. Effectivefield H(?)\nmegeneratedbythemagnetoelastic\ngeneration of phonons in a YIG film on a semi-infinite GGG\nsubstrate, as given by Eq. (26).\naccumulation that, when allowed to thermalize, gener-\nates a phonon chemical potential and/or non-equilibrium\ntemperature. The non-equilibrium thermodynamics of\nphonons in magnetic nanostructures is subject of our on-\ngoing research.\nThe damping enhancement by phonons may be com-\npared with that from electronic spin pumping [6–8],\n\u000bsp=\r~\n4\u0019dMsh\ne2g; (39)\nwhich is inversely proportional to the thickness dof the\nmagnetic film and does not depend on the FMR fre-\nquency, i.e. obeys the LLG phenomenology. Here, g\nis the spin mixing conductance per unit area at the in-\nterface. While phonons can be pumped into any elastic\nmaterial, spin pumping requires an electrically conduct-\ning contact. With a typical value of hg=e2\u00181018m\u00002\nthe damping enhancement of YIG on platinum is \u000bsp\u0018\n10\u00002nm=d:The physics is quite different, however, since\n\u000bsp;in contrast to \u000bme, does not require coherence over\nthe interface.\nIn conclusion, the pumping of phonons by magnetic\nanisotropy and magnetostriction causes a frequency-\ndependent contributions to the damping and effective\nfield of the magnetization dynamics. The generation\nof phonons by magnetic precession can cause significant\ndamping in a magnetic film when grown on an insulating,\nnon-magnetic substrate and partly explains the increased\ndampinginvariablyobservedforthinnerfilms. Theimpli-\ncations of further reaching ramifications, such as phonon-\ninduced dynamic exchange interactions, phonon accumu-\nlations and phonon spin Seebeck effect require additional\nresearch.\nThis work is financially supported by the Nederlandse\nOrganisatievoorWetenschappelijkOnderzoek(NWO)as\nwell as by the Grant-in-Aid for Scientific Research on In-\nnovative Area, ”Nano Spin Conversion Science” (Grant\nNo. 26103006). H. K. acknowledges support from theIran Science Elites Federation. We acknowledge use-\nful discussions with Yaroslav Blanter, Rembert Duine,\nAkashdeep Kamra, Eiji Saitoh, and Sanchar Sharma.\n[1] S. Bader and S. Parkin, Annu. Rev. Condens. Matter\nPhys. 1, 71 (2010).\n[2] V. V. Kruglyak, S. O. Demokritov, and D. Grundler, J.\nPhys. D: Appl. Phys. 43, 264001 (2010).\n[3] F. Pulizzi, Nat. Mater. 11, 367 (2012).\n[4] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and\nB. Hillebrands, Nat. Phys. 11, 453 (2015).\n[5] S. A. Nikitov, D. V. Kalyabin, I. V. Lisenkov, A. 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Bodzenta, Phys. Status Solidi B\n146, 467 (1988).\n[54] H. Chang, P. Li, W. Zhang, T. Liu, A. Hoffmann,\nL. Deng, and M. Wu, IEEE Magn. Lett. 5, 1 (2014).\n[55] E. B. Magrab, Vibrations of Elastic Systems (Springer,\nDordrecht, 2012).1\nSupplemental Material\nInSecs.IandIIofthisSupplementwederivetheangu-\nlar and transverse momentum of transverse elastic waves\nand the corresponding momentum currents. In Sec. III\nwe give results for a magnet sandwiched between two\nnon-magnets and in Sec. IV we present a theory for the\nmagnetizationdampingenhancementfrompumpingflex-\nural waves into a thin beam.\nI. ANGULAR MOMENTUM\nThe magnetization M=Msmof an uniformly mag-\nnetized magnet with saturation magnetization Msand\nvolumeVis associated with the angular momentum\nS=\u0000MsV\n\rm; (S1)\nwhere\u0000\ris the gyromagnetic ratio. The angular mo-\nmentum density relative to the origin of an elastic body\nwith displacement field u(r;t)and constant mass density\n\u001areads\nl(t) =\u001a(r+u(r;t))\u0002_u(r;t): (S2)\nWith uniaxial anisotropy axis along z, FMR generates\nthe transverse elastic wave\nu(z;t) = Re2\n40\n@ux\nuy\n01\nAeikz\u0000i!t3\n5; (S3)\nwith dispersion relation !=ctk. Defining the time av-\nerage\nhf(t)i= lim\nT!11\nTZT\n0dtf(t); (S4)\nhli= (0;0;hlzi)can be expressed as\nhlzi=\u001ahux_uy\u0000_uxuyi\n=\u0000\u001a!\n4\u0010\nju+j2\u0000ju\u0000j2\u0011\n; (S5)\nwhereu\u0006=ux\u0006iuyand where we used\n\nRe(ae\u0000i!t)Re(be\u0000i!t)\u000b\n=1\n2Re (a\u0003b):(S6)\nThe non-magnet harbors a constant phonon angular mo-\nmentum density in the z-direction, which implies pres-\nence of a phonon angular momentum current Ahjz\nliat\nthe interface to the magnet with area Athat is ab-\nsorbed at the phonon sink. In our model the angular\nmomentum loss rate of the magnet by phonon pumping\nh_Szjmei=\u0000Ahjz\nliand\nD\n_Sz\f\f\f\nmeE\n=hfSz;Ez\nmegi=MsV\u000b(?)\nme!\n4\r\u0010\njm+j2\u0000jm\u0000j2\u0011\n(S7)wheref;gis the Poisson bracket and Ez\nmethe magnetoe-\nlastic energy Eq. (19) in the main text. From Eq. (31)\nand\nu\u0006=Cm\u0006; (S8)\nwith\nC=(B?\u0000K1)h\ncos(~kd)\u00001i\nik\u0016cos(~kd) +~k~\u0016sin(~kd); (S9)\nwe obtain the relation\nhjz\nli=cthlzi; (S10)\nwhichagreeswiththesimplephysicalpictureofanelastic\nwave carrying away its angular momentum density hlzi\nwith transverse sound velocity ct.\nII. TRANSVERSE MOMENTUM\nFor the transverse elastic wave (S3) in a magnet ex-\ntending from z=z0toz=z1(withz1> z0), the time\nderivative of the transverse momentum P\u0006=Px\u0006iPy\nreads\n_P\u0006=\u001aZ\nVd3ru\u0006(z;t)\n=\u0016A\u0002\nu0\n\u0006(z1;t)\u0000u0\n\u0006(z0;t)\u0003\n:(S11)\nThe change of momentum can be interpreted as a trans-\nverse momentum current density j\u0006(z0) =\u0000\u0016u0\n\u0006(z0)\nflowing into the magnet at z0and a current j\u0006(z1) =\n\u0000\u0016u0\n\u0006(z1)flowing out at z1. The momentum current\nis related to the transverse momentum density p\u0006(z) =\n\u001a_u\u0006(z)by\n_p\u0006(z) =\u0000@\n@zj\u0006(z); (S12)\nwhich confirms that\nj\u0006(z;t) =\u0000\u0016u0\n\u0006(z;t): (S13)\nThe instantaneous conservation of transverse momentum\nisaboundaryconditionsattheinterface. Itstimeaverage\nhj\u0006i= 0, but the associated angular momentum along z\nis finite, as shown above.\nIII. SANDWICHED MAGNET\nWhen a non-magnetic material is attached at both\nsides of the magnet and elastic waves leave the magnet\natz= 0andz=\u0000d, the boundary condition are\nj\u0006(\u0000d\u0000)\u0000j\u0006(\u0000d+) = (B?\u0000K1)m\u0006;(S14)\nj\u0006(0+)\u0000j\u0006(0\u0000) =\u0000(B?\u0000K1)m\u0006;(S15)\nu\u0006(0+) =u\u0006(0\u0000); (S16)\nu\u0006(\u0000d+) =u\u0006(\u0000d\u0000); (S17)2\nwithd\u0006=d\u00060+. Since the Hamiltonian is piece-wise\nconstant\nu\u0006(z;t) =8\n><\n>:C\u0006eikz\u0000i!t; z> 0\nD\u0006ei~kz\u0000i!t+E\u0006e\u0000i~kz\u0000i!t;\u0000d<z< 0\nF\u0006e\u0000ikz\u0000i!t: z< \u0000d;\n(S18)\nUsing the boundary conditions\nv=u\u0006(0)\u0000u\u0006(\u0000d)\ndm\u0006=Ms!c\n!\rd~\u001a~ct2\ni\u001act\n~\u001a~ct\u0000cot(~kd\n2);(S19)\nleading to the damping coefficient\n\u000b(?)\nme=\u0010!c\n!\u00112Ms\n\rd~\u001a~ct2\n\u001act\n~\u001a~ct+~\u001a~ct\n\u001actcot2\u0010~kd\n2\u0011:(S20)\nWhen ~\u001a~ct=\u001act,\n\u000b(?)\nme=\u0010!c\n!\u001122Ms\n\rd\u001actsin2 ~kd\n2!\n;(S21)\nwhich differs from the sin4\u0010\n~kd=2\u0011\ndependence obtained\nfor the magnet|non-magnet bilayer. This result can be\nexplained by the phonon angular momentum leaking at\ntwo interfaces that should enhance the damping for thin\nmagneticfilms. However, thephononpumpingisacoher-\nent process that couples both interfaces, so the damping\nis not increased simply by a factor of 2 as in case of in-\ncoherent spin pumping of a magnetic film sandwiched by\nmetals. The position of the resonances, ~kd= (2n+1)\u0019=2\nwithn2N0, are independent of the ratio \u001act=~\u001a~ctwith\n\u000bme=\u0010!c\n!\u001122Ms\n\rd\u001act: (S22)\n\u000b(?)\nmeand the effective magnetic field for YIG sandwiched\nbetween two infinitely thick GGG layers are shown in\nFigs. S1 and S2.\nIV. FLEXURAL (BENDING) WAVES IN THIN\nBEAMS\nIn the main text we focus on the generation of trans-\nverseorlongitudinalsoundwaves. Infree-standingstruc-\ntured samples such as cantilevers, additional modes be-\ncome important that can be excited by magnetization\ndynamics as well. This can be illustrated by a thin\ncylindrical elastic beam (see Fig. S3) with cross section\nareaA=\u0019r2, in which flexural waves are generated by\nthe magnet of volume V=Adattached to the top of\nthe beam. The elastic energy according to the Euler-\nBernoulli beam theory [44]\nEel=ZL\n0dz\u00141\n2\u001aA_u2(z;t) +1\n2EYI?u002(z;t)\u0015\n;(S23)\n0 200 400 600 800 1000\nmagnetic film thickness d[nm]0246810FMR frequency ν[GHz]\n<10−510−410−3\nα(⊥)\nmeFigure S1. Phonon pumping-enhanced \u000b(?)\nmein a YIG film\nsandwiched between two infinitely thick GGG layers.\n0 200 400 600 800 1000\nmagnetic film thickness d[nm]0246810FMR frequency ν[GHz]\n−40−30−20−10010203040\nµ0H(⊥)\nme[µT]\nFigure S2. Phonon pumping effective field H(?)\nmein a YIG film\nsandwiched between two infinitely thick GGG layers.\nleads to the equation of motion for the flexural waves [44]\n\u001aAu\u0006(z;t) +EYI?u(4)\n\u0006(z;t) = 0;(S24)\nwhereI?=R\ndAx2=\u0019r4=4and elastic modulus EY=\n\u0016(3\u0015+ 2\u0016)=(\u0015+\u0016). The dispersion relation of the flex-\nural waves is quadratic,\n!=s\nEYI?\n\u001aAk2: (S25)\nWhen the dimensions of the magnet are much smaller\nthan the wavelength of the elastic waves, the magne-\ntoelastic coupling is suppressed and magnetization and\nlattice are coupled exclusively by the magnetocrystalline\nand, in contrast to the bulk magnet, also the shape\nanisotropies,\nEA=EK+ED; (S26)\nwith\nEK=K1V(m\u0002n)2; (S27)\nED=1\n2\u00160VMTDM: (S28)3\nm\nH\nz 0magnet non-magnet\nflexural wave\nFigure S3. Thin film magnet (shaded) with magnetization m\nattached to a thin semi-infinite elastic beam.\nFor a thin magnetic film Dzz=D3= 1. When the\nmagnet become very thick ( d\u001dr), i.e. a needle with\nits point forming the contact, and a coordinate system\nwithzaxis along the surface normal n,Dxx=Dyy=\nD?=\u00001=2. All otherD\u000b\fvanish. In contrast to the\nextended bilayer treated in the main text nis now a\ndynamic variable with n\u0006=\u0000u0\n\u0006(0;t). The mechanical\ntorque exerted by the magnet on the elastic beam reads\n\u001c=_L=VMs\n\r_m+_J; (S29)\nwhere _J=\u00160VMsm\u0002Hextis the torque exerted by the\nexternal magnetic field on the total angular momentum.\nFor a magnet with equilibrium magnetization mkn0\n\u001c\u0006=\u0006if(m\u0006\u0000n\u0006); (S30)\nwheref=VMs!A=\rand\n!A=(\n2\rK1=Ms\u0000\r\u00160Ms;thin film\n2\rK1=Ms+1\n2\r\u00160Ms;needle:(S31)\nIn order to compute the angular momentum current\npumped into the attached non-magnet, _L\u0006=\u001c\u0006, we\nhave to specify four boundary conditions. Two are pro-\nvided by the assumption that the beam is infinitely long\nso that the are no reflections. The absence of shear forces\nat the boundary is expressed by u000(0;t) = 0, while the\nbending by the torque follows from the principle of least\naction [55]\nu00\n\u0006(0;t) =\u0006i\u001c\u0006\nEYI?: (S32)The general solution for the differential equation can be\nwritten\nu\u0006(z;t) =e\u0000i!t\u0000\nA\u0006eikz+B\u0006e\u0000kz\u0001\n;(S33)\nbecause there are no back-reflections in the semi-infinite\nbeam. We find\nn\u0006=wm\u0006\nwith\nw=\u00002f\nEYI?k\u0012\n1 +i\u00002f\nEYI?k\u0013\u00001\n;(S34)\nand the following source term in the LLG equation,\n_m\u0006jan=\u0007i!An\u0006\n=\u0006i!ARe(w)m\u0006\u0007!AIm(w)m\u0006;(S35)\nThe first term on the right-hand-side is a field-like torque\nequivalent to the effective field\n\u00160Han=!A\n\rRe(w)ez; (S36)\nand the second one a damping-like torque with damping\ncoefficient\n\u000ban=\u0000!A\n!Imw: (S37)\nSince for weak magnetoelastic coupling we expect \u000ban\u001c\n1and thereforejwj \u001c 1, which is equivalent to\n2f=(EYI?k)\u001c1, we may approximate\nw\u0019f(1\u0000i)\nEYI?k; (S38)\n\u000ban\u0019VMs!2\nA\n!k\rI?EY; (S39)\n\u00160Han\u0019\u0000VMs!2\nA\n\r2EYI?kez: (S40)\nThe damping enhancement scales as\n\u000ban/V\nA2!3\n2: (S41)\nFor a needle-shaped YIG magnet attached to GGG with\nEY= 2:5\u00021011Paand!A= 1:4\u00021010s\u00001\n\u000ban\u00198:6\u000210\u00006d=nm\n(\u0017=GHz)3\n2(r=nm)2;(S42)\n\u00160jHanj\u00193:1\u000210\u00007d=nm\n(\u0017=GHz)1\n2(r=nm)2T;(S43)\nwhich are very small numbers even at nanoscale dimen-\nsions."    },    {        "title": "2203.06312v1.Stability_for_nonlinear_wave_motions_damped_by_time_dependent_frictions.pdf",        "content": "Stability for nonlinear wave motions damped by time-dependent\nfrictions?\nZhe Jiaoa, Yong Xua,b, Lijing Zhaoa,\u0003\naDepartment of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710129, People's Republic of\nChina\nbMIIT key Laboratory of Dynamics and Control of Complex Systems, Northwestern Polytechnical University,\nXi'an 710072, People's Republic of China\nAbstract\nWe are concerned with the dynamical behavior of solutions to semilinear wave systems with time-\nvarying damping and nonconvex force potential. Our result shows that the dynamical behavior of\nsolution is asymptotically stable without any bifurcation and chaos. And it is a sharp condition\non the damping coe\u000ecient for the solution to converge to some equilibrium. To illustrate our\ntheoretical results, we provide some numerical simulations for dissipative sine-Gordon equation\nand dissipative Klein-Gordon equation.\nKeywords: dissipative wave system, nonautonomous damping, convergence to equilibria, decay\nrate\n2010 MSC: 35L70, 35B35\n1. Introduction\nIn this paper, we consider the initial-boundary value problem for semilinear wave systems with\ntime-varying dissipation:\n8\n<\n:utt\u0000\u0001u+h(t)ut+f(u) = 0 ( t;x)2R+\u0002\n;\nu= 0 ( t;x)2R+\u0002@\n;\n(u(0;x);ut(0;x)) = (u0(x);u1(x))x2\n;(1)\nwhere \n is a bounded domain in Rd,d\u00151, with smooth boundary @\n, and the initial data\nu02H2(\n)\\H1\n0(\n) andu12H1\n0(\n). The main purpose of the paper is to study the dynamical\nbehavior of the solution of dissipative system (1) damped by time-dependent frictions. It is clear\nthat the nonlinearity fand the coe\u000ecient h(t) play signi\fcant roles in the analysis. In particular,\nthe set of equilibria associated to system (1) depends on the assumptions on the nonlinearity\nf. And whether h(t) is asymptotically vanishing or becoming larger too rapidly as time goes to\nin\fnity, convergence to some equilibrium for the solution of system (1) may fail.\nLet us \frst begin the introduction with a short survey in the literature. The dynamical behavior\nof solution to system (1) and its ODE version\nx(t) +h(t) _x(t) +r\b(x(t)) = 0; t\u00150\nwith a potential \b 2W2;1\nloc(Rd;R), which models the heavy ball with friction, have been studied by\nmany authors under various assumptions on the damping and potential terms (cf., e.g., [1{6] and\nreferences therein). Recently, assuming the nonlinearity fis monotone and conservative, which is\nequivalent to the convexity of the potential, the authors in [7, 8] proved that the solution converges\nweakly to an equilibrium if the damping coe\u000ecient behaves likeC\nt\u000bfor someC > 0 and\u000b2(0;1).\nHowever, in the case of nonconvex force potential, the question of convergence to an equilibrium\nfor the solution of the system (1) is left open. The new estimates in this paper allow to solve\n?This research was partially supported by the National Natural Science Foundation of China (11802236) and the\nFundamental Research Funds for the Central Universities (310201911CX033).\n\u0003Corresponding author\nEmail addresses: zjiao@nwpu.edu.cn (Zhe Jiao), hsux3@nwpu.edu.cn (Yong Xu), zhaolj@nwpu.edu.cn (Lijing\nZhao)\nPreprint submitted to CNSNS March 15, 2022arXiv:2203.06312v1  [math.AP]  12 Mar 2022this question and to propose necessary and su\u000ecient conditions on the damping coe\u000ecient for\nconvergence, see main theorem and remarks below.\nBefore giving the detailed statement of our main theorem, we make assumptions on the non-\nlinearityfand the damping coe\u000ecient h:\n(I-1)f2W1;1\nloc(R) satis\fes\nlim inf\njsj!+1f(s)\ns\u0015\u0000\u00161; (2)\nwhere the constant \u00161< \u0016 0, and\u00160is the \frst eigenvalue of \u0000\u0001 in \n with zero Dirichlet\nboundary condition; and that\njf0(s)j\u0014C(1 +jsjp); s2R; (3)\nwhereC > 0 andp\u00150, with (d\u00002)p<2, are constants.\n(I-2) Put\nG(v) :=\u0000\u0001v+f(v); E 0(v) :=1\n2krvk2\nL2+Z\n\nF(v)dx;\nwhereF(s) :=Rs\n0f(t)dt, and\nS:=f 2H2(\n)\\H1\n0(\n) :G( ) = 0g\n(the set of equilibria). There exists a number \u00122(0;1\n2] such that for each  2S,\nkG(v)kH\u00001\u0015c jE0(v)\u0000E0( )j1\u0000\u0012; (4)\nwheneverv2H1\n0(\n),kv\u0000 kH1\u0014\u001b ; here,c and\u001b are constants depending on  .\n(I-3) The damping coe\u000ecient h(t)2W1;1\nloc(R+) is a nonnegative function, and there exist con-\nstantsc;C > 0, and\n\u000b2[0;\u0012(1\u0000\u0012)\u00001) (5)\nsuch that\nc\n(t+ 1)\u000b\u0014h(t)\u0014C\n(t+ 1)\u000b;8t\u00150; (6)\nor\nct\u000b\u0014h(t)\u0014Ct\u000b;8t\u00150: (7)\nIn [9], the authors proved the convergence result under the conditions (I-1), (I-2), (I-3) without\n(7), and the following assumption (I-4).\n(I-4) For any a>0,\ninf\nt>0Zt+a\nth(s)ds> 0:\nCondition (I-4) is a technical assumption, which is only used to show decay to zero of utinL2\nand then the precompactness of the trajectories of system (1). However, condition (I-4) implies\nthathdoes not tend to 0 at in\fnity. Stimulated by all the work above, the major contribution of\nthis paper is to present an e\u000bective method to prove the convergence to equilibrium of solutions\nof system (1) without the assumption (I-4). More precisely, we will prove the main theorem as\nfollows.\nTheorem 1. Assume Conditions (I-1), (I-2) and (I-3). Let u2W1;1\nloc(R+;H1\n0)\\W2;1\nloc(R+;L2)\nbe a solution of (1). Then f(0; ) : 2Sgis the attracting set for system (1), that is,\nlim\nt!+1(kut(t;\u0001)kL2+ku(t;\u0001)\u0000 kH1) = 0:\nMoreover, there exist positive constants c, andCsuch that\nku\u0000 kL2\u0014\u001a\nc(1 +t)\u0000\u0015; \u00122(0;1\n2);\nCexp(\u0000ct1\u0000\u000b); \u0012=1\n2;\nwhere\u00152(0;\u0012\u0000(1\u0000\u0012)\u000b\n1\u00002\u0012).\n2Remark 1. It follows from [10] that (I-2) holds true if fis analytic. For example, the analytic\nfunctionf(u) =bsinu,bsome positive constant, with Lojasiewicz exponent \u0012=1\n2. Then equation\n(1) is a damped sine-Gordon equation.\nAs we know in [11], (I-2) is also suitable for some non-analytic functions, for instance,\nf(u) =au+jujp\u00001u; p> 1:\nAnd ifa>\u0000\u00160, then (I-2) is satis\fed with the Lojasiewicz exponent \u0012=1\n2; otherwise, (I-2) holds\nwith\u0012=1\np+1. Thus, equation (1) becomes a damped nonlinear Klein-Gordon equation.\nRemark 2. The condition (I-3) on the damping coe\u000ecient is optimal, which prevent the damping\nterm from being either too small, or too large as t!+1.\nFrom Theorem 1, we obtain the convergence results for equation (1) with a small damping\ncoe\u000ecienth(t)\u00181\n(t+1)\u000b, asymptotically vanishing, or a large damping coe\u000ecient h(t)\u0018t\u000b,\n\u000b2[0;\u0012(1\u0000\u0012)\u00001). Here, the notation \u0018means that the coe\u000ecient grows like a polynomial\nfunction.\nAs forh(t) = (t+ 1)\u000b,j\u000bj>1, Theorem 1 do not apply, and the solution u(t;x) may oscillate\nor approach to some functions (not an equilibrium) as time goes to in\fnity.\nIndeed, if\u000b > 1, there exist oscillating solution that do not approach zero as t!1 . For\nexample, we consider the problem\n8\n<\n:utt\u0000\u0001u+1\n(t+1)\u000but+bu= 0 (t;x)2R+\u0002\n;\nu= 0 ( t;x)2R+\u0002@\n;\n(u(0;x);ut(0;x)) = (u0(x);u1(x))x2\n;\nwhere\u0016is an eigenvalue of \u0000\u0001+b, having the corresponding eigenfunction  (x). Takingu(t;x) =\n!(t) (x), we have\n!tt+1\n(t+ 1)\u000b!t+\u0016!= 0:\nWhen\u000b<\u00001, solutions again do not in general approach zero, though their behavior is quite\ndi\u000berent from the case \u000b<\u00001. Note that an interesting solution\nu(t;x) = (1\u0000\u0016(t+ 1)1+\u000b\n1 +\u000b) (x)\nsolves\n\u001autt\u0000\u0001u+h(t)ut+bu= 0 (t;x)2R+\u0002\n;\nu= 0 ( t;x)2R+\u0002@\n;\nwhere the damping coe\u000ecient is as follows\nh(t) =t\u0000\u000b\u0000\u0016\n1 +\u000bt\u0000\u000bt\u00001;\nand the initial data\n(u(0;x);ut(0;x)) = (1 +\u000b\u0000\u0016\n1 +\u000b (x);\u0000\u0016\n1 +\u000b (x)); x2\n:\nBut it is easy to see that u(t;x) approaches to  (x), which is not an equilibrium, as t!1 .\nThe plan of this paper is as follows. In Section 2, we make some estimates of solutions. A key\nlemma in this section is to prove a generalized type of Lojasiewicz-Simon inequality, which is \frstly\ngiven in the literature. In Section 3, we prove our main results. And we will give some numerical\nanalysis to illustrate our results in Section 4. Throughout the paper, c,c1,c2\u0001\u0001\u0001,C,C1,C2\u0001\u0001\u0001\ndenote corresponding constants.\n32. Preliminary\nBy the semigroup theory (e.g., [12], section 6.1) and regard h(t)ut+f(u) as a perturbation\nterm), we know that under Condition (3) and for u02H2(\n)\\H1\n0(\n) andu12H1\n0(\n), Problem\n(1) has a unique solution\nu(t;x)2W1;1\nloc(R+;H1\n0(\n))\\W2;1\nloc(R+;L2(\n)):\nThe solution energy is de\fned by\nEu(t) :=1\n2(kutk2\nL2+kruk2\nL2) +Z\n\nF(u)dx:\nNotice that Eu(\u0001) is non-increasing by E0\nu(t) =\u0000h(t)kutk2\nL2\u00140.\nLemma 2. Letu2W1;1\nloc(R+;H1\n0)TW2;1\nloc(R+;L2)be a solution to (1). There is a positive\nconstantcdepending only on the norm of initial data in the energy space H1\n0(\n)\u0002L2(\n), such\nthat\nkutkL2+kukH1+kf(u)kL2\u0014cfort\u00150: (8)\nAnd the functional E0(u)has at least a minimizer v2H1(\n).\nProof. For every\u000e>0, it follows from (2) that there exists M(\u000e)\u001d1 such that forjsj>M (\u000e)\nf(s)\ns\u0015\u0000(\u00161+\u000e);\nand so\nF(s)\u0015\u0000\u00161+\u000e\n2s2:\nThen we haveZ\n\nF(u)dx=Z\njuj>M(\u000e)F(u)dx+Z\njuj\u0014M(\u000e)F(u)dx\n\u0015\u0000\u00161+\u000e\n2Z\n\njuj2dx+j\njsup\njuj\u0014M(\u000e)jF(s)j;\nwhich implies that\nEu(t)\u00151\n2kutk2\nL2+1\n2\u0000\n1\u0000\u00161+\u000e\n\u00160\u0001\nkruk2\nL2+j\njsup\njuj\u0014M(\u000e)jF(s)j: (9)\nSinceEu(\u0001) is non-increasing, then by using the Poincar\u0013 e inequality, it implies from (9) that\nkutk2\nL2+kuk2\nH1\u0014c1;\nwherec1depends on the norm of initial data in the energy space. Accordingly, kf(u)kL2is also\nbounded by (I-1). Then (8) is proved.\nAnd also we know from (9) that E0(u) is bounded below. Thus, there exists a minimizing\nsequenceun2H1(\n) such that E0(un) = infu2H1E0(u). By (3) H older inequality and Sobolev\ninequalities, we have\nkf(un)\u0000f(v)k2\nL2\u0014c2Z\n\n(junj2p+jvj2p+ 1)jun\u0000vj2dx\n\u0014c3k(junj2p+jvj2p+ 1)k\nL3p\n1\u0000qkjun\u0000vj2k\nL6\n1+2q\n\u0014c4(kunk2p\nH1\u0000q+kvk2p\nH1\u0000q+ 1)kun\u0000vk2\nH1\u0000q\nwithq=2\u0000p\n2(1+p)2(0;1). From (8) and the Aubin's compactness theorem, we know that unis\nrelatively compact in L1([0;T];H1\u0000q(\n)). Then there is a subsequence, denoted still by un, such\nthatun!vinH1\u0000q, andv2H1. It implies\njZ\n\nF(un)\u0000F(v)dxj=jZ\n\n[Z1\n0f(sun)un\u0000f(sv)vds]dxj\n=jZ\n\n[Z1\n0f(sun)un\u0000f(sun)v+f(sun)v\u0000f(sv)vds]dxj\n\u0014c5fkun\u0000vkL2+kf(sun)\u0000f(sv)kL2g!0\n4asngoes to in\fnity. Because kukH1is weakly lower semicontinuous, we obtain E0(v) = infu2H1E0(u),\nthat is,vis a minimizer of the functional E0(u). \u0003\nRemark 3. From this Lemma, we know the following static problem associated to system (1)\n\u001a\u0000\u0001 +f( ) = 0x2\n;\n = 0 x2@\nadmits at least a classical solution, which means that the set of equilibria is nonempty.\nThe following inequality is a generalized type of Simon-Lojasiewicz inequality.\nLemma 3. Assume that u2W1;1\nloc(R+;H1\n0)TW2;1\nloc(R+;L2)is a solution to (1) and  2S.\nThere exist constants c>0,T >0,\u00122(0;1\n2]and\f >0depending on  such that for\nkG(u)kH\u00001\u0015cjE0(u)\u0000E0( )j1\u0000\u0012\nprovidedku\u0000 kH1\u0000q(\n)<\f,q=2\u0000p\n2(1+p).\nProof. Ifku\u0000 kH1(\n)\u0015\f, then forz=u\u0000 , we have\n\u001a\u0000\u0001z=utt+h(t)ut+f(u)\u0000f( )x2\n;\nz= 0 x2@\nFrom the regularity theory for elliptic problem we know\nkzkH1\u0014c1k\u0001zkH\u00001; (10)\nwherec1is a constant independent of u. From (3), we have\nkf(u)\u0000f( )kL2\u0014c2\u0000\nkukp\nH1\u0000q+k kp\nH1\u0000q+ 1\u0001\nku\u0000 kH1\u0000q:\nThere exists ~\f >0 depending on  such that for any v,ku\u0000 kH1\u0000q<~\f,\nkf(u)\u0000f( )kL2<\f\n2c1: (11)\nThen it infers from (10), (11) and Lemma 2 that\nk\u0000\u0001u+f(u)kH\u00001=k\u0000\u0001z+f(u)\u0000f( )kH\u00001\n\u0015k\u0001zkH\u00001\u0000kf(u)\u0000f( )kH\u00001\n\u00151\nc1kzkH1\u0000kf(u)\u0000f( )kH\u00001\n\u00151\n2c1kzkH1\n\u0015c3jE0(u)\u0000E0( )j1\u0000\u0012:\nFor the other case kv\u0000 kH1(\n)<\f, it follows from (I-2) that the estimate in this Lemma holds.\nThus, our proof is completed. \u0003\n3. Main Result\nIn this section, we give the proof of Theorem 1.\nProof. The subsequent proof consists of several steps.\nStep 1. From (8) and the Aubin's compactness theorem, we know that uis relatively compact in\nL1([0;T];H1\u0000q(\n)). It follows that there exist a sequence tn!+1and 2H1\u0000qsuch that\nku(tn;\u0001)\u0000 kH1\u0000q!0; n!+1: (12)\nLet\u00112(0;1), we de\fne\nH(t) =Eu(t)\u0000E0( ) +\u0011\n(t+ 1)\u000bhG(u);uti;\n5whereh\u0001;\u0001iisH\u00001inner product. Then we have\nH0(t) =\u0000h(t)kutk2\nL2\u0000\u0011\u000b\n(t+ 1)\u000b+1hG(u);uti+\u0011\n(t+ 1)\u000bhG0(u)ut;uti\n\u0000\u0011\n(t+ 1)\u000bkG(u)kH\u00001\u0000\u0011\n(t+ 1)\u000bhG(u);h(t)uti:\nIfh(t) satis\fes (6), then we obtain\nH0(t)\u0014\u0000c\n(t+ 1)\u000bkutk2\nL2\u0000\u0011\n(t+ 1)\u000bkG(u)kH\u00001+\u0011\n(t+ 1)\u000bhG0(u)ut;uti\n+ (\u0011\u000b\n(t+ 1)\u000b+1+\u0011C\n(t+ 1)2\u000b)jhG(u);utij:\nAnd whenh(t) satis\fes (7), then we obtain\nH0(t)\u0014\u0000ct\u000bkutk2\nL2\u0000\u0011\n(t+ 1)\u000bkG(u)kH\u00001+\u0011\n(t+ 1)\u000bhG0(u)ut;uti\n+ (\u0011\u000b\n(t+ 1)\u000b+1+\u0011Ct\u000b\n(t+ 1)\u000b)jhG(u);utij\n\u0014\u0000ct\u000bkutk2\nL2\u0000\u0011\n(t+ 1)\u000bkG(u)kH\u00001+\u0011\n(t+ 1)\u000bhG0(u)ut;uti\n+ (\u0011\u000b\n(t+ 1)\u000b+1+\u0011C)jhG(u);utij:\nSince we know\njhG(u);utij\u0014kG(u)kH\u00001kutkH\u00001\u00141\n2kG(u)k2\nH\u00001+1\n2kutk2\nL2;\nor\njhG(u);utij\u0014kG(u)kH\u00001kutkH\u00001\u0014C1\n2(1 +t)\u000bkG(u)k2\nH\u00001+t\u000b\n2C2kutk2\nL2;\nmoreover,\nk(\u0000\u0001)\u00001(f0(u))utk\u0014C3(1 +kukH1)kutkL2\nby (3), and so\njhG0(u)ut;utij\u0014C4kutk2\nL2;\nthen it implies from choosing \u0011small enough that\nH0(t)\u0014\u0000C5\n(t+ 1)\u000b(kutk2\nL2+kG(u)k2\nH\u00001) (13)\nregardless of whether h(t) satis\fes (6) or (7). And (13) implies that H(t) is non-increasing. It\nfollows that H(t) has a \fnite limit as time goes to + 1.\nStep 2. Due to the Lemma 2, uis weakly compact in H1. And then we note that\nkG(u(t;\u0001))\u0000G(u(s;\u0001))kH\u00001\n= sup\n\u00102ZfjhG(u(t;\u0001))\u0000G(u(s;\u0001));\u0010ijg\n\u0014sup\n\u00102Zfjh\u0001u(t;\u0001)\u0000\u0001u(s;\u0001);\u0010ijg+ sup\n\u00102Zfjhf(u(t;\u0001))\u0000f(u(s;\u0001));\u0010ijg\n!0; t!1:\nwhereZ=f\u00102H1\n0:k\u0010kH1= 1g. Then we have\nkG(u)\u0000Zt+1\ntG(u)dskH\u00001\u0014Zt+1\ntkG(u(t;\u0001))\u0000G(u(s;\u0001))kH\u00001ds\n!0; t!+1:(14)\nBecausekutkH\u00001is uniformly continuous, we deduce that\nkut(t+ 1;\u0001)\u0000ut(t;\u0001)kH\u00001!0; t!+1: (15)\n6Ifh(t) satis\fes (6), we have\nkZt+1\nth(s)usdskH\u00001\u0014kZt+1\nth(s)usdskL2\u0014Zt+1\nth(s)kuskL2ds\n\u0014(Zt+1\nth(s)ds)1\n2(Zt+1\nth(s)kusk2\nL2ds)1\n2\n\u0014C6(Zt+1\nth(s)kusk2\nL2ds)1\n2\n\u0014C6(E(t)\u0000E(t+ 1))1\n2!0; t!+1:(16)\nLetJ(t) =kutk2\nH\u00001\n(t+1)\u000b. We know\nd\ndtJ(t) =\u0000\u000b\nt+ 1J(t) +2\n(t+ 1)\u000bhut;utti\n=\u0000\u000b\nt+ 1J(t)\u00002h(t)J(t) +2\n(t+ 1)\u000bhut;\u0000G(u)i\n\u0014\u0000(2h(t)\u00001 +\u000b\nt+ 1)J(t) +1\n(t+ 1)\u000bkG(u)k2\nH\u00001\n\u0014\u0000(2h(t)\u00001 +\u000b\nt+ 1)J(t)\u00001\nC5H0(t)\nby (13). Note that\nsup\nt\u00150Zt+1\ntH0(s)ds= 0;\nand fort >1, 2h(t)\u00001 +\u000b\nt+1>0 if h(t) satis\fes (7). Then it follows from a Grownwall type\ninequality (see Lemma 2.2 in [13]) that\nkutk2\nH\u00001\u0014C7(t+ 1)\u000be\u0000\u000ft; t> 1:\nFurthermore, h(t) satis\fes (7), we have\nkZt+1\nth(s)usdskH\u00001\u0014kZt+1\nth(s)usdskL2\u0014Zt+1\nth(s)kuskL2ds\n!0; t!+1:(17)\nNow we examine the term kG(u)kH\u00001. Since we know by (1) that\nG(u) =G(u)\u0000Zt+1\ntG(u)ds\u0000Zt+1\nt(uss+h(s)us)ds\n=G(u)\u0000Zt+1\ntG(u)ds\u0000(ut(t+ 1;\u0001)\u0000ut(t;\u0001))\u0000Zt+1\nth(s)usds\nthen we have\nkG(u)kH\u00001\u0014kG(u)\u0000Zt+1\ntG(u)dskH\u00001+kut(t+ 1;\u0001)\u0000ut(t;\u0001)kH\u00001\n+kZt+1\nth(s)usdskH\u00001:\nThanks to (14), (15), (16) and (17), we have\nlim\nt!1kG(u)kH\u00001= 0\nTherefore, we know  2H1and\n\u0000\u0001 +f( ) =G( ) = lim\nn!+1G(u(tn;\u0001)) = 0;\n7which means that  is an equilibrium.\nStep 3. For any 0<\u000e<\f , there exists an integer Nsuch that for any n\u0015N\n[H(tn)]\u0012(1\u0000\r)\u0000[H(t)]\u0012(1\u0000\r)\u0014\u000e\n2; t>tn\u00150; (18)\nwhere\fis the constant in Lemma 3. De\fne\n^tn= supft>tn:ku(s;\u0001)\u0000 (\u0001)kH1\u0000q<\f;8s2[tn;t]g:\nDue to 2(1\u00002\u0012)\u00150 and the uniform boundedness in Lemma 2, we deduce from Lemma 3 that\nfort2[tn;^tn),n>N ,\nkutk2\nL2+kG(u)k2\nH\u00001\n\u0015kutk2\nL2+1\n2kG(u)k2\nH\u00001+1\n2jE0(u)\u0000E0( )j2(1\u0000\u0012)\n\u0015kutk2(1\u00002\u0012)\nL2\nC8kutk2\nL2+kG(u)k2(1\u00002\u0012)\nH\u00001\nC9kG(u)k2\nH\u00001d+1\n2jE0(u)\u0000E0( )j2(1\u0000\u0012)\n\u0015C10fkutk4(1\u0000\u0012)\nL2 +kG(u)k4(1\u0000\u0012)\nH\u00001+jE0(u)\u0000E0( )j2(1\u0000\u0012)g\n\u0015C11fkutk4(1\u0000\u0012)\nL2 +kutk2(1\u0000\u0012)\nL2kG(u)k2(1\u0000\u0012)\nH\u00001+jE0(u)\u0000E0( )j2(1\u0000\u0012)g\n\u0015C12H(t)2(1\u0000\u0012):(19)\nThen we deduce from (13) that for t2[tn;^tn),n>N ,\nH0(t)\u0014\u0000C13\n(t+ 1)\u000bH(t)2(1\u0000\u0012);\nso that\nH(t)\u0014\u001a\nC14(1 +t)\u00001\u0000\u000b\n1\u00002\u0012; \u00122(0;1\n2);\nC15exp(\u0000C11t1\u0000\u000b)\u0012=1\n2:(20)\nTaking\r2(0;1), from (13) and (19) we also have\n\u0000d\ndt[H(t)\u0012(1\u0000\r)] =\u0000\u0012(1\u0000\r)H0(t)[H(t)]\u0012(1\u0000\r)\u00001\n\u0014\u0000C16\u0012(1\u0000\r)(kutkL2+kG(u)kH\u00001)2H(t)\u0012(1\u0000\r)\u00001\n\u0014\u0000C17\n(t+ 1)\u000bH(t)\u0000\u0012\r(kutkL2+kG(u)kH\u00001)\nfort2[tn;^tn),n>N . Note that (1\u0000\u000b)(1\u00002\u0012)\u00001\u0012\r >\u000b . By (20), we obtain\nC17\n(t+ 1)\u000bH(t)\u0000\u0012\r!+1; t!+1:\nTherefore,\nkutkL2\u0014\u0000C18d\ndt[H(t)\u0012(1\u0000\r)] (21)\nfort2[tn;^tn),nlarge enough. Then we see that\nsup\nt2[tn;^tn)ku(t;\u0001)\u0000 kL2\u0014ku(tn;\u0001)\u0000 kL2+C19jZ^tn\ntnd\ndt[H(t)\u0012(1\u0000\r)]dtj;\nwhich implies that ^tn= +1whennis large enough. Then we assert that u(t;\u0001) converges to  in\nL2astgoes to +1. Sinceuis relatively compact in H1\u0000q(\n) (see in Lemma 2), then we have\nku(t;\u0001)\u0000 kH1\u0000q!0; t!+1: (22)\n8From (22), we can deduce\njZ\n\nF(u)\u0000F( )dxj!0; t!+1:\nDue to\nlim\nt!1H(t) = lim\nt!1fkut(t;\u0001)kL2+E0(v)\u0000E0( )g= 0;\nthen we knowf(0; ) : 2Sgis the attracting set for system (1), that is,\nlim\nt!+1(kut(t;\u0001)kL2+ku(t;\u0001)\u0000 kH1) = 0:\nStep 4. From (21) and (22), it infers that\nkv(t;\u0001)\u0000 kL2\u0014Z1\ntkvtkL2ds\u0014C20[H(t)]\u0012(1\u0000\r);\nwhich implies the required speed estimates from (20). \u0003\n4. Numerical simulation\nIn this section, we present several examples to illustrate the evolution of the solutions to the\nsystem (1). We will show how the role of damping coe\u000ecient h(t) a\u000bects the dynamical behaviors\nof the solution. We consider the following one-dimensional equations\n8\n<\n:utt+h(t)ut\u0000uxx+G(u) = 0 (t;x)2(0;T)\u0002(\u0000L;L)\nu(t;\u0000L) =u(t;L) = 0 t2(0;T)\n(u(0;x);ut(0;x) = (f(x);g(x))x2(\u0000L;L)(23)\nwith the initial condition\nf(x) = 0; g(x) =4p\n1\u0000c2\ncosh(xp\n1\u0000c2):\nTakeh(t) =hi(t),i= 0, 1, ..., 6 with\nh0= 0; h 1= 1; h 2=1pt+ 1; h 3=1\nt+ 1;\nh4=p\nt; h 5=t; h 6=t3\n2;\nalso, we consider two cases of the function G(u): one is\nG(u) =G1(u) =au+jujp\u00001u; p\u00151;\nand the other is\nG(u) =G2(u) =bsinu; b> 0:\nThe system 23 with the nonlinear term G1(u) is so-called dissipative Klein-Gordon equation, and\nwithG2(u) is dissipative sine-Gordon equation. Here, the Lojasiewicz exponent for sine-Gordon\nequation is1\n2. And ifa>\u0000\u00192\nL2, which is the \frst eigenvalue for one-dimensional Laplace operator\nwith Dirichlet boundary condition, then the Lojasiewicz exponent for Klein-Gordon equation is1\n2.\nIfa <\u0000\u00192\nL2, the Lojasiewicz exponent is1\np+1. We will use central di\u000berences for both time and\nspace derivatives and the following parameters set:\nSpace interval L= 20\nSpace discretization \u0001x= 0:1\nTime discretization \u0001t= 0:05\nAmount of time steps T= 200\nVelocity of initial wave c= 0:2\n9As for the non-damping case h(t) =h0= 0, the numerical results can be seen in \fgure 1 (a)\nand (b). As can be seen easily, The solutions to these two equations does not decay as time goes\nto in\fnity. This is because the systems are conservative, that is, the energy of these two systems\nEu(t) keep to be some constant Eu(0) depended fully on the initial data.\nIf the systems are damped by time-dependent damping, the numerical results can be seen in\n\fgure 2 (a) and (b). It is clearly that the damping coe\u000ecients a\u000bect the dynamical behaviors of\nthe solutions to these two systems. From the \fgures, we also see that L2-norms of the solutions\nto these two equations converge to some equilibrium, when h(t) =hi(t),i= 1, 2, 4, 5. For\nh(t) =h3=1\nt+1,L2-norms of the solutions keep on oscillating as time goes to in\fnity. And if\nh(t) =h6=t3\n2,L2-norms of the solutions also converge to some constant, but which is not the\nvalue of some equilibrium. These numerical results are in accordance with the conclusion given in\nTheorem 1.\nHow the Lojasiewicz exponent a\u000bect the convergence speeds of the solutions to these two\nequations? In Theorem 1, whether the convergence for the systems is exponential or polynomial\ndepends on the Lojasiewicz exponent \u0012, but the convergence rate is in terms of the damping\ncoe\u000ecienth(t). These can be seen from \fgure 3 (a) and (b).\nRemark 4. Whenh(t)\u0018t, whether convergence results hold in unknown theoretically. From the\nnumerical simulation as in \fgure 2 or 3, we know the solution converges to some equilibrium in\nL2-norm.\nAuthors contributions\nAll the authors contributed equally and signi\fcantly in writing this paper. All authors read\nand approved the \fnal manuscript.\nDeclaration of Competing Interest\nThe authors declare that they have no competing interests.\nAcknowledgements\nThe authors thank the anonymous referees very much for the helpful suggestions.\nReferences\n[1] P. Pucci, J. Serrin, Asymptotic stability for nonautonomous dissipative wave systems, Com-\nmunications on Pure and Applied Mathematics 49 (1996) 177{216.\n[2] A. Cabot, H. Engler, S. Gadat, On the long time behavior of second order di\u000berential equations\nwith asymptotically small dissipation, Transactions of the American Mathematical Society\n361 (2009) 5983{6017.\n[3] H. Attouch, X. Goudou, P. Reont, The heavy ball with friction method. i: The continuous\ndynamical system, Commun. Contemp. Math. 2 (2000) 1{34.\n[4] M. Daoulatli, Rates of decay for the wave systems with time dependent damping, Discrete\nand Continuous Dynamical Systems-Series A 31 (2011) 407{443.\n[5] A. Haraux, M. A. Jendoubi, Asymptotics for a second order di\u000berential equation with a linear,\nslowly time-decaying damping term, Evolution Equations and Control Theory 2 (2013) 461{\n470.\n[6] Y. Gao, J. Liang, T. Xiao, A new method to obtain uniform decay rates for multidimensional\nwave equations with nonlinear acoustic boundary conditions, SIAM Journal of Control and\nOptimization 56 (2018) 1303{1320.\n[7] A. Cabot, P. Frankel, Asymptotics for some semilinear hyperbolic equations with non-\nautonomous damping, Journal of Di\u000berential Equations 252 (2012) 294{322.\n10[8] R. May, Long time behavior for a semilinear hyperbolic equation with asymptotics vanishing\ndamping term and convex potential, Journal of Mathematical Analysis and Applications 430\n(2015) 410{416.\n[9] Z. Jiao, X.-T. Xiao, Convergence and speed estimates for semilinear wave systems with\nnonautonomous damping, Mathematical Methods in the Applied Sciences 39 (2016) 5456{\n5474.\n[10] A. Haraux, M. A. Jendoubi, The Convergence problem for Dissipative Autonomous Systems,\nSpringer, 2013.\n[11] R. Chill, On the lojasiewicz-simon gradient inequality, Journal of Functional Analysis 201\n(2003) 572{601.\n[12] A. Pazy, Semigroups of Linear Operators and Applications to Partial Di\u000berential Equations,\nSpringer-Verlag: New York, 1983.\n[13] M. Grasselli, V. Pata, Asymptotic behavior of a parabolic-hyperbolic system, Commun. Pure\nAppl. Anal. 3 (2004) 849{881.\n11(a) sine-Gordon equation\n(b) Klein-Gordon equaiton\nFigure 1: Dynamical behaviors for two equations without damping. (a) h(t) =h0,b= 1 The tendency of the\nsolution to sine-Gordon equation. (b) h(t) =h0,a= 1, p= 3 The tendency of the solution to Klein-Gordon\nequation.\n120 50 100 150 200\ntime0510L2 normh1\n0 50 100 150 200\ntime0510L2 normh2\n0 50 100 150 200\ntime0510L2 normh3\n0 50 100 150 200\ntime0510L2 normh4\n0 50 100 150 200\ntime0510L2 normh5\n0 50 100 150 200\ntime010L2 normh6(a) sine-Gordon equation\n0 50 100 150 200\ntime05L2 normh1\n0 50 100 150 200\ntime05L2 normh2\n0 50 100 150 200\ntime0510L2 normh3\n0 50 100 150 200\ntime0510L2 normh4\n0 50 100 150 200\ntime0510L2 normh5\n0 50 100 150 200\ntime0510L2 normh6\n(b) Klein-Gordon equaiton\nFigure 2: Dynamical behaviors for two equations with time-dependent damping. (a) h(t) =hi,b= 1 The tendency\nofL2-norms of the solution to sine-Gordon equation. (b) h(t) =hi,a= 1,p= 3 The tendency of L2-norms of the\nsolution to Klein-Gordon equation.\n130 10 20 30 40 50 60 70 80 90 100012345678910\na=0\na=-0.01\na=-0.03\na=-0.04\na=-0.1(a)h(t) =h2,p= 3\n0 20 40 60 80 100\ntime01020L2 normh1\n0 20 40 60 80 100\ntime01020L2 normh2\n0 20 40 60 80 100\ntime01020L2 normh3\n0 20 40 60 80 100\ntime01020L2 normh4\n0 20 40 60 80 100\ntime01020L2 normh5\n0 20 40 60 80 100\ntime01020L2 normh6\n(b)h(t) =hi,a=\u00000:1,p= 3\nFigure 3: Convergence rates for dissipative Klein-Gordon equation.\n14"    },    {        "title": "1902.08700v1.Strongly_Enhanced_Gilbert_Damping_in_3d_Transition_Metal_Ferromagnet_Monolayers_in_Contact_with_Topological_Insulator_Bi2Se3.pdf",        "content": "1 \n Strongly Enhanced Gilbert Damping in 3 d Transition Metal \nFerromagnet Monolayers in Contact with Topological Insulator Bi 2Se3 \nY. S. Hou1, and R. Q. Wu1 \n1 Department of Physics and Astronomy, University of California, Irvine, California \n92697 -4575, USA  \n \nAbstract  \nEngineering Gilbert damping of ferromagnetic metal films is of great importance to \nexploit and design spintronic devices that are operated with an ultrahigh speed. Based on \nscattering theory of Gilbert damping, we extend the torque method originally  used in \nstudies of magnetocrystalline anisotropy to theoretically determine Gilbert dampings of \nferromagnetic metals. This method is utilized to investigate Gilbert dampings of 3 d \ntransition metal ferromagnet iron, cobalt and nickel monolayers that are co ntacted by the \nprototypical topological insulator Bi 2Se3. Amazingly, we find that their Gilbert dampings \nare strongly enhanced by about one order in magnitude, compared with dampings of their \nbulks and free -standing monolayers, owing to the strong spin -orbit coupling of Bi 2Se3. \nOur work provides an attractive route to tailoring Gilbert damping of ferromagnetic \nmetallic films by putting them in contact with topological insulators.  \n  \n \n \n \n \nEmail: wur@uci.edu   \n \n \n \n \n \n 2 \n I. INTRODUCTION  \nIn ferromagnets, the time -evolution of their magnetization M can be described by the \nLandau -Lifshitz -Gilbert (LLG) equation [1-3]  \n1Meff\nSdd\ndt dt   MMM H M\n, \nwhere \n0B g  \n  is the gyromagnetic ratio, and \nMSM  is the saturation \nmagnetization. The first term describes the precession motion of magnetization M about \nthe effective magnetic field, Heff, which  includes contributions from external field, \nmagnetic anisotropy, exchange, dipole -dipole and Dzyaloshinskii -Moriya  interactions [3]. \nThe second term represents the decay of magnetization prece ssion with a dimensionless \nparameter \n , known as the Gilbert damping [4-8]. Gilbert  damping is known to be \nimportant for the performance of various spintronic devices such as hard drives, magnetic \nrandom access memories, spin filters, and magnetic sensors [3, 9, 10]. For example, \nGilbert damping in the free layer of reader head in a magnetic hard drive determines its \nresponse speed and signal -to-noise ratio [11, 12]. The bandwidth, insertion loss , and \nresponse time of a magnetic thin film microwave device also critically depend on the \nvalue of \n  in the film [13]. \n \nThe rapid developm ent of spintronic technologies calls for the ability of tuning Gilbert  \ndamping in a wide range. Several approaches have been proposed for the engineering of \nGilbert damping in ferromagnetic (FM) thin films, by using non -magnetic or rare earth \ndopants, addi ng differ ent seed layers for growth, or adjusting composition ratios in the \ncase of alloy films [9, 14-16]. In par ticular, tuning \n  via contact with other materials \nsuch as heavy metals, topological insulators (TIs), van der Waals monolayers or magnetic \ninsulators is promising as the selection of material combinations is essentially unlimited.  \nSome of these materials may have fundamentally different damping mechanism and offer \nopportunity for studies of new phenomena such as spin -orbit torque, spin -charge \nconversion, and thermal -spin-behavior [17, 18].  \n \nIn this work, we systematically investigate the effect of Bi 2Se3 (BS), a prototypical TI, on \nthe Gilbert damping of 3d transitio n metal (TM) Fe, Co and Ni monolayers (MLs) as they 3 \n are in contacted with each other. We find that the Gilbert dampings in the TM/TI \ncombinations are enhanced by about an order of magnitude than their counterparts in \nbulk Fe, Co and Ni as well as in the fr ee-standing TM MLs.  This drastic enhancement \ncan be attributed to the strong spin -orbit coupling (SOC) of the TI substrate and might \nalso be related to its topological nature . Our work introduces an appealing way to \nengineer Gilbert dampings of FM metal fi lms by using the peculiar physical properties of \nTIs.  \n \nII. COMPUTATIONAL DETAILS  \nOur density functional theory (DFT) calculations are carried out using the Vienna Ab-\ninitio  Simulation Package (VASP) at the level of the generalized gradien t approximation \n[19-22]. We treat Bi -6s6p, Se -4s4p, Fe -3d4s, Co -3d4s and Ni -3d4s as valence electrons \nand employ the projector -augmented wave pseudopotentials to d escribe core -valence \ninteractions [23, 24].  The energy cutoff of plane -wave expansion is 450 eV [22]. The BS \nsubstrate is simulated by five quintuple layers ( QLs), with an in -plane lattice constant of \naBS = 4.164 Å and a vacuum space of 13 Å between slabs along the normal axis. For the \ncomputational convenience, we put Fe, Co and Ni MLs on both sides of the BS slab. For \nthe structural optimization of the BS/TM slabs, a 6× 6× 1 Gamma -centered k -point grid is \nused, and the positions of all atoms except those of the three central BS QLs are fully \nrelaxed with a criterion that the force on each atom is less than 0.01 eV/Å. The van der \nWaals (vdW) correction in the form of the nonlocal vdW functional (optB86b -vdW) [25, \n26] is included in all calculations.  \n \nThe Gilbert dampings are determined by extending the torque method that we developed \nfor the study of magnetocrystalline anisotropy [27, 28]. To ensure the numerical \nconvergence, we use very dense Gamma -centered k -point grids and, furthermore, large \nnumbers of unoccupied bands. For example, the first Bri llouin zone of BS/Fe is sampled \nby a 37× 37× 1 Gamma -centered k -point grid, and the number of bands for the second -\nvariation step is set to 396, twice of the number (188)  of the total valence electrons. More \ncomputational details are given in Appendix A. Mag netocrystalline anisotropy energies \nare determined by computing total energies with different magnetic orientations [29].  4 \n  \nIII. TORQUE METHOD OF DETERMINING GILBERT DAMPING  \nAccording to the scattering theory of Gilbert damping [30, 31], the energy dissipation \nrate of the electroni c system with a Hamiltonian, H(t), is determined by  \n dis 2i j j i F i F j\nijE E E E Euu\n         \nHHuu\n. \nHere, EF is the Fermi level and \nu  is the deviation of  normalized magnetic moment away \nfrom its equilibrium, i.e., \n0m m u  with \n00 MsM m . On the other hand, the time \nderivative of the magnetic energy in the LLG equation is [32] \n mag 3S\neffM dEdt\n MH\n mm\n. \nBy taking \ndis magEE\n  , one obtains the Gilbert damping as:  \n4i j j i F i F j\nij SE E E EM u u\n        \nHH\n. \nNote that, to obtain Eq. (4), we use \n mu  since the eq uilibrium normalized \nmagnetization m0 is a constant. In practical numerical calculations, \nFEE  is \ntypically substituted by the Lorentzian function \n 22\n0 0.5 0.5 L          . \nThe half maximum parameter, \n1 , is adjusted to reflect different scattering rates of \nelectron -hole pairs created by the precession of magnetization M [10]. This procedure  \nhas been already used in several ab initio  calculations for Gilbert dampings of metallic \nsystems [8, 9, 32-35], where the electronic responses play the major role for energy \ndissipation .  \n \nIn this work, we focus on the primary Gilbert damping in FM metals that arises from \nSOC [10, 36-38]. There are two important effects in a uniform precession of \nmagnetization M, when SOC is taken into consideration. The first is the F ermi surface \nbreathing as M rotates, i.e., some occupied states shift to above the Fermi level and some \nunoccupied states shift to below the Fermi level. The second is the transition between \ndifferent states across the Fermi level as the precession can be viewed as a perturbation to 5 \n the system. These two effects generate electron -hole pairs near the Fermi level and their \nrelaxation through lattice scattering leads to the Gilbert damping.  \n \nNow we demonstrate how to obtain the Gilbert damping due to SOC thro ugh extending \nour previous torque method [27]. The general Hamiltonian in Eq. (4) can be replaced by \n SOC j j jr\n H l s\n [4, 27] where  the index j refers to atoms, and \njji lr and s \nare orbital and spin operators, respectively. This is in the same spirit for the determination \nof the magnetocrystal line anisotropy [27], for which  our torque meth od is recognized as a \npowerful tool in the framework of spin -density theory [27]. When m points at the \ndirection of \n , , ,x y zm m m n , the term \nls  in HSOC is written as follows:   \n22\n2211cos sin sin22\n1sin sin cos 52 2 2\n1sin cos sin2 2 2ii\nz\nii\nz\nii\nzs l l e l e\ns l l e l e\ns l l e l e\n\n  \n\n\n\n\n  \n\n     \n   \n  \nn ls\n \nTo obtain the derivatives of H in Eq. (4), we assume that the magnitude of M is constant \nas its direction changes [36]. The processes of getting angular derivatives of H are \nstraightforward and the results are given by Eq. (A1) -(A5) in Appendix B.  \n \nIV. RESULTS AND DISCUSSION  \nIn this section, we first show that our approach of determining  Gilbert damping works \nwell for  FM metals such as 3d TM Fe, Co and Ni bulks. Following that, we demonstrate \nthe strongly enhanced Gilbert dampings of Fe, Co and Ni MLs due to the contact with BS \nand then discuss the underlying physical mechanism of these enhancements.  \n \nA. Gilbe rt dampings of 3d TM Fe, Co and Ni bulks  \nGilbert dampings of 3d TM bcc Fe, hcp Co and fcc Ni bulks calculated by means of our \nextended torque method are consistent with previous theoretical results [10]. As shown in \nFig. 1, the intraband contributions decrease whereas the interband contributions increase \nas the scattering rate \n  increases. The minimum values of  \n have the same magnitude 6 \n as those in Ref. [10] for all three metals, showing the applicability of our approach for the \ndetermination of Gilbert dampings of FM metals.  \n \n \nFigure 1 (color online) Gilbert dampings of (a) bcc Fe, (b) hcp Co and (c) fcc Ni bulks. Black \ncurves give the total Gilbert damping. Red and blue curves give the intraband and interband \ncontributi ons to the total Gilbert damping, respectively.  \n \nB. Strongly enhanced Gilbert dampings of Fe, Co and Ni MLs in contact with BS  \nWe now investigate the magnetic properties of heterostructures of BS and Fe, Co and Ni \nMLs. BS/Fe is taken as an example and its atom arrangement is shown in Fig. 2a. From \nthe spatial distribution of charge density difference \nBS+Fe-ML BS Fe-ML        in Fig. \n2b, we see that there is fairly obvious charge transfer between Fe and the topmost Se \natoms. By taking  the average of \n  in the xy plane, we find that charge transfer mainly \ntakes place near the interface (Fig.2c). Furthermore, the charge transfer induces non -\nnegligible magnetization  in the topmost QL of BS (Fig. 2b). Similar charge  transfers and \ninduced magnetization are also found in BS/Co and BS/Ni (Fig. A1 and Fig. A2 in \n7 \n Appendix C). These suggest that interfacial interactions between BS and 3 d TMs are very \nstrong.  Note that BS/Fe and BS/Co have in -plane easy axes whereas the BS/ Ni has an \nout-of-plane one.     \n \n \nFigure 2 (color online) (a) Top view of atom arrangement  in BS/Fe. (b) Charge density difference \n\n near the interface in BS/Fe. Numbers give the induced magnetic moments (in units of \nB ) in \nthe top most  QL BS. Color bar indicates the weight of negative (blue) and positive (red) charge \ndensity differences. (c) Planer -averaged charge density difference \n in BS/Fe. In (a), (b), (c), \ndark green, light gra y and red balls represent Fe, Se and Bi atoms, respectively.  \n \nFig. 3a and 3b show the \n  dependent Gilbert dampings of BS/Fe, BS/Co and BS/Ni. It is \nstriking that Gilbert dampings of BS/Fe, BS/Co and BS/Ni are enhanced by about one  or \ntwo order s in magnitude  from the counterparts of Fe, Co and Ni bulks as well as their \nfree-standing MLs, depending on the choice of scattering rate in the range from 0. 001 to \n1.0 eV.  Similar to Fe, Co and Ni bulks, the intraband contributions monotonically  \ndecrease while the interband contributions increase as the scattering rate \n  gets larger \n(Fig. A3 in Appendix D).  Note that our calculations indicate that there is no obvious \ndifference between the Gilbert dampings of BS/Fe when f ive- and six -QL BS slabs are \nused (Fig. A4 in Appendix E). This is consistent with the experimental observation that \nthe interaction  between the top and bottom topological surface states is negligible  in BS \nthicker than five QLs [39].  \n \n8 \n  \nFigure 3 (color online) Scattering rate \n  dependent Gilbert dampings of (a) Fe ML, bcc Fe bulk, \nBS/Fe and PbSe/Fe, (b) Co ML, hcp Co bulk, BS/Co, Ni ML, fcc Ni bulk and BS/Ni. (c) \nDependence of the Gilbert dampin g of BS/Fe on the scaled SOC \nBS  of BS in the range from \nzero (\nBS0 ) to full strength (\nBS1 ). Solid lines show the fitting of Gilbert damping \nBS/Fe  \nto Eq. (6). The inse t shows Gilbert damping comparisons between BS/Fe at \nBS0 , bcc Fe bulk \nand Fe ML.  \n \nAs is well -known, TIs are characterized by their strong SOC and topologically nontrivial \nsurface states. An important issue is how they affect the Gilbert damping s in BS/TM \nsystems. Using BS/Fe as an example, we artificially tune the SOC parameter \nBS of BS \nfrom zero to full strength and fit the Gilbert damping \nBS/Fe  in powers of \nBS  as \n2\nBS/Fe 2 BS BS/Fe BS 0 (6)       \n. \nAs shown in Fig. 3c, we obtain two interesting results: (I) when \nBS  is zero, the \ncalculated residual Gilbert damping \nBS/Fe BS 0    is comparable to Gilbert dampings of \nbcc Fe bulk and Fe free -standin g ML (see the inset in Fig. 3c) ; (II) Gilbert damping \nBS/Fe\n increases almost linearly with \n2\nBS  , simi lar to previous results [36]. These reveal \nthat the strong SOC of BS is crucial for the enhancement of Gilbert damping.    \n  \nTo gain insight i nto how the strong SOC of BS affects the damping of BS/Fe, we explore \nthe k-dependent contributions to Gilbert damping, \nBS/Fe . As shown in Fig. 4a , many \nbands near the Fermi  level show strong intermixing between Fe and BS orbitals (mar ked \nby black arrows ). Accordingly, these k-points have large contributions to the Gilbert \n9 \n damping  (marked by red arrows in Fig. 4b). However, if the hybridized states are far \naway from the Fermi level, they make almost zero contribution to the Gilbert damp ing. \nTherefore, we conclude that only hybridizations at or close to Fermi level have dominant \ninfluence on the Gilbert damping. This is understandable, since energy differences EF-Ei \nand EF-Ej are important in the Lorentzian functions in Eq. (4).   \n \n \nFigu re 4 (color online) (a) DFT+SOC calculated band structure of BS/Fe. Color bar indicates \nthe weights of BS (red) and Fe ML (blue). Black dashed line indicates the Fermi level. (b) k -\ndependent contributions to Gilbert damping \nBS/Fe at sc attering rate \n26meV . Inset shows \nthe first Brillouin zone and high -symmetry k -points \n , \nK and \n .  \n \nIt appears that there is no direct link between the topologic al nature of BS and the strong \nenhancement of Gilbert damping. The main contributions to Gilbert damping are not \nfrom the vicinity around the \n -point,  where the topological nature of BS manifests.  \nBesides,  BS should undergo a topol ogical phase transition from trivial to topological as \nits SOC \nBS  increases [40]. If the topological nature of BS dictates the e nhancement of \nGilbert damping, one should expect a kink in the \nBS  curve at this phase transition \npoint but this is obviously absent i n Fig. 3c.  \n \n10 \n To dig deeper into this interesting issue, we replace the topologically nontrivial BS with a \ntopologically trivial insulator PbSe, because the latter has a nearly the same SOC as the \nformer. As shown in Fig. 3a, the Gilbert damping of PbSe/Fe is noticeably smaller than \nthat of BS/Fe, although both are significantly enhanced from the values of \n  of Fe bulk \nand Fe free -standing ML. Taking the similar SOC and surface geometry between BS and \nPbSe  (Fig. A5 in Appendix F) , the large difference between the Gilbert dampings of \nBS/Fe and PbSe/Fe suggests that the topological nature of BS still has an influence on \nGilbert damping. One possibility is that the BS surface is metallic with the presence of \nthe time -reversal protected t opological surface states and hence the interfacial \nhybridization is stronger.      \n \n \nFigure 5 (color online) Comparisons between Gilbert damping \n  of BS/Fe at \n26meV  and \n(a) total DOS, (b) Fe projected DOS and (c) BS projected DOS. Red arrows and light cyan \nrectangles highlight the energy windows where Gilbert damping \n  and the total DOS and Fe \nPDOS have a strong correlation . In (a), (b) and (c), all DOS are in units of state per eV and \nFermi level EF indicated by the vertical green lines is set to be zero.  \n \nA previous study of Fe, Co and Ni bulks suggested a strong correlation between Gilbert \ndamping and total density of states (DOS) around the Fermi level [36]. To attest if this is \n11 \n applicable here, we show the total DOS and Gilbert damping \nBS/Fe  of BS/Fe as a \nfunction of the Fermi level based on the rigid band a pproximation. As shown in Fig. 5a, \nGilbert damping \nBS/Fe and the total DOS behave rather differently in most energy \nregions. From the Fe projected DOS (PDOS) and BS projected PDOS (Fig. 5b and 5c), \nwe see a better  correlation between G ilbert damping \nBS/Fe and Fe -projected DOS, \nespecially in regions highlighted by the cyan rectangles . We perceive that although the \n\n-DOS correlation might work for simple systems, it doesn’t hold when hybridiza tion and \nSOC are complicated as the effective SOC strength may vary from band to band.  \n \nV. SUMMARY  \nIn summary, we extend our previous torque method from determining magnetocrystalline \nanisotropy energies [27, 28] to calculating Gilbert da mping of FM metals and apply this  \nnew approach to Fe, Co and Ni MLs in contact with TI  BS. Remarkably, the presence of \nthe TI BS substrate causes order of magnitude enhancements in their Gilbert dampings. \nOur studies demonstrate such strong enhancement is mainly due to the strong SOC of TI \nBS substrate . The topological nature of BS may also play a role by facilitati ng the \ninterfacial hybridiz ation and leaving more states around the Fermi level . Although \nalloying with heavy elements also enhances Gilbert dampings  [32], the use of  TIs pushes \nthe enhancement into a much wide r range. Our work thus establishes an attractive way \nfor tuning the Gilbert damping of FM metallic films, especially in the ultrathin reg ime.  \n \nACKNOWLEDGMENTS  \nWe thank Prof. A. H. MacDonald and Q. Niu at University Texas, Austin, for insightful \ndiscussions. We also thank Prof. M. Z. Wu at Colorado State University and Prof. J. Shi \nat University of California, Riverside for sharing their ex perimental data before \npublication. Work was supported by DOE -BES (Grant No. DE -FG02 -05ER46237). \nDensity functional theory calculations were performed on parallel computers at NERSC \nsupercomputer centers.   \n \n 12 \n Appendix A: Details of Gilbert damping calcula tions  \nTo compare Gilbert dampings of Fe, Co and Ni free -standing MLs with BS/Fe, BS/Co, \nand BS/Ni, we use \n33  supercells containing three atoms and set their lattice \nconstants to 4.164 Å, same as that of the BS substrate. This means that the lattice \nconstant of their primitive unit cell (containing one atom) is fixed at 2.40 Å. The relaxed \nlattice constants of Fe (2.42 Å), Co (2.35 Å) and Ni (2.36 Å) free -standing MLs are close \nto this value.   \nSystems  a (Å)  b (Å) c (Å) k-point grid            \nFe bulk  2.931  2.931  2.931  35× 35× 35  16 36 2.25 \n Co bulk  2.491  2.491  4.044  37× 37× 23  18 40 2.22 \nNi bulk  3.520  3.520  3.520  31× 31× 31  40 80 2.00 \nFe ML  4.164  4.164  -- 38× 38× 1  24 56 2.33 \nCo ML  4.164  4.164  -- 37× 37× 1  27 64 2.37    \nNi ML  4.164  4.164  -- 39× 39× 1  30 72 2.40 \nBS/Fe  4.164  4.164  -- 37× 37× 1  188 396 2.11 \nBS/Co  4.164  4.164  -- 37× 37× 1  194 408 2.10 \nBS/Ni  4.164  4.164  -- 37× 37× 1  200 432 2.16 \nPbSe/Fe  4.265  4.265  -- 37× 37× 1  174 376 2.16 \n \nTable A1. Here are details of Gilbert damping calculations of all systems that are studied \nin this work.     is abbreviated for the number of  valence electrons and     stands for the \nnumber of total bands.   is the ratio between     and    , namely,          . Note that \nfive QLs of BS are used in calculations for BS/Fe, B S/Co and BS/Ni.   \n \n \n \n \n \nAppendix B: Derivatives of SOC Hamiltonian HSOC with respect to the \nsmall deviation \nu  of magnetic moments  \nBased on the SOC Hamiltonian HSOC in Eq. (5) in the main text, derivatives of the term \nls\n against the polar angle \n  and azimuth angle \n  are    13 \n \n11sin cos cos22\n1 1 1cos sin sin A1 ,2 2 2\n1 1 1cos sin sin2 2 2ii\nnz\nii\nz\nii\nzs l l e l e\ns l l e l e\ns l l e l e\n\n  \n  \n  \n\n\n  \n\n       \n   \n  \nls \nand \n \n  \n 22\n22110 sin sin22\n10 sin cos A2 .2 2 2\n10 cos sin2 2 2ii\nn\nii\niis i l e i l e\ns i l e i l e\ns i l e i l e\n\n\n\n\n\n\n  \n\n         \n     \n    \nls\n \nNote that magnetization M is assumed to have a constant magnitude when it precesses , so \nwe have \n0SOC SOC    H M H m . When the normalized magnetization  m points \nalong the direction of \n , , ,x y zm m m n , we have: \nsin cosxm , \nsin sinym  \nand \ncoszm . Taking \n0m m u  and the chain rule together, we obtain derivatives of \nSOC Hamiltonian HSOC with respect to the small deviation  of magnetic moments as \nfollows:   \nsincos cos A3 ,sinSOC SOC SOC SOC SOC\nx x x x x\nSOC SOCu m m m m\n\n                   \nH H H H H m\nm\nHH\n \ncoscos sin A4 ,sinSOC SOC SOC SOC SOC\ny y y y y\nSOC SOCu m m m m\n\n                   \nH H H H H m\nm\nHH\n \nand  \nu14 \n \n sin A5 .SOC SOC SOC SOC SOC\nz z z z z\nSOCu m m m m\n\n                 \nH H H H H m\nm\nH \nCombining Eq. (5) and Eq. (A1 -A6), we can easily obtain the final formulas of \nderivatives of SOC Hamiltonian HSOC of magnetization m. \n \n \n \n \nAppendix C: Charge transfers and induced magnetic moments in BS/Fe, \nBS/Co and BS/Ni  \n \nFigure A1 (color online) Planar -averaged char ge difference \nBS TM ML BS TM-ML         \n(TM = Fe, Co and Ni) of  (a) BS/Fe, (b) BS/Co and (c) BS/Ni . The atoms positions are given along \nthe z axis.  \n \n15 \n  \nFigure A2 (Color online) Charge density difference \nBS TM ML BS TM ML          (TM = Fe, \nCo and Ni) nea r the interface betwee n the TM monolayer and the top most  QL BS of (a) BS/Fe, (b) \nBS/Co and (c) BS/Ni. The color bar shows the weights of the negative (blue) and positive (red) \ncharge density differences. Numbers give the induced magnetic moments (in units  of \nB ) in the \ntopmost  QL BS. Bi and Se atoms are shown by the purple and light green balls, respectively.  \n \n \nAppendix D: Contributions of intraband and interband to the Gilbert \ndampings of BS/Fe, BS/Co and BS/Ni  \n \nFigure A3 (color  online) Calculated Gilbert dampings of (a) BS/Fe, (b) BS/Co and (c) BS/Ni. \nBlack curves give the total damping. Red and blue curves give the intraband and interband \ncontributions, respectively.  \n16 \n Appendix E: Gilbert dampings of BS/Fe with five - and six -QLs of BS slabs  \n8  \nFigure A4 (color online). Gilbert dampings of BS/Fe with five (red) and six (black) QLs of BS \nslabs. In the calculations of the Gilbert damping of BS/Fe with six QLs of BS, we use a 39 ×39×1 \nGamma -centered k -point grid, and the number of the  total bands is 448 which is twice \nlarger than the number of the total valence electrons (216).    \n \nAppendix F: Structural c omparisons between BS/Fe and PbSe/Fe  \n \n17 \n Figure A5 (color online) (a) Top view and (c) side view of atom arrangement  in BS/Fe. (b) Top  \nview and (d) side view of atom arrangement  in PbSe/Fe. In (a) and (c), the xyz -coordinates are \nshown by the red arrows. In (b) and (d), the rectangles with blue dashed lines highlight the most \ntop QL BS in BS/Fe which is similar to the Pb and Se atom laye rs in PbSe/Fe. 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Karis1\n1Department of Physics and Astronomy,\nUppsala University, Box 516, SE- 75120 , Uppsala, Sweden\n2School of Science and Technology,\nOrebro University, SE- 70182 Orebro, Sweden\n3Department of Materials Science and Engineering,\nUppsala University, Box 534, SE- 75121 , Uppsala, Sweden\nAbstract\nWe report on optically induced, ultrafast magnetization dynamics in the Heusler alloy Co 2FeAl,\nprobed by time-resolved magneto-optical Kerr e\u000bect. Experimental results are compared to results\nfrom electronic structure theory and atomistic spin-dynamics simulations. Experimentally, we \fnd\nthat the demagnetization time ( \u001cM) in \flms of Co 2FeAl is almost independent of varying structural\norder, and that it is similar to that in elemental 3d ferromagnets. In contrast, the slower process\nof magnetization recovery, speci\fed by \u001cR, is found to occur on picosecond time scales, and is\ndemonstrated to correlate strongly with the Gilbert damping parameter ( \u000b). Our results show\nthat Co 2FeAl is unique, in that it is the \frst material that clearly demonstrates the importance of\nthe damping parameter in the remagnetization process. Based on these results we argue that for\nCo2FeAl the remagnetization process is dominated by magnon dynamics, something which might\nhave general applicability.\n1arXiv:2002.12255v1  [cond-mat.mtrl-sci]  27 Feb 2020Studies of ultrafast demagnetization was pioneered by Beaurepaire et al. [1], who demon-\nstrated that the optical excitation of a ferromagnetic material - using a short pulsed laser\ncould quench the magnetic moment on sub-picosecond timescales. The exact underlying mi-\ncroscopic mechanisms responsible for the transfer of angular momentum have been strongly\ndebated for more than 20 years [2{4]. Ultrafast laser-induced demagnetization has now be-\ncome an intense \feld of research not only from fundamental point of view but also from\na technological aspect, due to an appealing possibility to further push the limits of oper-\nation of information storage and data processing devices [5]. Both experiment [4{15] and\ntheory [16{23] report that all of the 3d ferromagnets (Fe, Ni and Co) and their alloys, show\ncharacteristic demagnetization times in the sub-picosecond range, while 4f metals exhibit a\ncomplicated two-step demagnetization up to several picoseconds after the excitation pulse\n[4, 24].\nIn this work, we have made element speci\fc investigations of the ultrafast magnetiza-\ntion dynamics of a half-metallic Heusler alloy. This class of alloys has been investigated\nintensively, especially concerning the magnetic properties, ever since the discovery in 1903,\nwhen Heusler et al. reported that alloys like Cu 2MnAl exhibit ferromagnetic properties,\neven though none of its constituent elements was in itself ferromagnetic [25]. The ferro-\nmagnetic properties were found to be related to the chemical ordering [26]. One of the key\nfeatures of several Heusler alloys is their unique electronic structure, where the majority\nspin band-structure has a metallic character while the minority spin band is semiconducting\nwith a band gap. Such materials are also referred to as half-metallic ferromagnets (HMFs)\nand were initially predicted by de Groot et al. [27], based on electronic structure theory.\nHalf-metals ideally exhibit 100% spin-polarization at the Fermi level. This exclusive prop-\nerty makes them candidates to be incorporated in spintronic devices, e.g. spin \flters, tunnel\njunctions and giant magneto-resistance (GMR) devices [28{31]. One of the advantages of\nHeusler alloys with respect to other half-metallic system, like CrO 2and Fe 3O4, are their\nrelatively high Curie temperature ( Tc) and low coercivity ( Hc) [32, 33]. Heusler alloys are\nalso appealing for spintronic applications due to the low Gilbert damping, which allows for\na long magnon di\u000busion length [34{39]. It has been shown that the low value of the Gilbert\ndamping constant is related with the half-metallicity [37, 38]. The origin of the band gap\nand the mechanism of half-metallicity in these materials have been studied by using \frst\nprinciple electronic structure calculations [40{43]. The half-metallic property is furthermore\n2known to be very sensitive to structural disorder [41{45]. From a fundamental point of view,\nit is intriguing to ask, how the band gap in the minority spin channel e\u000bects the ultrafast\nmagnetization dynamics of Heusler alloys [46, 47]. It has already been reported that some\nof the half-metals like CrO 2and Fe 3O4exhibit very slow dynamics, involving time-scales of\nhundreds of picoseconds, [46, 47] while several Co-based Heusler alloys show a much faster\ndemagnetization, similar to the time-scales of the elemental 3d-ferromagnets [48{50]. The\nfaster dynamics of these Heuslers has been discussed in Ref.[46] to be due to the fact that\nthe band gap in the minority spin channel is typically around 0 :3\u00000:5 eV, which is smaller\nthan the photon energy (1 :5 eV) of the exciting laser. It is also smaller than the band gap\nof CrO 2and Fe 3O4. Importantly, the Heusler alloys o\u000ber the possibility to study magneti-\nzation dynamics, as a function of structural order, since they normally can be prepared to\nhave a fully ordered L21phase, a partially ordered B2 phase, and a completely disordered\nA2 phase. The structural relationships of these phases are described in the Supplemental\nMaterial (SM) [51].\nWe have here studied the optically induced, ultrafast magnetization dynamics of Co 2FeAl\n(CFA) \flms, using time-resolved magneto-optical Kerr e\u000bect (TR-MOKE) as described in\nRef. [52]. By control of the growth temperature, CFA alloy forms with varying degree of\nstructural order, in a continuous way between the A2 and B2 phases, as well as between\ntheB2 andL21phases [53, 54]. We present data from four CFA samples, grown at 300 K,\n573 K, 673 K, and 773 K respectively. We henceforth denote each sample by its growth\ntemperature as a subscript, e.g. CFA 300K. As evidenced by X-ray di\u000braction, the sample\ngrown at 300 K is found to exhibit the A2 phase, while the samples grown at 573 K and\n673 K predominantly exhibit the B2 phase. The sample grown at 773 K is found to exhibit a\npure B2 phase [54]. The value of the Gilbert damping \u000bis found to monotonously decrease\nwith annealing temperature and is thus lowest for the sample grown at 773 K [55].\nCalculations based on density functional theory (DFT) of the magnetic moment, Heisen-\nberg exchange interaction and the Gilbert damping parameter are described in detail in\n(SM) [51]. These parameters were used in a multiscale approach to perform atomistic mag-\nnetization dynamics simulations, described in Sec.S1 of (SM) [51]. Here we employed the\ntwo temperature model (2TM) for the temperature pro\fle of the spin-system. In the 2TM,\nthe spin temperature increases due to the coupling to the hot-electron bath, that is excited\nby the external laser pulse. In the simulations we used a peak temperature in the 2TM of\n30.70.750.80.850.90.951.01.051.1M/M(0)05 1 0 1 5A2CoFe05 1 0 1 5t( p s )B2\n05 1 0 1 5 2 0L21FIG. 1. (Color online) Simulations of ultrafast dynamics of Co 2FeAl in the di\u000berent structural\nphasesA2 (left panel), B2 (central panel), and L21(right panel). The demagnetization is shown\nelement resolved (blue line - Co, red line - Fe). The peak temperature is 1200 K. The dotted line\nindicates the equilibrium magnetization at T= 300 K.\n1200 K. A full description of the 2TM and the details of all spin-dynamics simulations are\ndescribed in Sec.S2 of SM [51].\nThe results of the simulations are shown in Fig. 1, for the A2,B2 andL21phases. It can\nbe seen that the di\u000berent phases react di\u000berently to the external stimulus. In general, this\nmodel provides a dynamics that is controlled by i)the temperature of the spin-subsystem,\nii)the strength of the magnetic exchange interaction and iii)the dissipation of angular\nmomentum and energy during the relaxation of the atomic magnetic moments (Gilbert\ndamping) [56]. Before continuing the discussion, we note that the average magnetization,\nM, of element Xis calculated as MX=P\nicX\niMX\ni=4P\nicX\ni, wherecX\niis the concentration of the\nparticular element Xin the particular phase and iruns over the four nonequivalent sites of\nthe unit cell. After the material demagnetizes, the spin temperature eventually drops and\nthe average magnetization returns to its initial value after 10 \u000020 ps (cf. Fig. 1).\nTo estimate the time constants of the demagnetization \u001cMand remagnetization ( \u001cR)\n4processes, in an element-speci\fc way, we \ft both the theoretical and experimental transient\nmagnetizations by a double exponential function [57]. We show results of \u001cMand\u001cRin\nFig. 2 for the A2 and B2 phase, as well as for alloys with intermediate degree of disorder\n(described in Sec.S2 of SM [51]).\n4.5\n4.0\n3.5\n3.0\n2.5\n2.0R (ps)\n0 20 50 80 100\nAmount of B2 order [%] Fe\n Co\n Remagnetization (b)2.0\n1.5\n1.0\n0.5\n0.0M (ps)\n Fe\n CoDemagnetization (a)\nFIG. 2. (Color online) Element resolved relaxation times of Co 2FeAl, from simulations of alloys\nwith varying amounts of A2!B2 phase. 0 corresponds to pure A2 phase while 100 corresponds to\npureB2 phase. Panel (a) shows the demagnetization time and panel (b) shows the remagnetization\ntime. Both time constants are obtained from \ftting the time trajectory of MX(t) by a double-\nexponential function (see text).\nThe theoretical demagnetization time is seen from Fig. 2 to typically be around 1 ps,\nwhereas the remagnetization time is 2 \u00005 ps. Going from the A2 to the B2 alloy, both\ntimes increase, albeit the simulations show a stronger increase of the remagnetization time\n5as function of alloy composition. We also note that the relevant time scale is somewhat\nlarger for Fe than for Co, and the ratio between them,\u001cFe=\u001cCo, grows when going from A2 to\nB2 phase.\nFigures 3 (a-d) shows the measured magnetization dynamics of CFA \flms that were\ngrown at di\u000berent temperatures (see SM, Sec.S3 for thin \flms synthesis, and Sec.S4 for\ndetails on the experimental measurements [51]). The inset shows the observed magnetization\ndynamics up to\u00181 ps. For all samples, the data for Fe (red) and Co (blue) show similar\ndemagnetization dynamics in the \frst few hundred femtoseconds, whereupon di\u000berences in\nthe magnetization dynamics become visible, especially on the picosecond timescale.\n1.000.950.900.850.800.75 Normalized Asymmetry (arb.units)1086420 Time Delay (ps) A2(a)\n1.000.950.900.850.800.75A/A00.80.40.0FeCo1086420 Time Delay (ps) 90% B2(b)\n1.000.950.900.850.800.75A/A00.80.40.0FeCo  1086420 Time Delay (ps) 90% B2(c)\n1.00.90.80.7A/A00.80.40.0 Fe Co1086420 Time Delay (ps) 100% B2(d)\n1.00.90.80.7A/A00.80.40.0FeCo\nFIG. 3. Measured element-speci\fc Fe (red) and Co (blue) magnetization dynamics of Co 2FeAl.\nSamples are denoted by the growth temperature in each case. The red and blue lines correspond\nto \ftted data (see text). (a) 300 K (100% A2phase), (b) 573 K (90% B2 phase), (c) 673 K (90%\nB2 phase), and (d) 773 K (100% B2 phase). The insets show the demagnetization dynamics up to\n\u00181 ps. All of the measurements were performed with similar pump-\ruence (for details, see Sec.S4\nof SM).\nFigure 4 (a-b) shows the measured values of the demagnetization and remagnetization\ntime constants, for the four di\u000berent growth temperatures, representing di\u000berent degree of\ndisorder in Co 2FeAl, along the alloy path A2!B2. It may be seen that the \u001cMfor Fe\nand Co is the same within the error bars for all four samples, regardless of the degree of\nstructural ordering (Fig. 4a). It may also be noted that the measured \u001cMfor CFA is similar\nto that of 3d transition metals [48, 49] and very much shorter than that of CrO 2or Fe 3O4.\nDemagnetization times that are independent on degree of structural ordering is interest-\n6ing, since it can be expected that the presence of structural disorder in Heusler alloys ought\nto result in a lower degree of spin polarization of the electronic states (i.e. an increased\ndensity of states (DOS) at the Fermi level in the minority band). This is expected to en-\nhance spin-\rip scattering, with an accompanying speed-up of the demagnetization dynamics\n[46, 47]. The electronic structure calculation of CFA also shows that the DOS at the Fermi\nlevel varies with di\u000berent structural phases (analyzed in the Sec.S1 of SM [51]). The A2\nphase has a large number of states at the Fermi level, while the L21phase, and to some\nextent the B2 phase, has a low amount [54]. Despite these di\u000berences in the electronic struc-\nture, the measured demagnetization dynamics shown in Fig. 4(a) is essentially independent\non degree of structural ordering.\nOn longer time-scales, there is a signi\fcant e\u000bect of structural ordering on the observed\nmagnetization dynamics, which becomes particularly relevant for the remagnetization pro-\ncess. As seen in Fig. 4b, there is a monotonous increase of remagnetization time, \u001cR, with\nincreasing growth temperature and hence the degree of ordering along the A2!B2 path.\nThe sample grown at 300 K with A2 phase, exhibits the fastest remagnetization dynam-\nics (\u001cR). With increasing growth temperature and corresponding increase in the structural\nordering along the A2!B2 path, a distinct trend of increasingly slower remagnetization\ndynamics is observed.\nThe most conspicuous behaviour of the measured magnetization dynamics, and its depen-\ndence on the degree of ordering, concerns the remagnetization time (Fig. 4b). The time-scale\nof the remagnetization process is su\u000eciently long to allow for an interpretation based on\natomistic spin-dynamics. Two materials speci\fc parameters should be the most relevant to\ncontrol this dynamics; the exchange interaction, as revealed by the local Weiss \feld, and\nthe damping parameter. In the Sec.S2 of SM [51], we report on the calculated Weiss \felds\nand damping parameters. It is clear from these results that the trend in the experimental\ndata shown in Fig. 4b, can not be understood from the Weiss \feld alone, whereas an expla-\nnation based on the damping is more likely. In order to illustrate this, we show in Fig. 5\nthe inverse of the measured remagnetizatiom time compared to the theoretically calculated\ndamping and experimental measured damping through ferromagnetic resonance (FMR) (de-\nscribed in Sec.S6 of SM)[51]. The \fgure shows that the damping is large in the completely\ndisordered A2 phase and for a large range of structural orderings, which comes out from\nboth theory and experiment. The \fgure also demonstrates that the inverse of the measured\n7300250200150100500 τM (fs)  Fe Co(a)\n3.02.52.01.5 τR (ps)CFA300 K CFA573 K CFA673 K CFA773 K (b)FeCo FIG. 4. Measured magnetization times for the investigated Co 2FeAl alloys. In (a) the demagneti-\nzation time, \u001cM, is shown and in (b) the remagnetization time, \u001cR, is plotted.\nremagnetization time scales very well with both the calculated damping and experimentally\nmeasured damping . According to the \fgure, a large damping parameter corresponds to\nfaster remagnetization dynamics in the measurements.\nCo2FeAl is, to the best of our knowledge, the \frst system where experimental observations\nand theory point to the importance of damping in the process of ultrafast magnetization\ndynamics. We note that this primarily is relevant for the remagnetization process; the\ninitial part of the magnetization dynamics (\frst few hundred fs) is distinctly di\u000berent. In\nthe demagnetization we observe a similar behaviour for Fe and Co in all samples, and\nan insensitivity of the demagnetization times in relation to structural ordering. Also, the\n80.550.500.450.400.350.300.25 (τr )-1 (1/ps) 43210Damping ( x 10-3)\n100806040200Amount of B2 ordering (%)Remagnetization timeTheoretical dampingExperimental dampingFIG. 5. The relationship of inverse of the measured remagnetization time (right y-axis) and the-\noretically calculated and experimentally measured Gilbert damping (left y-axis) in Co 2FeAl for\nvarying amount of B2 order along the A2!B2 path, i.e. 0 corresponds to pure A2 phase while\n100 corresponds to pure B2 phase.\nmeasured and theoretical demagnetization times evaluated from atomistic spin-dynamics\nsimulations, do not agree. Other mechanisms, of electronic origin, most likely play role in\nthis temporal regime.\nThe remagnetization process of Co 2FeAl alloys with varying degree of structural order,\nhighlights clearly the importance of the Gilbert damping and that magnon dynamics domi-\n9nates the magnetization at ps time-scales. The relevance of the Gilbert damping parameter\nfor ps dynamics is natural, since this controls angular momentum (and energy) transfer to\nthe surrounding. What is surprising with Co 2FeAl is the fact that other interactions (e.g.\nthe Weiss \feld) show such a weak dependence on the amount of structural diorder. This\nis fortuitous, since it allows to identify the importance of the Gilbert damping. A picture\nemerges from the results presented here, that the magnetization dynamics in general have\ntwo regimes; one which is primarily governed by electronic processes, and is mainly active\nin the \frst few hundered fs ( \u001cM), and a second regime where it is primarily magnons that\ngovern the remagnetiztion dynamics ( \u001cR).\nWe acknowledge support from the Swedish Research Council (VR, contracts 2019-\n03666,201703799, 2016-04524 and 2013-08316), the Swedish Foundation for Strategic Re-\nsearch, project SSF Magnetic materials for green energy technology under Grant No. EM16-\n0039, the Knut and Alice Wallenberg foundation, STandUP and eSSENCE, for \fnancial\nsupport. 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B\n81, 174401 (2010).\n13"    },    {        "title": "2312.04202v2.Probing_levitodynamics_with_multi_stochastic_forces_and_the_simple_applications_on_the_dark_matter_detection_in_optical_levitation_experiment.pdf",        "content": "Probing levitodynamics with multi-stochastic forces and the simple applications on\nthe dark matter detection in optical levitation experiment\nXi Cheng, Ji-Heng Guo, Wenyu Wang1,∗and Bin Zhu2,†\n1Beijing University of Technology, Beijing 100124, China\n2School of Physics, Yantai University, Yantai 264005, China\nIf the terrestrial environment is permeated by dark matter, the levitation experiences damping\nforces and fluctuations attributed to dark matter. This paper investigates levitodynamics with\nmultiple stochastic forces, including thermal drag, photon recoil, feedback, etc., assuming that\nall of these forces adhere to the fluctuation-dissipation theorem. The ratio of total damping to\nthe stochastic damping coefficient distinguishes the levitodynamics from cases involving only one\nsingle stochastic force. The heating and cooling processes are formulated to determine the limits of\ntemperature change. All sources of stochastic forces are comprehensively examined, revealing that\ndark matter collisions cannot be treated analogously to fluid dynamics. Additionally, a meticulous\nanalysis is presented, elucidating the intricate relationship between the fundamental transfer cross-\nsection and the macroscopic transfer cross-section. While the dark damping coefficient is suppressed\nby the mass of the levitated particle, scattering can be coherently enhanced based on the scale\nof the component microscopic particle, the atomic form factor, and the static structure factor.\nHence, dark damping holds the potential to provide valuable insights into the detection of the\nmacroscopic strength of fundamental particles. We propose experimental procedures for levitation\nand employ linear estimation to extract the dark damping coefficient. Utilizing current levitation\nresults, we demonstrate that the fundamental transfer cross section of dark matter can be of the\norder σD\nT∼<O(10−26)cm2.\nPACS numbers: 37.10.Mn, 95.35.+d, 84.71.Ba, 05.40.Ca, 05.10.Gg\nI. INTRODUCTION\nObservations in cosmology and astrophysics have con-\nfirmed the presence of dark matter (DM) [1]. One of\nthe most promising experimental approaches to detect\nDM on Earth is to search for small energy depositions\nresulting from elastic scattering of DM in sensitive de-\ntectors. Searches for weakly interacting massive parti-\ncles (WIMPs) represent some of the most advanced tech-\nniques in this field. However, stringent constraints exist\non the cross section for DM particles with masses ex-\nceeding 1 GeV. DM particles may have eluded detection\nif the resulting energy deposits fall below the threshold\nof current detectors. Therefore, it is imperative to ex-\nplore alternative strategies and develop novel detection\nmethods to investigate the remaining parameter space of\nDM i.e., light DM [2–19].\nIn recent years, a fundamentally different approach has\nbeen proposed to increase the sensitivity to the energy\ntransfer from DM particles to specific microscopic inter-\nnal degrees of freedom within a detector. This approach\ninvolves optically monitoring the center-of-mass (COM)\nmotion of a levitated macroscopic object. The levitation\nof micrometer-sized objects in vacuum was first demon-\nstrated in the pioneering work of Arthur Ashkin in the\n1970s [20]. Levitated nanoparticles have since been uti-\nlized in various fields, including trapping and cooling of\natoms, control mechanisms based on optical, electrical,\n∗guojiheng@buaa.edu.cn, wywang@bjut.edu.cn,\n†zhubin@mail.nankai.edu.cnand magnetic forces, and investigations of light-matter\ninteractions [21–31]\nLaboratory-scale levitation experiments aiming to\nreach the quantum regime require a substantial degree\nof isolation and control, making them highly sensitive to\nforces or phenomena that couple to mass. Recently, a\nstudy by Ref. [32] demonstrated that levitated nanoscale\nmechanical devices operated around the standard quan-\ntum limit for impulse sensing can detect the deposition of\nenergy transfer from DM particles. Levitation devices of-\nfer a unique opportunity for directional searches for DM\nmasses in the keV-GeV regime, which currently have lim-\nited direct detection constraints. Moreover, they exhibit\nremarkable complementarity to other proposals due to\ncoherent interaction at low momentum detection thresh-\nolds. [33] The results of Ref. [33] demonstrate the poten-\ntial of optomechanical sensors in searching for DM. Large\narrays of sensors could facilitate the reconstruction of\ntrack-like signals from DM particles while possessing suf-\nficient sensitivity to detect and count individual collisions\nof latent gas in ultra-high vacuum environments. Cavity\noptomechanical systems aim to operate in the shot-noise\ndominant regime to observe macroscopic quantum phe-\nnomena.\nIn this study, we focus on the detection of DM us-\ning levitated particles and investigate the thermal dy-\nnamics of Langevin systems subjected to multi-stochastic\nforces. The presence of distinct stochastic forces can\ninduce fluctuation-dissipation from an unknown source,\nmaking it challenging to detect the DM-induced collision\nsignal amidst the noise. By examining the properties of\nthese stochastic forces and studying the recoil of levi-arXiv:2312.04202v2  [cond-mat.stat-mech]  16 Dec 20232\ntated particles, we aim to unveil their characteristics and\nprovide insights into the presence of additional unique\nsources. The rest of this paper is organized as follows:\nSec. II presents the general formulation of the Langevin\nsystem with multi-stochastic forces. Sec. III examines\nthe properties of various stochastic forces. Sec. IV dis-\ncusses the application of these forces in detecting dark\nmatter. Finally, Sec. V concludes the paper.\nII. LEVITODYNAMICS WITH\nMULTI-STOCHASTIC FORCES\nA. Why we go beyond the framework of\nfluctuation theorem\nNanoparticle levitation or trapping can be achieved\nthrough laser cooling techniques, as illustrated in panel\nA of Fig. 1 [21]. In an atomic beam, collimation selects\natoms moving in a specific direction that can be deceler-\nated using a single laser beam molasses. Atoms in a gas\nmove in random directions, and reducing their tempera-\nture necessitates laser cooling in all three spatial dimen-\nsions by using three orthogonal standing waves. How-\never, for a moving atom, the Doppler effect introduces\nan imbalance in the forces. The Doppler shift brings\nthe light closer to the atom’s resonance, increasing the\nrate of absorption from the laser beam and resulting in\na force that decelerates the atom. This Doppler cool-\ning, also known as the feedback mechanism in levitation,\ncan be controlled using a switch, which is depicted in\npanel A of Fig. 1. By enabling or disabling the feed-\nback loop, the heating (relaxation) or cooling processes\nof the levitated particle can be measured. Apart from\nthe molasses forces, the levitated particle can interact\nwith the environment through various other forces, as\nillustrated in panel B of Fig. 1 [34]. Noise in the trap-\nping potentials generates fluctuations and dissipation in\nparticle motion. In the quantum regime, these mecha-\nnisms induce decoherence in the motional quantum state\nby leaking information about the particle’s position to\nthe environment. The primary sources of noise in lev-\nitodynamics include (i) collisions with gas molecules in\nthe vicinity; (ii) emission, absorption, and scattering of\nthermal electromagnetic radiation; (iii) noise in the trap\ncenter due to vibrations and/or stray fields; (iv) noise\nin the trap frequency caused by fluctuations in the elec-\ntromagnetic fields generating the trapping potentials; (v)\ncross-coupling to other thermal degrees of freedom in the\nparticle’s motion along orthogonal directions, rotation,\nor internal excitation; and (vi) the unknown interaction\nwhich can be dark interaction talked about in this pa-\nper. Further analysis of these noise sources can provide\na deep understanding of levitated particles and enhance\nsensor accuracy. In the next section, we will explore the\nthermal dynamics of this noise.\nAlthough the system may have numerous degrees of\nfreedom, a good approximation is that these freedomsare well decoupled. Each degree of freedom exhibits in-\ndependent harmonic oscillator motion with frequency ω0\nand damping coefficient γ. The system can be described\nby a one-dimensional Brownian particle with mass mfol-\nlowing the equation of motion\nd2x\ndt2+γdx\ndt+ω2\n0x=F(t)\nm. (1)\nIt is known as the Langevin system. The force F(t) can\nbe a time-dependent external force used to explore re-\nsponse functions or a time-dependent random force re-\nsulting from molecular collisions with the environment.\nFor example, in the case of Brownian motion in a fluid,\nγrepresents the damping coefficient of spherical parti-\ncles. The stochastic force F(t) has an expectation value\n⟨F⟩= 0. Its correlation function is given by\n⟨F(t)F(t′)⟩= 2mγk BTδ(t−t′), (2)\nwhere kBis the Boltzmann constant and it is propor-\ntional to the temperature of the environment TE. The\nsystem establishes a fundamental relationship between\nfluctuation and dissipation. The fluctuating force is re-\nlated to the coefficient of viscosity of the fluid, represent-\ning dissipative forces in the system, and it depends on the\ntemporal characteristics of molecular fluctuations. The\nenergy of the levitated particle changes due to its inter-\naction with the environment, and the time evolution of\nthe average energy ⟨E⟩is described by the Fokker-Planck\nequation. The energy flow can be understood as fluc-\ntuations driving the levitated particle while dissipation\ndampens it. In the equilibrium state, the energy from\nboth sides compensates each other. An effective temper-\nature, equal to the temperature of the environment, can\nbe assigned to the system.\nHowever, as mentioned earlier, the levitated particle\ninteracts with multiple stochastic forces simultaneously,\nresulting in the presence of multiple force sources. In\nparticular, these different stochastic forces don’t need to\nbe in equilibrium, which leads to the failure of the the-\noretical formulation mentioned above. Nevertheless, not\nevery damping coefficient needs to correspond to a single\nfluctuation source, thereby altering the thermal dynam-\nics of the levitated particle. While it is possible to achieve\na steady state for the levitated particle, this state does\nnot indicate the temperature of the environment due to\nthe absence of a uniform temperature. For instance, in\na cold atom system, the temperature in the correlation\nfunction of the feedback cooling force (as described by\nEq. (2)) must be significantly lower than the environ-\nment temperature due to collisions with surrounding gas\nmolecules. Otherwise, the cooling process will not be\neffective. Consider the possible dark matter interaction\nstudied in this paper, where the dark matter decoupled\nfrom the heat bath during the early epoch of the universe.\nThere is no reason for the correlation function of the dark\nforce to be identical to that of the gas environment. The\ndetails of the Langevin system with multiple stochastic3\nA\nB\nGas\nCxy\nzFeedback\nDark\ninteraction\nrad\nFB\nDM\nGas\nrad\n FB\nDM\nFluctuation\nDissipation\nField\nfluctuationsCoupling to\nother degrees\nof freedomGas moleculesBlack body\nradiation\nFIG. 1. A: Sketch map of the levitation of particle conducted by laser cooling techniques, The feedback loop can be turned\non or off in the experiments. B: The noise sources in levitodynamicsa: (i) collisions with molecules of the surrounding gas;\n(ii) emission, absorption, and scattering of thermal electromagnetic radiation; (iii) noise in the trap center from vibrations\nand/or stray fields; (iv) noise in the trap frequency from fluctuations of the electromagnetic fields that generate the trapping\npotentials; (v) cross-coupling to other thermal degrees of freedom in the particle either motion along orthogonal directions,\nrotation, or internal excitation; and (vi) the unknown interaction which can be dark interaction talked about in this paper. C:\nThe fluctuations and dissipation of the Langevin system with multi-stochastic sources. The arrows show the energy flow.\naThe figure is adapted from Fig. 4 of Ref. [34]\nforces should be investigated. In such scenarios, a steady\nstate can be achieved, and the fluctuation and dissipa-\ntion can be observed, as shown in panel C of Fig. 1.\nThe levitated particle is driven by fluctuations from var-\nious sources and dissipates through similar multi-sources.\nWhile the total energy flow in the fluctuations is equal\nto that of the dissipation in steady states, this relation-\nship may not hold for individual pairs of fluctuations and\ndissipation.\nB. New framework\nIn this work, we extend the framework of fluctuation\ntheorems, as introduced in Ref. [35], to investigate the\nLangevin system with multiple stochastic forces. This\nframework enables the study of relaxation from a non-\nequilibrium state towards equilibrium. The trapped par-\nticle experiences a trap force generated by the laser aswell as stochastic forces arising from random impacts\nof gas molecules, radiation, and dark interactions. As\ndemonstrated in Ref. [35], the platform established in\nthis study enables experimental investigations of non-\nequilibrium fluctuation theorems for arbitrary steady\nstates. Furthermore, it can be extended to explore quan-\ntum fluctuation theorems and systems that do not adhere\nto detailed balance. To simplify the analysis, we focus on\nfour specific types of forces:\n1. The force arises from collisions between the levi-\ntated particle and the thermally moving molecules\nin the atmosphere. This force is characterized by\na damping coefficient γTHand a stochastic driv-\ning force FTH, as shown in Eq. (2). The damping\ncoefficient can be obtained from the collisions of\nmicroscopic components in fluid dynamics.\n2. The heating rate is attributed to the stochastic\ndriving force FREarising from photon recoil kicks.4\nAdditionally, a damping coefficient γRE(denoted\nasγϵin Ref. [36]) arises from the back-reaction of\nthe scattered field on the motion of the levitated\nparticle and can be evaluated accordingly. [37] Pho-\ntons, the quanta of the radiation potential, trap the\nparticle and form the optical position sensor. In-\ncreasing the optical power raises the kick rate from\nindividual photons, resulting in a force caused by\nshot noise of radiation pressure. As demonstrated\nin Ref. [36], the damping coefficient can be directly\nmeasured from the photon recoil during reheating.\n3. We consider the feedback optical force FFBand the\ncorresponding damping coefficient γFB. The optical\nforce, along with the Doppler effect, decelerates the\nlevitated particle, making it a crucial mechanism\nin levitation and cold atom physics. Importantly,\nthe optical force FFBis a deterministic force that\ndoes not fluctuate, but rather damps the motion of\nthe levitated particle. This is analogous to other\ndamping effects. The main advancement of this\npaper lies in distinguishing between stochastic and\ndeterministic forces in thermal dynamics.\n4. An additional stochastic force can be generated\nthrough collisions with the surrounding dark mat-\nter. Within the Langevin system, the interaction\nwith dark matter is described by the damping co-\nefficient γDMand the driving force FDM.\nIt should be noted that laser trapping techniques gener-\nate a force where the driven force acts as the restoring\nforce for the particle’s oscillation. Furthermore, it is im-\nportant to note that the damping coefficient γREdiffers\nfrom γFBsince they arise from different optical trapping\nsetups. Dissipation and fluctuations can also occur due\nto self-emission, absorption, and scattering with ambient\nradiation from the environment. These dissipative and\nfluctuating effects are not considered in this paper. Nat-\nurally, the formulation in this section can be straightfor-\nwardly extended to include systems with other additional\nstochastic forces. It is important to keep in mind that\nthe correlation function of the added stochastic force does\nnot depend on a single uniform temperature.\nWe consider a scenario where a levitated particle with\nmass mis created using a strongly focused laser beam of\nfrequency Ω under ambient temperature and gas pressure\nconditions. In this setup, the particle is kicked by the\nincident photons from the environment, each carrying an\nenergy of ℏΩ, resulting in the formation of a harmonic\noscillator with a spring constant kand a frequency ω0= p\nk/m. Typically, the frequency of the incident laser\nbeam, Ω, is significantly higher than the frequency of the\nlevitation potential, ω0. The three spatial dimensions can\nbe treated as decoupled from one another. Consequently,\nthe motion of the Langevin system can be described as.\nd2q\ndt2+γTdq\ndt+ω2\n0q=FTH(t) +FRE+FDM(t) +FFB(t)\nm.\n(3)The total damping coefficient γT, represented by the sec-\nond term on the left side of the above equation, accounts\nfor the sum of damping coefficients arising from various\nsources\nγT=γTH+γRE+γFB+γDM. (4)\nThe right-hand side represents the driven forces in the\nsystem. Generally, each damping coefficient corresponds\nto a term on the right-hand side representing the driven\nforces. However, as mentioned earlier, the feedback force\nin the laser cooling technique is deterministic. This\nforce FFB=−ω0ηq2pis a time-varying, non-conservative\ndamping optical force introduced through parametric\nfeedback with a strength η, where p=m˙qrepresents\nthe momentum.\nIn Langevin systems, the average value of any stochas-\ntic force Fiis zero. The correlation functions should fol-\nlow a similar form as Eq. (2), which is proportional to the\ndamping coefficients and temperature if they satisfy the\nfluctuation dissipation theorem. Here, Tidoes not rep-\nresent a uniform temperature but should be regarded as\nan effective parameter that characterizes the correlation\nfunction of each individual stochastic force. Naturally,\nTTHcorresponds to the temperature of the ambient en-\nvironment TE. The sum of all stochastic forces is denoted\nas.\nFS(t) =FTH(t) +FRE(t) +FDM(t). (5)\nIt is evident that the equilibrium of the steady state of\nthe levitated particle does not imply that its temperature\nis equal to that of the environment. A new stochastic\ntemperature, denoted as TS, can be defined as\nTS=γTHTTH+γRETRE+γDMTDM\nγTH+γRE+γDM, (6)\nwhich accounts for the temperature of the system in the\ncorrelation function\n⟨FS(t)FS(t′)⟩= 2mγSkBTSδ(t−t′). (7)\nIn the above equation, γSrepresents the sum of all damp-\ning coefficients associated with the respective stochastic\nforces\nγS=γTH+γRE+γDM. (8)\nIt is important to note that the system temperature TS\ndoes not describe the thermal state of the Langevin sys-\ntem since the levitated particle represents a single de-\ngree of freedom. The parameter TS, along with all the\nTiin Eq. (6), characterizes the nature of the stochas-\ntic forces. In the absence of a feedback loop, TSrepre-\nsents the temperature of the environment in an ordinary\nLangevin system. In experiments, the dynamical relax-\nation can be measured by manipulating the feedback loop\nswitch. [36] The heating or cooling trajectories can pro-\nvide detailed information about the stochastic forces. As5\nmentioned in the introduction, the Langevin system con-\ntains multi-stochastic forces from different sources. The\nphenomenology of heating or cooling processes may devi-\nate from predictions if only two parameters, γTandTS,\nare used to account for all the stochastic forces. There-\nfore, the following section discusses parametric feedback\ncooling with multiple stochastic forces.\nAs demonstrated in Ref. [35], the energy of the\nLangevin system can be expressed in a form that resem-\nbles under-damped Brownian motion in the energy space.\nExpressing Eq. (3) as a stochastic differential equation\n(SDE) is more convenient.\ndq=p\nmdt ,\ndp=\u0000\n−mω2\n0q−ω0ηq2p−γTHp\n−γREp−γFBp−γDMp) dt (9)\n+p\n2mγTHkBTTHdWTH\n+p\n2mγREkBTREdWRE\n+p\n2mγDMkBTDMdWDM,\nwhere Wi(t) are Wiener processes with\n⟨Wi(t)⟩= 0,⟨Wi(t)Wj(t′)⟩=δij(t′−t). (10)\nHere only the thermal and additional dark stochastic\nforces are taken into account. For a short (infinitesimal)\ntime interval d twe have\n⟨dWi⟩= 0,\n(dWi)2\u000b\n= dt. (11)\nNext, the energy change d Eis determined by\ndE=\u0012∂E\n∂q\u0013\ndq+\u0012∂E\n∂p\u0013\ndp+1\n2\u0012∂2E\n∂p2\u0013\n(dp)2.(12)\nIt is important to note that the Wiener process implies\nthat (d p)2is of the same order as d t. Therefore, the\nsecond-order partial differential of the energy Eneeds to\nbe considered, while (d q)2and d qdpcan be neglected.\nBy taking the derivatives of the energy with respect to q\nandp, we obtain\ndE=mω2\n0qdq+p\nmdp+1\n2m(dp)2. (13)\nInserting d q,dpand neglecting all terms of order (d t)3/2\nor higher yields\ndE=\u0012\n−γTp2\nm−ηE2\n2mω0\u0013\ndt+γSkBTS\n2dt\n+p\n2EγTHkBTTHdWTH (14)\n+p\n2EγREkBTREdWRE\n+p\n2EγDMkBTDMdWDM.\nNote that an additional factor of 1 /2 arises in the second\nterm on the right-hand side when performing stochas-\ntic integration over the time evolution of the harmonicoscillation. This formula elucidates the fundamental dis-\ntinction between energy increments driven by single and\nmultiple stochastic forces. For a single stochastic force,\nthe formulation comprises a single damping coefficient\nand its corresponding temperature. However, the pres-\nence of multiple stochastic forces considerably compli-\ncates the formulation, impacting the energy distribu-\ntion, dissipation, relaxation, and other factors. Each\nterm γikBTidtarises in association with a correspond-\ning stochastic force. However, additional contributions\nto the damping terms, γip2dt/m, arise from the feedback\nmechanism. This additional damping force can, in fact,\nbe any dissipative force that does not involve fluctua-\ntions to drive the levitated particle. Consequently, the\ntotal damping coefficient γT, which incorporates the ad-\nditional contribution of γFB, is not equal to the stochastic\ndamping coefficient γS. As demonstrated below, this dif-\nference in damping coefficients gives rise to distinct ther-\nmal dynamics compared to those resulting from a single\nstochastic force, even when TSis erroneously assumed to\nbe the environmental temperature TE.\nFollowing a similar procedure outlined in the appendix\nof Ref. [35], we adopt ϵ=√\nEto study the energy incre-\nment and prevent the emergence of multiplicative noise\nin Eq. (14)\ndϵ=\u0012\n−γTϵ\n2−ηϵ3\n4mΩ0+γSkBTS\n4ϵ\u0013\ndt (15)\n+r\nγTHkBTTH\n2dWTH+r\nγREkBTRE\n2dWRE\n+r\nγDMkBTDM\n2dWDM.\nThe variable ϵevolves at temperature TSunder the in-\nfluence of an external force f(ϵ) in the presence of high\nfriction characterized by ν= 4/γS\ndϵ=1\nνf(ϵ)dt+r\nγTHkBTTH\n2dWTH (16)\n+r\nγREkBTRE\n2dWRE+r\nγDMkBTDM\n2dWDM.\nWe can denote a new parameter\nζ=γT\nγS(17)\nto account for the difference between stochastic damping\nand the total damping. Then the external force\nf(ϵ) =−2ζϵ−ηϵ3\nmω0γS+kBTS\nϵ\n=−dU(ϵ)\ndϵ. (18)\nThen the potential U(ϵ) should be\nU(ϵ) =ζϵ2+η\n4mω0γSϵ4−kBTElnϵ . (19)6\n-12-10-8-6-4-20\n0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5\nEη=1.000η=0.100η=0.010η=0.001ζ = 0\n0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5\nEη=1.000η=0.100\nη=0.010 η=0.001ζ = 2\n0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5\nEη=1.000η=0.100\nη=0.010η=0.001ζ = 5\nFIG. 2. The logarithm of the energy distribution ρ(E, α, ζ ) is compared for different ζvalues. The distributions are normalized\nwith ln ρ(0, α, ζ ) = 0. The parameters are set as kSTS= 1,m= 1,k= 1, and γS= 0.01. Each curve corresponds to a different\nfeedback strength ηshown in the plot. The plots for ζ= 2 and ζ= 5 depict the distributions with an additional damping\nfactor.\nDenote\nα=η\nmω0γS, β S=1\nkBTS. (20)\nThen the distribution of ϵcan be easily derived by\nBoltzmann-Gibbs distribution\nρ(ϵ, α, ζ )∝exp\u001a\n−U(ϵ)\nkBTS\u001b\n∝ϵexpn\n−βS\u0010\nζϵ2+α\n4ϵ4\u0011o\n.\n(21)\nThe distribution function of energy can be got by recov-\nering variables from ϵtoE=ϵ2\nρ(E, α, ζ ) =1\nZαζexpn\n−βSζE−βSα\n4E2o\n, (22)\nwhere the normalisation factor Zαζ=R\ndEρ(E, α) is\ngiven by\nZαζ=rπ\nαβSeζ2βS/αErfc r\nζ2βS\nα!\n. (23)\nThe distribution clearly reverts to the form of the single\nstochastic force when ζ= 1. However, the tempera-\ntureTSremains determined by multiplicative forces. To\ndemonstrate the modification of the energy distribution,\nwe numerically calculated the logarithm of the distribu-\ntionρ(E, α, ζ ) for different ζvalues. The results are de-\npicted in Fig. 2. From the figure, it is evident that in-\ncreasing the ζparameter sharpens the distribution as the\nenergy increases. This can be easily understood as a con-\nsequence of the additional damping factor ( γFB) leading\nto increased energy dissipation.\nThe modification of the distribution can be observed\nthrough the average energy or the effective tempera-\nture of the levitated particle. The average energy is ob-\ntained by integrating over the energy distribution given\nby Eq. (22).\n⟨E⟩=Z−1\nαZ\ndEEexp\u001a\n−βSζE−βEα\n4E2\u001b\n.(24)Evaluation of the integral yields\n⟨E⟩=2\nβS\nr\nβS\nαe−ζ2βS/α\n√πErfc\u0012q\nζ2βS\nα\u0013−ζβS\nα\n.(25)\nIf defined an effective temperature obtained by applying\nthe feedback mechanism ⟨E⟩=kBTeff, then\nTeff=TS\n\n2√αkBTSe−ζ2/αkBTS\n√πErfc\u0010\nζ√αkBTS\u0011−2ζ\nαkBTS\n\n.\n(26)\n00.10.20.30.40.50.60.70.80.91\n0 1 2 3 4 5 6 7 8 9 10\nαTeff / TS\nζ=5ζ=2ζ=1\nFIG. 3. The ratio of the effective temperature to the stochas-\ntic temperature Teff/TSversus α.ζ= 2 and ζ= 5 curves show\nthe depression of effective temperature with additional damp-\ning factor.\nFig. (3) shows the numerical results of the ratio Teff/TS\nas a function of the parameter α. It can be observed7\nthat the effective temperature decreases with increasing\nα. Moreover, the effective temperature is further reduced\nby the additional damping factor introduced by the feed-\nback mechanism. This yields a highly interesting result.\nlim\nα→0Teff\nTS=1\nζ. (27)\nWhen α→0, it signifies the absence of the driven force\nfrom the feedback. Hence, the term 1 /ζquantifies the\ndampening effect of the additional damping factor. This\nmechanism effectively cools the levitated particle by in-\nteracting with the environment. Additionally, a non-zero\nfeedback strength enhances the cooling effect.\nThe averages over the last two terms of Eq. (14) vanish\n⟨dWTH⟩= 0,⟨dWDM⟩= 0. Thus the time evolution of\nthe average energy is\nd⟨E⟩\ndt=−γT⟨E⟩ −η\n2mω0⟨E2⟩+γSkBTS\n2(28)\n=−γT⟨E⟩ −γSα\n2⟨E2⟩+γSkBTS\n2.This equation describes the cooling process in which the\naverage energy decreases with the evolution due to the\npresence of feedback. In the steady state, the time deriva-\ntive on the left side of Eq. (28) becomes zero.\nα⟨E2⟩=kBTS−2ζ⟨E⟩. (29)\nThe average dissipation rate ¯Pof the feedback mecha-\nnism can be formulated as\n¯P=γFB⟨E⟩+γSα⟨E2⟩\n2. (30)\nSince the second moment of the energy ⟨E2⟩in Eq. (28)\ncannot be expressed in terms of the average energy ⟨E⟩,\nthe equation cannot be easily solved for ⟨E⟩. However,\nin the equilibrium state for the exponential distribution,\nwe have ⟨E2⟩= 2⟨E⟩2. Therefore, the cooling process\ncan be approximately derived.\n⟨E(t)⟩=ζ\n2α−1 +p\n1 + 2 αkBTS/ζ2+\u0010\n1 +p\n1 + 2 αkBTS/ζ2\u0011\nCexp\u0010\n−γTp\n1 + 2 αkBTS/ζ2t\u0011\n1−Cexp\u0010\n−γTp\n1 + 2 αkBTS/ζ2t\u0011 . (31)\nHere, Cis an arbitrary constant determined by the ini-\ntial condition. It is reasonable to assume that the cool-\ning process initiates when the levitated particle reaches\na steady state with the ambient stochastic forces in the\nabsence of feedback. In the levitation experiment, the\nfeedback force can be toggled on or off, allowing for the\nmeasurement of both the cooling and heating processes of\nthe levitated particle. The heating process is conducted\nby turning off the feedback loop, resulting in the evolu-\ntion of the average energy according to\nd⟨E⟩\ndt=−γS⟨E⟩+γSkBTS\n2. (32)\nIt is important to note that when the feedback is turned\noff, the damping and fluctuation arise from the collec-\ntive ambient stochastic forces discussed earlier. There-\nfore, in Eq. (28), γTmust be replaced by γS. However,\nthe presence of multiple sources implies that the system\ntemperature does not necessarily equal the environmen-\ntal temperature. The relaxation evolution can be easily\nsolved\n⟨E(t)⟩=kBTS\n2+\u0014\n⟨E(0)⟩ −kBTS\n2\u0015\ne−γSt\n=kBTS\n2+\u0014kBTL\n2−kBTS\n2\u0015\ne−γSt.(33)\nThe temperature TLin the above equation represents the\nnewly introduced effective temperature that character-izes the initial heating state. By examining Eq. (33),\nit becomes apparent that as time progresses, the levi-\ntated particle will tend to reach the ambient temperature\nTH=TS. This result can also be obtained from Eq. (32),\nwhere the time derivative on the left-hand side vanishes\nin the equilibrium state. At this point, the energy flow\nfrom fluctuations compensates for the dissipation.\nIf the feedback is turned on, the system will be cooled\nand will no longer be in equilibrium with the ambient\nstochastic forces. By requiring ⟨E(0)⟩=kBTH/2, we\nobtain\nC=αkBTH+ζ−p\nζ2+ 2αkBTH\nαkBTH+ζ+p\nζ2+ 2αkBTH. (34)\nIt is straightforward to determine that 0 < C < 1 based\non the condition ζ >1. By examining Eq. (31), we can\ndetermine the final steady state of the cooling process,\nwhere the energy flow from the oscillator to the stochastic\nsource is counterbalanced by the energy extracted from\nthe feedback mechanism\n⟨E(t→ ∞ )⟩=ζ\n2α s\n1 +2αkBTH\nζ2−1!\n. (35)\nTherefore, the approximate effective temperature at the8\ncooling limit should be.\nTlimit\nL=ζ\nαkB s\n1 +2αkBTH\nζ2−1!\n. (36)\nIt can be observed that the ratio β≡2αkBTH/ζ2de-\ntermines the simple expression for the cooling limits. If\nβ≪1, the cooling limit is expected to be\nTlimit\nL≃TH\nζ. (37)\nThis indicates that the cooling process is dominated by\nthe feedback damping γFB. The cooling limit is in agree-\nment with the effective temperature given by Eq. (27).\nHowever, in the case of β≫1,\nTlimit\nL≃r\n2TH\nαkB, (38)\nThis is independent of γFB. This implies that the cool-\ning process is dominated by the feedback optical force,\nand it also necessitates 2 /αk B< TH. Otherwise, cooling\nwill not be achieved. These limits differ from the results\nin Ref. [35], which are solely determined by the balance\nbetween single fluctuation and feedback optical force.\n00.20.40.60.81\n-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6ζ=1.1\nheatingζ=2\nζ=5\nζ=10\nLog10(t+0.1 τ)T(t) / TS\nFIG. 4. The numerical results for the heating and cooling\nprocesses are presented. The parameters αkBandγSare nor-\nmalized to unity in the numerical calculation. The dashed\nline represents the heating process, which evolves exponen-\ntially as exp( −γSt) with time. The solid lines represent the\ncooling processes for different values of the parameter ζ. The\ntime evolution is scaled by a factor of 0 .1τto enable a loga-\nrithmic representation.\nFig. 4 illustrates the numerical results for the heating\nand cooling processes. Parameters αkBandγSare nor-\nmalized to unity in the numerical calculation, and the\nratio of the effective temperature to the ambient tem-\nperature TSis presented. The heating process evolves\nexponentially as exp( −γSt) with time, while the coolingrate is determined by the parameter ζand evolves as\nexp(−γTp\n1 + 2 αkBTS/ζ2t). A larger value of ζimplies\nfaster cooling. Here, it is important to address the dif-\nferences between our work and the theory proposed in\nRef. [35]. In our work, cooling can be achieved through\nboth feedback damping and feedback optical force. When\nγFB≫γS, the feedback damping from the Doppler effect\ndominates the dissipation process. However, in the limit\nγFB→0 or equivalently ζ→1, the cooling process is the\nsame as the one proposed in Ref. [35], where the optical\nforce cools the system. In the following section, we will\ndiscuss the distinction between the optical feedback force\nand the Doppler damping force.\nIII. THE DAMPING FORCES FROM\nDIFFERENT SOURCES\nBased on the thermal dynamics presented in the pre-\nvious section, it is evident that the damping factor and\ntemperature are the two fundamental parameters for de-\nscribing properties, such as correlation functions of the\nstochastic forces. The damping coefficients can be viewed\nas macroscopic phenomena that arise from collective in-\nteractions among the microscopic particles comprising\nboth the levitated particle and the environment. These\ninteractions can be attributed to mechanical or electro-\nmagnetic dynamics in the microscopic realm. To gain\na comprehensive understanding of the physics of levita-\ntion and enhance the sensor’s sensitivity, it is crucial to\nthoroughly investigate the collective interactions. This\nwill allow us to modify experimental setups and unveil\nthe underlying mechanism. Therefore, in this section, we\nexplore the microscopic origins of the three damping co-\nefficients discussed in the previous section. Since γTH,\nγRE, and γFBhave been extensively studied in the liter-\nature, and this paper aims to explore new methods for\ndark matter detection, we will provide a brief introduc-\ntion to γTH,γRE, and γFB. The formulation of γDMand\nthe distinctions between this coefficient and the other\nthree will be discussed in detail.\nA. The isothermal drag force\nThe first damping rate we need to consider is the\nisothermal drag force, γTH, which arises from the in-\nteraction between a spherical particle and rarefied gas\nmolecules. A single aerosol particle suspended in a non-\nequilibrium gas can experience various forces due to\nnon-uniformity, including isothermal drag, thermal force,\nphotophoretic force, diffusion force, and other forces re-\nsulting from combined flows of heat, mass, and momen-\ntum. Detailed investigations of these forces are beyond\nthe scope of this paper, but they can be found in stan-\ndard fluid dynamics textbooks. [38] Clearly, the isother-\nmal drag force is the primary contributor to the damping\nforce experienced by the levitated particle in the fluid. As9\nA\n𝒗𝑥𝑦𝑧𝑅!𝑈\"𝑂B\nFIG. 5. (A) Figure illustrating the Knudsen number in fluid dynamics: a stationary flow past a particle in its own saturated\nvapor. The damping is determined by the mean free path of the molecules and the length scale of the particle. (B) Calculation\nscheme for isothermal drag: a spherical ball with a radius R0moving through the fluid with a vapor velocity of U∞.\ndepicted in panel A of Fig. 5, a particle in its own sat-\nurated vapor experiences a stationary flow. The phys-\nical phenomena are attributed to phase changes occur-\nring on the particle surface. Theoretical fluid dynamics,\nas a subdivision of continuum mechanics, does not aim\nto describe the molecular structure of a medium or the\nmotion of individual molecules. Continuum models rep-\nresent matter that is sufficiently dense to allow averaging\nover a large number of molecules, enabling the definition\nof meaningful macroscopic quantities. However, this ap-\nproach has inherent limitations, which can be expressed\nin terms of the Knudsen number, denoted as Kn\nKn=λ\nL. (39)\nHere, λrepresents the mean free path of the microscopic\ncomponent particles in the surrounding matter, while L\nrefers to the characteristic length scale of the levitated\nparticle. The drag force can be analyzed by examin-\ning the low-speed motion of a volatile spherical parti-\ncle in its saturated vapor, regardless of the Knudsen\nnumber value. It was assumed that the particle main-\ntains its spherical shape, compensating for any defor-\nmations caused by evaporation-condensation processes\nthrough the surface-tension force that tends to maintain\na spherical shape. This assumption holds true for slow\nevaporation-condensation processes, allowing the parti-\ncle radius to be treated as a constant. The schematic\ndiagram can be observed in plot B of Fig. 5, illustrat-\ning a spherical particle with radius R0surrounded by a\nrarefied gas consisting of molecules with mass mM. The\nnumber density of molecules is denoted as n∞, while the\nvapor velocity is denoted as U∞. Here, rrepresents the\nradius vector from the particle’s center, and vrepresents\nthe velocity vector of the molecules. Typically, the vapor\nvelocity U∞is significantly smaller than the velocity ofthe molecules, resulting in a ratio u∞.\nu∞=U∞\u0012mM\n2kBTTH\u0013\n≪1. (40)\nis denoted and the molecular distribution can be lin-\nearized as\nf(r,v) =f∞[1 + 2 c·u∞+h(r,v)], (41)\nwhere f∞is the Maxwell-Boltzmann distribution\nf∞=n∞\u0012mM\n2πkBTTH\u00133\n2\nexp\u0000\n−c2\u0001\n, (42)\nwithc=v\u0012mM\n2kBTTH\u00131\n2\n.\nh(r,v) represents the disturbance in the distribution\nfunction near the particle. It is important to empha-\nsize that the linearization of the velocity distribution is\na crucial step in the analysis. However, when the levi-\ntated particle collides with the surrounding dark matter,\nthe condition given by Eq. (40) fails, and the lineariza-\ntion operation is not applicable. The temperature TTH\nin Eq. (40) has the same meaning as described in the pre-\nvious section, referring to the temperature of the heating\nbath.\nA comprehensive investigation of the isothermal drag\nforce can be found in Ref. [39]. Here, we provide a sum-\nmary of the coefficient results for different cases as fol-\nlows. When Kn≪1, the molecules in the surrounding\nmatter can be treated as material fragments of the mov-\ning medium. This region corresponds to the fundamental\nconcept of continuum description in fluid flows, known as\nthe viscous slip-flow regime. The analytical expression10\nfor the damping coefficients can be written as\nγTH=FV\nmU∞(43)\n=6π\nmηgR0(1 +aKn +bKn2+O(Kn3)),\nwhere Kn=λ/R 0.ηgrepresents the gas viscosity at\ntemperature TTH, while aandbare numerical coefficients\nthat depend on the momentum accommodation during\ncollisions between the molecules and the spherical parti-\ncle. The damping coefficient can be related to the Stokes\nformula used in statistical physics. When Kn≫1, the\nsurrounding molecules can be treated as freely moving,\nleading to what is known as the free-molecular regime.\nIn this regime, the damping coefficients are\nγTH=FF\nmU∞(44)\n=8π1\n3\n3mR2\n0PTH\u0012mM\n2kBTTH\u00131\n2\n×\u001a\n2 +ατ−αn32−π(9−αE)\n32−π(1−αn) (9−αE)\u001b\n,\nwhere PTHdenotes the pressure of the surrounding gas.\nThe momentum and energy accommodation coefficients\nαr, αn, αEcan be found in tables listed in Ref. [39] and\nthe accompanying references. It can be observed that ad-\njusting the pressure leads to different thermal damping\ncoefficients. Since nanoparticle levitation experiments\nare typically conducted in ultra-high vacuum, this prop-\nerty can be utilized to mitigate or eliminate thermal\ndamping and fluctuations in the levitation, as demon-\nstrated in the next section.\nB. The photon recoil in the optical levitation\nThe trapping potential in the levitation is achieved\nthrough the use of a strongly focused laser, which im-\nparts a photon kick to the levitated particle. Subse-\nquently, fluctuations occur due to classical noise in the\nlaser intensity, causing modulation of the trapping po-\ntential. Additionally, a damping force arises during colli-\nsions to prevent the energy of the levitated particle from\ndiverging. The damping coefficient can be deduced from\nthe fluctuations of the trapping potential. The Hamilto-\nnian for the trapped particle is given by [40]\nH=p2\n2m+1\n2mω2\n0(1 +ϵ(t))q2, (45)\nwhere ϵ(t) represents the newly introduced time-\ndependent factor that accounts for the fluctuation of\nthe laser intensity. The rate is reduced by the ratio of\nthe parametric resonance linewidth, ϵ0ω0, to the band-\nwidth of the fluctuations, ∆ ω0. Here, ϵ0denotes the\nroot-mean-square fractional fluctuation in the spring con-\nstant k. Consequently, the rate scales as ω2\n0S, withS≃ϵ2\n0/∆ω0representing the noise spectral density in\nfractions squared per rad/sec.\nThe damping rate can be computed classically, which\nis consistent with expectations for a harmonic-oscillator\npotential. First-order time-dependent perturbation the-\nory can be employed to calculate the average transition\nrates between quantum states of the trap. The time-\nevolving perturbation of the quantum-mechanical Hamil-\ntonian given by Eq. (45) is\nH′(t) =1\n2ϵ(t)m2ω2\n0q2. (46)\nThe damping of the levitated particle can be understood\nas the transition from a higher energy level |n⟩to a lower\nlevel|m⟩induced by the perturbation. The rate can be\ncalculated by taking the average over a time interval T\nRn→m=1\nT\f\f\f\f\f−i\nℏZT\n0dtH′\nmn(t)eiωmnt\f\f\f\f\f2\n(47)\n=\u0012mω2\n0\n2ℏ\u00132Z∞\n−∞dτeiωmnτ⟨ϵ(t)ϵ(t+τ)⟩\f\f⟨m|q2|n⟩\f\f2.\nHere, we assume that the averaging time Tis short com-\npared to the time scale over which the level populations\nvary, but large compared to the correlation time of the\nfluctuations. This allows the range of τto formally ex-\ntend to ±∞.\n⟨ϵ(t)ϵ(t+τ)⟩=1\nTZT\n0dtϵ(t)ϵ(t+τ). (48)\nUsing the transition matrix elements of q2andωn±2,n\nin Eq. (47), we can get the transition rate\nRn±2→n=π2ω2\n0\n16Sk(2ω0)(n+ 1±1)(n±1),(49)\nin which Skis the one-sided power spectrum of the frac-\ntional fluctuation in the spring constant\nSk(ω0) =2\nπZ∞\n0dτcos(ω0τ)⟨ϵ(t)ϵ(t+τ)⟩. (50)\nThe one-sided power spectrum is defined so that\nZ∞\n0dωSk(ω) =⟨ϵ(t)2⟩=ϵ2\n0. (51)\nAssuming that the levitated particle occupies the state\n|n⟩with a probability P(n, t) at time t, the average damp-\ning rate can be calculated as\nd⟨E(t)⟩\ndt=X\nnP(n, t)2ℏω0(Rn→n−2−Rn→n+2)\n=π\n2ω2\n0Sk(ω0)⟨E(t)⟩. (52)\nThe average energy of the oscillator, denoted as ⟨E(t)⟩,\ncan be expressed as ⟨E(t)⟩=P\nn(n+1\n2)P(n, t)2ℏω0. It11\nis evident that the damping coefficient can be extracted\nfrom the above equation in the form,\nd⟨E(t)⟩\ndt=−γRE⟨E(t)⟩, (53)\nwhere\nγRE=π\n2ω2\n0Sk(ω0). (54)\nNote that the derivation of the damping coefficient in\nthis work differs from that of Ref. [40], where the heat-\ning rate is calculated. The rationale behind this work is\nbased on the principle that every damping and fluctua-\ntion adhere to the fluctuation and dissipation theorem.\nBy applying Eq. (2), the fluctuation correlation function,\none can determine the fluctuation (or the heating rate\nas in Ref. [40]). Consequently, the damping coefficient\nshould be calculated first, which in turn determines the\nfluctuation. Here, we will refer to the damping coefficient\nas\nγRE=ω2\n0SRE. (55)\nin which SREis a newly defined parameter that absorbs\nall the constants, including ϵ2\n0present in the expression.\nIn the levitation experiment, we assume that SREcan\nbe finely adjusted to remain nearly constant within the\nvicinity of ω0. This allows for the straightforward extrac-\ntion of the photon recoil contribution to the signature.\nThe recoil temperature TREcan be derived as follows.\nThe force fluctuations acting on the levitated particle\nand their spectral density, as given in Eq. (2), can be\ndetermined using the Wiener-Khinchin theorem\nSFiFj(ω) =Z∞\n−∞D\nˆFi(ω)ˆF∗\nj(ω′)E\ndω′(56)\n=1\n2πZ∞\n−∞⟨Fi(t)Fj(t+t′)⟩eiωt′dt′\n=δijmγREkBTRE\nπ,\nwhere ˆFi(ω) represents the Fourier transform of Fi(t).\nThe force exerted on the particle by a focused laser with\nfrequency Ω is given by Fi(Ω) = Pscatti(Ω)/c, where\nPi\nscatt represents the power scattered in the direction i.\nTherefore,\nD\nˆFi(Ω)ˆF∗\nj(Ω′)E\n=1\nc2D\nˆPi(Ω)ˆP∗\nj(Ω′)E\n. (57)\nThe dominant source of the fluctuation is the short noise,\nthus the power spectral density is [41]\nSFiFj(Ω) =ℏΩ\n2πc2Pij\nscatt(Ω). (58)\nCompared it with the density Eq. (56), we can get the\neffective temperature of the recoil of the quanta ℏΩ in idirection\nTRE=ℏΩ\n2mc2kBγREPscatt(Ω) (59)\n=ℏΩ\n2mc2kBω2\n0SREPscatt(Ω).\nThere exists a natural linear relationship between TRE\nandPscatt. As discussed earlier, TREdoes not necessarily\nequal the temperature of the environment.\nC. Differences between feedback optical force and\nDoppler damping force\nThis subsection discusses the differences between the\nwork of Ref. [35] and Ref. [36]. This serves as one\nof the motivations for our work. Ref. [35] studied the\nlevitodynamics and relaxation of a levitated particle,\nwhere cooling is achieved through the equilibrium be-\ntween stochastic fluctuations and the optical damping\nforce FFB=−ω0ηq2p. However, as shown in Ref. [36],\nwhen the feedback loop is activated, an additional damp-\ning coefficient γFBappears in the Langevin system. It is\nconceivable that cooling can be achieved even without\nthe presence of the optical damping force FFB. The de-\ntails of the cooling are already presented in Fig. 4 and the\ncorresponding discussion in the preceding section. Here,\nwe provide a unified description of the two cooling ef-\nfects. In fact, the Doppler effect is caused by the motion\nof the levitated particle, which also alters its position. If\na feedback loop is set up in an experiment, neglecting\nthe higher-order O(p2) terms, the damping force can be\ngenerally expressed as\nFdamping =−G(q2)p . (60)\nG(q2) is a function of the square of the coordinate, q2.\nThis is due to the motion-induced Doppler effect. It\nshould be noted that the linear term in qdoes not con-\ntribute to damping. Therefore, the leading two terms in\nthe Taylor expansion are\nFdamping ≃ −G(0)p−G′(0)q2p . (61)\nThe first term represents the feedback damping coeffi-\ncient, while the second term corresponds to the optical\ndamping force. Both the feedback damping and optical\ndamping forces exist and are deterministic. As discussed\nin the previous section on levitodynamics, it is impor-\ntant to note that the total damping γTis not necessarily\nequal to the stochastic damping γS. However, the feed-\nback damping can dominate the cooling process of the\nlevitation.\nIn this section, we provide a brief introduction to\nDoppler cooling, which can also occur in the interac-\ntion between a levitated particle and the radiation back-\nground. The feedback mechanism, also known as optical\nmolasses, is a theoretical framework for the laser cooling12\n𝜔𝜔𝜔−𝑘𝑣𝜔+𝑘𝑣⟩|1⟩|2⟩|1⟩|2\n𝐹=−𝛼𝑣On resonance(a)(b)\nFIG. 6. The schematic diagram of Doppler cooling involves a two-level atom interacting with a pair of counter-propagating\nbeams, forming a laser with a frequency below the atomic resonance frequency. (a) The atom is stationary. (b) The Doppler\neffect causes an increase in the frequency of the laser beam propagating in the direction opposite to the atom’s velocity.\ntechnique that utilizes the configuration of three orthog-\nonal pairs of counter-propagating laser beams along the\nCartesian axes. As shown in plot (a) of Fig. 6, a station-\nary atom in a pair of counter-propagating laser beams\nexperiences no resultant force due to identical scatter-\ning from each laser beam. However, for a moving atom,\nas depicted in plot (b), the Doppler effect causes an in-\ncrease in the frequency of the laser beam propagating op-\nposite to the atom’s velocity. This Doppler shift brings\nthe light closer to resonance with the atom, increasing\nthe rate of absorption from this beam and resulting in a\nforce that decelerates the atom. The damping coefficient\nof the feedback can be derived from the molasses force\nFmol\nFmol=Fscatt(ω−ω0−kv)−Fscatt(ω−ω0+kv)\n≃Fscatt(ω−ω0)−kv∂F\n∂ω−\u0014\nFscatt(ω−ω0)−kv∂F\n∂ω\u0015\n≃ −2∂Fscatt\n∂ωkv , (62)\nwhere Fscattrepresents the scattering force exerted by the\nincident laser, and kdenotes the wavevector. Therefore,\nthe damping coefficient can be expressed simply as\nγFB=2k\nm∂Fscatt\n∂ω. (63)\nThe differentiation of the molasses force provides the\nprecise value of the coefficient, which can be found in\nRef. [42]. In this paper, we focus on extracting the dark\ndamping coefficient in the steady state without a feed-\nback loop. Therefore, we do not present the detailed\ndifferentiation of the molasses force here.D. The collision between levitated particle and\ndark matter\nEvidence of the existence of dark matter arises from the\ngravitational effects observed in the behavior of galax-\nies and clusters, as demonstrated by various astrophysi-\ncal observations. The accumulation of evidence increas-\ningly clarifies that a significant portion of the universe’s\nmatter exists in a non-luminous form, which could be\nweakly interacting with elements of the standard model\n(SM) and thus challenging to detect in terrestrial labo-\nratories. Direct detection aims to identify signatures of\ndark matter scattering off a terrestrial target in labora-\ntory settings. Leveraging the levitation of nanoparticles\nprovides a means to employ macroscopic force sensors for\nprobing long-range interactions between dark and visible\nmatter, including gravitational interactions. However,\nrealizing such ambitious experiments would require sub-\nstantial advancements beyond the current state of the\nart. Nevertheless, similar concepts for searching for dark\nmatter that might interact through stronger long-range\ninteractions are already feasible.\nNaively, one might expect the damping rate caused by\ndark matter collisions to be the same as the Kn≫1 case\nof the isothermal drag force since the mean free path of\ndark matter is much greater than the size of the levi-\ntated particle. However, a detailed analysis reveals that\ndamping from dark matter differs from the isothermal\ndrag force. Although the dark matter distribution can\nbe approximated as having a similar Maxwell-Boltzmann\ndistribution, the linearization condition in Eq. (41) fails\nas the vapor velocity of the dark matter becomes com-\nparable to the dark matter particle velocity. More de-\ntails regarding the velocity distribution can be found in\nRef. [43, 44].13\nLevitated particle𝒖\nFIG. 7. The sketch map of the dark matter scattering on a levitated particle from in one direction. The red ball is the final\nscattering state of the DM. Counting the transfer momentum to the levitated particle gives the damping force of the incident\nDM flux.\nNext, we provide a brief summary of the kinetic se-\ntups of galactic dark matter particles, which are grav-\nitationally bound to the halo of our galaxy. The local\ndistribution of dark matter is formulated using the sub-\nhalo model of the galaxy, with a density of approximately\nρ= 0.3,GeV/cm3. Assuming only one type of dark mat-\nter particles, the numerical density of these particles near\nEarth decreases as the dark matter mass mχincreases.\nThe scattering rate of dark matter depends on time due\nto variations in the dark matter flux on Earth caused by\nthe Earth’s motion around the Sun. Consequently, the\ndark matter signals are expected to exhibit annual mod-\nulation. The velocity of dark matter at Earth’s location\nis anticipated to be a few hundred km/s, limited by the\ngalactic escape velocity. In the galactic rest frame, the ve-\nlocity distribution follows the Maxwell-Boltzmann form,\nwith the most probable velocity v0typically chosen as\n220 km/s. While the circular velocity of the Sun around\nthe galaxy’s center is approximately 240 km/s, and the\ncircular velocity of the Earth around the Sun is about\n30 km/s. Although the relative velocity of the levitated\nparticle with respect to Earth can be much smaller, the\ndrift velocity of the dark matter flux is several hundred\nkilometers per second. Therefore, the motion of the solar\nsystem implies that the linearization of the expansion of\nthe drift wind U∞in fluid dynamics, as in Eq. (41), is no\nlonger applicable. Defining the pressure of the dark mat-\nter flux also becomes challenging. Consequently, we need\nto find an alternative approach to calculate the damping\ncoefficient caused by dark matter.\nIndeed, the picture of a saturated particle in a liquid,\nalong with the concept of isothermal drag, belongs to the\nrealm of macroscopic physical systems within fluid dy-\nnamics. On the other hand, when considering dark mat-\nter scattering on the levitated particle, we need to delve\ninto concepts from fundamental particle physics, such asscattering cross sections, particle masses, and coupling\nstrengths. These concepts are formulated within the\nframework of quantum scattering theory. While there are\nsimilarities between these two physical pictures, a direct\ncalculation from quantum scattering theory to the macro-\nscopic levitated particle is required. Therefore, we move\naway from fluid approaches in the subsequent study.\nThe damping coefficient can be straightforwardly de-\nrived from the collisions between the dark matter par-\nticle and the levitated particle. Let’s consider a one-\ndimensional collision as an example to illustrate the de-\ntails. As depicted in Fig. 7, both the levitated particle\nand the ambient dark matter are moving along the x\naxis. The velocity of the dark matter particle in the flux\nfollows a Maxwell-Boltzmann distribution.\nf(vx, TDM) =\u0012mχ\n2πkBTDM\u00131\n2\nexp\u0012\n−mχv2\nx\n2kBTDM\u0013\n.(64)\nHere the effective temperature corresponds to the average\nvelocity v0of the DM\nTDM=πmχ\n8kBv2\n0. (65)\nThe levitated particle will experience a damping force\nresulting from the transfer of momentum in the opposite\ndirection from the dark matter particles. The strength\nof this force should be proportional to the velocity of\nthe levitated particle, denoted as u, assuming it is much\nsmaller than the velocities of the dark matter particles.\nHowever, it is important to note that in this context, u\ncannot be equated to U∞as in Eq. (40) of fluid dynamics.\nIn Fig. 7, it is evident that the momentum transferred\nto the levitated particle can be computed by considering\nthe scattering of dark matter particles within the final\nsolid angle dΩ. The number density of the incident dark14\nmatter particles is given by ρχ/mχ. Utilizing the differ-\nential elastic cross section d σ/dΩ, the momentum trans-\nferred to the levitated particle during a time interval ∆ t\ncan be expressed as\n∆P=Z Zρχ\nmχ\u0012mχ\n2πkBTDM\u00131\n2\nexp\u0012\n−mχv2\nx\n2kBTDM\u0013\n×dσ\ndΩ|vx−u|∆t×mχvx(1−cosθ) dvxdΩ\n=−ρχ∆t\u0012mχ\n2kBTDM\u0013−11\n4\u0012\n1 + Erf\u0012rmχ\n2kBTDMu\u0013\n−Erf\u0012rmχ\n2kBTDMu\u0013\u0013Zdσ\ndΩ(1−cosθ) dΩ\n≃ −ρχ∆t\u0012mχ\n2kBTDM\u0013−1q\nmχ\n2kBTDMu\n√πσT\n=−ρχu∆t\u0012πmχ\n2kBTDM\u0013−1\n2\nσT.\n(66)\nIn the last line of Eq. (66), we employ Taylor expansion to\nobtain the dominant term of the transferred momentum.\nHere, σTrepresents the transfer cross section defined for\ncollisions between dark matter particles and the levitated\nparticle\nσT=Zdσ\ndΩ(1−cosθ) dΩ. (67)\nWe can see that the damping coefficient in one dimension\nis\nγ1D\nDM=−∆P\n∆t1\nmu=ρχ\nm\u0012πmχ\n2kBTDM\u0013−1\n2\nσT. (68)\nThe 3D result of γDMcan be obtained by using the afore-\nmentioned result. The momentum transfer from the dark\nmatter particles to the levitated particle within an in-\nfinitesimal solid angle dΩ′/2πon a hemispherical surface\ncan be approximated using the one-dimensional result\ngiven by Eq. (68). By integrating over dΩ′/2πfor all\nmomentum components along the direction of motion of\nthe levitated particle, we can derive the 3D result as fol-\nlows\nγDM=Z\nγ1D\nDMcos2θ′dΩ′\n2π(69)\n=γ1D\nDM\n3=ρχ\n3m\u0012πmχ\n2kBTDM\u0013−1\n2\nσT.\nSubstituted the effective temperature TDMEq. (65), we\ncan get a very simple expression of the damping coeffi-\ncient\nγDM=ρχv0\n6mσT. (70)\nThe damping coefficient mentioned above is suppressed\nby the mass of the levitated particle. However, as men-\ntioned earlier, the transfer scattering cross section σTrepresents the scattering between dark matter and a\nmacroscopic particle. The connection between this inter-\naction and the interaction between DM and fundamental\nparticles can be elucidated as follows.\nWhen comparing with the isothermal drag force dis-\ncussed in the previous subsection, we observe that the\ntransfer cross section σTin Eq. (70) corresponds to the\nareaR2\n0in Eq. (44). This area provides a macroscopic de-\nscription of the scattering process. The scattering cross\nsection and the mass of the DM are typically the two most\nsignificant parameters in dark matter detection. How-\never, in the case of the levitation experiment, there are\nsome misunderstandings regarding the interactions be-\ntween the DM and the levitated particle due to the com-\nposite nature of the levitation system. The levitated par-\nticle is composed of millions of microscopic molecules or\natoms, depending on the scale of the levitation. The de-\ntection of levitation aims to observe the collective motion\nof these composite particles. Therefore, the differential\ncross section in Eq. (66) and the transfer cross section σT\nin Eq. (67) do not represent fundamental dark matter\ninteractions as commonly understood in the literature,\nwhich refer to interactions between elementary particles.\nTo establish the relationship between fundamental in-\nteractions and levitation interactions, we need to derive\nthis connection. The key aspect of this derivation lies\nin the coherence that exists from the fundamental inter-\naction to the levitation level. The collective movement\nof the levitated particles can enhance the scattering rate\nthrough coherent effects. This enhanced scattering rate\ngives rise to a characteristic scattering pattern known as\nthe static structure factor. The static structure factor\nresults from the collective interference of waves scattered\nby particles in the system. This interference is sensitive\nto the relative separation between the particles, and the\nstatic structure factor can be expressed as the spatial\nFourier transform of the particle structure, represented\nby the density-density correlation function.\nLet’s consider the dark interaction with the nucleon\nas an example. The scattering rate can be derived from\nthe amplitude Mχn, which originates from the funda-\nmental dark interaction. The scattering of the levitated\nparticle involves atoms, which are composites composed\nof nucleons and electrons. Thus, the total scattering is\ndetermined at two levels: the distribution of nucleons\nwithin the atom and the configuration of atoms within\nthe levitated particle. This physical picture resembles X-\nRay diffraction in a crystal, as depicted in Fig. 8 (Ref.\n[45]). Firstly, the spatial distribution of matter can be\nformulated as the nucleus form factor f(P) in momen-\ntum space, which is obtained by performing the Fourier\ntransformation of the spatial distribution.\nf(P) =Z\nρ(r)e−iP·rd3r . (71)\nIt should be noted that an isotropic matter distribution\nis assumed in this case, simplifying the situation. There-\nfore, f(P) is solely dependent on P. Typically, f(P) is15\n𝑃!\"≫𝑠𝑦𝑠𝑡𝑒𝑚𝑃!\"≈∆𝑟𝑃!\"≪∆𝑟\n(a)(b)(c)\n∆𝑟\n∆𝑟\n∆𝑟\nFIG. 8. S(P) exhibits three distinct regimes depending on the particle spacing in relation to the scattering length. In the\ncase where the scattering length is significantly larger than the particle collection, interference manifests as a sum of nearly\nequivalent phases, resulting in a proportional relationship between S(P) and the number of particles. When the scattering\nlength is of comparable magnitude to the particle spacing, significant angular variations arise in the scattered intensity. In the\nscenario where the scattering length is significantly smaller than the particle spacing, the phases become randomized, resulting\nin interference that causes S(P) to approach unity, with the P-dependence solely determined by the particle form factor.\nnormalized to unity to facilitate the calculation of the to-\ntal scattering amplitude with the atom. This calculation\ninvolves counting the atomic number A.\nMχA=MχnAf(P). (72)\nNext, we proceed to calculate the scattering on the lev-\nitated particles. In this context, the term “levitated par-\nticle” represents a generic term encompassing the objects\ncomprising condensed matter. Regardless, the atoms\nwithin the collection occupy distinct relative positions,\ngiving rise to interference during the levitation process,\nparticularly in the context of DM scattering. The dis-\nplacement of the atom can be expressed as an ampli-\ntude, incorporating an additional unitary transformation\nˆU= exp ( −ir·P), yielding the total amplitude as a re-\nsult\nMχL=all the atomsX\niMχnAfi(P)e−iP·ri. (73)\nHere, rirepresents the position of the center of the ith\nnucleus. From the formulations above, we can establish\na relation between the fundamental interaction and the\nmacroscopic cross section as utilized in Eq. (69) from the\ndamping\nσT∝ |M χn|2A2(74)\n× X\nifi(P)e−iP·ri!\nX\njfj(P)e−iP·rj\n∗\n.\nIntegrating |Mχn|2yields the standard scattering cross\nsection σD\nTcommonly studied in the literature on darkmatter. Assuming all atoms are identical and possess\nidentical form factors.\nσT=σD\nT|f(P)|2A2N\n\n1\nNX\niX\nje−iP·(ri−rj)\n\n\n=σD\nTf(P)|2A2NS(P), (75)\nwhere Nis the total number of the component particle\nof the levitated particle. The static structure factor is\ndefined by\nS(P)≡1\nN*X\niX\nje−iP·(ri−rj)+\n, (76)\nwhere the angled brackets indicate an average taken over\nappropriate ensembles of the structure.\nThe quantity S(P) in Eq. (76) serves as an overall\nmeasure of the phase differences in the scattered field,\nwhich arise due to the relative separation of the parti-\ncles. In this context, the scattering wave vector plays a\ncrucial role. The reciprocal of the scattering wave vec-\ntor represents a significant scattering length scale, given\nbyl= 2π/P, which determines the severity of interfer-\nence effects in comparison to the mean particle spacing.\nThis is illustrated in Fig. 8, which displays three distinct\nregimes. When the scattering length scale is significantly\nlarger than the distances between atoms within the sys-\ntem (as shown in the left panel of Fig. 8), the phase\ndifferences, P·(ri−rj), between neighboring scattered\nwaves are nearly identical, leading to constructive inter-\nference.\nS(P) =1\nN*X\ni,je−iP·(ri−rj)+\n≈N2\nN=N. (77)16\nConversely, when the scattering length scale is small com-\npared to the particle spacing, (right panel of Fig. 8) the\nphase differences, P·(ri−rj), between waves scattered\nby neighboring atoms are randomized and produce\nS(P) =1\nN*X\ni,je−iP·(ri−rj)+\n≈N\nN= 1. (78)\nIn any case, the levitation experiments exhibit significant\nenhancements of the microscopic transfer cross section\nσD\nTby factors of A2N2orA2N. This sensitivity to the\nmicroscopic forces is the reason behind the high sensitiv-\nity of levitation. Additionally, it should be noted that\nthe measured momentum pin the levitation corresponds\nto the collective motion of the levitated particles.\nP=p\nNA∼p\nm⟨E⟩\nNA. (79)\nIn this scenario, the measured energy should be signif-\nicantly larger than the energy scale ℏΩ. Consequently,\nthe atomic form factor f(P) can be safely approximated\nasf(0), and the static structure factor can also be ap-\nproximated as N. Further studies on DM detection are\ndiscussed in the following section.\nSubstituted Eq. (75) to Eq. (70), the dark damping\ncoefficient is\nγDM=ρχv0\n6mσD\nTA2N|f(P)|2S(P). (80)\nIt is evident that the damping coefficient, despite being\nsuppressed by the mass of the levitated particle, is en-\nhanced by the number of fundamental particles. When\nboth f(P)→1 and S(P)→Noccur, the coefficient is\nenhanced to the point of resembling a macroscopic inter-\naction. This distinction arises from the coherence of the\nscattering and sets apart dark damping from all other\ntypes of damping discussed in the preceding subsections.\nComparing all the discussed damping coefficients in\nthis section, we can observe that the distinct properties\nof these coefficients provide valuable clues for extract-\ning each damping effect and its corresponding fluctua-\ntion from specific experimental setups and appropriate\nconduction in levitation experiments. In the next sec-\ntion, we revisit the initial motivation behind our work,\nwhich is to extract the dark interaction from the afore-\nmentioned damping effects. We aim to provide an esti-\nmation of the DM parameters, including the cross-section\nand interaction strength, among others.\nIV. EXPLORATION ON THE DARK DAMPING\nIN THE LEVITATION EXPERIMENTS\nA. The linear response and the extraction of the\ndamping coefficient\nBefore delving into the experimental exploration, it is\ncrucial to ascertain the reactions that occur in the lab-\noratory. Typically, the linear response of the levitatedparticle to driving forces is assumed. Subsequently, the\nresponse function can be derived as a perturbation of\nthe system. The analyticity, causality, and the Kramers-\nKronig relation in the response can be found in standard\ntextbooks on general kinetic theory. [46] In this section,\nwe provide a brief introduction to the reaction and dissi-\npation based on Eq. (1), incorporating a general damping\ncoefficient ( γ) and temperature ( T). Assuming that the\nLangevin system was in equilibrium in the distant past,\nthe position at time tis determined by the following equa-\ntion\nq(t) =Zt\n−∞χt(t−t′)F(t′)dt′, (81)\nwhere\nχ(t) =1\n2πZ∞\n−∞e−iωt˜χ(ω)dω . (82)\nThen\nq(t) =Z∞\n−∞dt′Z∞\n−∞1\n2πe−iω(t−t′)F(t′) ˜χ(ω) dω .(83)\nSubstituted it into the equation of motion Eq. (1)\nF(t) =Z∞\n−∞dt′Z∞\n−∞1\n2π\u0000\n−ω2−iωγ+ω2\n0\u0001\n×e−iω(t−t′)F(t′) ˜χ(ω) dω . (84)\nWe can see that the response function is\n˜χ(ω) =1\n(ω2\n0−ω2)−iωγ(85)\nχ(t) =1\n2πZ∞\n−∞e−iωt\n(ω2\n0−ω2)−iωγdω (86)\n=1\nω∗e−γt\n2sin (ω∗s),\nwhere ω∗=q\nω2\n0−γ2\n4. The real part of the response\nfunction ˜ χ(ω) is the reactive part of the system\nRe˜χ(ω) =ω2\n0−ω2\n(ω2\n0−ω2)2+γ2ω2. (87)\nThe higher of this function, the more the system will\nrespond to a given frequency. While the imaginary part\nof the response function is the dissipative part of the\nsystem\nIm˜χ(ω) =γω\n(ω2\n0−ω2)2+γ2ω2, (88)\nwhich is proportional to the damping coefficient.\nFor the case of natural fluctuations in the position and\nvelocity of the particle in equilibrium, the stochastic force\naverages to zero and is assumed to have delta-function\ntime correlation, as depicted in Eq. (2). The average of17\nthe position and velocity squares obeys the equipartition\ntheorem\n⟨q2(t)⟩=kBT\nmω2\n0,⟨p2(t)⟩=mkBT . (89)\nThe power spectrum of q2is the Lorentzian peak\n˜Sq2(ω) =2γkBT\nm1\n(ω2\n0−ω2)2+γ2ω2. (90)\nThis spectrum can be measured by counting the occupa-\ntion number in a levitation experiment. The dissipative\npart of the response function (Eq. (88)) is related to\nIm˜χ(ω) =m\n2kBT˜Sq2(ω). (91)\nThis result suggests that the dissipation caused by driv-\ning a system out of equilibrium with an external force is\nproportional to the power spectrum of the natural fluc-\ntuations that arise in equilibrium.\n∆ω2\nωp\nωSqq(ω)\nFIG. 9. The power spectrum ˜Sq2(ω) is characterized by the\npeak frequency ωpand the half width ∆ ω2.\nThe power spectrum ˜Sq2(ω) depicted in Fig. 9 can be\nmeasured in levitation experiments by counting the oc-\ncupation number of quanta ℏω0inω-space. It provides\ncomprehensive information that allows for the precise ex-\ntraction of the damping coefficient γand temperature of\nthe levitated particle. Various approaches exist for this\npurpose, and we adopt the simplest one. The primary\nmeasurable parameter is the peak frequency ωp, from\nwhich one can readily derive the relation.\nω2\np=ω2\n0−γ2\n2. (92)\nHowever, the damping coefficient cannot be obtained\ndirectly from this relation since ω0is not a precisely pre-\ndicted or measured variable in real experimental studies.\nTherefore, in this paper, we propose measuring the half-\nwidth ∆ ω2of the peak, which represents the differencebetween the squared frequencies at half the peak height.\nThis allows us to establish the relation.\nω2\n0=ω2\nps\n1 +\u0012∆ω2\n2ω2p\u00132\n, (93)\nto determine ω0more precisely. This relationship reveals\nthat, in the presence of a very sharp peak, we can obtain\nan approximate damping coefficient.\nγ=∆ω2\n2ωp. (94)\nWith the precisely determined coefficient, the tempera-\nture can be derived at the peak\nT=˜Sq2(ωp)\u0000\nω4\n0−ω4\np\u0001γm\nkB. (95)\nBy utilizing this measured damping coefficient γand\ntemperature T, we can conduct a comprehensive study\nof the correlation function and the fluctuation dissipation\ntheorem. Additionally, we can search for evidence of dark\ninteraction through the investigation of dark damping.\nB. The verification of the multi-stochastic force\ntheory and the searching for the dark damping\nIn this subsection, we present the experimental explo-\nration of the theory proposed in this work and the search\nfor dark interactions hidden in the noise of the levitation.\nFirstly, we list all the new ideas proposed in the above\ncontext:\n1. Multiple sources of stochastic force exist, each inde-\npendently adhering the fluctuation-dissipation the-\norem. These sources collectively interact with the\nlevitated particle.\n2. The cooling feedback mechanism provides a damp-\ning coefficient and an optical force that also damps\nthe levitated particle. Consequently, the forces\nwithin the feedback mechanism are deterministic.\n3. The levitated particle is assigned an effective tem-\nperature. When the feedback is turned off, the ef-\nfective temperature differs from the temperature of\nthe environment and is denoted as TS, which may\ninclude contributions from dark interactions.\n4. The motion of the levitated particle arises from the\ncollective motion of microscopic particles, thereby\nenabling the dark damping to unveil interactions\nbetween elementary and macroscopic particles.\nThe first two items provide physical insights into levita-\ntion physics, while the other two items represent phys-\nical relations and quantities that can be experimentally\nverified or measured. Fortunately, most of the relations18\ndiscussed in the previous section are linear, allowing the\nuse of linear estimation methods in experimental studies.\nTo demonstrate the effectiveness of linear estimation,\nthe inequality between the stochastic temperature TS\nand the environment temperature TEis initially verified.\nIn the actual levitation experiment, we can disable the\nfeedback at various environment temperatures TEiwhile\nkeeping all other levitation setups, such as laser intensity,\nunchanged. Subsequently, we can measure Nsamples of\nthe stochastic temperature TSiby analyzing the power\nspectrum and simulating the relation given by Eq. (95).\nBy utilizing Eq. (6), we can define a linear relationship\nTS=θ1+θ2TE. (96)\nIf only TEivaries in the experiment, we can treat θ1and\nθ2as constants. Non-zero θ1and the inequality θ2̸= 1\nserve as verification for our proposal. We define χ2\nTas\nχ2\nT=NX\ni(TSi−θ1−θ2TEi)2. (97)\nThe minimum χ2\nTcorresponds to the best fit with the\nexperimental data. Therefore, it is necessary to solve\npartial differential equations\n∂χ2\nT\n∂θ1=NX\ni(−2) (TSi−θ1−θ2TEi) = 0 , (98)\n∂χ2\nT\n∂θ2=NX\ni(−2TEi) (TSi−θ1−θ2TEi) = 0 .(99)\nThe Least Square Method gives the best matching result\nθ1=\u0010PN\niT2\nEi\u0011\u0010PN\niTSi\u0011\n−\u0010PN\niTEiTSi\u0011\u0010PN\niTEi\u0011\nN\u0010PN\niT2\nEi\u0011\n−\u0010PN\niTEi\u00112,\nθ2=N\u0010PN\niTEiTSi\u0011\n−\u0010PN\niTEi\u0011\u0010PN\niTSi\u0011\nN\u0010PN\niT2\nEi\u0011\n−\u0010PN\niTEi\u00112.\nIt is evident that θ1= 0 and θ2= 1 when every environ-\nment temperature is equal to the stochastic temperature\nTEi=TSi. Therefore, the agreement of θ1andθ2pro-\nvides a direct test of the proposal in this work.\nOur objective is to search for the dark interaction in\nthe damping coefficient. However, detecting the dark\ninteraction requires precise tuning of the experimental\nsetup and conduct. We propose two procedures for con-\nducting the experiment and exploring dark damping,\nconsidering all the analyzed damping coefficients in the\nprevious section.\n1. Firstly, we propose measuring the steady state\nwithout feedback in a high vacuum environment.\nAlthough cooling and heating measurements are\npossible, measuring the steady relaxation state\n012345678910\n012345678910P𝟏\nPTH𝛾𝛾!\"S𝟏S𝟐S𝟑S = 0P𝟐P𝟑FIG. 10. The sketch map using linear estimation to extract\nthe dark damping γDM.\nshould be easier and more precise. In a high vac-\nuum environment, where the Knudsen number Kn\nis much greater than 1, the damping coefficient\nγTHin Eq. (44) is proportional to the pressure of\nthe ambient gas. The zero pressure limit can be\napproached using linear estimation. Additionally,\nby varying the noise spectral density of the opti-\ncal trapping potential, denoted as SREin Eq. (55),\nit is possible to control the range of γRE. Then,\nthe damping coefficient and temperature can be\nmeasured, and linear estimation can be used to ap-\nproach the zero optical fluctuation limit.\n2. In the absence of optical fluctuation and pressure,\nnon-zero damping coefficients provide upper limits\non the transfer cross section between dark matter\nand the levitated particle. Subsequently, by uti-\nlizing the static structure factor S(P) and atomic\nfactor f(P), we can obtain constraints on the mi-\ncroscopic transfer cross-section between dark mat-\nter and fundamental particles.\nFig. 10 illustrates a schematic diagram showcasing the\nuse of linear estimation to extract the dark damping γDM\nfrom the precisely measured γS. The actual experiment\ncan be conducted under different vacuum pressures, al-\nlowing for the measurement of γSat various SREvalues.\nLinear estimation can then be employed to obtain the\nvalue of γSin the limit SRE→0. By performing linear\nestimation of lim PTH→0γSRE→0\nS , the value of γSat zero\npressure can be determined. Alternatively, if SREcan be\nadjusted to the same value, the linear estimation can be\ninitially performed on the vacuum pressure, as depicted\nin Fig. 10. These two different procedures can serve as\na cross-check to validate the final results of the damping\ncoefficient.\nThe final step involves extracting constraints on the\ndark matter side, such as the transfer cross section σD\nT,19\nbased on the previously derived dark damping coefficient\nγDMfrom Eq. (80). The levitation experiment offers\na high-precision sensor that enables the measurement\nof the coupling between a fundamental particle and a\nmacroscopic particle. Since the focus of this paper is\non studying levitodynamics and determining the dark\ndamping based on sensor measurements, the detailed\nconstraints on the cross-section and dark matter mass are\nbeyond the scope of this work. For simplicity, we assume\nthatσD\nTis constant, the atomic form factor f(P) = 1, and\nthe static structure factor S(P) =N. Additionally, we\nassume identical couplings between the dark matter and\nnucleons. By substituting the DM density, solar system\nvelocity, and mass of the levitated particle into Eq. (80),\nthe transfer cross section can be obtained.\nσD\nT∼<6γDM\n220×105s−1m2\nproton\n0.3GeV ×mlevitated particlecm2.\n(100)\nThough our proposal has not been applied in the exper-\niments, the error of current measured results in levitation\nexperiments could give us an estimation on the order of\nthe non-zero value of dark damping γDM. Thus we ex-\namine the constraints on the transfer cross-section from\ncurrent levitation experiments by utilizing the results of\ntwo experiments. Fused silica (SiO 2) particles are consis-\ntently employed in these experiments. In the experiment\ndescribed in Ref. [36], the levitated SiO 2particles have\nradii on the order of 50 nm, enabling direct measurement\nof the photon recoil. The rate Γ RE=γREn∞represents\nthe product of the damping coefficient and the total num-\nber of quanta in the final steady state. The error in Γ RE\nis approximately O(0.1) kHz, while n∞is on the order\nofO(105). Therefore, an estimation of the limit on γDM\ncan be made, placing it on the order of O(10−3) Hz. By\ninputting the corresponding parameters for fused silica,\nwe can derive the limit σD\nT∼<O(10−20)cm2. In the ex-\nperiments described in Ref. [33], the particle radii are on\nthe order of 5 µm, and the error of the Voigt profile is also\non the order of O(10−3) Hz. Consequently, the limit on\nthe transfer cross section could be σD\nT∼<O(10−26)cm2.\nHowever, conducting an actual experiment to search for\ndark damping would yield a much more precise limit.\nThe linear estimation of the damping coefficient would\nextend to a further non-zero damping limit, potentially\nserving as evidence of dark matter.\nV. CONCLUSION\nIf the terrestrial environment is filled with dark mat-\nter, the levitation experiences contributions from damp-\ning forces and fluctuations. In this paper, we study\nlevitodynamics with multiple stochastic forces, includ-\ning thermal collision, photon recoil, feedback, and oth-\ners. We assume that all stochastic forces are indepen-\ndent of each other, following the fluctuation dissipation\ntheorem. We observe that the fluctuation term does notnecessarily correspond one-to-one with the damping co-\nefficient when the feedback loop is active. Therefore, we\nintroduce two coefficients, γTandγSto differentiate be-\ntween total damping and stochastic damping. The ra-\ntio of these coefficients distinguishes our levitodynamics\nfrom those with only single stochastic forces. Both feed-\nback damping and optical damping can cool down the\nlevitated particle. We analyze the energy distribution,\neffective temperature, and heating and cooling behav-\niors. The cooling limit differs from the result presented\nin Ref. [35], where the limit is determined by the balance\nbetween single fluctuation and feedback optical force. In\nour theory, feedback damping can dominate the cooling\nlimit. Importantly, even in an equilibrium state with-\nout feedback, the effective temperature of the system is\nnot equal to the ambient or thermal temperature. It is\na complex value influenced by thermal collision, photon\nrecoil, and possibly dark matter.\nWe investigate the stochastic sources of forces in our\nstudy, including thermal drag, recoil, and feedback.\nThese coefficients are derived from various theories such\nas fluid dynamics and quantum optics. We discover that\nDM collisions cannot be treated in the same way as\nin fluid dynamics due to the failure of linearizing the\nvelocity distribution. It becomes necessary to extract\nthe fundamental interaction between DM and standard\nmodel particles. The paper provides a detailed analysis of\nthe relationship between the fundamental transfer cross\nsection and the macroscopic transfer cross section. Al-\nthough the dark damping coefficient is suppressed by the\nmass of the levitated particle, scattering can be coher-\nently enhanced based on the scale of the component mi-\ncroscopic particle, the atomic form factor, and the static\nstructure factor. Therefore, dark damping may offer in-\nsights into detecting the macroscopic strength of funda-\nmental particles.\nFinally, we propose the operations of the levitation ex-\nperiment. By precisely measuring the Lorentzian peak, a\nmuch more accurate damping coefficient can be obtained.\nAs the thermal drag and recoil can be linearly estimated\nto approach zero limits, non-zero results at zero pressure\nand optical fluctuation can test our theory. Based on the\ncurrent levitation results, we demonstrate that the fun-\ndamental transfer cross section can be on the order of\nσD\nT∼<O(10−26)cm2. If our proposal is implemented in\nactual experiments, we believe that it can provide deeper\nconstraints and reveal more details about the dark inter-\naction through levitation experiments.\nAlthough this paper only focuses on four types of\nstochastic forces, our theory can be readily extended\nto scenarios involving additional stochastic forces. 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Pathria, Statistical mechanics (Elsevier, 2016)."    },    {        "title": "1810.11016v2.Time_retarded_damping_and_magnetic_inertia_in_the_Landau_Lifshitz_Gilbert_equation_self_consistently_coupled_to_electronic_time_dependent_nonequilibrium_Green_functions.pdf",        "content": "Time-retarded damping and magnetic inertia in the Landau-Lifshitz-Gilbert equation\nself-consistently coupled to electronic time-dependent nonequilibrium Green functions\nUtkarsh Bajpai and Branislav K. Nikoli´ c∗\nDepartment of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA\nThe conventional Landau-Lifshitz-Gilbert (LLG) equation is a widely used tool to describe dy-\nnamics of local magnetic moments, viewed as classical vectors of fixed length, with their change\nassumed to take place simultaneously with the cause. Here we demonstrate that recently devel-\noped [M. D. Petrovi´ c et al. , Phys. Rev. Applied 10, 054038 (2018)] self-consistent coupling of\nthe LLG equation to time-dependent quantum-mechanical description of electrons—where nonequi-\nlibrium spin density from time-dependent nonequilibrium Green function (TDNEGF) calculations\nis inserted within a torque term into the LLG equation while local magnetic moments evolved by\nthe LLG equation introduce time-dependent potential in the quantum Hamiltonian of electrons—\nmicroscopically generates time-retarded damping in the LLG equation described by a memory kernel\nwhich is also spatially dependent. For sufficiently slow dynamics of local magnetic moments on the\nmemory time scale, the kernel can be expanded into power series to extract the Gilbert damping\n(proportional to first time derivative of magnetization) and magnetic inertia (proportional to second\ntime derivative of magnetization) terms whose parameters, however, are time-dependent in contrast\nto time-independent parameters used in the conventional LLG equation. We use examples of single\nor multiple local magnetic moments precessing in an external magnetic field, as well as field-driven\nmotion of a magnetic domain wall (DW), to quantify the difference in their time evolution computed\nfrom conventional LLG equation vs. TDNEGF+LLG quantum-classical hybrid approach. The faster\nDW motion predicted by TDNEGF+LLG approach reveals that important quantum effects, stem-\nming essentially from a finite amount of time which it takes for conduction electron spin to react\nto the motion of classical local magnetic moments, are missing from conventional classical micro-\nmagnetics simulations. We also demonstrate large discrepancy between TDNEGF+LLG-computed\nnumerically exact and, therefore, nonperturbative result for charge current pumped by a moving\nDW and the same quantity computed by perturbative spin motive force formula combined with the\nconventional LLG equation.\nI. INTRODUCTION\nThe conventional Landau-Lifshitz-Gilbert (LLG)\nequation [1–3] is the cornerstone of numerical micro-\nmagnetics [4] and atomistic spin dynamics [5] where one\nsimulates the classical time evolution of many magnetic\nunits coupled by exchange or magnetostatic interactions.\nThe LLG equation\n∂m(r,t)\n∂t=−gm(r,t)×Beff(r,t)+λGm(r,t)×∂m(r,t)\n∂t,\n(1)\ndescribes time evolution of m(r,t) as the unit vector\n|m|= 1 of constant length representing the direction\nof the local magnetization. Here gis the gyromag-\nnetic ratio and Beffis the sum of an external mag-\nnetic field and effective magnetic fields due to magnetic\nanisotropy and exchange coupling (additional stochastic\nmagnetic field can contribute to Beffto take into ac-\ncount finite temperature effects [6]). The second term\non the right-hand side of Eq. (1) is introduced phe-\nnomenologically to break the time-inversion symmetry,\nthereby generating a damping mechanism. The conven-\ntional intrinsic Gilbert damping λGis assumed to be ma-\nterials specific and, therefore, time-independent parame-\n∗bnikolic@udel.eduter. It is typically computed using the so-called breath-\ning Fermi surface [7] or torque-torque correlation formu-\nlas [8] within single-particle quantum-mechanical frame-\nwork (additional many-body processes have to be taken\ninto account to make λGfinite in the clean limit at low\ntemperatures [9]). In the original form, Eq. (1) is written\nfor a bulk material as a highly nonlinear partial differen-\ntial equation. It can also be re-written for a macrospin or\na lattice of atomic spins leading to a system of nonlinear\nordinary differential equations [5].\nIn the case of conducting ferromagnets, LLG equation\nhas to be extended by including additional terms, such as:\n(i) spin-transfer torque [10] T∝/angbracketleftˆs/angbracketright×mdue to injected\nelectrons generating nonequilibrium spin density /angbracketleftˆs/angbracketrightthat\nis noncollinear to local magnetization; ( ii) additional\nGilbert damping, ( g↑↓/4π)m×∂m/∂t, due to pumping\nof spin currents by the dynamics of m(t) whereg↑↓is the\nso-called spin-mixing conductance [11]; ( iii) additional\nnonlocal Gilbert damping [12–18], m×(D·∂m/∂t), due\nto spin pumping by noncollinear magnetic textures where\nDαβ=η/summationtext\ni(m×∂im)α(m×∂im)βis the 3×3 damping\ntensor,∂i=∂/∂iandα,β,i∈{x,y,z}; and ( iv) mag-\nnetic inertia [19–25], Im×∂2m/∂t2, of relevance to ul-\ntrafast magnetization dynamics. Like the original Gilbert\ndamping parameter λGin Eq. (1), T,g↑↓,DandIrequire\nmicroscopic quantum-mechanical calculations which are\noften combined [8, 9, 26–31] with first-principles Hamil-\ntonians of realistic materials.\nFurthermore, generalizations of LLG equation havearXiv:1810.11016v2  [cond-mat.mes-hall]  6 Dec 20182\nbeen considered to take into account the retardation ef-\nfects [32, 33]\n∂m(r,t)\n∂t=\u0002t\n0dt/prime\u0002\nd3r/primeΓ(r,t;r/prime,t/prime)m(r/prime,t/prime)\n×/bracketleftbigg\n−gBeff(r/prime,t/prime) +λG∂m(r/prime,t/prime)\n∂t/prime/bracketrightbigg\n,(2)\nby introducing a memory kernel Γ( r,t;r/prime,t/prime). The mem-\nory kernel models space-time correlation between local\nmagnetic moments, i.e., the fact that the cause for the\nchange of local magnetization occurs at time t−t/primeand\nat position r−r/primewhile the effect at position ris vis-\nible at the later time t. It has been specified phe-\nnomenologically, such as the sum of an instantaneous\nand time-dependent part which exponentially decays on\na characteristic time scale defining the strength of mem-\nory [32, 33]. It is also often simplified [33] by considering\ntime-retardation only, Γ( r,t;r/prime,t/prime)→Γ(t,t/prime), so that any\nspace-retardation effects are included only through the\neffective field Beff(r,t/prime).\nThe time-retardation described by Γ( t,t/prime) is a damping\nmechanism in addition to well-established mechanisms—\nthe combined effects of spin-orbit coupling and electron-\nphonon interaction [7, 8]—which govern λGin Eq. (1).\nHowever, the magnitude of Γ( t,t/prime) cannot be deduced\nfrom purely phenomenological considerations [32, 33].\nInstead, the introduction of the memory kernel Γ( t,t/prime)\ncan be justified microscopically [6, 22, 34–37] by us-\ningquantum-classical hybrid approaches, where time-\ndependent quantum formalism is used to compute /angbracketleftˆs/angbracketright(t)\nwhich is then fed into the LLG equation, while in turn, lo-\ncal magnetization from the LLG equation generates time-\ndependent field in the quantum Hamiltonian of electrons.\nAlthough electron dynamics is assumed to be much faster\nthan that of local magnetic moments, it still takes fi-\nnite time for electron spin to react to new position of\nm(r,t). This is the fundamental reason for time-retarded\ndamping effects encoded by Eq. (2), which are present\neven if the intrinsic Gilbert damping in Eq. (1) is van-\nishingly small due to small spin-orbit coupling (nonzero\nλGrequires spin-orbit coupling [7–9] and scales quadrati-\ncally with it [38]). Since classical micromagnetics simula-\ntions typically use only the conventional intrinsic Gilbert\ndamping term in Eq. (1), while not considering explicitly\nthe flow of conduction electrons in the presence of mag-\nnetization dynamics, the question arises about the mag-\nnitude of neglected effects like time-retarded damping in\nstandard simulations of magnetic-field- or current-driven\ndynamics of noncollinear magnetic textures such as mag-\nnetic domain walls (DWs) [39–47] and skyrmions [48, 49].\nAlthough quantum-classical approaches which auto-\nmatically include time-retardation effects have been dis-\ncussed previously [6, 22, 34–37], they have been focused\non the simples examples where one or two local magnetic\nmoments (pertinent to, e.g., magnetic molecules) inter-\nact with either closed electronic quantum system [22, 35]\n(i.e., not attached to macroscopic reservoirs to allow elec-\ntron spin and charge currents to flow into and from an\nFIG. 1. Schematic view of two-terminal devices where an infi-\nnite 1D TB chain, describing electrons quantum-mechanically,\nis attached to two macroscopic reservoirs while its middle part\nhosts: (a) single local magnetic moment, initially oriented in\nthe +x-direction, placed in an external magnetic field point-\ning along the + z-direction; (b) 11 local magnetic moments\n(illustration shows 7 of them), initially oriented in the + x-\ndirection, placed in an external magnetic field pointing along\nthe +z-direction; (c) three-site-wide head-to-head magnetic\nDW whose motion is driven by an external magnetic field\npointing in the + x-direction. Electrons within 1D TB chain\nand classical local magnetic moments interact via the s-dex-\nchange coupling of strength Jsd, and classical local magnetic\nmoments within the DW in (c) additionally interact with each\nother via the Heisenberg exchange coupling of strength J.\nexternal circuit), or open electronic quantum system but\nemploying approximations [6, 34, 36, 37] to obtain analyt-\nical solution. Thus, these approaches are not suitable for\nsimulations of spintronic devices containing large number\nof noncollinear local magnetic moments.\nHere we employ recently developed [50] numerically\nexact and, therefore, nonperturbative algorithm combin-\ning time-dependent nonequilibrium Green function for-\nmalism [51, 52] with the conventional LLG Eq. (1) (TD-\nNEGF+LLG) to demonstrate how it effectively generates\ntime-retardation effects, whose memory kernel can be ex-\nplicitly extracted in terms of TDNEGFs only in some lim-\nits (such as weak electron-spin/local-magnetic-moment\ninteraction and weak coupling of the active region to\nmacroscopic reservoirs). The paper is organized as fol-\nlows. Section II A introduces model quantum Hamilto-\nnian for electronic subsystem and classical Hamiltonian\nfor the subsystem comprised of local magnetic moments.\nIn Sec. II B, we show how the nonequilibrium expectation3\nvalue of spin density\n/angbracketleftˆs/angbracketrighti(t) =~\n2Tr [(\u001aneq(t)−\u001aeq)|i/angbracketright/angbracketlefti|⊗\u001b], (3)\ninserted into the LLG Eq. (1) generates a memory ker-\nnel because of the structure of the nonequilibrium time-\ndependent density matrix \u001aneq(t) obtained from TD-\nNEGF calculations. Here \u001aeqis the grand canonical equi-\nlibrium density matrix; \u001b= (ˆσx,ˆσy,ˆσz) is the vector of\nthe Pauli matrices; |i/angbracketrightelectron orbital centered on site i;\nand the operator |i/angbracketright/angbracketlefti|⊗\u001bacts in the composite Hilbert\nspaceH=Horb⊗Hspinof electronic orbital and spin\ndegrees of freedom. In this Section, we also discuss how\nin the limit of slow magnetization dynamics the memory\nkernel can be expanded in a Taylor series in order to ex-\ntract conventional Gilbert damping and magnetic inertia\nterms, but with time- and spatially-dependent parame-\ntersλD\ni(t) andID\ni(t). In Secs. III A–III C we compare the\ndynamics of local magnetic moments driven by an exter-\nnal magnetic field as computed by TDNEGF+LLG vs.\nconventional LLG simulations for three one-dimensional\n(1D) examples depicted in Fig. 1(a)–(c), respectively.\nSec. III C also compares pumped charge current due to\nthe DW motion as computed by TDNEGF+LLG vs. the\nwidely-used spin motive force (SMF) theory [12, 53] com-\nbined [54–56] with the conventional LLG equation. We\nconclude in Sec. IV.\nII. MODELS AND METHODS\nA. Coupled quantum and classical Hamiltonians\nThe conduction electron subsystem is modeled by a\nquantum Hamiltonian\nH(t) =−γ/summationdisplay\n/angbracketleftij/angbracketrightˆc†\niˆci−Jsd/summationdisplay\niˆc†\ni\u001b·Mi(t)ˆci, (4)\nwhich is (assumed to be 1D for simplicity) tight-binding\n(TB) model where electron interacts with magnetic mo-\nments localized at sites iand described by the classical\nvector Mi(t) of unit length. Here ˆ c†\ni= (ˆc†\ni↑,ˆc†\ni↓) is a row\nvector containing operators ˆ c†\niσwhich create an electron\nof spinσ=↑,↓at sitei; ˆciis a column vector that con-\ntains the corresponding annihilation operators; γ= 1 eV\nis the nearest neighbor hopping; and Jsdis thes-dex-\nchange coupling parameter between conduction electrons\nand local magnetic moments. The active region of de-\nvices depicted in Fig. 1(a)–(c) consists of 1, 11 and 21\nTB sites, respectively. These are attached to the left (L)\nand right (R) semi-infinite ideal leads modeled by the\nsame Hamiltonian in Eq. (4) but with Jsd= 0 eV. The\nleads are assumed to terminate into macroscopic reser-\nvoirs kept at the same chemical potential since we do not\napply any bias voltage to the devices in Fig. 1(a)–(c).The classical Hamiltonian describing the local mag-\nnetic moments is given by\nH=−J/summationdisplay\nijMi·Mj−µM/summationdisplay\niMi·Bi\next−K/summationdisplay\ni(Mx\ni)2\n−Jsd/summationdisplay\ni/angbracketleft^s/angbracketrighti·Mi,(5)\nwhereJis the Heisenberg exchange coupling parame-\nter;Bi\nextis the applied external magnetic field; Kis the\nmagnetic anisotropy (in the x-direction) and/angbracketleftˆs/angbracketrightiis the\nnonequilibrium electronic spin density computed from\nEq. (3).\nB. Time-retarded damping and magnetic inertia in\nthe LLG equation self-consistently coupled to\nTDNEGF\nThe quantum equation of motion for the nonequilib-\nrium density matrix of electrons [57, 58]\ni~∂\u001aneq(t)\n∂t= [H(t),\u001aneq(t)] +/summationdisplay\np=L,Ri[Πp(t) +Π†\np(t)].\n(6)\nis an example of a master equation for an open (i.e., con-\nnected to macroscopic reservoirs) quantum system [59]\ndue to the presence of the second term on the right hand\nside, in addition to standard terms of the von Neumann\nequation. This term and the density matrix itself can be\nexpressed using TDNEGF formalism [51, 52] as\n\u001aneq(t) =1\niG<(t,t/prime)|t=t/prime, (7)\nΠp(t/prime) =\u0002t/prime\n−∞dt1[G>(t/prime,t1)Σ<\np(t1,t/prime)−\nG<(t/prime,t1)Σ>\np(t1,t/prime)].(8)\nThe central quantities of the TDNEGF formalism are\nthe retarded Gr,σσ/prime\nii/prime(t,t/prime) =−iΘ(t−t/prime)/angbracketleft{ˆciσ(t),ˆci/primeσ/prime(t)}/angbracketright\nand the lesser G<,σσ/prime\nii/prime(t,t/prime) =i/angbracketleftˆc†\ni/primeσ/prime(t/prime)ˆciσ(t)/angbracketrightGreen\nfunctions (GFs) which describe the available density of\nstates and how electrons occupy those states, respec-\ntively. In addition, it is also useful to introduce the\ngreater GF, G>(t,t/prime) = [G<(t/prime,t)]†, and the advanced\nGF,Ga(t,t/prime) = [Gr(t,t/prime)]†. The current matrices Πp(t)\nmake it possible to compute directly [57, 58] charge cur-\nrent\nIp(t) =e\n~Tr[Πp(t)], (9)\nand spin current\nISα\np(t) =e\n~Tr[ˆσαΠp(t)], (10)4\nin the L and R semi-infinite leads. The equation of mo-\ntion for the lesser and greater GFs is given by\ni~∂G>,<(t,t1)\n∂t=H(t)G>,<(t,t1)+\n+∞\u0002\n−∞dt2/bracketleftbigg\nΣr\ntot(t,t2)G>,<(t2,t) +Σ>,<\ntot(t,t2)Ga(t2,t)/bracketrightbigg\n,\n(11)\nwhere Σr,>,<\ntot (t,t2) =/summationtext\np=L,RΣr,>,<\np (t,t2) and\nΣr,>,<\np (t,t2) are the lead self-energy matrices [52, 57, 58].\nThe classical equation of motion for the magnetic mo-\nment localized at site iis the Landau-Lifshitz equation\n∂Mi(t)\n∂t=−gMi(t)×Beff\ni(t), (12)\nwhere the effective magnetic field is\nBeff\ni(t) =−1\nµM∂H/∂MiandµMis the magnitude\nof the magnetic moment [5].\nThe full TDNEGF+LLG framework [50], which we\nalso denote as TDNEGF \u001cLLG, consists of self-\nconsistent combination of Eq.(6) and (12) where one first\nsolves for the nonequilibrium electronic spin density in\nEq.(3), which is then fed into Eq. (12) to propagate local\nmagnetic moments Mi(t) in the next time step. Evolving\n\u001aneq(t) via Eq. (6) requires time step δt= 0.1 fs for nu-\nmerical stability, and we use the same time step to evolve\nLLG or Landau-Lifshitz equations for Mi(t). These up-\ndated local magnetic moments are fed back into the quan-\ntum Hamiltonian of conduction electron subsystem in\nEq. (6). Thus obtained solutions for Mi(t),/angbracketleftˆs/angbracketrighti(t),Ip(t)\nandISαp(t) are numerically exact. For testing the im-\nportance of the self-consistent feedback loop, we also use\nTDNEGF←LLG where TDNEGF is utilized to obtain\nIp(t) andISαp(t) while the local magnetic moments are\nevolved solely by the conventional LLG Eq. (1), i.e., by\nusingJsd≡0 in Eq. (5) but Jsd/negationslash= 0 is used in Eq. (4).\nIn the weak-coupling limit [34, 60] (i.e., small Jsd)\nfor electron-spin/local-magnetic-moment interaction it is\npossible to extract explicitly the generalized LLG equa-\ntion with a memory kernel. For this purpose we use the\nfollowing expansions in the powers of small Jsd\n\u001aneq(t) =∞/summationdisplay\nn=0\u001an(t)Jn\nsd, (13)\nΠp(t/prime) =∞/summationdisplay\nn=0Π(n)\np(t/prime)Jn\nsd, (14)\nGr,a,>,<(t/prime,t1) =∞/summationdisplay\nn=0Gr,a,>,<\nn (t/prime,t1)Jn\nsd. (15)In Appendix A, we show how to combine Eqs.(6), (11),\n(13), (14) and (15) to obtain the perturbative equation\n∂Mi(t)\n∂t=−g/bracketleftbigg\nMi(t)×Beff,0\ni(t)+\nJ2\nsd\nµM/summationdisplay\np=L,RMi(t)×+∞\u0002\n−∞dt/prime/primeMi(t/prime/prime){Kp\ni(t/prime/prime,t)+Kp∗\ni(t/prime/prime,t)}/bracketrightbigg\n,\n(16)\nfor the dynamics of each local magnetic moment at site i,\nby retaining only the terms linear in Jsdin Eqs. (13)–(15).\nHere Beff,0\ni≡−1\nµM∂H0/∂Mi,H0is the classical Hamil-\ntonian in Eq. (5) with Jsd≡0 and Kp\ni(t/prime/prime,t) is defined in\nAppendix A. The physical origin [35] of time-retardation\neffects described by the second term in Eq. (16) is that,\neven though electron dynamics is much faster than the\ndynamics of local magnetic moments, the nonequilibrium\nspin density in Eq. (3) is always behind Mi(t) and, there-\nfore, never parallel to it which introduces spin torque\nterm into the Landau-Lifshitz Eq. (12). In other words it\ntakes finite amount of time for conduction electron spin\nto react to the motion of classical local magnetic mo-\nments, so that nonequilibrium electrons effectively me-\ndiate interaction of Mi(t) with the same local magnetic\nmoment at time t/prime< t. In the full TDNEGF+LLG,\nsuch retardation effects are mediated by the nonequilib-\nrium electrons starting at site iat timet/primeand returning\nback to the same site at time t > t/prime, while in the per-\nturbative limit the same effect is captured by the second\nterm in Eq. (16). The perturbative formula Eq. (16) is\nexpected [35] to breakdown after propagation over time\nt∼~/Jsd.\nFurther approximation to Eq. (16) can be made by\nconsidering sufficiently slow dynamics of local magnetic\nmoments so that higher order terms in the Taylor series\nMi(t/prime/prime)≈Mi(t)+∂Mi(t)\n∂t(t/prime/prime−t)+1\n2∂2Mi(t)\n∂t2(t/prime/prime−t)2+...,\n(17)\ncan be neglected. By defining the following quantities\nλD\np,i(t)≡+∞\u0002\n−∞dt/prime/prime(t/prime/prime−t)[Kp\ni(t/prime/prime,t) + K∗p\ni(t/prime/prime,t)],(18)\nand\nID\np,i(t)≡1\n2+∞\u0002\n−∞dt/prime/prime(t/prime/prime−t)2[Kp\ni(t/prime/prime,t) + K∗p\ni(t/prime/prime,t)],(19)\nand by retaining terms up to the second order in Eq. (17)5\nwe obtain the conventionally looking LLG equation\n∂Mi(t)\n∂t=−g/bracketleftbigg\nMi(t)×Beff,0\ni(t)+\nJ2\nsd\nµM/braceleftbigg/summationdisplay\np=L,RλD\np,n(t)/bracerightbigg\nMi(t)×∂Mi(t)\n∂t+\nJ2\nsd\nµM/braceleftbigg/summationdisplay\np=L,RID\np,i(t)/bracerightbigg\nMi(t)×∂2Mi(t)\n∂t2/bracketrightbigg\n.(20)\nHowever, the Gilbert damping term prefactor\nλD\ni(t) =J2\nsd\nµM/summationdisplay\np=L,RλD\np,i(t), (21)\nand the magnetic inertia term prefactor\nID\ni(t) =J2\nsd\nµM/summationdisplay\np=L,RID\np,i(t), (22)\nin Eq. (20) are now time- and position-dependent. This\nis in sharp contrast to conventional LLG Eq. (1) em-\nployed in classical micromagnetics where Gilbert damp-\ning and magnetic inertia prefactors are material specific\nconstants.\nIII. RESULTS AND DISCUSSION\nA. Single local magnetic moment in an external\nmagnetic field\nTo compare the dynamics of local magnetic moments\nin full TDNEGF+LLG quantum-classical simulations vs.\nconventional LLG classical simulations, we first consider\na well-known example [5] for which the conventional LLG\nequation can be analytically solved—a single local mag-\nnetic moment which at t= 0 points along the + x-\ndirection and then starts to precesses due to an external\nmagnetic field pointing in the + z-direction. Its trajectory\nis given by [5]\nMx(t) = sech/parenleftbigggλGB\n1 +λGt/parenrightbigg\ncos/parenleftbigggB\n1 +λ2\nGt/parenrightbigg\n,(23a)\nMy(t) = sech/parenleftbigggλGB\n1 +λGt/parenrightbigg\nsin/parenleftbigggB\n1 +λ2\nGt/parenrightbigg\n,(23b)\nMz(t) = tanh/parenleftbigggλGB\n1 +λGt/parenrightbigg\n, (23c)\nwhere B= (0,0,B) is the applied external mag-\nnetic field. Thus, if the conventional intrinsic Gilbert\ndamping parameter is set to zero, λG= 0, then\nthe local magnetic moment precesses steadily around\nthez-axis with Mz≡0. On the other hand,\nfor nonzero λG>0, the local magnetic moment\nFIG. 2. (a) Time dependence of tanh−1(Mz) for a sin-\ngle local magnetic moment in Fig. 1(a) obtained from TD-\nNEGF+LLG simulations. Colors red to blue indicate in-\ncreasings-dexchange coupling in steps of 0 .1 eV, ranging\nfromJsd= 0 eV toJsd= 1.9 eV. (b) The dynamical Gilbert\ndamping parameter in Eq. (21) extracted from panel (a) as\na function of Jsd. (c) Time dependence of Mzcomponent\nfor a single local magnetic moment in Fig. 1(a) at large\nJsd= 2.0 eV exhibits nutation as a signature of magnetic in-\nertia. To generate fast magnetization dynamics and reduce\nsimulation time, we use an unrealistically large external mag-\nnetic field of strength B= 1000 T. The conventional intrinsic\nGilbert damping parameter is set to zero, λG= 0, and the\nFermi energy is EF= 0 eV.\nwill relax towards the direction of magnetic field,\ni.e., lim\nt→∞(Mx(t),My(t),Mz(t)) = (0,0,1). Thus, such\ndamped dynamics is signified by a linear tanh−1(Mz)\nvs. time dependence. Figure 2 plots results of TD-\nNEGF+LLG simulations for the same problem. Even\nthough we set conventional intrinsic Gilbert damping\nto zero,λG= 0, Fig. 2(a) shows linear tanh−1(Mz) vs.\ntime, independently of the strength of s-dexchange cou-\npling as long as Jsd.2 eV. This means that the lo-\ncal magnetic moment is experiencing (time-independent)\ndynamical Gilbert damping λD∝J2\nsd, in accord with\nEq. (21) and as shown in Fig. 2(b), which is generated\nsolely by the TDNEGF part of the self-consistent loop\nwithin the full TDNEGF+LLG scheme.\nForJsd&2 eV, the dynamics of the local magnetic mo-\nment also exhibits nutation [35], as shown in Fig. 2(c),\nwhich is the signature of the magnetic inertia [19–24]\nterm∝Mi×∂2Mi/∂t2in Eq. (20). Thus, nutation be-\ncomes conspicuous when the dynamics of the local mag-\nnetic moments is sufficiently fast, so that ∂2Mi/∂t2is\nlarge, as well as when the interaction between the itiner-\nant and localized spins is sufficiently large.6\nFIG. 3. TDNEGF+LLG-computed trajectories\n(Mx(t),My(t),Mz(t)) on the Bloch sphere of local magnetic\nmoment in the setup of Fig. 1(b) at: (a) site 1; and (c) site\n6. The total number of local magnetic moments is N= 11,\nand they do not interact with each other via exchange\ncoupling [i.e., J= 0 eV in Eq. (5)]. Panels (b) and (d) show\nthe corresponding time dependence of Mzcomponent from\npanels (a) and (c), respectively. The external magnetic\nfield isB= 1000 T, and the s-dexchange coupling strength\nJsd= 0.1 eV is nonperturbative in this setup, therefore,\nnotallowing us to extract explicitly the dynamical Gilbert\ndamping parameter from Eq. (21). The conventional intrinsic\nGilbert damping parameter is set to zero, λG= 0, and the\nFermi energy is EF= 0 eV.\nB. Multiple exchange-uncoupled local magnetic\nmoments in an external magnetic field\nIn order to examine possible spatial dependence of the\ndynamical Gilbert damping parameter or emergence of\ndynamical exchange coupling [61, 62] between local mag-\nnetic moments, we consider a chain of N= 11 magnetic\nmoments which do not interact with each other ( J= 0)\nbut interact with conduction electron spin ( Jsd/negationslash= 0), as\nillustrated in Fig. 1(b). At t= 0, all magnetic moments\npoint in the + x-direction while the external magnetic\nfield is in the + zdirection, and the conventional intrin-\nsic Gilbert damping is set to zero, λG= 0.\nFigures 3(a) and 3(c) show the trajectory of selected\nlocal magnetic moments ( i= 1 and 6) on the Bloch\nsphere forJsd= 0.1 eV. In contrast to single local mag-\nnetic moment in Fig. 2(a), for which tanh−1(Mz) vs.\ntime is linear using Jsd= 0.1 eV, we find that in case of\nmultiple exchange-uncoupled magnetic moments this is\nno longer the case, as demonstrated by Figs. 3(b) and\n3(d). Hence, the trajectory followed by these local mag-\nnetic moments cannot be described by Eq. (23) so that\nFIG. 4. (a) TDNEGF+LLG-computed time dependence of\nMzcomponent of local magnetic moment on sites 1, 3 and 6 in\nthe setup of Fig. 1(b) with a total of N= 11 moments. (b) Po-\nsition dependence of the dynamical Gilbert damping param-\neter in Eq. (21). The external magnetic field is B= 1000 T,\nand thes-dexchange coupling strength Jsd= 0.01 eV is per-\nturbative in this setup, therefore, allowing us to extract the\ndynamical Gilbert damping explicitly from Eq. (21). The con-\nventional intrinsic Gilbert damping parameter is set to zero,\nλG= 0, and the Fermi energy is EF= 0 eV.\nthe conventional-like Gilbert damping parameter cannot\nbe extracted anymore. Thus, such a nonstandard damp-\ning of the dynamics of local magnetic moments originates\nfrom time-dependence of the dynamical damping param-\neterλD\niin Eq. (21).\nFigure 4(a) shows tanh−1(Mz) vs. time for selected\nlocal magnetic moments ( i= 1,3 and 6) and smaller\nJsd= 0.01 eV. Although all local magnetic moments fol-\nlow linear tanh−1(Mz) vs. time, as predicted by the\nsolution in Eq. (23c) of the conventional LLG equation,\nthe dynamical Gilbert damping extracted from Eq. (23)\nchanges from site to site as shown in Fig. 4(b). Further-\nmore, the linear tanh−1(Mz) vs. time relation breaks\ndown for times t&50 ps at specific sites, which then pre-\nvents extracting time-independent λD\niat those sites.\nC. Magnetic field-driven motion of a domain wall\ncomposed of multiple exchange-coupled local\nmagnetic moments\nIn order to examine difference in predicted dynam-\nics of exchange-coupled local magnetic moments by TD-\nNEGF+LLG framework vs. conventional LLG equa-\ntion, we consider the simplest example of 1D head-to-\nhead magnetic DW depicted in Fig. 1(c). Its motion is\ndriven by applying an external magnetic field in the + x-\ndirection. Some type of damping mechanism is crucial\nfor the DW to move, as demonstrated by solid lines in\nFig. 5(e)–(h), obtained by solving the conventional LLG\nequation with λG= 0, which show how local magnetic\nmoments precess around the magnetic field but without\nnet displacement of the center of the DW.\nOn the other hand, even though we set λG= 0 in TD-\nNEGF+LLG simulations in Fig. 5(a)–(d), the center of\nthe DW moves to the right due to dynamically generated7\nFIG. 5. (a)–(d) TDNEGF+LLG-computed snapshots of head-to-head DW in the setup of Fig. 1(c) driven by an external\nmagnetic field of strength B= 100 T pointing in the + x-direction, in the absence ( λG= 0) or presence ( λG= 0.01) of the\nconventional intrinsic Gilbert damping. Panels (e)–(h) show the corresponding snapshots computed solely by the conventional\nLLG Eq. (1) where in the absence ( λG= 0) of the conventional intrinsic Gilbert damping the DW does not move at all. The\nHeisenberg exchange coupling between local magnetic moments is J= 0.01 eV;s-dexchange coupling between electrons and\nlocal magnetic moments is Jsd= 0.1 eV; magnetic anisotropy (in the x-direction) is K= 0.01 eV; and the Fermi energy of\nelectrons is EF=−1.9 eV. The magnetic field is applied at t= 2 ps, while prior to that we evolve the conduction electron\nsubsystem with TDNEGF until it reaches the thermodynamic equilibrium where all transient spin and charge currents have\ndecayed to zero.\ntime-retarded damping encoded by the memory kernel\nin Eq. (16). Including the conventional intrinsic Gilbert\ndamping,λG= 0.01 as often used in micromagnetic sim-\nulations of DW along magnetic nanowires [43, 44, 63],\nchanges only slightly the result of TDNEGF+LLG sim-\nulations which demonstrates that the effective dynam-\nical Gilbert damping (which is also time-dependent) is\nabout an order of magnitude larger than λG. This is also\nreflected in the DW velocity being much larger in TD-\nNEGF+LLG simulations with λG= 0 in Fig. 5(a)–(d)\nthan in the conventional LLG equation simulations with\nλG= 0.01 in Fig. 5(e)–(h).\nIt has been predicted theoretically [12, 53, 64–68] and\nconfirmed experimentally [69] that a moving DW will\npump charge current even in the absence of any applied\nbias voltage. The corresponding open circuit pumping\nvoltage in the so-called spin motive force (SMF) the-\nory [12, 53] is given by\nVSMF=1\nG0\u0002\njxdx, (24a)\njα(r) =Pσ0~\n2e[∂tm(r,t)×∂αm(r,t)]·m(r,t),(24b)\nwherejxis the pumped local charge current along the\nx-axis. Here σ0=σ↑+σ↓is the total conductivity;\nP= (σ↑−σ↓)/(σ↑+σ↓) is the spin polarization of the\nferromagnet; and ∂t=∂/∂t. Equation (24) is typicallycombined [54–56] with classical micromagnetics which\nsupplies Mi(t) that is then plugged into the discretized\nversion [50]\njx(i)∝1\na[∂tMi(t)×(Mi+1(t)−Mi(t))]·Mi(t)\n∝1\na[∂Mi(t)×Mi+1(t)]·Mi(t). (25)\nof Eq. (24b). We denote this approach as SMF ←LLG,\nwhich is perturbative in nature [67, 70] since it considers\nonly the lowest temporal and spatial derivatives.\nOn the other hand, the same pumping voltage can be\ncomputed nonperturbatively\nVTDNEGF =Ip(t)\nG(t), (26)\nusing TDNEGF expression for charge current in lead p\nin Eq. 9, where TDNEGF calculations are coupled to\nLLG calculations either self-consistently (i.e., by using\nTDNEGF\u001cLLG) or non-self-consistently (i.e., by us-\ning TDNEGF←LLG). Here, G(t) is the conductance\ncomputed using the Landauer formula applied to two-\nterminal devices with a frozen at time ttexture of local\nmagnetic moments.\nFigures 6(a) and 6(b) plot the pumping voltage cal-\nculated by TDNEGF \u001cLLG for DW motion shown in\nFig. 5(a)–(d) in the absence or presence of conventional8\nFIG. 6. Time dependence of pumping voltage generated by\nthe DW motion depicted in Fig. 5(a)–(d) for: (a) λG= 0;\n(b)λG= 0.01. In panels (a) and (b) local magnetic mo-\nments evolve in time by the full TDNEGF+LLG framework\nwhere the arrows indicate how TDNEGF sends nonequilib-\nrium electronic spin density into the LLG equation which, in\nturn, sends trajectories of local magnetic moments into TD-\nNEGF. Time dependence of pumping voltage generated by\nDW motion depicted in Fig. 5(e)–(h) for: (c) λG= 0; (d)\nλG= 0.01. In panels (c) and (d) local magnetic moments\nevolve in time using the conventional LLG equation which\nsends their trajectories into either TDNEGF (green) or SMF\nformulas (blue) in Eq. (26) or Eq. (24), respectively, to obtain\nthe corresponding pumping voltage.\nGilbert damping, respectively. The two cases are virtu-\nally identical due to an order of magnitude larger dynam-\nical Gilbert damping that is automatically generated by\nTDNEGF\u001cLLG in both Figs. 6(a) and 6(b). The\nnonperturbative results in Figs. 6(a) and Fig. 6(b) are\nquite different from SMF ←LLG predictions in Figs. 6(c)\nand Fig. 6(d), respectively. This is due to both failure\nof Eqs. (24) and (25) to describe noncoplanar and non-\ncollinear magnetic textures with neighboring local mag-\nnetic moments tilted by more than 10◦[50] and lack\nof dynamical Gilbert damping in SMF ←LLG simula-\ntions [54–56]. The latter effect is also emphasized by the\ninability of TDNEGF ←LLG in Figs. 6(c) and Fig. 6(d)\nto reproduce the results of self-consistent TDNEGF \u001c\nLLG in Figs. 6(a) and Fig. 6(b), respectively.\nIV. CONCLUSIONS\nIn conclusion, we delineated a hierarchy of theoret-\nical descriptions of a nonequilibrium quantum many-\nbody system in which conduction electron spins inter-\nact with local magnetic moments within a ferromagneticlayer sandwiched between normal metal electrodes. On\nthe top of the hierarchy is a fully quantum approach,\nfor both electrons and local magnetic moments, whose\ncomputational complexity (using either original spin op-\nerators [71, 72] for local magnetic moments, or their\nmapping to bosonic operators in order to enable ap-\nplication of many-body perturbation theory within the\nNEGF formalism [73]) makes it impractical for systems\ncontaining large number of local magnetic moments.\nThe next approach in the hierarchy is computation-\nally much less expensive quantum-classical hybrid [74]\nbased on self-consistent coupling [50] of TDNEGF (which\ncan be implemented using algorithms that scale linearly\nwith both system size and simulation time [52, 58, 75])\nwith classical LLG equation for local magnetic moments.\nSuch TDNEGF+LLG approach is numerically exact and,\ntherefore, nonperturbative in the strength of electron-\nspin/local-magnetic-moment interaction, speed of local\nmagnetic moment dynamics and degree of noncollinearity\nbetween them. Even though electron dynamics is much\nfaster than localized spin dynamics, the most general sit-\nuation cannot be handled by integrating out [6, 34] the\nconduction electron degrees of freedom and by focusing\nonly on the LLG-type equation where a much larger time\nstep can be used to propagate spins only.\nNevertheless, in the limit [34, 60] of weak electron-\nspin/local-magnetic-moment interaction [i.e., small Jsd\nin Eqs. (4) and (5)] one can derive analytically a type\nof generalized LLG equation [34–37] for each local mag-\nnetic moment which is next approach in the hierarchy\nthat sheds light onto different effects included in the nu-\nmerically exact TDNEGF+LLG scheme. Instead of the\nconventional Gilbert damping term in Eq. (1), the gen-\neralized LLG equation we derive as Eq. (16) contains\na microscopically determined memory kernel which de-\nscribes time-retardation effects generated by the coupling\nto TDNEGF. Fundamentally, the memory kernel is due\nto the fact that electron spin can never follow instanta-\nneously change in the orientation of the local magnetic\nmoments [35]. In the limit of slow dynamics of local\nmagnetic moments, one can further expand the memory\nkernel into a Taylor series to obtain the final approach\nwithin the hierarchy whose LLG Eq. (20) is akin to the\nconventional one, but which contains both Gilbert damp-\ning (proportional to first time derivative of local mag-\nnetization) and magnetic inertia terms (proportional to\nsecond time derivative of local magnetization) with time-\ndependent parameters instead of usually assumed mate-\nrials specific constants.\nUsing three simple examples—single or multiple local\nmagnetic moments precessing in an external magnetic\nfield or magnetic-field-driven magnetic DW motion—\nwe demonstrate the importance of dynamically induced\ndamping which operates even if conventional static\nGilbert damping is set to zero. In the case of field-\ndriven magnetic DW motion, we can estimate that the\nstrength of dynamical damping is effectively an order of\nmagnitude larger than typically assumed [43, 44, 63] con-9\nventional static Gilbert damping λG/similarequal0.01 in classical\nmicromagnetic simulations of magnetic nanowires. In ad-\ndition, we show that charge pumping by the dynamics of\nnoncoplanar and noncollinear magnetic textures, which\nis outside of the scope of pure micromagnetic simulations\nbut it is often described by combining [54–56] them with\nthe SMF theory formula [12, 53], requires to take into ac-\ncount both the dynamical Gilbert damping and possiblylarge angle between neighboring local magnetic moments\nin order to reproduce numerically exact results of TD-\nNEGF+LLG scheme.\nACKNOWLEDGMENTS\nThis work was supported by NSF Grant No. ECCS\n150909.\nAppendix A: Derivation of Memory Kernel in LLG equation self-consistently coupled to TDNEGF\nIn this Appendix, we provide a detailed derivation of the memory kernel in Eq. (16). To obtain the perturbative\nequation of motion for local magnetic moments we start from Landau-Lifshitz Eq. (12) where the effective magnetic\nfield can be written as\nBeff\ni(t) =Beff,0\ni(t) +Jsd/angbracketleftˆs/angbracketrighti(t). (A.1)\nThe nonequilibrium spin density is expanded up to terms linear in Jsdusing Eq. (13)\n/angbracketleftˆs/angbracketrighti(t) =~\n2Tr[\u001aneq(t)|i/angbracketright/angbracketlefti|⊗\u001b]−/angbracketleftˆs/angbracketrighti\neq≈~\n2Tr/bracketleftbigg\n{\u001a0(t)+Jsd\u001a1(t)}|i/angbracketright/angbracketlefti|⊗\u001b/bracketrightbigg\n−/angbracketleftˆs/angbracketrighti\neq=Jsd~\n2Tr[\u001a1(t)|i/angbracketright/angbracketlefti|⊗\u001b]−/angbracketleftˆs/angbracketrighti\neq.\n(A.2)\nHere/angbracketleftˆs/angbracketrighti\neqis the equilibrium electronic spin density i.e., /angbracketleftˆs/angbracketrighti\neq= (~/2) Tr [ \u001aeq|i/angbracketright/angbracketlefti|⊗\u001b]. Furthermore, the electronic\nspin density in the zeroth order must vanish, i.e., Tr [ \u001a0(t)|i/angbracketright/angbracketlefti|⊗\u001b] = 0 since for Jsd= 0 electrons are not spin-\npolarized. Hence, we can write Eq. (12) as\n∂Mi(t)\n∂t=−gMi(t)×/bracketleftbigg\nBeff,0\ni(t) +J2\nsd~\n2Tr[\u001a1(t)|i/angbracketright/angbracketlefti|⊗\u001b]−Jsd/angbracketleftˆs/angbracketrighti\neq/bracketrightbigg\n. (A.3)\nTo obtain analytical results, we assume that the equilibrium spin density follows the direction of local magnetic\nmoments, so that Mi(t)×/angbracketleftˆs/angbracketrighti\neq= 0. By expanding Eq. (6) we obtain\ni~∂\u001a0(t)\n∂t= [H0(t),\u001a0(t)] +/summationdisplay\np=L,Ri[Π(0)\np(t) +Π(0)†\np(t)], (A.4)\nand\ni~∂\u001a1(t)\n∂t= [H1(t),\u001a0(t)] +i/summationdisplay\np=L,R[Π(1)\np(t) +Π(1)†\np(t)], (A.5)\nwhere H1(t) =−/summationtext\ni|i/angbracketright/angbracketlefti|⊗\u001b·Mi(t). One can formally integrate Eq. (A.5) which leads to\n~\n2Tr[\u001a1(t)|i/angbracketright/angbracketlefti|⊗\u001b] =/summationdisplay\np=L,R1\n2t\u0002\n−∞dt/primeTr/bracketleftbigg\n{Π(1)\np(t/prime) +Π(1)†\np(t/prime)}|i/angbracketright/angbracketlefti|⊗\u001b/bracketrightbigg\n. (A.6)\nwhich requires to find an expression for Π(1)\np(t/prime). Using Eq. (8) and the fact that lead self-energy matrices do not\ndepend onJsdleads to\nΠ(1)\np(t/prime) =t/prime\u0002\n−∞dt1[G>\n1(t/prime,t1)Σ<\np(t1,t/prime)−G<\n1(t/prime,t1)Σ>\np(t1,t/prime)]. (A.7)\nEquations (11) and (15) can be formally integrated to yield lesser and greater GFs in Eq. (A.7)\nG>,<\n1(t/prime,t1) =1\ni~/parenleftbiggt/prime\u0002\n−∞dt/prime/primeH1(t/prime/prime)G>,<\n0(t/prime/prime,t1) +t/prime\u0002\n−∞dt/prime/prime+∞\u0002\n−∞dt2/bracketleftbigg\nΣr\ntot(t/prime/prime,t2)G>,<\n1(t2,t/prime/prime) +Σ>,<\ntot(t/prime/prime,t2)Ga\n1(t2,t/prime/prime)/bracketrightbigg/parenrightbigg\n.\n(A.8)10\nWe further assume that the active region in Fig. 1 is weakly coupled with semi-infinite leads and, therefore, macroscopic\nreservoirs into which they terminate. This means that after we substitute Eq. (A.8) into Eq. (A.7) we can keep only\nthose terms that are linear in the self-energy\nΠ(1)\np(t/prime) =1\n2it/prime\u0002\n−∞dt/prime/primeH1(t/prime/prime)t/prime\u0002\n−∞dt1/bracketleftbigg\nG>\n0(t/prime/prime,t1)Σ<\np(t1,t/prime)−G<\n0(t/prime/prime,t1)Σ>\np(t1,t/prime)/bracketrightbigg\n(A.9)\n=i\n2/summationdisplay\nit/prime\u0002\n−∞dt/prime/prime|i/angbracketright/angbracketlefti|⊗\u001b·Mi(t/prime/prime)t/prime\u0002\n−∞dt1/bracketleftbigg\nG>\n0(t/prime/prime,t1)Σ<\np(t1,t/prime)−G<\n0(t/prime/prime,t1)Σ>\np(t1,t/prime)/bracketrightbigg\n(A.10)\n=i/summationdisplay\nit/prime\u0002\n−∞dt/prime/prime|i/angbracketright/angbracketlefti|⊗\u001b·Mi(t/prime/prime)A0\np(t/prime/prime,t/prime), (A.11)\nwhere A0\np(t/prime/prime,t/prime) is an operator constructed out of the zeroth order terms in the expansion of GFs shown in Eq. (15)\nA0\np(t/prime/prime,t/prime)≡i\n2t/prime\u0002\n−∞dt1/bracketleftbigg\nG>\n0(t/prime/prime,t1)Σ<\np(t1,t/prime)−G<\n0(t/prime/prime,t1)Σ>\np(t1,t/prime)/bracketrightbigg\n. (A.12)\nBy plugging in Eqs. (A.11) and (A.12) into Eq. (A.6) we obtain\n~\n2Tr[\u001a1(t)|i/angbracketright/angbracketlefti|⊗ˆσµ] =/summationdisplay\np=L,R/summationdisplay\nj/summationdisplay\nνt\u0002\n−∞dt/primet/prime\u0002\n−∞dt/prime/primeMν\nj(t/prime/prime) Tr/bracketleftbigg\n|j/angbracketright/angbracketleftj|⊗ˆσν{A0\np(t/prime/prime,t/prime)+A0†\np(t/prime/prime,t/prime)}|i/angbracketright/angbracketlefti|⊗σµ/bracketrightbigg\n.(A.13)\nSince A0\np(t/prime/prime,t/prime) is an operator constructed from the zeroth order GFs, it can be written in the followin form\nA0\np(t/prime/prime,t/prime) =1\n2/summationdisplay\nmnAp\nmn(t/prime/prime,t/prime)|m/angbracketright/angbracketleftn|⊗12, (A.14)\nwhere 12is a 2×2 identity matrix. Using this it is easy to show that\n~\n2Tr[\u001a1(t)|i/angbracketright/angbracketlefti|⊗\u001b] =/summationdisplay\np=L,R+∞\u0002\n−∞Θ(t−t/prime)dt/prime+∞\u0002\n−∞dt/prime/primeMi(t/prime/prime){Ap\nii(t/prime/prime,t/prime) + Ap∗\nii(t/prime/prime,t/prime)}Θ(t/prime−t/prime/prime), (A.15)\nand\nKp\ni(t/prime/prime,t) =+∞\u0002\n−∞dt/primeΘ(t−t/prime)Θ(t/prime−t/prime/prime)Ap\nii(t/prime/prime,t/prime). (A.16)\nBy plugging in Eq. (A.15) into the second term on the right hand side of Eq. (A.3), we finally obtain Eq. (16) of the\nmain text.\n[1] G. Bertotti, I. D. Mayergoyz, and C. Serpico, Nonlinear\nmagnetization dynamics in nanosystems (Elsevier, Ams-\nterdam, 2009).\n[2] R. Wieser, Phys. Rev. Lett. 110, 147201 (2013).\n[3] R. Wieser, Euro. Phys. J. B 88, 77 (2015).\n[4] D. Kumar and A. O. Adeyeye, J. Phys. D: Appl. Phys.50, 343001 (2017).\n[5] R. F. L. Evans, W. J. Fan, P. Chureemart, T. A. Ostler,\nM. O. A. Ellis, and R. W. Chantrell, J. 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B 93, 134506\n(2016)."    },    {        "title": "1408.6261v2.Stability_of_an_abstract_wave_equation_with_delay_and_a_Kelvin_Voigt_damping.pdf",        "content": "arXiv:1408.6261v2  [math.AP]  15 May 2015Stability of an abstract–wave equation with\ndelay and a Kelvin–Voigt damping\nKa¨ ıs AMMARI∗, Serge NICAISE†and Cristina PIGNOTTI‡\nAbstract. In this paper we consider a stabilization problem for an abstract wav e equation\nwith delay and a Kelvin–Voigt damping. We prove an exponential stabilit y result for appropri-\nate damping coefficients. The proofofthe main result is based on a fr equency–domainapproach.\n2010 Mathematics Subject Classification : 35B35, 35B40, 93D15, 93D20.\nKeywords : Internal stabilization, Kelvin-Voigt damping, abstract wave equation with\ndelay.\n1 Introduction\nOur main goal is to study the internal stabilization of a dela yed abstract wave equation\nwith a Kelvin–Voigt damping. More precisely, given a consta nt time delay τ >0,we\nconsider the system given by:\nu′′(t)+aBB∗u′(t)+BB∗u(t−τ) = 0, in (0,+∞),(1.1)\nu(0) =u0, u′(0) =u1, (1.2)\nB∗u(t−τ) =f0(t−τ), in (0,τ), (1.3)\nwherea >0 is a constant, B:D(B)⊂H1→His a linear unbounded operator from\na Hilbert space H1into another Hilbert space Hequipped with the respective norms\n|| · ||H1,|| · ||Hand inner products ( ·,·)H1, (·,·)H, andB∗:D(B∗)⊂H→H1is the\nadjoint of B. The initial datum ( u0,u1,f0) belongs to a suitable space.\nWe supposethat the operator B∗satisfies the following coercivity assumption: there\nexistsC >0 such that\n/ba∇dblB∗v/ba∇dblH1≥C/ba∇dblv/ba∇dblH,∀v∈D(B∗). (1.4)\nFor shortness we set V=D(B∗) and we assume that it is closed with the norm /ba∇dblv/ba∇dblV:=\n/ba∇dblB∗v/ba∇dblH1and that it is compactly embedded into H.\n∗UR Analysis and Control of Pde, UR 13ES64, Department of Math ematics, Faculty of Sciences of\nMonastir, University of Monastir, 5019 Monastir, Tunisia, e-mail : kais.ammari@fsm.rnu.tn\n†Universit´ e de Valenciennes et du Hainaut Cambr´ esis, LAMA V, FR CNRS 2956, 59313 Valenciennes\nCedex 9, France, e-mail: snicaise@univ-valenciennes.fr\n‡Dipartimento di Ingegneria e Scienze dell’Informazione e M atematica, Universit` a di L’Aquila, Via\nVetoio, Loc. Coppito, 67010 L’Aquila, Italy, e-mail : pigno tti@univaq.it\n1Delay effects arise in many applications and practical proble ms and it is well–known\nthat an arbitrarily small delay may destroy the well–posedn ess of the problem [16,\n12, 19, 20] or destabilize a system which is uniformly asympt otically stable in absence\nof delay (see e.g. [9, 11], [17], [20]). Different strategies w ere recently developed to\nrestitute either the well–posedness or the stability. In th e first case, one idea is to\nadd a non–delay term, see [7, 19] for the heat equation. In the second case, we refer\nto [2, 5, 10, 17, 18] for stability results for systems with ti me delay where a standard\nfeedback compensating the destabilizing delay effect is intr oduced. Nevertheless recent\npapers reveal that particular choices of the delay may resti tute exponential stability\nproperty, see [13, 4].\nNote that the above system is exponentially stable in absenc e of time delay, and if\na >0. On the other hand if a= 0 and −BB∗corresponds to the Laplace operator with\nDirichlet boundary conditions in a bounded domain of Rn, problem (1.1)–(1.3) is not\nwell–posed, see [16, 12, 19, 20]. Therefore in this paper in o rder to restitute the well-\nposedness character and its stability we propose to add the K elvin–Voigt damping term\naBB∗u′. Hence the stabilization of problem (1.1)–(1.3) is perform ed using a frequency\ndomain approach combined with a precise spectral analysis.\nThe paper is organized as follows. The second section deals w ith the well–posedness\nof the problem while, in the third section, we perform the spe ctral analysis of the\nassociated operator. In section 4, we prove the exponential stability of the system\n(1.1)–(1.3) if τ≤a. In the last section we give an example of an application.\n2 Existence results\nIn this section we will give a well–posedness result for prob lem (1.1)–(1.3) by using\nsemigroup theory.\nInspired from [17], we introduce the auxiliary variable\nz(ρ,t) =B∗u(t−τρ), ρ∈(0,1), t >0. (2.1)\nThen, problem (1.1)–(1.3) is equivalent to\nu′′(t)+aBB∗u′(t)+Bz(1,t) = 0, in (0,+∞), (2.2)\nτzt(ρ,t)+zρ(ρ,t) = 0 in (0 ,1)×(0,+∞),(2.3)\nu(0) =u0, u′(0) =u1, (2.4)\nz(ρ,0) =f0(−ρτ), in (0,1), (2.5)\nz(0,t) =B∗u(t), t > 0. (2.6)\nIf we denote\nU:=/parenleftbig\nu,u′,z/parenrightbig⊤,\nthen\nU′:=/parenleftbig\nu′,u′′,zt/parenrightbig⊤=/parenleftbig\nu′,−aBB∗u′−Bz(1,t),−τ−1zρ/parenrightbig⊤.\nTherefore, problem (2.2)–(2.6) can be rewritten as\n/braceleftbiggU′=AU,\nU(0) = (u0,u1,f0(−·τ))⊤,(2.7)\n2where the operator Ais defined by\nA\nu\nv\nz\n:=\nv\n−aBB∗v−Bz(·,1)\n−τ−1zρ\n,\nwith domain\nD(A) :=/braceleftBig\n(u,v,z)⊤∈D(B∗)×D(B∗)×H1(0,1;H1) :aB∗v+z(1)∈D(B),\nB∗u=z(0)/bracerightBig\n,\n(2.8)\nin the Hilbert space\nH:=D(B∗)×H×L2(0,1;H1), (2.9)\nequipped with the standard inner product\n((u,v,z),(u1,v1,z1))H= (B∗u,B∗u1)H1+(v,v1)H+ξ/integraldisplay1\n0(z,z1)H1dρ,\nwhereξ >0 is a parameter fixed later on.\nWe will show that Agenerates a C0semigroup on Hby proving that A −cIdis\nmaximal dissipative for an appropriate choice of cin function of ξ,τanda. Namely we\nprove the next result.\nLemma 2.1. Ifξ >2τ\na, then there exists a∗>0such that A −a−1\n∗Idis maximal\ndissipative in H.\nProof.TakeU= (u,v,z)T∈D(A).Then we have\n(A(u,v,z),(u,v,z))H= (B∗v,B∗u)H1−((B(aB∗v+z(1)),v)H\n−ξτ−1/integraldisplay1\n0(zρ,z)H1dρ.\nHence, we get\n(A(u,v,z),(u,v,z))H= (B∗v,B∗u)H1−(aB∗v+z(1),B∗v)H1\n−ξ\n2τ/ba∇dblz(1)/ba∇dbl2\nH1+ξ\n2τ/ba∇dblz(0)/ba∇dbl2\nH1.\nHence reminding that z(0) =B∗uand using Young’s inequality we find that\nℜ(A(u,v,z),(u,v,z))H\n≤(ε−a)/ba∇dblB∗v/ba∇dbl2\nH1+(1\n2ε−ξ\n2τ)/ba∇dblz(1)/ba∇dbl2\nH1+(1\n2ε+ξ\n2τ)/ba∇dblB∗u/ba∇dbl2\nH1.\nChosing ε=a\n2, we find that\nℜ(A(u,v,z),(u,v,z))H≤ −a\n2/ba∇dblB∗v/ba∇dbl2\nH1+(1\na−ξ\n2τ)/ba∇dblz(1)/ba∇dbl2\nH1+(1\na+ξ\n2τ)/ba∇dblB∗u/ba∇dbl2\nH1.\n3The choice of ξis equivalent to1\na−ξ\n2τ<0, and therefore for a∗=/parenleftBig\n1\na+ξ\n2τ/parenrightBig−1\n,\nℜ(A(u,v,z),(u,v,z))H≤ −a\n2/ba∇dblB∗v/ba∇dbl2\nH1+(1\na−ξ\n2τ)/ba∇dblz(1)/ba∇dbl2\nH1+a−1\n∗/ba∇dblB∗u/ba∇dbl2\nH1.(2.10)\nAs/ba∇dblB∗u/ba∇dbl2\nH1≤ /ba∇dbl(u,v,z)/ba∇dbl2\nH, we get\nℜ((A−a−1\n∗Id)(u,v,z),(u,v,z))H≤ −a\n2/ba∇dblB∗v/ba∇dbl2\nH1+(1\na−ξ\n2τ)/ba∇dblz(1)/ba∇dbl2\nH1≤0,\nwhich directly leads to the dissipativeness of A−a−1\n∗Id.\nLet us go on with the maximality, namely let us show that λI−Ais surjective for\na fixedλ >0.Given (f,g,h)T∈ H,we look for a solution U= (u,v,z)T∈D(A) of\n(λI−A)\nu\nv\nz\n=\nf\ng\nh\n, (2.11)\nthat is, verifying\n\nλu−v=f,\nλv+B(aB∗v+z(1)) =g,\nλz+τ−1zρ=h.(2.12)\nSuppose that we have found uwith the appropriate regularity. Then,\nv=λu−f (2.13)\nand we can determine z.Indeed, by (2.8),\nz(0) =B∗u, (2.14)\nand, from (2.12),\nλz(ρ)+τ−1zρ(ρ) =h(ρ) forρ∈(0,1). (2.15)\nThen, by (2.14) and (2.15), we obtain\nz(ρ) =B∗ue−λρτ+τe−λρτ/integraldisplayρ\n0h(σ)eλστdσ. (2.16)\nIn particular, we have\nz(1) =B∗ue−λτ+z0, (2.17)\nwithz0∈H1defined by\nz0=τe−λτ/integraldisplay1\n0h(σ)eλστdσ. (2.18)\nThis expression in (2.12) shows that the function uverifies formally\nλ2u+B(aB∗(λu−f)+B∗ue−λτ+z0) =g+λf,\nthat is,\nλ2u+(λa+e−λτ)BB∗u=g+λf+B(aB∗f)−Bz0. (2.19)\n4Problem (2.19) can be reformulated as\n(λ2u+(λa+e−λτ)BB∗u,w)H= (g+λf+B(aB∗f)−Bz0,w)H,∀w∈V.(2.20)\nUsing the definition of the adjoint of B, we get\nλ2(u,w)H+(λa+e−λτ)(B∗u,B∗w)H1= (g+λf,w)H+(aB∗f−z0,B∗w)H1,∀w∈V.\n(2.21)\nAs the left-hand sideof (2.21) is coercive on D(B∗), the Lax–Milgram lemma guarantees\nthe existence and uniqueness of a solution u∈Vof (2.21). Once uis obtained we define\nvby (2.13) that belongs to Vandzby (2.16) that belongs to H1(0,1;H1). Hence we\ncan setr=aB∗v+z(1), it belongs to H1but owing to (2.21), it fulfils\nλ(v,w)H+(r,B∗w)H1= (g,w)H,∀w∈D(B∗),\nor equivalently\n(r,B∗w)H1= (g−λv,w)H,∀w∈D(B∗).\nAsg−λv∈H, this implies that rbelongs to D(B) with\nBr=g−λv.\nThis shows that the triple U= (u,v,z) belongs to D(A) and satisfies (2.11), hence\nλI−Ais surjective for every λ >0.\nWe have then the following result.\nProposition 2.2. The system (1.1)–(1.3)is well–posed. More precisely, for every\n(u0,u1,f0)∈ H, there exists a unique solution (u,v,z)∈C(0,+∞,H)of(2.7). More-\nover, if(u0,u1,f0)∈D(A)then(u,v,z)∈C(0,+∞,D(A))∩C1(0,+∞,H)withv=u′\nanduis indeed a solution of (1.1)–(1.3).\n3 The spectral analysis\nAsD(B∗) is compactly embedded into H, the operator BB∗:D(BB∗)⊂H→H\nhas a compact resolvent. Hence let ( λk)k∈N∗be the set of eigenvalues of BB∗repeated\naccording to their multiplicity (that are positive real num bers and are such that λk→\n+∞ask→+∞) and denote by ( ϕk)k∈N∗the corresponding eigenvectors that form an\northonormal basis of H(in particular for all k∈N∗,BB∗ϕk=λkϕk).\n3.1 The discrete spectrum\nWe have the following lemma.\nLemma 3.1. Ifτ≤a, then any eigenvalue λofAsatisfies ℜλ <0.\n5Proof.Letλ∈CandU= (u,v,z)⊤∈D(A) be such that\n(λI−A)\nu\nv\nz\n= 0,\nor equivalently\n\nv=λu,\n−B(aB∗v+z(·,1)) =λv,\n−τ−1zρ=λz.(3.1)\nBy (2.14), we find that\nz(ρ) =λ−1B∗ve−λρτ. (3.2)\nUsing this property in (3.1), we find that u∈D(B∗) is solution of\nλ2u+(aλ+e−λτ)BB∗u= 0.\nHence a non trivial solution exists if and only if there exist sk∈N∗such that\nλ2\naλ+e−λτ=−λk. (3.3)\nThis condition implies that λdoes not belong to\nΣ :={λ∈C:aλ+e−λτ= 0}, (3.4)\nand that\ne−λτ+λ2\nλk+aλ= 0. (3.5)\nWritingλ=x+iy, withx,y∈R, we see that this identity is equivalent to\ne−τxcos(τy)+x2−y2\nλk+ax= 0, (3.6)\n−e−τxsin(τy)+2xy\nλk+ay= 0. (3.7)\nThe second equation is equivalent to\neτx/parenleftBig2x\nλk+a/parenrightBig\ny= sin(τy).\nHence if y/\\e}atio\\slash= 0, we will get\neτx\nτ/parenleftBig2x\nλk+a/parenrightBig\n=sin(τy)\nτy.\nAs the modulus of the right-hand side is ≤1, we obtain\n/vextendsingle/vextendsingle/vextendsingleeτx\nτ/parenleftBig2x\nλk+a/parenrightBig/vextendsingle/vextendsingle/vextendsingle≤1,\n6or equivalently/vextendsingle/vextendsingle/vextendsingle2x\nλk+a/vextendsingle/vextendsingle/vextendsingle≤τe−τx.\nTherefore if x≥0, we find that\n2x\nλk+a≤τe−τx≤τ,\nwhich implies that\n2x\nλk≤τ−a.\nForτ < a, we arrive to a contradiction. For τ=a, the sole possibility is x= 0 and by\n(3.7), we find that\nsin(τy) =τy,\nwhich yields y= 0 and again we obtain a contradiction.\nIfy= 0, we see that (3.7) always holds and (3.6) is equivalent to\ne−τx=−x(x\nλk+a).\nThis equation has no non–negative solutions xsince for x≥0, the left hand side is\npositive while the right–hand side is non positive, hence ag ain if a solution xexists, it\nhas to be negative.\nThe proof of the lemma is complete.\nIfa < τ, we now show that there exist some pairs of ( a,τ) for which the system\n(1.1)–(1.3) becomes unstable. Hence the condition τ≤ais optimal for the stability of\nthis system.\nLemma 3.2. There exist pairs of (a,τ)such that 0< a < τand for which the associated\noperator Ahas a pure imaginary eigenvalue.\nProof.We look for a purely imaginary eigenvalue iyofA, hence system (3.6)–(3.7)\nreduces to\ncos(τy) =y2\nλk, (3.8)\nsin(τy) =ay. (3.9)\nSuch a solution exists if\ny4\nλ2\nk+a2y2= 1. (3.10)\nOne solution of this equation is\nyk=\n−a2λ2\nk+/radicalBig\na4λ4\nk+4λ2\nk\n2\n1\n2\n.\n7We now take any τ∈(0,π\n2yk) anda=sin(τyk)\nyk. Then (3.9) automatically holds, while\n(3.8) is valid owing to (3.10) (as cos( τyk)>0). Finally a < τbecause\na\nτ=sin(τyk)\nτyk<1.\nTherefore with such a choice of aandτ, the operator Ahas a purely imaginary eigen-\nvalue equal to iyk.\n3.2 The continuous spectrum\nInspired from section 3 of [1], by using a Fredholm alternati ve technique, we perform\nthe spectral analysis of the operator A.\nRecall that an operator Tfrom a Hilbert space Xinto itself is called singular if there\nexists a sequence un∈D(T) with no convergent subsequence such that /ba∇dblun/ba∇dblX= 1 and\nTun→0 inX, see [21]. According to Theorem 1.14 of [21] Tis singular if and only if\nits kernel is infinite dimensional or its range is not closed. Let Σ be the set defined in\n(3.4). The following results hold:\nTheorem 3.3. 1. Ifλ∈Σ, thenλI−Ais singular.\n2. Ifλ/\\e}atio\\slash∈Σ, thenλI−Ais a Fredholm operator of index zero.\nProof.For the proof of point 1, let us fix λ∈Σ and for all k∈N∗set\nUk= (uk,λuk,B∗uke−λτ·)⊤,\nwithuk=1√λkϕk. ThenUkbelongs to D(A) and easy calculations yield (due to the\nassumption λ∈Σ)\n(λI−A)Uk=λ2(0,uk,0)⊤.\nTherefore we deduce that\n/ba∇dbl(λI−A)Uk/ba∇dblH→0,ask→ ∞.\nMoreover due to the property /ba∇dblB∗uk/ba∇dblH1= 1, there exist positive constants c,C,such\nthat\nc≤ /ba∇dblUk/ba∇dblH≤C,∀k∈N∗.\nThis shows that λI−Ais singular.\nFor allλ∈C, introduce the (linear and continuous) mapping AλfromVinto its\ndual by\n/a\\}b∇acketle{tAλv,w/a\\}b∇acket∇i}htV′−V=λ2(v,w)H+(aλ+e−λτ)(B∗v,B∗w)H1,∀v,w∈D(B∗).\nThen from the proof of Lemma 2.1, we know that for λ >0,Aλis an isomorphism.\nNow for λ∈C\\Σ, we can introduce the operator\nBλ= (aλ+e−λτ)−1Aλ.\n8Hence for λ∈C\\Σ,Aλis a Fredholm operator of index 0 if and only if Bλis a Fredholm\noperator of index 0. Furthermore for λ,µ∈C\\Σ asBλ−Bµis a multiple of the identity\noperator, due to the compact embedding of VintoV′, and asBµis an isomorphism for\nµ >0, we finally deduce that Aλis a Fredholm operator of index 0 for all λ∈C\\Σ.\nNow we readily check that, for any λ∈C\\Σ, we have the equivalence\nu∈kerAλ⇐⇒(u,λu,B∗ue−λτ·)⊤∈ker(λI−A). (3.11)\nThis equivalence implies that\ndim ker( λI−A) = dim ker Aλ,∀λ∈C\\Σ. (3.12)\nFor the range property for all λ∈C\\Σ introduce the inner product\n(u,z)λ,V:=/parenleftBig\n(u,λu,B∗ue−λτ·)⊤,(z,λz,B∗ze−λτ·)⊤/parenrightbig\nH,\nonVwhose associated norm is equivalent to the standard one.\nDenote by {y(i)}N\ni=1an orthonormal basis of ker Aλfor this new inner product (for\nshortness the dependence of λis dropped), i.e.\n(y(i),y(j))λ,V=δij,∀i,j= 1,...,N.\nFinally, for all i= 1,...,N, we set\nZ(i)= (y(i),λy(i),B∗y(i)e−λτ·)⊤,\nthe element of ker( λI− A) associated with y(i)that are orthonormal with respect to\nthe inner product of H.\nLet us now show that for all λ∈C\\Σ, the range R(λI− A) ofλI− Ais closed.\nIndeed, let us consider a sequence Un= (un,vn,zn)⊤∈D(A) such that\n(λI−A)Un=Fn= (fn,gn,hn)⊤→F= (f,g,h)⊤inH. (3.13)\nWithout loss of generality we can assume that\n(Un,Z(i))H=−αn,i,∀i= 1,...,N, (3.14)\nwhere\nαn,i:= ((0,fn,−τe−λτ·/integraldisplay·\n0hn(σ)eλστdσ)⊤,Z(i))H.\nIndeed, if this is not the case, we can consider\n˜Un=Un−N/summationdisplay\ni=1βiZ(i)\nthat still belongs to D(A) and satisfies\n(λI−A)˜Un=Fn,\n9as well as\n(˜Un,Z(i))H=−αn,i,∀i= 1,...,N,\nby setting\nβi= (Un,Z(i))H+αn,i,∀i= 1,...,N.\nNote that the condition (3.14) is equivalent to\n(un,y(i))λ,V= 0,∀i= 1,...,N.\nIn other words,\nun∈(kerAλ)⊥λ,V, (3.15)\nwhere⊥λ,Vmeans that the orthogonality is taken with respect to the inn er product\n(·,·)λ,V.\nReturning to (3.13), the arguments of the proof of Lemma 2.1 i mply that\nAλun=LFninV′,\nwhereLFis defined by\nLF(w) := (g,w)H−τe−λτ(/integraldisplay1\n0h(σ)eλστdσ,B∗w)H1+(λf+aB∗f,w)H1,\nwhenF= (f,g,h)⊤. But it is easy to check that\nLFn→LFinV′.\nMoreover, as λ∈C\\Σ,Aλis an isomorphism from (ker Aλ)⊥λ,VintoR(Aλ), hence by\n(3.15) we deduce that there exists a positive constant C(λ) such that\n/ba∇dblun−um/ba∇dblV≤C(λ)/ba∇dblLFn−LFm/ba∇dblV′,∀n,m∈N.\nHence, (un)nis a Cauchy sequence in V, and therefore there exists u∈Vsuch that\nun→uinV,\nas well as\nAλu=LFinV′.\nThen defining vby (2.13) and zby (2.16), we deduce that U:= (u,v,z)⊤belongs to\nD(A) and\n(λI−A)U=F.\nIn other words, Fbelongs to R(λI−A). The closedness of R(λI−A) is thus proved.\nAt this stage, for any λ∈C\\Σ, we show that\ncodimR(Aλ) = codim R(λI−A), (3.16)\nwhere for W⊂ H, codimWis the dimension of the orthogonal in HofW, while for\nW′⊂V′, codimW′is the dimension of the annihilator\nA:={v∈V:/a\\}b∇acketle{tv,w/a\\}b∇acket∇i}htV−V′= 0,∀w∈W′},\n10ofW′inV.\nIndeed, let us set N= codim R(Aλ), then there exist Nelements ϕi∈V, i=\n1,...,N,such that\nf∈R(Aλ)⇐⇒f∈V′and/a\\}b∇acketle{tf,ϕi/a\\}b∇acket∇i}htV′−V= 0,∀i= 1,...,N.\nConsequently, for F∈ H, ifLF(that belongs to V′) satisfies\nLF(ϕi) = 0,∀i= 1,...,N, (3.17)\nthen there exists a solution u∈Vof\nAλu=LFin V’,\nand the arguments of the proof of Lemma 2.1 imply that Fis inR(λI−A). Hence, the\nNconditions on F∈ Hfrom (3.17) allow to show that it belongs to R(λI− A), and\ntherefore\ncodimR(λI−A)≤N= codim R(Aλ). (3.18)\nThis shows that λI−Ais a Fredholm operator.\nConversely, set M= codim R(λI−A),thenthereexist MelementsΨ i= (ui,vi,zi)∈\nH, i= 1,...,M, such that\nF∈R(λI−A)⇐⇒F∈ Hand (F,Ψi)H= 0,∀i= 1,...,M.\nThen, for any g∈H, if\n(g,vi)H= ((0,g,0)⊤,Ψi)H= 0,∀i= 1,...,M, (3.19)\nthere exists U= (u,v,z)⊤∈D(A) such that\n(λI−A)U= (0,g,0),\nwhich implies that\nAλu=g.\nThis shows that\nR(Aλ)⊃H0,\nwhereH0:={g∈Hsatisfying (3 .19)}. This inclusion implies that (here ⊥means the\nannihilator of the set in V)\nR(Aλ)⊥⊂H⊥\n0.\nTherefore\nR(Aλ)⊥⊂ {v∈V:/a\\}b∇acketle{tv,g/a\\}b∇acket∇i}htV−V′= 0,∀g∈H0}\n={v∈V: (v,g)H= 0,∀g∈H0}\n⊂Span{vi}M\ni=1∩V.\nHence,\ncodimR(Aλ)≤M= codim R(λI−A). (3.20)\nThe inequalities (3.18) and (3.20) imply (3.16).\n11Lemma 3.4. Ifτ≤a, then\nΣ⊂ {λ∈C:ℜλ <0}.\nProof.Letλ=x+iy∈Σ, withx,y∈Rwe deduce that\nax+e−τxcos(τy) = 0,\nay−e−τxsin(τy) = 0.\nThis corresponds to the system (3.6)–(3.7) with k=∞, hence the arguments as in the\nproof of Lemma 3.1 yield the result.\nCorollary 3.5. It holds\nσ(A) =σpp(A)∪Σ,\nand therefore if τ≤a\nσ(A)⊂ {λ∈C:ℜλ <0}.\nProof.By Theorem 3.3,\nC\\Σ⊂σpp(A)∪ρ(A).\nThe first assertion directly follows.\nThe second assertion follows from Lemmas 3.1 and 3.4.\n4 Asymptotic behavior\nIn this section, we show that if τ≤aandξ >2τ\na, the semigroup etAdecays to the null\nsteady state with an exponential decay rate. To obtain this, our technique is based on\na frequency domain approach and combines a contradiction ar gument to carry out a\nspecial analysis of the resolvent.\nTheorem 4.1. Ifξ >2τ\naandτ≤a, then there exist constants C,ω >0such that the\nsemigroup etAsatisfies the following estimate\n/vextenddouble/vextenddoubleetA/vextenddouble/vextenddouble\nL(H)≤Ce−ωt,∀t >0. (4.21)\nProof of theorem 4.1. We will employ the following frequency domain theorem for un i-\nform stability from [15, Thm 8.1.4] of a C0semigroup on a Hilbert space:\nLemma 4.2. AC0semigroup etLon a Hilbert space Hsatisfies\n||etL||L(H)≤Ce−ωt,\nfor some constant C >0and forω >0if and only if\nℜλ <0,∀λ∈σ(L), (4.22)\nand\nsup\nℜλ≥0/ba∇dbl(λI−L)−1/ba∇dblL(H)<∞. (4.23)\nwhereσ(L)denotes the spectrum of the operator L.\n12According to Corollary 3.5 the spectrum of Ais fully included into ℜλ <0, which\nclearly implies (4.22). Then the proof of Theorem 4.1 is base d on the following lemma\nthat shows that (4.23) holds with L=A.\nLemma 4.3. The resolvent operator of Asatisfies condition\nsup\nℜλ≥0/ba∇dbl(λI−A)−1/ba∇dblL(H)<∞. (4.24)\nProof.Suppose that condition (4.24) is false. By the Banach-Stein haus Theorem (see\n[8]), there exists a sequence of complex numbers λnsuch that ℜλn≥0,|λn| →+∞\nand a sequence of vectors Zn= (un,vn,zn)t∈D(A) with\n/ba∇dblZn/ba∇dblH= 1 (4.25)\nsuch that\n||(λnI−A)Zn||H→0 asn→ ∞, (4.26)\ni.e.,\nλnun−vn≡fn→0 inD(B∗), (4.27)\nλnvn+aB(B∗vn+zn(1))≡gn→0 inH, (4.28)\nλnzn+τ−1∂ρzn≡hn→0 inL2((0,1);H1). (4.29)\nOur goal is to derive from (4.26) that ||Zn||Hconverges to zero, that furnishes a\ncontradiction.\nWe notice that from (2.10) and (4.27) we have\n||(λnI−A)Zn||H≥ |ℜ((λnI−A)Zn,Zn)H|\n≥ ℜλn−a−1\n∗/ba∇dblB∗un/ba∇dbl2\nH1+/parenleftbiggξ\n2τ−1\na/parenrightbigg\n/ba∇dblzn(1)/ba∇dbl2\nH1+a\n2/ba∇dblB∗vn/ba∇dbl2\nH1\n=ℜλn−a−1\n∗/vextenddouble/vextenddouble/vextenddouble/vextenddoubleB∗vn+B∗fn\nλn/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nH1+/parenleftbiggξ\n2τ−1\na/parenrightbigg\n/ba∇dblzn(1)/ba∇dbl2\nH1+a\n2/ba∇dblB∗vn/ba∇dbl2\nH1.\nHence using the inequality\n/ba∇dblB∗vn+B∗fn/ba∇dbl2\nH1≤2/ba∇dblB∗vn/ba∇dbl2\nH1+2/ba∇dblB∗fn/ba∇dbl2\nH1,\nwe obtain that\n||(λnI−A)Zn||H≥ ℜλn−2a−1\n∗|λn|−2/ba∇dblB∗fn/ba∇dbl2\nH1+/parenleftbiggξ\n2τ−1\na/parenrightbigg\n/ba∇dblzn(1)/ba∇dbl2\nH1\n+(a\n2−2a−1\n∗|λn|−2)/ba∇dblB∗vn/ba∇dbl2\nH1.\nHence for nlarge enough, say n≥n∗, we can suppose that\na\n2−2a−1\n∗|λn|−2≥a\n4.\n13and therefore for all n≥n∗, we get\n||(λnI−A)Zn||H≥ ℜλn−2a−1\n∗|λn|−2/ba∇dblB∗fn/ba∇dbl2\nH1+/parenleftbiggξ\n2τ−1\na/parenrightbigg\n/ba∇dblzn(1)/ba∇dbl2\nH1\n+a\n4/ba∇dblB∗vn/ba∇dbl2\nH1.\nBy this estimate, (4.26) and (4.27), we deduce that\nzn(1)→0, B∗vn→0,inH1,asn→ ∞, (4.30)\nand in particular, from the coercivity (1.4), that\nvn→0,inH,asn→ ∞.\nThis implies according to (4.27) that\nun=1\nλnvn+1\nλnfn→0,inD(B∗),asn→ ∞, (4.31)\nas well as\nzn(0) =B∗un→0,inH1,asn→ ∞. (4.32)\nBy integration of the identity (4.29), we have\nzn(ρ) =zn(0)e−τλnρ+τ/integraldisplayρ\n0e−τλn(ρ−γ)hn(γ)dγ. (4.33)\nHence recalling that ℜλn≥0\n/integraldisplay1\n0/ba∇dblzn(ρ)/ba∇dbl2\nH1dρ≤2/ba∇dblzn(0)/ba∇dbl2\nH1+2τ2/integraldisplay1\n0/integraldisplayρ\n0/ba∇dblhn(γ)/ba∇dbl2\nH1dγρdρ→0,asn→ ∞.\nAll together we have shown that /ba∇dblZn/ba∇dblHconverges to zero, that clearly contradicts\n/ba∇dblZn/ba∇dblH= 1.\nThe two hypotheses of Lemma 4.2 are proved, then (4.21) holds . The proof of\nTheorem 4.1 is then finished.\n5 Application to the stabilization of the wave equation with delay and\na Kelvin–Voigt damping\nWe study the internal stabilization of a delayed wave equati on. More precisely, we\nconsider the system given by :\nutt(x,t)−a∆ut(x,t)−∆u(x,t−τ) = 0,in Ω×(0,+∞), (5.1)\nu= 0, on∂Ω×(0,+∞),(5.2)\nu(x,0) =u0(x), ut(x,0) =u1(x), in Ω, (5.3)\n∇u(x,t−τ) =f0(t−τ), in Ω×(0,τ), (5.4)\n14where Ω is a smooth open bounded domain of Rnanda,τ >0 are constants.\nThisproblementersinourabstractframeworkwith H=L2(Ω),B=−div:D(B) =\nH1(Ω)n→L2(Ω),B∗=∇:D(B∗) =H1\n0(Ω)→H1:=L2(Ω)n, the assumption (1.4)\nbeing satisfied owing to Poincar´ e’s inequality. The operat orAis then given by\nA\nu\nv\nz\n:=\nv\na∆v+ divz(·,1)\n−τ−1zρ\n,\nwith domain\nD(A) :=/braceleftBig\n(u,v,z)⊤∈H1\n0(Ω)×H1\n0(Ω)×L2(Ω;H1(0,1)) :a∇v+z(·,1)∈H1(Ω),\n∇u=z(·,0) in Ω/bracerightBig\n,\n(5.5)\nin the Hilbert space\nH:=H1\n0(Ω)×L2(Ω)×L2(Ω×(0,1)). (5.6)\nAccording to Lemma 3.5 and Theorem 4.1 we have:\nCorollary 5.1. 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Pure Appl. Math., 12(1959), 211–228.\n17"    },    {        "title": "1406.6225v2.Interface_enhancement_of_Gilbert_damping_from_first_principles.pdf",        "content": "arXiv:1406.6225v2  [cond-mat.mtrl-sci]  16 Nov 2014Interface enhancement of Gilbert damping from first-princi ples\nYi Liu,1,∗Zhe Yuan,1,2,†R. J. H. Wesselink,1Anton A. Starikov,1and Paul J. Kelly1\n1Faculty of Science and Technology and MESA+Institute for Nanotechnology,\nUniversity of Twente, P.O. Box 217, 7500 AE Enschede, The Net herlands\n2Institut f¨ ur Physik, Johannes Gutenberg–Universit¨ at Ma inz, Staudingerweg 7, 55128 Mainz, Germany\n(Dated: June 6, 2018)\nThe enhancement of Gilbert damping observed for Ni 80Fe20(Py) films in contact with the non-\nmagnetic metals Cu, Pd, Ta and Pt, is quantitatively reprodu ced using first-principles scattering\ncalculations. The “spin-pumping” theory that qualitative ly explains its dependence on the Py thick-\nness is generalized to include a number of extra factors know n to be important for spin transport\nthrough interfaces. Determining the parameters in this the ory from first-principles shows that inter-\nface spin-flipping makes an essential contribution to the da mping enhancement. Without it, a much\nshorter spin-flip diffusion length for Pt would be needed than the value we calculate independently.\nPACS numbers: 85.75.-d, 72.25.Mk, 76.50.+g, 75.70.Tj\nIntroduction. —Magnetizationdissipation, expressedin\ntermsofthe Gilbert dampingparameter α, is akeyfactor\ndetermining the performance of magnetic materials in a\nhost of applications. Of particular interest for magnetic\nmemorydevicesbasedupon ultrathin magneticlayers[ 1–\n3] is the enhancement of the damping of ferromagnetic\n(FM) materials in contact with non-magnetic (NM) met-\nals [4] that can pave the way to tailoring αfor particu-\nlar materials and applications. A “spin pumping” theory\nhas been developed that describes this interface enhance-\nment in terms of a transverse spin current generated by\nthe magnetization dynamics that is pumped into and ab-\nsorbed by the adjacent NM metal [ 5,6]. Spin pumping\nsubsequently evolved into a technique to generate pure\nspin currents that is extensively applied in spintronics\nexperiments [ 7–9].\nA fundamental limitation of the spin-pumping the-\nory is that it assumes spin conservation at interfaces.\nThis limitation does not apply to a scattering theoret-\nical formulation of the Gilbert damping that is based\nupon energy conservation, equating the energy lost by\nthe spin system through damping to that parametrically\npumped out of the scattering region by the precessing\nspins [10]. In this Letter, we apply a fully relativistic\ndensity functional theory implementation [ 11–13] of this\nscattering formalism to the Gilbert damping enhance-\nment in those NM |Py|NM structures studied experimen-\ntally in Ref. 4. Our calculated values of αas a function\nof the Py thickness dare compared to the experimental\nresults in Fig. 1. Without introducing any adjustable pa-\nrameters, we quantitatively reproduce the characteristic\n1/ddependence aswellasthe dependenceofthe damping\non the NM metal.\nTo interpret the numerical results, we generalize the\nspin pumping theory to allow: (i) for interface [ 14–16]\nas well as bulk spin-flip scattering; (ii) the interface mix-\ning conductance to be modified by spin-orbit coupling;\n(iii) the interface resistance to be spin-dependent. An\nimportant consequence of our analysis is that withoutinterface spin-flip scattering, the value of the spin-flip\ndiffusion length lsfin Pt required to fit the numerical\nresults is much shorter than a value we independently\ncalculate for bulk Pt. A similar conclusion has recently\nbeen drawn for Co |Pt interfaces from a combination of\nferromagnetic resonance, spin pumping and inverse spin\nHall effect measurements [ 17].\nGilbert damping in NM |Py|NM.—We focus on the\nNM|Py|NM sandwiches with NM = Cu, Pd, Ta and Pt\nthat were measured in Ref. 4. The samples were grown\non insulating glass substrates, the NM layer thickness\nwas fixed at l=5 nm, and the Py thickness dwas vari-\nable. To model these experiments, the conventional NM-\nlead|Py|NM-lead two-terminal scattering geometry with\nsemi-infinite ballistic leads [ 10–13] has to be modified\nbecause: (i) the experiments were carried out at room\n0 2 4 6 8 10 \nd (nm) 00.02 0.04 0.06 0.08 0.10 _Pt |Py|Pt \nPd|Py|Pd \nTa|Py|Ta \nCu|Py|Cu Calc. Expt. NM \r\n(l)NM \r\n(l)\nLead \n Lead Py \r\n(d)\nFIG. 1. (color online). Calculated (solid lines) Gilbert da mp-\ning of NM |Py|NM (NM = Cu, Pd, Ta and Pt) compared to\nexperimental measurements (dotted lines) [ 4] as a function of\nthe Py thickness d. Inset: sketch of the structure used in the\ncalculations. The dashed frame denotes one structural unit\nconsisting of a Py film between two NM films.2\ntemperature so the 5 nm thick NM metals used in the\nsamples were diffusive; (ii) the substrate |NM and NM |air\ninterfaces cannot transmit charge or spin and behave ef-\nfectively as “mirrors”, whereas in the conventional scat-\ntering theory the NM leads are connected to charge and\nspin reservoirs.\nWe start with the structural NM( l)|Py(d)|NM(l) unit\nindicated by the dashed line in the inset to Fig. 1that\nconsists of a Py film, whose thickness dis variable, sand-\nwichedbetween l=5nm-thick diffusiveNM films. Several\nNM|Py|NM units are connected in series between semi-\ninfinite leads to calculate the total magnetization dissi-\npation of the system [ 10–13] thereby explicitly assuming\na “mirror” boundary condition. By varying the number\nof these units, the Gilbert damping for a single unit can\nbe extracted [ 18], that corresponds to the damping mea-\nsured for the experimental NM( l)|Py(d)|NM(l) system.\nAs shown in Fig. 1, the results are in remarkably\ngood overall agreement with experiment. For Pt and\nPd, where a strong damping enhancement is observed for\nthin Py layers, the values that we calculate are slightly\nlower than the measured ones. For Ta and Cu where\nthe enhancement is weaker, the agreement is better. In\nthe case of Cu, neither the experimental nor the calcu-\nlated data shows any dependence on dindicating a van-\nishinglysmalldampingenhancement. Theoffsetbetween\nthe two horizontal lines results from a difference between\nthe measured and calculated values of the bulk damping\nin Py. Acareful analysisshowsthat the calculated values\nofαare inversely proportional to the Py thickness dand\napproach the calculated bulk damping of Py α0=0.0046\n[11] in the limit of large dfor all NM metals. However,\nextrapolation of the experimental data yields values of\nα0ranging from 0.004 to 0.007 [ 19]; the spread can be\npartly attributed to the calibration of the Py thickness,\nespecially when it is very thin.\nGeneralized spin-pumping theory. —In spite of the very\ngood agreement with experiment, our calculated re-\nsults cannot be interpreted satisfactorily using the spin-\npumping theory [ 5] that describes the damping enhance-\nment in terms of a spin current pumped through the\ninterface by the precessing magnetization giving rise to\nan accumulation of spins in the diffusive NM metal,\nand a back-flowing spin current driven by the ensuing\nspin-accumulation. The pumped spin current, Ipump\ns=\n(/planckover2pi12A/2e2)Gmixm×˙m, is described using a “mixing con-\nductance” Gmix[20] that is a property of the NM |FM\ninterface [ 21,22]. Here, mis a unit vector in the di-\nrection of the magnetization and Ais the cross-sectional\narea. The theory only takes spin-orbit coupling (SOC)\ninto account implicitly via the spin-flip diffusion length\nlsfof the NM metal and the pumped spin current is con-\ntinuous across the FM |NM interface [ 5].\nWith SOC included, this boundary condition does not\nhold. Spin-flip scattering at an interface is described by\nthe “spin memory loss” parameter δdefined so that thespin-flip probability of a conduction electron crossing the\ninterface is 1 −e−δ[14,15]. It alters the spin accumula-\ntion in the NM metal and, in turn, the backflow into the\nFM material. To take δand the spin-dependence of the\ninterface resistance into account, the FM |NM interface\nis represented by a fictitious homogeneous ferromagnetic\nlayerwithafinitethickness[ 15,16]. Thespincurrentand\nspin-resolved chemical potentials (as well as their differ-\nenceµs, the spin accumulation) are continuous at the\nboundaries of the effective “interface” layer. We impose\nthe boundary condition that the spin current vanishes at\nNM|air or NM |substrate interfaces. Then the spin accu-\nmulation in the NM metal can be expressed as a function\nof the net spin-current Isflowing out of Py [ 23], which\nis the difference between the pumped spin current Ipump\ns\nand the backflow Iback\ns. The latter is determined by the\nspin accumulation in the NM metal close to the inter-\nface,Iback\ns[µs(Is)]. Following the original treatment by\nTserkovnyak et al. [ 5],Isis determined by solving the\nequation Is=Ipump\ns−Iback\ns[µs(Is)] self-consistently. Fi-\nnally, the total damping of NM( l)|Py(d)|NM(l) can be\ndescribed as\nα(l,d) =α0+gµB/planckover2pi1\ne2MsdGmix\neff=α0+gµB/planckover2pi1\ne2Msd\n×/bracketleftbigg1\nGmix+2ρlsfR∗\nρlsfδsinhδ+R∗coshδtanh(l/lsf)/bracketrightbigg−1\n.(1)\nHere,R∗=R/(1−γ2\nR) is an effective interface spe-\ncific resistance with Rthe total interface specific resis-\ntance between Py and NM and its spin polarization,\nγR= (R↓−R↑)/(R↓+R↑) is determined by the con-\ntributions R↑andR↓from the two spin channels [ 16].ρ\nis the resistivity of the NM metal. All the quantities in\nEq. (1) can be experimentally measured [ 16] and calcu-\nlated from first-principles [ 24]. If spin-flip scattering at\nthe interface is neglected, i.e., δ= 0, Eq. ( 1) reduces to\nthe original spin pumping formalism [ 5]. Eq. (1) is de-\nrived using the Valet-Fert diffusion equation [ 25] that is\nstill applicable when the mean free path is comparable\nto the spin-flip diffusion length [ 26].\nMixing conductance. —Assuming that SOC can be\nneglected and that the interface scattering is spin-\nconserving, the mixing conductance is defined as\nGmix=e2\nhA/summationdisplay\nm,n/parenleftbig\nδmn−r↑\nmnr↓∗\nmn/parenrightbig\n, (2)\nin terms of rσ\nmn, the probability amplitude for reflection\nofaNMmetalstate nwithspin σintoaNM state mwith\nthesamespin. UsingEq.( 2), wecalculate GmixforPy|Pt\nand Py|Cu interfaces without SOC and indicate the cor-\nresponding damping enhancement gµB/planckover2pi1Gmix/(e2MsA)\non the vertical axis in Fig. 2with asterisks.\nWhen SOC is included, Eq. ( 2) is no longer applicable.\nWecanneverthelessidentify aspin-pumpinginterfaceen-\nhancement Gmixas follows. We artificially turn off the3\n0 2 4 6 8 10\nd (nm)00.050.100.15αd (nm)Pt\nCuWithout backflow\nWith backflow\nFIG. 2. (color online). Total damping calculated for Pt |Py|Pt\nand Cu|Py|Cu as a function of the Py thickness. The open\nsymbols correspond to the case without backflow while the\nfull symbols are the results shown in Fig. 1where backflow\nwas included. The lines are linear fits to the symbols. The as-\nterisks on the yaxis are the values of Gmixcalculated without\nSOC using Eq. ( 2).\nbackflow by connecting the FM metal to ballistic NM\nleads so that any spin current pumped through the in-\nterface propagatesawayimmediately and there is no spin\naccumulation in the NM metal. The Gilbert damping αd\ncalculated without backflow (dashed lines) is linear in\nthe Py thickness d; the intercept Γ at d= 0 represents\nan interface contribution. As seen in Fig. 2for Cu, Γ\ncoincides with the orange asterisk meaning that the in-\nterface damping enhancement for a Py |Cu interface is,\nwithin the accuracy of the calculation, unchanged by in-\ncluding SOC because this is so small for Cu, Ni and Fe.\nBy contrast, Γ and thus Gmix=e2MsAΓ/(gµB/planckover2pi1) for the\nPy|Pt interface is 25% larger with SOC included, con-\nfirming the breakdown of Eq. ( 2) for interfaces involving\nheavy elements.\nThe data in Fig. 1for NM=Pt and Cu are replotted\nas solid lines in Fig. 2for comparison. Their linearity\nmeans that we can extract an effective mixing conduc-\nTABLE I. Different mixing conductances calculated for\nPy|NM interfaces. Gmixis calculated using Eq. ( 2) without\nSOC.Gmixis obtainedfrom theinterceptofthetotal damping\nαdcalculated as a function of the Py thickness dwith SOCfor\nballistic NM leads. The effective mixing conductance Gmix\neffis\nextracted from the effective αin Fig.1in the presence of 5 nm\nNM on either side of Py. Sharvin conductances are listed for\ncomparison. All values are given in units of 1015Ω−1m−2.\nNM GSh GmixGmixGmix\neff\nCu 0.55 0.49 0.48 0.01\nPd 1.21 0.89 0.98 0.57\nTa 0.74 0.44 0.48 0.34\nPt 1.00 0.86 1.07 0.95tanceGmix\neffwith backflow in the presence of 5 nm dif-\nfusive NM metal attached to Py. For Py |Pt,Gmix\neffis\nonly reduced slightly compared to Gmixbecause there is\nvery little backflow. For Py |Cu, the spin current pumped\ninto Cu is only about half that for Py |Pt. However, the\nspin-flipping in Cu is so weak that spin accumulation in\nCu leads to a backflow that almost exactly cancels the\npumped spin current and Gmix\neffis vanishingly small for\nthe Py|Cu system with thin, diffusive Cu.\nThe values of Gmix,GmixandGmix\neffcalculated for all\nfour NM metals are listed in Table I. Because Gmix(Pd)\nandGmix(Pt) are comparable, Py pumps a similar spin\ncurrent into each of these NM metals. The weaker spin-\nflipping and larger spin accumulation in Pd leads to a\nlarger backflow and smaller damping enhancement. The\nrelatively low damping enhancement in Ta |Py|Ta results\nfrom a small mixing conductance for the Ta |Py interface\nrather than from a large backflow. In fact, Ta behaves\nas a good spin sink due to its large SOC and the damp-\ning enhancement in Ta |Py|Ta can not be significantly\nincreased by suppressing the backflow.\nThickness dependence of NM. —In the following we fo-\ncus on the Pt |Py|Pt system and examine the effect of\nchanging the NM thickness lon the damping enhance-\nment, a procedure frequently used to experimentally de-\ntermine the NM spin-flip diffusion length [ 27–31].\nThe total damping calculated for Pt |Py|Pt is plotted\nin Fig.3as a function of the Pt thickness lfor two thick-\nnessesdof Py. For both d= 1 nm and d= 2 nm,\nαsaturates at l=1–2 nm in agreement with experiment\n0 10 20 30 40 50 l (nm)0.00.51.0\n0 1 2 3 4 5\nl (nm)0.000.050.100.15 αd=1 nm\nd=2 nmPt(l)|Py(d)|Pt( l)G↑↑/G↑\nG↑↓/G↑Pt@RT\nl↑=7.8±0.3 nm\nFIG. 3. αas a function of the Pt thickness lcalculated for\nPt(l)|Py(d)|Pt(l). The dashed and solid lines are the curves\nobtained by fitting without and with interface spin memory\nloss, respectively. Inset: fractional spin conductances G↑↑/G↑\nandG↑↓/G↑when a fully polarized up-spin current is injected\ninto bulk Pt at room temperature. Gσσ′is (e2/htimes) the\ntransmission probability of a spin σfrom the left hand lead\ninto a spin σ′in the right hand lead; G↑=G↑↑+G↑↓. The\nvalue of the spin-flip diffusion length for a single spin chann el\nobtained by fitting is lσ= 7.8±0.3 nm.4\n[17,28–31]. AfitofthecalculateddatausingEq.( 1)with\nδ≡0 requires just three parameters, Gmix,ρandlsf. A\nseparate calculation gives ρ= 10.4µΩcm at T=300 K in\nvery good agreement with the experimental bulk value of\n10.8µΩcm [32]. Using the calculated Gmixfrom Table I\nleaves just one parameter free; from fitting, we obtain\na valuelsf=0.8 nm for Pt (dashed lines) that is consis-\ntent with values between0.5and 1.4nm determined from\nspin-pumping experiments [ 28–31]. However, the dashed\nlines clearly do not reproduce the calculated data very\nwell and the fit value of lsfis much shorter than that\nextracted from scattering calculations [ 11]. By injecting\na fully spin-polarized current into diffusive Pt, we find\nl↑=l↓= 7.8±0.3nm, asshownin theinsettoFig. 3, and\nfrom [25,33],lsf=/bracketleftbig\n(l↑)−2+(l↓)−2/bracketrightbig−1/2= 5.5±0.2 nm.\nThis value is confirmed by examining how the current\npolarization in Pt is distributed locally [ 34].\nIf we allow for a finite value of δand use the in-\ndependently determined Gmix,ρandlsf, the data in\nFig.3(solid lines) can be fit with δ= 3.7±0.2 and\nR∗/δ= 9.2±1.7 fΩm2. The solid lines reproduce the\ncalculateddatamuch better than when δ= 0 underlining\nthe importance of including interface spin-flip scattering\n[17,35]. The large value of δwe find is consistent with a\nlow spin accumulation in Pt and the corresponding very\nweak backflow at the Py |Pt interface seen in Fig. 2.\nConductivity dependence. —Many experiments deter-\nmining the spin-flip diffusion length of Pt have reported\nPt resistivities that range from 4.2–12 µΩcm at low tem-\nperature [ 35–38] and 15–73 µΩcm at room temperature\n[17,39–41]. The large spread in resistivity can be at-\ntributed to different amounts of structural disorder aris-\ning during fabrication, the finite thickness of thin film\nsamples etc. We can determine lsfandρ≡1/σfrom\nfirst principles scattering theory [ 11,12] by varying the\ntemperature in the thermal distribution of Pt displace-\nments in the range 100–500 K. The results are plot-\nted (black solid circles) in Fig. 4(a).lsfshows a lin-\near dependence on the conductivity suggesting that the\nElliott-Yafet mechanism [ 42,43] dominates the conduc-\ntion electron spin relaxation. A linear least squares fit\nyieldsρlsf= 0.61±0.02fΩm2that agrees very well with\nbulk data extracted from experiment that are either not\nsensitive to interface spin-flipping [ 37] or take it into ac-\ncount [17,35,38]. For comparison, we plot values of lsf\nextracted from the interface-enhanced damping calcula-\ntions assuming δ= 0 (empty orange circles). The result-\ning values of lsfare very small, between 0.5 and 2 nm, to\ncompensate for the neglect of δ.\nHaving determined lsf(σ), we can calculate the\ninterface-enhanced damping for Pt |Py|Pt for different\nvalues of σPtand repeat the fitting of Fig. 3using Eq. ( 1)\n[44]. The parameters R∗/δandδare plotted as a func-\ntion of the Pt conductivity in Fig. 4(b). The spin mem-\nory lossδdoes not show any significant variation about0102030R*/δ (fΩ m2)\n0 0.1 0.2 0.3\nσ (108Ω-1m-1)024\nδ1020lsf (nm)Rojas-Sánchez\nNiimi\nNguyen\nKurt50 20 10 7 54ρ (µΩ cm)\nδ=0(a)\n(b)\nFIG. 4. (a) lsffor diffusive Pt as a function of its conductivity\nσ(solid black circles) calculated by injecting a fully polar ized\ncurrent into Pt. The solid black line illustrates the linear\ndependence. Bulk values extracted from experiment that are\neithernotsensitivetointerface spin-flipping[ 37]ortakeitinto\naccount [ 17,35,38] are plotted (squares) for comparison. The\nempty circles are values of lsfdetermined from the interface-\nenhanced damping using Eq. ( 1) withδ= 0. (b) Fit values of\nR∗/δandδas a function of the conductivity of Pt obtained\nusing Eq. ( 1). The solid red line is the average value (for\ndifferent values of σ) ofδ=3.7.\n3.7, i.e., it does not appear to depend on temperature-\ninduced disorder in Pt indicating that it results mainly\nfrom scattering of the conduction electrons at the abrupt\npotential change of the interface. Unlike δ, the effective\ninterfaceresistance R∗decreaseswithdecreasingdisorder\nin Pt and tends to saturate for sufficiently ordered Pt. It\nsuggests that although lattice disorder at the interface\ndoes not dissipate spin angular momentum, it still con-\ntributestotherelaxationofthemomentumofconduction\nelectrons at the interface.\nConclusions. —We have calculated the Gilbert damp-\ning for Py |NM-metal interfaces from first-principles and\nreproduced quantitatively the experimentally observed\ndamping enhancement. To interpret the numerical re-\nsults, we generalized the spin-pumping expression for\nthe damping to allow for interface spin-flipping, a mix-\ning conductance modified by SOC, and spin dependent\ninterface resistances. The resulting Eq. ( 1) allows the\ntwo main factors contributing to the interface-enhanced\ndamping to be separated: the mixing conductance that\ndeterminesthespincurrentpumpedbyaprecessingmag-\nnetization and the spin accumulation in the NM metal\nthat induces a backflow of spin current into Py and low-\ners the efficiency of the spin pumping. In particular, the\nlatter is responsible for the low damping enhancement\nfor Py|Cu while the weak enhancement for Py |Ta arises\nfrom the small mixing conductance.\nWe calculate how the spin-flip diffusion length, spin5\nmemory loss and interface resistance depend on the con-\nductivity of Pt. It is shown to be essential to take ac-\ncount of spin memory loss to extract reasonable spin-\nflip diffusion lengths from interface damping. This has\nimportant consequences for using spin-pumping-related\nexperiments to determine the Spin Hall angles that char-\nacterize the Spin Hall Effect [ 17].\nAcknowledgments. —We are grateful to G.E.W. Bauer\nfor a critical reading of the manuscript. Our work was\nfinancially supported by the “Nederlandse Organisatie\nvoor Wetenschappelijk Onderzoek” (NWO) through the\nresearch programme of “Stichting voor Fundamenteel\nOnderzoek der Materie” (FOM) and the supercomputer\nfacilities of NWO “Exacte Wetenschappen (Physical Sci-\nences)”. It was also partly supported by the Royal\nNetherlands Academy of Arts and Sciences (KNAW). Z.\nYuan acknowledges the financial support of the Alexan-\nder von Humboldt foundation.\n∗Present address: Institut f¨ ur Physik, Johannes\nGutenberg–Universit¨ at Mainz, Staudingerweg 7, 55128\nMainz, Germany\n†zyuan@uni-mainz.de\n[1] See the collection of articles in Handbook of Spin Trans-\nport and Magnetism , edited by E. Y. Tsymbal and\nI.ˇZuti´ c (Chapman and Hall/CRC Press, Boca Raton,\n2011).\n[2] G. E. W. Bauer, E. Saitoh, and B. J. van Wees,\nNature Materials 11, 391 (2012) .\n[3] A. Brataas, A. D. Kent, and H. Ohno,\nNature Materials 11, 372 (2012) .\n[4] S. Mizukami, Y. Ando, and T. Miyazaki,\nJpn. J. Appl. Phys. 40, 580 (2001) ;\nJ. Magn. & Magn. Mater. 226–230 , 1640 (2001) .\n[5] Y. Tserkovnyak, A. Brataas, and G. E. W.\nBauer, Phys. Rev. Lett. 88, 117601 (2002) ;\nPhys. Rev. B 66, 224403 (2002) .\n[6] Y. Tserkovnyak, A. Brataas, G. E. W. Bauer, and B. I.\nHalperin, Rev. Mod. 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Wei,6\nT. Tanaka, H. Kontani, T. Kimura, and Y. Otani,\nPhys. Rev. B 83, 174405 (2011) .\n[37] Y. Niimi, D. Wei, H. Idzuchi, T. Wakamura, T. Kato,\nand Y. C. Otani, Phys. Rev. Lett. 110, 016805 (2013) .\n[38] H. Kurt, R. Loloee, K. Eid, W. P. Pratt Jr., and J. Bass,\nAppl. Phys. Lett. 81, 4787 (2002) .\n[39] M. Althammer, S. Meyer, H. Nakayama, M. Schreier,\nS. Altmannshofer, M. Weiler, H. Huebl, S. Gepr¨ ags,\nM. Opel, R. Gross, D. Meier, C. Klewe, T. Kuschel, J.-M.\nSchmalhorst, G. Reiss, L. Shen, A. Gupta, Y.-T. Chen,\nG. E. W. Bauer, E. Saitoh, and S. T. B. Goennenwein,\nPhys. Rev. B 87, 224401 (2013) .[40] V. Castel, N. Vlietstra, J. Ben Youssef, and B. J. van\nWees,Appl. Phys. Lett. 101, 132414 (2012) .\n[41] K. Ando, S. Takahashi, K. Harii, K. Sasage,\nJ. Ieda, S. Maekawa, and E. Saitoh,\nPhys. Rev. Lett. 101, 036601 (2008) .\n[42] R. J. Elliott, Phys. Rev. 96, 266 (1954) .\n[43] Y.Yafet,in Solid State Physics ,Vol.14,editedbyF.Seitz\nand D. Turnbull (Academic, New York, 1963) pp. 1–98.\n[44] Experiment [ 8] and theory [ 45] indicate at most a weak\ntemperature dependence of the spin mixing conductance.\n[45] K.Nakata, J. Phys.: Condens. Matter 25, 116005 (2013) ."    },    {        "title": "1606.04483v2.Anomalous_Damping_of_a_Micro_electro_mechanical_Oscillator_in_Superfluid___3_He_B.pdf",        "content": "arXiv:1606.04483v2  [cond-mat.supr-con]  7 Oct 2016Anomalous Damping of a Micro-electro-mechanical Oscillat or in Superfluid3He-B\nP. Zheng,1W.G. Jiang,1C.S. Barquist,1Y. Lee,1,∗and H.B. Chan2\n1Department of Physics, University of Florida, Gainesville , Florida 32611-8440, USA\n2Department of Physics, The Hong Kong University of Science a nd Technology, Clear Water Bay, Hong Kong\n(Dated: June 4, 2021)\nThe mechanical resonanceproperties ofamicro-electro-me chanical oscillator with agapof1.25 µm\nwas studied in superfluid3He-B at various pressures. The oscillator was driven in the l inear damping\nregime where the damping coefficient is independent of the osc illator velocity. The quality factor\nof the oscillator remains low ( Q≈80) down to 0.1 Tc, 4 orders of magnitude less than the intrinsic\nquality factor measured in vacuum at 4 K. In addition to the Bo ltzmann temperature dependent\ncontribution to the damping, a damping proportional to temp erature was found to dominate at low\ntemperatures. We propose a multiple scattering mechanism o f the surface Andreev bound states to\nbe a possible cause for the anomalous damping.\nOver several decades, different families of unconven-\ntional superconductors have been discovered. Many\nof these possess high transition temperatures, which\ngenerates much interest from the community pursuing\nroom temperature superconductors. However, the com-\nplete microscopic understanding of them still remains a\nchallenge1. Superfluid3He with p-wave spin-triplet pair-\ning is a prime model system to study the unconventional\nnature of Cooper pairs because the symmetry of the con-\ndensate is clearly identified and the properties of the\nintrinsically pure bulk system are well understood to a\nquantitative level2. The early theoretical works3,4have\nrevealedthe extremefragilityofCooperpairsagainstany\ntype of impurity scattering in unconventional supercon-\nductors. Interfaces and surfaces also serve as effective\npair-breaking agents in these systems, which results in\nmany intriguing surface properties5,6. The surface scat-\ntering in unconventional superfluids/superconductors in-\nduces quasiparticle mid-gap bound states spatially local-\nized near the surface within the coherence length, ξ0,\noften called surface Andreev bound states (SABS), ac-\ncompanying selective suppression of the order param-\neter components7–10. The detailed structure of SABS\nhas been theoretically investigated for various boundary\nconditions9,10.\nIn superconductors, tunneling spectroscopyhasproven\nto be a powerful tool for studying the pairing symmetry\nand surface states11. However, the detection of SABS in\nsuperfluid3He has been difficult due to the lack of an\nappropriate probe for the uncharged fluid. Nevertheless,\nvarious workshavesuggested the existence ofSABS12–20.\nMeasurements of transverse acoustic impedance using\nquartztransducershavebeenusedtoinvestigateSABS12.\nThe measured transverseacoustic impedance agreeswith\ntheoretical calculations, which provides indirect confir-\nmation of SABS13,14. The high resolution heat capacity\nmeasurement of3He in a silver heat exchanger was able\nto identify the contribution from the SABS near the sil-\nver surface15. In the recent experiment by the Lancaster\ngroup16, they linked the absence of the critical velocity\nof a wire moving without acceleration to the presence\nFIG. 1. A schematic side-view of the MEMS device. A mobile\ncenter plate is suspended above the bottom plate by springs\n(not shown). The gap Dbetween the mobile plate and the\nbottom plate is 1.25 µm. The thickness of each layer is shown\nto scale. The horizontal arrow represents the direction of t he\noscillation of the shear mode.\nof SABS. Recent theoretical studies provide a fresh in-\nsight into the nature of SABS21,22. They suggest the\nanisotropic magnetic response of the film or surface of\n3He-B with specular boundaries as a direct indicator of\nMajorana fermions in surface bound states.\nVarious resonators in direct contact with liquid3He,\nsuch as torsional oscillators23, vibrating wires24,25, tun-\ning forks26,27, and movingwires16, have been successfully\nutilized to investigatethe propertiesofits normaland su-\nperfluid phases. A new direction in the development of\nthe mechanical probes is based on the nanolithography\ntechnology, such as micro- and nano-electro-mechanical\nsystem (MEMS and NEMS) devices28,29. We have devel-\noped MEMS devices to study superfluid3He films30,31.\nTheses devices have also been successfully exploited to\nstudy the viscosityof normal liquid3He below 800mK32.\nIn this paper, we report the measurement of the damp-\ning of a MEMS device in superfluid3He-B which exhibits\nanomalous low temperature behavior. A plausible phys-\nical mechanism involving SABS is conjectured to be re-\nsponsible for the observed behavior.\nThe MEMS device used in this measurement has a\nmobile plate with 2 µm thickness and 200 µm lateral\nsize. The plate is suspended above the substrate by\nfour serpentine springs, maintaining a gap of 1.25 µm.\nA schematic side-view of the device is shown in Fig.1.2\nWhen the device is submerged in the fluid, a film is\nformedbetween the mobile plate and the substrate, while\nthe bulk fluid is in direct contact with the top surface\nof the plate. Its in-plane oscillation, called the shear\nmode, can be actuated and detected by the comb elec-\ntrodes fabricated on either side of the plate. The details\nof the devices and the measurement scheme can be found\nelsewhere28,33,34.\nThe MEMS device was studied in liquid3He at pres-\nsures of 9.2, 18.2, 25.2, and 28.6 bars and cooled down\nto a base temperature of about 250 µK by a dilution re-\nfrigerator and a copper demagnetization stage. The res-\nonance spectrum of the shear mode was obtained contin-\nuously upon warming from the base temperature with a\ntypical warming rate of 30 µK/hr. The temperature was\ndetermined by calibrated tuning fork thermometers26,27\nbelow 0.6 mK and by a3He melting curve thermometer\nabove. The PLTS-2000 was adopted as the temperature\nscale35. The uncertainty oftemperature measured by the\ntuning forks is mainly from the calibration process and\nis represented by error bars in Fig.2. A magnetic field of\n14 mT was applied in the direction perpendicular to the\nplane of the film except for one of the 28.6 bar measure-\nments. The full width at half maximum (FWHM), γ,\nand the resonance frequency, f0, were obtained by fitting\nthe spectrum to the Lorentzian:\nx=Aγf0/radicalbig\n(f2\n0−f2)2+(γf)2, (1)\nwherexis the vibration amplitude of the plate, A=\nF0/4π2mf0γis the amplitude of the Lorentzian peak,\nF0is the amplitude of the driving force applied on the\nplate,misthe effectivemassoftheplate, and fisthe fre-\nquency of the driving force. The FWHM is proportional\nto the damping coefficient in the equation of motion of\na damped driven harmonic oscillator. The uncertainty\nfrom fitting is represented by error bars in Fig.2. The\nresonance feature of the MEMS device is sensitive to the\ntemperature36. Forinstance, at28.6baritsqualityfactor\nreaches around 80 when the liquid is cooled to 300 µK,\nanddecreasesrapidlytoorderofunitynearthe A-Btran-\nsition.\nThe mean free path, ℓ, of the3He quasiparticles is of\nthe order of 10 µm at the transition temperature and in-\ncreases exponentially when the temperature approaches\nzero due to the isotropic energy gap of3He-B2. For\nT<∼0.4Tc,ℓbecomes larger than any length scale of the\nMEMS devices, and the MEMS-superfluid system tran-\nsitions into the ballistic regime. This aspect is verified\nby the temperature independent resonance frequency ob-\nserved in this temperature range. At low velocities, the\ndamping has a temperature dependence solely from the\ndensity ofthe quasiparticleswhich decreasesrapidlywith\ntemperature as exp( −∆/kBT)25. Below 0.4 Tc, the en-\nergy gap ∆ develops fully to the zero-temperature value,/s48/s46/s49 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56/s49/s48/s50/s49/s48/s51/s49/s48/s52\n/s50 /s52 /s54 /s56/s49/s48/s50/s49/s48/s51/s49/s48/s52\n/s32/s32/s77/s69/s77/s83/s32 /s32/s40/s72/s122/s41\n/s84/s99/s32/s47/s32/s84\n/s32/s32/s57/s46/s50/s32/s98/s97/s114\n/s32/s49/s56/s46/s50/s32/s98/s97/s114\n/s32/s50/s53/s46/s50/s32/s98/s97/s114\n/s32/s50/s56/s46/s54/s32/s98/s97/s114\n/s32/s50/s56/s46/s54/s32/s98/s97/s114/s32/s40/s122/s101/s114/s111/s32/s102/s105/s101/s108/s100/s41/s77/s69/s77/s83/s32 /s32/s40/s72/s122/s41\n/s84/s32/s47/s32/s84/s99/s126/s32/s84\nFIG. 2. ( Color online ) The FWHM of the MEMS as a func-\ntion of the reduced temperature at various pressures in a log -\nlog scale. For clarity, the error bars are only shown for the\ndata of 9.2 and 18.2 bar. The straight line corresponds to\na linear temperature dependence. ( Inset) The same data in\nan Arrhenius scale. A straight line in this scale represents a\nBoltzmann dependence.\n∆0. The FWHM of the MEMS is expected to follow\nγ=Bexp(−∆0/kBT), (2)\nwhereBis the damping amplitude determined by the\ngeometry of the device and the properties of the fluid.\nThe intrinsic FWHM measured in vacuum at 4 K,\nwhich is 0.071 Hz, is subtracted from the fitted FWHM\nto yield the FWHM due to the fluid only. This FWHM is\nplotted as a function of the reduced temperature at vari-\nous pressures in Fig.2. At the lowest attainable temper-\nature for 28.6 bar, 280 µK, the FWHM is around 270 Hz,\nwhich is 4 orders of magnitude larger than the intrinsic\nFWHM and 2 orders of magnitude larger than the TF\nFWHM in the same condition37. In contrast, the FWHM\nof MEMS in superfluid4He below 200 mK is weakly tem-\nperature dependent and approaches the intrinsic value38.\nTherefore, the anomalously large damping observed in\n3He-B is believed to stem from some mechanism other\nthan the scattering of thermal quasiparticles from the\nbulk. As shown in the inset of Fig.2, the FWHM does\nnot follow Eqn.(2). Furthermore, the FWHM becomes\nlinear in temperature below ∼0.15Tcfor the three high-\nest pressures. For 9.2 bar, the linear dependence is not\nfully developed, probably because of the relatively low\nTcat this pressure. This linear temperature dependent\nterm emerging at low temperatures keeps the FWHM of\nthe MEMS from decreasing exponentially as expected.\nThe coefficient of the linear temperature dependent con-\ntribution can be extracted from the ratio of FWHM to\ntemperature in the low temperature limit39.\nThe acquired linear term is then subtracted from the\ntotal FWHM. The residual FWHM is plotted against the\ntemperature in an Arrhenius scale in Fig.3. The linear\nbehavior of the three highest pressures demonstrates a3\n/s50 /s52 /s54 /s56/s49/s48/s48/s49/s48/s49/s49/s48/s50/s49/s48/s51/s49/s48/s52\n/s32/s32\n/s32/s57/s46/s50/s32/s98/s97/s114\n/s32/s49/s56/s46/s50/s32/s98/s97/s114\n/s32/s50/s53/s46/s50/s32/s98/s97/s114\n/s32/s50/s56/s46/s54/s32/s98/s97/s114\n/s32/s50/s56/s46/s54/s32/s98/s97/s114/s32/s40/s122/s101/s114/s111/s32/s102/s105/s101/s108/s100/s41/s77/s69/s77/s83/s32\n/s66/s32/s40/s72/s122/s41\n/s84\n/s67/s32/s47/s32/s84\nFIG. 3. ( Color online ) The FWHM of the Boltzmann damp-\ning,γB, against thereducedtemperature for various pressures\nin an Arrhenius scale. A linear fit ( straight lines ) in this scale\ngives the measured energy gap, ∆ m.\nBoltzmann exponential temperature dependence follow-\ning Eqn.(2) and justifies the assumption that the damp-\ning in addition to the expected thermal quasiparticles in\nthe bulk is linear. Since its temperature dependence is\nnot fully developed at low temperatures for 9.2 bar, the\nlinear term is estimated by requiringthe residualFWHM\nto obey Eqn.(2). It was found that the residual FWHM\ndata are sensitive to the choice of the linear coefficient.\nA 5% variation in the value is sufficient to skew the de-\npendence of the residual FWHM. Therefore, the total\nFWHM can be expressed by\nγ=γA+γB=αT+Bexp(−∆0/kBT),(3)\nwhereγAis the linear temperature dependent term that\ndominates at low temperatures, and γBis the Boltzmann\nexponentialtemperaturedependent termduetothe ther-\nmal quasiparticles in the bulk region. Hereafter, the ex-\nponential term is called the Boltzmann damping and the\nlinear term the additional damping.\nThe coefficient of the additional damping, α, decreases\nby a factor of two as the pressure increases from 9.2 bar\nto 28.6 bar (Fig.4). The linear coefficient seems to have\na linear dependence on the coherence length. For the\nBoltzmann damping, the data in Fig.3 can be fitted to\nstraight lines according to Eqn.(2) to get ∆ m, the mea-\nsuredenergygap. It wasfoundthat ∆ mismuchlessthan\nthe known bulk value at the correspondingpressure. The\npressure dependence of ∆ mis presented in terms of the\nBCS coherence length ξ0= ¯hvF/π∆0, wherevFis the\nFermi velocity (Fig.4). The measured energy gap is sup-\npressed from the bulk value and shows a strong pressure\ndependence. It decreases monotonically with the scaled\nfilm thickness, D/ξ0, since a larger effective thickness\ngives more space for the order parameter to recoverto its\nbulk value, hence a larger overall energy gap for the film.\nAlso shown in the plot is a calculation which evaluates/s50/s48 /s50/s53 /s51/s48 /s51/s53/s55/s48/s48/s49/s48/s48/s48/s49/s53/s48/s48/s50/s48/s48/s48\n/s32/s32/s40/s72/s122/s47/s109/s75/s41\n/s67/s111/s104/s101/s114/s101/s110/s99/s101/s32/s108/s101/s110/s103/s116/s104/s32/s40/s110/s109/s41\n/s50/s48 /s52/s48 /s54/s48/s49/s46/s48/s49/s46/s53/s49/s46/s57/s32\n/s32/s77/s101/s97/s115/s117/s114/s101/s100\n/s32/s66/s117/s108/s107\n/s32/s86/s111/s114/s111/s110/s116/s115/s111/s118/s32/s107\n/s66/s84\n/s67\n/s68/s32/s47/s32\n/s48\nFIG. 4. ( Top) The coefficient of the linear temperature term,\nα, against the coherence length of the bulk superfluid. ( Bot-\ntom) The measured energy gap, ∆ m, against the scaled film\nthickness, D/ξ0. The error bars associated with ∆ mare from\nthe linear fitting in Fig.3. Also plotted are the bulk en-\nergy gap and the average energy gap in a film calculated by\nVorontsov. The bulk gap used here is from the weak coupling\nplus model40.\nthe energy gap by averaging the parallel and perpendic-\nular components of the order parameter for a superfluid\nfilm with both boundaries diffusive41. Our measurement\nshows a much stronger suppression and pressure depen-\ndence of the energy gap than the theoretical estimation.\nOne might argue the local heating from the MEMS de-\nvice is responsible for the additional damping. However,\nthe film formed in our MEMS device is in good thermal\ncontact with the surrounding bulk because of the open\ngeometry. Furthermore, the temperature rise due to the\nheat dissipation is negligible. For instance, at 0.2 Tcand\n9.2 bar, when the center plate is oscillating at a veloc-\nity of 1.4 mm/s, the damping force on the plate is about\n2 nN, which results in a dissipation powerofabout 3 pW.\nIn addition, all the measurements were performed in the\nlinear regime where the damping coefficient was indepen-\ndent of the excitation42. Any heating effect would have\nresulted in the increase of the FWHM at higher excita-\ntions.\nIt is also unlikely that the additional damping comes\nfrom the vortices around the MEMS devices or other\ntopological objects as suggested by Winkelmann et al.43,\nbecause multiple independent cooldowns produced con-\nsistentspectraatagiventemperatureandpressure. Dur-\ning each thermal cycle the MEMS device was driven at\nhigh velocities beyond the linear regime where heating\neffect was clearly observed. The severe heating and the\nhigh velocity should have altered vortex lines or other\ntopological objects around the device. But after a rea-\nsonable relaxation time, the spectrum always recovers to4\nthe shape right before the heating.\nHowever, it is possible that the mobile plate dissipates\nthrough the surface bound states in the vicinity of the\nplate, leading to the additional damping. The atomic\nforce microscopystudy of the MEMS surfaces shows that\nthe average height variation of the polysilicon surface is\n≈10 nm, while their lateral size is ≈150 nm28. Since\nthese length scales are much larger than the Fermi wave-\nlength of the3He quasiparticles, the surface of the plate\nis diffusive. The density of states, D(E), for the surface\nbound states is almost independent of energy for a diffu-\nsive boundary9(Fig.5). It is reasonable to project that\nthe number of quasiparticles excited in the bound states\nshould be proportional to temperature. Therefore, the\nscatteringofthe quasiparticlesoff the movingplate could\nlead to a linear temperature dependence of the damping,\nif the transverse momentum transfer occurs.\nThe perpendicular component of the order parameter\nis completely suppressed at either specular or diffusive\nboundary9. One can expect that quasiparticles will be\ngenerated with an infinitesimally small amount of energy\ninthesurfaceboundstates, whichareconfinedbythegap\npotential around the boundary within a distance char-\nacterized by the coherence length ξ0(Fig.5). Parallel\nto the plane of the plate with a specular boundary, the\nbound quasiparticles move with a slow velocity v/bardbl≈vL,\nwherevLis the Landau critical velocity. In the direction\nperpendicular to the plane, however, the quasiparticle\nmoves with a fast velocity v⊥≈vF, since the energy gap\nis closed in this direction. The Fermi velocity of the su-\nperfluid varies from 60 to 35 m/s as the pressure changes\nfrom0to30bar, while thecoherencelengthchangesfrom\n90 to 18 nm in the pressure range. Therefore it takes ap-\nproximately 1 ns for a quasiparticle to travel from the\nsurface of the plate to the edge of the potential well,\nwhere it is then retroreflected due to the Andreev scat-\ntering. Thequasiparticlebecomesaquasiholeandfollows\nits previous path, moving towards the plate. It is scat-\ntered normally off the plate and Andreev scattered off\nthe gap potential again before returning to its original\nposition (Fig.5). This completes an entire loop involving\nthe normal and Andreev scattering. Considering the res-\nonance frequency of the MEMS device ( ≈20 kHz), one\nestimates that about 104such scatterings occur within\none cycle of the oscillation. However, for the normal\nscattering at the specular boundary, there is no momen-\ntum transfer in the parallel direction between the plate\nand the quasiparticles, hence no damping for the shear\nmotion of the plate. Therefore, one expects a very small\ndamping force for the specular boundary. This may be\nverified by coating the MEMS plate with a couple of lay-\ners of4He atoms, since the4He atoms drastically alter\nthe boundary conditions44,45.\nFor a diffusive boundary, it is difficult to trace the tra-\njectory of a particular quasiparticle, though the process\nof the multiple Andreev scattering is still valid. Those\nFIG. 5. (A) A schematic picture showing the surface den-\nsity of states of superfluid3He at a diffusive boundary9. The\nquasiparticles excited in the mid-gap band are promoted by\nthe MEMS up to the edge, ∆∗. (B) The SABS confined by\na potential well near the boundary at z= 0. (C) A complete\nscattering cycle of a quasiparticle at a specular boundary i n-\nvolving two normal scatterings and two Andreev scatterings .\nhavingananti-parallelgroupvelocitycomponentwithre-\nspect to the plate velocity vpwill have a higher chance of\nscattering, resulting in a net flux proportional to vp. The\ntiny difference between the momentum of quasiparticles\nandquasiholesaroundtheFermimomentumaccumulates\ndue to the high number of scattering during one cycle of\nthe plate motion. This multiple scattering process leads\nto a net momentum transfer between the plate and the\nbound states which are then promoted to higher energy\nstates until the mid-gap edge ∆∗is reached9,14. We pro-\nposethatthisprocesscouldbetheunderlyingmechanism\nfor the large and linear temperature dependent damping.\nFurthermore, our measurements in the nonlinear regime,\nwhich will be reported elsewhere, can be coherently un-\nderstood with this mechanism. Nonetheless, this model\nneither requires the presence of a film nor involves sur-\nfacebound stateson the otherside ofthe film. Currently,\nwe do not understand the influence of another surface in\nclose proximity on the damping of the plate. To clarify\nthis, we have designed MEMS devices with the substrate\netched away so that both sides of the plate are exposed\nto bulk fluid.\nIn conclusion, a superfluid3He film with a thickness\nof 1.25µm was studied by a MEMS device at various\npressures. At low temperatures, an anomalously large\ndamping on the MEMS was measured in addition to the\nordinary Boltzmann damping. It was attributed to a\nmultiple scattering picture of the interaction between the\nMEMS devices and the surface bound states on the film.\nWewouldliketoacknowledgePeterHirschfeldandAn-\ntonVorontsovforhelpful discussionandcalculations. We\nalsowanttothanktheLancasterLowTemperaturegroup5\nfor providing quartz tuning forks as one of the TF ther-\nmometers. 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Soulen, and\nE. D. Adams, J. Low Temp. Phys. 126, 633 (2002).\n[36] See Supplemental Material at [URL] for the spectrum at\nvarious temperatures.\n[37] See Supplemental Material at [URL] for the plot of the\nMEMSγagainst the TF γ.\n[38] M. Gonz´ alez, P. Zheng, B. H. Moon, E. Garcell, Y. Lee,\nand H. B. Chan, J. Low Temp. Phys. 171, 200 (2013).\n[39] See Supplemental Material at [URL] for the plot of\nγ/TagainstT. The Supplemental Material also includes\nRef.46.\n[40] W. P. Halperin and E. Varoquaux, “Order-parameter\ncollective modes in superfluid3he,” (Elsevier, 1990)\nChap. 7, p. 353.\n[41] Private communication with A. B. Vorontsov.\n[42] See Supplemental Material at [URL] for the linear\nvelocity-force curve.\n[43] C. B. Winkelmann, J. Elbs, Y. M. Bunkov, and H. God-\nfrin, Phys. Rev. Lett. 96, 205301 (2006).\n[44] M. R. Freeman, R. S. Germain, E. V. Thuneberg, and\nR. C. Richardson, Phys. Rev. Lett. 60, 596 (1988).\n[45] D. Kim, M. Nakagawa, O. Ishikawa,\nT. Hata, T. 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However, it is still open whether there ex ists a blow-up\nsolution if the power of nonlinearity is smaller than the exp ected exponent.\n1.Introduction\nWe consider the Cauchy problem for the semilinear damped wave equa tion\n/braceleftBigg\nutt−∆u+a(x)b(t)ut=f(u),(t,x)∈(0,∞)×Rn,\nu(0,x) =u0(x), ut(0,x) =u1(x), x∈Rn,(1.1)\nwhere the coefficients of damping are\na(x) =a0/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−α, b(t) = (1+t)−β,witha0>0,α,β≥0,α+β <1,\nwhere/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht= (1+|x|2)1/2. Hereuis a real-valued unknown function and ( u0,u1) is\ninH1(Rn)×L2(Rn). We note that u0andu1need not be compactly supported.\nThe nonlinear term f(u) is given by\nf(u) =±|u|por|u|p−1u\nand the power psatisfies\n1<p≤n\nn−2(n≥3),1<p<∞(n= 1,2).\nOur aim is to determine the critical exponent pc, which is a number defined by the\nfollowing property:\nIfpc<p, all small data solutions of (1.1) are global; if 1 <p≤pc, the time-local\nsolution cannot be extended time-globally for some data.\nIt is expected that the critical exponent of (1.1) is given by\npc= 1+2\nn−α.\nIn this paper we shall prove the existence of global solutions with sm all data when\np>1+2/(n−α). However, it is still open whether there exists a blow-up solution\nwhen 1<p≤1+2/(n−α).\nKey words and phrases. semilinear damped wave equations; critical exponent; smal l data\nglobal existence.\n12 YUTA WAKASUGI\nWhen the damping term is missing and f(u) =|u|p, that is\n/braceleftBigg\nutt−∆u=|u|p,(t,x)∈(0,∞)×Rn,\nu(0,x) =u0(x), ut(0,x) =u1(x), x∈Rn,(1.2)\nitiswellknownthatthecriticalexponent pw(n)isthepositiverootof( n−1)p2−(n+\n1)p−2 = 0 forn≥2 (pw(1) =∞). This is the famous Strauss conjecture and the\nproofis completed by the effort of many mathematicians (see [2,3,1 3,21,30–34,41]).\nFor the linear wave equation with a damping term/braceleftBigg\nutt−∆u+c(t,x)ut= 0,(t,x)∈(0,∞)×Rn,\nu(0,x) =u0(x), ut(0,x) =u1(x), x∈Rn,(1.3)\nthere are many results about the asymptotic behavior of the solut ion. When\nc(t,x) =c0>0 and (u0,u1)∈(H1∩L1)×(L2∩L1), Matsumura [22] showed\nthat the energy of solutions decays at the same rate as the corre sponding heat\nequation. When the space dimension is 3, using the exact expression of the solu-\ntion, Nishihara[24] discoveredthat the solutionof(1.3) with c(t,x) = 1is expressed\nasymptotically by\nu(t,x)∼v(t,x)+e−t/2w(t,x),\nwherev(t,x) is the solution of the corresponding heat equation\n/braceleftBigg\nvt−∆v= 0,(t,x)∈(0,∞)×R3,\nv(0,x) =u0(x)+u1(x), x∈R3\nandw(t,x) is the solution of the free wave equation\n/braceleftBigg\nwtt−∆w= 0,(t,x)∈(0,∞)×R3,\nw(0,x) =u0(x), wt(0,x) =u1(x),∈R3.\nThese results indicate a diffusive structure of damped waveequatio ns. On the other\nhand, Mochizuki [23] showed that if 0 ≤c(t,x)≤C(1+|x|)−1−δ, whereδ>0, then\nthe energy of solutions of (1.3) does not decay to 0 for nonzero da ta and the solu-\ntion is asymptotically free. We can interpret this result as (1.3) loses its “parabolic-\nity”and recoverits “hyperbolicity”. Wirth [38,39] treated time-d ependent damping\ncase, that is c(t,x) =b(t) in (1.3). By the Fourier transform method, he got sev-\neral sharpLp−Lqestimates of the solution and showed that there exists diffusive\nstructure for general b(t) including b(t) =b0(1+t)−β(−1<β <1). Todorova and\nYordanov [37] considered the case c(t,x) =a(x) =a0/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−αwithα∈[0,1) and\nJ. S. Kenigson and J. J. Kenigson [16] considered space-time depen dent coefficient\ncasec(t,x) =a(x)b(t),a(x) =a0/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−α,b(t) = (1 +t)−β,(0≤α+β <1). They\nestablished the energy decay estimate that also implies diffusive stru cture even in\nthe decaying coefficient cases. From these results, the decay rat e−1 of the coeffi-\ncient of the damping term is the threshold of parabolicity. This is the r eason why\nwe assume α+β <1 for (1.1). We mention that recently, Ikehata, Todorova and\nYordanov[12] treated the case c(t,x) =a0/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−1and obtained almost optimal decay\nestimates.\nThere are also many results for the semilinear damped wave equation with ab-\nsorbing semilinear term:/braceleftBigg\nutt−∆u+a(x)b(t)ut+|u|p−1u= 0,(t,x)∈(0,∞)×Rn,\nu(0,x) =u0(x), ut(0,x) =u1(x), x∈Rn,(1.4)SEMILINEAR DAMPED WAVE EQUATIONS 3\nIt is well known that there exists a unique global solution even for lar ge initial data.\nWhena(x)b(t) = 1, that is constant coefficient case, Kawashima, Nakao and Ono\n[15], Karch[14], Hayashi, KaikinaandNaumkin [7], Ikehata, Nishiharaan dZhao[9]\nand Nishihara [25] showed global existence of solutions and that the ir asymptotic\nprofile is given by a constant multiple of the Gauss kernel for 1+2 /n<pandn≤4.\nFor 1< p≤1 + 2/n, Nishihara and Zhao [28], Ikehata, Nishihara and Zhao [9],\nNishihara [25] proved that the decay rate of the solution agrees wit h that of a self-\nsimilarsolutionofthe correspondingheatequation. Hayashi,Kaikina andNaumkin\n[4–7] proved the large time asymptotic formulas in terms of the weigh ted Sobolev\nspaces. These results indicate the critical exponent for (1.4) with a(x)b(t) = 1 is\ngiven bypc= 1+2\nn. In this case the critical exponent means the turning point of\nthe asymptotic behavior of the solution. When b(t) = 1,a(x) =/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−α(0≤α<1),\nnamely space-dependent damping case, Nishihara [26] established d ecay estimates\nof solutions and conjectured the critical exponent is given by pc= 1+2/(n−α).\nWhena(x) = 1,b(t) = (1 +t)−β(−1< β <1), Nishihara and Zhai [29] proved\ndecay estimates of solutions and conjectured the critical expone nt ispc= 1+2/n.\nFinally in the case a(x) =/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−α,b(t) = (1+t)−β(0≤α+β <1), Lin, Nishihara\nand Zhai [19,20] showed decay estimates of the solution and conje ctured the critical\nexponent is pc= 1 + 2/(n−α). They used a weighted energy method, which is\noriginally developed by Todorova and Yordanov [35,36]. In this paper we shall\nessentially use the techniques and method that they used.\nLi and Zhou [17] considered the semilinear damped wave equation\nutt−∆u+ut=|u|p. (1.5)\nThey proved that if n≤2,1<p≤1+2\nnand the data are positive on average, then\nthe local solution of (1.5) must blow up in a finite time. Todorova and Yo rdanov\n[35,36] developed a weighted energy method using the function whic h has the form\ne2ψand determined that the critical exponent of (1.5) is\npc= 1+2\nn,\nwhich is well known as Fujita’s critical exponent for the heat equatio nut−∆u=\nup(see [1]). More precisely, they proved small data global existence in the case\np>1+2/nand blow-up for all solutions of (1.5) with positive on average data in\nthe case 1< p <1+2/n. Later on Zhang [40] showed that the critical exponent\np= 1+2/nbelongstotheblow-upregion. WementionthatTodorovaandYorda nov\n[35,36] assumed data have compact support and essentially used t his property.\nHowever, Ikehata and Tanizawa [10] removed this assumption. Ik ehata, Todorova\nand Yordanov [11] investigated the space-dependent coefficient c ase:\nutt−∆u+a(x)ut=|u|p, (1.6)\nwhere\na(x)∼a0/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−α,|x| → ∞,radially symmetric and 0 ≤α<1.\nThey proved that the critical exponent of (1.5) is given by\npc= 1+2\nn−α\nby using a refined multiplier method. Their method also depends on the finite\npropagation speed property. Recently, Nishihara [27] and Lin, Nis hihara and Zhai4 YUTA WAKASUGI\n[20] considered the semilinear wave equation with time-dependent da mping\nutt−∆u+b(t)ut=|u|p, (1.7)\nwhere\nb(t) =b0(1+t)−β, β∈(−1,1).\nThey proved that the critical exponent of (1.7) is\npc= 1+2\nn.\nThis shows that, roughly speaking, time-dependent coefficients of damping term do\nnot influence the critical exponent. Therefore we expect that th e critical exponent\nof the semilinear wave equation (1.1) is\npc= 1+2\nn−α.\nTo state our results, we introduce an auxiliary function\nψ(t,x) :=A/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht2−α\n(1+t)1+β(1.8)\nwith\nA=(1+β)a0\n(2−α)2(2+δ), δ>0 (1.9)\nThis type of weight function is first introduced by Ikehata and Taniz awa [10]. We\nhave the following result:\nTheorem 1.1. If\np>1+2\nn−α,\nthen there exists a small positive number δ0>0such that for any 0< δ≤δ0the\nfollowing holds: If\nI2\n0:=/integraldisplay\nRne2ψ(0,x)(u2\n1+|∇u0|2+|u0|2)dx\nis sufficiently small, then there exists a unique solution u∈C([0,∞);H1(Rn))∩\nC1([0,∞);L2(Rn))to(1.1)satisfying/integraldisplay\nRne2ψ(t,x)|u(t,x)|2dx≤Cδ(1+t)−(1+β)n−2α\n2−α+ε,(1.10)\n/integraldisplay\nRne2ψ(t,x)(|ut(t,x)|2+|∇u(t,x)|2)dx≤Cδ(1+t)−(1+β)(n−α\n2−α+1)+ε,\nwhere\nε=ε(δ) :=3(1+β)(n−α)\n2(2−α)(2+δ)δ (1.11)\nandCδis a constant depending on δ.\nRemark 1.2. When1<p≤1+2/(n−α), it is expected that no matter how small\nthe data are, if the data have some shape, then the correspond ing local solution\nblows up in finite time. However, we have no result.\nRemark 1.3. We do not assume that the data are compactly supported. Hence\nour result is an extension of the results of Ikehata, Todorov a and Yordanov [11]\nto noncompactly supported data cases. However, we prove onl y the case a(x) =\na0/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−α.SEMILINEAR DAMPED WAVE EQUATIONS 5\nAs a consequence of the main theorem, we have an exponential dec ay estimate\noutside a parabolic region.\nCorollary 1.4. If\np>1+2\nn−α,\nthen there exists a small positive number δ0>0such that for any 0< δ≤δ0the\nfollowing holds: Take ρandµso small that\n0<ρ<1−α−β,and0<µ<2A,\nand put\nΩρ(t) :={x∈Rn;/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht2−α≥(1+t)1+β+ρ}.\nThen, for the global solution uin Theorem 1.1, we have the following estimate/integraldisplay\nΩρ(t)(u2\nt+|∇u|2+u2)dx≤Cδ,ρ,µ(1+t)−(1+β)(n−2α)\n2−α+εe−(2A−µ)(1+t)ρ,(1.12)\nhereεis defined by (1.11)andCδ,ρ,µis a constant depending on δ,ρandµ.\nNamely, the decay rate of solution in the region Ω ρ(t) is exponential. We note\nthat the support of u(t) and the region Ω ρ(t) can intersect even if the data are\ncompactly supported. This phenomenon was first discovered by To dorova and\nYordanov [36]. We can interpret this result as follows: The support o f the solution\nis strongly suppressed by damping, so that the solution is concentr ated in the\nparabolic region much smaller than the light cone.\n2.Proof of Theorem 1.1\nIn this section we prove our main result. At first we prepare some no tation and\nterminology. We put\n/ba∇dblf/ba∇dblLp(Rn):=/parenleftbigg/integraldisplay\nRn|f(x)|pdx/parenrightbigg1/p\n,/ba∇dblu/ba∇dbl:=/ba∇dblu/ba∇dblL2(Rn).\nByH1(Rn) we denote the usual Sobolev space. For an interval Iand a Banach\nspaceX, we define Cr(I;X) as the Banach space whose element is an r-times\ncontinuously differentiable mapping from ItoXwith respect to the topology in\nX. The letter Cindicates the generic constant, which may change from line to the\nnext line.\nTo prove Theorem 1.1, we use a weighted energy method which was or iginally\ndeveloped by Todorova and Yordanov [35,36]. We first describe th e local existence:\nProposition 2.1. For anyδ >0, there exists Tm∈(0,+∞]depending on I2\n0\nsuch that the Cauchy problem (1.1)has a unique solution u∈C([0,Tm);H1(Rn))∩\nC1([0,Tm);L2(Rn)), and ifTm<+∞then we have\nliminf\nt→Tm/integraldisplay\nRneψ(t,x)(u2\nt+|∇u|2+u2)dx= +∞.\nWe can prove this proposition by standard arguments (see [10]). W e prove a\npriori estimate for the following functional:\nM(t) := sup\n0≤τ<t/braceleftbigg\n(1+τ)B+1−ε/integraldisplay\nRne2ψ(u2\nt+|∇u|2)dx\n+(1+τ)B−ε/integraldisplay\nRne2ψa(x)b(t)u2dx/bracerightbigg\n, (2.1)6 YUTA WAKASUGI\nwhere\nB:=(1+β)(n−α)\n2−α+β\nandεis given by (1.11). From (1.8), (1.9), it is easy to see that\n−ψt=1+β\n1+tψ, (2.2)\n∇ψ=A(2−α)/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−αx\n(1+t)1+β, (2.3)\n∆ψ=A(2−α)(n−α)/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−α\n(1+t)1+β+A(2−α)α/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−2−α\n(1+t)1+β\n≥(1+β)(n−α)\n(2−α)(2+δ)a(x)b(t)\n1+t\n=:/parenleftbigg(1+β)(n−α)\n2(2−α)−δ1/parenrightbigga(x)b(t)\n1+t. (2.4)\nHere and after, δi(i= 1,2,...) is a positive constant depending only on δsuch that\nδi→0+asδ→0+.\nWe also have\n(−ψt)a(x)b(t) =Aa0(1+β)/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht2−2α\n(1+t)2+2β\n≥a0(1+β)\n(2−α)2AA2(2−α)2/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−2α|x|2\n(1+t)2+2β\n= (2+δ)|∇ψ|2. (2.5)\nBy multiplying (1.1) by e2ψut, it follows that\n∂\n∂t/bracketleftbigge2ψ\n2(u2\nt+|∇u|2)/bracketrightbigg\n−∇·(e2ψut∇u)\n+e2ψ/parenleftbigg\na(x)b(t)−|∇ψ|2\n−ψt−ψt/parenrightbigg\nu2\nt+e2ψ\n−ψt|ψt∇u−ut∇ψ|2\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nT1\n=∂\n∂t/bracketleftbig\ne2ψF(u)/bracketrightbig\n+2e2ψ(−ψt)F(u), (2.6)\nwhereFis the primitive of fsatisfyingF(0) = 0, namely F′(u) =f(u). Using the\nSchwarz inequality and (2.5), we can calculate\nT1=e2ψ\n−ψt(ψ2\nt|∇u|2−2ψtut∇u·∇ψ+u2\nt|∇ψ|2)\n≥e2ψ\n−ψt/parenleftbigg1\n5ψ2\nt|∇u|2−1\n4u2\nt|∇ψ|2/parenrightbigg\n≥e2ψ/parenleftbigg1\n5(−ψt)|∇u|2−a(x)b(t)\n4(2+δ)u2\nt/parenrightbigg\n.SEMILINEAR DAMPED WAVE EQUATIONS 7\nFrom this and (2.5), we obtain\n∂\n∂t/bracketleftbigge2ψ\n2(u2\nt+|∇u|2)/bracketrightbigg\n−∇·(e2ψut∇u)\n+e2ψ/braceleftbigg/parenleftbigg1\n4a(x)b(t)−ψt/parenrightbigg\nu2\nt+−ψt\n5|∇u|2/bracerightbigg\n≤∂\n∂t/bracketleftbig\ne2ψF(u)/bracketrightbig\n+2e2ψ(−ψt)F(u). (2.7)\nBy multiplying (2.7) by ( t0+t)B+1−ε, heret0≥1 is determined later, it follows\nthat\n∂\n∂t/bracketleftbigg\n(t0+t)B+1−εe2ψ\n2(u2\nt+|∇u|2)/bracketrightbigg\n−(B+1−ε)(t0+t)B−εe2ψ\n2(u2\nt+|∇u|2)\n−∇·((t0+t)B+1−εe2ψut∇u)\n+e2ψ(t0+t)B+1−ε/braceleftbigg/parenleftbigg1\n4a(x)b(t)−ψt/parenrightbigg\nu2\nt+−ψt\n5|∇u|2/bracerightbigg\n≤∂\n∂t/bracketleftbig\n(t0+t)B+1−εe2ψF(u)/bracketrightbig\n−(B+1−ε)(t0+t)B−εe2ψF(u)\n+2(t0+t)B+1−εe2ψ(−ψt)F(u). (2.8)\nWe put\nE(t) :=/integraldisplay\nRne2ψ(u2\nt+|∇u|2)dx, Eψ(t) :=/integraldisplay\nRne2ψ(−ψt)(u2\nt+|∇u|2)dx,\nJ(t;g) :=/integraldisplay\nRne2ψgdx, J ψ(t;g) :=/integraldisplay\nRne2ψ(−ψt)gdx.\nIntegrating (2.8) over the whole space, we have\n1\n2d\ndt/bracketleftbig\n(t0+t)B+1−εE(t)/bracketrightbig\n−1\n2(B+1−ε)(t0+t)B−εE(t)\n+1\n4(t0+t)B+1−εJ(t,a(x)b(t)u2\nt)+1\n5(t0+t)B+1−εEψ(t)\n≤d\ndt/bracketleftbigg\n(t0+t)B+1−ε/integraldisplay\ne2ψF(u)dx/bracketrightbigg\n+C(t0+t)B+1−εJψ(t;|u|p+1)+C(t0+t)B−εJ(t;|u|p+1) (2.9)8 YUTA WAKASUGI\nTherefore, we integrate (2.9) on the interval [0 ,t] and obtain the estimate for ( t0+\nt)B+1−εE(t), which is the first term of M(t):\n(t0+t)B+1−εE(t)−C/integraldisplayt\n0(t0+τ)B−εE(τ)dτ\n+/integraldisplayt\n0(t0+τ)B+1−εJ(τ;a(x)b(t)u2\nt)+(t0+τ)B+1−εEψ(τ)dτ\n≤CI2\n0+C(t0+t)B+1−εJ(t;|u|p+1)\n+C/integraldisplayt\n0(t0+τ)B+1−εJψ(τ;|u|p+1)dτ\n+C/integraldisplayt\n0(t0+t)B−εJ(τ;|u|p+1)dτ. (2.10)\nIn orderto complete a priori estimate, however, we have to manag e the second term\nof the inequality above whose sign is negative, and we also have to est imate the\nsecond term of M(t). The following argument, which is little more complicated,\ncan settle both these problems.\nAt first, we multiply (1.1) by e2ψuand have\n∂\n∂t/bracketleftbigg\ne2ψ/parenleftbigg\nuut+a(x)b(t)\n2u2/parenrightbigg/bracketrightbigg\n−∇·(e2ψu∇u)\n+e2ψ\n\n|∇u|2+/parenleftbigg\n−ψt+β\n2(1+t)/parenrightbigg\na(x)b(t)u2+2u∇ψ·∇u/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nT2−2ψtuut−u2\nt\n\n\n=e2ψuf(u). (2.11)\nWe calculate\ne2ψT2= 4e2ψu∇ψ·∇u−2e2ψu∇ψ·∇u\n= 4e2ψu∇ψ·∇u−∇·(e2ψu2∇ψ)+2e2ψu2|∇ψ|2+e2ψ(∆ψ)u2\nand by (2.4) we can rewrite (2.11) to\n∂\n∂t/bracketleftbigg\ne2ψ/parenleftbigg\nuut+a(x)b(t)\n2u2/parenrightbigg/bracketrightbigg\n−∇·(e2ψ(u∇u+u2∇ψ))\n+e2ψ/braceleftBig\n|∇u|2+4u∇u·∇ψ+(−ψt)a(x)b(t)+2|∇ψ|2)u2\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nT3\n+(B−2δ1)a(x)b(t)\n2(1+t)u2−2ψtuut−u2\nt/bracerightBig\n≤e2ψuf(u). (2.12)SEMILINEAR DAMPED WAVE EQUATIONS 9\nIt follows from (2.5) that\nT3=|∇u|2+4u∇u·∇ψ\n+/braceleftbigg/parenleftbigg\n1−δ\n3/parenrightbigg\n(−ψt)a(x)b(t)+2|∇ψ|2/bracerightbigg\nu2+δ\n3(−ψt)a(x)b(t)u2\n≥ |∇u|2+4u∇u·∇ψ\n+/parenleftbigg\n4+δ\n3−δ2\n3/parenrightbigg\n|∇ψ|2u2+δ\n3(−ψt)a(x)b(t)u2\n=/parenleftbigg\n1−4\n4+δ2/parenrightbigg\n|∇u|2+δ2|∇ψ|2u2\n+/vextendsingle/vextendsingle/vextendsingle/vextendsingle2√4+δ2∇u+/radicalbig\n4+δ2u∇ψ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n+δ\n3(−ψt)a(x)b(t)u2\n≥δ3(|∇u|2+|∇ψ|2u2)+δ\n3(−ψt)a(x)b(t)u2,\nwhere\nδ2:=δ\n6−δ2\n6, δ3:= min(1 −4\n4+δ2,δ2).\nThus, we obtain\n∂\n∂t/bracketleftbigg\ne2ψ/parenleftbigg\nuut+a(x)b(t)\n2u2/parenrightbigg/bracketrightbigg\n−∇·(e2ψ(u∇u+u2∇ψ))\n+e2ψδ3|∇u|2\n+e2ψ/parenleftbigg\nδ3|∇ψ|2+δ\n3(−ψt)a(x)b(t)+(B−2δ1)a(x)b(t)\n2(1+t)/parenrightbigg\nu2\n+e2ψ(−2ψtuut−u2\nt)\n≤e2ψuf(u). (2.13)\nFollowing Nishihara [19], related to the size of 1+ |x|2and the size of (1+ t)2, we\ndivide the space Rninto two different zones Ω( t;K,t0) and Ωc(t;K,t0), where\nΩ = Ω(t;K,t0) :={x∈Rn;(t0+t)2≥K+|x|2},\nand Ωc= Ωc(t;K,t0) :=Rn\\Ω(t;K,t0) withK≥1 determined later. Since\na(x)b(t)≥a0(t+t0)−(α+β)in the domain Ω, we multiply (2.7) by ( t0+t)α+βand\nobtain\n∂\n∂t/bracketleftbigge2ψ\n2(t0+t)α+β(u2\nt+|∇u|2)/bracketrightbigg\n−∇·(e2ψ(t0+t)α+βut∇u)\n+e2ψ/bracketleftbigg/parenleftbigga0\n4−α+β\n2(t0+t)1−α−β/parenrightbigg\n+(t0+t)α+β(−ψt)/bracketrightbigg\nu2\nt\n+e2ψ/bracketleftbigg−ψt\n5(t0+t)α+β−α+β\n2(t0+t)1−α−β/bracketrightbigg\n|∇u|2\n≤∂\n∂t[(t0+t)α+βe2ψF(u)]−α+β\n(t0+t)1−α−βe2ψF(u)\n+2(t0+t)α+βe2ψ(−ψt)F(u). (2.14)10 YUTA WAKASUGI\nLetνbe a small positive number depends on δ, which will be chosen later. By\n(2.14)+ν(2.13), we have\n∂\n∂t/bracketleftbigg\ne2ψ/parenleftbigg(t0+t)α+β\n2u2\nt+νuut+νa(x)b(t)\n2u2+(t0+t)α+β\n2|∇u|2/parenrightbigg/bracketrightbigg\n−∇·(e2ψ(t0+t)α+βut∇u+νe2ψ(u∇u+u2∇ψ))\n+e2ψ/bracketleftbigg/parenleftbigga0\n4−α+β\n2(t0+t)1−α−β−ν/parenrightbigg\n+(t0+t)α+β(−ψt)/bracketrightbigg\nu2\nt\n+e2ψ/bracketleftbigg\nνδ3−α+β\n2(t0+t)1−α−β+−ψt\n5(t0+t)α+β/bracketrightbigg\n|∇u|2\n+e2ψ/bracketleftbigg\nν(δ3|∇ψ|2+δ\n3(−ψt)a(x)b(t)+(B−2δ1)a(x)b(t)\n2(1+t)/bracketrightbigg\nu2\n+2νe2ψ(−ψt)uut\n≤∂\n∂t[(t0+t)α+βe2ψF(u)]−α+β\n(t0+t)1−α−βe2ψF(u)\n+2(t0+t)α+βe2ψ(−ψt)F(u)+νe2ψuf(u). (2.15)\nBythe Schwarzinequality, the last term ofthe left hand side in the ab oveinequality\ncan be estimated as\n|2ν(−ψt)uut| ≤νδ\n3(−ψt)a(x)b(t)u2+3ν\na0δ(−ψt)(t0+t)α+βu2\nt.\nThus, we have\n∂\n∂t/bracketleftbigg\ne2ψ/parenleftbigg(t0+t)α+β\n2u2\nt+νuut+νa(x)b(t)\n2u2+(t0+t)α+β\n2|∇u|2/parenrightbigg/bracketrightbigg\n−∇·(e2ψ(t0+t)α+βut∇u+νe2ψ(u∇u+u2∇ψ))\n+e2ψ/bracketleftbigg/parenleftbigga0\n4−α+β\n2(t0+t)1−α−β−ν/parenrightbigg\n+/parenleftbigg\n1−3ν\na0δ/parenrightbigg\n(t0+t)α+β(−ψt)/bracketrightbigg\nu2\nt\n+e2ψ/bracketleftbigg\nνδ3−α+β\n2(t0+t)1−α−β+−ψt\n5(t0+t)α+β/bracketrightbigg\n|∇u|2\n+e2ψ/bracketleftbigg\nν/parenleftbigg\nδ3|∇ψ|2+(B−2δ1)a(x)b(t)\n2(1+t)/parenrightbigg/bracketrightbigg\nu2\n≤∂\n∂t[(t0+t)α+βe2ψF(u)]−α+β\n(t0+t)1−α−βe2ψF(u)\n+2(t0+t)α+βe2ψ(−ψt)F(u)+νe2ψuf(u). (2.16)\nNow we choose the parameters νandt0such that\na0\n4−α+β\n2(t0+t)1−α−β−ν≥c0,1−3ν\na0δ≥c0,\nνδ3−α+β\n2(t0+t)1−α−β≥c0, νδ3≥c0,1\n5≥c0,\nhold for some constant c0>0. This is possible because we first determine ν\nsufficiently small depending on δand then we choose t0sufficiently large depending\nonν. Therefore, integrating (2.16) on Ω, we obtain the following energy inequality:\nd\ndtEψ(t;Ω(t;K,t0))−N1(t)−M1(t)+Hψ(t;Ω(t;K,t0))≤P1,(2.17)SEMILINEAR DAMPED WAVE EQUATIONS 11\nwhere\nEψ(t;Ω) =Eψ(t;Ω(t;K,t0))\n:=/integraldisplay\nΩe2ψ/parenleftbigg(t0+t)α+β\n2u2\nt+νuut+νa(x)b(t)\n2u2+(t0+t)α+β\n2|∇u|2/parenrightbigg\ndx,\nN1(t) :=/integraldisplay\nSn−1e2ψ/parenleftbigg(t0+t)α+β\n2u2\nt+νuut+νa(x)b(t)\n2u2\n+(t0+t)α+β\n2|∇u|2/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n|x|=√\n(t0+t)2−K\n×[(t0+t)2−K](n−1)/2dθ·d\ndt/radicalbig\n(t0+t)2−K,\nM1(t) :=/integraldisplay\n∂Ω(e2ψ(t0+t)α+βut∇u+νe2ψ(u∇u+u2∇ψ))·/vector ndS,\nHψ(t;Ω) =Hψ(t;Ω(t;K,t0))\n:=c0/integraldisplay\nΩe2ψ(1+(t0+t)α+β(−ψt))(u2\nt+|∇u|2)dx\n+ν(B−2δ1)/integraldisplay\nΩe2ψa(x)b(t)\n2(1+t)u2dx,\nP1:=d\ndt/bracketleftbigg\n(t0+t)α+β/integraldisplay\nΩe2ψF(u)dx/bracketrightbigg\n−/integraldisplay\nSn−1(t0+t)α+βe2ψF(u)/vextendsingle/vextendsingle/vextendsingle\n|x|=√\n(t0+t)2−K\n×[(t0+t)2−K](n−1)/2dθ·d\ndt/radicalbig\n(t0+t)2−K\n+C/integraldisplay\nΩe2ψ(1+(t0+t)α+β(−ψt))|u|p+1dx.\nHere/vector ndenotes the unit outer normal vector of ∂Ω. We note that by ν≤a0/4 and\n|νuut| ≤νa(x)b(t)\n4u2+ν(t0+t)α+β\na0u2\nt,\nit follows that\nc/integraldisplay\nΩe2ψ(t0+t)α+β(u2\nt+|∇u|2)dx+c/integraldisplay\nΩe2ψa(x)b(t)u2dx\n≤Eψ(t;Ω(t;K,t0))\n≤C/integraldisplay\nΩe2ψ(t0+t)α+β(u2\nt+|∇u|2)dx+C/integraldisplay\nΩe2ψa(x)b(t)u2dx\nfor some constants c>0 andC >0.\nNext, we derive an energy inequality in the domain Ωc. We use the notation\n/a\\}b∇acketle{tx/a\\}b∇acket∇i}htK:= (K+|x|2)1/2.12 YUTA WAKASUGI\nSincea(x)b(t)≥a0/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht−(α+β)\nKin Ωc(t,;K,t0), we multiply (2.7) by /a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nKand\nobtain\n∂\n∂t/bracketleftbigge2ψ\n2/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK(u2\nt+|∇u|2)/bracketrightbigg\n−∇·(e2ψ/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nKut∇u)\n+e2ψ/parenleftBiga0\n4+(−ψt)/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK/parenrightBig\nu2\nt+1\n5e2ψ(−ψt)/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK|∇u|2\n+(α+β)e2ψ/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β−2\nKx·ut∇u\n≤∂\n∂t[e2ψ/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nKF(u)]+2e2ψ/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK(−ψt)F(u). (2.18)\nBy (2.18) + ˆν×(2.13), here ˆνis a small positive parameter determined later, it\nfollows that\n∂\n∂t/bracketleftBigg\ne2ψ/parenleftBigg\n/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK\n2u2\nt+ ˆνuut+ˆνa(x)b(t)\n2u2+/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK\n2|∇u|2/parenrightBigg/bracketrightBigg\n−∇·(e2ψ/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nKut∇u+ ˆνe2ψ(u∇u+u2∇ψ))\n+e2ψ/bracketleftBiga0\n4−ˆν+(−ψt)/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK/bracketrightBig\nu2\nt+e2ψ/bracketleftbigg\nˆνδ3+−ψt\n5/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK/bracketrightbigg\n|∇u|2\n+e2ψ/bracketleftbigg\nˆν/parenleftbigg\nδ3|∇ψ|2+δ\n3(−ψt)a(x)b(t)+(B−2δ1)a(x)b(t)\n2(1+t)/parenrightbigg/bracketrightbigg\nu2\n+e2ψ[(α+β)/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β−2\nKx·ut∇u−2ˆνψtuut/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nT4]\n≤∂\n∂t/bracketleftBig\ne2ψ/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nKF(u)/bracketrightBig\n+2e2ψ/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK(−ψt)F(u)+ ˆνe2ψuf(u).(2.19)\nThe termsT4can be estimated as\n|(α+β)/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β−2\nKx·ut∇u| ≤ˆνδ3\n2|∇u|2+(α+β)2\n2ˆνδ3K2(1−α−β)u2\nt,\n|2ˆν(−ψt)uut| ≤ˆνδ\n3(−ψt)a(x)b(t)u2+3ˆν\na0δ(−ψt)/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nKu2\nt.\nFrom this we can rewrite (2.19) as\n∂\n∂t/bracketleftBigg\ne2ψ/parenleftBigg\n/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK\n2u2\nt+ ˆνuut+ˆνa(x)b(t)\n2u2+/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK\n2|∇u|2/parenrightBigg/bracketrightBigg\n−∇·(e2ψ/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nKut∇u+ ˆνe2ψ(u∇u+u2∇ψ))\n+e2ψ/bracketleftbigg/parenleftbigga0\n4−ˆν−(α+β)2\n2ˆνδ3K2(1−α−β)/parenrightbigg\n+/parenleftbigg\n1−3ˆν\na0δ/parenrightbigg\n(−ψt)/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK/bracketrightbigg\nu2\nt\n+e2ψ/bracketleftbiggˆνδ3\n2+−ψt\n5/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK/bracketrightbigg\n|∇u|2\n+e2ψ/bracketleftbigg\nˆν/parenleftbigg\nδ3|∇ψ|2+(B−2δ1)a(x)b(t)\n2(1+t)/parenrightbigg/bracketrightbigg\nu2\n≤∂\n∂t/bracketleftBig\ne2ψ/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nKF(u)/bracketrightBig\n+2e2ψ/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK(−ψt)F(u)+ ˆνe2ψuf(u).(2.20)\nNow we choose the parameters ˆ νandKin the same manner as before. Indeed\ntaking ˆνsufficiently small depending on δand then choosing Ksufficiently largeSEMILINEAR DAMPED WAVE EQUATIONS 13\ndepending on ˆ ν, we can obtain\na0\n4−ˆν−(α+β)2\n2ˆνδ3K2(1−α−β)≥c1,1−3ˆν\na0δ≥c1, νδ3≥c1,1\n5≥c1\nfor some constant c1>0. Consequently, By integrating (2.20) on Ωc, the energy\ninequality on Ωcfollows:\nd\ndtEψ(t;Ωc(t;K,t0))+N2(t)+M2(t)+Hψ(t;Ωc(t;K,t0))≤P2,(2.21)\nwhere\nEψ(t;Ωc) =Eψ(t;Ωc(t;K,t0))\n:=/integraldisplay\nΩce2ψ/parenleftBigg\n/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK\n2u2\nt+ ˆνuut+ˆνa(x)b(t)\n2u2+/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK\n2|∇u|2/parenrightBigg\ndx,\nN2(t) :=/integraldisplay\nSn−1e2ψ/parenleftBigg\n/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK\n2u2\nt+ ˆνuut+ˆνa(x)b(t)\n2u2\n+/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK\n2|∇u|2)/vextendsingle/vextendsingle/vextendsingle\n|x|=√\n(t0+t)2−K\n×[(t0+t)2−K](n−1)/2dθ·d\ndt/radicalbig\n(t0+t)2−K,\nM2(t) :=/integraldisplay\n∂Ωc(e2ψ/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nKut∇u+ ˆνe2ψ(u∇u+u2∇ψ))·/vector ndS,\nHψ(t;Ωc) =Hψ(t;Ωc(t;K,t0))\n:=c1/integraldisplay\nΩe2ψ(1+/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK(−ψt))(u2\nt+|∇u|2)dx\n+ˆν(B−2δ1)/integraldisplay\nΩce2ψa(x)b(t)\n2(1+t)u2dx,\nP2:=d\ndt/bracketleftbigg/integraldisplay\nΩce2ψ/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nKF(u)dx/bracketrightbigg\n+/integraldisplay\nSn−1/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nKe2ψF(u)/vextendsingle/vextendsingle/vextendsingle\n|x|=√\n(t0+t)2−K\n×[(t0+t)2−K](n−1)/2dθ·d\ndt/radicalbig\n(t0+t)2−K\n+C/integraldisplay\nΩce2ψ(1+/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK(−ψt))|u|p+1dx.\nIn a similar way as the case in Ω, we note that\nc/integraldisplay\nΩce2ψ(t0+t)α+β(u2\nt+|∇u|2)dx+c/integraldisplay\nΩce2ψa(x)b(t)u2dx\n≤Eψ(t;Ωc(t;K,t0))\n≤C/integraldisplay\nΩce2ψ(t0+t)α+β(u2\nt+|∇u|2)dx+C/integraldisplay\nΩce2ψa(x)b(t)u2dx\nfor some constants c>0 andC >0.14 YUTA WAKASUGI\nWe add the energy inequalities on Ω and Ωc. We note that replacing νand ˆνby\nν0:= min{ν,ˆν}, we can still have the inequalities (2.17) and (2.21), provided that\nwe retaket0andKlarger.\nBy((2.17)+(2.21))×(t0+t)B−ε, we have\nd\ndt[(t0+t)B−ε(Eψ(t;Ω)+Eψ(t;Ωc))]\n−(B−ε)(t0+t)B−1−ε(Eψ(t;Ω)+Eψ(t;Ωc))/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nT5\n+ (t0+t)B−ε(Hψ(t;Ω)+Hψ(t;Ωc))/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nT6\n≤(t0+t)B−ε(P1+P2), (2.22)\nhere we note that\nN1(t) =N2(t), M1(t) =M2(t)\non∂Ω. Since\n|ν0uut| ≤ν0δ4\n2a(x)b(t)u2+ν0\n2δ4a0(t0+t)α+βu2\nt\non Ω and\n|ν0uut| ≤ν0δ4\n2a(x)b(t)u2+ν0\n2δ4a0/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nKu2\nt\non Ωc, we have\n−T5+T6≥(t0+t)B−εI1+(t0+t)B−εI2, (2.23)\nwhere\nI1:=/integraldisplay\nΩe2ψ/braceleftBigc0\n2(1+(t0+t)α+β(−ψt))−B−ε\n2(t0+t)/parenleftbigg\n1+2ν0\nδ4a0/parenrightbigg\n(t0+t)α+β/bracerightbigg\nu2\nt\n+e2ψ/braceleftbiggc0\n2(1+(t0+t)α+β(−ψt))−B−ε\n2(t0+t)(t0+t)α+β/bracerightbigg\n|∇u|2dx\n+/integraldisplay\nΩce2ψ/braceleftbiggc1\n2(1+/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK(−ψt))−B−ε\n2(t0+t)/parenleftbigg\n1+2ν0\nδ4a0/parenrightbigg\n/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK/bracerightbigg\nu2\nt\n+e2ψ/braceleftbiggc1\n2(1+/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK(−ψt))−B−ε\n2(t0+t)/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK/bracerightbigg\n|∇u|2dx\n=:I11+I12,\nI2:=ν0(B−2δ1−(1+δ4)(B−ε))/parenleftbigg/integraldisplay\nΩ+/integraldisplay\nΩc/parenrightbigg\ne2ψa(x)b(t)\n2(1+t)u2dx\n+c2\n2/integraldisplay\nRne2ψ(u2\nt+|∇u|2)dx,\nwherec2:= min(c0,c1). Recall the definition of εandδ1(i.e. (1.11) and (2.4)). A\nsimple calculation shows ε= 3δ1. Choosing δ4sufficiently small depending on ε,\nwe have\n(t0+t)B−εI2≥c3(t0+t)B−1−ε/integraldisplay\nRne2ψa(x)b(t)u2dx+c2\n2(t0+t)B−εE(t)\nfor some constant c3>0. Next, we prove that I1≥0. By noting that α+β <1,\nit is easy to see that I11≥0 if we retake t0larger depending on c0,ν0andδ4. To\nestimateI12, we further divide the region Ωcinto\nΩc(t;K,t0) = (Ωc(t;K,t0)∩ΣL)∪(Ωc(t;K,t0)∩Σc\nL),SEMILINEAR DAMPED WAVE EQUATIONS 15\nwhere\nΣL:={x∈Rn;/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht2−α≤L(1+t)1+β},Σc\nL:=Rn\\ΣL\nwithL≫1 determined later. First, since K+|x|2≤K(1+|x|2)≤KL2/(2−α)(1+\nt)2(1+β)/(2−α)on Ωc∩ΣL, we have\nc1\n2(1+/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK(−ψt))−B−ε\n2(t0+t)/parenleftbigg\n1+2ν0\nδ4a0/parenrightbigg\n/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK\n≥c1\n2−B−ε\n2(t0+t)/parenleftbigg\n1+2ν0\nδ4a0/parenrightbigg\nK(α+β)/2L(α+β)/(2−α)(1+t)(1+β)(α+β)\n2−α.\nWe note that −1+(1+β)(α+β)\n2−α<0 byα+β <1. Thus, we obtain\nc1\n2−B−ε\n2(t0+t)/parenleftbigg\n1+2ν0\nδ4a0/parenrightbigg\nK(��+β)/2L(α+β)/(2−α)(1+t)(1+β)(α+β)\n2−α≥0\nfor larget0depending on LandK. Secondly, on Ωc∩Σc\nL, we have\nc1\n2(1+/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK(−ψt))−B−ε\n2(t0+t)/parenleftbigg\n1+2ν0\nδ4a0/parenrightbigg\n/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK\n≥/braceleftbiggc1\n2(1+β)/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht2−α\n(1+t)2+β−B−ε\n2(t0+t)/parenleftbigg\n1+2ν0\nδ4a0/parenrightbigg/bracerightbigg\n/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK\n≥/braceleftbiggc1\n2(1+β)L\n1+t−B−ε\n2(t0+t)/parenleftbigg\n1+2ν0\nδ4a0/parenrightbigg/bracerightbigg\n/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK.\nTherefore one can obtain I12≥0, provided that L≥B−ε\nc1(1+β)(1 +2ν0\nδ4a0). Conse-\nquently, we have I1≥0. By (2.23) and that we mentioned above, it follows that\n−T5+T6≥c3(t0+t)B−1−ε/integraldisplay\nRne2ψa(x)b(t)u2dx+c2\n2(t0+t)B−εE(t).\nTherefore, we have\nd\ndt[(t0+t)B−ε(Eψ(t;Ω)+Eψ(t;Ωc)]+c2\n2(t0+t)B−εE(t)\n+c3(t0+t)B−1−εJ(t;a(x)b(t)u2)\n≤(t0+t)B−ε(P1+P2). (2.24)\nIntegrating (2.24) on the interval [0 ,t], one can obtain the energy inequality on the\nwhole space:\n(t0+t)B−ε(Eψ(t;Ω)+Eψ(t;Ωc))+c2\n2/integraldisplayt\n0(t0+τ)B−εE(τ)dτ\n+c3/integraldisplayt\n0(t0+τ)B−1−εJ(τ;a(x)b(τ)u2)dτ\n≤CI2\n0+/integraldisplayt\n0(t0+τ)B−ε(P1+P2)dτ. (2.25)16 YUTA WAKASUGI\nBy (2.25) +µ×(2.10), hereµis a small positive parameter determined later, it\nfollows that\n(t0+t)B−εEψ(t;Ω)+(t0+t)B−εEψ(t;Ωc)\n+/integraldisplayt\n0c2\n2(t0+τ)B−εE(τ)−µC(t0+τ)B−εE(τ)dτ\n+c3/integraldisplayt\n0(t0+τ)B−1−εJ(τ;a(x)b(τ)u2)dτ+µ(t0+t)B+1−εE(t)\n+µ/integraldisplayt\n0(t0+τ)B+1−εJ(τ;a(x)b(τ)u2\nt)+(t0+τ)B+1−εEψ(τ)dτ\n≤CI2\n0+P\n+C(t0+t)B+1−εJ(t;|u|p+1)\n+C/integraldisplayt\n0(t0+τ)B+1−εJψ(τ;|u|p+1)dτ\n+C/integraldisplayt\n0(t0+τ)B−εJ(τ;|u|p+1)dτ, (2.26)\nwhere\nP=/integraldisplayt\n0(t0+τ)B−ε(P1+P2)dτ.\nNow we choose µsufficiently small, then we can rewrite (2.26) as\n(t0+t)B+1−εE(t)+(t0+t)B−εJ(t;a(x)b(t)u2)\n≤CI2\n0+P+C(t0+t)B+1−εJ(t;|u|p+1)\n+C/integraldisplayt\n0(t0+τ)B+1−εJψ(τ;|u|p+1)dτ\n+C/integraldisplayt\n0(t0+τ)B−εJ(τ;|u|p+1)dτ. (2.27)\nWe shall estimate the right hand side of (2.27). We need the following le mma.\nLemma 2.2 (Gagliardo-Nirenberg) .Letp,q,r(1≤p,q,r≤ ∞)andσ∈[0,1]\nsatisfy\n1\np=σ/parenleftbigg1\nr−1\nn/parenrightbigg\n+(1−σ)1\nq\nexcept forp=∞orr=nwhenn≥2. Then for some constant C=C(p,q,r,n)>\n0, the inequality\n/ba∇dblh/ba∇dblLp≤C/ba∇dblh/ba∇dbl1−σ\nLq/ba∇dbl∇h/ba∇dblσ\nLr,for anyh∈C1\n0(Rn)\nholds.\nWe first estimate ( t0+t)B+1−εJ(t;|u|p+1). From the above lemma, we have\nJ(t;|u|p+1)≤C/parenleftbigg/integraldisplay\nRne4\np+1ψu2dx/parenrightbigg(1−σ)(p+1)/2\n×/parenleftbigg/integraldisplay\nRne4\np+1ψ|∇ψ|2u2dx\n+/integraldisplay\nRne4\np+1ψ|∇u|2dx/parenrightbiggσ(p+1)/2\n(2.28)SEMILINEAR DAMPED WAVE EQUATIONS 17\nwithσ=n(p−1)\n2(p+1). Since\ne4\np+1ψu2= (e2ψa(x)b(t)u2)a(x)−1b(t)−1e(4\np+1−2)ψ\n≤C(e2ψa(x)b(t)u2)/bracketleftBigg/parenleftbigg/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht2−α\n(1+t)1+β/parenrightbiggα\n2−α\ne(4\np+1−2)ψ/bracketrightBigg\n×(1+t)β+(1+β)α/(2−α)\n≤C(1+t)β+(1+β)α/(2−α)e2ψa(x)b(t)u2\nand\ne4\n(p+1)ψ|∇ψ|2u2≤C/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht2−2α\n(1+t)2+2βe1\n2(4\np+1−2)ψe1\n2(4\np+1−2)ψe2ψu2\n≤Ce1\n2(4\np+1−2)ψe2ψ/bracketleftBigg/parenleftbigg/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht2−α\n(1+t)1+β/parenrightbigg2−2α\n2−α\ne1\n2(4\np+1−2)ψ/bracketrightBigg\n×(1+t)−2(1+β)+(1+β)(2−2α)/(2−α)u2\n≤C(1+t)−2(1+β)/(2−α)e1\n2(4\np+1−2)ψe2ψu2\n≤C(1+t)−2(1+β)/(2−α)(1+t)β+(1+β)α/(2−α)e2ψa(x)b(t)u2,\nwe can estimate (2.28) as\nJ(t;|u|p+1)≤C(1+t)[β+(1+β)α/(2−α)](1−σ)(p+1)/2J(t;a(x)b(t)u2)(1−σ)(p+1)/2\n×[(1+t)−1J(t;a(x)b(t)u2)+E(t)]σ(p+1)/2\nand hence\n(t0+t)B+1−εJ(t;|u|p+1)≤C/parenleftBig\n(t0+t)γ1M(t)(p+1)/2+(t0+t)γ2M(t)(p+1)/2/parenrightBig\n,\nwhere\nγ1=B+1−ε+/bracketleftbigg\nβ+(1+β)α\n2−α/bracketrightbigg1−σ\n2(p+1)−σ\n2(p+1)\n−(B−ε)p+1\n2,\nγ2=B+1−ε+/bracketleftbigg\nβ+(1+β)α\n2−α/bracketrightbigg1−σ\n2(p+1)−(B−ε)1−σ\n2(p+1)\n−(B+1−ε)σ\n2(p+1).\nBy a simple calculation it follows that if\np>1+2\nn−α,\nthen by taking εsufficiently small (i.e. δsufficiently small) both γ1andγ2are\nnegative. We note that\nJψ(t;|u|p+1) =/integraldisplay\nRne2ψ(−ψt)|u|p+1dx\n≤C\n1+t/integraldisplay\nRne(2+ρ)ψ|u|p+1dx,18 YUTA WAKASUGI\nwhereρis a sufficiently small positive number. Therefore, we can estimate th e\nterms/integraldisplayt\n0(t0+τ)B+1−εJψ(τ;|u|p+1)dτand/integraldisplayt\n0(t0+τ)B−εJ(τ;|u|p+1)dτ\nin the same manner as before. Noting that\nP1+P2=d\ndt/bracketleftbigg\n(t0+t)α+β/integraldisplay\nΩe2ψF(u)dx+/integraldisplay\nΩce2ψ/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nKF(u)dx/bracketrightbigg\n+C/integraldisplay\nΩe2ψ(1+(t0+t)α+β(−ψt))|u|p+1dx\n+C/integraldisplay\nΩce2ψ(1+/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK(−ψt))|u|p+1dx,\nwe have\nP=/integraldisplayt\n0(t0+τ)B−ε(P1+P2)dτ\n≤CI2\n0+C(t0+t)B−ε/integraldisplay\nΩe2ψ(t0+t)α+βF(u)dx\n+C(t0+t)B−ε/integraldisplay\nΩce2ψ/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nKF(u)dx\n+C/integraldisplayt\n0(t0+τ)B−1−ε/integraldisplay\nΩe2ψ(t0+τ)α+βF(u)dxdτ\n+C/integraldisplayt\n0(t0+τ)B−1−ε/integraldisplay\nΩce2ψ/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nKF(u)dxdτ\n+C/integraldisplayt\n0(t0+τ)B−ε/integraldisplay\nΩe2ψ(1+(t0+τ)α+β(−ψt))|u|p+1dxdτ\n+C/integraldisplayt\n0(t0+τ)B−ε/integraldisplay\nΩce2ψ(1+/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK(−ψt))|u|p+1dxdτ.\nWe calculate\ne2ψ/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK=e2A/angbracketleftx/angbracketright2−α\n(1+t)1+β/a\\}b∇acketle{tx/a\\}b∇acket∇i}htα+β\nK\n≤Ce2A/angbracketleftx/angbracketright2−α\n(1+t)1+β/parenleftbigg/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht2−α\n(1+t)1+β/parenrightbiggα+β\n2−α\n(1+t)(α+β)(1+β)\n2−α\n≤Ce(2+ρ)ψ(1+t)(α+β)(1+β)\n2−α\nfor smallρ>0. Noting that(α+β)(1+β)\n2−α<1 and taking ρsufficiently small, we can\nestimate the terms Pin the same manner as estimating ( t0+t)B+1−εJ(t;|u|p+1).\nConsequently, we have a priori estimate for M(t):\nM(t)≤CI2\n0+CM(t)(p+1)/2.\nThis shows that the local solution of (1.1) can be extended globally. W e note that\ne2ψa(x)b(t)≥c(1+t)−(1+β)α\n2−α−β\nwith some constant c>0. Then we have/integraldisplay\nRne2ψa(x)b(t)u2dx≥c(1+t)−(1+β)α\n2−α−β/integraldisplay\nRnu2dx. (2.29)SEMILINEAR DAMPED WAVE EQUATIONS 19\nThis implies the decay estimate of global solution (1.10) and completes the proof\nof Theorem 1.1.\nProof of Corollary 1.4. In a similar way to derive (2.29), we have\n/integraldisplay\nRne2ψa(x)b(t)u2dx≥c(1+t)−(1+β)α\n2−α−β/integraldisplay\nRne(2A−µ)/angbracketleftx/angbracketright2−α\n(1+t)βu2dx.\nBy noting that\n/a\\}b∇acketle{tx/a\\}b∇acket∇i}ht2−α\n(1+t)1+β≥(1+t)ρ\non Ωρ(t) and Theorem 1.1, it follows that\n(1+t)−(1+β)α\n2−α−β/integraldisplay\nΩρ(t)e(2A−µ)(1+t)ρ(u2\nt+|∇u|2+u2)dx\n≤C(1+t)−(1+β)α\n2−α−β/integraldisplay\nΩρ(t)e(2A−µ)/angbracketleftx/angbracketright2−α\n(1+t)β(u2\nt+|∇u|2+u2)dx\n≤C/integraldisplay\nRne2ψ(u2\nt+|∇u|2+a(x)b(t)u2)dx\n≤C(1+t)−B+ε.\nThus, we obtain\n/integraldisplay\nΩρ(t)(u2\nt+|∇u|2+u2)dx≤C(1+t)−(1+β)(n−2α)\n2−α+εe−(2A−µ)(1+t)ρ.\nThis proves Corollary 1.4. /square\nAcknowledgement\nThe author is deeply grateful to Professors Ryo Ikehata, Kenji Nishihara, Tatsuo\nNishitani, Akitaka Matsumura and Michael Reissig. They gave me cons tructive\ncomments and warm encouragement again and again.\nReferences\n[1]H. 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Wirth ,Wave equations with time-dependent dissipation. II. Effect ive dissipation , J. Dif-\nferential Equations 232(2007), 74-103.\n[40]Qi S. Zhang ,A blow-up result for a nonlinear wave equation with damping: the critical case ,\nC. R. Acad. Sci. Paris S´ er. I Math., 333(2001), 109-114.\n[41]Y. Zhou ,Cauchy problem for semilinear wave equations in four space d imensions with small\ninitial data , J. Partial Defferential Equations, 8(1995), 135-144.\nDepartment of Mathematics, Graduate School of Science, Osa ka University, Osaka,\nToyonaka, 560-0043, Japan\nE-mail address :y-wakasugi@cr.math.sci.osaka-u.ac.jp"    },    {        "title": "1911.09400v1.Low_damping_and_microstructural_perfection_of_sub_40nm_thin_yttrium_iron_garnet_films_grown_by_liquid_phase_epitaxy.pdf",        "content": " 1 Low damping and microstructural perfection of sub-4 0nm-thin \nyttrium iron garnet films grown by liquid phase epi taxy \n \nCarsten Dubs, 1* Oleksii Surzhenko, 1 Ronny Thomas, 2 Julia Osten, 2 Tobias Schneider, 2 Kilian Lenz, 2 \nJörg Grenzer, 2 René Hübner, 2 Elke Wendler 3 \n \n1 INNOVENT e.V. Technologieentwicklung, Prüssingstr.  27B, 07745 Jena, Germany \n2 Institute of Ion Beam Physics and Materials Resear ch, Helmholtz-Zentrum Dresden-Rossendorf, \nBautzner Landstr. 400, 01328 Dresden, Germany \n3 Institut für Festkörperphysik, Friedrich-Schiller- Universität Jena, Helmholtzweg 3, 07743 Jena, \nGermany \n* Correspondence: cd@innovent-jena.de  \n \nThe field of magnon spintronics is experiencing an increasing interest in the development of \nsolutions for spin-wave-based data transport and pr ocessing technologies that are complementary or \nalternative to modern CMOS architectures. Nanometer -thin yttrium iron garnet (YIG) films have \nbeen the gold standard for insulator-based spintron ics to date, but a potential process technology tha t \ncan deliver perfect, homogeneous large-diameter fil ms is still lacking. We report that liquid phase \nepitaxy (LPE) enables the deposition of nanometer-t hin YIG films with low ferromagnetic \nresonance losses and consistently high magnetic qua lity down to a thickness of 20 nm. The obtained \nepitaxial films are characterized by an ideal stoic hiometry and perfect film lattices, which show \nneither significant compositional strain nor geomet ric mosaicity, but sharp interfaces. Their \nmagneto-static and dynamic behavior is similar to t hat of single crystalline bulk YIG. We found, \nthat the Gilbert damping coefficient α is independent of the film thickness and close to 1 × 10 -4, and \nthat together with an inhomogeneous peak-to-peak li newidth broadening of ∆H0||  = 0.4 G, these \nvalues are among the lowest ever reported for YIG f ilms with a thickness smaller than 40 nm. These \nresults suggest, that nanometer-thin LPE films can be used to fabricate nano- and micro-scaled \ncircuits with the required quality for magnonic dev ices. The LPE technique is easily scalable to YIG \nsample diameters of several inches. \n \n \nI. INTRODUCTION  \n \nYttrium iron garnet (Y 3Fe 5O12 ; YIG) in the micrometer thickness range is the mat erial of choice in \nradio-frequency (RF) engineering for decades (see, e.g., Refs. [1-5]). Especially the lowest spin \nwave loss of all known magnetic materials and the f act, that it is a dielectric are of decisive \nimportance. Since one has learned how to grow YIG f ilms in the nanometer thickness range, there \nhas been a renaissance of this material, as its mag netic and microwave properties are in particular \ndemand in many areas of modern physics.  \nA growing field of application for magnetic garnets  is (i) magnonics, which deals with future \npotential devices for data transfer and processing using spin waves [1,6-9]. The significant thickness  \nreduction achieved today allows reducing the circui t sizes from classical millimeter dimensions [1] \ndown to 50 nm [10-12] . Another important field is (ii) spintronics: By in creasing the YIG surface-\nto-volume ratio as much as possible (while keeping its magnetic properties), physical phenomena, \nsuch as the inverse spin Hall effect [13], spin-tra nsfer torque [14], and the spin Seebeck effect [15]  \n(generated by a spin angular momentum transfer at t he interfaces between YIG and a nonmagnetic \nmetallic conductor layer) become much more efficien t [7,16-29].  Also (iii) the field of terahertz \nphysics, which uses ultrafast spin dynamics to cont rol ultrafast magnetism, for example for potential  2 terahertz spintronic devices [30,31,32], and (iv) t he field of low-temperature physics, which deals \nwith magnetization dynamics at cryogenic temperatur es [33 ] for prospective quantum computer \nsystems, are possible fields of applications for na nometer-thin iron garnet films. \nThere are several different techniques to grow YIG on different substrates. (i) Pulsed laser \ndeposition (PLD) is an excellent technique for fabr icating small samples of nanometer-thin YIG \nfilms with narrow ferromagnetic resonance (FMR) lin ewidths [17,19,21,22,28,34-36] whereas its \nup-scaling to larger sample dimensions of several i nches is challenging. (ii) Magnetron sputtered \nYIG usually yields wider FMR linewidths, and inhomo geneous line broadening is frequently \nobserved [37-40]. (iii) For large-scale, low-cost c hemical solution techniques, such as spin coating, \nstrongly broadened FMR linewidths and increased Gil bert damping parameters were reported \n[41,42]. (iv) Liquid phase epitaxy (LPE) from high- temperature solutions (flux melts), is a well-\nestablished technique. Since nucleation and crystal  growth take place under almost thermodynamic \nequilibrium conditions, this guarantees high qualit y with respect to narrow absolute FMR linewidths \nand a small Gilbert damping coefficient [43-45] at the same time, making LPE comparable or \nsuperior to the other growth techniques. In additio n, LPE allows YIG to be deposited in the required \nquality on 3- or 4-inch wafers [46]. This is import ant for possible applications mentioned above. \nSo far, classical LPE was applied to grow micromete r-thick samples used for magneto-static \nmicrowave devices [47,48] or for magneto-optical im aging systems [49]. The typical shortcomings \nof the LPE technology making thin-film growth so di fficult lie in the fact, that, due to high growth \nrates, nanometer-thin films were technologically di fficult to access. The etch-back processes in high-\ntemperature solutions or interdiffusion processes a t the substrate/film interface at high temperatures  \nusually prevent sharp interfaces. In addition, film  contamination by flux melt constituents (if it is not \na self-flux without foreign components) is unavoida ble in most cases. Nevertheless, it was recently \ndemonstrated, that epitaxial films of 100 nm or thi nner are also accessible with this technique \n[50,51].  \nIn this study, we will show that we are able to dep osit nanometer-thin YIG LPE films with low FMR \nlosses and consistently high magnetic quality down to a thickness of 20 nm. There is no thinnest \n\"ultimate\" thickness for iron garnet LPE films, as it is sometimes claimed. \nIt should be pointed out, that, in addition to the damping properties, magnetic anisotropy \ncontributions as a function of the sample stoichiom etry and film/substrate pairing are also of great \nimportance, since they determine the static and dyn amic magnetization of the epitaxial iron garnet \nfilms and thus their possible applications. For exa mple, large negative uniaxial anisotropy fields \nwere usually observed for garnet films under compre ssion, such as for YIG on gadolinium gallium \ngarnet (Gd 3Ga 5O12 ; GGG) or other suitable substrates with smaller latt ice parameters grown by gas \nphase deposition techniques (see e.g. Refs. [35,36, 52-57]), which favors in-plane magnetization. \nLarge perpendicular magnetic anisotropies, on the o ther hand, can be found for films under tensile \nstrain, e.g. on substrates with larger lattice para meter or for rare earth iron garnet films with smal ler \nlattice parameter than GGG (see e.g. Refs. [58-62] ). Between these two extremes are YIG LPE \nfilms, which are usually grown on standard GGG subs trates and exhibit small tensile strain if no \nlattice misfit compensation, e.g. by La ion substit ution [50,63], has been performed. Such films are \ncharacterized  by a small uniaxial magnetic anisotropy and dominan t shape anisotropy when no \nlarger growth–induced anisotropy contributions due to Pb or Bi substitution occurs [64]. \nHowever, only little information about the structur al properties and the thickness-dependent \nmagnetic anisotropy contributions of nanometer-thin  LPE films has been published so far, which is \nwhy we are concentrating on these properties for YI G films with thicknesses down to 10 nm. This \nallowed us to describe the intrinsic damping behavi or over a wide frequency range and to determine \na set of magnetic anisotropy parameters for all inv estigated films. \n \n  3 II. EXPERIMENTAL DETAILS \n \nNanometer-thin YIG films were deposited on 1-inch ( 111) GGG substrates by LPE from PbO-B 2O3-\nbased high-temperature solutions (HTL) at about 865 °C using the isothermal dipping method (see \ne.g. [65]). Nominally pure Y 3Fe 5O12  films with smooth surfaces were obtained within on e minute \ndeposition time on horizontally rotated substrates with rotation rates of 100 rpm. The only variable \ngrowth parameter for all samples in this study was the degree of undercooling ( ∆T = TL-Tepitaxy ) that \nwas restricted to ∆T ≤ 5 K to obtain films with thicknesses between 10 an d 110 nm. Here TL is the \nliquidus temperature of the high-temperature soluti on and Tepitaxy  is the deposition temperature. After \ndeposition, the samples were pulled out of the solu tion followed by a spin-off of most of the liquid \nmelt remnants at 1000 rpm, pulled out of the furnac e and cooled down to room temperature. \nSubsequently, the sample holder had to be stored wi th the sample in a diluted, hot nitric-acetic-acid \nsolution to remove the rest of the solidified solut ion residues. Finally, the reverse side YIG film of  \nthe doubled-sided grown samples was removed by mech anical polishing and samples were cut into \nchips of different sizes by a diamond wire saw. The  film thicknesses were determined by X-ray \nreflectometry (XRR) and by high-resolution X-ray di ffraction (HR-XRD) analysis, and the latter \ndata were used to calculate anisotropy and magnetiz ation values. \nAtomic force microscopy (AFM) using a Park Scientif ic M5 instrument was carried out for each \nsample at three different regions over 400 µm2 ranges to determine the root-mean-square (RMS) \nsurface roughness. \nThe XRR measurements were carried out using a PANan alytical/X-Pert Pro system. For the HR-\nXRD investigations, a Seifert-GE XRD3003HR diffract ometer using a point focus was equipped \nwith a spherical 2D Göbel mirror and a Bartels mono chromator on the source side. Both systems use \nCu Kα1 radiation. Reciprocal space maps (RSMs) were measu red with the help of a position-sensitive \ndetector (Mythen 1k) at the symmetric (444) and (88 8) as well as the asymmetric (088), (624), and \n(880) reflections. To obtain the highest possible a ngular resolution for symmetric θ−2 θ line scans, a \ntriple-axis analyzer in front of a scintillation co unter was installed on the detector. Using a recurs ive \ndynamical algorithm implemented in the commercial p rogram RC_REF_Sim_Win [66], the vertical \nlattice misfits were calculated.  \nRutherford backscattering spectrometry (RBS) was ap plied to investigate the composition of the \ngrown YIG films using 1.8 MeV He ions and a backsca ttering angle of 168°. Backscattering events \nwere registered with a common Si detector. The ener gy calibration of the multichannel analyzer \nrevealed 3.61 keV per channel. A thin carbon layer was deposited on top of the samples to avoid \ncharging during analysis. The samples were tilted b y 5° with respect to the incoming He ion beam \nand rotated around the axis perpendicular to the sa mple surface in order to obtain reliable random \nspectra. The analysis of the measured spectra was p erformed by a home-made software [67] based \non the computer code NDF [68] and then enables the calculation of the RBS spectra. The measured \ndata were fitted by calculated spectra to extract t he film composition. In this way, the Fe-to-Y ratio  \nof the films was determined. Because of the low mas s of oxygen, the O signal of the deposited films \nis too low for quantitative analysis. \nHigh-resolution transmission electron microscopy (H R-TEM) investigations were performed with \nan image C s-corrected Titan 80-300 microscope (FEI) operated a t an accelerating voltage of 300 kV. \nHigh-angle annular dark-field scanning transmission  electron microscopy (HAADF-STEM) imaging \nand spectrum imaging analysis based on energy-dispe rsive X-ray spectroscopy (EDXS) were done \nat 200 kV with a Talos F200X microscope equipped wi th a Super-X EDXS detector system (FEI). \nPrior to TEM analysis, the specimen mounted in a hi gh-visibility low-background holder was placed \nfor 10 s into a Model 1020 Plasma Cleaner (Fischion e) to remove possible contaminations. Classical \ncross-sectional TEM-lamella preparation was done by  sawing, grinding, polishing, dimpling, and  4 final Ar-ion milling. Quantification of the element  maps including Bremsstrahlung background \ncorrection based on the physical TEM model, series fit peak deconvolution, and application of \ntabulated theoretical Cliff-Lorimer factors as well  as absorption correction was done for the \nelements Y (K α line), Fe (K α line), Gd (L α line), Ga (K α line), O (K line), and C (K line) using the \nESPRIT software version 1.9 (Bruker). \nThe ferromagnetic resonance (FMR) absorption spectr a were taken on two different setups. The \nfrequency-swept measurements were recorded on a Roh de & Schwarz ZVA 67 vector network \nanalyzer attached to a broadband stripline. The YIG /GGG sample was mounted face-down on the \nstripline, and the transmission signals S21  and S12  were recorded using a source power of -10 dBm (= \n0.1 mW). The microwave frequency was swept across t he resonance frequency fres , while the in-\nplane magnetic field H remained constant. Each recorded frequency spectru m was fitted by a \nLorentz function and allowed us to define the reson ance frequency fres  and the frequency linewidth \nΔfFWHM  corresponding to the applied field H = Hres . \nIn addition, field-swept measurements were carried out with another setup using an Agilent E8364B \nvector network analyzer and an 80-µm-wide coplanar waveguide. Again, the microwave \ntransmission parameter S21  was recorded as the FMR signal. This time, the mic rowave frequency \nwas kept constant and the external magnetic field w as swept through resonance. This facilitates \ntracking the FMR signals over large frequency range s. The microwave power was set to 0 dBm (= \n1 mW). In addition, this setup allowed for azimutha l and polar angle-dependent measurements to \ndetermine the anisotropy and damping contributions in detail. The FMR spectra were fitted by a \ncomplex Lorentz function to retrieve the resonance field Hres  and field-swept peak-to-peak linewidth \nΔHpp . By fitting the four sets of resonance field data,  i.e. (i) the in-plane and (ii) the perpendicular-\nto-plane frequency dependence as well as (iii) the azimuthal and (iv) polar angular dependences at f \n= 10 GHz, with the resonance equation for the cubic (111) garnet system, a consistent set of \nanisotropy parameters was determined for each sampl e. In addition, the damping parameters and \ncontributions were determined from the frequency- a nd angle-dependent linewidth data. \nThe vibrating sample magnetometer (VSM, MicroSense LLC, EZ-9) was used to measure the \nmagnetic moments of the YIG/GGG samples magnetized along the YIG film surface. The external \nmagnetic field H was controlled within an error of ≤0.01 Oe. To est imate the volume magnetization \nM of the YIG films, the raw VSM signal was corrected  from background contributions (due to the \nsample holder and the GGG substrate) and normalized  to the YIG volume. The Curie temperatures \nTC for the YIG samples were determined by zero-extrap olation of the temperature dependencies M \n(H=const, T) measured in small in-plane magnetic fields. In or der to verify the Curie temperatures \nmeasured by VSM, a differential thermal analysis of  a 0.55 mm thick YIG single crystal slice was \ncarried out and then used as a reference sample for  the VSM temperature calibration. \n \n \nIII. RESULTS AND DISCUSSION \n \nA. Microstructural properties of nanometer-thin YIG  films \n \nThe thickness values reported in this study are der ived from the Laue oscillations observed in the θ-\n2θ patterns of the high-resolution X-ray diffraction (HR-XRD) measurements and are confirmed by \nX-ray reflectivity (XRR) measurements (see Fig. 1).  The differences between both methods for \ndetermining the film thickness are in the range of ±1 nm. The surface roughness of the films, \nmeasured by atomic force microscopy (AFM) reveals R MS values ranging between 0.2 and 0.4 nm, \nindependent of the film thickness. Sometimes, howev er, partial remnants of dendritic overgrowth  5 increase the surface roughness to RMS values above 0.4 nm for inspection areas larger than 400 µm2 \n(see, e.g., the disturbance in the top right corner  of the AFM image inset in Fig. 1). \n-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 10 010 110 210 310 410 510 610 710 810 910 10 \n  \n t = 10.6 nm Intensity (arb.units) \nIncidence angle [deg] 20 \nµm15 \n10 \n5\n010 15 20 5 0µmnm  1.5 \n 1.0 \n 0.5 \n 0.0 \n-0.5 \n-1.0 \n-1.5 \n t = 21.5 nm  t = 30.9 nm  t = 43.2 nm \n \nFIG. 1. XRR plots of LPE-grown YIG films of differe nt thicknesses. Solid lines correspond to the \nexperimental data, while dashed lines represent the  fitted curves. The spectra shifted vertically for \nease of comparison. The inset shows an AFM image of  the surface topography of the 11 nm YIG \nfilm with a RMS roughness of 0.4 nm. \n \n \n1. Epitaxial perfection studied by high-resolution X-ray diffraction  \n \nCombined high-resolution reciprocal space map (HR-R SM) investigations around asymmetric and \nsymmetric Bragg reflections are useful to evaluate the intergrowth relations of epitaxial films on \nsingle-crystalline substrates as well as to disting uish between lattice strain induced by the film \nlattice distortion or compositional changes due to stoichiometric deviations. \n33.10 33.15 33.20 46.0 46.5 47.0 47.5 48.0 48.5 49.0 \nQx (nm -1 )Qz (nm -1 )\n(088) \nYIG (GGG) \nt = 21 nm 5E0 5E1 5E2 5E3 5E4 5E5 5E6 Intensity (a) \n-0.1 0 0.1 34.5 35.0 35.5 36.0 \nQx (nm -1 )Qz (nm -1 ) (444) \n  6 0 20 40 60 80 100 46810 \n118.5 119.0 119.5 120.0 10 110 210 310 410 510 6\n  t  =     9 nm \n  t   =   11 nm \n  t   =   21 nm \n  t   =   30 nm \n  t   =   42 nm \n  t   = 106 nm     (888) \nYIG / GGG \nScattering angle θ− 2θ (deg) Intensity (counts) (b) \nOut-of-plane misfit \n-δd⊥\nfilm  [10 -4]\nFilm thickness (nm) \n \nFIG. 2. (a) Combined high-resolution reciprocal spa ce maps around the asymmetric YIG/GGG \n(088) Bragg reflection of a 21-nm-thin single-cryst alline YIG LPE film. The inset shows the \ncorresponding symmetric YIG/GGG (444) peak: measure ments were carried out using a position-\nsensitive detector. (b) HR-XRD triple-axis θ–2θ scans around the symmetric YIG/GGG (888) peak \nfor various film thicknesses. The inset shows the ( vertical) out-of-plane misfit vs. film thickness (t he \nsolid line is a guide to the eyes).  \n \nFigure 2(a) shows the HR-RSM of the 21 nm YIG film grown on GGG (111) substrate, measured at \nthe asymmetric (088) reflection in steep incidence,  indicating that both the film and the substrate \nBragg peak positions are almost identical. Besides the nearly symmetrical intensity distribution \nalong the [111]  out-of-plane direction (i.e. the Qz axis), there is only very weak diffuse scattering \nclose to the Bragg peak visible, pointing towards a  nearly perfect crystal lattice without significant  \ncompositional strain or geometric mosaicity. In add ition, no shift of in-plane (the Qx axis) film \nBragg peak position with respect to the substrate i s observed. This behavior indicates a fully straine d \npseudomorphic film growth with a perfect coherent i n-plane lattice match with the GGG substrate. \nThe pattern of the diffuse scattering observed alon g the Qx axis of the symmetric (444) reflection \n(inset in Fig. 2(a)) is very similar to the one fou nd for a comparable GGG substrate (not shown), \nindicating that the defect structure of the system is mainly defined by the substrate and/or substrate  \nsurface. Within the experimental error of ∆Q/Q ∼ 5×10 −6 nm -1 of the high-resolution diffractometer, \nthe same performance was found for all investigated  LPE films with thicknesses below 100 nm, \nclearly demonstrating coherent YIG film growth with out signs of film relaxation. \nHigh-resolution triple-axis coupled θ–2θ scans at the (888) and (444) symmetrical reflectio ns \n(angular accuracy better than 1.5\") were carried ou t to define the strain and film thicknesses of the \nYIG films. Figure 2(b) shows the results obtained a t the (888) reflection. Under these conditions, the  \nBragg reflection of the 106 nm thick YIG layer is c learly visible as a shoulder of the (888) GGG \nsubstrate reflection at higher diffraction angles a nd this indicates a smaller out-of-plane value for the \nlattice parameter d888  than for the GGG substrate. This is characteristic  for tensely stressed “pure” \nYIG LPE films [50,63]. For LPE films with a thickne ss significantly less than 100 nm, however, \nonly simulations can provide the structural paramet ers. For this reason, the diffracted signals shown \nin Fig. 2(b) were simulated and fitted. Using the b est fit of both, the (444) and (888) reflections, t he \nout-of-plane lattice misfit values ( )⊥ ⊥ ⊥ ⊥ ⊥ ⊥∆ − = − = QQ d d d d / /substrate substrate film filmδ  were determined (see \nTable 1). Assuming a fully pseudomorphic [111]-orie nted system, the in-plane stress of the YIG \nfilm can be calculated by σ′|| = -2c 44 δd⊥\nfilm  (see the Supplemental Material [69 ] for a detailed \nderivation and references therein [61,70,71]). The in-plane biaxial ε|| and out-of-plane uniaxial ε⊥  7 strains can be calculated as well using the stiffne ss tensor components c11 , c12 , and c44  for which we \nuse averaged values taken from [72,73] (see also Su pplemental Material [69 ]). The resulting \nparameters are listed in Table I. \nThe inset in Fig. 2(b) shows the out-of-plane misfi t as a function of the film thickness. A weak \nmonotonous increase of ⊥\nfilmdδ with decreasing film thickness is observed between 106 nm and \n21 nm. The same behavior was reported by Ortiz et al . [61] for compressively strained EuIG and \nTbIG PLD-grown films with film thicknesses down to 4 nm and 5 nm, respectively. However, for \nour thinnest LPE films with t ∼ 10 nm, the out-of-plane misfit rapidly drops. Such  a significant \nchange of the misfit with respect to the film thick ness was only mentioned for considerably \ncompressively strained YIG PLD films by O. d’Allivy  Kelly et al.  [17]. They assume, that this \neffect indicated a critical film thickness (below 1 5 nm) for strain relaxation, but did not explain it  in \ntheir letter. \nFor semiconductor LPE films, however, it is known, that interdiffusion processes at the \nfilm/substrate interfaces can generate continuous c omposition profiles in the diffusion zone without \nabrupt changes in the lattice parameters, which lea d to modified stress profiles depending on the \nthickness of the epilayers (see e.g. [74]). A possi ble explanation for the observed behavior could \ntherefore be the presence of a smoothly changing la ttice parameter value in the interface region.  \nSuch composition profiles have recently been discus sed for YIG films grown on GGG substrates by \nhigh-temperature and long-time laser MBE deposition  experiments [75] , and transition layer \nthicknesses have been modeled based on polarized ne utron and X-ray reflectometry techniques. The \nprobability of the existence of such a thin continu ous transition layer and its influence on the \nmagneto-static film properties will be discussed be low. \n \nTABLE I. Structural parameters of the YIG LPE films  grown on GGG (111) substrates: film \nthickness t measured by HR-XRD, RMS roughness obtained by AFM,  vertical lattice misfit ⊥\nfilmdδ  \nobtained by HR-XRD, in-plane strain ε|| and out-of-plane strain ε⊥ and the resulting  in-plane stress σ′||. \n \nt \n(nm)  roughness  \n(nm)  δd⊥\nfilm   \n×10 -4 ε|| \n×10 -4 ε⊥ \n×10 -4 σ′|| \n×10 8 Pa  \n9 - -4.3 2.3 -2.0 0.7 \n11 0.4 -6.1 3.3 -2.8 0.9 \n21  0.2 -10.4 5.6 -4.8 1.6 \n30 0.2 -9.4 5.1 -4.3 1.4 \n42  0.3 -9.2 5.0 -4.2 1.4 \n106  0.4 -8.5 4.6 -3.9 1.3 \n±1 ±0.1 ±0.7 ±0.4 ±0.4 ±0.1 \n \n \n2. Chemical composition studied by Rutherford Backs cattering spectrometry \n \nBesides the epitaxial perfection, the chemical comp osition of the films is of interest to estimate \ndeviations from the ideal Y 3Fe 5O12  stoichiometry and to detect impurity elements. The refore, RBS \nmeasurements were performed for selected LPE films.  As an example, Fig. 3 shows the random \nspectrum of a 30 nm thick YIG film on GGG substrate . The inset presents the main part of the \nspectrum. Applying the NDF software, the computed c urve (solid line) matches perfectly the \nexperimental one (symbols). This enables us to dete rmine the Fe:Y ratio. As for all investigated LPE \nfilms, the Fe:Y ratio was determined to be R = 1.67, which corresponds to the ideal iron garnet   8 stoichiometry with Fe:Y = 5:3. At higher magnificat ions of the backscattering yield in Fig. 3, a very \nlow intensity signal can be observed at ion energie s higher than for backscattering on gadolinium \natoms from the GGG substrate. Although the intensit y is rather low, it can be attributed to heavy \nimpurity elements present to a very low amount over  all in the YIG film. We assign this signal to  \nlead and platinum. These elements may come from the  solvent and the crucible during the \ndeposition of the YIG film. After background correc tion a total quantity of (0.08 ± 0.02) at.% for the \nsum of both elements could be determined. This corr esponds to 0.01 < x + y < 0.02 formula units of \nthe nominal film composition (Y 3-x-yPb xPt y)(Fe 5-x-yPb xPt y)O 12 . In a first approximation, for the \ncalculation of the RBS spectra, it was assumed, tha t both elements contribute in equal parts to the \nhigh-energy signal. So, the calculated spectrum tak es into account the existence of 0.04 at.% lead \nand 0.04 at.% platinum within the YIG layer. This y ields a good representation of the separated \nsignal for these two elements. \n1550 1600 1650 1700 1750 1800 0100 200 300 400 \n  \n measurement       simulation \n separated Pb + Pt signal Backscattering yield  (counts) \nIon energy  (keV) YIG/GGG  \nt = 30 nm \nGd signal \nfrom GGG \nsubstrate 1000 1500 02000 4000 6000 8000 \n   \n Gd Ga YFe \n \nFIG. 3. Energy spectrum of 1.8 MeV He ions backscat tered on the YIG/GGG sample with a YIG \nfilm thickness of t = 30 nm. The inset shows the main part of the spec trum with the edges of the \nsubstrate elements Gd and Ga and the Fe and Y peak from the YIG film. \n \n \n3. Crystalline perfection studied by high-resolutio n transmission electron microscopy \n \nTo analyze the film lattice perfection as well as t he heteroepitaxial intergrowth behavior, HR-TEM \ninvestigations were performed. A cross-sectional im age of an 11 nm thin YIG film on a GGG \nsubstrate makes it possible to visualize both, the entire YIG film volume up to the film surface and \nthe interface in a magnified HR-TEM microscope imag e (see Fig. 4(a)). Besides the perfect \nfilm/substrate interface, neither structural lattic e defects nor significant misalignment could be \nobserved in the coherently strained YIG film lattic e up to the film surface. \nTo prove the homogeneity of the bulk composition an d the performance of the film/substrate \ninterface, HAADF-STEM imaging (Fig. 4(b)) together with element mapping, based on EDXS \nanalysis (Figs. 4(c)-(g)), were performed. The corr esponding HAADF-STEM image in Fig. 4(b) \nallows clearly resolving the film/interface region due to the significant difference of the atomic \nnumber contrast. Because of the uniform spatial dis tribution of both, the film (Y, Fe, O) and the \nsubstrate elements (Gd, Ga, O), which are independe ntly represented by different colors in Figs.  9 4(c)-(g), a homogeneous composition over the entire  YIG film can be confirmed. Small brightness \nvariations within the element maps (on the right ha nd side) result from slight thickness variations of  \nthe classically prepared TEM lamella. Neither an in termixing of the substrate nor of the film \nelements at the YIG/GGG interface is observed in th e element maps within the EDXS detection \nlimit, which is estimated to be slightly below 1 at .-% for the measuring conditions used. For that \nreason, tiny Pb and Pt contributions in the YIG fil m, as shown by RBS (see Fig. 3), where not \ndetected here. \nTo evaluate the lateral element distributions acros s the film near the film/substrate interface, \nquantified line scans were performed as presented i n Fig. 4(h). Using the 10%-to-90% edge \nresponse criterion, it shows a  transition width of (1.9 ± 0.4) nm at the interface . This is lower than \nthe observed 4-6 nm non-magnetic dead layer reporte d for YIG films deposited by RF magnetron \nsputtering [76], and the about 4 nm or the 5–7 nm d eep Ga diffusion observed for PLD [77] or laser \nmolecular beam epitaxy (MBE) [75], respectively . However, at some positions of the sample’s \ncross-section we found a reduced YIG film thickness  on a wavy GGG surface (not shown), which \nwe attribute to a possible etch-back of the substra te at the beginning of film growth or an already \nexisting wavy substrate surface. For further growth  experiments, a careful characterization of the \nsubstrate surfaces by AFM should, therefore, be per formed. The TEM investigations show, that the \nLPE technology is suitable for growing nanometer-th in YIG films without lattice defects and \nwithout significant interdiffusion at the film/subs trate interface, which are necessary preconditions \nfor undisturbed spin-wave propagation and low ferro magnetic damping losses. \n \n \nGGG YIG resist (a) \n  10  0 2 4 6 8 10 12 14 16 18 Composition \n(at.-%) \nDistance (nm) \n(c) Y \n(d) Fe \n(e) Gd \n(f) Ga \n(g) O \n(h) \nGd 3Ga 5O12 Y3Fe 5O12 surface \ninterface \n(b) HAADF Line scan \nY\nFe \nGd \nGa \nO\n \n \nFIG. 4. (a) Cross-sectional high-resolution TEM ima ge of the 11-nm-thin YIG/GGG (111) film. The \narrows mark the YIG/GGG interface. (b) HAADF-STEM i mage highlighting the well-separated \nYIG/GGG interface. (c-g) EDXS element maps of the 1 1-nm-thin YIG/GGG (111) film cross-\nsection. (h) Line scan as marked in (b) of the elem ental concentrations across the film thickness.  \n \n \nB. Static and dynamic magnetization characterizatio n of nanometer-thin YIG films \n \nAfter gaining insight into the YIG film microstruct ure, we want to link these properties to the FMR \nperformance to find out, which of them plays an ess ential role in the observed magneto-static and \ndynamic behavior. Therefore, FMR measurements were carried out within a frequency range of 1 to \n40 GHz, with the external magnetic field either par allel to the surface plane of the sample along the \nH || [11-2] film direction or perpendicular to it ( H || [111 ]). In addition, angle-dependent \nmeasurements, i.e., varying the angle θH of the external magnetic field (polar angular depe ndence, \nwhere θH = 0 is the sample’s normal [111 ] direction) or the azimuth angle φH (in-plane angular \ndependence, where φH = 0 is the sample’s horizontal  [1-10] direction), were performed at \nf = 10 GHz. These four measurement ‘geometries’ allo w to determine Landé’s g-factor, effective \nmagnetization 4π Meff , and anisotropy fields from the resonance field de pendence and to disentangle \nthe damping contributions from the linewidth depend ence [78,79].  \nThe FMR resonance equations to fit the angle- and f requency-dependencies (see eqs. (S22), (S23) in \nthe Supplemental Material [69] for in-plane and out -of-plane bias field conditions after Baselgia et \nal.  [80 ])  are derived from the free energy density of a cubi c (111) system [81]: \n \n ( ) [ ]\n( ) ( )\n\n\n\n− + +− − − ++ − ⋅ −=\n⊥\nϕ θ θ θ θϕϕ θ θ πθ θ ϕϕ θ θ\n3sin cos sin32sin41cos31cos sin cos 2cos cos cos sin sin\n3 4 4\n42 2\n|| 22\n22\nKK K MH M F\nu sH H H s\n, (1) \n  11  where K2⊥, K2|| , and K4 are the uniaxial out-of plane, uniaxial in-plane, and cubic anisotropy \nconstants, respectively.  φ and θ are the angles of the magnetization. Angle φu allows for a rotation of \nthe uniaxial anisotropy direction with respect to t he cubic anisotropy direction.  \n \n \n1. Frequency-dependent FMR linewidth analysis \n \nTo investigate the influence of different contribut ions on the overall magnetic damping, we model \nthe field-swept peak-to-peak linewidth Δ Hpp  of our YIG (111) films as a sum of four contributi ons \n[79,82]:  \n TMS 0 mos G pp H H H H H ∆ + ∆ + ∆ + ∆ = ∆ , (2) \n \nwhere Δ HG is the Gilbert damping, Δ Hmos  the mosaicity, Δ H0 the inhomogeneous broadening, and \nΔHTMS  is the two-magnon scattering contribution, respect ively. Note, that all linewidths in this paper \nare peak-to-peak linewidths, even if not explicitly  stated. \nThe intrinsic Gilbert damping is given by \n  \n f H\nΞ= ∆\nγπα\n34\nG , (3) \n \nwhere γ = gµBħ is the gyromagnetic ratio and Ξ is the dragging function. The dragging function is  a \ncorrection factor to the linewidth needed in field- swept FMR measurements if H and M are not \ncollinear (see e.g . [82]). For H || M follows Ξ = 1. \nThe inhomogeneity term ∆Hmos  accounts for a spread (distribution) of the effect ive magnetization \n4π Meff  [82,83] given by the parameter δ4πMeff : \n \n eff\neffres\nmos 4432MMHH πδπ∂∂= ∆ . (4) \n \n∆H0, i.e. the zero-frequency linewidth, is a general b roadening term accounting for other \ninhomogeneities of the sample, such as the microwav e power dependence of the linewidth in YIG \n(see, e.g., [84]) and systematic fit errors: for ex ample, consistently narrower total full-width at ha lf-\nmaximum linewidths ∆HFWHM  of up to 0.5 Oe were determined by additional freq uency-swept \nmeasurements at a microwave power of -10 dBm compar ed to the field-swept measurements at \n0 dBm discussed here. \nAll kinds of inhomogeneous broadening (including ∆Hmos ) are caused by slightly different \nresonance fields in parts of the sample. These indi vidual resonance lines might be still resolvable at  \nlow frequencies, where Gilbert damping is not large  enough yet–especially for YIG. However, at \nhigher frequencies, these lines become broader and eventually coalesce to a single (apparently \nbroadened) line, which even might exhibit small sho ulders or other kinds of asymmetry. Hence, \nwhat might be nicely fit with a single line at high  frequencies might cause difficulties at low \nfrequencies and sub-mT linewidths. The effect on fi tting the anisotropy constants from the \nresonance fields is not so sensitive. If the resona nce lines cannot be disentangled or the line is not  \nentirely Lorentzian-shaped anymore, the fit might o verestimate the true linewidth resulting in a \nsystematic broader line accounted for by ∆H0.  12  The last term in Eq. (2), ∆HTMS , covers the two-magnon scattering contribution, wh ich is an \nextrinsic damping mechanism due to randomly distrib uted defects. For the in-plane frequency-\ndependence it reads [78,79,82,85-87]:  \n \n \n2 22 2sin\n32\n02\n0 202\n0 2\n1\nTMS\nf fff ff\nH\n+\n\n+−\n\n+\n⋅ Γ\nΞ= ∆−, (5) \n \nwhere f0 = γ4π Meff  and Γ is the two-magnon scattering strength. \nEach of the contributions has a characteristic angl e and frequency dependence. Overall, the \nlinewidth vs. frequency dependencies and the linewi dth vs. angle dependencies can be described \nwith one set of parameters. \nAs we will see, the applied model fits very well to  the experimental results and allows for \ndisentangling the contributions that are responsibl e for the frequency dependence of the linewidth. \nAt first, we discuss the different damping contribu tions. Then, we go into details for the individual \nmagneto-static parameters, the relevant anisotropy contributions mentioned above, which provided \nalso the base input for the fit parameters for the frequency-dependent FMR linewidth of our YIG \nfilms. \nIn Figure 5, the obtained frequency-dependent peak- to-peak linewidths ∆Hpp  (symbols) for the four \nthicknesses 11 nm, 21 nm, 30 nm, and 42 nm are pres ented. The red (solid) curves represent fits \nusing Eq. (2). Figure 5(a) shows data and fits for the out-of-plane bias field configuration ( θH = 0°) \nand Fig. 5(b) for field-in-plane ( θH = 90°), respectively. As mentioned above, due to a  quite complex \nshape of the resonance lines below ~15 GHz for θH = 0° (with more absorption lines needed to \nreflect the shape of the spectrum than for higher f requencies) the linewidths could not anymore be \nevaluated unambiguously with the required precision  for films with thicknesses above 11 nm. \nHowever, for the thinnest film, the evaluation was possible and the overall fit exhibits a linear \nbehavior down to 1 GHz. This means, in the field-ou t-of-plane geometry, the main contribution to \nthe damping is the Gilbert damping α, which can be determined from the linear slope according to \nEq. (3). As it is known from two-magnon scattering (TMS) theory [85,86], there is no TMS \ncontribution if M is perpendicular to the sample plane. The only rem aining contribution is the \ninhomogeneous broadening given by the zero-frequenc y offset \n  13  0369\n036\n036\n0 5 10 15 20 25 30 35 40 036θ H= 0 deg \n 11 nm \n 21 nm ∆Hpp  (Oe) \n 30 nm \n \nf (GHz) 42 nm (a) \n \n \nFIG. 5: Frequency dependence of the linewidth with magnetic field (a) perpendicular-to-plane and \n(b) in-plane. The red (solid) lines are fits to the  data. For the 11-nm sample the individual \ncontributions to the total linewidth are shown in t he top-right panel. Note the different y-axis scaling \nfor the 11 nm sample in the top-left panel. \n \nFrom these out-of-plane measurements, the Gilbert d amping coefficients could be determined, \nranging from α = 0.9 × 10 -4 for the 42-nm-thick sample to α = 2.0 × 10 -4 for the 21 nm sample, \nwhich is about twice the value obtained from in-pla ne measurements (as discussed below). For the \nultrathin 11 nm film, a slightly increased Gilbert damping coefficient of α = 2.7 × 10 -4 and a \nsignificantly enlarged zero-frequency linewidth of 2.8 Oe were found. As mentioned above, the \nreason for the larger offset might be an apparent u nresolvable broadening due to inhomogeneity. For \nthe 21 and 30 nm sample, the zero-frequency interce pt is about ∆H0 = 0.5 Oe, in contrast to ∆H0 = \n1.5 Oe for the 42 nm sample. This indicates, that t he 42 nm sample, in contrast to the thinner \nsamples, seems to have additional microstructural d efects, leading to a superposition of lines. This i s \nvery likely, because the inhomogeneous broadening p reviously reported for 100 nm YIG LPE films \nwas also in the range of ∆H0  = 0.5-0.7 Oe [50].   \nIn Fig. 5(b), the results of the corresponding in-p lane field configuration are given. For the 11 nm \nsample, the four individual fit contributions consi dered in the fit according to Eq. (2) are depicted by \nsolid curves. This sample shows a significant curva ture. The 42 nm sample also shows a small \ncurvature, whereas the other two samples only have a weak curvature at lower frequencies. This \ncurvature usually hints to a contribution from two- magnon scattering, but can also be due to a spread \nof the effective magnetization. Note, the frequency -dependence of the mosaicity and TMS term look \nquite similar at higher frequencies, but show a dif ferent curvature at lower frequencies. Hence, the \nshape of the curve and, thus, the fit reveal, that it is due to a spread of the effective magnetizatio n, \nδ4πMeff  as given by Eq. (4), which lies in the range of 0. 4 to 0.9 G. For the 11 nm sample, this value 036\n036\n036\n0 5 10 15 20 25 30 35 40 03611 nm  Exp.  ∆H  ∆HG ∆Hmos  ∆HTMS  ∆H0\nθ H= 90 deg \n21 nm \n30 nm \nf (GHz) 42 nm ∆Hpp  (Oe) (b) \n 036\n036\n036\n0 5 10 15 20 25 30 35 40 03611 nm  Exp.  ∆H  ∆HG ∆Hmos  ∆HTMS  ∆H0\nθ H= 90 deg \n21 nm \n30 nm \nf (GHz) 42 nm ∆Hpp  (Oe) (b) \n  14  is larger, i.e., δ4π Meff  = 3.2 G, and in addition one needs a small TMS dam ping contribution of Γ = \n1.5×10 7 Hz for a proper fit (see Table II). This is again a distinctive sign, that the 11 nm sample has \nsignificantly different structural and/or magnetic properties, leading to the additional linewidth \ncontributions. The Gilbert damping coefficients of all four samples in in-plane configuration are \nα ≤ 1.3 × 10 -4 and correspond to the best values reported earlier  for 100 nm YIG LPE films [50]. \nThese are also lower than for a recently reported 1 8 nm YIG LPE film [51]. Thus, at room \ntemperature, no significant increase in Gilbert dam ping could be observed for LPE films down to \n10 nm with decreasing thickness. This contrasts wit h various references for PLD and RF-sputtered \nYIG films grown on (111) GGG substrates [88-92].  \n \nTABLE II. Magnetic damping parameters of the LPE (1 11) YIG films: film thickness t, in-plane \nGilbert damping parameter α||, inhomogeneous broadening ∆H0|| , spread of effective magnetization  \nδ4πMeff   and two-magnon scattering contribution Γ. \n \nt \n(nm)  α||  \n (×10 -4) ∆H0||  \n (G)  δ4π Meff   \n(Oe)  Γ \n(10 7 Hz)  \n11 1.2  0.4  3.2  1.5  \n21  1.3  0.6  0.4  0 \n30  1.2  0.4  0.7  0 \n42  1.0  0.4  0.9  0 \naccuracy  ±0.2  ±0.2  ±0.3  ±0.3  \n \n \nAll field-in-plane linewidth parameters of the inve stigated samples are summarized in Table II. It is \nobvious, that inhomogeneous contributions, i.e., th ose originating from magnetic mosaicity δ4πMeff , \nare very small for the samples without two-magnon s cattering. This confirms the high \nmicrostructural perfection and homogeneity of the v olume and interfaces of the LPE-grown films \nwith film thicknesses larger than 11 nm. Contributi ons to two-magnon scattering appear to occur \nonly for LPE films with a thickness of less than 21  nm thick. \n \n \n2. Analysis of magnetic anisotropy contributions \n \nIn the following, we will discuss the anisotropy co ntributions, which provided the base input for the \nfit parameters used for the frequency-dependent FMR  linewidth curves shown above. All curves \nwere fitted iteratively with the respective resonan ce equation (see Eqs. (S22) and (S23) in the \nSupplemental Material [69]) to retrieve a coherent set of fit parameters. The fit parameters are liste d \nin table III. Since the saturation magnetization an d the in-plane stress are known from VSM \nmeasurements and HR-XRD investigations, the anisotr opy constants K can be calculated from the \nanisotropy fields determined by FMR. \n \nTABLE III. Magneto-static parameters of the YIG LPE films of t hickness t: Landé’s g-factor, \neffective magnetization 4 πMeff exp , cubic anisotropy field 2 K4/Ms, and uniaxial in-plane anisotropy \nfield 2 K2||/Ms determined from FMR, saturation magnetization 4 πMs determined from VSM, stress-\ninduced anisotropy field 2 Kσ/Ms calculated from X-ray diffraction data, resulting o ut-of-plane \nuniaxial anisotropy field 2 K2⊥/Ms and effective magnetization 4 πMeff cal , cubic anisotropy constant \nK4, stress-induced anisotropy constant Kσ, and out-of-plane uniaxial anisotropy constant K2⊥. \n  15  t \n(nm) g 4πMeff exp \n(G) 2K4/Ms \n(Oe) 2K2||/Ms \n(Oe) 4πMs \n(G) \n11  2.015 1566 -93 2.0 1494 \n21 2.016 1647 -79 0.8 1819 \n30 2.015 1677 -79 0.6 1830 \n42 2.014 1699 -86 1.1 1860 \naccuracy ±0.002 ±13 ±2 ±3 ±41 \n \nt \n(nm) 2Kσ/Ms \n(G) 2K2⊥/Ms \n(G) 4πMeff cal  \n(G) K4 \n(10 3 erg/cm 3) Kσ \n(10 3 erg/cm 3) K2⊥ \n(10 3 erg/cm 3) \n11 65 127 1368 -5.5 3.9   7.5 \n21 91 143 1676 -5.7 6.6 10.4 \n30 82 135 1696 -5.8 6.0   9.8 \n42 79 136 1724 -6.4 5.8 10.1 \naccuracy ±7 ±4 ±40 ±0.3 ±0.4 ±0.5 \n \n \nThe g-factor of the samples was determined from the freq uency dependencies of the resonance field. \nThere was no significant thickness dependence obser ved yielding a value of g = 2.015(1) for all \nsamples. The cubic anisotropy field 2 K4/Ms was found to be nearly constant, and the average v alue \nis -84(2) Oe, which is in good agreement to reporte d values of -85 Oe for a 120 micrometer thick \nLPE film [81] and of about -80 Oe for a 18 nm thin LPE film [51]. Our calculated anisotropy \nconstants K4 are almost always in the range between -5.7×10 3 and -6.4×10 3 erg/cm 3, which \ncorresponds to YIG single crystal bulk values at 29 5 K [93] . Furthermore, a rather weak in-plane \nuniaxial anisotropy field 2 K2|| /Ms of about 0.6–2 Oe was found, which  had already be en determined \nfor 100 nm YIG LPE films [50].   \nThe stress-induced anisotropy constant Kσ and anisotropy field 2 Kσ/Ms are calculated according to \nRef. [94 ] (for details, see Eqs. (S14), (S15), (S18) in the Supplemental Material [69]). 2 Kσ/Ms is \nsmall and in the same order of magnitude as the cub ic anisotropy field 2 K4/Ms, but with opposite \nsign. Due to the observed monotonous increase of th e out-of-plane lattice misfit (see inset in Fig. \n2(b)), 2 Kσ/Ms grows with decreasing film thickness until it decl ines significantly at a film thickness \nbelow 21 nm. However, the observed stress values ar e almost an order of magnitude smaller than, \ne.g., for as-deposited YIG PLD films on GGG (111) u nder compressive strain (see, e.g., Refs. \n[17,23,35,36]). Only by a complex procedure, applyi ng mid-temperature deposition, cooling, and \npost-annealing treatment, authors of Ref. [95] succ eeded in a change from compressively to tensely \nstrained YIG films. These samples then exhibited th e same stress-induced anisotropy constant as it \nwas observed for our YIG LPE films. \nIn the following, we take a closer look to the cont ributions to the out-of-plane uniaxial anisotropy  \nfield H2⊥ = 2 K2⊥/Ms. A general description for magnetic garnets has been  given for example by \nHansen [94]. Applied to thick [43,64]  as well as to thin epitaxial iron garnet films (see , e.g., \n[37,56,59,60,62]) , the out-of-plane uniaxial anisotropy field H2⊥ is mainly determined by the \nmagnetocrystalline and  uniaxial anisotropy contributions. While the former  refers to the direction of \nmagnetization to preferred crystallographic directi ons in the cubic garnet lattice, the latter origina tes \nfrom lattice strain and growth conditions. Due to t he very low supercooling ( ≤5K), growth-induced \ncontributions, usually observed for micrometer YIG films with larger Pb impurity contents, can be \nneglected in the case of our nanometer-thin YIG LPE  films (see e.g. [64]). Thus,  H 2⊥ can be  16  determined quantitatively by summing the cubic magn etocrystalline anisotropy (first term, \ndetermined by FMR) and the stress-induced anisotrop y (second term, determined by XRD), \n \n \ns sMK\nMKHσ2\n344\n2 + − =⊥ , (6) \n \nor expressed for the (111) substrate orientation (s ee also SM [69 ] and Ref. [93,96]) by:  \n \n \nsMKH39 4111|| 4\n2λσ′ − −=⊥ . (7) \n \nUsing the experimentally determined first-order cub ic anisotropy constant K4 and the in-plane stress \ncomponent ||σ′from Tables I and III along with the room-temperatu re magnetostriction coefficient \nλ111  [94 ], the uniaxial anisotropy field H2⊥ can be calculated, if the saturation magnetization  Ms is \nknown. Ms can be obtained with appropriate accuracy for exam ple from VSM or SQUID \nmeasurements, if the sample volume is exactly known .  \nMagnetic hysteresis loops of YIG LPE films recorded  at room-temperature by VSM measurements \nwith in-plane applied magnetic field are shown in F ig. 6. The paramagnetic contribution of the GGG \nsubstrate was subtracted as described in Ref. [50]. Extremely small coercivity fields with Hc values \nof ∼ 0.2 Oe were obtained for all YIG/GGG samples with the exception of the 21 nm film. These \nvalues are comparable with the best gas phase epita xial films [17,39,76], but the measured saturation \nfields with Hs < 2.0 Oe are significantly smaller. All films exhi bit nearly in-plane magnetization due \nto the dominant contribution of form anisotropy. Ap art from the thinnest sample, the saturation \nmoments determined are not thickness-dependent (see  Table III and Fig. 6) and are very close to \nYIG volume values determined for YIG single crystal s at room temperature (4 πMs ∼ 1800 G) \n[93,97]. However, the observed decrease of the satu ration magnetization in such films with a \nthickness of about 10 nm is significant and will be  discussed below. \n-20 -15 -10 -5 0 5 10 15 20 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 \n10 100 1.4 1.5 1.6 1.7 1.8 1.9 \n  \n H (Oe)  4 πM (10 4 G)  106 nm \n 43 nm \n 30 nm \n 21 nm \n 11 nm \n   4 πM (10 4 G) \n d (nm) \n \nFIG.6: Magnetization loops M(H) of YIG films at room temperature as a function of  the in-plane \nmagnetic field. The inset shows the thickness-depen dent saturation magnetization (the solid line is a \nguide to the eyes). \n  17  However, for nanometer-thin films, it is a big chal lenge to determine Ms precisely enough, because \ntoo large errors can arise from the film’s volume c alculation. While the surface area of the sample \ncan be determined with sufficient precision by opti cal microscopy, thickness measurements with X-\nray or ellipsometry methods can lead to thickness e rrors in the range of ±1 nm due to very small \nmacroscopic morphology or roughness fluctuations. T herefore, for films with thicknesses below \n20 nm, for example, uncertainties up to a maximum o f 10 percent must be considered. This could \nsignificantly affect the effective magnetization 4 πMeff , which can be calculated based on the \nmeasured Ms values by \n ⊥ − =2 eff 4 4 H M Msπ π . (8) \nThis fact can explain the large difference between the calculated 4 πMeff cal  and the measured \n4πMeff exp  values for the 11 nm thin film discussed below, wh ile a much better agreement was \nachieved for the thicker films (see Table III). \nAs expected from micrometer-thick YIG LPE films gro wn on GGG (111) substrates [81], the out-\nof-plane uniaxial anisotropy field H2⊥ and the out-of-plane uniaxial anisotropy constants K2⊥ show, \nthat completely pseudomorphically strained, nanomet er-thin LPE films exhibit no pronounced \nmagnetic anisotropy. Small changes of the in-plane stress σ′|| (see Table I) and thus also in the \nstress-induced anisotropy 2 Kσ/Ms (or Kσ) have no significant influence on the out-of-plane  uniaxial \nanisotropy H2⊥ (see Table III). A comparable H2⊥ value is also expected for films thicker than \n42 nm, since the out-of-plane lattice misfit δd⊥\nfilm  tends to a constant value (see inset in Fig. 2 (b) ). \nThis is in contrast to Ref. [51], where the uniaxia l anisotropy field of YIG LPE films becomes \nnegative above a film thickness of about 50 nm. \n \n \n3. Thickness-dependent analysis of the effective ma gnetization field \n \nTo verify the trend of the calculated 4 πMeff cal  values for decreasing film thicknesses, one can \ncompare the effective magnetization with the experi mentally determined one. This was done for 18 \nYIG films with thicknesses ranging from 10 to 120 n m, including the four samples from above. All \nfilms were grown during the same run under nearly i dentical conditions. Only the growth \ntemperature was varied within a range of 5 K.  This time, the FMR was measured with a constant \nexternal magnetic field applied in-plane and sweepi ng the frequency.  \n0 20 40 60 80 100 120 1500 1550 1600 1650 1700 1750 1800 \n frequency field sweep \n magnetic field sweep Effective Magnetization (G) \nFilm thickness (nm) (a) \n  18  10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 535 540 545 550 555 560 \n fitted VSM values \n Outlier values Curie temperature (K) \nYIG thickness (m) (b) \n \nFig. 7. (a) Thickness dependence of the effective m agnetization 4 πMeff . Blue circles denote \nmeasurements taken by field-sweep and red squares d enote frequency-swept measurements, \nrespectively. (b) Thickness dependence of the Curie  temperature Tc. The dashed line is a guide to the \neyes. \n \n \nIn Fig. 7(a), the obtained thickness dependence of the effective magnetization 4 πMeff  (squares) is \npresented and a monotonous decrease of 4 πMeff  with film thickness reduction can be observed. \nBelow 40 nm, the slope of the curve increases and, for the thinnest films, there is a significant drop  \nof about 100 G. This behavior has been confirmed by  in-plane FMR magnetic field-sweep \nmeasurements for selected samples (circles), as dis cussed before. The values are listed in Table III. \nSimilar results have been reported for YIG PLD film s by Kumar et al. [53]. \nIf we compare the experimental values with the calc ulated ones in Table III, then the same trend of a \nsteady reduction of the effective saturation magnet ization with decreasing film thickness can be \nobserved. The deviation between both 4 πMeff  values is approximately 1–2 %, except for the 11 n m \nfilm. Hence, the saturation magnetization used to c alculate the effective saturation (according to \nequation (8)) does not appear to be as error-prone as it could be due to an inaccuracy in the film \nthickness determination. Therefore, we speculate, t hat the significant drop of 4 πMeff  for the 11 nm \nthin film can be explained by the observed reductio n of 4 πMs (see Table III). \nA similar behavior for 4 πMs was reported for thin PLD or magnetron-sputtered Y IG films, and \ndifferent explanations were given [76,56,91]. One r eason for a reduced saturation magnetization \ncould be an intermixing of substrate and film eleme nts at the GGG/YIG interface, whereby a gradual \nchange of the film composition is assumed [75,77]. In particular, gallium ion diffusion into the first  \nYIG atomic layers will lead to magnetically diluted  ferrimagnetic layers at the interface, due to the \nfact, that magnetic Fe ions are replaced by diamagn etic Ga ions in the various magnetic sublattices. \nThis assumption is supported by recent reports of Y IG films on GGG substrates. One reports about a \n5–7 nm deep Ga penetration found in laser-MBE films  [75 ]. Another group found a Ga penetration \nthroughout a 13-nm-thin PLD film [98]. In these cas es, high-temperature film growth above 850°C \nor prolonged post-annealing at temperatures of 850° C could promote such diffusion processes. In \ncontrast, the deposition time during which the LPE samples were exposed to high temperatures \nabove 860°C was only 5 minutes. Though, the assumed  Ga diffusion depth in our YIG films should \nnot exceed more than 2 nm according to the EDXS ele ment maps in Fig. 4(h). In addition, the Gd  19  diffusion in YIG films, as discussed for RF-magnetr on sputtered [76,99]  or  PLD films [98], could \nlead to the incorporation of paramagnetic ions into  the diamagnetic rare earth sublattice sites, which  \nwould also alter the magnetization [98]. However, n o extended interdiffusion layer was observed at \nthe film/substrate interface for our LPE films, so that the presence of a ‘separate, abrupt’ gadoliniu m \niron garnet interface layer, as reported by Ref. [9 8], is not expected. Therefore, due to possible \ninterdiffusion effects at temperatures of about 860 °C, a gradual reduction of Ms at a postulated \ninterface layer could be the reason for the observe d low saturation value for the thinnest LPE film, \nlisted in Tab. III. \nTo further rule out a discrete magnetic dead layer,  Curie temperature ( Tc) measurements were \nperformed by VSM. It is known from literature, that  Tc remains constant up to a film thickness of \napproximately four YIG unit cells [100], i.e. 2.8 n m, since one YIG unit cell length along the [111 ] \ndirection amounts to d111  ∼ 0.7 nm. Accordingly, the Tc of “pure” YIG films with abrupt interfaces \nand a film thickness of ∼10 nm should be equal to that of bulk material. In order to check this, \ntemperature-dependent VSM measurements (see Fig. 7( b)) were carried out for our LPE films as \nwell as for a bulk YIG single crystal slice, which was used as a reference. We found almost constant \nvalues of Tc = (551±2) K for sample thicknesses between 46 nm ( thin film) and 0.55 mm (bulk). \nThis is in good agreement with the literature, in w hich a Tc of ∼550 K has been reported, e.g. for a \n100-nm-thin sputtered YIG film [76],  while 559 K has been reported for YIG single crysta ls [97]. \nHowever, for our about 10-nm-thin YIG films, Tc decreased significantly to ∼534±1 K (Fig 7(b)), \nwhich is consistent with the observed reduction of 4 πMs listed in Table III. \nHence, the most likely explanation for the observed  reduction of 4 πMs is that the YIG layers at the \nsubstrate/film interface exhibit a reduced saturati on magnetization due to a magnetically diluted iron  \nsublattice, resulting from high-temperature diffusi on of gallium ions from the GGG substrate into \nthe YIG film . While nearly zero gallium content at the film surfa ce leads to a bulk-like value of \n4πMs ∼ 1800 G [93], an increased content of gallium at th e film/substrate interface should, therefore, \nresult in significantly reduced 4 πMs values. In this case, the average saturation magne tization for the \nentire film volume should be reduced and that could  explain the observed decrease in 4 πMs to about \n1500 G for the 11 nm thin LPE film. For thicker fil ms, however, the influence of thin gallium-\nenriched interface layers on the entire film magnet ization decreases, which explains the fast \nachievement of a constant Curie temperature, and th us, a constant Ms with increasing thickness of \nthe YIG volume. In order to confirm our assumptions , additional analyses, such as detailed \nsecondary ion mass spectroscopy (SIMS) investigatio ns, are necessary which, however, go beyond \nthe scope of this report. \n \n \nIV. CONCLUSIONS AND OUTLOOK \n \nIn summary, we have demonstrated that LPE can be us ed to fabricate sub-40 nm YIG films with \nhigh microstructural perfection, smooth surfaces an d sharp interfaces as well as excellent microwave \nproperties down to a minimum film thickness of 11 n m. All LPE films with ≥21 nm thickness \nexhibit extremely narrow FMR linewidths of ∆Hpp  <1.5 Oe at 15 GHz and very low magnetic \ndamping coefficients of α ≤1.3 × 10 -4 which are the lowest values reported within an ext ended \nfrequency range of 1 to 40 GHz. We were able to sho w that LPE-grown YIG films down to a \nthickness of 21 nm have the same magnetization dyna mics influenced by small cubic and stress-\ninduced anisotropy fields. The deviating magnetizat ion dynamics of ultrathin LPE films with \nthicknesses of ∼10 nm are probably caused by an increased inhomogen eous damping and by small \ntwo-magnon scattering contributions, and we specula te that possible inhomogeneities of the \ncomposition in the vicinity of the film/substrate i nterface might be the reason for this. Therefore, i n  20  further studies we will address detailed investigat ions of the composition of the film/substrate \ninterface by high-resolution SIMS measurements and advanced STEM analyses to confirm a gradual \nchange of the LPE film composition at the interface . \nThe results presented here encourage us to take the  next step towards nano- and microscaled \nmagnonic structures, such as directional couplers, logic gates, transistors etc. for a next-generation  \nof computing circuits. The development of nanoscopi c YIG waveguides and nanostructures is \nalready underway and the first circuits are current ly being fabricated [10,12,29]. With its scalabilit y \nto large wafer diameters of up to 3 and 4 inches, L PE technology opens up an alternative way for \nefficient circuit manufacturing for a future YIG pl anar technology on a wafer scale. \n \n \nACKNOWLEDGMENTS \n \nWe thank P. Landeros and R. Gallardo for fruitful d iscussions and A. Khudorozhkov for his help \nduring the measurements. C. D. and O. S. thank R. K öcher for AFM measurements, A. Hartmann \nfor the DSC measurements and R. Meyer and B. Wenzel  for technical support. J. G. thanks A. \nScholz for the support during the XRD measurements.  We would like to thank Romy Aniol for the \nTEM specimen preparation. The use of HZDR’s Ion Bea m Center TEM facilities and the funding of \nTEM Talos by the German Federal Ministry of Educati on of Research (BMBF), Grant No. \n03SF0451 in the framework of HEMCP are gratefully a cknowledged. \nThis research was financially supported by the Deut sche Forschungsgemeinschaft (DFG), via Grant \nNo. DU 1427/2-1. \n \nReferences  \n \n[1] A. A. Serga, A.V. Chumak, and B. Hillebrands, YIG magnonics , J. Phys. D: Appl. Phys. 43 , \n264002 (2010). \n[2] I. V. Zavislyak and M. A. Popov, Microwave Properties and Applications of Yttrium Ir on \nGarnet, in: Yttrium: Compounds, Production, Ed. by. B. D. Volkerts , Nova Science \nPublishers, Frankfurt 2009 . \n[3] F. Schloemann, Circulators for Microwave and Millimeter-Wave Integ rated Circuits,  Proc. \nIEEE 76 , 188 (1988). \n[4] W. S. Ishak, Magnetostatic Wave Technology: A Review , Proc. IEEE 76 , 171 (1988). \n[5] J. D. Adam, Analog Signal Processing with Microwave Magnetics, Proc. 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Mater. 170 , L243 (1997).  \n[101 ] Strictly speaking this is valid only for [001] ori ented systems for all others, if the cubic \nlattice constant values of afilm  and asubstrate  are almost identical.  27  Supplemental Material: \nLow damping and microstructural perfection of sub-4 0nm-thin yttrium iron garnet films \ngrown by liquid phase epitaxy  \n \nCarsten Dubs, 1 Oleksii Surzhenko, 1 Ronny Thomas, 2 Julia Osten, 2 Tobias Schneider, 2 Kilian Lenz, 2 \nJörg Grenzer, 2 René Hübner,2 Elke Wendler 3 \n \n1 INNOVENT e.V. Technologieentwicklung, Prüssingstr.  27B, 07745 Jena, Germany \n2 Institute of Ion Beam Physics and Materials Resear ch, Helmholtz-Zentrum Dresden-Rossendorf, \nBautzner Landstr. 400, 01328 Dresden, Germany \n3 Institut für Festkörperphysik, Friedrich-Schiller- Universität Jena, Helmholtzweg 3, 07743 Jena, \nGermany \n \n \nI. STRAIN CALCULATIONS  \n \nIn the following we will derive the in-plane (horiz ontal) stress σ||  as a function of the out-of-plane \n(vertical) lattice misfit ⊥\nfilmdδ  for a [111] oriented cubic system. These calculati ons are based on the \nelasticity theory following mainly Hinckley [70 ], Sander [71 ] and Ortiz et al . [61 ]. Assuming a fully \npseudomorphic system there is only one parameter to  be determined: The out-of-plane lattice misfit \n⊥\nfilmdδ . This value can be directly  obtained from the data of the HR-XRD measurements a nd/or from \nthe corresponding simulations of the symmetrical (4 44) and (888) reflections. \nThe vertical and parallel lattice misfits can be ca lculated by \n , 0||\nsubstrate||\nsubstrate||\nfilm ||\nfilm\nsubstratesubstrate film\nfilm =−=−=⊥⊥ ⊥\n⊥\ndd dddd dd δ δ  (S1) \nwhere dk\ni with i = [substrate, (pseudomorph) film or (cubic) relaxe d film] is the (measured) net \nplane distances for the k = ⊥ (vertical) or || (parallel) direction with respect  to the substrate surface. \nIgnoring any dynamical diffraction effects, the out -of-plane lattice misfit can be directly determined  \nfrom the measurement as follows: \n \nB qq\nqq qq dθθδ δtansubstrate substratesubstrate film\nfilm film∆− =∆− =−− = − =⊥⊥\n⊥⊥ ⊥\n⊥ ⊥, (S2) \nwhere q⊥film  and q⊥substrate are the derived peak positions in the Q-space of th e thin film and the \nsubstrate, respectively. ∆θB is the (kinematical) Bragg angular difference “thi n film – substrate” and \nθB is the Bragg Peak position of the substrate, respe ctively. These formulas follow directly from the \nderivative of Bragg’s law. \nThe in- and out-of-plane strains are given as follo ws: \n .||\nfilm  relaxed||\nfilm  relaxed||\nfilm ||\nfilm  relaxedfilm  relaxed film\ndd d\ndd d −=−=⊥⊥ ⊥\n⊥ε ε  (S3) \nThe general relationship between stress and strain is defined as follows: \n ,kl ijkl ij ε σ C=  (S4) \nwhere σij  are the stress, ε kl  the strain and Cijkl  the second order stiffness tensors and the summati on is \ndone over the repeated indices. The subscripts ij and  kl  refer to the axes of the coordinates system of  28  the unit cell (1,2,3 = x,y,z ). The samples under investigations have a [111] ou t-of-plane orientation; \ntherefore, the corresponding rotation matrices have  to be applied: \n ijkll k j i CUUUU Cδ γ β α αβγδ=′ . (S5) \nFor the [111] oriented surfaces U 111  yields to \n \n\n\n\n −\n=\n310\n3231\n21\n6131\n21\n61\n111U . (S6) \n \n \n \nFigure S1 : Representation of the cubic unprimed and the rota ted, primed coordinate system for an \n<111> oriented thin film; where x´, y´ correspond to the in-plane (||) directions and z´ to the out-of-\nplane ( ⊥) direction; after [71 ]. \n \n \nThe in-plane stress ||σ′ can be expressed in terms of the out-of-plane latt ice misfit ⊥\nfilmdδ obtained by \nXRD measurements.  \n \nFor a cubic pseudomorphic system we can write:  \n 1313 1212 1111 11 ε ε ε σ c c c + + =  (S7) \n ( )⊥′+′+′=′ ε ε σ12 || 12 11 || c c c  (S8) \nwith  \n ⊥′− =ε ν ε||  (S9) \nresulting in: \n ||\n44 12 1112 11\n44 ||4 226 ε σc c cc cc+ ++=′ . (S10) \n \n  29   \nTaking the corresponding rotation matrices and rela tionships into account  [101 ]: \n ⊥ ⊥ ⊥\n+− =+=film 111111\n||\nfilm 1111and11d d δννε δνε , (S11) \nwhere \n .4 4 24 2\n44 12 1144 12 11 111\nc c cc c c\n− ++ +=ν  (S12) \n \nThe in-plane stress can be now expressed in terms o f the out-of-plane lattice misfit by: \n ⊥− =′film 44 || 2 dcδ σ . (S13) \nHere, c44  is the component from the stiffness tensor and ⊥\nfilmdδ is the out-of-plane lattice misfit as \ndefined above. \n \n \nII. ANISOTROPY CALCULATIONS FOR (111) ORIENTED EPIT AXIAL GARNET FILMS  \n \nThe stress-induced anisotropy parameter for the cub ic (111) orientation can be calculated according \nto [94] by \n 111||23λ =Kσ σ′ − , (S14) \nwhere σ´||  is the above calculated in-plane stress for {111} oriented thin films, and λ 111  is the \ncorresponding magnetostriction constant. \nThe stress-induced anisotropy parameter is therefor e given by: \n 111 film 443 λdc=Kσ⊥δ . (S15) \nThe perpendicular magnetic anisotropy field can be calculated according to [43]: \n growth cub 2 H+ H+ H= Hstress ⊥ . (S16) \nAssuming negligible growth-induced contributions Hgrowth  and applying the cubic anisotropy field \nfor (111) film orientation obtained by FMR measurem ents \n \nscubMK= H4\n34− , (S17) \nand taking into account the stress-induced anisotro py field \n \ns sσ\nstressMλ=MK= H111||3 2 σ′\n− , (S18) \nthe effective perpendicular anisotropy field result s in \n \nsMλ +KH39 4111|| 4\n2σ′\n− =⊥ . (S19)  30  From the resonance conditions for the perpendicular  (M ||  [111 ]) magnetized epitaxial thin film, the \neffective saturation magnetization can be obtained by [64 ] \n \n+ − − =⊥\ns ssMK\nMKM Hf2 4\neff2\n344πω,  (S20) \nfrom which the effective saturation magnetization c an be calculated by \n ⊥ −2 eff eff 4 4 H πM= πM=Hs . (S21) \n \nIII. FERROMAGNETIC RESONANCE  \n \nFrom the free energy density given by Eq. (1) of th e main text, the resonance equations have been \ncalculated applying the approach of Baselgia et al.  [80 ]. The resonance conditions for the frequency-\ndependences with field out-of-plane ( f⊥) and in-plane ( f|| ) read: \n \n \n\n\n+ − − \n\n− − =⊥MK\nMKM HMKM H fe e|| 2 4\nff4\nff 23443442π ππγ, (S22)  \n ( ) ( ) ( )\n\n\n\n\n\n− − − − + ×\n\n\n− − = ϕ ϕϕ π ϕϕπγ3 cos 2 cos24 2cos2\n222\n4 2 || 2 4\nff|| 2\n||MK\nMK\nMKM HMKH fu e u\n  (S23). \n \nExamples of angle- and frequency-dependent FMR meas urements with out- and in-plane \nconfiguration of the magnetic bias field are shown in Figure S2.  31  -30° 0° 30° 60° 90° 120° 345f = 10 GHz \n11 nm \n21 nm \n30  nm \n42 nm \n Fits Hres (kOe)\nθH\n0 5 10 15 010 20 30 40 f (GHz)\nH (kOe) \n0 5 10 15 010 20 30 40 \nθH=0° f(GHz)\nH (kOe)  11 nm \n 21 nm \n 30 nm \n 42 nm \n Fit (11 nm) θH=90° \n 11 nm \n 21 nm \n 30 nm \n 42 nm (a) \n(b) \n(c) 2.76 2.77 2.78 2 .7 9 2 .8 0 FMR- S igna l  (arb . units)\nH (kOe) 42 nm \nf = 10 GHz \n4.30 4.32 4.34 4.36 4.38 4 .4 0 FMR-Signa l  (arb . units)\nH (kOe) 11 nm \nf = 8 GHz θH\nφHH→\n[110] _ [112] _M→\nY IG(111) φθ\n \nFigure S2. (a) Polar angular dependencies of the FM R measured at f = 10 GHz. The inset shows the \nFMR coordinate system. Solid lines are fits accordi ng to the resonance equation. (b) Frequency \ndependencies of the resonance field measured with f ield in-plane and (c) out-of-plane. The solid \nblack line is a fit to the 11 nm dataset. Other fit  curves have been omitted for visual clarity. Inset s \nshow FMR spectra and the indicated positions includ ing Lorentzian fits. "    },    {        "title": "1210.6879v1.Decay_rates_for_the_damped_wave_equation_on_the_torus.pdf",        "content": "arXiv:1210.6879v1  [math.AP]  25 Oct 2012Decay rates for the damped wave equation on the torus\nWith an appendix by St´ ephane Nonnenmacher∗\nNalini Anantharaman†and Matthieu L´ eautaud‡,\nUniversit´ e Paris-Sud 11, Math´ ematiques, Bˆ atiment 425, 91405 Orsay Cedex, France\nOctober 26, 2012\nAbstract\nWe address the decay rates of the energy for the damped wave eq uation when the damping\ncoefficient bdoes not satisfy the Geometric Control Condition (GCC). Fir st, we give a link with\nthe controllability of the associated Schr¨ odinger equati on. We prove in an abstract setting that\nthe observability of the Schr¨ odinger group implies that th e semigroup associated to the damped\nwave equation decays at rate 1 /√\nt(which is a stronger rate than the general logarithmic one\npredicted by the Lebeau Theorem).\nSecond, we focus on the 2-dimensional torus. We prove that th e best decay one can expect\nis 1/t, as soon as the damping region does not satisfy GCC. Converse ly, for smooth damping\ncoefficients b, we show that the semigroup decays at rate 1 /t1−ε, for allε >0. The proof relies\non a second microlocalization around trapped directions, a nd resolvent estimates.\nIn the case where the damping coefficient is a characteristic f unction of a strip (hence dis-\ncontinuous), St´ ephane Nonnenmacher computes in an append ix part of the spectrum of the\nassociated damped wave operator, proving that the semigrou p cannot decay faster than 1 /t2/3.\nIn particular, our study shows that the decay rate highly dep ends on the way bvanishes.\nKeywords\nDampedwaveequation, polynomialdecay, observability, Sc hr¨ odingergroup, torus, two-microlocal\nsemiclassical measures, spectrum of the damped wave operat or.\nContents\nI The damped wave equation 2\n1 Decay of energy: a survey of existing results 2\n2 Main results of the paper 5\n2.1 The damped wave equation in an abstract setting . . . . . . . . . . . . . . . . . . . . . 5\n2.2 Decay rates for the damped wave equation on the torus . . . . . . . . . . . . . . . . . 7\n2.3 Some related open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\nII Resolvent estimates and stabilization in the abstract se tting 10\n3 Proof of Theorem 2.3 assuming Proposition 2.4 10\n∗snonnenmacher@cea.fr\n†Nalini.Anantharaman@math.u-psud.fr\n‡Matthieu.Leautaud@math.u-psud.fr\n14 Proof of Proposition 2.4 10\nIII Proof of Theorem 2.6: smooth damping coefficients on the to rus 15\n5 Semiclassical measures 16\n6 Zero-th and first order informations on µ 17\n7 Geometry on the torus and decomposition of invariant measu res 19\n7.1 Resonant and non-resonant vectors on the torus . . . . . . . . . . . . . . . . . . . . . . 19\n7.2 Decomposition of invariant measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20\n7.3 Geometry of the subtori TΛandTΛ⊥. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21\n8 Change of quasimode and construction of an invariant cutoff function 22\n9 Second microlocalization on a resonant affine subspace 23\n10 Propagation laws for the two-microlocal measures νΛandρΛ 28\n10.1 Propagation of νΛ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28\n10.2 Propagation of ρΛ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30\n11 The measures νΛandρΛvanish identically. End of the proof of Theorem 2.6 32\n12 Proof of Proposition 8.2 33\n13 Proof of Proposition 8.3: existence of the cutoff function 35\nIV An a priori lower bound for decay rates on the torus: proof of\nTheorem 2.5 41\nA Pseudodifferential calculus 42\nB Spectrum of P(z)for a piecewise constant damping\n(by St´ ephane Nonnenmacher) 42\nPart I\nThe damped wave equation\n1 Decay of energy: a survey of existing results\nLet (M,g) be a smooth compact connected Riemannian d-dimensional manifold with or without\nboundary∂M. We denote by ∆ the (non-positive) Laplace-Beltrami operator on Mfor the metric\ng. Given a bounded nonnegative function, b∈L∞(M),b(x)≥0 onM, we want to understand the\nasymptotic behaviour as t→+∞of the solution uof the problem\n\n\n∂2\ntu−∆u+b(x)∂tu= 0 inR+×M,\nu= 0 on R+×∂M(if∂M/\\e}atio\\slash=∅),\n(u,∂tu)|t=0= (u0,u1) inM.(1.1)\nThe energy of a solution is defined by\nE(u,t) =1\n2(/ba∇dbl∇u(t)/ba∇dbl2\nL2(M)+/ba∇dbl∂tu(t)/ba∇dbl2\nL2(M)). (1.2)\n2Multiplying (1.1) by ∂tuand integrating on Myields the following dissipation identity\nd\ndtE(u,t) =−/integraldisplay\nMb|∂tu|2dx,\nwhich, asbis nonnegative, implies a decay of the energy. As soon as b≥C >0 on a nonempty open\nsubset ofM, the decay is strict and E(u,t)→0 ast→+∞. The question is then to know at which\nrate the energy goes to zero.\nThe first interesting issue concerns uniform stabilization: under wh ich condition does there exist\na functionF(t),F(t)→0, such that\nE(u,t)≤F(t)E(u,0) ? (1.3)\nThe answer was given by Rauch and Taylor [RT74] in the case ∂M=∅and by Bardos, Lebeau and\nRauch [BLR92] in the general case (see also [BG97] for the necessit y of this condition): assuming\nthatb∈C0(M), uniform stabilisation occurs if and only if the set {b >0}satisfies the Geometric\nControl Condition (GCC). Recall that a set ω⊂Mis said to satisfy GCC if there exists L0>0\nsuch that every geodesic γ(resp. generalised geodesic in the case ∂M/\\e}atio\\slash=∅) ofMwith length larger\nthanL0satisfiesγ∩ω/\\e}atio\\slash=∅. Under this condition, one can take F(t) =Ce−κt(for some constants\nC,κ>0) in (1.3), and the energy decays exponentially. Finally, Lebeau give s in [Leb96] the explicit\n(and optimal) value of the best decay rate κin terms of the spectral abscissa of the generator of the\nsemigroup and the mean value of the function balong the rays of geometrical optics.\nIn the case where {b >0}does not satisfy GCC, i.e. in the presence of “trapped rays” that d o\nnot meet {b>0}, what can be said about the decay rate of the energy? As soon as b≥C >0 on\na nonempty open subset of M, Lebeau shows in [Leb96] that the energy (of smoother initial data )\ngoes at least logarithmically to zero (see also [Bur98]):\nE(u,t)≤C/parenleftbig\nf(t)/parenrightbig2/parenleftBig\n/ba∇dblu0/ba∇dbl2\nH2(M)∩H1\n0(M)+/ba∇dblu1/ba∇dbl2\nH1\n0(M)/parenrightBig\n,for allt>0, (1.4)\nwithf(t) =1\nlog(2+t)(whereH2(M)∩H1\n0(M) andH1\n0(M) have to be replaced by H2(M) andH1(M)\nrespectively if ∂M=∅). Note that here,/parenleftbig\nf(t)/parenrightbig2characterizes the decay of the energy, whereas f(t)\nis that of the associated semigroup. Moreover, the author const ructed a series of explicit examples\nof geometries for which this rate is optimal, including for instance the case where M=S2is the\ntwo-dimensional sphere and {b >0} ∩Nε=∅, whereNεis a neighbourhood of an equator of S2.\nThis result is generalised in [LR97] for a wave equation damped on a (sm all) part of the boundary.\nIn this paper, the authors also make the following comment about th e result they obtain:\n“Notons toutefois qu’une ´ etude plus approfondie de la localisation s pectrale et des taux de\nd´ ecroissance de l’´ energie pour des donn´ ees r´ eguli` eres doit faire intervenir la dynamique globale du\nflot g´ eod´ esique g´ en´ eralis´ esur M. Les th´ eor` emes [LR97, Th´ eor` eme 1] et [LR97, Th´ eor` eme 2] ne four-\nnissent donc que les bornes a prioriqu’on peut obtenir sans aucune hypoth` ese sur la dynamique, en\nn’utilisant que les in´ egalit´ es de Carleman qui traduisent “l’effet tunn el”.”\nIn all examples where the optimal decay rate is logarithmic, the trap ped ray is a stable trajectory\nfrom the point of view of the dynamics of the geodesic flow. This mean s basically that an important\namount of the energy can stay concentrated, for a long time, in a n eighbourhood of the trapped ray,\ni.e. away from the damping region.\nIf the trapped trajectories are less stable, or unstable, one can expect to obtain an intermediate\ndecay rate, between exponential and logarithmic. We shall say tha t the energy decays at rate f(t)\nif (1.4) is satisfied (more generally, see Definition 2.2 below in the abstr act setting). This problem\nhas already been adressed and, in some particular geometries, sev eral different behaviours have been\nexhibited. Two main directions have been investigated.\nOn the one hand, Liu and Rao considered in [LR05] the case where Mis a square and the set\n{b >0}contains a vertical strip. In this situation, the trapped trajecto ries consist in a family\n3of parallel vertical geodesics; these are unstable, in the sense th at nearby geodesics diverge at a\nlinear rate. They proved that the energy decays at rate/parenleftBig\nlog(t)\nt/parenrightBig1\n2(i.e., that (1.4) is satisfied with\nf(t) =/parenleftBig\nlog(t)\nt/parenrightBig1\n2). This was extended by Burq and Hitrik [BH07] (see also [Nis09]) to the case of\npartially rectangular two-dimensional domains, if the set {b >0}contains a neighbourhood of the\nnon-rectangular part. In [Phu07], Phung proved a decay at rate t−δfor some (unprecised) δ >0 in\na three-dimensional domain having two parallel faces. In all these s ituations, the only obstruction\nto GCC is due to a “cylinder of periodic orbits”. The geometry is flat an d the unstabilities of the\ngeodesic flow around the trapped rays are relatively weak (geodes ics diverge at a linear rate).\nIn [BH07], the authors argue that the optimal decay in their geomet ry should be of the form1\nt1−ε,\nfor allε>0. They provide conditions on the damping coefficient b(x) under which one can obtain\nsuch decay rates, and wonder whether this is true in general. Our m ain theorem (see Theorem 2.6\nbelow) extends these results to more general damping functions bon the two-dimensional torus.\nOn the other hand, Christianson [Chr10] proved that the energy d ecays at rate e−C√\ntfor some\nC >0, in the case where the trapped set is a hyperbolic closed geodesic. Schenck [Sch11] proved an\nenergy decay at rate e−Cton manifolds with negative sectional curvature, if the trapped set is “small\nenough” in terms of topological pressure (for instance, a small ne ighbourhood of a closed geodesic),\nand if the damping is “large enough” (that is, starting from a damping functionb,βbwill work for\nanyβ >0 sufficiently large). In these two papers, the geodesic flow near th e trapped set enjoys\nstrong instability properties: the flow on the trapped set is uniform ly hyperbolic, in particular all\ntrajectories are exponentially unstable.\nThese cases confirm the idea that the decay rates of the energy s trongly depends on the stability\nof trapped trajectories.\nOne may now want to compare these geometric situations to situatio ns where the Schr¨ odinger\ngroup is observable (or, equivalently, controllable), i.e. for which th ere existC >0 andT >0 such\nthat, for all u0∈L2(M), we have\n/ba∇dblu0/ba∇dbl2\nL2(M)≤C/integraldisplayT\n0/ba∇dbl√\nb e−it∆u0/ba∇dbl2\nL2(M)dt. (1.5)\nThe conditions under which this property holds are also known to be r elated to stability of the\ngeodesic flow. In particular, the works [BLR92], [LR05], [BH07, Nis09] and [Chr10, Sch11] can be\nseen as counterparts for damped wave equations of the articles [L eb92], [Har89a, Jaf90], [BZ04] and\n[AR10], respectively, in the context of observation of the Schr¨ od inger group.\nOur main results are twofold. First, we clarify (in an abstract settin g) the link between the ob-\nservability (or the controllability) of the Schr¨ odinger equation and polynomial decay for the damped\nwave equation. This follows the spirit of [Har89b], [Mil05], exploring the lin ks between the different\nequations and their control properties (e.g. observability, contr ollability, stabilization...). More pre-\ncisely, we prove that the controllability of the Schr¨ odinger equatio n implies a polynomial decay at\nrate1√\ntfor the damped wave equation (Theorem 2.3).\nSecond, we study precisely the damped wave equation on the flat to rusT2in case GCC fails.\nWe give the following a priori lower bound on the decay rate, revisiting the argument of [BH07]:\n(1.1) is not stable at a better rate than1\nt, provided that GCC is not satisfied. In this situation, the\nSchr¨ odingergroupisknowntobecontrollable(see[Jaf90], [Kom92] andthemorerecentworks[AM11]\nand [BZ11]). Thus, one cannot hope to have a decay better than po lynomial in our previous result,\ni.e. under the mere assumption that the Schr¨ odinger flow is observ able.\nThe remainder of the paper is devoted to studying the gap between thea priorilower and upper\nbounds given respectively by1\ntand1√\nton flat tori. For smoothnonvanishing damping coefficient\nb(x), we prove that the energy decays at rate1\nt1−εfor allε >0. This result holds without making\nany dynamical assumption on the damping coefficient, but only on the order of vanishing of b. It\ngeneralises a result of [BH07], which holds in the case where bis invariant in one direction. Our\n4analysis is, again, inspired by the recent microlocal approach propo sed in [AM11] and [BZ11] for\nthe observability of the Schr¨ odinger group. More precisely, we fo llow here several ideas and tools\nintroduced in [Mac10] and [AM11].\nIn the situation where bis a characteristic function of a vertical strip of the torus (hence discon-\ntinuous), St´ ephane Nonnenmacher proves in Appendix B that the decay rate cannot be faster than\n1\nt2/3. This is done by explicitly computing the high frequency eigenvalues of the damped wave oper-\nator which are closest to the imaginary axis (see for instance the fig ures in [AL03, AL12]). The fact\nthat the decay rate 1 /tis not achieved in this situation was observed in the numerical comput ations\npresented in [AL12].\nIn contrast to the control problem for the Sch¨ odinger equation , this result shows that the stabi-\nlization of the wave equation is not only sensitive to the global proper ties of the geodesic flow, but\nalso to the rate at which the damping function vanishes.\n2 Main results of the paper\nOur first result can be stated in a general abstract setting that w e now introduce. We come back to\nthe case of the torus afterwards.\n2.1 The damped wave equation in an abstract setting\nLetHandYbe two Hilbert spaces (resp. the state space and the observation /control space) with\nnorms/ba∇dbl·/ba∇dblHand/ba∇dbl·/ba∇dblY, and associated inner products ( ·,·)Hand (·,·)Y.\nWe denote by A:D(A)⊂H→Hanonnegative selfadjoint operatorwith compact resolvent, and\nB∈ L(Y;H) a control operator. We recall that B∗∈ L(H;Y) is defined by ( B∗h,y)Y= (h,By)H\nfor allh∈Handy∈Y.\nDefinition 2.1. We say that the system\n∂tu+iAu= 0, y=B∗u, (2.1)\nis observable in time Tif there exists a constant KT>0 such that, for all solution of (2.1), we have\n/ba∇dblu(0)/ba∇dbl2\nH≤KT/integraldisplayT\n0/ba∇dbly(t)/ba∇dbl2\nYdt.\nWe recall that the observability of (2.1) in time Tis equivalent to the exact controllability in\ntimeTof the adjoint problem\n∂tu+iAu=Bf, u(0) =u0, (2.2)\n(see for instance [Leb92] or [RTTT05]). More precisely, given T >0, the exact controllability in time\nTis the ability of finding for any u0,u1∈Ha control function f∈L2(0,T;Y) so that the solution\nof (2.2) satisfies u(T) =u1.\nWe equip H=D(A1\n2)×Hwith the graph norm\n/ba∇dbl(u0,u1)/ba∇dbl2\nH=/ba∇dbl(A+Id)1\n2u0/ba∇dbl2\nH+/ba∇dblu1/ba∇dbl2\nH,\nand define the seminorm\n|(u0,u1)|2\nH=/ba∇dblA1\n2u0/ba∇dbl2\nH+/ba∇dblu1/ba∇dbl2\nH.\nOf course, if Ais coercive on H,|·|His a norm on Hequivalent to /ba∇dbl·/ba∇dblH.\nWe also introduce in this abstract setting the damped wave equation on the space H,\n/braceleftBigg\n∂2\ntu+Au+BB∗∂tu= 0,\n(u,∂tu)|t=0= (u0,u1)∈ H,(2.3)\n5which can be recast on Has a first order system\n/braceleftbigg\n∂tU=AU,\nU|t=0=t(u0,u1),U=/parenleftbigg\nu\n∂tu/parenrightbigg\n,A=/parenleftbigg\n0 Id\n−A−BB∗/parenrightbigg\n, D(A) =D(A)×D(A1\n2).(2.4)\nThe compact injections D(A)֒→D(A1\n2)֒→Himply that D(A)֒→ Hcompactly, and that the\noperator Ahas a compact resolvent.\nWe define the energy of solutions of (2.3) by\nE(u,t) =1\n2/parenleftbig\n/ba∇dblA1\n2u/ba∇dbl2\nH+/ba∇dbl∂tu/ba∇dbl2\nH/parenrightbig\n=1\n2|(u,∂tu)|2\nH2.\nDefinition 2.2. Letfbe a function such that f(t)→0 whent→+∞. We say that System (2.3)\nis stable at rate f(t) if there exists a constant C >0 such that for all ( u0,u1)∈D(A), we have\nE(u,t)1\n2≤Cf(t)|A(u0,u1)|H,for allt>0.\nIf it is the case, for all k >0, there exists a constant Ck>0 such that for all ( u0,u1)∈D(Ak), we\nhave (see for instance [BD08, page 767])\nE(u,t)1\n2≤Ck/parenleftbig\nf(t)/parenrightbigk/ba∇dblAk(u0,u1)/ba∇dblH,for allt>0.\nTheorem 2.3. Suppose that there exists T >0such that System (2.1)is observable in time T. Then\nSystem(2.3)is stable at rate1√\nt.\nNote that the gain of the log( t)1\n2with respect to [LR05, BH07] is not essential in our work. It is\ndue to the optimal characterization of polynomially decaying semigro ups obtained by Borichev and\nTomilov [BT10].\nThis Theorem may be compared with the works (both presented in a s imilar abstract setting)\n[Har89b]byHaraux,provingthatthe controllabilityofwave-typee quationsinsometimeisequivalent\nto uniform stabilization of (2.3), and [Mil05] by Miller, showing that the c ontrollability of wave-type\nequations in some time implies the controllability of Schr¨ odinger-type equations in any time.\nNote that the link between this abstract setting and that of Proble m (1.1) isH=Y=L2(M),\nA=−∆ withD(A) =H2(M) if∂M=∅andH2(M)∩H1\n0(M) otherwise, Bis the multiplication in\nL2(M) by the bounded function√\nb.\nAs a first application of Theorem 2.3 we obtain a different proof of the polynomial decay results\nfor wave equations of [LR05] and [BH07] as consequences of the as sociated control results for the\nSchr¨ odinger equation of [Har89a] and [BZ04] respectively.\nMoreover, Theorem 2.3 provides also several new stability results f or System (1.1) in particu-\nlar geometric situations; namely, in all following situations, the Schr¨ odinger group is proved to be\nobservable, and Theorem 2.3 gives the polynomial stability at rate1√\ntfor (1.1):\n•For any nonvanishing b(x)≥0 in the 2-dimensional square (resp. torus), as a consequence of\n[Jaf90] (resp. [Mac10, BZ11]); for any nonvanishing b(x)≥0 in thed-dimensional rectangle\n(resp.d-dimensional torus) as a consequence of [Kom92] (resp. [AM11]);\n•IfMis the Bunimovich stadium and b(x)>0 on the neighbourhood of one half disc and on\none point of the opposite side, as a consequence of [BZ04];\n•IfMis ad-dimensional manifold of constant negative curvature and the set of trapped tra-\njectories (as a subset of S∗M, see [AR10, Theorem 2.5] for a precise definition) has Hausdorff\ndimension lower than d, as a consequence of [AR10];\n6Moreover, Lebeau gives in [Leb96, Th´ eor` eme 1 (ii)] several 2-dim ensional examples for which the\ndecay rate1\nlog(2+t)is optimal. For all these geometrical situations, Theorem 2.3 implies th at the\nSchr¨ odinger group is not observable.\nThe proof of Theorem 2.3 relies on the following characterization of p olynomial decay for Sys-\ntem (2.3). For z∈C, we define on Hthe operator P(z) =A+z2Id+zBB∗, with domain\nD(P(z)) =D(A). We prove in Lemma 4.2 below that P(is) is invertible for all s∈R,s/\\e}atio\\slash= 0.\nProposition 2.4. Suppose that\nfor any eigenvector ϕofA, we haveB∗ϕ/\\e}atio\\slash= 0. (2.5)\nThen, for all α>0, the five following assertions are equivalent:\nThe system (2.3)is stable at rate1\ntα, (2.6)\nThere exist C >0ands0≥0such that for all s∈R,|s| ≥s0,/ba∇dbl(isId−A)−1/ba∇dblL(H)≤C|s|1\nα,(2.7)\nThere exist C >0ands0≥0such that for all z∈C,satisfying |z| ≥s0,\nand|Re(z)| ≤1\nC|Im(z)|1\nα,we have /ba∇dbl(zId−A)−1/ba∇dblL(H)≤C|Im(z)|1\nα,(2.8)\nThere exist C >0ands0≥0such that for all s∈R,|s| ≥s0,/ba∇dblP(is)−1/ba∇dblL(H)≤C|s|1\nα−1,(2.9)\nThere exists C >0ands0≥0such that for all s∈R,|s| ≥s0andu∈D(A),\n/ba∇dblu/ba∇dbl2\nH≤C/parenleftbig\n|s|2\nα−2/ba∇dblP(is)u/ba∇dbl2\nH+|s|1\nα/ba∇dblB∗u/ba∇dbl2\nY/parenrightbig\n.(2.10)\nThispropositionisprovedasaconsequenceofthecharacterizatio nofpolynomialdecayforgeneral\nsemigroups in terms of resolvent estimates given in [BT10], providing t he equivalence between (2.6)\nand (2.7). See also [BD08] for generaldecay rates in Banach space s. Note in particular that the proof\nof a decay rate is reduced to the proof of a resolvent estimate on t he imaginary axes. By the way,\nthis estimate implies the existence of a “spectral gap” between the spectrum of Aand the imaginary\naxis, given by (2.8).\nNote finally that the estimates (2.7), (2.9) and (2.10) can be equivale ntly restricted to s >0,\nsinceP(−is)u=P(is)u.\n2.2 Decay rates for the damped wave equation on the torus\nThe main results of this article deal with the decay rate for Problem ( 1.1) on the torus T2:=\n(R/2πZ)2. In this setting, as well as in the abstract setting, we shall write P(z) =−∆+z2+zb(x).\nFirst, we give an a priori lower bound for the decay rate of the damped wave equation, on\nT2, when GCC is “strongly violated”, i.e. assuming that supp( b) does not satisfy GCC (instead of\n{b>0}). This theorem is proved by constructing explicit quasimodes for the operator P(is).\nTheorem 2.5. Suppose that there exists (x0,ξ0)∈T∗T2,ξ0/\\e}atio\\slash= 0, such that\n{b>0}∩{x0+τξ0,τ∈R}=∅.\nThen there exist two constants C >0andκ0>0such that for all n∈N,\n/ba∇dblP(inκ0)−1/ba∇dblL(L2(T2))≥C. (2.11)\nAs a consequence of Proposition 2.4, polynomial stabilization at rate1\nt1+εforε >0 is not\npossible if there is a strongly trapped ray (i.e. that does not interse ct supp(b)). More precisely,\nin such geometry, Theorem 2.5 combined with Lemma 4.6 and [BD08, Pro position 1.3] shows that\nm1(t)≥C\n1+t, for someC >0 (with the notation of [BD08] where m1(t) denotes the best decay rate).\nThen, the main goal of this paper is to explore the gap between the a prioriupper bound1√\ntfor\nthe decay rate, given by Theorem 2.3, and the a priorilower bound1\ntof Theorem 2.5. Our results\nare twofold (somehow in two opposite directions) and concern eithe r the case of smooth damping\nfunctionsb, or the case b=1U, withU⊂T2.\n72.2.1 The case of smooth damping coefficients\nOur main result deals with the case of smooth damping coefficients. Wit hout any geometric assump-\ntion, but with an additional hypothesis on the order of vanishing of t he damping function b, we prove\na weak converse of Theorem 2.5.\nTheorem 2.6. LetM=T2with the standard flat metric. There exists ε0>0satisfying the following\nproperty. Suppose that bis a nonnegative nonvanishing function on T2satisfying√\nb∈C∞(T2)and\nthat there exist ε∈(0,ε0)andCε>0such that\n|∇b(x)| ≤Cεb1−ε(x),forx∈T2. (2.12)\nThen, there exist C >0ands0≥0such that for all s∈R,|s| ≥s0,\n/ba∇dblP(is)−1/ba∇dblL(L2(T2))≤C|s|δ,withδ= 8ε (2.13)\nAs a consequence of Proposition 2.4, in this situation, the d amped wave equation (1.1)is stable at\nrate1\nt1\n1+δ.\nFollowing carefully the steps of the proof, one sees that ε0=1\n76works, but the proof is not\noptimized with respect to this parameter, and it is likely that it could be much improved.\nOne of the main difficulties in understanding the decay rates is that th ere exists no general\nmonotonicity property of the type “ b1(x)≤b2(x) for allx=⇒the decay rate associated to the\ndampingb2is larger (or smaller) than the decay rate associated to the damping b1”. This makes a\nsignificant difference with observability or controllability problems of t he type (1.5).\nAssumption (2.12) is only a local assumption in a neighbourhood of ∂{b>0}(even if it is stated\nhere globally on T2). Far from this set, i.e. on each compact set {b≥b0}forb0>0, the constant\nCεcan be choosen uniformly, depending only on b0, and not on ε. Hence,εsomehow quantifies the\nvanishing rate of the damping function b.\nAn interesting situation is when the smooth function bvanishes like e−1\nxαin smooth local coordi-\nnates, for some α>0. In this case, Assumption (2.12) is satisfied for any ε>0, and the associated\ndamped wave equation (1.1) is stable at rate1\nt1−δfor anyδ >0. This shows that the lower bound\ngiven by Theorem 2.5, as well as the decay rate1\nt, are sharp in general. This phenomenon had\nalready been remarked by Burq and Hitrik in [BH07] in the case where bis invariant in one direction.\nTypical smooth functions not satisfying Assumption (2.12) are for instance functions vanishing\nlike sin(1\nx)2e−1\nx. We do not have any idea of the decay rate achieved in this case (exc ept for the a\nprioribounds1√\ntand1\nt).\nTheorem 2.6 generalises the result of [BH07], which only holds if bis assumed to be invariant\nin one direction. Our proof is based on ideas and tools developped in [Ma c10, AM11] and especially\non two-microlocal semiclassical measures. One of the key technica l points appears in Section 13:\nwe have to construct, for each trapped direction, a cutoff funct ion invariant in that direction and\nadapted to the damping coefficient b. We do not know how to adapt this technical construction\nto tori of higher dimension, d >2; hence we do not know whether Theorem 2.6 holds in higher\ndimension (although we have no reason to suspect it should not hold) . Only in the particular case\nwherebis invariant in d−1 directions can our methods (or those of [BH07]) be applied to prove the\nanalogue of Theorem 2.6.\nNote that if GCC is satisfied, one has (on a general compact manifold M) for someC >1 and\nall|s| ≥s0the estimate\n/ba∇dblP(is)−1/ba∇dblL(L2(M))≤C|s|−1. (2.14)\ninstead of (2.13). Estimate (2.14) is in turn equivalent to uniform sta bilization (see [Hua85] together\nwith Lemma 4.6 below).\n8Remark 2.7. As a consequence of Theorem 2.6 on the torus, we can deduce that the decay rate\nt−1\n1+δalso holds for Equation (1.1) if M= (0,π)2is the square, one takes with Dirichlet or Neumann\nboundary conditions, and the damping function bis smooth, vanishes near ∂Mand satisfies As-\nsumption (2.12). First, we extend the function bas an even (with respect to both variables) smooth\nfunction on the larger square ( −π,π)2, and using the injection ı: (−π,π)2→T2, as a smooth func-\ntion onT2, still satisfying (2.12). Moreover, D(∆D) (resp.D(∆N)) on (0,π)2can be identified as\nthe closed subspace of odd (resp. even) functions of D(∆D) (resp.D(∆N)) on (−π,π)2. Using again\nthe injection ı, it can also be identified with a closed subspace of H2(T2). The estimate\n/ba∇dblu/ba∇dblL2(T2)≤C|s|δ/ba∇dblP(is)u/ba∇dblL2(T2)for allu∈H2(T2),\nis thus also true on the square (0 ,π)2for Dirichlet or Neumann boundary conditions. In particular,\nthis strongly improves the results of [LR05].\nThe lower bound of Theorem 2.5 can be similarly extended to the case o f a square with Dirichlet\nor Neumann boundary conditions, implying that the rate1\ntis optimal if GCC is strongly violated.\n2.2.2 The case of discontinuous damping functions\nAppendix B (by St´ ephane Nonnenmacher) deals with the case wher ebis the characteristic function\nof a vertical strip, i.e. b=/tildewideB1U, for some/tildewideB >0 andU= (a,b)×T⊂T2. Due to the invariance of b\nin one direction, the spectrum of the damped waveoperator Asplits into countably many “branches”\nof eigenvalues. This structure of the spectrum is illustrated in the n umerics of [AL03, AL12].\nThe branch closest to the imaginary axis is explicitly computed, it cont ains a sequence of eigen-\nvalues (zi)i∈Nsuch that Im zi→ ∞and|Rezi| ≤C0\n(Imzi)3/2. This result is in agreement with the\nnumerical tests given in [AL12].\nAs a consequence, for any ε >0 andC >0, the strip/braceleftbig\n|Rez| ≤C|Im(z)|−3/2+ε/bracerightbig\ncontains in-\nfinitely many poles of the resolvent ( zId−A)−1, so item (2.8) in Proposition 2.4 implies the following\nobstruction to the stability of this damped system :\nCorollary 2.8. For anyε >0, the damped wave equation (1.1)onT2with the damping function\n(B.1)cannot be stable at the rate1\nt2/3+ε.\nThe same result holds on the square with Dirichlet or Neumann boundary conditions.\nMore precisely, in this situation, Lemma 4.6 and [BD08, Proposition 1.3] yield thatm1(t)≥\nC\n(1+t)2/3, for someC >0 (with the notation of [BD08] where m1(t) denotes the best decay rate).\nThis corollary shows in particular that the regularity conditions in The orem 2.6 cannot be com-\npletely disposed of if one wants a stability at the rate 1 /t1−εfor smallε.\n2.3 Some related open questions\nThe various results obtained in this article lead to several open ques tions.\n1. In the case where bis the characteristic function of a vertical strip, our analysis show s that the\nbest decay rate lies somewhere between1\nt1\n2and1\nt2\n3, but the “true” decay rate is not yet clear.\n2. It would also be interesting to investigate the spectrum and the d ecay rates for damping\nfunctionsbinvariant in one direction, but having a less singular behaviour than a c haracteristic\nfunction. In particular, is it possible to give a precise link between the vanishing rate of band\nthe decay rate?\n3. In the general setting of Section 2.1 (as well as in the case of the damped wave equation on the\ntorus), is the a prioriupper bound1\nt1\n2for the decay rate optimal?\n4. For smooth damping functions vanishing like e−1\nxα, Theorem 2.6 yields stability at rate1\nt1−δ\nfor allδ>0. Is the decay rate1\ntreached in this situation? Can one find a damping function b\nsuch that the decay rate is exactly1\nt?\n95. The lower bound of of Theorem 2.5 is still valid in higher dimensional to ri. Is there an analogue\nof Theorem 2.6 (i.e. for general “smooth” damping functions) for Td, withd≥3?\nPart II\nResolvent estimates and stabilization in\nthe abstract setting\n3 Proof of Theorem 2.3 assuming Proposition 2.4\nTo prove Theorem 2.3, we express the observability condition as a re solvent estimate (also known as\nthe Hautus test), as introduced by Burq and Zworski [BZ04], and f urther developed by Miller [Mil05]\nand Ramdani, Takahashi, Tenenbaum and Tucsnak [RTTT05]. For a su rvey of this notion, we refer\nto the book [TW09, Section 6.6].\nIn particular [Mil05, Theorem 5.1] (or [TW09, Theorem 6.6.1]) yields that System (2.1) is ob-\nservable in some time T >0 if and only if there exists a constant C >0 such that we have\n/ba∇dblu/ba∇dbl2\nH≤C/parenleftbig\n/ba∇dbl(A−λId)u/ba∇dbl2\nH+/ba∇dblB∗u/ba∇dbl2\nY/parenrightbig\n,for allλ∈Randu∈D(A).\nAs a first consequence, Assumption (2.5) is satisfied and Propositio n 2.4 applies in this context.\nMoreover, we have, for all s∈Randu∈D(A),\n/ba∇dblu/ba∇dbl2\nH≤C/parenleftbig\n/ba∇dbl(A−s2Id+isBB∗−isBB∗)u/ba∇dbl2\nH+/ba∇dblB∗u/ba∇dbl2\nY/parenrightbig\n≤C/parenleftbig\n/ba∇dblP(is)u/ba∇dbl2\nH+s2/ba∇dblBB∗u/ba∇dbl2\nH+/ba∇dblB∗u/ba∇dbl2\nY/parenrightbig\n(3.1)\nSinceB∈ L(Y;H), we obtain for s≥1 and for some C >0,\n/ba∇dblu/ba∇dbl2\nH≤C/parenleftbig\n/ba∇dblP(is)u/ba∇dbl2\nH+s2/ba∇dblB∗u/ba∇dbl2\nY/parenrightbig\n≤C/parenleftbig\ns2/ba∇dblP(is)u/ba∇dbl2\nH+s2/ba∇dblB∗u/ba∇dbl2\nY/parenrightbig\n.\nProposition 2.4 then yields the polynomial stability at rate1√\ntfor (2.3). This concludes the proof of\nTheorem 2.3.\n4 Proof of Proposition 2.4\nOur proof strongly relies on the characterization of polynomially sta ble semigroups, given in [BT10,\nTheorem 2.4], which can be reformulated as follows.\nTheorem 4.1 ([BT10], Theorem 2.4) .Let(et˙A)t≥0be a bounded C0-semigroup on a Hilbert space\n˙H, generated by ˙A. Suppose that iR∩Sp(˙A) =∅. Then, the following conditions are equivalent:\n/ba∇dblet˙A˙A−1/ba∇dblL(˙H)=O(t−α),ast→+∞, (4.1)\n/ba∇dbl(isId−˙A)−1/ba∇dblL(˙H)=O(|s|1\nα),ass→ ∞. (4.2)\nLet us first describe some spectral properties of the operator Adefined in (2.4).\nLemma 4.2. The spectrum of Acontains only isolated eigenvalues and we have\nSp(A)⊂/parenleftbigg/parenleftbig\n−1\n2/ba∇dblB∗/ba∇dbl2\nL(H;Y),0/parenrightbig\n+iR/parenrightbigg\n∪/parenleftBig\n[−/ba∇dblB∗/ba∇dbl2\nL(H;Y),0]+0i/parenrightBig\n,\nwithker(A) = ker(A)×{0}.\n10Moreover, the operator P(z)is an isomorphism from D(A)ontoHif and only if z /∈Sp(A). If\nthis is satisfied, we have\n(zId−A)−1=/parenleftbigg\nP(z)−1(BB∗+zId)P(z)−1\nP(z)−1(zBB∗+z2Id)−IdzP(z)−1/parenrightbigg\n. (4.3)\nThe localization properties for the spectrum of A, stated in the first part of this lemma are\nillustrated for instance in [AL03] or [AL12].\nThis Lemma leads us to introduce the spectral projector of Aon ker(A), given by\nΠ0=1\n2iπ/integraldisplay\nγ(zId−A)−1dz∈ L(H),\nwhereγdenotes a positively oriented circle centered on 0 with a radius so sma ll that 0 is the single\neigenvalue of Ain the interior of γ. We set ˙H= (Id−Π0)Hand equip this space with the norm\n/ba∇dbl(u0,u1)/ba∇dbl2\n˙H:=|(u0,u1)|2\nH=/ba∇dblA1\n2u0/ba∇dbl2\nH+/ba∇dblu1/ba∇dbl2\nH,\nand associated inner product. This is indeed a norm on ˙Hsince/ba∇dbl(u0,u1)/ba∇dbl˙H= 0 is equivalent to\n(u0,u1)∈ker(A)×{0}= Π0H.\nBesides, we set ˙A=A|˙Hwith domain D(˙A) =D(A)∩˙H. A first remark is that Sp( ˙A) =\nSp(A)\\{0}, so that Sp( ˙A)∩iR=∅.\nThe remainder of the proof consists in applying Theorem 4.1 to the op erator˙Ain˙H. We first\ncheck the assumptions of Theorem 4.1 and describe the solutions of the evolution problem (2.4) (or\nequivalently (2.3)).\nLemma 4.3. The operator ˙Agenerates a contraction C0-semigroup on ˙H, denoted (et˙A)t≥0. More-\nover, for all initial data U0∈ H, Problem (2.4)(or equivalently (2.3)) has a unique solution\nU∈C0(R+;H), issued from U0, that can be decomposed as\nU(t) =et˙A(Id−Π0)U0+Π0U0,for allt≥0. (4.4)\nAs a consequence, we can apply Theorem 4.1 to the semigroup gener ated by ˙A. The proof of\nProposition 2.4 will be achieved when the following lemmata are proved.\nLemma 4.4. Conditions (2.6)and(4.1)are equivalent.\nLemma 4.5. Conditions (2.9)and(2.10)are equivalent. Conditions (2.7)and(2.8)are equivalent.\nLemma 4.6. There exist C >1ands0>0such that for s∈R,|s| ≥s0,\n/ba∇dbl(isId−˙A)−1/ba∇dblL(˙H)−C\n|s|≤ /ba∇dbl(isId−A)−1/ba∇dblL(H)≤ /ba∇dbl(isId−˙A)−1/ba∇dblL(˙H)+C\n|s|,(4.5)\nand\nC−1|s|/ba∇dblP(is)−1/ba∇dblL(H)≤ /ba∇dbl(isId−A)−1/ba∇dblL(H)≤C/parenleftbig\n1+|s|/ba∇dblP(is)−1/ba∇dblL(H)/parenrightbig\n. (4.6)\nIn particular this implies that (4.2),(2.7)and(2.9)are equivalent.\nThe proof of Lemma 4.6 is more or less classical and we follow [Leb96, BH 07].\nProof of Lemma 4.2. AsAhas compact resolvent, its spectrum contains only isolated eigenva lues.\nSuppose that z∈Sp(A), then we have, for some ( u0,u1)∈D(A)\\{0},\n/braceleftbiggu1=zu0,\n−Au0−BB∗u1=zu1,\nand in particular\nAu0+z2u0+zBB∗u0= 0, (4.7)\n11withu0∈D(A)\\{0}.\nSuppose that z∈iR, then, this yields Au0−Im(z)2u0+iIm(z)BB∗u0= 0. Following [Leb96],\ntakingtheinnerproductofthisequationwith u0yieldsiIm(z)/ba∇dblB∗u0/ba∇dbl2\nY= 0. Hence, eitherIm( z) = 0,\norB∗u0= 0. In the first case, Au0= 0, i.e.u0∈ker(A), andu1= 0. This yields ker( A)⊂\nker(A)×{0}(and the otherinclusion is clear). In the second case, u0is an eigenvectorof Aassociated\nto the eigenvalue Im( z)2and satisfies B∗u0= 0, which is absurd, according to Assumption (2.5).\nThus, Sp( A)∩iR⊂ {0}.\nNow, for a general eigenvalue z∈C, taking the inner product of (4.7) with u0yields\n/braceleftbigg(Au0,u0)H+(Re(z)2−Im(z)2)/ba∇dblu0/ba∇dbl2\nH+Re(z)/ba∇dblB∗u0/ba∇dbl2\nY= 0,\n2Re(z)Im(z)/ba∇dblu0/ba∇dbl2\nH+Im(z)/ba∇dblB∗u0/ba∇dbl2\nY= 0.(4.8)\nIf Im(z)/\\e}atio\\slash= 0, then, the second equation of (4.8) together with Sp( ˙A)∩iR⊂ {0}gives\n0>Re(z) =−1\n2/ba∇dblB∗u0/ba∇dbl2\nY\n/ba∇dblu0/ba∇dbl2\nH≥ −1\n2/ba∇dblB∗/ba∇dbl2\nL(H;Y).\nIf Im(z) = 0, then, the first equation of (4.8) together with ( ˙Au0,u0)H≥0 gives−Re(z)/ba∇dblB∗u0/ba∇dbl2\nY≥\nRe(z)2/ba∇dblu0/ba∇dbl2\nH, which yields\n0≥Re(z)≥ −/ba∇dblB∗/ba∇dbl2\nL(H;Y).\nFollowing [Leb96], we now give the link between P(z)−1and (zId−A)−1forz /∈Sp(A). Taking\nF= (f0,f1)∈ H, andU= (u0,u1), we have\nF= (zId−A)U⇐⇒/braceleftbiggu1=zu0−f0,\nP(z)u0=f1+(BB��+zId)f0.(4.9)\nAs a consequence, we obtain that P(z) :D(A)→His invertible if and only if ( zId−A) :D(A)→ H\nis invertible, i.e. if and only if z /∈Sp(A). Moreover, for such values of z, System (4.9) is equivalent\nto /braceleftbiggu0=P(z)−1f1+P(z)−1(BB∗+zId)f0,\nu1=zP(z)−1f1+zP(z)−1(BB∗+zId)f0−f0,\nwhich can be rewritten as (4.3). This concludes the proof of Lemma 4 .2.\nProof of Lemma 4.3. Let us check that ˙Ais a maximal dissipative operator on ˙H[Paz83]. First, it\nis dissipative since, for U= (u0,u1)∈D(˙A),\n(˙AU,U)˙H= (A1\n2u1,A1\n2u0)H−(Au0,u1)H−(BB∗u1,u1)H=−/ba∇dblB∗u1/ba∇dbl2\nY≤0.\nNext, the fact that A −Id is onto is a consequence of Lemma 4.2. Hence, for all F∈˙H ⊂ H,\nthere exists U∈D(A) such that ( A −Id)U=F. Applying (Id −Π0) to this identity yields ( ˙A −\nId)(Id−Π0)U=F, sothat ˙A−Id :D(˙A)→˙Hisonto. AccordingtotheLumer-PhillipsTheorem(see\nfor instance [Paz83, Chapter 1, Theorem 4.3]) ˙Agenerates a contraction C0-semigroup on ˙H. Then,\nFormula (4.4) directly comes from the linearity of Equation (2.4) (or e quivalently (2.3)) together\nwith the decomposition of the initial condition U0= (I−Π0)U0+Π0U0.\nProof of Lemma 4.4. Condition (4.1) is equivalent to the existence of C >0 such that for all t>0,\nand˙U0∈˙H, we have\n/ba∇dblet˙A˙A−1˙U0/ba∇dbl˙H≤C\ntα/ba∇dbl˙U0/ba∇dbl˙H.\nThis can be rephrased as\n/ba∇dblet˙A˙U0/ba∇dbl˙H≤C\ntα/ba∇dbl˙A˙U0/ba∇dbl˙H, (4.10)\nfor allt >0, and˙U0∈D(˙A). Now, take any U0= (u0,u1)∈D(A), and associated projection\n˙U0= (Id−Π0)U0∈D(˙A). According to (4.4), we have\nE(u,t) =1\n2/parenleftbig\n/ba∇dblA1\n2u(t)/ba∇dbl2\nH+/ba∇dbl∂tu(t)/ba∇dbl2\nH/parenrightbig\n=1\n2|et˙A˙U0+Π0U0|2\nH=1\n2/ba∇dblet˙A˙U0/ba∇dbl2\n˙H,\n12and\n|AU0|H=|˙A˙U0+AΠ0U0|H=/ba∇dbl˙A˙U0/ba∇dbl˙H.\nThis shows that (4.10) is equivalent to (2.6), and concludes the proo f of Lemma 4.4.\nProof of Lemma 4.5. First, (2.9) clearly implies (2.10). To prove the converse, for u∈D(A), we\nhave\n(P(is)u,u)H=/parenleftbig\n(A−s2Id)u,u/parenrightbig\nH+is/ba∇dblB∗u/ba∇dbl2\nY.\nTaking the imaginarypartof this identity gives s/ba∇dblB∗u/ba∇dbl2\nY= Im(P(is)u,u)H, so that, usingthe Young\ninequality, we obtain for all ε>0,\n|s|1\nα/ba∇dblB∗u/ba∇dbl2\nY=|s|1\nα−1|Im(P(is)u,u)H| ≤|s|2\nα−2\n4ε/ba∇dblP(is)u/ba∇dbl2\nH+ε/ba∇dblu/ba∇dbl2\nH.\nPlugging this into (2.10) and taking εsufficiently small, we obtain that for some C >0 ands0≥0,\nfor anys∈R,|s| ≥s0,\n/ba∇dblu/ba∇dbl2\nH≤C|s|2\nα−2/ba∇dblP(is)u/ba∇dbl2\nH,\nwhich yields (2.9). Hence (2.9) and (2.10) are equivalent.\nSecond, Condition (2.8) clearlyimplies (2.7) and it only remains to prove the converse. For z∈C,\nwe writer= Re(z) ands= Im(z). We have the identity\n((r+is)Id−A)−1= (isId−A)−1/parenleftbig\nId+r(isId−A)−1/parenrightbig−1. (4.11)\nHence, assuming\n/ba∇dblr(isId−A)−1/ba∇dblL(H)≤1\n2, (4.12)\nthis gives\n/vextenddouble/vextenddouble/vextenddouble/parenleftbig\nId+r(isId−A)−1/parenrightbig−1/vextenddouble/vextenddouble/vextenddouble\nL(H)=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble∞/summationdisplay\nk=0(−1)k/parenleftbig\nr(isId−A)−1/parenrightbigk/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL(H)≤2.\nAs a consequence of (4.11) and (2.7), we then obtain\n/vextenddouble/vextenddouble((r+is)Id−A)−1/vextenddouble/vextenddouble\nL(H)≤2/ba∇dbl(isId−A)−1/ba∇dblL(H)≤2C|s|1\nα,\nfor alls≥s0, under Condition (4.12). Finally, (2.7) also yields\n/ba∇dblr(isId−A)−1/ba∇dblL(H)≤ |r|C|s|1\nα,\nso that Condition (4.12) is realised as soon as |r| ≤1\n2C|s|1\nα. This proves (2.8) and concludes the\nproof of Lemma 4.5.\nProof of Lemma 4.6. To prove (4.5), we first remark that the norms /ba∇dbl · /ba∇dbl˙Hand/ba∇dbl · /ba∇dblHare equiv-\nalent on ˙H, so that the norms /ba∇dbl · /ba∇dblL(˙H)and/ba∇dbl · /ba∇dblL(H)are equivalent on L(˙H). Next, we have\n(isId−˙A)−1(Id−Π0) = (isId−A)−1(Id−Π0) and\n/ba∇dbl(isId−˙A)−1/ba∇dblL(H)=/ba∇dbl(isId−˙A)−1(Id−Π0)/ba∇dblL(H)=/ba∇dbl(isId−A)−1(Id−Π0)/ba∇dblL(H)\n≤ /ba∇dbl(isId−A)−1/ba∇dblL(H)+/ba∇dbl(isId−A)−1Π0/ba∇dblL(H),\ntogether with\n/ba∇dbl(isId−A)−1/ba∇dblL(H)=/ba∇dbl(isId−˙A)−1(Id−Π0)+(isId−A)−1Π0/ba∇dblL(H)\n≤ /ba∇dbl(isId−˙A)−1/ba∇dblL(H)+/ba∇dbl(isId−A)−1Π0/ba∇dblL(H).\n13Moreover, for |s| ≥1, we have\n/ba∇dbl(isId−A)−1Π0/ba∇dblL(H)=/ba∇dbl(is)−1Π0/ba∇dblL(H)=1\n|s|/ba∇dblΠ0/ba∇dblL(H)=C\n|s|,\nwhich concludes the proof of (4.5).\nLet us now prove (4.6). For concision, we set H1=D(A1\n2) endowed with the graph norm\n/ba∇dblu/ba∇dblH1=/ba∇dbl(A+ Id)1\n2u/ba∇dblHand denote by H−1=D(A1\n2)′its dual space. The operator Acan be\nuniquely extended as an operator L(H1;H−1), still denoted Afo simplicity. With this notation, the\nspaceH−1can be equipped with the natural norm /ba∇dblu/ba∇dblH−1=/ba∇dbl(A+Id)−1\n2u/ba∇dblH.\nAs a consequenceof Formula(4.3), and using the fact that Sp( A)∩iR⊂ {0}, there exist constants\nC >1 ands0>0 such that for all s∈R,|s| ≥s0,\nC−1M(s)≤ /ba∇dbl(isId−A)−1/ba∇dblL(H)≤CM(s) (4.13)\nwith\nM(s) =/parenleftBig\n/ba∇dblP(is)−1(BB∗+isId)/ba∇dblL(H1)+/ba∇dblP(is)−1/ba∇dblL(H;H1)\n+/ba∇dblP(is)−1(isBB∗−s2Id)−Id/ba∇dblL(H1;H)+/ba∇dblsP(is)−1/ba∇dblL(H)/parenrightBig\n(4.14)\nOn the one hand, this direcly yields for s∈R,|s| ≥s0,\n|s|/ba∇dblP(is)−1/ba∇dblL(H)≤C/ba∇dbl(isId−A)−1/ba∇dblL(H).\nThis proves that (4.2) implies (2.9).\nOn the other hand, we have to estimate each term of (4.14). First, usingAu=P(is)u+s2u−\nisBB∗u, we have\n/ba∇dblu/ba∇dbl2\nH1=/ba∇dblA1\n2u/ba∇dbl2\nH+/ba∇dblu/ba∇dbl2\nH=/parenleftbig\nP(is)u+s2u−isBB∗u,u/parenrightbig\nH+/ba∇dblu/ba∇dbl2\nH\n= Re/parenleftbig\nP(is)u,u/parenrightbig\nH+(s2+1)/ba∇dblu/ba∇dbl2\nH≤C/parenleftbig\n/ba∇dblP(is)u/ba∇dbl2\nH+(s2+1)/ba∇dblu/ba∇dbl2\nH/parenrightbig\n≤C/parenleftBig\n1+(s2+1)/ba∇dblP(is)−1/ba∇dbl2\nL(H)/parenrightBig\n/ba∇dblP(is)u/ba∇dbl2\nH,\nso that\n/ba∇dblP(is)−1/ba∇dblL(H;H1)≤C/parenleftbig\n1+(|s|+1)/ba∇dblP(is)−1/ba∇dblL(H)/parenrightbig\n. (4.15)\nSecond, the same computation for ( P(is)−1)∗= (A−s2Id−isBB∗)−1(the adjoint of P(is)−1in\nthe spaceH) in place of P(is)−1leads to (P(is)−1)∗∈ L(H;H1), together with the estimate\n/ba∇dbl(P(is)−1)∗/ba∇dblL(H;H1)≤C/parenleftbig\n1+(|s|+1)/ba∇dblP(is)−1/ba∇dblL(H)/parenrightbig\n.\nBy transposition, we havet(P(is)−1)∗∈ L(H−1;H), together with the estimate\n/ba∇dblt(P(is)−1)∗/ba∇dblL(H−1;H)≤ /ba∇dbl(P(is)−1)∗/ba∇dblL(H;H1)≤C/parenleftbig\n1+(|s|+1)/ba∇dblP(is)−1/ba∇dblL(H)/parenrightbig\n.(4.16)\nMoreover,t(P(is)−1)∗is defined, for every u∈H,v∈H−1, by\n/parenleftbigt(P(is)−1)∗v,u/parenrightbig\nH=/angbracketleftbig\nv,(P(is)−1)∗u/angbracketrightbig\nH−1,H1=/parenleftBig\n(A+Id)−1\n2v,(A+Id)1\n2(P(is)−1)∗u/parenrightBig\nH.\nIn particular, taking v∈Hgives\n/parenleftbigt(P(is)−1)∗v,u/parenrightbig\nH=/parenleftbig\nP(is)−1v,u/parenrightbig\nH,\nwhich implies that the restriction of the operatort(P(is)−1)∗toHcoincides with P(is)−1. For\nsimplicity, we will denote P(is)−1fort(P(is)−1)∗.\n14Equation (4.16) can thus be rewritten\n/ba∇dblP(is)−1/ba∇dblL(H−1;H)≤C/parenleftbig\n1+(|s|+1)/ba∇dblP(is)−1/ba∇dblL(H)/parenrightbig\n. (4.17)\nThen, we have P(is)−1(isBB∗−s2Id)−Id =P(is)−1A, so that\n/ba∇dblP(is)−1(isBB∗−s2Id)−Id/ba∇dblL(H1;H)=/ba∇dblP(is)−1A/ba∇dblL(H1;H)≤ /ba∇dblP(is)−1/ba∇dblL(H−1;H)/ba∇dblA/ba∇dblL(H1;H−1)\n≤/parenleftbig\n1+(|s|+1)/ba∇dblP(is)−1/ba∇dblL(H)/parenrightbig\n(4.18)\nThird, for |s| ≥1 we write\nP(is)−1(BB∗+isId) =i\ns/parenleftbig\nP(is)−1A−Id/parenrightbig\n, (4.19)\nand it remains to estimate the term /ba∇dblP(is)−1A/ba∇dblL(H1)in (4.14). For f∈H1, we setu=P(is)−1Af.\nWe haveu∈H1, together with\n(A−s2Id+isBB∗)u=Af.\nTaking the real part of the inner product of this identity with u, we find\n/ba∇dblA1\n2u/ba∇dbl2\nH−s2/ba∇dblu/ba∇dbl2\nH= Re(Af,u)H≤ /ba∇dblAf/ba∇dblH−1/ba∇dblu/ba∇dblH1≤C/ba∇dblf/ba∇dblH1/ba∇dblu/ba∇dblH1,\nasA∈ L(H1,H−1). Hence\n/ba∇dblu/ba∇dbl2\nH1≤C(1+s2)/ba∇dblu/ba∇dbl2\nH+C/ba∇dblf/ba∇dbl2\nH1\nUsing (4.17), this gives\n/ba∇dblu/ba∇dbl2\nH1≤C(1+s2)/ba∇dblP(is)−1A/ba∇dbl2\nL(H1;H)/ba∇dblf/ba∇dbl2\nH1+C/ba∇dblf/ba∇dbl2\nH1\n≤C(1+s2)/ba∇dblP(is)−1/ba∇dbl2\nL(H−1;H)/ba∇dblf/ba∇dbl2\nH1+C/ba∇dblf/ba∇dbl2\nH1\n≤C(1+s2)/parenleftbig\n1+(|s|+1)/ba∇dblP(is)−1/ba∇dblL(H)/parenrightbig2/ba∇dblf/ba∇dbl2\nH1,\nand finally /ba∇dblP(is)−1A/ba∇dblL(H1)≤C(1 +|s|)/parenleftbig\n1+(|s|+1)/ba∇dblP(is)−1/ba∇dblL(H)/parenrightbig\n. Coming back to (4.19), we\nhave, for |s| ≥1,\n/ba∇dblP(is)−1(BB∗+isId)/ba∇dblL(H1)≤C/parenleftbig\n1+|s|/ba∇dblP(is)−1/ba∇dblL(H)/parenrightbig\n. (4.20)\nFinally, combining (4.15), (4.18) and (4.20), together with (4.13)-(4 .14), we obtain for |s| ≥1,\n/ba∇dbl(isId−A)−1/ba∇dblL(H)≤C/parenleftbig\n1+|s|/ba∇dblP(is)−1/ba∇dblL(H)/parenrightbig\n.\nThis concludes the proof of Lemma 4.6.\nPart III\nProof of Theorem 2.6: smooth damping\ncoefficients on the torus\nTo prove Theorem 2.6, we shall instead prove Estimate (2.9) with α=1\n1+δ(which, according to\nProposition2.4, is equivalent to the statement ofTheorem 2.6). Let us first recast (2.9) with α=1\n1+δ\nin the semiclassical setting : taking h=s−1, we are left to prove that there exist C >1 andh0>0\nsuch that for all h≤h0, for allu∈H2(T2), we have\n/ba∇dblu/ba∇dblL2(T2)≤Ch−δ/ba∇dblP(i/h)u/ba∇dblL2(T2) (4.21)\n15We prove this inequality by contradiction, using the notion of semiclas sical measures. The idea\nof developing such a strategy for proving energy estimates, toge ther with the associate technology,\noriginates from Lebeau [Leb96].\nWe assume that (4.21) is not satisfied, and will obtain a contradiction at the end of Section 11.\nHence, for all n∈N, there exists 0 <hn≤1\nnandun∈H2(T2) such that\n/ba∇dblun/ba∇dblL2(T2)>n\nhδn/ba∇dblP(i/hn)un/ba∇dblL2(T2).\nSettingvn=un//ba∇dblun/ba∇dblL2(T2), and\nPhn\nb=−h2\nn∆−1+ihnb(x) =h2\nnP(i/hn),\nwe then have, as n→ ∞,\n\n\nhn→0+,\n/ba∇dblvn/ba∇dblL2(T2)= 1,\nh−2−δ\nn/ba∇dblPhn\nbvn/ba∇dblL2(T2)→0.\nOur goal is now to associate to the sequence ( un,hn) a semiclassicalmeasure on the cotangent bundle\nµonT∗T2=T2×(R2)∗(where (R2)∗is the dual space of R2). To obtain a contradiction, we shall\nprove both that µ(T∗T2) = 1, and that µ= 0 onT∗T2.\nFrom now on, we drop the subscript nof the sequences above, and write hin place ofhnandvh\nin place ofvn. We study sequences ( h,vh) such that h→0+and\n/braceleftBigg\n/ba∇dblvh/ba∇dblL2(T2)= 1\n/ba∇dblPh\nbvh/ba∇dblL2(T2)=o(h2+δ),ash→0+.(4.22)\nIn particular, this last equation also yields the key information\n(bvh,vh)L2(T2)=h−1Im(Ph\nbvh,vh)L2(T2)=o(h1+δ),ash→0+.\nIn the following, it will be convenient to identify ( R2)∗andR2through the usual inner product.\nIn particular, the cotangent bundle T∗T2=T2×(R2)∗will be identified with T2×R2.\n5 Semiclassical measures\nWe denote by T∗T2the compactification of T∗T2obtained by adding a point at infinity to each fiber\n(i.e., the set T2×(R2∪ {∞})). A neighbourhood of ( x,∞)∈T∗T2is a setU×/parenleftbig\n{∞}∪R2\\K/parenrightbig\n,\nwhereUis a neighbourhood of xinT2andKa compact set in R2. Endowed with this topology, the\nsetT∗T2is compact.\nWe denote by S0(T∗T2),S0for short, the space of functions a(x,ξ) that satisfy the following\nproperties:\n1.a∈C∞(T∗T2).\n2. There exists a compact set K⊂R2and a constant k0∈Csuch thata(x,ξ) =k0for all\nξ∈R2\\K.\nNote that we have in particular C∞\nc(T∗T2)⊂S0(T∗T2).\nTo a symbol a∈S0(T∗T2), we associate its semiclassical Weyl quantization Oph(a) by For-\nmula (A.1), which, according to the Calder´ on-Vaillancourt Theorem (see Appendix A) defines a\nuniformly bounded operator on L2(T2).\nFrom the sequence ( vh,h) (see for instance [GL93]), we can define (using again the Calder´ on -\nVaillancourt Theorem) the associated Wigner distribution Vh∈(S0)′by\n/angbracketleftbig\nVh,a/angbracketrightbig\n(S0)′,S0= (Oph(a)vh,vh)L2(T2),for alla∈S0(T∗T2). (5.1)\n16Decomposing vhandain Fourier series,\nˆvh(k) =1\n2π/integraldisplay\nT2e−ik·xvh(x)dx,ˆa(h,k,ξ) =1\n2π/integraldisplay\nT2e−ik·xa(h,x,ξ)dx,\nthe expression (5.1) can be more explicitly rewritten as\n/angbracketleftbig\nVh,a/angbracketrightbig\n(S0)′,S0=1\n2π/summationdisplay\nk,j∈Z2ˆa/parenleftbigg\nh,j−k,h\n2(k+j)/parenrightbigg\nˆvh(k)ˆvh(j).\nProposition 5.1. The family (Vh)is bounded in (S0)′. Hence, there exists a subsequence of the\nsequence (h,vh)and an element µ∈(S0)′, such that Vh⇀µweakly in (S0)′, i.e.\n(Oph(a)vh,vh)L2(T2)→ /a\\}b∇acketle{tµ,a/a\\}b∇acket∇i}ht(S0)′,S0for alla∈S0(T∗T2). (5.2)\nIn addition, /a\\}b∇acketle{tµ,a/a\\}b∇acket∇i}ht(S0)′,S0is nonnegative if ais; in other words, µmay be identified with a nonnegative\nRadon measure on T∗T2.\nNotation: in what follows we shall denote by M+(T∗T2) the set of nonnegativeRadon measures\nonT∗T2.\nProof.The proof is an adaptation from the original proof of G´ erard [G´ er 91] (see also [GL93] in the\nsemiclassical setting).\nThe fact that the Wigner distributions Vhare uniformly bounded in ( S0)′follows from the\nCalder´ on-Vaillancourt theorem (see Appendix A), and from the bo undedness of ( vh) inL2(T2).\nThe sharp G˚ arding inequality gives the existence of C >0 such that, for all a≥0 andh>0,\n(Oph(a)vh,vh)L2(T2)≥ −Ch/ba∇dblvh/ba∇dbl2\nL2(T2),\nso that the distribution µis nonnegative (and is hence a measure).\n6 Zero-th and first order informations on µ\nTo simplify the notation, we set\nPh\nb=Ph\n0+ihb(x),withPh\n0=−h2∆−1 = Oph(|ξ|2−1).\nThe geodesic flow on the torus φτ:T∗T2→T∗T2forτ∈Ris the flow generated by the\nHamiltonian vector field associated to the symbol1\n2(|ξ|2−1), i.e. by the vector field ξ·∂xonT∗T2.\nExplicitely, we have\nφτ(x,ξ) = (x+τξ,ξ), τ∈R,(x,ξ)∈T∗T2.\nNotethatφτpreservesthe ξ-component,and,inparticulareveryenergylayer {|ξ|2=C >0} ⊂T∗T2.\nNow, we describe the first properties of the measure µimplied by (4.22).\nWe recall that for ν∈D′(T∗T2), (φτ)∗ν∈D′(T∗T2) is defined by /a\\}b∇acketle{t(φτ)∗ν,a/a\\}b∇acket∇i}ht=/a\\}b∇acketle{tν,a◦φτ/a\\}b∇acket∇i}htfor\nalla∈C∞\nc(T∗T2). In particular, ( φτ)∗νis a measure if νis. We shall say that νis aninvariant\nmeasure if it is invariant by the geodesic flow, i.e. ( φτ)∗ν=νfor allτ∈R.\nProposition 6.1. Letµbe as in Proposition 5.1. We have\n1.supp(µ)⊂ {|ξ|2= 1}(hence is compact in T∗T2),\n2.µ(T∗T2) = 1,\n3.µis invariant by the geodesic flow, i.e. (φτ)∗µ=µ,\n4./a\\}b∇acketle{tµ,b/a\\}b∇acket∇i}htMc(T∗T2),C0(T∗T2)= 0, whereMc(T∗T2)denotes the space of compactly supported mea-\nsures onT∗T2.\n17In other words, µis an invariant probability measure on T∗T2vanishing on {b>0}.\nThese are standard arguments, that we reproduce here for the reader’s comfort. In particular,\nwe recover all informations required to prove the Bardos-Lebeau -Rauch-Tayloruniform stabilization\ntheorem under GCC. But we do not use here the second order infor mations of (4.22); this will be\nthe key point to prove Theorem 2.6.\nProof.First, we take χ∈C∞(T∗T2) depending only on the ξvariable, such that χ≥0,χ(ξ) = 0\nfor|ξ| ≤2, andχ(ξ) = 1 for |ξ| ≥3. Hence,χ(ξ)\n|ξ|2−1∈C∞(T∗T2) and we have the exact composition\nformula\nOph(χ) = Oph/parenleftbiggχ(ξ)\n|ξ|2−1/parenrightbigg\nPh\n0,\nsince both operators are Fourier multipliers. Moreover, Oph/parenleftBig\nχ(ξ)\n|ξ|2−1/parenrightBig\nis a bounded operator on\nL2(T2). As a consequence, we have\n/angbracketleftbig\nVh,χ/angbracketrightbig\n(S0)′,S0→ /a\\}b∇acketle{tµ,χ/a\\}b∇acket∇i}htM(T∗T2),C0(T∗T2),\ntogether with\n/angbracketleftbig\nVh,χ/angbracketrightbig\n(S0)′,S0=/parenleftbigg\nOph/parenleftbiggχ(ξ)\n|ξ|2−1/parenrightbigg\nPh\n0vh,vh/parenrightbigg\nL2(T2)\n=/parenleftbigg\nOph/parenleftbiggχ(ξ)\n|ξ|2−1/parenrightbigg\nPh\nbvh,vh/parenrightbigg\nL2(T2)−ih/parenleftbigg\nOph/parenleftbiggχ(ξ)\n|ξ|2−1/parenrightbigg\nbvh,vh/parenrightbigg\nL2(T2).\nSince/ba∇dblPh\nbvh/ba∇dblL2(T2)=o(1) and/ba∇dblvh/ba∇dblL2(T2)= 1, both terms in this expression vanish in the limit\nh→0+. This implies that /a\\}b∇acketle{tµ,χ/a\\}b∇acket∇i}htM(T∗T2),C0(T∗T2)= 0. Since this holds for all χas above, we have\nsupp(µ)⊂ {|ξ|2= 1}, which proves Item 1.\nIn particular, this implies that µ/parenleftBig\nT∗T2\\T∗T2/parenrightBig\n= 0. Now, Item 2 is a direct consequence of 1 =\n/ba∇dblvh/ba∇dbl2\nL2(T2)→ /a\\}b∇acketle{tµ,1/a\\}b∇acket∇i}htM(T∗T2),C0(T∗T2)and Item 1. Item 4 is a direct consequence of ( bvh,vh)L2(T2)=\no(1).\nFinally, for a∈C∞\nc(T∗T2), we recall that\n/bracketleftbig\nPh\n0,Oph(a)/bracketrightbig\n=h\niOph({|ξ|2−1,a}) =2h\niOph(ξ·∂xa),\nis a consequence of the Weyl quantization (any other quantization would have left an error term of\norderO(h2)). Hence, (5.1) yields\n/angbracketleftbig\nVh,ξ·∂xa/angbracketrightbig\nD′(T∗T2),C∞c(T∗T2)→ /a\\}b∇acketle{tµ,ξ·∂xa/a\\}b∇acket∇i}htM(T∗T2),C0c(T∗T2), (6.1)\ntogether with\n/angbracketleftbig\nVh,ξ·∂xa/angbracketrightbig\nD′(T∗T2),C∞c(T∗T2)=i\n2h/parenleftbig/bracketleftbig\nPh\n0,Oph(a)/bracketrightbig\nvh,vh/parenrightbig\nL2(T2)\n=i\n2h/parenleftbig\nOph(a)vh,Ph\n0vh/parenrightbig\nL2(T2)−i\n2h/parenleftbig\nOph(a)Ph\n0vh,vh/parenrightbig\nL2(T2)\n=i\n2h/parenleftbig\nOph(a)vh,Ph\nbvh/parenrightbig\nL2(T2)−i\n2h/parenleftbig\nOph(a)Ph\nbvh,vh/parenrightbig\nL2(T2)\n−1\n2(Oph(a)vh,bvh)L2(T2)−1\n2(Oph(a)bvh,vh)L2(T2).(6.2)\nIn this expression, we have1\nh/parenleftbig\nOph(a)vh,Ph\nbvh/parenrightbig\nL2(T2)→0 and1\nh/parenleftbig\nOph(a)Ph\nbvh,vh/parenrightbig\nL2(T2)→0 since\n/ba∇dblPh\nbvh/ba∇dblL2(T2)=o(h). Moreover, the last two terms can be estimated by\n|(Oph(a)bvh,vh)L2(T2)| ≤ /ba∇dbl√\nbvh/ba∇dblL2(T2)/ba∇dbl√\nbOph(a)vh/ba∇dblL2(T2)=o(1), (6.3)\n18since (bvh,vh)L2(T2)=o(1). This yields/angbracketleftbig\nVh,ξ·∂xa/angbracketrightbig\nD′(T∗T2),C∞c(T∗T2)→0, so that, using (6.1),\n/a\\}b∇acketle{tµ,ξ·∂xa/a\\}b∇acket∇i}htM(T∗T2),C0\nc(T∗T2)= 0 for all a∈C∞\nc(T∗T2). Replacing abya◦φτand integrating with\nrespect to the parameter τgives (φτ)∗µ=µ, which concludes the proof of Item 3.\n7 Geometry on thetorus and decomposition of invariant mea-\nsures\n7.1 Resonant and non-resonant vectors on the torus\nIn this section, we collect several facts concerning the geometry ofT∗T2and its resonant subspaces.\nMost of the setting and the notation comes from [AM11, Section 2].\nWe shall say that a submodule Λ ⊂Z2is primitive if /a\\}b∇acketle{tΛ/a\\}b∇acket∇i}ht∩Z2= Λ, where /a\\}b∇acketle{tΛ/a\\}b∇acket∇i}htdenotes the linear\nsubspace of R2spanned by Λ. The family of all primitive submodules will be denoted by P.\nLet us denote by Ω j⊂R2, forj= 0,1,2, the set of resonant vectors of order exactly j, i.e.,\nΩj:={ξ∈R2such that rk(Λ ξ) = 2−j},with Λ ξ:=/braceleftbig\nk∈Z2such thatξ·k= 0/bracerightbig\n=ξ⊥∩Z2.\nNote that the sets Ω jform a partition of R2, and that we have\n•Ω0={0};\n•ξ∈Ω1if and only if the geodesic issued from any x∈T2in the direction ξis periodic;\n•ξ∈Ω2if and only if the geodesic issued from any x∈T2in the direction ξis dense in T2.\nFor each Λ ∈ Psuch that rk(Λ) = 1, we define\nΛ⊥:=/braceleftbig\nξ∈R2such thatξ��k= 0 for allk∈Λ/bracerightbig\n,\nTΛ:=/a\\}b∇acketle{tΛ/a\\}b∇acket∇i}ht/2πΛ,\nTΛ⊥:= Λ⊥/(2πZ2∩Λ⊥).\nNote that TΛandTΛ⊥are two submanifolds of T2diffeomorphic to one-dimensional tori. Their\ncotangent bundles admit the global trivialisations T∗TΛ=TΛ×/a\\}b∇acketle{tΛ/a\\}b∇acket∇i}htandT∗TΛ⊥=TΛ⊥×Λ⊥.\nFor a function fonT2with Fourier coefficients ( ˆf(k))k∈Z2, and Λ∈ P, we shall say that fhas\nonly Fourier modes in Λ if ˆf(k) = 0 fork /∈Λ. This means that fis constant in the direction Λ⊥,\nor, equivalently, that σ·∂xf= 0 for all σ∈Λ⊥. We denote by Lp\nΛ(T2) the subspace of Lp(T2)\nconsisting of functions having only Fourier modes in Λ. For a function f∈L2(T2) (resp. a symbol\na∈S0(T∗T2)), we denote by /a\\}b∇acketle{tf/a\\}b∇acket∇i}htΛits orthogonal projection on L2\nΛ(T2), i.e. the average of falong\nΛ⊥:\n/a\\}b∇acketle{tf/a\\}b∇acket∇i}htΛ(x) :=/summationdisplay\nk∈Λeik·x\n2πˆf(k)/parenleftBigg\nresp./a\\}b∇acketle{ta/a\\}b∇acket∇i}htΛ(x,ξ) :=/summationdisplay\nk∈Λeik·x\n2πˆa(k,ξ)/parenrightBigg\n.\nIf rk(Λ) = 1 and vis a vector in Λ⊥\\{0}, we also have\n/a\\}b∇acketle{tf/a\\}b∇acket∇i}htΛ(x) = lim\nT→∞1\nT/integraldisplayT\n0f(x+tv)dt. (7.1)\nIn particular, note that /a\\}b∇acketle{tf/a\\}b∇acket∇i}htΛ(resp./a\\}b∇acketle{ta/a\\}b∇acket∇i}htΛ) is nonnegative if f(resp.a) is, and that /a\\}b∇acketle{tf/a\\}b∇acket∇i}htΛ∈C∞(T2)\n(resp./a\\}b∇acketle{ta/a\\}b∇acket∇i}htΛ∈S0(T∗T2)) iff∈C∞(T2) (resp.a∈S0(T∗T2)).\nFinally, given f∈L∞\nΛ(T2), we denote by mfthe bounded operator on L2\nΛ(T2), consisting in the\nmultiplication by f.\n197.2 Decomposition of invariant measures\nWe denote by M+(T∗T2) the set of finite, nonnegative measures on T∗T2. With the definitions\nabove, we have the following decomposition Lemmata, proved in [Mac1 0] or [AM11, Section 2].\nThese properties are given for general measures µ∈ M+(T∗T2). Of course, they apply in particular\nto the measure µdefined by Proposition 5.1.\nLemma 7.1. Letµ∈ M+(T∗T2). Thenµdecomposes as a sum of nonnegative measures\nµ=µ|T2×{0}+µ|T2×Ω2+/summationdisplay\nΛ∈P,rk(Λ)=1µ|T2×(Λ⊥\\{0}) (7.2)\nGivenµ∈ M+(T∗T2), we define its Fourier coefficients by the complex measures on R2:\nˆµ(k,·) :=/integraldisplay\nT2e−ik·x\n2πµ(dx,·), k∈Z.\nOne has, in the sense of distributions, the following Fourier inversion formula:\nµ(x,ξ) =/summationdisplay\nk∈Z2eik·x\n2πˆµ(k,ξ).\nLemma 7.2. Letµ∈ M+(T∗T2)andΛ∈ P. Then, the distribution\n/a\\}b∇acketle{tµ/a\\}b∇acket∇i}htΛ(x,ξ) :=/summationdisplay\nk∈Λeik·x\n2πˆµ(k,ξ),\nis inM+(T∗T2)and satisfies, for all a∈C∞\nc(T∗T2),\n/a\\}b∇acketle{t/a\\}b∇acketle{tµ/a\\}b∇acket∇i}htΛ,a/a\\}b∇acket∇i}htM(T∗T2),C0c(T∗T2)=/a\\}b∇acketle{tµ,/a\\}b∇acketle{ta/a\\}b∇acket∇i}htΛ/a\\}b∇acket∇i}htM(T∗T2),C0c(T∗T2).\nLemma 7.3. Letµ∈ M+(T∗T2)be aninvariant measure. Then, for all Λ∈ P,µ|T2×(Λ⊥\\{0})is\nalso a nonnegative invariant measure and\nµ|T2×(Λ⊥\\{0})=/a\\}b∇acketle{tµ/a\\}b∇acket∇i}htΛ|T2×(Λ⊥\\{0}).\nLet us now come back to the measure µgiven by Proposition 5.1, which satisfies all properties\nlisted in Proposition 6.1. In particular, this measure vanishes on the n on-empty open subset of T2\ngiven by {b >0}(see Item 4 in Proposition 6.1). As a consequence of Proposition 6.1, and of the\nthree lemmata above, this yields the following lemma.\nLemma 7.4. We haveµ=/summationtext\nΛ∈P,rk(Λ)=1µ|T2×(Λ⊥\\{0}).\nAs a consequence of Proposition 6.1, we have indeed that the measu reµis supported in {|ξ|= 1},\nwhich implies µ|T2×{0}= 0. In addition, Lemma 7.3 applied with Λ = {0}implies that µ|T2×Ω2is\nconstant in x– and thus vanishes everywhere since it vanishes on {b>0}.\nRemark 7.5. Since the measure µis supported in {|ξ|= 1}(Proposition 6.1, Item 1), we have\nµ|T2×Λ⊥=µ|T2×(Λ⊥\\{0})\n(which simplifies the notation).\nAs a consequence of these lemmata and the last remark, the study of the measure µis now\nreduced to that of all nonnegative invariant measures µ|T2×Λ⊥with rk(Λ) = 1.\nThe aim of the next sections is to prove that the measure µ|T2×Λ⊥vanishes identically, for each\nperiodic direction Λ⊥.\n207.3 Geometry of the subtori TΛandTΛ⊥\nTo study the measure µ|T2×(Λ⊥\\{0}), we need to describe briefly the geometry of the subtori TΛand\nTΛ⊥ofT2, and introduce adapted coordinates.\nWe defineχΛthe linear isomorphism\nχΛ: Λ⊥×/a\\}b∇acketle{tΛ/a\\}b∇acket∇i}ht →R2: (s,y)/ma√sto→s+y,\nand denote by ˜ χΛ:T∗Λ⊥×T∗/a\\}b∇acketle{tΛ/a\\}b∇acket∇i}ht →T∗R2its extension to the cotangent bundle. This application\ncan be defined as follows: for ( s,σ)∈T∗Λ⊥= Λ⊥×(Λ⊥)∗and (y,η)∈T∗/a\\}b∇acketle{tΛ/a\\}b∇acket∇i}ht=/a\\}b∇acketle{tΛ/a\\}b∇acket∇i}ht×/a\\}b∇acketle{tΛ/a\\}b∇acket∇i}ht∗, we can\nextendσto a covector of R2vanishing on /a\\}b∇acketle{tΛ/a\\}b∇acket∇i}htandηto a covector of R2vanishing on Λ⊥. Remember\nthat we identify ( R2)∗withR2through the usual inner product; thus we can also see σas an element\nof Λ⊥andηas an element of /a\\}b∇acketle{tΛ/a\\}b∇acket∇i}ht. Then, we have\n˜χΛ(s,σ,y,η) = (s+y,σ+η)∈T∗R2=R2×(R2)∗.\nConversely, any ξ∈(R2)∗can be decomposed into ξ=σ+ηwhereσ∈Λ⊥andη∈ /a\\}b∇acketle{tΛ/a\\}b∇acket∇i}ht. We\ndenote byPΛthe orthogonal projection of R2onto/a\\}b∇acketle{tΛ/a\\}b∇acket∇i}ht, i.e.PΛξ=η.\nNext, the map χΛgoes to the quotient, giving a smooth Riemannian covering of T2by\nπΛ:TΛ⊥×TΛ→T2: (s,y)/ma√sto→s+y.\nWe shall denote by ˜ πΛits extension to cotangent bundles:\n˜πΛ:T∗TΛ⊥×T∗TΛ→T∗T2.\nAs the map πΛis not an injection (because the torus TΛ⊥×TΛcontains several copies of T2), we\nintroduce its degree pΛ, which is also equal toVol(TΛ⊥×TΛ)\nVol(T2).\nThen, the application\nTΛu:=1√pΛu◦χΛ,\ndefines a linear isomorphism L2\nloc(R2)→L2\nloc(Λ⊥× /a\\}b∇acketle{tΛ/a\\}b∇acket∇i}ht). Note that because of the factor1√pΛ,\nTΛmapsL2(T2) isometrically into a subspace of L2(TΛ⊥×TΛ). Moreover, TΛmapsL2\nΛ(T2) into\nL2(TΛ)⊂L2(TΛ⊥×TΛ), since the nonvanishing Fourier modes of u∈L2\nΛ(T2) correspond only to\nfrequencies k∈Λ. This reads\nTΛu(s,y) =1√pΛu(y) for (s,y)∈TΛ⊥×TΛ. (7.3)\nSince ˜χΛis linear, we have, for any a∈C∞(T∗R2)\nTΛOph(a) = Oph(a◦˜χΛ)TΛ, (7.4)\nwhere on the left Ophis the Weyl quantization on R2(A.1), and on the right Ophis the Weyl\nquantization on Λ⊥× /a\\}b∇acketle{tΛ/a\\}b∇acket∇i}ht. Next, we denote by OpΛ⊥\nhand OpΛ\nhthe Weyl quantization operators\ndefined on smooth test functions on T∗Λ⊥×T∗/a\\}b∇acketle{tΛ/a\\}b∇acket∇i}htand acting only on the variables in T∗Λ⊥and\nT∗/a\\}b∇acketle{tΛ/a\\}b∇acket∇i}htrespectively, leaving the other frozen. For any a∈C∞\nc(T∗Λ⊥×T∗/a\\}b∇acketle{tΛ/a\\}b∇acket∇i}ht), we have :\nOph(a) = OpΛ⊥\nh◦OpΛ\nh(a) = OpΛ\nh◦OpΛ⊥\nh(a). (7.5)\nNow, if the symbol a∈C∞\nc(T∗T2) has only Fourier modes in Λ, we remark, in view of (7.3), that\na◦˜πΛdoes not depend on s∈TΛ⊥. Therefore, we sometimes write a◦˜πΛ(σ,y,η) fora◦˜πΛ(s,σ,y,η)\nand (7.4)-(7.5) give\nTΛOph(a) = OpΛ\nh◦OpΛ⊥\nh(a◦˜πΛ)TΛ= OpΛ\nh(a◦˜πΛ(hDs,·,·))TΛ. (7.6)\nNote that for every σ∈Λ⊥, the operator OpΛ\nh(a◦˜πΛ(σ,·,·)) mapsL2(TΛ) into itself. More precisely,\nit maps the subspace TΛ(L2\nΛ(T2)) into itself.\n218 Change of quasimode and construction of an invariant cut-\noff function\nIn this section, we first construct from the quasimode vhanother quasimode wh, that will be easier\nto handle when studying the measure µ|T2×Λ⊥. Indeedwhis basically a microlocalization of vhin\nthe direction Λ⊥at a precise concentration rate.\nMoreover, we introduce a cutoff function χΛ\nh(x) =χΛ\nh(y,s), well-adapted to the damping coeffi-\ncientband to the invariance of the measure µ|T2×Λ⊥in the direction Λ⊥(this cutoff function plays\nthe role of the function χ(b/h) used in [BH07] in the case where bis itself invariant in the direction\nΛ⊥). Its construction is a key point in the proof of Theorem 2.6.\nLetχ∈C∞\nc(R) be a nonnegative function such that χ= 1 in a neighbourhood of the origin. We\nfirst define\nwh:= Oph/parenleftbigg\nχ/parenleftbigg|PΛξ|\nhα/parenrightbigg/parenrightbigg\nvh,\nwhich implicitely depends on α∈(0,1). The following lemma implies that, for δandαsufficiently\nsmall,whis as well a o(h2+δ)-quasimode for Ph\nb.\nLemma 8.1. For anyα>0such that\nδ+ε\n2+α≤1\n2,3α+2δ<1, (8.1)\nwe have\n/ba∇dblPh\nbwh/ba∇dblL2(T2)=o(h2+δ).\nAs a consequence of this lemma, the semiclassical measures associa ted towhsatisfy in particular\nthe conclusions of Proposition 6.1. Moreover, the following proposit ion implies that the sequence wh\ncontains all the information in the direction Λ⊥.\nProposition 8.2. For anya∈C∞\nc(T∗T2)and anyα∈(0,3/4)satisfying the assumptions of\nLemma 8.1, we have\n/a\\}b∇acketle{tµ|T2×Λ⊥,a/a\\}b∇acket∇i}htM(T∗T2),C0c(T∗T2)= lim\nh→0(Oph(a)wh,wh)L2(T2).\nNext, we state the desired properties of the cutoff function χΛ\nh. The proof of its existence is a\ncrucial point in the proof of Theorem 2.6.\nProposition 8.3. Forδ= 8ε, andε<1\n76, there exists αsatisfying (8.1), such that for any constant\nc0>0, there exists a cutoff function χΛ\nh∈C∞(T2)valued in [0,1], such that\n1.χΛ\nh=χΛ\nh(y)does not depend on the variable s(i.e.χΛ\nhisΛ⊥-invariant),\n2./ba∇dbl(1−χΛ\nh)wh/ba∇dblL2(T2)=o(1),\n3.b≤c0honsupp(χΛ\nh),\n4./ba∇dbl∂yχΛ\nhwh/ba∇dblL2(T2)=o(1),\n5./ba∇dbl∂2\nyχΛ\nhwh/ba∇dblL2(T2)=o(1).\nNote that the function χΛ\nhimplicitely depends on the constant c0, that will be taken arbitrarily\nsmall in Section 10.\nIn the particular case where the damping function bis invariant in one direction, this proposition\nis not needed. In this case, one can take as in [BH07] χΛ\nh=χ(b\nc0h). In thed-dimensional torus, this\ncutoff functions works as well if bis invariant in d−1 directions, and an analogue of Theorem 2.6\ncan be stated in this setting. Unfortunately, our construction of the function χΛ\nh(see the proof of\nProposition 8.3 in Section 13) strongly relies on the fact that all trap ped directions are periodic, and\nfails in higher dimensions.\nWe give here a proof of Lemma 8.1. Because of their technicality, we p ostpone the proofs of\nPropositions 8.2 and 8.3 to Sections 12 and 13 respectively.\n22Proof of Lemma 8.1. First, we develop\nPh\nbwh=Ph\nbOph/parenleftbigg\nχ/parenleftbigg|PΛξ|\nhα/parenrightbigg/parenrightbigg\nvh= Oph/parenleftbigg\nχ/parenleftbigg|PΛξ|\nhα/parenrightbigg/parenrightbigg\nPh\nbvh+ih/bracketleftbigg\nb,Oph/parenleftbigg\nχ/parenleftbigg|PΛξ|\nhα/parenrightbigg/parenrightbigg/bracketrightbigg\nvh,\n(8.2)\nsincePh\n0and Oph/parenleftBig\nχ/parenleftBig\n|PΛξ|\nhα/parenrightBig/parenrightBig\nare both Fourier mutipliers. We know that\n/vextenddouble/vextenddouble/vextenddouble/vextenddoubleOph/parenleftbigg\nχ/parenleftbigg|PΛξ|\nhα/parenrightbigg/parenrightbigg\nPh\nbvh/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2(T2)≤ /ba∇dblPh\nbvh/ba∇dblL2(T2)=o(h2+δ).\nIt only remains to study the operator\n/bracketleftbigg\nb,Oph/parenleftbigg\nχ/parenleftbigg|PΛξ|\nhα/parenrightbigg/parenrightbigg/bracketrightbigg\n=ih1−αOph/parenleftbigg\n∂yb χ′/parenleftbigg|PΛξ|\nhα/parenrightbigg/parenrightbigg\n+OL(L2)(h2(1−α)) (8.3)\naccording to the symbolic calculus. Moreover, using Assumption (2.1 2), we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂yb χ′/parenleftbigg|PΛξ|\nhα/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cb1−ε.\nThe sharp G˚ arding inequality applied to the nonnegative symbol\nC2b2(1−ε)−/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂yb χ′/parenleftbigg|PΛξ|\nhα/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n,\nthen yields/parenleftBigg\nOph/parenleftBigg\nC2b2(1−ε)−/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂yb χ′/parenleftbigg|PΛξ|\nhα/parenrightbigg/vextendsingle/vextendsingle/vextendsingle/vextendsingle2/parenrightBigg\nvh,vh/parenrightBigg\nL2(T2)≥ −Ch1−α,\nand hence\n/vextenddouble/vextenddouble/vextenddouble/vextenddoubleOph/parenleftbigg\n∂yb χ′/parenleftbigg|PΛξ|\nhα/parenrightbigg/parenrightbigg\nvh/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nL2(T2)≤C2(b2(1−ε)vh,vh)L2(T2)+O(h1−α). (8.4)\nWhen using the inequality/integraltext\nf1−εdν≤/parenleftbig/integraltext\nfdν/parenrightbig1−εfor nonnegative functions (with dν=|vh(x)|2dx),\nwe obtain\n(b2(1−ε)vh,vh)L2(T2)≤(b2vh,vh)(1−ε)\nL2(T2)≤C/ba∇dblbvh/ba∇dbl2(1−ε)\nL2(T2)=o(h1−ε).\nCombining this estimate together with (8.3) and (8.4) gives\n/vextenddouble/vextenddouble/vextenddouble/vextenddoubleih/bracketleftbigg\nb,Oph/parenleftbigg\nχ/parenleftbigg|PΛξ|\nhα/parenrightbigg/parenrightbigg/bracketrightbigg\nvh/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL2(T2)=o(h5\n2−α−ε\n2)+O(h5−3α\n2).\nComing back to the expression of Ph\nbwhgiven in (8.2), this concludes the proof of Lemma 8.1.\n9 Second microlocalization on a resonant affine subspace\nWe want to analyse precisely the structure of the restriction µ|T2×(Λ⊥\\{0}), using the full information\ncontained in o(h2+δ)-quasimodes like vhandwh.\nFrom now on, we want to take advantage of the family whofo(h2+δ)-quasimodes constructed in\nSection 8, which are microlocalised in the direction Λ⊥. Hence, we define the Wigner distribution\nWh∈D′(T∗T2) associated to the functions whand the scale h, by\n/angbracketleftbig\nWh,a/angbracketrightbig\n(S0)′,S0= (Oph(a)wh,wh)L2(T2)for alla∈S0(T∗T2).\n23According to Proposition 8.2, we recover in the limit h→0,\n/angbracketleftbig\nWh,a/angbracketrightbig\n(S0)′,S0→ /a\\}b∇acketle{tµ|T2×Λ⊥,a/a\\}b∇acket∇i}htM(T∗T2),C0\nc(T∗T2),\nfor anya∈C∞\nc(T∗T2) (andαsatisfying (8.1)).\nTo provide a precise study of µ|T2×Λ⊥, we shall introduce as in [Mac10, AM11] two-microlocal\nsemiclassicalmeasures,describingat afinerlevel the concentrat ionofthe sequence vhonthe resonant\nsubspace\nΛ⊥={ξ∈R2such thatPΛξ= 0}.\nThese objects have been introduced in the local Euclidean case by N ier [Nie96] and Fermanian-\nKammerer [FK00b, FK00a]. A specific concentration scale may also be chosen in the in the two-\nmicrolocal variable, giving rise to the two-scales semiclassical measu res studied by Miller [Mil96,\nMil97] and Fermanian-Kammerer and G´ erard [FKG02].\nWe first have to describe the adapted symbol class (inspired by [FK0 0a] and used in [AM11]).\nAccording to Lemma 7.3 (see also Remark 7.5), it suffices to test the m easureµ|T2×Λ⊥with functions\nconstant in the direction Λ⊥(or equivalently, having only x-Fourier modes in Λ, in the sense of the\nfollowing definition).\nDefinition 9.1. Given Λ ∈ P, we shall say that a∈S1\nΛifa=a(x,ξ,η)∈C∞(T∗T2×/a\\}b∇acketle{tΛ/a\\}b∇acket∇i}ht) and\n1. there exists a compact set Ka⊂T∗T2such that, for all η∈ /a\\}b∇acketle{tΛ/a\\}b∇acket∇i}ht, the function ( x,ξ)/ma√sto→a(x,ξ,η)\nis compactly supported in Ka;\n2.ais homogeneous of order zero at infinity in the variable η∈ /a\\}b∇acketle{tΛ/a\\}b∇acket∇i}ht; i.e., if we denote by SΛ:=\nS1∩/a\\}b∇acketle{tΛ/a\\}b∇acket∇i}httheunit spherein /a\\}b∇acketle{tΛ/a\\}b∇acket∇i}ht, thereexists R0>0(depending on a)andahom∈C∞\nc(T∗T2×SΛ)\nsuch that\na(x,ξ,η) =ahom/parenleftbigg\nx,ξ,η\n|η|/parenrightbigg\n,for|η| ≥R0and (x,ξ)∈T∗T2;\nforη/\\e}atio\\slash= 0, we will also use the notation a(x,ξ,∞η) :=ahom/parenleftBig\nx,ξ,η\n|η|/parenrightBig\n.\n3.ahas onlyx-Fourier modes in Λ, i.e.\na(x,ξ,η) =/summationdisplay\nk∈Λeik·x\n2πˆa(k,ξ,η).\nNote that this last assumption is equivalent to saying that σ·∂xa= 0 for any σ∈Λ⊥. We denote\nbyS1\nΛ′the topological dual space of S1\nΛ.\nLetχ∈C∞\nc(R) be a nonnegative function such that χ= 1 in a neighbourhood of the origin. Let\nR>0. The previous remark allows us to define, for a∈S1\nΛthe two following elements of S1\nΛ′:\n/angbracketleftBig\nWh,Λ\nR,a/angbracketrightBig\nS1\nΛ′,S1\nΛ:=/angbracketleftbigg\nWh,/parenleftbigg\n1−χ/parenleftbigg|PΛξ|\nRh/parenrightbigg/parenrightbigg\na/parenleftbigg\nx,ξ,PΛξ\nh/parenrightbigg/angbracketrightbigg\nD′(T∗T2),C∞\nc(T∗T2),(9.1)\n/angbracketleftbig\nWh\nR,Λ,a/angbracketrightbig\nS1\nΛ′,S1\nΛ:=/angbracketleftbigg\nWh,χ/parenleftbigg|PΛξ|\nRh/parenrightbigg\na/parenleftbigg\nx,ξ,PΛξ\nh/parenrightbigg/angbracketrightbigg\nD′(T∗T2),C∞c(T∗T2). (9.2)\nIn particular, for any R>0 anda∈S1\nΛ, we have\n/angbracketleftbigg\nWh,a/parenleftbigg\nx,ξ,PΛξ\nh/parenrightbigg/angbracketrightbigg\nD′(T∗T2),C∞c(T∗T2)=/angbracketleftBig\nWh,Λ\nR,a/angbracketrightBig\nS1\nΛ′,S1\nΛ+/angbracketleftbig\nWh\nR,Λ,a/angbracketrightbig\nS1\nΛ′,S1\nΛ.(9.3)\nThe following two propositions are the analogues of [FK00a] in our con text. They state the\nexistence of the two-microlocal semiclassical measures, as the limit objects ofWh,Λ\nRandWh\nR,Λ.\n24Proposition 9.2. There exists a subsequence (h,wh)and a nonnegative measure νΛ∈ M+(T∗T2×\nSΛ)such that, for all a∈S1\nΛ, we have\nlim\nR→∞lim\nh→0/angbracketleftBig\nWh,Λ\nR,a/angbracketrightBig\nS1\nΛ′,S1\nΛ=/angbracketleftbigg\nνΛ,ahom/parenleftbigg\nx,ξ,η\n|η|/parenrightbigg/angbracketrightbigg\nM(T∗T2×SΛ),C0c(T∗T2×SΛ).\nTo define the limit of the distributions Wh\nR,Λ, we need first to introduce operator spaces and\noperator-valued measures, following [G´ er91]. Given a Hilbert space H(in the following, we shall\nuseH=L2(TΛ)), we denote respectively by K(H),L1(H) the spaces of compact and trace class\noperators on H. We recall that they are both two-sided ideals of the ring L(H) of bounded operators\nonH. We refer for instance to [RS80, Chapter VI.6] for a description of the space L1(H) and its\nbasic properties. Given a Polish space T(in the following, we shall use T=T∗TΛ⊥), we denote by\nM+(T;L1(H)) the space of nonnegative measures on T, taking values in L1(H). More precisely, we\nhaveρ∈ M+(T;L1(H)) ifρis a bounded linear form on C0\nc(T) such that, for every nonnegative\nfunctiona∈C0\nc(T),/a\\}b∇acketle{tρ,a/a\\}b∇acket∇i}htM(T),C0\nc(T)∈ L1(H) isanonnegativehermitian operator. As aconsequence\nof [RS80, Theorem VI.26], these measures can be identified in a natur al way to nonnegative linear\nfunctionals on C0\nc(T;K(H)).\nProposition 9.3. There exists a subsequence (h,wh)and a nonnegative measure\nρΛ∈ M+(T∗TΛ⊥;L1(L2(TΛ))),\nsuch that, for all K∈C∞\nc(T∗TΛ⊥;K(L2(TΛ))),\nlim\nh→0(K(s,hDs)TΛwh,TΛwh)L2(TΛ⊥;L2(TΛ))= tr/braceleftBigg/integraldisplay\nT∗TΛ⊥K(s,σ)ρΛ(ds,dσ)/bracerightBigg\n.(9.4)\nMoreover (for the same subsequence), for all a∈S1\nΛ, we have\nlim\nR→∞lim\nh→0/angbracketleftbig\nWh\nR,Λ,a/angbracketrightbig\nS1\nΛ′,S1\nΛ= tr/braceleftBigg/integraldisplay\nT∗TΛ⊥OpΛ\n1/parenleftbig\na(˜πΛ(σ,y,0),η)/parenrightbig\nρΛ(ds,dσ)/bracerightBigg\n.(9.5)\nIn the left hand-side of (9.4), the inner product actually means\n(K(s,hDs)TΛwh,TΛwh)L2(TΛ⊥L2(TΛ))\n=/integraldisplay\ns∈TΛ⊥,s′∈Λ⊥,σ∈Λ⊥ei\nh(s−s′)·σ/parenleftbigg\nK/parenleftbigs+s′\n2,σ/parenrightbig\nTΛwh(s′,y),TΛwh(s,y)/parenrightbigg\nL2y(TΛ)ds ds′dσ.\nIn the expression (9.5), remark that for each σ∈Λ⊥, the operator OpΛ\n1/parenleftbig\na(˜πΛ(σ,y,0),η) is in\nL(L2(TΛ)). Hence, its product with the operator ρΛ(ds,dσ) defines a trace-class operator.\nBefore proving Propositions 9.2 and 9.3, we explain how to reconstru ct the measure µ|T2×Λ⊥\nfrom the two-microlocal measures νΛandρΛ. This reduces the study of the measure µto that of all\ntwo-microlocal measures νΛandρΛ, for Λ∈ P.\nWe denote by M+\nc(T) the set of compactly supported measures on T, and by /a\\}b∇acketle{t·,·/a\\}b∇acket∇i}htMc(T),C0(T)\nthe associated duality bracket.\nProposition 9.4. For alla∈C∞\nc(T∗T2)having only x-Fourier modes in Λ(i.e. for all a∈S1\nΛ\nindependent of the third variable η∈ /a\\}b∇acketle{tΛ/a\\}b∇acket∇i}ht), we have\n/a\\}b∇acketle{tµ,a/a\\}b∇acket∇i}htM(T∗T2),C0c(T∗T2)=/angbracketleftbig\nνΛ,a/angbracketrightbig\nM(T∗T2×SΛ),C0c(T∗T2×SΛ)+tr/braceleftBigg/integraldisplay\nT∗TΛ⊥ma◦˜πΛ(σ)ρΛ(ds,dσ)/bracerightBigg\n,(9.6)\n25and\n/a\\}b∇acketle{tµ|T2×Λ⊥,a/a\\}b∇acket∇i}htM(T∗T2),C0\nc(T∗T2)=/angbracketleftbig\nνΛ|T2×Λ⊥×SΛ,a/angbracketrightbig\nM(T∗T2×SΛ),C0c(T∗T2×SΛ)\n+tr/braceleftBigg/integraldisplay\nT∗TΛ⊥ma◦˜πΛ(σ)ρΛ(ds,dσ)/bracerightBigg\n, (9.7)\nwhere forσ∈Λ⊥,ma◦˜πΛ(σ)denotes the multiplication in L2(TΛ)by the function y/ma√sto→a◦˜πΛ(σ,y).\nMoreover, we have νΛ∈ M+\nc(T∗T2×SΛ)andρΛ∈ M+\nc(T∗TΛ⊥;L2(TΛ))(i.e. both measures are\ncompactly supported).\nFormula (9.7) follows immediately from (9.6) by restriction. By the defi nition of the measure ρΛ,\nwe see that it is already supported on T2×Λ⊥(see expression (9.2)).\nThe end of this section is devoted to the proofs of the three propo sitions, inspired by [FK00a,\nAM11].\nProof of Proposition 9.2. The Calder´ on-Vaillancourt theorem implies that the operators\nOph/parenleftbigg/parenleftbigg\n1−χ/parenleftbigg|PΛξ|\nRh/parenrightbigg/parenrightbigg\na/parenleftbigg\nx,ξ,PΛξ\nh/parenrightbigg/parenrightbigg\n= Op1/parenleftbigg/parenleftbigg\n1−χ/parenleftbigg|PΛξ|\nR/parenrightbigg/parenrightbigg\na(x,hξ,P Λξ)/parenrightbigg\nare uniformly bounded as h→0 andR→+∞. It follows that the family Wh,Λ\nRis bounded in S1\nΛ′,\nand thus there exists a subsequence ( h,wh) and a distribution ˜ µΛsuch that\nlim\nR→∞lim\nh→0/angbracketleftBig\nWh,Λ\nR,a/angbracketrightBig\nS1\nΛ′,S1\nΛ=/angbracketleftbig\n˜µΛ,a(x,ξ,η)/angbracketrightbig\nS1\nΛ′,S1\nΛ.\nBecause of the support properties of the function χ, we notice that/angbracketleftbig\n˜µΛ,a/angbracketrightbig\nS1\nΛ′,S1\nΛ= 0 as soon as the\nsupport of ais compact in the variable η. Hence, there exists a distribution νΛ∈D′(T∗T2×SΛ)\nsuch that/angbracketleftbig\n˜µΛ,a(x,ξ,η)/angbracketrightbig\nS1\nΛ′,S1\nΛ=/angbracketleftbigg\nνΛ,ahom/parenleftbigg\nx,ξ,η\n|η|/parenrightbigg/angbracketrightbigg\nD′(T∗T2×SΛ),C∞c(T∗T2×SΛ).\nNext, suppose that a >0 (and that√1−χis smooth). Then, using [AM11, Corollary 35], and\nsetting\nbR(x,ξ) =/parenleftbigg/parenleftbigg\n1−χ/parenleftbigg|PΛξ|\nRh/parenrightbigg/parenrightbigg\na/parenleftbigg\nx,ξ,PΛξ\nh/parenrightbigg/parenrightbigg1\n2\n,\nthere exists C >0 such that for all h≤h0andR≥1, we have\n/vextenddouble/vextenddouble/vextenddouble/vextenddoubleOph/parenleftbigg/parenleftbigg\n1−χ/parenleftbigg|PΛξ|\nRh/parenrightbigg/parenrightbigg\na/parenleftbigg\nx,ξ,PΛξ\nh/parenrightbigg/parenrightbigg\n−Oph(bR)2/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nL(L2(T2))≤C\nR.\nAs a consequence, we have,\n/angbracketleftBig\nWh,Λ\nR,a/angbracketrightBig\nS1\nΛ′,S1\nΛ≥ /ba∇dblOph(bR)wh/ba∇dbl2\nL2(T2)−C\nR/ba∇dblwh/ba∇dbl2\nL2(T2),\nso that the limit/angbracketleftBig\nνΛ,ahom/parenleftBig\nx,ξ,η\n|η|/parenrightBig/angbracketrightBig\nD′(T∗T2×SΛ),C∞c(T∗T2×SΛ)is nonnegative. The distribution νΛ\nis nonnegative, and is hence a measure. This concludes the proof of Proposition 9.2.\nProof of Proposition 9.3. First, the proof of the existence of a subsequence ( h,wh) and the measure\nρΛsatisfying (9.4) is the analogue of Proposition 5.1 in the context of op erator valued measures,\nviewing the sequence whas a bounded sequence of L2(TΛ⊥;L2(TΛ)). It follows the lines of this\nresult, after the adaptation of the symbolic calculus to operator v alued symbols (or more precisely,\nof [G´ er91] in the semiclassical setting).\n26Second, using the definition (9.2) together with (7.6), we have\n/angbracketleftbig\nWh\nR,Λ,a/angbracketrightbig\nS1\nΛ′,S1\nΛ=/parenleftbigg\nOph/parenleftbigg\nχ/parenleftbigg|PΛξ|\nRh/parenrightbigg\na/parenleftbigg\nx,ξ,PΛξ\nh/parenrightbigg/parenrightbigg\nwh,wh/parenrightbigg\nL2(T2)\n=/parenleftbigg\nOpΛ⊥\nh◦OpΛ\nh/parenleftbigg\nχ/parenleftbigg|η|\nRh/parenrightbigg\na/parenleftBig\n˜πΛ(σ,y,η),η\nh/parenrightBig/parenrightbigg\nTΛwh,TΛwh/parenrightbigg\nL2(TΛ⊥×TΛ).\nHence, setting\nah\nR,Λ(σ,y,η) =χ/parenleftbigg|η|\nR/parenrightbigg\na(˜πΛ(σ,y,hη),η),\nwe obtain\n/angbracketleftbig\nWh\nR,Λ,a/angbracketrightbig\nS1\nΛ′,S1\nΛ=/parenleftBig\nOpΛ⊥\nh◦OpΛ\n1/parenleftbig\nah\nR,Λ(σ,y,η)/parenrightbig\nTΛwh,TΛwh/parenrightBig\nL2(TΛ⊥×TΛ).\nWe also notice that OpΛ\n1/parenleftbig\nah\nR,Λ/parenrightbig\n∈ K(L2(TΛ)), for anyσ∈Λ⊥sinceah\nR,Λhas compact support with\nrespect toη. Moreover, for any R>0 fixed and a∈S1\nΛ, the Calder´ on-Vaillancourt theorem yields\nOpΛ\n1/parenleftbig\nah\nR,Λ/parenrightbig\n= OpΛ\n1/parenleftbig\na0\nR,Λ/parenrightbig\n+hB\nfor someB∈ L(L2(TΛ)), uniformly bounded with respect to h. Using (9.4), this implies that for\nanyR>0 fixed and a∈S1\nΛ, we have\nlim\nh→0/angbracketleftbig\nWh\nR,Λ,a/angbracketrightbig\nS1\nΛ′,S1\nΛ= tr/braceleftBigg/integraldisplay\nT∗TΛ⊥OpΛ\n1/parenleftbig\na0\nR,Λ/parenrightbig\nρΛ(ds,dσ)/bracerightBigg\n.\nMoreover, we have\nlim\nR→+∞OpΛ\n1/parenleftbig\na0\nR,Λ/parenrightbig\n= OpΛ\n1/parenleftbig\na0\n∞,Λ/parenrightbig\n= OpΛ\n1/parenleftbig\na(˜πΛ(σ,y,0),η)/parenrightbig\n,\nin the strong topology of C∞\nc(T∗TΛ⊥;L(L2(TΛ))). This proves (9.5) and concludes the proof of\nProposition 9.3.\nProof of Proposition 9.4. Takinga∈S1\nΛ, independent of the third variable η∈ /a\\}b∇acketle{tΛ/a\\}b∇acket∇i}htgives\n/angbracketleftbig\nWh,a(x,ξ)/angbracketrightbig\nD′(T∗T2),C∞c(T∗T2)→ /a\\}b∇acketle{tµ|T2×Λ⊥,a/a\\}b∇acket∇i}htM(T∗T2),C0c(T∗T2),\ntogether with/angbracketleftBig\nWh,Λ\nR,a/angbracketrightBig\nS1\nΛ′,S1\nΛ→/angbracketleftbig\nνΛ,a/angbracketrightbig\nM(T∗T2×SΛ),C0c(T∗T2×SΛ),\n(according to Proposition 9.2) and\n/angbracketleftbig\nWh\nR,Λ,a/angbracketrightbig\nS1\nΛ′,S1\nΛ→tr/braceleftBigg/integraldisplay\nT∗TΛ⊥OpΛ\n1/parenleftbig\na(˜πΛ(σ,y,0))/parenrightbig\nρΛ(ds,dσ)/bracerightBigg\n= tr/braceleftBigg/integraldisplay\nT∗TΛ⊥ma◦˜πΛ(σ)ρΛ(ds,dσ)/bracerightBigg\n,\n(according to Proposition 9.3). Now, using the last three equations together with Equation (9.3)\ndirectly gives (9.7).\nAs both terms in the right hand-side of (9.7) are nonnegative measu res and the left-hand side\nis a compactly supported nonnegative measure, this implies that νΛandρΛare both compactly\nsupported.\n2710 Propagation laws for the two-microlocal measures νΛand\nρΛ\nIn this section, we study the propagation properties of νΛandρΛ. The key point here is the use of\nthe cutoff function introduced in Proposition 8.3.\nWe will use repeatedly the following fact, which follows from Item 2 in Pr oposition 8.3: if Ais a\nbounded operator on L2(T2), we have\n(Awh,wh)L2(T2)=/parenleftbig\nAχΛ\nhwh,χΛ\nhwh/parenrightbig\nL2(T2)+/ba∇dblA/ba∇dblL(L2)o(1). (10.1)\nTo simplify the notation, we shall write Ac0,hforχΛ\nhAχΛ\nh.\n10.1 Propagation of νΛ\nWe define for ( x,ξ,η)∈T∗T2×/a\\}b∇acketle{tΛ/a\\}b∇acket∇i}htandτ∈Rthe flows\nφ0\nτ(x,ξ,η) := (x+τξ,ξ,η),\ngenerated by the vector field ξ·∂xand, forη/\\e}atio\\slash= 0,\nφ1\nτ(x,ξ,η) :=/parenleftbigg\nx+τη\n|η|,ξ,η/parenrightbigg\n,\ngenerated by the vector fieldη\n|η|·∂x. With these definitions, we have the following propagation laws\nfor the two-microlocal measure νΛ.\nProposition 10.1. The measure νΛisφ0\nτ- andφ1\nτ-invariant, i.e.\n(φ0\nτ)∗νΛ=νΛand(φ1\nτ)∗νΛ=νΛ,for everyτ∈R.\nThe key result here is the additional “transverse propagation law” given by the flow φ1\nτ. The\nmeasureνΛnot only propagates along the geodesic flow φ0\nτ, but also along directions transverse to\nΛ⊥.\nProof.Fixa∈S1\nΛ. Thecomputationdonein(6.2)isstillvalidreplacing aby/parenleftBig\n1−χ/parenleftBig\n|PΛξ|\nRh/parenrightBig/parenrightBig\na/parenleftBig\nx,ξ,PΛξ\nh/parenrightBig\n,\nsinceitonlyusesthefactthatOph/parenleftBig/parenleftBig\n1−χ/parenleftBig\n|PΛξ|\nRh/parenrightBig/parenrightBig\na/parenleftBig\nx,ξ,PΛξ\nh/parenrightBig/parenrightBig\nisboundedandthat /ba∇dblPh\nbwh/ba∇dblL2(T2)=\no(h) and (bwh,wh)L2(T2)=o(1). This yields\nlim\nh→0/angbracketleftBig\nWh,Λ\nR,ξ·∂xa/angbracketrightBig\nS1\nΛ′,S1\nΛ\n= lim\nh→0/angbracketleftbigg\nWh,ξ·∂x/braceleftbigg/parenleftbigg\n1−χ/parenleftbigg|PΛξ|\nRh/parenrightbigg/parenrightbigg\na/parenleftbigg\nx,ξ,PΛξ\nh/parenrightbigg/bracerightbigg/angbracketrightbigg\nD′(T∗T2),C∞\nc(T∗T2)= 0,\nand hence, in the limit R→+∞, we obtain\n/angbracketleftbigg\nνΛ,ξ·∂xahom/parenleftbigg\nx,ξ,η\n|η|/parenrightbigg/angbracketrightbigg\nM(T∗T2×SΛ),C0c(T∗T2×SΛ)= 0.\nReplacingahombyahom◦φ0\nτand integrating with respect to the parameter τgives (φ0\nτ)∗νΛ=νΛ,\nwhich concludes the first part of the proof.\nSecond, to prove the φ1\nτ-invariance of νΛwe compute\n/angbracketleftbigg\nνΛ,η\n|η|·∂xahom/parenleftbigg\nx,ξ,η\n|η|/parenrightbigg/angbracketrightbigg\nM(T∗T2×SΛ),C0c(T∗T2×SΛ)= lim\nR→∞lim\nh→0/angbracketleftbigg\nWh,Λ\nR,η\n|η|·∂xa/angbracketrightbigg\nS1\nΛ′,S1\nΛ.(10.2)\n28Setting\naR(x,ξ,η) =1\n|η|/parenleftbigg\n1−χ/parenleftbigg|η|\nR/parenrightbigg/parenrightbigg\na(x,ξ,η),\nand\nAR:= Oph/parenleftbigg\naR/parenleftBig\nx,ξ,PΛξ\nh/parenrightBig/parenrightbigg\n(10.3)\nwe have the relation\n/angbracketleftbigg\nWh,Λ\nR,η\n|η|·∂xa/angbracketrightbigg\nS1\nΛ′,S1\nΛ=−i\n2([∆Λ,AR]wh,wh)L2(T2)\nwhere ∆ Λ=∂2\nyis the laplacian in the direction Λ.\nLemma 10.2. For any given c0>0andR>0, we have\n([∆Λ,AR]wh,wh)L2(T2)= ([∆Λ,AR\nc0,h]wh,wh)L2(T2)+o(1).\nWe postpone the proof of Lemma 10.2 and first indicate how it allows to prove Proposition 10.1.\nWe now know that\n/angbracketleftbigg\nνΛ,η\n|η|·∂xahom/parenleftbigg\nx,ξ,η\n|η|/parenrightbigg/angbracketrightbigg\nM(T∗T2×SΛ),C0\nc(T∗T2×SΛ)= lim\nR→∞lim\nh→0−i\n2([∆Λ,AR\nc0,h]wh,wh)L2(T2).\nRecall that a∈S1\nΛimplies that ahas onlyx-Fourier modes in Λ, i.e. PΛξ·∂xa=ξ·∂xa. We have\nalso assumed in this section that bhas onlyx-Fourier modes in Λ. As a consequence, we have\n−i\n2([∆Λ,AR\nc0,h]wh,wh)L2(T2)=−i\n2([∆,AR\nc0,h]wh,wh)L2(T2)\n=i\n2h2/parenleftbig/bracketleftbig\nPh\n0,AR\nc0,h/bracketrightbig\nwh,wh/parenrightbig\nL2(T2). (10.4)\nDeveloping the last expression of (10.4), we obtain\ni\n2h2/parenleftbig/bracketleftbig\nPh\n0,AR\nc0,h/bracketrightbig\nwh,wh/parenrightbig\nL2(T2)=i\n2h2/parenleftbig\nAR\nc0,hwh,Ph\nbwh/parenrightbig\nL2(T2)−i\n2h2/parenleftbig\nAR\nc0,hPh\nbwh,wh/parenrightbig\nL2(T2)\n−1\n2h/parenleftbig\nAR\nc0,hwh,bwh/parenrightbig\nL2(T2)−1\n2h/parenleftbig\nAR\nc0,hbwh,wh/parenrightbig\nL2(T2).(10.5)\nSinceAR\nc0,his bounded in L(L2(T2)), its adjoint AR\nc0,his also bounded so that the first two terms in\nthe last expression vanish in the limit h→0, using/ba∇dblPh\nbwh/ba∇dblL2(T2)=o(h2). To estimate the last two\nterms, we use again the boundedness of ARand (AR)∗and write\n|/parenleftbig\nAR\nc0,hwh,bwh/parenrightbig\nL2(T2)| ≤ /ba∇dblAR/ba∇dbl/ba∇dblχΛ\nhbwh/ba∇dblL2(T2)≤2c0h/ba∇dblAR/ba∇dbl,\naccording to Item 3 in Proposition 8.3. It follows that\nlimsup\nh→0/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\n2h/parenleftbig\nAR\nc0,hwh,bwh/parenrightbig\nL2(T2)+1\n2h/parenleftbig\nAR\nc0,hbwh,wh/parenrightbig\nL2(T2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤2c0sup/ba∇dblAR/ba∇dbl.\nComing back to the expression (10.2), we obtain\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/angbracketleftbigg\nνΛ,η\n|η|·∂xahom/parenleftbigg\nx,ξ,η\n|η|/parenrightbigg/angbracketrightbigg\nM(T∗T2×SΛ),C0c(T∗T2×SΛ)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤2c0sup/ba∇dblAR/ba∇dbl\nand sincec0was arbitrary,\n/angbracketleftbigg\nνΛ,η\n|η|·∂xahom/parenleftbigg\nx,ξ,η\n|η|/parenrightbigg/angbracketrightbigg\nM(T∗T2×SΛ),C0c(T∗T2×SΛ)= 0\nReplacingahombyahom◦φ1\nτand integrating with respect to the parameter τgives (φ1\nτ)∗νΛ=νΛ,\nwhich concludes the proof of Proposition 10.1.\n29Proof of Lemma 10.2. We are going to show that\n([∆Λ,AR\nc0,h]wh,wh)L2(T2)= ([∆Λ,AR]c0,hwh,wh)L2(T2)+o(1). (10.6)\nThen, usingthefactthat[∆ Λ,AR]isaboundedoperator(itssymbolis/parenleftBig\n1−χ/parenleftBig\n|η|\nR/parenrightBig/parenrightBig\nη\n|η|·∂xa(x,ξ,η)),\ntogether with (10.1), this is also ([∆ Λ,AR]wh,wh)L2(T2)+o(1).\nTo prove (10.6), we develop the difference [∆ Λ,AR\nc0,h]−[∆Λ,AR]c0,has\n[∆Λ,AR\nc0,h]−[∆Λ,AR]c0,h=/bracketleftbig\n∂2\ny,χΛ\nh/bracketrightbig\nARχΛ\nh+χΛ\nhAR/bracketleftbig\n∂2\ny,χΛ\nh/bracketrightbig\n. (10.7)\nThen, writing/bracketleftbig\n∂2\ny,χΛ\nh/bracketrightbig\n=∂2\nyχΛ\nh+2∂yχΛ\nh∂y,\nwe have\n/parenleftbig/bracketleftbig\n∂2\ny,χΛ\nh/bracketrightbig\nARχΛ\nhwh,wh/parenrightbig\nL2(T2)=/parenleftbig\nARχΛ\nhwh,∂2\nyχΛ\nhwh/parenrightbig\nL2(T2)+/parenleftbig\n∂y◦ARχΛ\nhwh,2∂yχΛ\nhwh/parenrightbig\nL2(T2).\nRecalling that the operator ∂y◦ARis bounded, and using Items 4 and 5 in Proposition 8.3, we obtain\n/vextendsingle/vextendsingle/vextendsingle/parenleftbig/bracketleftbig\n∂2\ny,χΛ\nh/bracketrightbig\nARχΛ\nhwh,wh/parenrightbig\nL2(T2)/vextendsingle/vextendsingle/vextendsingle≤C/ba∇dbl∂2\nyχΛ\nhwh/ba∇dblL2(T2)++C/ba∇dbl∂yχΛ\nhwh/ba∇dblL2(T2)=o(1).\nThe last term in (10.7) is handled similarly. This finally implies (10.6) and con cludes the proof of\nLemma 10.2.\n10.2 Propagation of ρΛ\nWe denote by ( ωj\nΛ,ej\nΛ)j∈Nthe eigenvalues and associated eigenfunctions of the operator −∆Λ=−∂2\ny\nforming a Hilbert basis of L2(TΛ). We shall use the projector onto low frequencies of −∆Λ, i.e., for\nanyω∈R+, the operator\nΠω\nΛ:=/summationdisplay\nωj\nΛ≤ω(·,ej\nΛ)L2(TΛ)ej\nΛ,\nwhich has finite rank.\nWe have the following propagation laws for the two-microlocal measu reρΛ.\nProposition 10.3. 1. For any K∈C∞\nc/parenleftbig\nT∗TΛ⊥;K(L2(TΛ))/parenrightbig\n, independent of s(i.e.K(s,σ) =\nK(σ)), and any ω>0, we have\ntr/braceleftBigg/integraldisplay\nT∗TΛ⊥[∆Λ,Πω\nΛK(σ)Πω\nΛ]ρΛ(ds,dσ)/bracerightBigg\n= 0.\n2. Moreover, defining\nMΛ:=/integraldisplay\nTΛ⊥×Λ⊥ρΛ(ds,dσ)∈ L1(L2(TΛ)),\nwe have\n[∆Λ,M��] = 0.\nRemark that for any σ∈Λ⊥, the operator\n[∆Λ,Πω\nΛK(σ)Πω\nΛ] = Πω\nΛ[∆Λ,K(σ)]Πω\nΛ,\nhas finite rank, so the right hand-side of Item 1 is well-defined. Note that the definition of MΛhas\na signification since ρΛhas a compact support, according to Proposition 9.4.\nThe commutation relations of Items 1 and 2 in this proposition corres pond to propagation laws\nat the operator level. They are formulated here in a “derivated for m”, which, for Item 2 for instance,\nis equivalent to\neiτ∆ΛMΛe−iτ∆Λ=MΛ,for allτ∈R,\nin the “integrated form”.\n30Proof.ForK∈C∞\nc(Λ⊥;K(L2(TΛ))) (in other words K∈C∞\nc(T∗TΛ⊥;K(L2(TΛ))) independent of\ns∈TΛ⊥), we denote\nKω(σ) := Πω\nΛK(σ)Πω\nΛ\nand we note that Kωis also in C∞\nc(Λ⊥;K(L2(TΛ))). Hence, we have\ntr/braceleftBigg/integraldisplay\nT∗TΛ⊥[∆Λ,Πω\nΛK(σ)Πω\nΛ]ρΛ(ds,dσ)/bracerightBigg\n=−lim\nh→0([−∆Λ,Kω(hDs)]TΛwh,TΛwh)L2(TΛ⊥;L2(TΛ))\nTo show that this limit vanishes, we proceed as in lines (10.4), (10.5) an d in the subsequent calcula-\ntion, replacing the operator ARbyKω(hDs).\nWith the notation ∆ Λ=∂2\nyand ∆ Λ⊥=∂2\ns, we first note that\n([−∆Λ,Kω(hDs)]TΛwh,TΛwh)L2(TΛ⊥;L2(TΛ))= ([−∆,Kω(hDs)]TΛwh,TΛwh)L2(TΛ⊥;L2(TΛ)),\nsince ∆ = ∆ Λ+∆Λ⊥and since [∆ Λ⊥,Kω(hDs)] = 0.As a matter of fact, Kω(hDs) = OpΛ\nh(Kω(σ))\nand ∆ Λ⊥=−h−2OpΛ\nh(|σ|2) are both Fourier multipliers.\nThe following lemma is proved the same way as Lemma 10.2\nLemma 10.4. For any given c0>0, we have\n([∆Λ,Kω(σ)]TΛwh,TΛwh)L2(T2)= ([∆Λ,Kω\nc0,h(hDs)]TΛwh,TΛwh)L2(T2)+o(1).\nHereKω\nc0,h(hDs) meansχΛ\nhKω(hDs)χΛ\nh.\nWriting\n−h2∆ =TΛPh\nbT∗\nΛ−ihb◦πΛ,\nwe have\n/parenleftbig\n[−∆,Kω\nc0,h(hDs)]TΛwh,TΛwh/parenrightbig\nL2(TΛ⊥;L2(TΛ))\n=1\nh2/parenleftbig\nKω\nc0,h(hDs)TΛwh,TΛPh\nbwh/parenrightbig\nL2(TΛ⊥;L2(TΛ))−1\nh2/parenleftbig\nKω\nc0,h(hDs)TΛPh\nbwh,TΛwh/parenrightbig\nL2(TΛ⊥;L2(TΛ))\n+i\nh/parenleftbig\nKω\nc0,h(hDs)TΛwh,TΛ(bwh)/parenrightbig\nL2(TΛ⊥;L2(TΛ))+i\nh/parenleftbig\nKω\nc0,h(hDs)TΛ(bwh),TΛwh/parenrightbig\nL2(TΛ⊥;L2(TΛ)).\nIt follows, as in (10.5), that\nlimsup\nh→0|/parenleftbig\n[−∆,Kω\nc0,h(hDs)]TΛwh,TΛwh/parenrightbig\nL2(TΛ⊥;L2(TΛ))| ≤2c0/ba∇dblK/ba∇dbl\nand sincec0was arbitrary, we can conclude that\nlim\nh→0([∆Λ,Kω(σ)]TΛwh,TΛwh)L2(T2)= 0,\nwhich concludes the proof of Item 1.\nItem 1 gives, for all K∈ K(L2(TΛ)) constant (which is possible since ρΛ(ds,dσ) has compact\nsupport),\n0 = tr/braceleftBigg/integraldisplay\nT∗TΛ⊥[∆Λ,Kω]ρΛ(ds,dσ)/bracerightBigg\n= tr/braceleftBigg\n[∆Λ,Kω]/integraldisplay\nT∗TΛ⊥ρΛ(ds,dσ)/bracerightBigg\n= tr{[∆Λ,Kω]MΛ}.\nUsing that tr( AB) = tr(BA) for allA∈ L1andB∈ Ltogether with the linearity of the trace\n(see [RS80, Theorem VI.25]), we now obtain, for all K∈ K(L2(TΛ)), and allω>0,\n0 = tr{[∆Λ,Πω\nΛKΠω\nΛ]MΛ}= tr{KΠω\nΛ[∆Λ,MΛ]Πω\nΛ}.\nConsequently, we have for all ω>0, Πω\nΛ[∆Λ,MΛ]Πω\nΛ= 0 (see [RS80, Theorem VI.26]). Letting ωgo\nto +∞, this yields [∆ Λ,MΛ] = 0 and concludes the proof of Item 2.\n3111 The measures νΛandρΛvanish identically. End of the\nproof of Theorem 2.6\nIn this section, we prove that both measures νΛandρΛvanish when paired with the function /a\\}b∇acketle{tb/a\\}b∇acket∇i}htΛ.\nThen, we deduce that these two measures vanish identically. In tur n, this implies that µ|T2×Λ⊥= 0,\nand finally that µ= 0, which will conclude the proof of Theorem 2.6.\nProposition 11.1. We have\n/angbracketleftbig\nνΛ|T2×Λ⊥×SΛ,/a\\}b∇acketle{tb/a\\}b∇acket∇i}htΛ/angbracketrightbig\nMc(T∗T2×SΛ),C0(T∗T2×SΛ)= 0,andtr{m/a\\}bracketle{tb/a\\}bracketri}htΛMΛ}= 0.\nAs a consequence, we prove that ρΛandνΛ|T2×Λ⊥×SΛvanish.\nProposition 11.2. We haveρΛ= 0andνΛ|T2×Λ⊥×SΛ= 0. Henceµ|T2×Λ⊥= 0.\nThis allows to conclude the proof of Theorem 2.6. Indeed, as a conse quence of the decomposition\nformula of Proposition 9.4, we obtain, for all Λ ∈ P, such that rk(Λ) = 1, µ|T2×Λ⊥= 0. Using the\ndecomposition of the measure µgiven in Lemma 7.1 together with Lemma 7.4, this yields µ= 0\nonT2. This is in contradiction with µ(T∗T2) = 1 (Proposition 6.1), and this contradiction proves\nTheorem 2.6.\nWe now prove Propositions 11.1 and 11.2\nProof of Proposition 11.1. First, (4.22) implies that ( bvh,vh)L2(T2)→0, and hence\n/a\\}b∇acketle{tµ,b/a\\}b∇acket∇i}htMc(T∗T2),C0(T∗T2)= 0.\nThen the decomposition given in Lemma 7.1 into a sum of nonnegative me asures yields that, for all\nΛ∈ P,\n/a\\}b∇acketle{tµ|T2×Λ⊥,b/a\\}b∇acket∇i}htMc(T∗T2),C0(T∗T2)= 0, (11.1)\nsincebis also nonnegative. Lemmata 7.2, 7.3 and 7.4 (see also Remark 7.5), th en give\n/a\\}b∇acketle{tµ|T2×Λ⊥,/a\\}b∇acketle{tb/a\\}b∇acket∇i}htΛ/a\\}b∇acket∇i}htMc(T∗T2),C0(T∗T2)=/angbracketleftbig\nµ|T2×(Λ⊥\\{0}),/a\\}b∇acketle{tb/a\\}b∇acket∇i}htΛ/angbracketrightbig\nMc(T∗T2),C0(T∗T2)\n=/a\\}b∇acketle{tµ|T2×Λ⊥,b/a\\}b∇acket∇i}htMc(T∗T2),C0(T∗T2)= 0, (11.2)\nwhere the function /a\\}b∇acketle{tb/a\\}b∇acket∇i}htΛis also nonnegative. The decomposition formula of Proposition 9.4 into the\ntwo-microlocal semiclassical measures then yields\n/a\\}b∇acketle{tµ|T2×Λ⊥,/a\\}b∇acketle{tb/a\\}b∇acket∇i}htΛ/a\\}b∇acket∇i}htMc(T∗T2),C0(T∗T2)=/angbracketleftbig\nνΛ|T2×Λ⊥×SΛ,/a\\}b∇acketle{tb/a\\}b∇acket∇i}htΛ/angbracketrightbig\nMc(T∗T2×SΛ),C0(T∗T2×SΛ)\n+tr/braceleftBigg/integraldisplay\nT∗TΛ⊥m/a\\}bracketle{tb/a\\}bracketri}htΛρΛ(ds,dσ)/bracerightBigg\n.\nBesides,themeasure νΛ|T2×Λ⊥×SΛisnonnegative,hence/angbracketleftbig\nνΛ|T2×Λ⊥×SΛ,/a\\}b∇acketle{tb/a\\}b∇acket∇i}htΛ/angbracketrightbig\nMc(T∗T2×SΛ),C0(T∗T2×SΛ)≥\n0. Similarly, ρΛ∈ M+\nc(T∗TΛ⊥;L1(TΛ)) and the operator m/a\\}bracketle{tb/a\\}bracketri}htΛ∈ L(L2(TΛ)) is selfadjoint and non-\nnegative, which gives tr/braceleftBig/integraltext\nT∗TΛ⊥m/a\\}bracketle{tb/a\\}bracketri}htΛρΛ(ds,dσ)/bracerightBig\n≥0. Using (11.1) and (11.2), this yields\n/angbracketleftbig\nνΛ|T2×Λ⊥×SΛ,/a\\}b∇acketle{tb/a\\}b∇acket∇i}htΛ/angbracketrightbig\nMc(T∗T2×SΛ),C0(T∗T2×SΛ)= 0,\nand\ntr/braceleftBigg/integraldisplay\nT∗TΛ⊥m/a\\}bracketle{tb/a\\}bracketri}htΛρΛ(ds,dσ)/bracerightBigg\n= 0.\nIn this expression, the operator m/a\\}bracketle{tb/a\\}bracketri}htΛdoes not depend on ( s,σ), so that\n0 = tr/braceleftBigg\nm/a\\}bracketle{tb/a\\}bracketri}htΛ/integraldisplay\nT∗TΛ⊥ρΛ(ds,dσ)/bracerightBigg\n= tr{m/a\\}bracketle{tb/a\\}bracketri}htΛMΛ},\nwhich concludes the proof of Proposition 11.1.\n32Proof of Proposition 11.2. Let us first prove that ρΛ= 0. We recall that the operator MΛis a\nselfadjoint nonnegative trace-class operator. Moreover, Prop osition 10.3 implies that the operators\nMΛand ∆ Λcommute. As a consequence, there exists a Hilbert basis (˜ ej\nΛ)j∈NofL2(TΛ) in which\nMΛand ∆ Λare simultaneously diagonal, i.e. such that\n−∆Λ˜ej\nΛ=ωj\nΛ˜ej\nΛ,andMΛ˜ej\nΛ=γj\nΛ˜ej\nΛ,\nwhere (γj\nΛ)j∈Nare the associated eigenvalues of MΛ. In particular, we have γj\nΛ≥0 for allj∈N(and\nγj\nΛ∈ℓ1). Note that the basis (˜ ej\nΛ)j∈Nis not necessarily the same as the basis ( ej\nΛ)j∈Nintroduced in\nSection 10.2.\nUsing Proposition 11.1, together with the definition of the trace (se e for instance [RS80, Theorem\nVI.18]) we have\n0 = tr{m/a\\}bracketle{tb/a\\}bracketri}htΛMΛ}=/summationdisplay\nj∈N/parenleftBig\nm/a\\}bracketle{tb/a\\}bracketri}htΛMΛ˜ej\nΛ,˜ej\nΛ/parenrightBig\nL2(TΛ)=/summationdisplay\nj∈Nγj\nΛ/parenleftBig\n/a\\}b∇acketle{tb/a\\}b∇acket∇i}htΛ˜ej\nΛ,˜ej\nΛ/parenrightBig\nL2(TΛ).\nSince all terms in this sum are nonnegative (because both γj\nΛand/a\\}b∇acketle{tb/a\\}b∇acket∇i}htΛare), we deduce that for all\nj∈N,\nγj\nΛ/parenleftBig\n/a\\}b∇acketle{tb/a\\}b∇acket∇i}htΛ˜ej\nΛ,˜ej\nΛ/parenrightBig\nL2(TΛ)= 0.\nSuppose that γj\nΛ/\\e}atio\\slash= 0 for some j∈N. Then,/parenleftBig\n/a\\}b∇acketle{tb/a\\}b∇acket∇i}htΛ˜ej\nΛ,˜ej\nΛ/parenrightBig\nL2(TΛ)= 0 where /a\\}b∇acketle{tb/a\\}b∇acket∇i}htΛis nonnegative and\nnot identically zero on TΛ. This yields ˜ ej\nΛ= 0 on the nonempty open set {/a\\}b∇acketle{tb/a\\}b∇acket∇i}htΛ>0}. Using a unique\ncontinuation property for eigenfunctions of the Laplace operato r onTΛ, we finally obtain that the\neigenfunction ˜ ej\nΛvanishes identically on TΛ. This is absurd, and thus we must have γj\nΛ= 0 for all\nj∈N, so thatMΛ= 0. SinceρΛ∈ M+(T∗TΛ⊥;L1(TΛ)), this directly gives ρΛ= 0.\nNext, we prove that νΛ= 0. This is a consequence of the additional propagation law of νΛwith\nrespect to the flow φ1\nτ(see Section 10.1). Indeed the torus TΛhas dimension one, ( φ1\nτ)∗νΛ=νΛ\n(according to Proposition 10.1) and, using Proposition 11.1, νΛvanishes on the (nonempty) set\n{/a\\}b∇acketle{tb/a\\}b∇acket∇i}htΛ>0}×R2×SΛ(with{/a\\}b∇acketle{tb/a\\}b∇acket∇i}htΛ>0}clearly satisfying GCC on TΛ). Hence,νΛ= 0.\nTo conclude the proof of Proposition 11.2, it only remains to use the d ecomposition formula (9.7)\nwhich directly yields µ|T2×Λ⊥= 0.\n12 Proof of Proposition 8.2\nIn this section, we prove Proposition 8.2. For this, we consider two- microlocal semiclassical measures\nat the scale hα. The setting is close to that of [FK05].\nWe shall see that the concentration rate of the sequence vhtowards the direction Λ⊥is of the\nformhαfor allα≤3+δ\n4.\nFirst, Lemma 7.3 yields µ|T2×Λ⊥=/a\\}b∇acketle{tµ/a\\}b∇acket∇i}htΛ|T2×Λ⊥(see also Remark 7.5), i.e.\n/a\\}b∇acketle{tµ|T2×Λ⊥,a/a\\}b∇acket∇i}htM(T∗T2),C0c(T∗T2)=/a\\}b∇acketle{tµ|T2×Λ⊥,/a\\}b∇acketle{ta/a\\}b∇acket∇i}htΛ/a\\}b∇acket∇i}htM(T∗T2),C0c(T∗T2),\nand it suffices to characterize the action of µ|T2×Λ⊥on Λ⊥-invariant symbols. Recall that, for all\na∈C∞\nc(T∗T2),\n/a\\}b∇acketle{tµ,a/a\\}b∇acket∇i}htM(T∗T2),C0\nc(T∗T2)= lim\nh→0(Oph(a)vh,vh)L2(T2).\nIn this section, the assumption√\nb∈C∞(T2) is used in an essential way for the propagation\nresult of Lemma 12.2 below. Like in (9.1) and (9.2), let us define :\n/angbracketleftBig\nVh,Λ\nR,a/angbracketrightBig\nS1\nΛ′,S1\nΛ:=/angbracketleftbigg\nVh,/parenleftbigg\n1−χ/parenleftbigg|PΛξ|\nRh/parenrightbigg/parenrightbigg\na/parenleftbigg\nx,ξ,PΛξ\nh/parenrightbigg/angbracketrightbigg\nD′(T∗T2),C∞c(T∗T2),(12.1)\n/angbracketleftbig\nVh\nR,Λ,a/angbracketrightbig\nS1\nΛ′,S1\nΛ:=/angbracketleftbigg\nVh,χ/parenleftbigg|PΛξ|\nRh/parenrightbigg\na/parenleftbigg\nx,ξ,PΛξ\nh/parenrightbigg/angbracketrightbigg\nD′(T∗T2),C∞\nc(T∗T2), (12.2)\n33fora∈S1\nΛ.\nWe takeR=R(h) =h−(1−α)for someα∈(0,1), so thatRh=hα. The proof of Proposition 9.2\napplies verbatim and shows the existence of a subsequence ( h,vh) and a nonnegative measure νΛ\nα∈\nM+(T∗T2×SΛ) such that, for all a∈S1\nΛ, we have\nlim\nh→0/angbracketleftBig\nVh,Λ\nR(h),a/angbracketrightBig\nS1\nΛ′,S1\nΛ=/angbracketleftbigg\nνΛ\nα,ahom/parenleftbigg\nx,ξ,η\n|η|/parenrightbigg/angbracketrightbigg\nM(T∗T2×SΛ),C0\nc(T∗T2×SΛ).\nProposition 12.1. LetR(h) =h−(1−α)withα≤3+δ\n4. Then\nνΛ\nα|T2×(Λ⊥\\{0})×SΛ= 0\nThe proof of Proposition 12.1 relies on the following propagation resu lt.\nLemma 12.2. Forα≤3+δ\n4the measure νΛ\nαisφ0\nτ- andφ1\nτ-invariant, i.e.\n(φ0\nτ)∗νΛ\nα=νΛ\nαand(φ1\nτ)∗νΛ\nα=νΛ\nα,for everyτ∈R.\nThe proof is very similar to that of Proposition 10.1 but does not use A ssumption (2.12).\nProof.The proof of φ0\nτ-invariance is strictly identical to what has been done for Propositio n 10.1 and\nthus we focus on the φ1\nτ-invariance. Equation (10.5) still holds with R(h) =h−(1−α), now reading\n/angbracketleftbigg\nVh,Λ\nR(h),η\n|η|·∂xa/angbracketrightbigg\nS1\nΛ′,S1\nΛ=i\n2h2/parenleftBig\nAR(h)vh,Ph\nbvh/parenrightBig\nL2(T2)−i\n2h2/parenleftBig\nAR(h)Ph\nbvh,vh/parenrightBig\nL2(T2)\n−1\n2h/parenleftBig\nAR(h)vh,bvh/parenrightBig\nL2(T2)−1\n2h/parenleftBig\nAR(h)bvh,vh/parenrightBig\nL2(T2)\nwhereARwas defined in (10.3). Using /ba∇dblPh\nbvh/ba∇dblL2(T2)=o(h1+δ) together with the boundedness of\nAR(h), it follows that\nlim\nh→0/angbracketleftbigg\nVh,Λ\nR(h),η\n|η|·∂xa/angbracketrightbigg\nS1\nΛ′,S1\nΛ= lim\nh→0/parenleftBig\n−1\n2h/parenleftbig\nAR(h)vh,bvh/parenrightbig\nL2(T2)−1\n2h/parenleftbig\nAR(h)bvh,vh/parenrightbig\nL2(T2)/parenrightBig\n.\nRecall from (4.22) that /ba∇dbl√\nbvh/ba∇dblL2(T2)=o(h1+δ\n2).In addition, it follows from standard microlocal\ncalculus that\n[AR(h),√\nb] =OL(L2)(R(h)−2).\nWe can thus write\n/angbracketleftbigg\nVh,Λ\nR(h),η\n|η|·∂xa/angbracketrightbigg\nS1\nΛ′,S1\nΛ=o(1)−1\nh/parenleftbig\nAR(h)√\nbvh,√\nbvh/parenrightbig\nL2(T2)+1\n2h/parenleftbig√\nb[AR(h),√\nb]vh,vh/parenrightbig\nL2(T2)\n+1\n2h/parenleftbig\n[√\nb,AR(h)]√\nbvh,vh/parenrightbig\nL2(T2)\n=o(1)+o(R(h)−2h−1+δ\n2) =o(1)+o(h3\n2+δ\n2−2α),\nwhich vanishes if we take α≤3+δ\n4.\nProof of Proposition 12.1. To prove Proposition 12.1, we first note that\n/angbracketleftbig\nνΛ\nα|T2×(Λ⊥\\{0})×SΛ,/a\\}b∇acketle{tb/a\\}b∇acket∇i}htΛ/angbracketrightbig\nMc(T∗T2×SΛ),C0(T∗T2×SΛ)= 0,\nsinceνΛ\nαis (φ0\nτ)-invariant and/angbracketleftbig\nνΛ\nα,b/angbracketrightbig\nMc(T∗T2×SΛ),C0(T∗T2×SΛ)= 0. Then, the φ1\nτ-invariance of νΛ\nα\nimplies that νΛ\nα|T2×(Λ⊥\\{0})×SΛvanishes.\n34Proof of Proposition 8.2. Proposition 12.1 implies that\n/a\\}b∇acketle{tµ|T2×Λ⊥,a/a\\}b∇acket∇i}htM(T∗T2),C0c(T∗T2)= lim\nh→0/parenleftbigg\nOph/parenleftbigg\nχ/parenleftbigg|PΛξ|\nhα/parenrightbigg\na(x,ξ)/parenrightbigg\nvh,vh/parenrightbigg\nL2(T2)\nfor allα≤3+δ\n4anda∈C∞\nc(T∗T2). The same holds if we replace χbyχ2:\n/a\\}b∇acketle{tµ|T2×Λ⊥,a/a\\}b∇acket∇i}htM(T∗T2),C0c(T∗T2)= lim\nh→0/parenleftbigg\nOph/parenleftbigg\nχ2/parenleftbigg|PΛξ|\nhα/parenrightbigg\na(x,ξ)/parenrightbigg\nvh,vh/parenrightbigg\nL2(T2).\nSince\nOph/parenleftbigg\nχ2/parenleftbigg|PΛξ|\nhα/parenrightbigg\na(x,ξ)/parenrightbigg\n= Oph/parenleftbigg\nχ/parenleftbigg|PΛξ|\nhα/parenrightbigg/parenrightbigg\nOph(a)Oph/parenleftbigg\nχ/parenleftbigg|PΛξ|\nhα/parenrightbigg/parenrightbigg\n+O(h1−α),(12.3)\nwe obtain\n/a\\}b∇acketle{tµ|T2×Λ⊥,a/a\\}b∇acket∇i}htM(T∗T2),C0c(T∗T2)= lim\nh→0/parenleftbigg\nOph(a)Oph/parenleftbigg\nχ/parenleftbigg|PΛξ|\nhα/parenrightbigg/parenrightbigg\nvh,Oph/parenleftbigg\nχ/parenleftbigg|PΛξ|\nhα/parenrightbigg/parenrightbigg\nvh/parenrightbigg\nL2(T2),\nfor allα≤3+δ\n4anda∈C∞\nc(T∗T2).\n13 Proof of Proposition 8.3: existence of the cutoff function\nGiven a constant c0>0, we define the following subsets of T2:\nEh=/a\\}b∇acketle{t{b>c0h}/a\\}b∇acket∇i}htΛ,Fh=/angbracketleftBigg/uniondisplay\nx∈{b>c0h}B(x,(c0h)2ε)/angbracketrightBigg\nΛ=/uniondisplay\nx∈EhB(x,(c0h)2ε),Gh=Fh\\Eh,\nwhere forU⊂T2, we denote /a\\}b∇acketle{tU/a\\}b∇acket∇i}htΛ:=/uniontext\nτ∈R{U+τσ}for someσ∈Λ⊥\\{0}. Remark that Eh⊂Fh\nand that T2=Eh∪Gh∪(T2\\Fh). Note also that the sets Eh,Fhare non-empty for hsmall enough,\nand that Ghis non empty (for hsmall enough) as soon as bvanishes somewhere on T2(this condition\nis assumed here since otherwise, GCC is satisfied).\nIn this section, we construct the cutoff function χΛ\nhneeded to prove the propagation results of\nSection 10. In particular, this function will be Λ⊥-invariant and will satisfy χΛ\nh= 0 on EhandχΛ\nh= 1\nonT2\\Fh.\nThe proof of Proposition 8.3 relies on three key lemmata. The first ke y lemma is a precised\nversion of Proposition 6.1 concerning the localization in T∗T2of the semiclassical measure µ. It is\nan intermediate step towards the propagation result stated in Lem ma 13.2.\nLemma 13.1. For anyχ∈C∞\nc(R), such that χ= 1in a neighbourhood of the origin, for all\na∈C∞\nc(T∗T2), andγ≤3+δ\n2, we have\n(Oph(a)wh,wh)L2(T2)=/parenleftbigg\nOph(a)Oph/parenleftbigg\nχ/parenleftbigg|ξ|2−1\nhγ/parenrightbigg/parenrightbigg\nwh,wh/parenrightbigg\nL2(T2)+o(h3+δ\n2−γ)/ba∇dblOph(a)/ba∇dblL(L2),\n(13.1)\nFor alla∈C∞\nc(T∗T2)and allτ∈R,\n(Oph(a◦φτ)wh,wh)L2(T2)= (Oph(a)wh,wh)L2(T2)+o(τh1+δ\n2)/ba∇dblOph(a◦φt)/ba∇dblL∞(0,τ;L(L2(T2)))\nIn this statement, we used the notation\n/ba∇dblOph(a◦φt)/ba∇dblL∞(0,τ;L(L2(T2))):= sup\nt∈(0,τ)/ba∇dblOph(a◦φt)/ba∇dblL(L2(T2)).\nIn turn, this lemma implies the following transport property.\n35Lemma 13.2. Suppose that the coefficients α,εsatisfy\n0<10ε≤α,andα+2ε≤1. (13.2)\nThen, for any time τ∈Runiformly bounded with respect to h, and any h-family of functions\nψ=ψh∈C∞\nc(T2)satisfying\n/ba∇dbl∂k\nxψ/ba∇dblL∞(T2)≤Ckh−2ε|k|,for allk∈N2, (13.3)\nwe have,\n(ψ(s,y)wh,wh)L2(T2)= (ψ(s+τ,y)wh,wh)L2(T2)+(ψ(s−τ,y)wh,wh)L2(T2)\n+O(hα−10ε)+O(h1−α−2ε)+o(h1+δ\n2), (13.4)\nwhere the coordinates (s,y)are the ones introduced in Section 7.3.\nIn view of Proposition 8.3, this lemma will allow us to propagate the smalln ess of the sequence\nwhabove the set {b>c0h}to all Eh.\nThe third key lemma states a property of the damping function b, as a consequence of Assump-\ntion 2.12.\nLemma 13.3. There exists b0=b0(ε)>0such that for all x∈T2satisfying 0<b(x)<b0and for\nallz∈B(x,b(x)2ε), we haveb(z)≥b(x)\n2.\nWith these three lemmata, we are now able to prove Proposition 8.3.\nProof of Proposition 8.3. In the coordinates ( s,y) of Section 7.3, we can write\nEh=TΛ⊥×Eh,Fh=TΛ⊥×Fh,withEh⊂Fh⊂TΛ.\nHere,Fhis a union of intervals and has uniformly bounded total length. We can hence cover Fhwith\nC1h−2εsubsets of length of order ( c0h)2ε/2, overlapping on intervals of length of order ( c0h)2ε/10.\nAssociated to this covering, we denote by ( ψj)j∈{1,...,J},J=J(h), a smooth partition of unity on\nEh, satisfying moreover\n•ψj∈C∞\nc(Fh);\n•/summationtextJ\nj=1ψj(y) = 1 fory∈Eh;\n• /ba∇dbl∂m\nyψj/ba∇dblL∞(TΛ)≤Cmh−2εm, for allm∈N;\n•J=J(h)≤Ch−2ε.\nSimilarly, we cover TΛ⊥withC2h−2εsubsets of length of order ( c0h)2ε/2, overlapping on intervals\nof length of order ( c0h)2ε/10, and define ( ψk)k∈{1,...,K}an associated partition of unity on TΛ⊥\nsatisfying\n•ψk∈C∞\nc(TΛ⊥);\n•/summationtextK\nk=1ψk(s) = 1 fors∈TΛ⊥;\n• /ba∇dbl∂m\nsψk/ba∇dblL∞(TΛ⊥)≤Cmh−2εm, for allm∈N;\n•K=K(h)≤Ch−2ε,\n•for anyk,k0∈ {1,...,K}2, there exists τksatisfying |τk| ≤Length(TΛ⊥)≤Candψk(s+τk) =\nψk0(s).\n36We set\nψkj(s,y) :=ψk(s)ψj(y),andχΛ\nh(s,y) = 1−J/summationdisplay\nj=1K/summationdisplay\nk=1ψkj(s,y)∈C∞(T2),\nwhich satisfies ∂sχΛ\nh(s,y) = 0, i.e.χΛ\nhis Λ⊥-invariant, together with\n•χΛ\nh= 0 on Ehand henceb≤c0hon supp(χΛ\nh);\n•χΛ\nh= 1 onT2\\Fh;\n•χΛ\nh∈[0,1] onGh, with|∂yχΛ\nh| ≤Ch−2εand|∂2\nyχΛ\nh| ≤Ch−4ε.\nTo conclude the proof of Proposition 8.3, it remains to check Item 2 ( /ba∇dbl(1−χΛ\nh)wh/ba∇dblL2(T2)=o(1)),\nItem 4 (/ba∇dbl∂yχΛ\nhwh/ba∇dblL2(T2)=o(1)) and Item 5 ( /ba∇dbl∂2\nyχΛ\nhwh/ba∇dblL2(T2)=o(1)).\nNow, letus fix j0∈ {1,...,J}. Becauseofthe definition ofthe set Eh, there exists k0∈ {1,...,K}\nandx0∈ {b > c0h}such that supp( ψk0j0)⊂B(x0,(c0h)2ε). According to Lemma 13.3, we have\nB(x0,(c0h)2ε)⊂ {b>c0h\n2}, so that supp( ψk0j0)⊂ {b>c0h\n2}. This yields\nc0h\n2(ψk0j0wh,wh)L2(T2)≤(bψk0j0wh,wh)L2(T2)=o(h1+δ),\nand hence ( ψk0j0wh,wh)L2(T2)=o(hδ). Moreover, for any k∈ {1,...,K}, there exists τksatisfying\n|τk| ≤C2with\nψkj0(s+τk,y) =ψk0j0(s,y).\nHence, using (13.4), we obtain\no(hδ) = (ψk0j0(s,y)wh,wh)L2(T2)= (ψkj0(s+τk,y)wh,wh)L2(T2)\n= (ψkj0(s+2τk,y)wh,wh)L2(T2)+(ψkj0(s,y)wh,wh)L2(T2)\n+O(hα−10ε)+O(h1−α−2ε)+o(h1+δ\n2). (13.5)\nSince both termsonthe righthand-sidearenonnegative,this implies (ψkj0(s,y)wh,wh)L2(T2)=o(hδ)\nas soon as \n\nα−10ε>δ,\n1−α−2ε>δ,\n1+δ\n2≥δ,\n(which implies (13.2)). From now on we will take δ= 8ε(this choice is explained in the following\nlines). The existence of αsatisfying this condition together with (8.1) and α<3/4, is equivalent to\nhavingε<1\n76.\nTo conclude the proof of Proposition 8.3, we first compute\n((1−χΛ\nh)wh,wh)L2(T2)=J/summationdisplay\nj=1K/summationdisplay\nk=1(ψkjwh,wh)L2(T2)=Ch−4εo(hδ) =o(1),\nsinceδ≥4ε. This proves Item 2. Next, we have by construction supp( ∂2\nyχΛ\nh)⊂supp(∂yχΛ\nh)⊂\nGhwith/ba∇dbl∂yχΛ\nh/ba∇dblL∞(T2)=O(h−2ε),/ba∇dbl∂2\nyχΛ\nh/ba∇dblL∞(T2)=O(h−4ε). Hence, covering supp( ∂yχΛ\nh)) by\nballs of radius ( c0h)2εand using a propagation argument similar to (13.5) shows that we hav e\n/ba∇dblwh/ba∇dblL2(supp(∂yχΛ\nh))=o(hδ\n2). We thus obtain\n/ba∇dbl∂yχΛ\nhwh/ba∇dblL2(T2)=o(hδ\n2−2ε) =o(1),/ba∇dbl∂2\nyχΛ\nhwh/ba∇dblL2(T2)=o(hδ\n2−4ε) =o(1),\n(sinceδ≥8ε) which concludes the proof of Items 4 and 5, and that of Propositio n 8.3.\n37To conclude this section, it remains to prove Lemmata 13.2, 13.1 and 1 3.3. In the following\nproofs, we shall systematically write ηin place of PΛξandσin place of (1 −PΛ)ξto lighten the\nnotation. Hence, ξ∈R2is decomposed as ξ=η+σwithη∈ /a\\}b∇acketle{tΛ/a\\}b∇acket∇i}htandσ∈Λ⊥, in accordance to\nSection 7.3.\nProof of Lemma 13.2 from Lemma 13.1. First, given a function ψ∈C∞\nc(T2) satisfying (13.3), we\nhave,\n(ψwh,wh)L2(T2)= (Oph(ψ◦φτ)wh,wh)L2(T2)+o(τh1+δ\n2)/ba∇dblOph(ψ◦φt)/ba∇dblL∞(0,τ;L(L2))\n=/parenleftbigg\nOph(ψ◦φτ)Oph/parenleftbigg\nχ/parenleftbigg|ξ|2−1\nhγ/parenrightbigg/parenrightbigg\nOph/parenleftBig\nχ/parenleftBigη\n2hα/parenrightBig/parenrightBig\nwh,wh/parenrightbigg\nL2(T2)\n+/parenleftbig\no(τh1+δ\n2)+o(τh3+δ\n2−γ)/parenrightbig\n/ba∇dblOph(ψ◦φt)/ba∇dblL∞(0,τ;L(L2)),\nwhen using Lemma 13.1 together with Oph/parenleftbig\nχ/parenleftbigη\n2hα/parenrightbig/parenrightbig\nwh=wh. Next, the pseudodifferential calculus\nyields\n(ψwh,wh)L2(T2)=/parenleftbigg\nOph/parenleftbigg\nψ◦φτχ/parenleftbigg|ξ|2−1\nhγ/parenrightbigg\nχ/parenleftBigη\n2hα/parenrightBig/parenrightbigg\nwh,wh/parenrightbigg\nL2(T2)+O(h2−γ−2ε)+O(h1−α−2ε)\n+/parenleftbig\no(τh1+δ\n2)+o(τh3+δ\n2−γ)/parenrightbig\n/ba∇dblOph(ψ◦φt)/ba∇dblL∞(0,τ;L(L2)). (13.6)\nA particular feature of the Weyl quantization in the Euclidean settin g is that the Egorov theorem\nprovides an exact formula (see for instance [DS99]): Oph(ψ◦φt) =e−ith∆\n2Oph(ψ)eith∆\n2, so that\n/ba∇dblOph(ψ◦φt)/ba∇dblL∞(0,τ;L(L2))≤C0uniformly with respect to h. Now, remark that the cutoff function\nχ/parenleftbigη\n2hα/parenrightbig\nχ/parenleftBig\n|ξ|2−1\nhγ/parenrightBig\ncan be decomposed (for hsmall enough) as\nχ/parenleftBigη\n2hα/parenrightBig\nχ/parenleftbigg|ξ|2−1\nhγ/parenrightbigg\n=χ/parenleftBigη\n2hα/parenrightBig/parenleftbig\n˜χh\nη(σ)+ ˜χh\nη(−σ)/parenrightbig\nfor some nonnegative function ˜ χh\nηsuch that (σ,η)/ma√sto→˜χh\nη(σ)∈C∞\nc(R2), such that ˜ χh\nη(σ) =χ/parenleftBig\n|ξ|2−1\nhγ/parenrightBig\nforη∈suppχ/parenleftbig·\n2hα/parenrightbig\nandσ>0, and ˜χh\nη(σ) = 0 forη /∈suppχ/parenleftbig·\n2hα/parenrightbig\norσ≤0.\nChoosingγ=α, we have in particular\n|σ−1| ≤Chαon supp/parenleftBig\nχ/parenleftBigη\n2hα/parenrightBig\n˜χh\nη(σ)/parenrightBig\n.\nNext, we recallthat ψ◦φτ(s,y,σ,η) =ψ(s+τσ,y+τη), and we focus on the first term (corresponding\ntoσ>0) in the right-hand side of the identity\nχ/parenleftbigg|ξ|2−1\nhα/parenrightbigg\nχ/parenleftBigη\n2hα/parenrightBig\nψ◦φτ=χ/parenleftBigη\n2hα/parenrightBig/parenleftbig\n˜χh\nη(σ)+ ˜χh\nη(−σ)/parenrightbig\nψ◦φτ. (13.7)\nWe set\nζ(1)\nτ(s,y,σ,η) =χ/parenleftBigη\n2hα/parenrightBig\n˜χh\nη(σ)ψ(s+τσ,y+τη),andζ(2)\nτ(s,y,σ,η) =χ/parenleftBigη\n2hα/parenrightBig\n˜χh\nη(σ)ψ(s+τ,y),\nandwewanttocompareOph(ζ(1)\nτ)andOph(ζ(2)\nτ). Forthis, letusestimate, formultiindices ℓ,m∈N2,\n/vextendsingle/vextendsingle/vextendsingle∂ℓ\n(s,y)∂m\n(σ,η)/parenleftBig\nζ(2)\nτ−ζ(1)\nτ/parenrightBig\n(s,y,σ,η)/vextendsingle/vextendsingle/vextendsingle\n≤Cm/summationdisplay\nν≤m/vextendsingle/vextendsingle/vextendsingle∂m−ν\n(σ,η)/parenleftBig\nχ/parenleftBigη\n2hα/parenrightBig\n˜χh\nη(σ)/parenrightBig\n∂ℓ\n(s,y)∂ν\n(σ,η)(ψ(s+τσ,y+τη)−ψ(s+τ,y))/vextendsingle/vextendsingle/vextendsingle.(13.8)\nOn the one hand, we have\n/vextendsingle/vextendsingle/vextendsingle∂m−ν\n(σ,η)/parenleftBig\nχ/parenleftBigη\n2hα/parenrightBig\n˜χh\nη(σ)/parenrightBig/vextendsingle/vextendsingle/vextendsingle≤Cm,νh−α|m−ν|. (13.9)\n38On the other hand, for |ν|>0 we can also write\n/vextendsingle/vextendsingle/vextendsingle∂ℓ\n(s,y)∂ν\n(σ,η)(ψ(s+τσ,y+τη)−ψ(s+τ,y))/vextendsingle/vextendsingle/vextendsingle=/vextendsingle/vextendsingle/vextendsingle∂ℓ\n(s,y)∂ν\n(σ,η)ψ(s+τσ,y+τη)/vextendsingle/vextendsingle/vextendsingle\n≤Cℓ,ν|τ||ν|h−2ε(|ℓ|+|ν|)≤Cℓ,νh−2ε(|ℓ|+|ν|),\nsince|τ| ≤C.\nFinally, for |ν|= 0, we apply the mean value theorem to the function\n(σ,η)/ma√sto→∂ℓ\n(s,y)ψ(s+τσ,y+τη)\nand write\n/vextendsingle/vextendsingle/vextendsingle∂ℓ\n(s,y)(ψ(s+τσ,y+τη)−ψ(s+τ,y))/vextendsingle/vextendsingle/vextendsingle\n≤(|η|+|σ−1|) sup\nT∗T2/vextendsingle/vextendsingle/vextendsingle∇(σ,η)∂ℓ\n(s,y)(ψ(s+τσ,y+τη))/vextendsingle/vextendsingle/vextendsingle.\nWith (13.3), this yields\n/vextendsingle/vextendsingle/vextendsingle∂ℓ\n(s,y)(ψ(s+τσ,y+τη)−ψ(s+τ,y))/vextendsingle/vextendsingle/vextendsingle≤(|η|+|σ−1|)Cℓh−2ε|ℓ||τ|h−2ε\n≤(|η|+|σ−1|)Cℓh−2ε(|ℓ|+1), (13.10)\nfor|τ| ≤C.\nUsingnowthat |η| ≤Chαand|σ−1| ≤Chαonsupp/parenleftbig\nχ/parenleftbigη\n2hα/parenrightbig\n˜χh\nη(σ)/parenrightbig\n,andcombining(13.8),(13.9)\nand (13.10), we obtain, for all m∈N2,ℓ∈N2and 0<h≤h0sufficiently small,\nh|m|/vextendsingle/vextendsingle/vextendsingle∂ℓ\n(s,y)∂m\n(σ,η)/parenleftBig\nζ(2)\nτ−ζ(1)\nτ/parenrightBig\n(s,y,σ,η)/vextendsingle/vextendsingle/vextendsingle≤Cℓ,mhα−2ε(|ℓ|+1)h|m|h−α|m|\n+Cℓ,m/summationdisplay\n0<ν≤mh|m|h−2ε(|ℓ|+|ν|)h−α|m−ν|\n≤Cℓ,m/parenleftBig\nh(1−α)|m|hα−2ε(|ℓ|+1)+|m|h|m|(1−α)h−2ε|ℓ|hα−2ε/parenrightBig\n≤Cℓ,mhα−2ε(|ℓ|+1).\nUsing a precised version of the Calder´ on-Vaillancourt theorem, as presented in Theorem A.1 below\n(in which only |ℓ|= 4 derivations are needed with respect to xin dimension two), we obtain\nOph(ζ(2)\nτ) = Oph(ζ(1)\nτ)+OL(L2)(hα−10ε).\nSimilarly, we have\nOph/parenleftBig\nχ/parenleftBigη\n2hα/parenrightBig\n˜χh\nη(−σ)ψ(s+τσ,y+τη)/parenrightBig\n= Oph/parenleftBig\nχ/parenleftBigη\n2hα/parenrightBig\n˜χh\nη(−σ)ψ(s−τ,y)/parenrightBig\n+OL(L2)(hα−10ε).\nComing back to (13.6) and using (13.7), we finally obtain, for all |τ| ≤C,\n(ψwh,wh)L2(T2)=/parenleftBig\nOph/parenleftBig\nχ/parenleftBigη\n2hα/parenrightBig\n˜χh\nη(σ)ψ(s+τ,y)/parenrightBig\nwh,wh/parenrightBig\nL2(T2)\n+/parenleftBig\nOph/parenleftBig\nχ/parenleftBigη\n2hα/parenrightBig\n˜χh\nη(−σ)ψ(s−τ,y)/parenrightBig\nwh,wh/parenrightBig\nL2(T2)\n+O(hα−10ε)+O(h1−α−2ε)+o(h1+δ\n2)+o(h3+δ\n2−α).\nWith the pseudodifferential calculus, this yields (13.4), which conclud es the proofofLemma 13.2.\nProof of Lemma 13.1. Here, we only have to make more precise some arguments in the proo f of\nLemma 6.1. Recall that according to Lemma 8.1, whsatisfiesPh\nbwh=o(h2+δ).\n39First, we take χ∈C∞\nc(R), such that χ= 1 in a neighbourhood of the origin. Hence,1−χ(r)\nr∈\nC∞(R) and we have the exact composition formula\nOph/parenleftbigg\n1−χ/parenleftbigg|ξ|2−1\nhγ/parenrightbigg/parenrightbigg\n= Oph/parenleftbigg/parenleftbigg\n1−χ/parenleftbigg|ξ|2−1\nhγ/parenrightbigg/parenrightbigghγ\n|ξ|2−1/parenrightbiggPh\n0\nhγ,\nsince both operators are Fourier multipliers. Moreover, Oph/parenleftBig/parenleftBig\n1−χ/parenleftBig\n|ξ|2−1\nhγ/parenrightBig/parenrightBig\nhγ\n|ξ|2−1/parenrightBig\nis uniformly\nbounded as an operator of L(L2(T2)). As a consequence, we have\n/parenleftbigg\nOph(a)Oph/parenleftbigg\n1−χ/parenleftbigg|ξ|2−1\nhγ/parenrightbigg/parenrightbigg\nwh,wh/parenrightbigg\nL2(T2)\n=/parenleftbigg\nOph(a)Oph/parenleftbigg/parenleftbigg\n1−χ/parenleftbigg|ξ|2−1\nhγ/parenrightbigg/parenrightbigghγ\n|ξ|2−1/parenrightbiggPh\n0\nhγwh,wh/parenrightbigg\nL2(T2)\n=/parenleftbigg\nAPh\nb\nhγwh,wh/parenrightbigg\nL2(T2)−/parenleftbigg\nAihb\nhγwh,wh/parenrightbigg\nL2(T2),\nwhereA= Oph(a)Oph/parenleftBig/parenleftBig\n1−χ/parenleftBig\n|ξ|2−1\nhγ/parenrightBig/parenrightBig\nhγ\n|ξ|2−1/parenrightBig\nis bounded on L2(T2). UsingPh\nbwh=o(h2+δ)\nand (bwh,wh)L2(T2)=o(h1+δ), this gives\n/parenleftbigg\nOph(a)Oph/parenleftbigg\n1−χ/parenleftbigg|ξ|2−1\nhγ/parenrightbigg/parenrightbigg\nwh,wh/parenrightbigg\nL2(T2)=o(h3+δ\n2−γ)/ba∇dblOph(a)/ba∇dblL(L2),\nwhich in turn implies (13.1).\nNext, Identity (6.2) yields, for all a∈C∞\nc(T2),\n(Oph(ξ·∂xa)wh,wh)L2(T2)=i\n2h/parenleftbig\nOph(a)wh,Ph\nbwh/parenrightbig\nL2(T2)−i\n2h/parenleftbig\nOph(a)Ph\nbwh,wh/parenrightbig\nL2(T2)\n−1\n2(Oph(a)wh,bwh)L2(T2)−1\n2(Oph(a)bwh,wh)L2(T2)\n=o(h1+δ)/ba∇dblOph(a)/ba∇dblL(L2)+o(h1+δ\n2)/ba∇dblOph(a)/ba∇dblL(L2),\nas a consequence of Ph\nbwh=o(h2+δ) and (bwh,wh)L2(T2)=o(h1+δ). Applying this identity to a◦φt\nin place ofa, and integrating on t∈[0,τ] finally gives\n(Oph(a◦φτ)wh,wh)L2(T2)= (Oph(a)wh,wh)L2(T2)+o(τh1+δ\n2)/ba∇dblOph(a◦φt)/ba∇dblL∞(0,τ;L(L2)),\nwhich concludes the proof of Lemma 13.1.\nProof of Lemma 13.3. Here,B:=B(x,b(x)2ε) denotes the euclidian ball in T2centered at xof\nradiusb(x)2ε. Setting\nM:= sup\nz∈Bb(z), m:= inf\nz∈Bb(z),\nwe have\n|∇b(z)| ≤Cεb1−ε(z)≤CεM1−ε,for allz∈B,\nas a consequence of Assumption 2.12. Moreover, the mean value th eorem yields\nb(x)−CεM1−εb(x)2ε≤b(z)≤b(x)+CεM1−εb(x)2ε,forz∈B,\nand, in particular,\nm≥b(x)−CεM1−εb(x)2ε,andM≤b(x)+CεM1−εb(x)2ε. (13.11)\n40Now, defining f(M) :=b(x) +CεM1−εb(x)2ε, we see that fis a strictly concave function with\nf(0) =b(x)>0. There exists a unique M0∈R+satisfyingf(M0) =M0. Moreover, we have\nM≤f(M) if and only if M≤M0. Takingb0sufficiently small so that b0+Cεb1+ε2\n0≤b1−ε\n0,\nwe obtainf(b(x)1−ε)≤b(x)1−ε. In particular, this gives M0≤b(x)1−εand hence M≤b(x)1−ε\naccording to the second estimate of (13.11). Coming back to the fir st estimate of (13.11), this yields\nm≥b(x)−Cεb(x)(1−ε)2b(x)2ε=b(x)−Cεb(x)1+ε2.\nTakingb0sufficiently small so that b0−Cεb1+ε2\n0≥b0\n2, we obtain m≥b(x)\n2, which concludes the proof\nof Lemma 13.3.\nPart IV\nAna priori lower bound for decay rates\non the torus: proof of Theorem 2.5\nUnder the assumption\n{b>0}∩{x0+τξ0,τ∈R}=∅, (13.12)\nfor some (x0,ξ0)∈T∗T2,ξ0/\\e}atio\\slash= 0, we construct in this section a constant κ0>0 and a sequence\n(ϕn)n∈NofO(1)-quasimodes in the limit n→+∞for the family of operators P(inκ0).\nWeusethenotationintroducedinSections7.1and9. First, notetha t, asaconsequenceof (13.12),\nξ0is necessarily a rational direction, and the set {x0+τξ0,τ∈R}is a one-dimensional subtorus of\nT2, given by\n{x0+τξ0,τ∈R}={x0+τξ0,τ∈R}=x0+TΛ⊥\nξ0,with Λ ξ0∈ P.\nLetχ∈C∞\nc(T2) such that χhas onlyx-Fourier modes in Λ ξ0,χ= 0 on a neighbourhood of\n{b>0}andχ= 1 onx0+TΛ⊥\nξ0.\nFrom Assumption (13.12), we have rk(Λ ξ0) = 1, so that one can find k∈Λ⊥\nξ0∩Z2\\{0}. Besides,\nfor alln∈Nwe havenk∈Λ⊥\nξ0∩Z2\\{0}.\nWe then define the sequence of quasimodes (ϕn)n∈Nby\nϕn(x) =χ(x)eink·x, n∈N, x∈T2.\nWe haveϕn∈C∞(T2), together with the decoupling\nϕn◦πΛξ0(s,y) =χ(y)eink·s, n∈N,(s,y)∈TΛ⊥\nξ0×TΛξ0.\nThis yields\n−/parenleftbig\nTΛξ0∆T∗\nΛξ0/parenrightbig\nϕn◦πΛξ0(s,y) =−/parenleftbig\n∆Λξ0+∆Λ⊥\nξ0/parenrightbig\nϕn◦πΛξ0(s,y)\n=−eink·s∆Λξ0χ(y)+n2|k|2χ(y)eink·s.\nMoerover,bϕn= 0, according to their respective supports. Hence, recalling that P(in|k|) =−∆−\nn2|k|2+in|k|b(x), we have\n/parenleftbig\nTΛξ0P(in|k|)T∗\nΛξ0/parenrightbig\nϕn◦πΛξ0=−eink·s∆Λξ0χ(y),\nand\n/ba∇dblP(in|k|)ϕn/ba∇dblL2(T2)=/ba∇dbl/parenleftbig\nTΛξ0P(in|k|)T∗\nΛξ0/parenrightbig\nϕn◦πΛξ0/ba∇dblL2(TΛ⊥\nξ0×TΛξ0)=C0/ba∇dbl∆Λξ0χ/ba∇dblL2(TΛξ0).\n41Since we also have /ba∇dblϕn/ba∇dblL2(T2)=/ba∇dblTΛξ0ϕn/ba∇dblL2(TΛ⊥\nξ0×TΛξ0)=C0/ba∇dblχ/ba∇dblL2(TΛξ0), we obtain, for all n∈N,\n/ba∇dblP−1(in|k|)/ba∇dblL(L2(T2))≥/ba∇dblϕn/ba∇dblL2(T2)\n/ba∇dblP(in|k|)ϕn/ba∇dblL2(T2)=/ba∇dblχ/ba∇dblL2(TΛξ0)\n/ba∇dbl∆Λξ0χ/ba∇dblL2(TΛξ0)=C >0,\nwhich concludes the proof of Theorem 2.5.\nAcknowledgments. The authors would like to thank Nicolas Burq for having found a signific ant\nerror in a previous version of this article, and for advice on how to fix it. The second author wishes\nto thank Luc Robbiano for several interesting discussions on the s ubject of this article.\nA Pseudodifferential calculus\nIn the main part of the article, we use the semiclassical Weyl quantiz ation, that associates to a\nfunctionaonT∗R2an operator Oph(a) defined by\n/parenleftbig\nOph(a)u/parenrightbig\n(x) :=1\n(2πh)2/integraldisplay\nR2/integraldisplay\nR2ei\nhξ·(x−y)a/parenleftbiggx+y\n2,ξ/parenrightbigg\nu(y)dy dξ. (A.1)\nFor smooth functions awith uniformly bounded derivatives, Oph(a) defines a continuous operator\nonS(R2), and also by duality on S′(R2). On a manifold, the quantization Ophmay be defined by\nworking in local coordinates with a partition of unity. On the torus, f ormula (A.1) still makes sense :\ntakinga∈C∞(T∗T2) is equivalent to taking a∈C∞(R2×R2), (2πZ)2-periodic with respect to the\nx-variable. Then the operator defined by (A.1) preserves the spac e of (2πZ)2-periodic distributions\nonR2, and hence D′(T2).\nWe sometimes write, with D:=1\ni∂,\na(x,hD) = Oph(a).\nWe also note that Op1(a) is the classical Weyl quantization, and that we have the relation\na(x,hD) = Oph(a(x,ξ)) = Op1(a(x,hξ)).\nTheorem A.1. There exists a constant C >0such that for any a∈C∞(T∗T2)with uniformly\nbounded derivatives, we have\n/ba∇dblOp1(a)/ba∇dblL(L2(T2))≤C/summationdisplay\nα,β∈{0,1,2}2/ba∇dbl∂α\nx∂β\nξa/ba∇dblL∞(T∗T2).\nEquivalently, this can be rewritten as\n/ba∇dblOph(a)/ba∇dblL(L2(T2))≤C/summationdisplay\nα,β∈{0,1,2}2h|β|/ba∇dbl∂α\nx∂β\nξa/ba∇dblL∞(T∗T2).\nThis precised version of the Calder´ on-Vaillancourt theorem is need ed in Section 13, and proved\nin [Cor75, Theorem Bρ] or [CM78, Th´ eor` eme 3]. Here in dimension two, this means that only\n|α|= 4 derivations are needed with respect to the space variable x.\nB Spectrum of P(z)for a piecewise constant damping\n(by St´ ephane Nonnenmacher)\nIn this Appendix we provide an explicit description of some part of the spectrum of the damped wave\nequation (1.1) on T2, for a damping function proportional to the characteristic funct ion of a vertical\n42strip. We identify the torus T2with the square {−1/2≤x <1/2,0≤y <1}. We choose some\nhalf-widthσ∈(0,1/2), and consider a vertical strip of width 2 σ. Due to translation symmetry of\nT2, we may center this strip on the axis {x= 0}. Choosing a damping strength /tildewideB >0, we then get\nthe damping function\nb(x,y) =b(x) =/braceleftBigg\n0,|x| ≤σ,\n/tildewideB, σ< |x| ≤1/2.(B.1)\nThe reason for centering the strip at x= 0 is the parity of the problem w.r.t. that axis, which greatly\nsimplifies the computations.\nWe areinterested in the spectrum ofthe operator Ageneratingthe equation(1.1), which amounts\nto solving the eigenvalue problem\nP(z)u= 0,forP(z) =−∆+zb(x)+z2, z∈C, u∈L2(T2), u/\\e}atio\\slash≡0.(B.2)\nThis spectrum consists in a discrete set {zj}, which is symmetric w.r.t. the horizontal axis: indeed,\nany solution ( z,u) admits a “sister” solution (¯ z,¯u). Furthermore, any solution with Im z/\\e}atio\\slash= 0 satisfies\nRez=−1\n2(u,bu)L2(T2)\n/ba∇dblu/ba∇dbl2\nL2(T2),and thus −/tildewideB/2≤Rez≤0. (B.3)\nWe may thus restrict ourselves to the half-strip {−/tildewideB/2≤Rez≤0,Imz>0}.\nOur aim is to find high frequency eigenvalues (Im z≫1) which are as close as possible to the\nimaginary axis. We will prove the following\nProposition B.1. There exists C0>0such that the spectrum (B.3)for the damping function (B.1)\ncontains an infinite subsequence {zi}such that Imzi→ ∞and|Rezi| ≤C0\n(Imzi)3/2.\nThe proof of the proposition will actually give an explicit value for C0, as a function of ˜B,σ.\nProof.To study the high frequency limit Im z→ ∞we will change of variables and take\nz=i(1/h+/tildewideζ),\nwhereh∈(0,1] will be a small parameter, while /tildewideζ∈Cis assumed to be uniformly bounded when\nh→0. The eigenvalue equation then takes the form\n(−h2∆+ih(1+h/tildewideζ)b)u=/parenleftbig\n1+2h/tildewideζ(1+h/tildewideζ/2)/parenrightbig\nu. (B.4)\nHaving chosen bindependent of y, we may naturally Fourier transform along this direction, that is\nlook for solutions of the form u(x,y) =e2iπnyv(x),n∈Z. For each n, we now have to solve the\n1-dimensional problem\n(−h2∂2/∂2\nx+ih(1+h/tildewideζ)b(x))v=/parenleftbig\n1−(2πhn)2+2h/tildewideζ(1+h/tildewideζ/2)/parenrightbig\nv. (B.5)\nLet us call\nBdef=/tildewideB(1+h/tildewideζ), ζdef=/tildewideζ(1+h/tildewideζ/2).\nIn terms of these parameters, the above equation reads:\n(−h2∂2/∂2\nx+ihB1l{σ<|x|≤1/2}(x))v=Ev, withE= 1−(2πhn)2+2hζ. (B.6)\nSince we will assume throughout that /tildewideζ=O(1), we will have in the semiclassical limit\nB=/tildewideB+O(h),/tildewideζ=ζ(1−hζ/2+O(h2)). (B.7)\nAt leading order we may forget that the variables B,ζare not independent from one another, and\nconsider (B.6) as a bona fide linear eigenvalue problem.\n43Since the function b(x) is even, we may separately search for even, resp. odd solutions v(x). Let\nus start with the even solutions. Since b(x) is piecewise constant, any even and periodic solution\nv(x) takes the following form on [ −1/2,1/2] (up to a global normalization factor):\nv(x) =/braceleftBigg\ncos(kx), |x| ≤σ,\nβcos/parenleftbig\nk′(1/2−|x|)/parenrightbig\n, σ<|x| ≤1/2,, (B.8)\nk=E1/2\nh, k′=(E−ihB)1/2\nh. (B.9)\nWe notice that k,k′are defined modulo a change of sign, so we may always assume that Re k≥0,\nRek′≥0. The factor βis obtained by imposing the continuity of vand of its derivative v′at the\ndiscontinuity point x=σ(we use the notation σ′def= 1/2−σ):\ncos(kσ) =βcos(k′σ′),\n−ksin(kσ) =βk′sin(k′σ′).\nThe ratio of these two equations provides the quantization conditio n for the even solutions:\ntan(kσ) =−k′\nktan(k′σ′). (B.10)\nSimilarly, any odd eigenfunction takes the form (modulo a global norm alization factor):\nv(x) =/braceleftBigg\nsin(kx), |x| ≤σ,\nβsgn(x) sin(k′(1/2−|x|)), σ<|x| ≤1/2,,\nso the associated eigenvalues should satisfy the condition\ntan(kσ) =−k\nk′tan(k′σ′). (B.11)\nWe will now study the solutions of the quantization conditions (B.10) a nd (B.11), taking into account\nthe relations (B.9) between the wavevectors k,k′and the energy E. To describe the full spectrum\n(which we plan to present in a separate publication), we would need to consider several r´ egimes,\ndepending on the relative scales of Eandh. However, since we are only interested here in proving\nProposition B.1, we will focus on the r´ egimeleading to the smallest pos sible values of |Im/tildewideζ|=|Rez|.\nWhat characterizes the corresponding eigenmodes v(x) ? From (B.3) we see that the mass of v(x) in\nthe damped region, 2/integraltext1/2\nσ|v(x)|2dx, should be small compared to its full mass. Intuitively, if such a\nmode were carrying a large horizontal “momentum” Re( hk) in the undamped region, it would then\nstrongly penetrate the damped region, because the boundary at x=σis not reflecting. As a result,\nthe mass in the damped region would be of the same order of magnitud e as the one in the undamped\none. This hand-waving argument explains why we choose to investiga te the eigenmodes for which hk\nis the smallest possible, namely of order O(h). This implies that E= (hk)2=O(h2), which means\nthat almost all of the energy is carried by the vertical momentum:\nhn= (2π)−1+O(h).\nThe study of the full spectrum actually confirms that the smallest v alues of Im/tildewideζare obtained in this\nr´ egime.\nEq.(B.9) implies that the wavevector k′in the damped region is then much larger than k:\nk′=(−ihB+(hk)2)1/2\nh=e−iπ/4(B/h)1/2+O(h1/2).\nImk′σ′≈ −σ′(B/2h)1/2is negative and large, so that tan( k′σ′) =−i+O(e2Im(k′σ′)), uniformly\nw.r.t. Re(k′σ′).\n44Even eigenmodes\nIn this situation the even quantization condition (B.10) reads\ntan(kσ) =ik′\nk/parenleftbig\n1+O(e−σ′(2B/h)1/2)/parenrightbig\n. (B.12)\nSince the r.h.s. is large, kσmust be close to a pole of the tangent function. Hence, for each int eger\nmin a bounded interval10≤m≤Mwe look for a solution of the form\nkm+1/2=π(m+1/2)\nσ+δkm+1/2,with|δkm+1/2| ≪1.\nThe quantization condition (B.12) then reads\nσδkm+1/2+O((δkm+1/2)2) =ikm+1/2\ne−iπ/4(B/h)1/2+O(h1/2)/parenleftbig\n1+O(e−σ′(2B/h)1/2)/parenrightbig\n=⇒km+1/2=π(m+1/2)\nσ/parenleftBig\n1+h1/2ei3π/4\nσB1/2+O(h)/parenrightBig\n.\nUsing (B.6), the corresponding spectral parameter ζis then given by\nζn,m+1/2=(hkm+1/2)2+(2πhn)2−1\n2h\n=(2πhn)2−1\n2h+h\n2/parenleftBigπ(m+1/2)\nσ/parenrightBig2\n+h3/2/parenleftBigπ(m+1/2)\nσ/parenrightBig2ei3π/4\nσB1/2+O(h2).\nFrom the assumptions on the quantum numbers n,m, we check that ζn,m+1/2=O(1). We may now\ngo back to the original variables /tildewideζ,/tildewideB, using the relations (B.7). The spectral parameter /tildewideζhas an\nimaginary part\nIm/tildewideζn,m+1/2= Imζn,m+1/2(1−hReζn,m+1/2)+O(h2) =h3/2(π(m+1/2))2\nσ3(2/tildewideB)1/2+O(h2).(B.13)\nReturningbacktothespectralvariable z,theaboveexpressiongivesastringofeigenvalues {zn,m+1/2}\nwith Imzn,m+1/2=h−1+O(1), Rezn,m+1/2=−Im/tildewideζn,m+1/2. These even-parity eigenvalues prove\nProposition B.1, and one can take for C0any value greater than(π/2)2\nσ3(2/tildewideB)1/2.\nWe remark that the leading order of km+1/2corresponds to the even spectrum of the operator\n−h2∂2/∂2\nxon the undamped interval [ −σ,σ], with Dirichlet boundary conditions. The eigenmode\nvn,m+1/2associated with /tildewideζn,m+1/2is indeed essentially supported on that interval, where it resembles\nthe Dirichlet eigenmode cos/parenleftbig\nxπ(1/2+m)/σ/parenrightbig\n. At the boundary of that interval, it takes the value\nvn,m+1/2(σ) = (−1)m+1ei3π/4h1/2π(m+1/2)\nσ/tildewideB1/2+O(h),\nand decays exponentially fast inside the damping region, with a “pene tration length” (Im k′)−1≈\n(2h//tildewideB)1/2. From (B.3) we see that the intensity |vn,m+1/2(σ)|2∼Chpenetrating on a distance\n∼h1/2exactly accounts for the size ∼h3/2=hh1/2of the Rezn,m+1/2.\nWe notice that the smallest damping occurs for the state vn,1/2resembling the ground state of\nthe Dirichlet Laplacian.\n1Recall that we only need to study values Re k≥0.\n45Odd eigenmodes\nFor completeness we also investigate the odd-parity eigenmodes wit hk=O(1). The computations\nare very similar as in the even-parity case. The odd quantization con dition reads in this r´ egime\ntan(kσ) =ik\nk′/parenleftbig\n1+O(e−(2B/h)1/2)/parenrightbig\n. (B.14)\nThe r.h.s. is then very small, showing that σkis close to a zero of the tangent, so we may take\nkm=πm/σ+δkmwith|δkm| ≪1 and 0≤m≤M. We easily see that the case m= 0 does not\nlead to a solution. For the case m>0 we get\nδkm=e3iπ/4h1/2πm\nσ2B1/2+O(h),\nand thus\nkm=πm\nσ/parenleftBig\n1+h1/2e3iπ/4\nσB1/2+O(h)/parenrightBig\n,1≤m≤M.\nThese values kmapproximately sit on the same “line” {s(1 +h1/2e3iπ/4\nσB1/2), s∈R}as the values\nkm+1/2corresponding to the even eigenmodes, both types of eigenvalues appearing successively. The\ncorresponding energy parameter /tildewideζn,msatisfies\nIm/tildewideζn,m=h3/2(πm)2\nσ3(2/tildewideB)1/2+O(h2). (B.15)\nAs in the even parity case, the eigenmodes vn,mare close to the odd eigenmodes sin/parenleftbig\nxπm/σ/parenrightbig\nof the\nsemiclassical Dirichlet Laplacian on [ −σ,σ], and penetrate on a length ∼h1/2inside the damped\nregion.\nThe case of the square\nIf the torus is replaced by the square [ −1/2,1/2]×[0,1] with Dirichlet boundary conditions, with\nthe same damping function (B.1), the eigenmodes P(z) can as well be factorized into u(x,y) =\nsin(2πny)v(x), withn∈1\n2N\\0, andv(x) must be an eigenmode of the operator (B.6) vanishing at\nx=±1/2. We notice that the odd-parity eigenstates (B) satisfy this boun dary conditions, so the\neigenvalues zn,m(with real parts given by (B.15)) belong to the spectrum of the dam ped Dirichlet\nproblem.\nSimilarly, in the case of Neumann boundary conditions the eigenmodes factorize as u(x,y) =\ncos(2πny)v(x), withn∈1\n2N. 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Res. Lett. , 18(2):379–388,\n2011.\n[TW09] M. Tucsnak and G. Weiss. Observation and control for operator semigroups . Birkh¨ auser Advanced\nTexts: Basel Textbooks. Birkh¨ auser Verlag, Basel, 2009.\n48"    },    {        "title": "2302.06402v2.Damping_of_gravitational_waves_in_f_R__gravity.pdf",        "content": "arXiv:2302.06402v2  [gr-qc]  24 Oct 2023Damping of gravitational waves in f(R)gravity\nHaiyuan Feng1,∗and LaiYuan Su1,†\n1Department of Physics, Southern University of Science and T echnology,\nShenzhen 518055, Guangdong, China\nAbstract\nWe study the damping of f(R) gravitational waves by matter in flat spacetime and in expan ding\nuniverse. Intheformercase, we findthat theLandaudampingo f scalar modein f(R) theoryexists,\nwhilethat of thetensor modein general relativity does not; wealso present theviscosity coefficients\nand dispersion relations of the two modes. In the later case, we investigate the evolution of tensor\nand scalar modes in Friedmann-Robertson-Walker (FRW) cosm ology with a matter distribution;\nby considering the case of f(R) =R+αR2, we analysis the influence of parameter αon wave\ndamping, and put restrictions on its magnitude.\n∗Email address: 406606114@qq.com\n†12131268@mail.sustech.edu.cn\n1I. INTRODUCTION\nThe detection of gravitational waves (GW) in the universe gives a sig nificant promotion\nto the development of modern astronomy and physics. Continuous observations provide\ncrucialdatastorestrict characteristicsofastrophysical sour ces[1–9], aswell astotestgeneral\nrelativity (GR) [10–15]. The interaction of gravitational waves with m atter, though in most\ncases ignored, has been investigated throughout the history. Ha wking first calculated the\ndamping rate of gravitational waves as γ= 16πGηby viewing matter as a perfect fluid, with\nηthe viscosity [16]. Subsequently, Ehlers et.al proved in general tha t gravitational waves\ntraveling through perfect fluids do not suffer from dispersion or dis sipation [17]. While in\nthe collisionless limit, the damping rate of gravitational waves by non- relativistic particles\nis shown to be related to the particle velocity and the number density [18]. By linearizing\nthe Boltzmann equation and taking into account the collision term, a u nified treatment for\ndamping from collision and the Landau damping is given [19]. Landau damp ing, firstly\nintroduced to investigate the dispersion relationship in the plasma sy stem [20–22], and then\ngeneralized to the research of large-scale galaxy clusters [23, 24], is proved to be vanish for\ngravitational waves in flat spacetime; while in some modified gravity th is is not always the\ncase for the extra modes, as we will shown in f(R) gravity. On the other hand, gravitational\nwaves played a significant role in the evolution of early universe. The o bservation of cosmic\ntensor fluctuations by measurements of microwave background p olarization is the finest\napproach to check an inflationary universe. Weinberg has already s ketched out the main\napproach for calculating the influence of collisionless three massless neutrinos [25]. The\nperturbations of neutrinos by GW was estimated using the collisionles s Boltzmann equation;\nthe result of the analysis showed that the damping effect of neutrin os on the GW’s spectrum\ncan be rather large. This approach was used to study cosmic gravit ational waves in the\nradiation-dominatedera. BasedonWeinberg’sconclusions, asetof analyticalsolutionsusing\nmodal expansion with spherical Bessel functions as bases were de veloped [26]. Following\nthat, the damping effect of gravitational waves in cold dark matter was studied when mass-\nrelativistic particles were included [27].\nAs an attempt to restore the non-renormalization of GR, as well as to alleviate the\ncosmological constant problem, f(R) theory is proposed. The model has two significant\nadvantages: the actions are sufficiently general to encompass so me of the fundamental prop-\n2erties of higher-order gravity while remaining simple enough to be eas ily handled; it appears\nto be the only ones capable of averting the long-known and catastr ophic Ostrogradski insta-\nbility [28]. Especially, choosing the simplest f(R) =R+αR2(α >0) alone can provide an\nexplanation for the universe’s accelerated expansion [29], and can b eregarded as a candidate\nfor an inflationary field.\nThe action of f(R) theory has the following form,\nS[gµν] =1\n2κ2/integraldisplay\nd4x√−gf(R)+/integraldisplay\nd4x√−gLm, (1)\nwithκ2= 8πG,Lmthe Lagrangian of matter. The field equations can be obtained by\nvarying the above action [30],\nF(R)Rµν−1\n2f(R)gµν−∇µ∇νF(R)+gµν/squareF(R) =κ2T(M)\nµν, (2)\nwhereF(R) =df(R)\ndR, and/squareis the d’Alembertian operator. The energy-momentum ten-\nsorT(M)\nµν≡ −2√−gδLm\nδgµνsatisfies the continuity equation ∇µT(M)µν= 0. We rearrange the\npreceding equations to get\n\n\nGµν=κ2/parenleftbig\nT(eff)\nµν+T(M)\nµν/parenrightbig\nκ2T(eff)\nµν≡gµν\n2(f(R)−R)+∇µ∇νF(R)−gµν/squareF(R)+(1−F(R))Rµν,(3)\nwith Einstein tensor Gµν=Rµν−1\n2gµνR, and fulfills the Bianchi identity ∇µGµν= 0. It can\nbe proved that the contribution of curvature T(eff)\nµνalso obey ∇µT(eff)µν= 0. It is possible\nto get the trace of (3) as\n3/squareF(R)+F(R)R−2f(R) =κ2T(M), (4)\nwhich clearly shows the difference from Einstein’s trace formula R=−κ2T. The presence\nof terms /squareF(R) leads to additional propagation degrees of freedom in the f(R) theory.\nTo investigate the equation of f(R) gravitational waves, the metric gµνand Riemann\ncurvature scalar Rin Minkowski spacetime is perturbed as\n\n\ngµν=ηµν+hµν\nR=R0+δR,(5)\nwhere tensor perturbation hµνis restricted by |hµν| ≪ |ηµν|. Background curvature\n(Minkowski) and scalar perturbation are denoted by R0andδR, respectively. As can\n3be shown, different from gravitational waves in GR, the perturbat ion has the form hµν=\n¯hTT\nµν+hS\nµν, where¯hTT\nµνrepresents the transverse-traceless (TT) part of the pertur bation. It\nsatisfies∂i¯hTT\nij= 0,¯hTT\nii= 0, and hS\nµν=−φηµν(φ≡F′(R0)δR\nF(R0)) represents the scalar degrees\nof freedom [30]. The linearized field equations [31–35] is given by\n\n\n/square¯hTT\nij=−2κ′2Π(1)\nij\n/squareφ−M2φ=κ′2\n3T(1),(6)\nwhere\nM2≡1\n3/parenleftbiggF(R0)\nF′(R0)−R0/parenrightbigg\n=F(0)\n3F′(0)(7)\nis the square of the effective mass, κ′2≡κ2\nF(0), and Π(1)\nijis the linear part of the anisotropic\npart of the spatial components of energy-momentum tensor Tij/parenleftbig\nTi\nj= Πi\nj+1\n3δi\nj/summationtext3\nk=1Tk\nk/parenrightbig\n.\nIt couples with gravitational waves and satisfies Π ii= 0,∂iΠij= 0. However, in f(R)\ngravity, since the emergence of the degrees of freedom R, there is the presence of an extra\nscalar mode, commonly known as the breathing mode. It is obvious fr om equation (6) that\nwhen the effective mass Mapproaches infinity, the system no longer has the excitation of\nthe scalar mode and returns to the tensor mode of GR gravity. As a result, the number of\npolarizations in f(R) gravity is three [36, 37].\nIn this paper, by taking into account contributions from the collision term, we investigate\nthe damping of f(R) gravitational waves in the presence of medium matter and determ ine\nthe dispersion relation. In Section II, we introduces linearized f(R) theory and provides\nwave equations for tensor and scalar modes in Minkowski spacetime . In section III, we ap-\nply kinetic theory to investigate the first-order approximation of t he relativistic Boltzmann\nequation. We calculate the anisotropic part of the spatial compone nts of energy-momentum\ntensor and derive the dispersion relation of two modes using the rela xation time approxima-\ntion. We also derive damping coefficients in collision-dominant case and L andau damping\nin the collisionless limit. In section IV, within the context of FRW inflation ary cosmology,\nwe present the wave equation of tensor and scalar disturbances in the early universe by\nusingf(R) theory, and investigate the specific case of f(R) =R+αR2. The damping of\ngravitational waves by neutrinos with mass is also investigated, and the collision term is\nproved to be negligible.\n4II. DISPERSION AND DAMPING OF GRAVITATIONAL WAVES BY RELA-\nTIVISTIC BOLTZMANN GASES IN FLAT SPACETIME\nTo calculate the anisotropic stress Π ijand the energy-momentum tensor trace Tinduced\nby gravitational waves, we consider the relativistic Boltzmann equa tion [38–42]\npµ∂f\n∂xm−gijΓi\nµν∂f\n∂pj=C[f], (8)\nwhere distribution function f(xi,pj) as the particle positions and canonical momenta de-\nscribestheprobabilityofthespatialdistribution. Γi\nµνistheconnectioncoefficientand pµrep-\nresents the four-momentum of a single particle with on-shell condit iongµνpµpν=−m2.C[f]\nis the collision term. To analyse the dynamics of the Boltzmann equatio n, we consider the\nrelaxation time approximation [43], therefore C[f] can be represented as\nC[f] =−pµuµ\nτc(fh−f), (9)\nτcis the particle’s average collision time, and uµdenotes themacroscopic fluid’s four-velocity\n[44]. Therefore, four-velocity could currently be written as uµ= (1,0,0,0) in the fluid’s\nrest reference frame. The distribution function of the local equilib rium in the presence of\ngravitational waves, denoted by fh, is given by\nfh=g\ne−pµuµ\nT±1, (10)\nwhere±corresponds to fermions or bosons, gis the number of degrees of freedom for the\nvarieties of single particles, and Tis the temperature. Using geodesic equation of particles,\n(8) can be simplified by\n∂f\n∂t+pm\npt∂f\n∂xm+dpm\ndt∂f\n∂pm=1\nτc(fh−f). (11)\nwith\ndpm\ndt=1\n2∂gµν\n∂xmpµpν\np0. (12)\nWe will apply the dynamic perturbation approach to determine the fo rmulation of the\ninduced energy-momentum tensor. Firstly, starting with hµν=¯hTT\nµν−φηµν, the perturbation\non-shell condition can be expressed as\n\n\nǫ=ǫ0+δǫ\nδǫ=hµνpµpν\n2ǫ0=−¯hTT\nijpipj−m2φ\n2ǫ0,(13)\n5with\np0≡ǫ0=/radicalbig\nm2+pipi, (14)\nwe have adopted the first-order perturbation hµνηµαηνβ=−hαβ. Substituting (13) into (12)\nto derive\ndpm\ndt=1\n2p0/parenleftbigg\npkpl∂¯hTT\nkl\n∂xm+m2∂φ\n∂xm/parenrightbigg\n. (15)\nSecondly, we handletheperturbeddistributionfunction f=f0(p)+δf(xi,pj,t)according\nto[19]. By ignoring all higher order terms, the linearized Boltzmann eq uation is obtained\n∂δf\n∂t+pm\np0∂δf\n∂xm+1\n2p0/parenleftbigg\npkpl∂¯hTT\nkl\n∂xm+m2∂φ\n∂xm/parenrightbigg∂f0(p)\n∂pm=−1\nτc(δf−δfh),(16)\nit is worth emphasizing that δfhrepresents the deviation between the distribution function\nafter local equilibrium and the absence of gravitational waves, whic h could be expanded\ninto a first-order small quantity as δfh=∂f0\n∂ǫδǫby Taylor formula. By Fourier transforming\n¯hTT\nij(/vector r,t) =ei/vectork·/vector r−iωt¯hTT\nij(ω,/vectork) andφ(/vector r,t) =ei/vectork·/vector r−iωtφ(ω,/vectork), we derive the solution of (16).\nδf(ω,/vectork) =f′(p)\n2p¯hTT\nijpipj/parenleftBig\n−1\nτc−i/vector p·/vectork\np0/parenrightBig\n−m2φ/parenleftBig\ni/vector p·/vectork\np0+1\nτc/parenrightBig\n/parenleftBig\n−iω+i/vector p·/vectork\np0+1\nτc/parenrightBig , (17)\nwheref′(p)meanstoderivationwithrespectto p. Conclusively, sincetheinducedanisotropic\nstress tensor is assessed in terms of the distribution function f, the dynamical system is fully\ncharacterised [19, 27, 45],\n\n\nΠ(1)\nij=/integraldisplayd3p\n(2π)3pipj\nǫ0¯δf\nT(1)=−m2/integraldisplayd3p\n(2π)31\nǫ0¯δf,(18)\nwhere¯δf≡δf−δfhshould be interpreted as the effect of the distribution function’s\nown variation, since the total shift is the sum of the distribution fun ction’s own and the\ntransformation caused by gravitational waves. We can determine the expression by inserting\n(17) into (18), which follows\nΠ(1)\nij=¯hTT\nkl/integraldisplayd3p\n(2π)3pkplpipjf′\n0(p)\n2pǫ0\n−iω\n−iω+i/vector p·/vectork\np0+1\nτc\n, (19)\n6and\nT(1)=−m2/integraldisplayd3p\n(2π)3f′\n0(p)\n2pǫ0\n¯hTT\nklpkpl\n−iω\n−iω+i/vector p·/vectork\np0+1\nτc\n−m2φ\niω\n−iω+i/vector p·/vectork\np0+1\nτc\n\n\n=m4φ(ω,/vectork)/integraldisplayd3p\n(2π)3f′\n0(p)\n2pǫ0\niω\n−iω+i/vector p·/vectork\np0+1\nτc\n,(20)\nbased on the angular integration, the contribution of gravitationa l waves in (20) is zero. The\nfirst term on the right side of (19) can be shown to be proportional to¯hTT\nij, it follows\nΠ(1)\nij=¯hTT\nij/integraldisplayd3p\n(2π)3(pipj)2f′\n0(p)\npǫ0\n−iω\n−iω+i/vector p·/vectork\np0+1\nτc\n, (21)\nwherei/negationslash=jand only x,ycan be taken. We could produce the dispersion relation of\ngravitational waves in relativistic particle flow by replacing (20) and ( 21) into (6).\n\n\nω2−k2+2κ′2/integraldisplayd3p\n(2π)3(pipj)2f′\n0(p)\npǫ0\n−iω\n−iω+i/vector p·/vectork\np0+1\nτc\n= 0\nω2−k2−M2+m4κ′2\n6/integraldisplayd3p\n(2π)3f′\n0(p)\npǫ0\n−iω\n−iω+i/vector p·/vectork\np0+1\nτc\n= 0.(22)\nTwo damping mechanisms must be addressed in order to get mode dam ping from dis-\npersion. Landau damping and collision-dominated hydrodynamic damp ing are two different\ntypesofgravitationalwavedampingthataredeterminedbytheima ginarypartofthesource.\nLandau damping is the excitation of two real particle-hole pairs caus ed by decay of the mode\nwithout considering thecollisioninto account. Fromthe(20), (21), andthecollisionless limit\n1\nτc→0 can be derived,\n\n\nℑ/parenleftBigg\nΠ(1)\nij\n¯hTT\nij/parenrightBigg\n=−πω/integraldisplayd3p\n(2π)3(pipj)2f′\n0(p)\npǫ0δ/parenleftBigg\nω−/vector p·/vectork\np0/parenrightBigg\nℑ/parenleftbiggT(1)\nφ/parenrightbigg\n=πωm4/integraldisplayd3p\n(2π)3f′\n0(p)\n2pǫ0δ/parenleftBigg\nω−/vector p·/vectork\np0/parenrightBigg\n,(23)\nThe preceding formula shows that the Landau damping phenomenon happens only when\np0=|p|cosθ, implying that the particles must be massless and move along the wave direc-\ntion to contribute. However, In the flat spacetime, gravitational waves will not encounter\nLandau damping because ( pipj)2= (pxpy)2. It is worth noting that the Landau damping\n7of scalar modes only contributes when the particle motion direction c orresponds with the\nwave propagation direction. To investigate another damping mecha nism, we focus at the\ncollision-dominated region ( ω≪1\nτc). (20) and (21) will be wirtten as\nℑΠ(1)\nij=¯hTT\nkl(ω,/vectork)/integraldisplayd3p\n(2π)3pkplpipjf′\n0(p)\n2pǫ0−ω\nτc/parenleftBig\nω−/vector p·/vectork\np0/parenrightBig2\n+1\nτ2c\n≈ −ωτc¯hTT\nij(ω,/vectork)\n15/integraldisplayd3p\n(2π)3p3f′\n0(p)\nǫ0=−τcωη1¯hTT\nij(ω,/vectork),(24)\nand\nℑT(1)=m4φ(ω,/vectork)/integraldisplayd3p\n(2π)3f′\n0(p)\n2pǫ0ω\nτc/parenleftBig\nω−/vector p·/vectork\np0/parenrightBig2\n+1\nτ2c\n≈m4ωτcφ(ω,/vectork)/integraldisplayd3p\n(2π)3f′\n0(p)\n2pǫ0=τcωη2φ(ω,/vectork),(25)\nThe collision-dominated viscosity coefficient under the relaxation time approximation are η1\nandη2, which follows\n\nη1≡τc\n15/integraldisplayd3p\n(2π)3p3f′\n0(p)\nǫ0\nη2≡m4τc/integraldisplayd3p\n(2π)3f′\n0(p)\n2pǫ0.(26)\nThe viscosity coefficient provided by the given equations is obviously b ased on the dis-\ntribution function of the equilibrium state and the collision relaxation t ime. These two\ncomponents are also the primary causes of the damping of tensor a nd scalar modes.\nIn this section, we calculate the solution of the anisotropic stress t ensor induced by\ngravitational waves by using Fourier transform of the linearized Bo ltzmann equation, and\nwe achieve the coefficient of damping through adopting two mechanis ms of gravitational\nwave damping.\nIII. DAMPING OF TENSOR AND SCALAR MODES IN COSMOLOGY\nThe detection of cosmic tensor fluctuations by measurements of m icrowave background\npolarisation is widely expected to provide a uniquely valuable check on t he validity of basic\ninflationary cosmology. Particularly, gravitational wave spectra g enerated by other nonin-\nflationary sources have also been proposed [46]. As a result, any fin ding of gravitational\nwave spectrum would be a tremendously valuable tool in studying the early cosmology. The\nf(R) wave equation will be discussed in this section.\n8A. The wave equations of tensor and scalar mode\nWe will use conformal coordinates to investigate the evolution of te nsor modes in the\nspatially flat FRW universe. The line element with a perturbation metric is represented by\nds2=a2(τ)/bracketleftbig\n−(1−φ)dτ2+/parenleftbig\nδij−δijφ+¯hTT\nij/parenrightbig\ndxidxj/bracketrightbig\n, (27)\nThe cosmological equation satisfied by the gravitational wave tens or mode could be deter-\nmined by[47, 48],\n¨¯hTT\nij+(2+aM)H(τ)˙¯hTT\nij−∇2¯hTT\nij=2κ2\nF(R0)a2(τ)Π(1)\nij, (28)\nand the scalar mode\n¨φ+/parenleftBigg\n2H(τ)+˙F\nF/parenrightBigg\n˙φ−∇2φ+/parenleftBigg\na2(τ)M2+4H(τ)˙F\nF+2¨F\nF/parenrightBigg\nφ=−κ2\n3F(R0)a2(τ)T(1).(29)\na(τ) represents the cosmic evolution factor,˙¯hTT\nijand˙φdenote the derivative with respect to\nthe conformal time τ, andHindicates the Hubble constant. aMis defined asF′(R0)˙R\nF(R0)H, and\nthe scenario of aM=φ= 0 is accompanied by GR gravitational waves. We concentrate on\nthe conclusions on the right-hand side of the above equation, as we did in the flat spacetime.\nThe disturbance of Boltzmann equation (16) is\n1\na(τ)∂δf\n∂τ+pm∂mδf\na(τ)pτ+1\na(τ)dpm\ndτ∂f0\n∂pm=−1\nτc(δf−δfh), (30)\nwhich can be simplify to\n/parenleftbigg∂\n∂τ+vm∂m+1\n¯τc/parenrightbigg\nδf=δfh\n¯τc−dpm\ndτ∂f0\n∂pm, (31)\nwherevm≡pm\npτ=pm√\npipi+m2a2corresponds to the three-velocity of particles. ¯ τc≡τc\na(τ)is\ncollision time in cosmology. Thedpm\ndτis expressed by\ndpm\ndτ=1\n2∂mgµνpµpν\npτ\n=∂m¯hTT\nijpipj+m2a2(τ)∂mφ\n2pτa2(τ).(32)\nSimilarly, the on-shell condition and its perturbation are denoted by\n\n\nǫ=ǫ0+δǫ\nǫ0≡pτ=/radicalBigg\nm2\na2(τ)+pipi\na4(τ)\nδǫ=δgµνpµpν\n2a2(τ)ǫ0=−¯hTT\nijpipj−m2a2(τ)φ\n2a4(τ)ǫ0,(33)\n9The spatial Fourier transform is used to reduce the ultimate Boltzm ann equation (30) to\n/parenleftbigg∂\n∂τ+Q(τ)/parenrightbigg\nδf(τ,/vectork) =−f′\n0(p)Q(τ)\n2p/bracketleftBig\n¯hTT\nij(τ,/vectork)pipj+m2a2(τ)φ(τ,/vectork)/bracketrightBig\n,(34)\nwith\nQ(τ)≡i/vector v·/vectork+1\n¯τc(35)\nwe can derive the particular solution of the first-order differential equation (34)\n\n\nδf(τ) =−/integraldisplayτ\nτ0/parenleftBig\n¯hTT\nij(τ′,/vectork)pipj+m2a2(τ′)φ(τ′,/vectork)/parenrightBigf′\n0(p)\n2p∂e−Λ(τ,τ′)\n∂τ′dτ′\nΛ(τ,τ′) = Λ1(τ,τ′)+icosθkΛ2(τ,τ′)≡/integraldisplayτ\nτ′1\n¯τc(τ′′)dτ′′+icosθk/integraldisplayτ\nτ′v(τ′′)dτ′′,(36)\nwhereτ0depicts the initial assertion at which the system is in equilibrium, and f0is the\npartial function in the equilibrium.\nf0=g\nept\nT±1=g\ne−pτ\na0T0±1, (37)\nwithpτ=−/radicalbig\nm2a2+p2, and the second equality holds for our normalization that the\npresent day scale factor is a0= 1.T0is the current background radiation temperature. The\nperturbed anisotropic part is a generalization of (18), which can be defined as [27]\n\n\nΠ(1)\nij=/integraldisplayd3p\n(2π)3pipj√−g(−pτ)¯δf=/integraldisplayd3p\n(2π)3pipj\na4/radicalbig\nm2a2+p2¯δf\nT(1)=−m2/integraldisplayd3p\n(2π)31√−gǫ0¯δf=−m2/integraldisplayd3p\n(2π)31\na2/radicalbig\nm2a2+p2¯δf,(38)\nwith\n¯δf=/integraldisplayτ\nτ0e−Λ(τ,τ′)∂\n∂τ′/bracketleftbiggf′\n0(p)\n2p/parenleftBig\n¯hTT\nij(τ′,/vectork)pipj+m2a2(τ′)φ(τ′,/vectork)/parenrightBig/bracketrightbigg\ndτ′.(39)\nThe anisotropic sress tensor Π(1)\nijandT(1), which follows\n\n\nΠ(1)\nij=/integraldisplayd3p\n(2π)3pipjpkpl\n2pa4(τ)/radicalbig\nm2a2+p2/integraldisplayτ\nτ0e−Λ(τ,τ′)∂\n∂τ′/bracketleftbig\nf′\n0(p)¯hTT\nkl(τ′)/bracketrightbig\ndτ′\nT(1)=−m4/integraldisplayd3p\n(2π)31\n2pa2/radicalbig\nm2a2+p2/integraldisplayτ\nτ0e−Λ(τ,τ′)∂\n∂τ′/bracketleftbig\nf′\n0(p)a2(τ′)φ(τ′)/bracketrightbig\ndτ′,(40)\nThe integral formula would be used to simplify the expression[26, 27],\n\n\n/integraldisplay2π\n0dφpipjpkpl\np4=π(1−cos2θ)2\n4(δijδkl+δikδjl+δilδjk)\nK(x)≡1\n16/integraldisplay1\n−1/parenleftbig\n1−cos2θ/parenrightbig2eixcosθdcosθ,(41)\n10As a consequence, Π(1)\nijandT(1)are ultimate able to represented as\n\n\nΠ(1)\nij=/integraldisplayp5dp\n2π2a4/radicalbig\nm2a2+p2/integraldisplayτ\nτ0K(kΛ2(τ,τ′))e−Λ1(τ,τ′)∂\n∂τ′/bracketleftbig\nf′\n0(p)¯hTT\nij(τ′)/bracketrightbig\ndτ′\nT(1)=−m4/integraldisplaypdp\n4π2a2/radicalbig\nm2a2+p2/integraldisplayτ\nτ0sinkΛ2(τ,τ′)\nkΛ2(τ,τ′)e−Λ1(τ,τ′)∂\n∂τ′/bracketleftbig\nf′\n0(p)a2(τ′)φ(τ′)/bracketrightbig\ndτ′.\n(42)\nK(x) =j0(x)\n15+2j2(x)\n21+j4(x)\n35is a linear combination of spherical Bessel functions [26]. When\nm= 0, the right side of the wave equation returns to the previous res ult, although with\nadditionalcollisioncontributions. Sincetheexistenceof (40),twom odesinvolvestheLandau\ndamping phenomena induced by the anisotropic tensor of matter in t he universe’s evolution.\nWe primarily figure out the contribution of the matter term on the rig ht-hand side of the\nwave equation by linearizing the Boltzmann equation in the FRW metric t o derive a general\nsolution. In the subsequent sections, we will go over the specific f(R) model and investgate\nthe numerical results of two modes travel through decoupled neu trino systems.\nB. Numerical solution of Damping from neutrinos\nNeutrinos are one of the fundamental Fermi particles participate d the original Big Bang’s\nweakandgravitationalinteractions. Sincethepositiveandnegativ eprocessesoftheneutrino\ninteraction before decoupling approach chemical equilibrium, neutr inos satisfy the equilib-\nrium state distribution function in earlier universe. Weinberg’s origina l study, as well as\nthe majority of the early research, concentrated on the effect o f three massless neutrinos.\nRecent cosmology investigations, nevertheless, have shown indica tions of departures from\nthe traditional cosmological value of three effective neutrino degr ees of freedom [49]. Ex-\nperiments on neutrino oscillations demonstrate that it has mass, wh ich will have an effect\non gravitational wave damping [50]. The Landau damping phenomenon occurs when the k\nof two modes are longer than the cosmic horizon keq≡a(eq)H(eq) (τeqrepresents the time\nwhen the proportion of radiation and matter is the same). This sect ion we focuses on the\nevolution of two modes as f(R) =R+αR2enters the period of matter-radiation dominance.\nAfter the Fourier transform of the spatial part, (28)(29) beco mes\n11\n\n¨¯hTT\nij(u)+2H(u)˙¯hTT\nij(u)+2α˙R\n1+2αR˙¯hTT\nij(u)+¯hTT\nij(u) =2κ2T4\n0\nk2(1+2αR)a2(u)Π(1)\nij\nΠ(1)\nij=/integraldisplay∞\n0x5dx\n2π2a4/radicalBig\nm2a2(u)\nT2\n0+x2/integraldisplayu\n0K(Λ2(u,u′))e−Λ1(u,u′)∂\n∂u′/bracketleftbiggdf0(x,u′)\ndx¯hTT\nkl(u′)/bracketrightbigg\ndu′,\n(43)\nand\n\n\n¨φ(u)+/parenleftBigg\n2H(u)+2α˙R\n1+2αR/parenrightBigg\n˙φ(u)+/parenleftBigg\n1+a2(u)\n6αk2+8αH(u)˙R\n1+2αR+4α¨R\n1+2αR/parenrightBigg\nφ(u)\n=−κ2a2(u)\n3k2(1+2αR)T(1)\nT(1)=−m4/integraldisplayxdx\n4π2a2/radicalBig\nm2a2\nT2\n0+x2/integraldisplayu\n0sinΛ2(u,u′)\nΛ2(u,u′)e−Λ1(u,u′)∂\n∂u′/bracketleftbiggdf0(x,u′)\ndxa2(u′)φ(u′)/bracketrightbigg\ndu′,\n(44)\nwith \n\nΛ1(u,u′) =/integraldisplayu\nu′1\nk¯τc(u′′)du′′\nΛ2(u,u′) =/integraldisplayu\nu′v(u′′)du′′,(45)\nwhere the dimensionless independent variable u≡kτ,x≡p\nT0. We record the initial moment\nkτ0as 0 when the wave enters the matter and the radiation. In the per iod dominated by\nmatter and radiation, a(u) is\na(u) =u2\nu2\n0+2u\nu0√aeq, (46)\nwith\nu0≡2k√ΩMH0. (47)\nIn standard cosmic evolution, there are three generations of neu trinos corresponding to\naeq=1\n3600, ΩM= 0.3 [51, 52]. According to the above equation, (45) can be stated as\n\n\nΛ1(u,u′) =u3−u′3\n3u2\n0kτc+(u2−u′2)√aeq\nu0kτc\nΛ2(u,u′)≈(u−u′)/parenleftbigg\n1−m2\n2x2T2\n0/parenrightbigg\n.(48)\nThe neutrino mass is Taylor extended as a small quantity, reverting to previous gravita-\ntional wave damping conclusions by m= 0. Nonzero neutrino masses will add an additional\nk dependence to damping since free streaming, and thus damping will be lowered when the\n12temperature is of order of the mass. While neutrinos are relativistic , the k modes that come\ninside the horizon and contribute considerably to the overall energ y density will be damped\nmore. The following figure depicts the numerical results of (43) and (44). It is seen that\nthere is a major difference in the varying trend of the χ(u)≡¯hTT\nij(u)\n¯hTT\nij(0)with respect to kτ.\nAskτgoes up, the χ(u) decreases. Note that the m=0 scenario drops faster than the m =\n1ev situation in the right panel of the figure; however, The figure o n the left displays the\ninfluence on τwhen the model parameters are scaled downwards by orders of ma gnitude,\nwith the oscillation being rapid shown by the blue line ( α= 0). When the magnitude of\nα >1018m2, there is no oscillation decay solution for this waveform, so the best limit range\nfor the parameters does not exceed 1018m2. In particular, we can clearly see from the third\nfigurethat theevolution of thescalar mode decays almost complete ly at u=1. When m=1ev,\nthe influence of neutrinos on it slows down its oscillation frequency.\n0 2 4 6 8 10 12 14 16 18\nu=k-0.200.20.40.60.81\n=0m2\n=1015m2\n=1018m2\n0 2 4 6 8 10 12 14 16 18 20\nu=k-0.200.20.40.60.81\nm=0ev\nm=1ev\nno-damping\nFIG. 1. The two diagrams depict the waveform’s evolution ove r time. We fix the parameters\nkτc= 100,u0= 100 and discuss the decay of gravitational waves. We find fro m the left figure\nthat the phase of the wave is translate and the attenuation of the wave is slowed down as the\nparameter αincreases. The figure on the right shows that m=0 ev decays fas ter than m=1ev, and\nonly when the temperature and mass are in the same order of mag nitude, the contribution of mass\nwill increase the decay.\n130.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9\nu=k00.20.40.60.81\nm=0ev\nm=1ev\nno-damping\nFIG. 2. The third picture mainly describes the evolution of t he scalar mode at α= 1018m2, it can\nbe seen that neutrinos with a mass of 1ev slow down the oscilla tion frequency of the wave. It also\nshow that the scalar mode is almost completely attenuated at u=1, which also makes the detection\nof this mode more difficult.\nIV. CONCLUSION AND DISCUSSION\nIn this work, through the application of kinetic theory, we were able to discover the\nLandau damping phenomenon of the scalar mode in the f(R) gravity in the flat space-time.\nWe also discuss how the early universe’s inflation dampened primordial gravitational waves,\nand we impose a restriction as to how the parameter αinf(R) =R+αR2could be. In\nsectionII, weintroducethelinearized f(R)model andprovides wave equationsforthetensor\nand scalar modes. We can clearly see that the background metric ha s absolute influence on\nthe mass of the scalar mode. The theory degenerates into genera l relativity when the effect\nmass term reaches infinity, while there is no excitation of this mode.\nIn section III, we calculate the perturbed form of the Boltzmann e quation, derive the\nperturbed solution of momentum space, and substitute it into the t ransverse-traceless part\nof the anisotropic stress tensor to establish the dispersion relatio n of scalar and tensor mode.\nWe also investigated the damping coefficient in the collision dominated re gion and Landau\ndamping in the collisionless limit. We discovered that the contribution of tensor mode’s\nLandau damping is zero, however, the scalar mode exists only when t he particle motion\ndirection coincides with the direction of wave propagation.\n14Finally, we examine the Boltzmann equation satisfied by the perturba tion in the FRW\nscenariooftheconformalcoordinatesystem. Weobtainthewave equationofthetensormode\nwith damping under the f(R) =R+αR2. Furthermore, after passing through neutrinos\ndecoupling, wenumerically solvethedecayofprimordialgravitationa lwaves createdbyearly\ninflation. We discover that the case of neutrinos oscillate more rapid ly than the undamped\ncase, but the variation between the two is demonstrated to be neg ligible in the case of\nneutrino’s mass m=0,1ev. We also took into account how the paramet erαaffected the\ngravitational wave under this model, and we found that as the magn itude of the parameter\nalbecomes larger, it shows faster damping and oscillation. When the pa rameter magnitude\nα= 1018m2, an upper limit is provided because the waveform does not match wha t is\navailable about physical reality in this scenario. In the future, we will continue to investigate\nthe power spectrum function of the primordial gravitational wave and anticipate its decay\nin modified gravity.\nACKNOWLEDGMENTS\n[1] Simeon Bird, Ilias Cholis, Julian B. 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Peremohy, Kyiv, 03056, Ukraine\n(Dated: 5 May 2020)\nWe propose and analyze theoretically a class of energy-efficient magneto-elastic devices for analogue signal processing.\nThe signals are carried by transverse acoustic waves while the bias magnetic field controls their scattering from a\nmagneto-elastic slab. By tuning the bias field, one can alter the resonant frequency at which the propagating acoustic\nwaves hybridize with the magnetic modes, and thereby control transmission and reflection coefficients of the acoustic\nwaves. The scattering coefficients exhibit Breit-Wigner/Fano resonant behaviour akin to inelastic scattering in atomic\nand nuclear physics. Employing oblique incidence geometry, one can effectively enhance the strength of magneto-\nelastic coupling, and thus countermand the magnetic losses due to the Gilbert damping. We apply our theory to discuss\npotential benefits and issues in realistic systems and suggest routes to enhance performance of the proposed devices.\nOptical and, more generally, wave-based computing\nparadigms gain momentum on a promise to replace and com-\nplement the traditional semiconductor-based technology.1The\nenergy savings inherent to non-volatile memory devices has\nspurred the rapid growth of research in magnonics,2,3in which\nspin waves4are exploited as a signal or data carrier. Yet, the\nprogress is hampered by the magnetic loss (damping).5,6In-\ndeed, the propagation distance of spin waves is rather short in\nferromagnetic metals while low-damping magnetic insulators\nare more difficult to structure into nanoscale devices. In con-\ntrast, the propagation distance of acoustic waves is typically\nmuch longer than that of spin waves at the same frequencies.7\nHence, their use as the signal or data carrier could reduce the\npropagation loss to a tolerable level. Notably, one could con-\ntrol the acoustic waves using a magnetic field by coupling\nthem to spin waves within magnetostrictive materials.8–10To\nminimize the magnetic loss, the size of such magneto-acoustic\nfunctional elements should be kept minimal. This implies\ncoupling propagating acoustic waves to confined spin wave\nmodes of finite-sized magnetic elements. As we show below\nthis design idea opens a route towards hybrid devices combin-\ning functional benefits of magnonics2,3with the energy effi-\nciency of phononics.7,11,12\nThe phenomena resulting from interaction between coher-\nent spin and acoustic waves have already been addressed\nin the research literature: the spin wave excitation of prop-\nagating acoustic waves7,13–15and vice versa,8,16–18acous-\ntic parametric pumping of spin waves,19–21magnon-phonon\ncoupling in cavities22–24and mode locking,25magnonic-\nphononic crystals,26,27Bragg scattering of spin waves from a\nsurface acoustic wave induced grating,28–30topological prop-\nerties of magneto-elastic excitations,15,31acoustically driven\nspin pumping and spin Seebeck effect,32,33and optical excita-\ntion and detection of magneto-acoustic waves.34–40However,\nstudies of the interaction between propagating acoustic waves\nand spin wave modes of finite-sized magnetic elements, which\nare the most promising for applications, have been relatively\nscarce to date.10,34,36,39\nHere, we explore theoretically the class of magneto-\na)Electronic mail: V .V .Kruglyak@exeter.ac.uk\nθHB\nM\nI\nTω\nRω\nM NM NM\nFIG. 1: The prototypical magneto-elastic resonator is a thin\nmagnetic slab (M) of width d, biased by an external field HB,\nand embedded into a non-magnetic (NM) matrix. The\nacoustic wave with amplitude Iincident at angle qinduces\nprecession of the magnetisation vector Mvia the\nmagneto-elastic coupling. As a result, the wave is partly\ntransmitted and reflected, with respective amplitudes Tw\nandRw.\nacoustic devices in which the signal is carried by acoustic\nwaves while the magnetic field controls its propagation via\nthe magnetoelastic interaction in thin isolated magnetic in-\nclusions as shown in Fig. 1. By changing the applied mag-\nnetic field, one can alter the frequency at which the incident\nacoustic waves hybridize with the magnetic modes of the in-\nclusions. Thereby, one can control the acoustic waves by the\nresonant behaviour of Breit-Wigner and Fano resonances in\nthe magnetic inclusion.41We find that the strength of the res-\nonances is suppressed by the ubiquitous magnetic damping\nin realistic materials, but this can be mitigated by employing\noblique incidence geometry. To compare magneto-acoustic\nmaterials for such devices, we introduce a figure of merit. The\nmagneto-elastic Fano resonance is identified as most promis-\ning in terms of frequency and field tuneability. To enhance res-\nonant behaviour, we explore the oblique incidence as a means\nby which to enhance the figure of merit.\nWe consider the simplest geometry in which magneto-arXiv:1906.07297v2  [physics.app-ph]  4 May 2020Controlling acoustic waves using magneto-elastic Fano resonances 2\nelastic coupling can affect sound propagation. A ferromag-\nnetic slab (\"magnetic inclusion\") of thickness d, of the or-\nder of 10 nm, is embedded within a non-magnetic medium\n(Fig.1). The slab is infinite in the Y\u0000Zplane, has satu-\nration magnetization Ms, and is biased by the applied field\nHB=HBˆz. Due to the magneto-elastic coupling, this equilib-\nrium configuration is perturbed by shear stresses in the xz- and\nyzplanes associated with the incident acoustic wave.\nTo derive the equations of motion, we represent the mag-\nnetic energy density Fof the magnetic material as a sum of the\nmagneto-elastic FMEand purely magnetic FMcontributions.42\nTaking into account the Zeeman and demagnetizing energies,\nwe write FM=\u0000m0HBM+m0\n2(NxM2\nx+NyM2\ny), where Nx(y)\nare the demagnetising coefficients, Nx+Ny=1,Mis the\nmagnetization and m0is the magnetic permeability. In a crys-\ntal of cubic symmetry, the magnetoelastic contribution takes\nthe form43\nFME=B\nM2så\ni6=jMiMjui j+B0\nM2så\niM2\niuii;i;j=x;y;z;(1)\nwhere B0and Bare the linear isotropic and anisotropic\nmagneto-elastic coupling constants, respectively.44The strain\ntensor is ujk=1\n2(¶jUk+¶kUj), where Ujare the displace-\nment vector components. To maximize the effect of the cou-\npling B, we consider a transverse acoustic plane wave incident\non the slab from the left and polarized along the bias field, so\nthatUx=Uy=0,Uz=U(x;y;t). The non-vanishing com-\nponents of the strain tensor are uxz=1\n2¶xUanduyz=1\n2¶yU,\nandFMEis linear in both MandU:\nFME=B\nMs(Mxuxz+Myuyz): (2)\nThe magnetization dynamics in the slab is due to the effec-\ntive magnetic field, m0Heff=\u0000dF=dM. We define mas\nthe small perturbation of the magnetic order, i.e. jmj\u001cMs.\nLinearizing the Landau-Lifshitz-Gilbert equation,4we write\n\u0000¶mx\n¶t=gm0(HB+NyMs)my+gB¶U\n¶y+a¶my\n¶t;(3)\n¶my\n¶t=gm0(HB+NxMs)mx+gB¶U\n¶x+a¶mx\n¶t;(4)\nwhere gis the gyromagnetic ratio and ais the Gilbert\ndamping constant. To describe the acoustic wave, we in-\nclude the magneto-elastic contribution to the stress, s(ME)\njk=\ndFME=dujk, into the momentum balance equation:\nr¶2U\n¶t2=¶\n¶x\u0012\nC¶U\n¶x+B\nMsmx\u0013\n+¶\n¶y\u0012\nC¶U\n¶y+B\nMsmy\u0013\n;\n(5)\nwhere C=c44is the shear modulus and ris the mass density.\nThe non-magnetic medium is described by Eq.(5) with B=0.\nSince the values of C,B, and Nx;yare constant within each\nindividual material, we shall seek solutions of the equations in\nthe form of plane waves U;mx(y)µexp[i(kw;xx+kw;yy\u0000wt)].\nFrom herein, we consider all variables in the Fourier domain.For the magnetization precession in the magnetic layer driven\nby the acoustic wave, we thus obtain\nmx=gB(wkw;y+iewykw;x)\nw2\u0000ewxewyU; (6)\nmy=igB(ewxkw;y+iwkw;x)\nw2\u0000ewxewyU; (7)\nwhere we have denoted wx(y)=gm0(HB+Nx(y)Ms)and\newx(y)=wx(y)\u0000iwa. The complex-valued wave number kw;x\nis given by the dispersion relation\nk2\nw;x=r\nCw2\u0000\nw2\u0000ewxewy\u0001\n\u0000k2\nw;y\u0010\nw2\u0000ewxewy+gB2\nMsCewx\u0011\nh\nw2\u0000ewxewy+gB2\nMsCewyi ;\n(8)\nwhere kw;yis equal to that of the incident wave, and the\nbranch with Im kw;x>0 describes a forward wave decaying\ninto the slab. Eq. (8) describes the hybridization between\nacoustic waves and magnetic precession at frequencies close\nto ferromagnetic resonance (FMR) at frequency wFMR, with\nlinewidth GFMR. The frequency at which the precession am-\nplitudes (Eqs. (6) and (7)) diverge is given by the condition\n(wFMR+iGFMR=2)2=ewxewy. In the limit of small a, this\nyields wFMR=wxwyandGFMR=a(wx+wy). Away from\nthe resonance, Eq. (8) gives the linear dispersion of acous-\ntic waves. In the non-magnetic medium ( B=0), one finds\nk2\n0=w2r0=C0. Here and below, the subscript ’0’ is used to\nmark quantities pertaining to the non-magnetic matrix.\nTo calculate the reflection and transmission coefficients, Rw\nandTw, for a magnetic inclusion, we introduce the mechanical\nimpedance as Z=isxz=wUw. Solution of the wave matching\nproblem can then be expressed via the ratio of load ( ZME) and\nsource ( Z0) impedances. For impedances in the forward (F)\nand backward (B) directions in the magnetic slab, we find\nZ(F=B)\nw;ME=Ckw;x\nw0\n@1+gB2\nCM sewy\u0007iwkw;y\nkw;x\nw2\u0000ewxewy1\nA: (9)\nHere, the ‘-’ and ‘+’ signs correspond to (F) and (B), re-\nspectively. For the non-magnetic material, Eq. (9) recov-\ners the usual acoustic impedance45Z0=cosqpr0C0. Due\nto magnon-phonon hybridization, Re Z(F=B)\nw;MEdiverges at wFMR\nand vanishes at a nearby frequency wME. For a=0, the latter\nis given by\nwME=s\nwxwy\u0000gB2\nMsCwy: (10)\nReflection Rwand transmission Twcoefficients are then\nfound via the well-known relations45as\nRw=(ehw+1)(1\u0000hw)sin(kw;xd)\n(ehwhw+1)sin(kw;xd)+i(hw+ehw)cos(kw;xd);\n(11)\nTw=i(hw+ehw)\n(ehwhw+1)sin(kw;xd)+i(hw+ehw)cos(kw;xd);\n(12)Controlling acoustic waves using magneto-elastic Fano resonances 3\nwhere dis the thickness of the magnetic inclusion, hw=\nZ(F)\nME=Z0andehw=Z(B)\nME=Z0.46In close proximity to the res-\nonance, the impedances changes rapidly. Expanding Eq. (11)\nnearwMEin the limit kwd\u001c1, we obtain\nRw=iGR=2\n(w\u0000wME)+iGR=2eif+R0; (13)\nf=\u00002 arctan\u0014C\nC0rwx\nwytanq\u0015\n;\nwhere R0represents a smooth non-resonant contribution due\nto elastic mismatch at the interfaces, while frepresents a res-\nonant phase, which is non-zero for finite qand approaches p\nrapidly. In a system with no magnetic damping, the hybridiza-\ntion yields a resonance of finite linewidth GR,\nGR=gB2\n2MsC2cosqp\nr0C0\u0012\nwycos2q+C2\nC2\n0wxsin2q\u0013\nd:\n(14)\nThe origin of this linewidth can be explained as follows. Due\nto the magneto-elastic coupling incident propagating acoustic\nmodes can be converted into localised magnon modes. These\nmodes in turn either decay due to the Gilbert damping or are\nre-emitted as phonons. The rates of these transitions are pro-\nportional to GFMR andGR, respectively, and the total decay\nrate is G=GR+GFMR. This is similar to resonant scattering\nin quantum theory47, such that GRandGFMRare analogous to\nthe the elastic (Ge)and inelastic (Gi)linewidths respectively.\nWhen a=0,GFMRvanishes, and G=GR.\nAcoustic waves in the geometry of Fig. 1 can be scattered\nvia several channels. E.g. in a non-magnetic system ( B=0),\nelastic mismatch can yield Fabry-Pérot resonance due to the\nquarter wavelength matching of dand the acoustic wave-\nlength. However, this occurs at very high frequencies, which\nwe do not consider here. To understand the resonant magneto-\nelastic response, it is instructive to consider first the case of\nnormal incidence ( q=0), when the demagnetising energy\ntakes a simplified form due to the lack of immediate inter-\nfaces to form surface poles in ythe direction, so that Nx=1\nandNy=0. Including magneto-elastic coupling ( B6=0), we\nplot the frequency dependence of RwandTwusing Eq. (11)\nand (12) in Fig.2. To gain a quantitative insight, we analysed\na magnetic inclusion made of cobalt ( r=8900kgm\u00003,B=\n10MPa, C=80GPa, g=176GHzT\u00001,M=1MAm\u00001), em-\nbedded into a non-magnetic matrix ( r0=3192kgm\u00003;C0=\n298GPa). To highlight the resonant behaviour, we first sup-\npress ato 10\u00004. The reflection coefficient exhibits an asym-\nmetric non-monotonic dependence, shown as a black curve in\nFig.2(a), characteristic of Fano resonance.27,41This line shape\ncan be attributed to coupling between the discrete FMR mode\nof the magnetic inclusion and the continuum of propagating\nacoustic modes in the surrounding non-magnetic material.41\nIf the two materials had matching elastic properties, Rwwould\nexhibit a symmetric Breit-Wigner lineshape.47The transmis-\nsion shown in Fig.2(b) exhibits an approximately symmetric\ndip near the resonance.48The absorbancejAwj2=1\u0000jRwj2\u0000\njTwj2, shown in Fig.2(c) exhibits a symmetric peak, since theacoustic waves are damped in our model only due to the cou-\npling with spin waves.\nTo consider how the magneto-elastic resonance is affected\nby the damping, we also plot the response for aof 10\u00003and\n10\u00002, red and blue curves in Fig.2, respectively. An increase\nofafrom 10\u00004to 10\u00003significantly suppresses and broad-\nens the resonant peak. For a more common, realistic value of\n10\u00002the resonance is quenched entirely. A stronger magne-\ntoelastic coupling (i.e. high values of B) could, in principle,\ncountermand this suppression. This, however, is also likely\nto enhance the phonon contribution to the magnetic damping,\nleading to a correlation between Bandaobserved in realistic\nmagnetic materials.49\nTo characterise the strength of the Fano resonance, we note\nthat the fate of the magnon excited by the incident acoustic\nwave is decided by the relation between the emission rate GR,\nsee Eq. (14), and absorption rate GFMR. Hence, we introduce\nthe respective figure of merit as ¡=GR=GFMR. This quan-\ntity depends upon the material parameters, device geometry,\nand bias field. As seen from the first terms on the l.h.s. of\nEqs. (6) and (7), the relation between the dynamic magnetisa-\ntion components mx;yare determined by the quantities wxand\nwy. Equating these terms, one finds mxµmyp\nwy=wx, i.e. the\nprecession of mis highly elliptical,50due to the demagnetis-\ning field along x. This negatively affects the phonon-magnon\ncoupling for normal incidence ( ky=0): the acoustic wave\ncouples only to mx, as given by the second term in Eqs. (6)\nand (7). One way to mitigate this is to increase HB, mov-\ning the ratio wy=wxcloser to 1 and thus improving the figure\nof merit. To compare different magneto-elastic materials, the\ndependence on the layer thickness dand elastic properties of\nthe non-magnetic matrix (i.e. r0andC0) can be eliminated by\ncalculating a ratio of the figures of merit for the compared ma-\nterials. The comparison can be performed either at the same\nvalue of the bias field, or at the same operating frequency. The\nlatter situation is more appropriate for a device application,\nbut to avoid unphysical parameters, we present our results for\nthe same m0HB. An example of such comparisons for yttrium\niron garnet (YIG), cobalt (Co) and permalloy (Py) is offered\nin Table I.\nAnother way to improve ¡is to employ the oblique inci-\ndence ( q6=0), in which the acoustic mode is also coupled to\nthe magnetisation component my. The latter is not suppressed\nby the demagnetisation effects if Ny\u001c1. The resulting en-\nhancement in ¡is reflected in the full equation by the inclu-\nsion of wxandwyfromGR,\n¡=GR\nGFMR=gdB2\n2p\nr0C0\u0010\nHBcos2q+C2\nC2\n0Mssin2q\u0011\naC2M2scosq;(15)\nwhere wx\u001dwyandHB\u001cMsis assumed. For small q, the\napproximation Nx'1 and Ny'0 still holds. As a result, non-\nzeroqincreases peak reflectivity, as seen in Fig.3. The evolu-\ntion of the curves in Fig.3 with qis explained by the variation\nof the phase fof the resonant scattering relative to that of the\nnon-resonant contribution R0. The latter changes its sign at in-\ncidence angle of about 30\u000e, which yields a nearly symmetric\ncurve (blue), and an inverted Fano resonance at larger anglesControlling acoustic waves using magneto-elastic Fano resonances 4\n7.10 7.12 7.14 7.16 7.180.000.050.100.150.200.25|R(f)|(a)Damping, α:\n10−4\n10−3\n10−2\n7.10 7.12 7.14 7.16 7.18\nFrequency, f (GHz)0.750.800.850.900.951.00|T(f)|(b)\n7.10 7.12 7.14 7.16 7.180.00.10.20.3|A(f)|2(c)\nFIG. 2: The frequency dependence of the absolute values of (a) reflection and (b) transmission coefficients and (c) absorbance\nis shown for a 20nm thick magnetic inclusion. The vertical dashed and solid black lines represent the ferromagnetic resonance\nfrequency wFMRand magneto-elastic resonance frequency wMErespectively. The non-magnetic and magnetic materials are\nassumed to be silicon nitride and cobalt, respectively, with parameters given in the text. The bias field is m0HB=50mT, which\nleads to fME\u00197:138 GHz.\n(green). Although larger incidence angles may be hard to im-\nplement in a practical device, the resonant scattering is still\nenhanced at smaller angles.\nAbove, we have focused on the simplest geometry that ad-\nmits full analytic treatment. To implement our idea exper-\nimentally, particular care should be taken about the acous-\ntic waves polarization and propagation direction relative to\nthe direction of the magnetization. Indeed, our choice max-\nimises magnetoelastic response. If however, the polariza-\ntion is orthogonal to the bias field HB, i.e. Uz=0, the cou-\npling would be second-order in magnetization components\nmx;y, and would not contribute to the linearized LLG equation.\nFurthermore, we have neglected the exchange and magneto-\ndipolar fields that could arise due to the non-uniformity of the\nmagnetization. To assess the accuracy of this approximation,\nwe note that the length scale of this non-uniformity is set by\nthe acoustic wavelength l, of about 420nm for our parame-\nters rather than by the magnetic slab thickness d. The asso-\n7.00 7.05 7.10 7.15 7.20 7.25\nFrequency, f (GHz)0.010.020.030.040.050.06|R(f)|0◦\n15◦\n30◦45◦\nFIG. 3: Peak R(f)is enhanced and slightly shifted to the left\nin the oblique incidence geometry ( q>0\u000e). Coloured curves\nrepresent specific incidence angles sweeping from 0\u000eto 45\u000e.\nModerate Gilbert damping of a=10\u00003is assumed. The\ndashed vertical line corresponds to the magnetoelastic\nresonance frequency.\n0 10 20 30 40\nAngle, θ (deg.)0.000.020.040.06Figure of Merit, Υ\n0.050.100.150.200.25\nΓR(10−1)/FMR(ns)−1ΓFMR\nΓRΥFIG. 4: Figure of merit ¡and radiative linewidth GRare both\nenhanced in the oblique incidence geometry ( q>0\u000e).\nFerromagnetic linewidth GFMRremains unchanged. Co is\nassumed with a=10\u00003:\nciated exchange field is m0Ms(klex)2'9mT. The k-dependent\ncontributions to the magneto-dipole field vanish at normal in-\nTABLE I: Comparison of the figure of merit ¡for different\nmaterials, assuming d=20nm, m0HB=50mT and\nC0=298GPa.\nParameters YIG Co Py\n¡(q=0\u000e) 4:3x10\u000021:7x10\u000032:7x10\u00004\nGR(ns\u00001) 1 :9x10\u000047:5x10\u000032:0x10\u00004\nGFMR (ns\u00001) 4 :4x10\u000034.3 0.74\n¡(q=30\u000e) 4:1x10\u000022:5x10\u000032:8x10\u00004\nGR(ns\u00001) 1 :8x10\u000041:1x10\u000022:1x10\u00004\nGFMR (ns\u00001) 4 :4x10\u000034.3 0.74\nfME=wME=2p(GHz) 2.97 7.14 6.26\nB(MJm\u00003) 0.55 10 -0.9\nC(GPa) 74 80 50\nr(kgm\u00003) 5170 8900 8720\na 0:9x10\u000041:8x10\u000024:0x10\u00003\nMs(kAm\u00001) 140 1000 760Controlling acoustic waves using magneto-elastic Fano resonances 5\ncidence but may become significant at oblique incidence, giv-\ningm0Mskyd'98mT at q=15\u000e. In principle, these could\nincrease the resonant frequency of the slab by a few GHz but\nwould complicate the theory significantly. 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Phys. 13, 448 (2017)] on magnetic-\feld-driven annihilation of two magnetic\ndomain walls (DWs) but, furthermore, we predict that this setup additionally generates highly un-\nusual pumping of electronic spin currents in the absence of any bias voltage. Prior to the instant of\nannihilation, their power spectrum is ultrabroadband , so they can be converted into rapidly changing\nin time charge currents, via the inverse spin Hall e\u000bect, as a source of THz radiation of bandwidth\n'27 THz where the lowest frequency is controlled by the applied magnetic \feld. The spin pump-\ning stems from time-dependent \felds introduced into the quantum Hamiltonian of electrons by the\nclassical dynamics of localized magnetic moments (LMMs) comprising the domains. The pumped\ncurrents carry spin-polarized electrons which, in turn, exert backaction on LMMs in the form of\nnonlocal damping which is more than twice as large as conventional local Gilbert damping. The\nnonlocal damping can substantially modify the spectrum of emitted SWs when compared to widely-\nused micromagnetic simulations where conduction electrons are completely absent . Since we use fully\nmicroscopic (i.e., Hamiltonian-based) framework, self-consistently combining time-dependent elec-\ntronic nonequilibrium Green functions with the Landau-Lifshitz-Gilbert equation, we also demon-\nstrate that previously derived phenomenological formulas miss ultrabroadband spin pumping while\nunderestimating the magnitude of nonlocal damping due to nonequilibrium electrons .\nIntroduction .|The control of the domain wall (DW)\nmotion1{3within magnetic nanowires by magnetic \feld\nor current pulses is both a fundamental problem for\nnonequilibrium quantum many-body physics and a build-\ning block of envisaged applications in digital memories.4\nlogic5and arti\fcial neural networks.6Since DWs will be\nclosely packed in such devices, understanding interaction\nbetween them is a problem of great interest.7For ex-\nample, head-to-head or tail-to-tail DWs|illustrated as\nthe left (L) or right (R) noncollinear texture of local-\nized magnetic moments (LMMs), respectively, in Fig. 1|\nbehave as free magnetic monopoles carrying topological\ncharge.8The topological charge (or the winding number)\nQ\u0011\u00001\n\u0019R\ndx@x\u001e, associated with winding of LMMs as\nthey interpolate between two uniform degenerate ground\nstates with \u001e= 0 or\u001e=\u0019, is opposite for adjacent\nDWs, such as QL=\u00001 andQR= +1 for DWs in Fig. 1.\nThus, long-range attractive interaction between DWs\ncan lead to their annihilation , resulting in the ground\nstate without any DWs.9{12This is possible because to-\ntal topological charge remains conserved, QL+QR= 0.\nThe nonequilibrium dynamics13triggered by annihilation\nof topological solitons is also of great interest in many\nother \felds of physics, such as cosmology,14gravitational\nwaves,15quantum13and string \feld16theories, liquid\ncrystals17and Bose-Einstein condensates.18,19\nThe recent experiment20has monitored annihilation\nof two DWs within a metallic ferromagnetic nanowire by\nobserving intense burst of spin waves (SWs) at the mo-\nment of annihilation. Thus generated large-amplitude\nSWs are dominated by exchange, rather than dipolar,\ninteraction between LMMs and are, therefore, of short\nwavelength. The SWs of \u001810 nm wavelength are cru-cial for scalability of magnonics-based technologies,21,22\nlike signal transmission or memory-in-logic and logic-\nin-memory low-power digital computing architectures.\nHowever, they are di\u000ecult to excite by other methods\ndue to the requirement for high magnetic \felds.23,24\nThe computational simulations of DW annihila-\ntion,9,10,20together with theoretical analysis of generic\nfeatures of such a phenomenon,11have been based exclu-\nsively on classical micromagnetics where one solves cou-\npled Landau-Lifshitz-Gilbert (LLG) equations25for the\ndynamics of LMMs viewed as rotating classical vectors of\n\fxed length. On the other hand, the dynamics of LMMs\ncomprising two DWs also generates time-dependent \felds\nwhich will push the surrounding conduction electrons out\nof equilibrium. The nonequilibrium electrons comprise\npumped spin current26{28(as well as charge currents if\nthe left-right symmetry of the device is broken28,29) in\nthe absence of any externally applied bias voltage. The\npumped spin currents \row out of the DW region into\nthe external circuit, and since they carry away excess an-\ngular momentum of precessing LMMs, the backaction of\nnonequilibrium electrons on LMMs emerges26as an ad-\nditional damping-like (DL) spin-transfer torque (STT).\nThe STT, as a phenomenon in which spin angu-\nlar momentum of conduction electrons is transferred\nto LMMs when they are not aligned with electronic\nspin-polarization, is usually discussed for externally in-\njected spin current.30But here it is the result of compli-\ncated many-body nonequilibrium state in which LMMs\ndrive electrons out of equilibrium which, in turn, ex-\nertbackaction in the form of STT onto LMMs to\nmodify their dynamics in a self-consistent fashion.27,31\nSuch e\u000bects are absent from classical micromagneticsarXiv:1908.03194v5  [cond-mat.mes-hall]  24 Jun 20212\nFIG. 1. Schematic view of a ferromagnetic nanowire modeled\nas a 1D tight-binding chain whose sites host classical LMMs\n(red arrows) interacting with spins (blue arrow) of conduc-\ntion electrons. The nanowire is attached to two NM leads\nterminating into the macroscopic reservoirs kept at the same\nchemical potential. The two DWs within the nanowire carry\nopposite topological charge,8QL=\u00001 for the L one and\nQR= +1 for the R one. They collide with the opposite ve-\nlocities VL\nDWandVR\nDWand annihilate, upon application of an\nexternal magnetic \feld Bextparallel to the nanowire, thereby\nmimicking the setup of the experiment in Ref. 20.\nor atomistic spin dynamics25because they do not in-\nclude conduction electrons. This has prompted deriva-\ntion of a multitude of phenomenological expressions32{39\nfor the so-called nonlocal (i.e., magnetization-texture-\ndependent) and spatially nonuniform (i.e., position-\ndependent) Gilbert damping that could be added into\nthe LLG equation and micromagnetics codes40{42to cap-\nture the backaction of nonequilibrium electrons while not\nsimulating them explicitly. Such expressions do not re-\nquire spin-orbit (SO) or magnetic disorder scattering,\nwhich are necessary for conventional local Gilbert damp-\ning,43{45but they were estimated33,36to be usually a\nsmall e\u000bect unless additional conditions (such as narrow\nDWs or intrinsic SO coupling splitting the band struc-\nture33) are present. On the other hand, a surprising\nresult40of Gilbert damping extracted from experiments\non magnetic-\feld-driven DW being several times larger\nthan the value obtained from standard ferromagnetic res-\nonance measurements can only be accounted by including\nadditional nonlocal damping.\nIn this Letter, we unravel complicated many-body\nnonequilibrium state of LMMs and conduction elec-\ntrons created by DW annihilation using recently de-\nveloped27,46{49quantum-classical formalism which com-\nbines time-dependent nonequilibrium Green function\n(TDNEGF)50,51description of quantum dynamics of con-\nduction electrons with the LLG equation description of\nclassical dynamics of LMMs on each atom.25Such TD-\nNEGF+LLG formalism is fully microscopic, since it re-\nquires only the quantum Hamiltonian of electrons and the\nclassical Hamiltonian of LMMs as input, and numerically\nexact . We apply it to a setup depicted in Fig. 1 where\ntwo DWs reside at time t= 0 within a one-dimensional\n(1D) magnetic nanowire attached to two normal metal\n(NM) leads, terminating into the macroscopic reservoirs\nwithout any bias voltage.\nOur principal results are: ( i) annihilation of two DWs\n[Fig. 2] pumps highly unusual electronic spin currents\nwhose power spectrum is ultrabroadband prior to the in-\nFIG. 2. (a) Sequence of snapshots of two DWs, in the course\nof their collision and annihilation in the setup of Fig. 1; and\n(b) the corresponding time-dependence of the z-component\nof LMMs where blue and orange line mark t= 6:9 ps (when\ntwo DWs start vanishing) and t= 7:2 ps (when all LMMs\nbecome nearly parallel to the x-axis) from panel (a). A movie\nanimating panels (a) and (b) is provided in the SM.58Spatio-\ntemporal pro\fle of: (c) angle \u000eeq\niand (d) \\nonadiabaticity\"\nangle\u000eneq\ni\u0000\u000eeq\ni, with the meaning of \u000eneq\niand\u000eeq\niillustrated in\nthe inset above panel (c); (e) DL STT [Eq. (3)] as electronic\nbackaction on LMMs; (f) ratio of DL STT to conventional\nlocal Gilbert damping [Eq. (2)]; and (g) ratio of the sum of\nDL STT to the sum of conventional local Gilbert damping\nover all LMMs.\nstant of annihilation [Fig. 3(d)], unlike the narrow peak\naround a single frequency for standard spin pumping;26\n(ii) because pumped spin currents carry away excess\nangular momentum of precessing LMMs, this acts as\nDL STT on LMMs which is spatially [Figs. 2(e) and\n4(b)] and time [Fig. 2(g)] dependent, as well as '2:4\ntimes larger [Fig. 2(f)] than conventional local Gilbert\ndamping [Eq. (2)]. This turns out to be remarkably\nsimilar to'2:3 ratio of nonlocal and local Gilbert\ndamping measured experimentally in permalloy,40but\nit is severely underestimated by phenomenological the-\nories32,33[Fig. 4(a),(b)].\nModels and methods .|The classical Hamiltonian for3\n≃ 27 THz\nFIG. 3. Time dependence of: (a){(c) electronic spin currents pumped into the right NM lead during DW collision and annihila-\ntion; (e){(g) SW-generated contribution to spin currents in panels (a){(c), respectively, after spin current carried by SW from\nFig. 2(b) is stopped at the magnetic-nanowire/nonmagnetic-NM-lead interface and converted (as observed experimentally20,61)\ninto electronic spin current in the right NM lead. Vertical dashed lines mark times t= 6:9 ps andt= 7:2 ps whose snapshots\nof LMMs are shown in Fig. 2(a). For easy comparison, gray curves in panels (f) and (g) are the same as the signal in panels (b)\nand (c), respectively, for post-annihilation times t\u00157:2 ps. Panels (d) and (h) plot FFT power spectrum of signals in panels\n(c) and (g), respectively, before (red curve) and after (brown curves) completed annihilation at t= 7:2 ps.\nLMMs, described by unit vectors Mi(t) at each site iof\n1D lattice, is chosen as\nH=\u0000JX\nhijiMi\u0001Mj\u0000KX\ni(Mx\ni)2\n+DX\ni(My\ni)2\u0000\u0016BX\niMi\u0001Bext; (1)\nwhereJ= 0:1 eV is the Heisenberg exchange coupling\nbetween the nearest-neighbor LMMs; K= 0:05 eV is the\nmagnetic anisotropy along the x-axis; andD= 0:007 eV\nis the demagnetizing \feld along the y-axis. The last term\nin Eq. (1) is the Zeeman energy ( \u0016Bis the Bohr magne-\nton) describing the interaction of LMMs with an external\nmagnetic \feld Bextparallel to the nanowire in Fig. 1 driv-\ning the DW dynamics, as employed in the experiment.20\nThe classical dynamics of LMMs is described by a system\nof coupled LLG equations25(using notation @t\u0011@=@t)\n@tMi=\u0000gMi\u0002Be\u000b\ni+\u0015Mi\u0002@tMi\n+g\n\u0016M\u0010\nTih\nIS\u000b\nexti\n+Ti[Mi(t)]\u0011\n: (2)\nwhere Be\u000b\ni=\u00001\n\u0016M@H=@Miis the e\u000bective magnetic\n\feld (\u0016Mis the magnitude of LMMs); gis the gy-\nromagnetic ratio; and the magnitude of conventional\nlocal Gilbert damping is speci\fed by spatially- and\ntime-independent \u0015, set as\u0015= 0:01 as the typi-\ncal value measured40in metallic ferromagnets. The\nspatial pro\fle of a single DW in equilibrium, i.e.,\nat timet= 0 as the initial condition, is given by\nMi(Q;X DW) =\u0000\ncos\u001ei(Q;X DW);0;sin\u001ei(Q;X DW)\u0001\n,\nwhere\u001ei(Q;X DW) =Qarccos [tanh ( xi\u0000XDW)];Q\nis the topological charge; and XDW is the positionof the DW. The initial con\fguration of two DWs,\nMi(t= 0) = Mi(QL;XL) +Mi(QR;XR), positioned at\nsitesXL= 15 andXR= 30 harbors opposite topological\nchargesQR=\u0000QL= 1 around these sites.\nIn general, two additional terms32,33,52in Eq. (2) ex-\ntend the original LLG equation|STT due to externally\ninjected electronic spin current,30which is actually ab-\nsentTih\nIS\u000b\nexti\n\u00110 in the setup of Fig. 1; and STT due to\nbackaction of electrons\nTi[Mi(t)] =Jsd(h^siineq(t)\u0000h^siieq\nt)\u0002Mi(t); (3)\ndriven out of equilibrium by Mi(t). HereJsd= 0:1 eV\nis chosen as the s-dexchange coupling (as mea-\nsured in permalloy53) between LMMs and electron\nspin. We obtain \\adiabatic\"54,55electronic spin density,\nh^siieq\nt= Tr [ \u001aeq\ntjiihij\n\u001b], from grand canonical equilib-\nrium density matrix (DM) for instantaneous con\fgura-\ntion of Mi(t) at timet[see Eq. (5)]. Here \u001b= (^\u001bx;^\u001by;^\u001bz)\nis the vector of the Pauli matrices. The nonequilibrium\nelectronic spin density, h^siineq(t) = Tr [ \u001aneq(t)jiihij\n\u001b],\nrequires to compute time-dependent nonequilibrium DM,\n\u001aneq(t) =~G<(t;t)=i, which we construct using TD-\nNEGF algorithms explained in Refs. 56 and 57 and com-\nbine27with the classical LLG equations [Eq. (2)] using\ntime step\u000et= 0:1 fs. The TDNEGF calculations require\nas an input a quantum Hamiltonian for electrons, which\nis chosen as the tight-binding one\n^H(t) =\u0000\rX\nhiji^cy\ni^ci\u0000JsdX\ni^cy\ni\u001b\u0001Mi(t)^ci: (4)\nHere ^cy\ni= (^cy\ni\";^cy\ni#) is a row vector containing operators\n^cy\ni\u001bwhich create an electron of spin \u001b=\";#at the sitei,4\nand ^ciis a column vector that contains the correspond-\ning annihilation operators; and \r= 1 eV is the nearest-\nneighbor hopping. The magnetic nanowire in the setup\nin Fig. 1 consists of 45 sites and it is attached to semi-\nin\fnite NM leads modeled by the \frst term in ^H. The\nFermi energy of the reservoirs is set at EF= 0 eV. Due\nto the computational complexity of TDNEGF calcula-\ntions,51we use magnetic \feld jBextj= 100 T to complete\nDW annihilation on \u0018ps time scale (in the experiment20\nthis happens within \u00182 ns).\nResults .|Figure 2(a) demonstrates that\nTDNEGF+LLG-computed snapshots of Mi(t)fully\nreproduce annihilation in the experiment,20including \f-\nnalewhen SW burst is emitted at t'7:2 ps in Fig. 2(b).\nThe corresponding complete spatio-temporal pro\fles\nare animated as a movie provided in the Supplemental\nMaterial (SM).58However, in contrast to micromagnetic\nsimulations of Ref. 20 where electrons are absent,\nFig. 2(d) shows that they generate spin density h^siineq(t)\nwhich is noncollinear with either Mi(t) orh^siieq\nt. This\nleads to \\nonadiabaticity\" angle ( \u000eneq\ni\u0000\u000eeq\ni)6= 0 in\nFig. 2(d) and nonzero STT [Eq. (3) and Fig. 2(e)] as\nself-consistent backaction of conduction electrons onto\nLMMs driven out of equilibrium by the dynamics of\nLMMs themselves. The STT vector, Ti=TFL\ni+TDL\ni,\ncan be decomposed [see inset above Fig. 2(e)] into: ( i)\neven under time-reversal or \feld-like (FL) torque, which\na\u000bects precession of LMM around Be\u000b\ni; and ( ii) odd\nunder time-reversal or DL torque, which either enhances\nGilbert term [Eq. (2)] by pushing LMM toward Be\u000b\nior\ncompetes with it as antidamping. Figure 2(f) shows that\nTDL\ni[Mi(t)] acts like an additional nonlocal damping\nwhile being'2:4 times larger than conventional local\nGilbert damping \u0015Mi\u0002@tMi[Eq. (2)].\nThe quantum transport signature of DW vanishing\nwithin the time interval t= 6:9{7:2 ps in Fig. 2(a) is the\nreduction in the magnitude of pumped electronic spin\ncurrents [Fig. 3(a){(c)]. In fact, ISx\nR(t)!0 becomes\nzero [Fig. 3(a)] at t= 7:2 ps at which LMMs in Fig. 2(a)\nturn nearly parallel to the x-axis while precessing around\nit. The frequency power spectrum [red curve in Fig. 3(d)]\nobtained from fast Fourier transform (FFT) of ISz\nR(t), for\ntimes prior to completed annihilation and SW burst at\nt= 7:2 ps, reveal highly unusual spin pumping over an\nultrabroadband frequency range. This can be contrasted\nwith the usual spin pumping26whose power spectrum\nis just a peak around a single frequency,59as also ob-\ntained [brown curve in Fig. 3(d)] by FFT of ISz\nR(t) at\npost-annihilation times t>7:2 ps.\nThe spin current in Fig. 3(a){(c) has contributions\nfrom both electrons moved by time-dependent Mi(t) and\nSW hitting the magnetic-nanowire/NM-lead interface.\nAt this interface, SW spin current is stopped and trans-\nmuted47,48,60into an electronic spin current \rowing into\nthe NM lead. The transmutation is often employed ex-\nperimentally for direct electrical detection of SWs, where\nan electronic spin current on the NM side is converted\ninto a voltage signal via the inverse spin Hall e\u000bect.20,61\n~10-7FIG. 4. Spatial pro\fle at t= 6:9 ps of: (a) locally pumped\nspin current ISx\ni!j47between sites iandj; and nonlocal damp-\ning due to backaction of nonequilibrium electrons . Solid lines\nin (a) and (b) are obtained from TDNEGF+LLG calcula-\ntions, and dashed lines are obtained from SMF theory phe-\nnomenological formulas.32,33,69(c){(e) FFT power spectra22\nofMz\ni(t) where (c) and (d) are TDNEGF+LLG-computed\nwith\u0015= 0:01 and\u0015= 0, respectively, while (e) is LLG-\ncomputed with backaction of nonequilibrium electrons re-\nmoved, Ti[Mi(t)]\u00110, in Eq. (2). The dashed horizontal\nlines in panels (c){(e) mark frequencies of peaks in Fig. 3(d).\nWithin the TDNEGF+LLG picture, SW reaching the\nlast LMM of the magnetic nanowire, at the sites i= 1\nori= 45 in our setup, initiates their dynamics whose\ncoupling to conduction electrons in the neighboring left\nand right NM leads, respectively, leads to pumping47of\nthe electronic spin current into the NM leads. The prop-\nerly isolated electronic spin current due to transmutation\nof SW burst, which we denote by IS\u000b;SW\np , is either zero\nor very small until the burst is generated in Fig. 3(e){\n(g), as expected. We note that detected spin current in\nthe NM leads was attributed in the experiment20solely\nto SWs, which corresponds in our picture to considering\nonlyIS\u000b;SW\np while disregarding IS\u000bp\u0000IS\u000b;SW\np .\nDiscussion .|A computationally simpler alternative to\nour numerical self-consistent TDNEGF+LLG is to \\in-\ntegrate out electrons\"31,62{65and derive e\u000bective expres-\nsions solely in terms of Mi(t), which can then be added\ninto the LLG Eq. (2) and micromagnetics codes.40{42\nFor example, spin motive force (SMF) theory69gives\nISx\nSMF(x) =g\u0016B~G0\n4e2[@M(x;t)=@t\u0002@M(x;t)=@x]xfor the\nspin current pumped by dynamical magnetic texture, so\nthatM\u0002D\u0001@tMis the corresponding nonlocal Gilbert\ndamping.32,33Here M(x;t) is local magnetization (as-\nsuming our 1D system); D\u000b\f=\u0011P\n\u0017(M\u0002@\u0017M)\u000b(M\u0002\n@\u0017M)\f(using notation \u000b;\f;\u00172 fx;y;zg) is 3\u00023\nspatially-dependent damping tensor; and \u0011=g\u0016B~G0\n4e2\nwithG0=G\"+G#being the total conductivity. We\ncompare in Fig. 4: ( i) spatial pro\fle of ISx\nSMF(x) to locally\npumped spin current ISx\ni!j47from TDNEGF+LLG calcu-5\nlations [Fig. 4(a)] to \fnd that the former predicts negli-\ngible spin current \rowing into the leads, thereby missing\nultrabroadband spin pumping predicted in Fig. 3(d); ( ii)\nspatial pro\fle of M\u0002D\u0001@tMto DL STT TDL\nifrom TD-\nNEGF+LLG calculations, to \fnd that the former has\ncomparable magnitude only within the DW region but\nwith substantially di\u000bering pro\fles. Note also that47\n[P\niTi(t)]\u000b=~\n2eh\nIS\u000b\nL(t) +IS\u000b\nR(t)i\n+P\ni~\n2@h^ s\u000b\niineq\n@t, which\nmakes the sum of DL STT plotted in Fig. 2(g) time-\ndependent during collision, in contrast to the sum of lo-\ncal Gilbert damping shown in Fig. 2(g). The backaction\nof nonequilibrium electrons viaTi[Mi(t)] can strongly\na\u000bect the dynamics of LMMs, especially for the case of\nshort wavelength SWs and narrow DWs,32,33,41,42as con-\n\frmed by comparing FFT power spectra of Mz\ni(t) com-\nputed by TDNEGF+LLG [Fig. 4(c),(d)] with those from\nLLG calculations [Fig. 4(e)] without any backaction .\nWe note that SMF theory69is derived in the \\adi-\nabatic\" limit,2,54which assumes that electron spin re-\nmains in the the lowest energy state at each time. \\Adi-\nabaticity\" is used in two di\u000berent contexts in spintron-\nics with noncollinear magnetic textures|temporal and\nspatial.2In the former case, such as when electrons in-\nteract with classical macrospin due to collinear LMMs,\none assumes that classical spins are slow and h^siineq(t)\ncan \\perfectly lock\"2to the direction Mi(t) of LMMs.\nIn the latter case, such as for electrons traversing thick\nDW, one assumes that electron spin keeps the lowest en-\nergy state by rotating according to the orientation of\nMi(t) at each spatial point, thereby evading re\rection\nfrom the texture.2The concept of \\adiabatic\" limit is\nmade a bit more quantitative by considering2ratio of\nrelevant energy scales, Jsd=~!\u001d1 orJsd=\u0016BjBextj\u001d1,\nin the former case; or combination of energy and spa-\ntial scales,JsddDW=~vF=JsddDW=\ra\u001d1, in the latter\ncase (where vFis the Fermi velocity, ais the lattice spac-\ning anddDWis the DW thickness). In our simulations,\nJsd=\u0016BjBextj\u001910 andJsddDW=\ra\u00191 fordDW\u001910a\nin Fig. 2(a). Thus, it seems that fair comparison of our\nresults to SMF theory requires to substantially increase\nJsd. However, Jsd= 0:1 eV (i.e.,\r=Jsd=\u001810, for typical\n\r\u00181 eV which controls how fast is quantum dynam-\nics of electrons) in our simulations is \fxed by measured\nproperties of permalloy.53\nLet us recall that rigorous de\fnition of \\adiabaticity\"\nassumes that conduction electrons within a closed quan-\ntum system54at timetare in the ground state j\t0i\nfor the given con\fguration of LMMs Mi(t),j\t(t)i=\nj\t0[Mi(t)]i; or in open system55their quantum state is\nspeci\fed by grand canonical DM\n\u001aeq\nt=\u00001\n\u0019Z\ndEImGr\ntf(E): (5)\nwhere the retarded GF, Gr\nt=\u0002\nE\u0000H[Mi(t)]\u0000\u0006L\u0000\u0006R\u0003\u00001, and \u001aeq\ntdepend parametrically66{68(or implic-\nitly, so we put tin the subscript) on time via instanta-\nneous con\fguration of Mi(t), thereby e\u000bectively assum-\ning@tMi(t) = 0. Here Im Gr\nt= (Gr\nt\u0000[Gr\nt]y)=2i;\u0006L;R\nare self-energies due to the leads; and f(E) is the Fermi\nfunction. For example, the analysis of Ref. 69 assumes\nh^siineq(t)kh^siieq\ntto reveal the origin of spin and charge\npumping in SMF theory|nonzero angle \u000eeq\nibetween\nh^siieq\ntandMi(t) with the transverse component scaling\njh^siieq\nt\u0002Mi(t)j=\u0000\nh^siieq\nt\u0001Mi(t)\u0001\n/1=Jsdas the signature\nof \\adiabatic\" limit. Note that our \u000eeq\ni.4\u000e[Fig. 2(c)]\nin the region of two DWs (and \u000eeq\ni!0 elsewhere). Ad-\nditional Figs. S1{S3 in the SM,58where we isolate two\nneighboring LMMs from the right DW in Fig. 1 and\nput them in steady precession with frequency !for sim-\nplicity of analysis, demonstrate that entering such \\adi-\nabatic\" limit requires unrealistically large Jsd&2 eV.\nAlso, our exact55result [Figs. S1(b), S2(b) and S3(b) in\nthe SM58] showsjh^siieq\nt\u0002Mi(t)j=\u0000\nh^siieq\nt\u0001Mi(t)\u0001\n/1=J2\nsd\n(instead of/1=Jsdof Ref. 69). Changing ~!|which,\naccording to Fig. 3(c), is e\u000bectively increased by the\ndynamics of annihilation from ~!'0:01 eV, set ini-\ntially by Bext, toward ~!'0:1 eV|only modi\fes scal-\ning of the transverse component of h^siineq(t) withJsd\n[Figs. S1(a), S2(a), S3(a), S4(b) and S4(d) in the SM58].\nThe nonadiabatic corrections55,66{68take into account\n@tMi(t)6= 0. We note that only in the limit Jsd!1 ,\u0000\nh^siineq(t)\u0000h^siieq\nt\u0001\n!0. Nevertheless, multiplication\nof these two limits within Eq. (3) yields nonzero geo-\nmetric STT,54,55as signi\fed by Jsd-independent STT\n[Figs. S1(c), S2(c) and S3(c) in the SM58]. Otherwise,\n\\nonadiabaticity\" angle is always present ( \u000eneq\ni\u0000\u000eeq\ni)6= 0\n[Fig. 2(d)], even in the absence of spin relaxation due to\nmagnetic impurities or SO coupling,70and it can be di-\nrectly related to additional spin and charge pumping48,70\n[see also Figs. S1(f), S2(f) and S3(f) in the SM58].\nConclusions and outlook .|The pumped spin current\nover ultrabroadband frequency range [Fig. 3(d)], as our\ncentral prediction, can be converted into rapidly chang-\ning transient charge current via the inverse spin Hall ef-\nfect.71{73Such charge current will, in turn, emit electro-\nmagnetic radiation covering \u00180:03{27 THz range (for\njBextj\u00181 T) or\u00180:3{27:3 THz range (for jBextj\u001810\nT), which is highly sought range of frequencies for variety\nof applications.72,73\nACKNOWLEDGMENTS\nM. D. P., U. B., and B. K. N. was supported by the\nUS National Science Foundation (NSF) Grant No. ECCS\n1922689. P. P. was supported by the US Army Research\nO\u000ece (ARO) MURI Award No. W911NF-14-0247.6\n\u0003bnikolic@udel.edu\n1G. Tatara, H. Kohno, and J. Shibata, Microscopic ap-\nproach to current-driven domain wall dynamics, Phys.\nRep.468, 213 (2008).\n2G. 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Then we\nshow that the global solution converges to that of the Schr¨ oding er maps in the\nnatural space as the Gilbert damping term vanishes. The proof is ba sed on some\nstudies on the derivative Ginzburg-Landau equations.\n1.Introduction\nInthis paperwe study theCauchy problemfortheLandau-Lifshitz -Gilbert (LLG)\nequation\n∂ts=as×∆s−εs×(s×∆s), s(x,0) =s0(x), (1.1)\nwheres(x,t) :Rn×R→S2⊂R3,×denotes the wedge product in R3,a∈R\nandε >0 is the Gilbert damping parameter. The equation (1.1) is one of the\nequations of ferromagnetic spin chain, which was proposed by Land au-Lifshitz [19]\nin studying the dispersive theory of magnetisation of ferromagnet s. Later on, such\nequations were also found in the condensed matter physics. The LL G equation has\nbeen studied extensively, see [17, 7] for an introduction on the equ ation.\nFormally, if a= 0, then (1.1) reduces to the heat flow equations for harmonic\nmaps\n∂ts=−εs×(s×∆s), s(x,0) =s0(x), (1.2)\nand ifε= 0, then (1.1) reduces to the Schr¨ odinger maps\n∂ts=as×∆s, s(x,0) =s0(x). (1.3)\nBoth special cases have been objects of intense research. The p urpose of this paper\nis to study the inviscid limit of (1.1), namely, to prove rigorously that t he solutions\nof (1.1) converges to the solutions of (1.3) as ε→0 under optimal conditions on\nthe initial data.\nThe inviscid limit is an important topic in mathematical physics, and has b een\nstudied in various settings, e.g. for hyperbolic-dissipative equation s such as Navier-\nStokes equation to Euler equation (see [11] and references ther ein), for dispersive-\ndissipative equations such as KdV-Burgers equation to KdV equatio n (see [9]) and\nGinzburg-Landau equation to Schr¨ odinger equations (see [23, 12 ]). The LLG equa-\ntion (1.1) is an equation with both dispersive and dissipative effects. T his can be\n2010Mathematics Subject Classification. 35Q55.\nKey words and phrases. Landau-Lifshitz-Gilbert equation, Schr¨ odinger maps, Inviscid limit ,\nCritical Besov Space.\n12 Z. GUO AND C. HUANG\nseen from the stereographic projection transform. It was know n that (see [18]) let\nu=P(s) =s1+is2\n1+s3, (1.4)\nwheres= (s1,s2,s3) is a solution to (1.1), then usolves the following complex\nderivative Ginzburg-Landau type equation\n(i∂t+∆−iε∆)u=2a¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2−2iε¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2\nu(x,0) =u0.(1.5)\nOn the other hand, the projection transform has an inverse\nP−1(u) =/parenleftbiggu+ ¯u\n1+|u|2,−i(u−¯u)\n1+|u|2,1−|u|2\n1+|u|2/parenrightbigg\n. (1.6)\nTherefore, (1.1) is equivalent to (1.5) assuming PandP−1is well-defined, and we\nwill focus on (1.5). The previous works [13, 14, 1, 2, 3] on the Schr¨ odinger maps\n(ε= 0) were also based on this transform. Note that (1.5) is invariant u nder the\nfollowing scaling transform: for λ >0\nu(x,t)→u(λx,λ2t), u0(x)→u0(λx).\nThus the critical Besov space is ˙Bn/2\n2,1in the sense of scaling.\nTo study the inviscid limit, the crucial task is to obtain uniform well-pos edness\nwith respect to the inviscid parameter. Energy method was used in [1 1]. For\ndispersive-dissipative equations, one needs to exploit the dispersiv e effect uniformly.\nStrichartz estimates and energy estimates were used in [23] for Gin zburg-Landau\nequations, andBourgainspacewasused in[9]forKdV-Burgersequ ations. In[24,10]\nthe inviscid limit for the derivative Ginzburg-Landau equations were s tudied by us-\ning the Strichartz estimates, local smoothing estimates and maxima l function esti-\nmates. However, these results requires high regularities when app lied to equation\n(1.5). In this paper we will use Bourgain-type space and exploit the n ull structure\nthat are inspired by the latest development for the Schr¨ odinger m aps (ε= 0) (see\n[4, 3, 1, 2, 13, 14, 8]) to study (1.5) with small initial data in the critica l Besov\nspace. In [2] and [14] it was proved independently that global well-p osedness for\n(1.3) holds for small data in the critical Besov space. We will extend t heir results\nto (1.1) uniformly with respect to ε. We exploit the Bourgain space in a differ-\nent way from both [2] and [14]. One of the novelties is the use of X0,1-structure\nthat results in many simplifications even for the Schr¨ odinger maps. The presence\nof dissipative term brings many technical difficulties, e.g. the lack of s ymmetry\nin time and incompatibility with Xs,bstructure. We need to overcome these dif-\nficulties when extending the linear estimates for the Schr¨ odinger e quation to the\nSchr¨ odinger-dissipative equation uniformly with respect to ε.\nBy scaling we may assume a=±1. From now on, we assume a= 1 since the\nother case a=−1 is similar. For Q∈S2, the space ˙Bs\nQis defined by\n˙Bs\nQ=˙Bs\nQ(Rn;S2) ={f:Rn→R3;f−Q∈˙Bs\n2,1,|f(x)| ≡1 a.e. in Rn},\nwhere˙Bs\n2,1is the standard Besov space. It was known the critical space is ˙Bn/2\nQ.\nThe main result of this paper isLANDAU-LIFSHITZ EQUATION 3\nTheorem 1.1. Assumen≥3. The LLG equation (1.1)is globally well-posed for\nsmall data s0∈˙Bn/2\nQ(Rn;S2),Q∈S2in a uniform way with respect to ε∈(0,1].\nMoreover, for any T >0, the solution converges to that of Schr¨ odinger map (1.3)\ninC([−T,T] :˙Bn/2\nQ)asε→0.\nAs we consider the inviscid limit in the strongest topology (same space as the\ninitial data), no convergence rate is expected. This can be seen fr om linear solutions\nfor (1.5). However, if assuming initial data has higher regularity, on e can have\nconvergence rate O(εT) (see (5.3) below).\n2.Definitions and Notations\nForx,y∈R,x/lessorsimilarymeans that there exists a constant Csuch that x≤Cy, and\nx∼ymeans that x/lessorsimilaryandy/lessorsimilarx. We use F(f),ˆfto denote the space-time Fourier\ntransform of f, andFxi,tfto denote the Fourier transform with respect to xi,t.\nLetη:R→[0,1] be an even, non-negative, radially decreasing smooth function\nsuch that: a) ηis compactly supported in {ξ:|ξ| ≤8/5}; b)η≡1 for|ξ| ≤5/4. For\nk∈Zletχk(ξ) =η(ξ/2k)−η(ξ/2k−1),χ≤k(ξ) =η(ξ/2k),/tildewideχk(ξ) =/summationtext9n\nl=−9nχk+l(ξ),\nand then define the Littlewood-Paley projectors Pk,P≤k,P≥konL2(Rn) by\n/hatwidestPku(ξ) =χk(|ξ|)/hatwideu(ξ),/hatwideP≤ku(ξ) =χ≤k(|ξ|)/hatwideu(ξ),\nandP≥k=I−P≤k−1,P[k1,k2]=/summationtextk2\nj=k1Pj. We also define /tildewidePku=F−1/tildewideχk(|ξ|)/hatwideu(ξ)\nLetSn−1be the unit sphere in Rn. Fore∈Sn−1, define /hatwidePk,eu(ξ) =/tildewideχk(|ξ·\ne|)χk(|ξ|)/hatwideu(ξ). Since for |ξ| ∼2kwe have\n5n/summationdisplay\nl=−5nχk+l(ξ1)+···+5n/summationdisplay\nl=−5nχk+l(ξn)∼1,\nthen let\nβj\nk(ξ) =/summationtext5n\nl=−5nχk+l(ξj)\n/summationtextn\nj=1/summationtext5n\nl=−5nχk+l(ξj)·1/summationdisplay\nl=−1χk+l(|ξ|), j= 1,···,n.\nDefine the operator Θj\nkonL2(Rn) by/hatwidestΘj\nkf(ξ) =βj\nk(ξ)ˆf(ξ), 1≤j≤n. Lete1=\n(1,0,···,0),···,en= (0,···,0,1). Then we have\nPk=n/summationdisplay\nj=1Pk,ejΘj\nk. (2.1)\nFor anyk∈Z, we define the modulation projectors Qk,Q≤k,Q≥konL2(Rn×R) by\n/hatwidestQku(ξ,τ) =χk(τ+|ξ|2)/hatwideu(ξ,τ),/hatwideQ≤ku(ξ,τ) =χ≤k(τ+|ξ|2)/hatwideu(ξ,τ),\nandQ≥k=I−Q≤k−1,Q[k1,k2]=/summationtextk2\nj=k1Qj.\nFor anye∈Sn−1, we can decompose Rn=λe⊕He, whereHeis the hyperplane\nwith normal vector e, endowed with the induced measure. For 1 ≤p,q <∞, we\ndefineLp,q\nethe anisotropic Lebesgue space by\n/ba∇dblf/ba∇dblLp,q\ne=/parenleftBigg/integraldisplay\nR/parenleftbigg/integraldisplay\nHe×R|f(λe+y,t)|qdydt/parenrightbiggp/q\ndλ/parenrightBigg1/p4 Z. GUO AND C. HUANG\nwith the usual definition if p=∞orq=∞. We write Lp,q\nej=Lp\nxjLq\n¯xj,t. We use\n˙Bs\np,qto denote the homogeneous Besov spaces on Rnwhich is the completion of the\nSchwartz functions under the norm\n/ba∇dblf/ba∇dbl˙Bsp,q= (/summationdisplay\nk∈Z2qsk/ba∇dblPkf/ba∇dblq\nLp)1/q.\nTo exploit the null-structure we also need the Bourgain-type space associated to\ntheSchr¨ odinger equation. Inthis paperwe use themodulation-ho mogeneousversion\nas in [2, 8]. We define X0,b,qto be the completion of the space of Schwartz functions\nwith the norm\n/ba∇dblf/ba∇dblX0,b,q= (/summationdisplay\nk∈Z2kbq/ba∇dblQkf/ba∇dblq\nL2\nt,x)1/q. (2.2)\nIfq= 2 we simply write X0,b=X0,b,2. By the Plancherel’s equality we have\n/ba∇dblf/ba∇dblX0,1=/ba∇dbl(i∂t+ ∆)f/ba∇dblL2\nt,x. SinceX0,b,qis not closed under conjugation, we also\ndefine the space ¯X0,b,qby the norm /ba∇dblf/ba∇dbl¯X0,b,q=/ba∇dbl¯f/ba∇dblX0,b,q, and similarly write ¯X0,b=\n¯X0,b,2. It’s easy to see that X0,b,qfunction is unique modulo solutions of the homo-\ngeneous Schr¨ odinger equation. For a more detailed description of theX0,b,pspaces\nwe refer the readers to [21] and [20]. We use X0,b,p\n+to denote the space restricted to\nthe interval [0 ,∞):\n/ba∇dblf/ba∇dblX0,b,p\n+= inf\n˜f:˜f=font∈[0,∞)/ba∇dbl˜f/ba∇dblX0,b,p.\nIn particular, we have\n/ba∇dblf/ba∇dblX0,1\n+∼ /ba∇dbl˜f/ba∇dblX0,1 (2.3)\nwhere˜f=f(t,x)1t≥0+f(−t,x)1t<0.\nLetL=∂t−i∆ and¯L=∂t+i∆. We define the main dyadic function space. If\nf(x,t)∈L2(Rn×R+) has spatial frequency localized in {|ξ| ∼2k}, define\n/ba∇dblf/ba∇dblFk=/ba∇dblf/ba∇dblX0,1/2,∞\n++/ba∇dblf/ba∇dblL∞\ntL2x+/ba∇dblf/ba∇dbl\nL2\ntL2n\nn−2\nx\n+2−(n−1)k/2sup\ne∈Sn−1/ba∇dblf/ba∇dblL2,∞\ne+2k/2sup\n|j−k|≤20sup\ne∈Sn−1/ba∇dblPj,ef/ba∇dblL∞,2\ne,\n/ba∇dblf/ba∇dblYk=/ba∇dblf/ba∇dblL∞\ntL2x+/ba∇dblf/ba∇dbl\nL2\ntL2n\nn−2\nx+2−(n−1)k/2sup\ne∈Sn−1/ba∇dblf/ba∇dblL2,∞\ne\n+2−kinf\nf=f1+f2(/ba∇dblLf1/ba∇dblL2\nt,x+/ba∇dbl¯Lf2/ba∇dblL2\nt,x),\n/ba∇dblf/ba∇dblZk=2−k/ba∇dblLf/ba∇dblL2\nt,x\n/ba∇dblf/ba∇dblNk= inf\nf=f1+f2+f3(/ba∇dblf1/ba∇dblL1\ntL2x+2−k/2sup\ne∈Sn−1/ba∇dblf2/ba∇dblL1,2\ne+/ba∇dblf3/ba∇dblX0,−1/2,1\n+)+2−k/ba∇dblf/ba∇dblL2\nt,x.\nThen we define the space Fs,Ys,Zs,Nswith the following norm\n/ba∇dblu/ba∇dblFs=/summationdisplay\nk∈Z2ks/ba∇dblPku/ba∇dblFk,/ba∇dblu/ba∇dblYs=/summationdisplay\nk∈Z2ks/ba∇dblPku/ba∇dblYk,\n/ba∇dblu/ba∇dblZs=/summationdisplay\nk∈Z2ks/ba∇dblPku/ba∇dblZk,/ba∇dblu/ba∇dblNs=/summationdisplay\nk∈Z2ks/ba∇dblPku/ba∇dblNk.\nObviously, Fk∩Zk⊂Yk, and thus Fs∩Zs⊂Ys. In the end of this section, we\npresent a standard extension lemma (See Lemma 5.4 in [22]) giving the r elation\nbetween Xs,band other space-time norm.LANDAU-LIFSHITZ EQUATION 5\nLemma 2.1. Letk∈ZandBbe a space-time norm satisfying with some C(k)\n/ba∇dbleit0eit∆Pkf/ba∇dblB≤C(k)/ba∇dblPkf/ba∇dbl2\nfor anyt0∈Randf∈L2. Then\n/ba∇dblPku/ba∇dblB/lessorsimilarC(k)/ba∇dblPku/ba∇dblX0,1/2,1.\n3.Uniform linear estimates\nIn this section we prove some uniform linear estimates for the equat ion (1.5) with\nrespect to the dissipative parameter. First we recall the known line ar estimates for\nthe Schr¨ odinger equation, see [16] and [14].\nLemma 3.1. Assumen≥3. For any k∈Zwe have\n/ba∇dbleit∆Pkf/ba∇dbl\nL2\ntL2n\nn−2\nx∩L∞\ntL2x/lessorsimilar/ba∇dblPkf/ba∇dbl2, (3.1)\nsup\ne∈Sn−1/ba∇dbleit∆Pkf/ba∇dblL2,∞\ne/lessorsimilar2(n−1)k\n2/ba∇dblPkf/ba∇dbl2, (3.2)\nsup\ne∈Sn−1/ba∇dbleit∆Pk,ef/ba∇dblL∞,2\ne/lessorsimilar2−k\n2/ba∇dblPkf/ba∇dbl2. (3.3)\nLemma 3.2. Assumen≥3,u,Fsolves the equation: for ε >0\nut−i∆u−ε∆u=F(x,t), u(x,0) =u0.\nThen for b∈[1,∞]\n/ba∇dblPku/ba∇dblX0,1/2,b\n+/lessorsimilar/ba∇dblPku0/ba∇dblL2+/ba∇dblPkF/ba∇dblX0,−1/2,b\n+, (3.4)\n/ba∇dblPku/ba∇dblZk/lessorsimilarε1/2/ba∇dblPku0/ba∇dblL2+2−k/ba∇dblPkF/ba∇dblL2\nt,x, (3.5)\nwhere the implicit constant is independent of ε.\nProof.First we show the second inequality. We have\nu=eit∆+εt∆u0+/integraldisplayt\n0ei(t−s)∆+ε(t−s)∆F(s)ds (3.6)\nand thus\n/ba∇dblPku/ba∇dblZk/lessorsimilarε2k/ba∇dblPku/ba∇dblL2\nt,x+2−k/ba∇dblPkF/ba∇dblL2\nt,x\n/lessorsimilarε1/2/ba∇dblPku0/ba∇dblL2+2−k/ba∇dblPkF/ba∇dblL2\nt,x.\nNow we show the first inequality. We only prove the case b=∞since the other\ncases aresimilar. First we assume F= 0. Then u=eit∆+εt∆u0. Let ˜u=eit∆+ε|t|∆u0,\nthen ˜uis an extension of u. Then\n/ba∇dblPku/ba∇dblX0,1/2,∞\n+/lessorsimilarsup\nj2j/2/ba∇dblFt(e−it|ξ|2−ε|t|·|ξ|2)(τ)ˆu0(ξ)χk(ξ)χj(τ+|ξ|2)/ba∇dblL2\nτ,ξ\n/lessorsimilarsup\nj2j/2/ba∇dbl(ε|ξ|2)−1/parenleftbigg\n1+|τ+|ξ|2|2\n(ε|ξ|2)2/parenrightbigg−1\nˆu0(ξ)χk(ξ)χj(τ+|��|2)/ba∇dblL2\nτ,ξ\n/lessorsimilar/ba∇dblPku0/ba∇dbl2.\nNext we assume u0= 0. Fix an extension ˜FofFsuch that\n/ba∇dbl˜F/ba∇dblX0,−1/2,∞≤2/ba∇dblF/ba∇dblX0,−1/2,∞\n+.6 Z. GUO AND C. HUANG\nThen define ˜ u=F−11\nτ+|ξ|2+iε|ξ|2F˜F. We see ˜ uis an extension of uand then\n/ba∇dblPku/ba∇dblX0,1/2,∞\n+/lessorsimilar/ba∇dbl˜u/ba∇dblX0,1/2,∞\n/lessorsimilarsup\nj2j/2/ba∇dblχk(ξ)χj(τ+|ξ|2)1\nτ+|ξ|2+iε|ξ|2F˜F/ba∇dblL2\nτ,ξ\n/lessorsimilar/ba∇dbl˜F/ba∇dblX0,−1/2,∞/lessorsimilar/ba∇dblF/ba∇dblX0,−1/2,∞\n+.\nThus we complete the proof. /square\nLemma 3.3. Letn≥3,k∈Z,ε≥0. Assume u,Fsolves the equation\nut−i∆u−ε∆u=F(x,t), u(x,0) =u0.\nThen for any e∈Sn−1we have\n/ba∇dblPku/ba∇dblL2,∞\ne/lessorsimilar2k(n−1)/2/ba∇dblu0/ba∇dblL2+2k(n−2)/2sup\ne∈Sn−1/ba∇dblF/ba∇dblL1,2\ne, (3.7)\n/ba∇dblPk,eu/ba∇dblL∞,2\ne/lessorsimilar2−k/2/ba∇dblu0/ba∇dblL2+2−ksup\ne∈Sn−1/ba∇dblF/ba∇dblL1,2\ne, (3.8)\nwhere the implicit constant is independent of ε.\nProof.Ifε= 0, then the inequalities were proved in [14]. By the scaling and\nrotational invariance, we may assume k= 0 and e= (1,0,···,0). Then the second\ninequality follows from Proposition 2.5, 2.7 in [10]. We prove the first ineq uality by\nthe following two steps.\nStep 1: F= 0.\nFrom the fact\n|eεt∆f(·,t)(x)| ≤(εt)−n/2/integraldisplay\ne−|x−y|2\n2εt|f(y,t)|dy\n/lessorsimilar(εt)−n/2/integraldisplay\ne−|x−y|2\n2εt/ba∇dblf(y,t)/ba∇dblL∞\ntdy\nwe get\n/ba∇dbleit∆+εt∆P0u0/ba∇dblL2x1L∞\n¯x1,t/lessorsimilar/ba∇dbleit∆P0u0/ba∇dblL2x1L∞\n¯x1,t/lessorsimilar/ba∇dblu0/ba∇dbl2.\nStep 2: u0= 0.\nDecompose P0u=U1+···Unsuch that FxUiis supported in {|ξ| ∼1 :|ξi| ∼\n1}×R. Thus it suffices to show\n/ba∇dblUi/ba∇dblL2x1L∞\n¯x,t/lessorsimilar/ba∇dblF/ba∇dblL1,2\nei. (3.9)\nWe only show the estimate for U1. We still write u=U1. We assume FxFis\nsupported in {|ξ| ∼1 :ξ1∼1}×R. We have\nu(t,x) =/integraldisplay\nRn+1eitτeixξ\nτ+|ξ|2+iε|ξ|2/hatwideF(ξ,τ)dξdτ\n=/integraldisplay\nRn+1eitτeixξ\nτ+|ξ|2+iε|ξ|2/hatwideF(ξ,τ)(1{−τ−|¯ξ|2∼1}c+1−τ−|¯ξ|2∼1,|τ+|ξ|2|/lessorsimilarε\n+1−τ−|¯ξ|2∼1,|τ+|ξ|2|≫ε)dξdτ\n:=u1+u2+u3.\nForu1, we simply use the Plancherel equality and get\n/ba∇dbl∆u1/ba∇dblL2+/ba∇dbl∂tu1/ba∇dbl2≤ /ba∇dblF/ba∇dbl2,LANDAU-LIFSHITZ EQUATION 7\nand thus by Sobolev embedding and Bernstein’s inequality we obtain th e desired\nestimate. For u2, using the Lemma 2.1, Lemma 3.1 and Bernstein’s inequality we\nget\n/ba∇dblu2/ba∇dblL2x1L∞\n¯x1,t/lessorsimilar/ba∇dblu2/ba∇dbl˙X0,1/2,1/lessorsimilarε−1/2/ba∇dbl/hatwideF/ba∇dblL2/lessorsimilar/ba∇dblF/ba∇dblL1,2\ne1.\nNow we estimate u3. Since|τ+|ξ|2| ≫ε, we have\n1\nτ+|ξ|2+iε|ξ|2=1\nτ+|ξ|2+∞/summationdisplay\nk=1(−iε|ξ|2)k\n(τ+|ξ|2)k+1.\nMoreover, let s= (−τ−|¯ξ|2)1/2. Thenτ+|ξ|2=−(s−ξ1)(s+ξ1), and thus we get\n|s−ξ1| ≫ε,|s+ξ1| ∼1\n(τ+|ξ|2)−1=−1\n2s(1\ns−ξ1+1\ns+ξ1) =−1\n2s(s−ξ1)(1+s−ξ1\ns+ξ1).\nHence\n1\nτ+|ξ|2+iε|ξ|2=1\nτ+|ξ|2+∞/summationdisplay\nk=1(−iε|ξ|2)k\n(2s(ξ1−s))k+1\n+∞/summationdisplay\nk=1(−iε|ξ|2)k\n(2s(ξ1−s))k+1[(1+s−ξ1\ns+ξ1)k+1−1]\n:=a1(ξ,τ)+a2(ξ,τ)+a3(ξ,τ).\nInserting this identity into the expression of u3, then we have u3=u1\n3+u2\n3+u3\n3,\nwhere\nuj\n3=/integraldisplay\nRn+1eitτeixξaj(ξ,τ)/hatwideF(ξ,τ)1−τ−|¯ξ|2∼1,|τ+|ξ|2|≫εdξdτ, j = 1,2,3.\nForu1\n3, this corresponds to the case ε= 0 which is proved in [14]. For u3\n3, we can\ncontrol it similarly as u1, since\n|a3(ξ,τ)|/lessorsimilar∞/summationdisplay\nk=1εk|ξ|2k\n(2|s(s−ξ1)|)k+1(k+1)|s−ξ1|\n|s+ξ1|/lessorsimilar1.\nIt remains to control u2\n3. LetGk(x1,¯ξ,τ) =F−1\nx11−τ−|¯ξ|2∼11|ξ|∼1|ξ|2k/hatwideF. Note that\n/ba∇dblGk/ba∇dblL1,2\ne1/lessorsimilarck/ba∇dblF/ba∇dblL1,2\ne1.\nThen\nu2\n3=∞/summationdisplay\nk=1(−iε)k/integraldisplay\nR/integraldisplay\nRn+1eitτeixξ1|s−ξ1|≫ε\n(2s(ξ1−s))k+1[e−iy1ξ1Gk(y1,¯ξ,τ)]dξdτdy 1\n=∞/summationdisplay\nk=1(−iε)k/integraldisplay\nRTk\ny1(G(y1,·))(t,x)dy1\nwhere\nTk\ny1(f)(t,x) =/integraldisplay\nRn+1eitτeixξ1|s−ξ1|≫ε\n(2s(ξ1−s))k+1[e−iy1ξ1f(y1,¯ξ,τ)]dξdτ.8 Z. GUO AND C. HUANG\nWe have\nTk\ny1(f)(t,x) =/integraldisplay\nRn/integraldisplay\nRei(x1−y1)ξ11|s−ξ1|≫ε\n(ξ1−s)k+1dξ1·(2s)−k−1eitτei¯x·¯ξ[f(y1,¯ξ,τ)]d¯ξdτ\n=/integraldisplay\nRei(x1−y1)ξ11|ξ1|≫ε\n(ξ1)k+1dξ1·/integraldisplay\nRnei(x1−y1)s(2s)−k−1eitτei¯x·¯ξ[f(y1,¯ξ,τ)]d¯ξdτ.\nThen we get\n|Tk\ny1(f)(t,x)|/lessorsimilarM−kε−k/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRnei(x1−y1)s·(2s)−k−1eitτei¯x·¯ξ[f(y1,¯ξ,τ)]d¯ξdτ/vextendsingle/vextendsingle/vextendsingle/vextendsingle.\nMaking a change of variable η1=s=/radicalbig\n−τ−|¯ξ|2,dτ=−2η1dη1, we obtain\n/integraldisplay\nRnei(x1−y1)s·(2s)−k−1eitτei¯x·¯ξ[f(y1,¯ξ,τ)]d¯ξdτ\n=/integraldisplay\nRnei(x1−y1)η1·(2η1)−keit(η2\n1+|¯ξ|2)ei¯x·¯ξ[f(y1,¯ξ,η2\n1+|¯ξ|2)]d¯ξdτ.\nThus, by the linear estimate (see Lemma 3.1) we get\n/ba∇dblTk\ny1(f)/ba∇dbl\nL2x1L∞\n¯x,t/lessorsimilarM−kε−k/ba∇dblf/ba∇dbl2,\nwhich suffices to give the estimate for u2\n3. We complete the proof of the lemma. /square\nLemma 3.4. Letn≥3,k∈Z,ε≥0. Assume u,Fsolves the equation\nut−i∆u−ε∆u=F(x,t), u(x,0) =u0.\nThen for any e∈Sn−1we have\n/ba∇dblPku/ba∇dbl\nL∞\ntL2x∩L2\ntL2n\nn−2\nx/lessorsimilar/ba∇dblu0/ba∇dblL2+/ba∇dblF/ba∇dblNk, (3.10)\nwhere the implicit constant is independent of ε.\nProof.Since we have for t >0\n/ba∇dbleit∆+εt∆u0/ba∇dblL∞x/lessorsimilart−n/2/ba∇dblu0/ba∇dblL1x\n/ba∇dbleit∆+εt∆u0/ba∇dblL2x/lessorsimilar/ba∇dblu0/ba∇dblL2x\nwhere the implicit constant is independent of ε, then by the abstract framework of\nKeel-Tao [16] we get the Strichartz estimates\n/ba∇dblPku/ba∇dbl\nL∞\ntL2x∩L2\ntL2n\nn−2\nx/lessorsimilar/ba∇dblu0/ba∇dblL2+/ba∇dblF/ba∇dblL1\ntL2x,\nwith the implicit constant independent of ε.\nBy the same argument as in Step 2 of the proof of Lemma 3.3, we get\n/ba∇dblPku/ba∇dbl\nL∞\ntL2x∩L2\ntL2n\nn−2\nx/lessorsimilar/ba∇dblu0/ba∇dblL2+2−k/2sup\ne∈Sn−1/ba∇dblF/ba∇dblL1,2\ne.\nOn the other hand, by Lemma 2.1, Lemma 3.1 and Lemma 3.2, we get\n/ba∇dblPku/ba∇dbl\nL∞\ntL2x∩L2\ntL2n\nn−2\nx/lessorsimilar/ba∇dblPku/ba∇dblX0,1/2,1\n+/lessorsimilar/ba∇dblu0/ba∇dblL2+/ba∇dblF/ba∇dblX0,−1/2,1\n+.\nThus we complete the proof. /square\nGathering the above lemmas, we can get the following linear estimates :LANDAU-LIFSHITZ EQUATION 9\nLemma 3.5 (Linear estimates) .Assumen≥3,u,Fsolves the equation: for ε >0\nut−i∆u−ε∆u=F(x,t), u(x,0) =u0.\nThen for s∈R\n/ba∇dblu/ba∇dblFs∩Zs/lessorsimilar/ba∇dblu0/ba∇dbl˙Bs\n2,1+/ba∇dblF/ba∇dblNs, (3.11)\nwhere the implicit constant is independent of ε.\n4.Nonlinear estimates\nIn this section we prove some nonlinear estimates. The nonlinear ter m in the\nLandau-Lifshitz equation is\nG(u) =¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2.\nBy Taylor’s expansion, if /ba∇dblu/ba∇dbl∞<1 we have\nG(u) =∞/summationdisplay\nk=0¯u(−1)k|u|2kn/summationdisplay\nj=1(∂xju)2.\nSo we will need to do multilinear estimates.\nLemma 4.1. (1) Ifj≥2k−100andXis a space-time translation invariant Banach\nspace, then Q≤jPkis bounded on Xwith bound independent of j,k.\n(2) For any j,k,Q≤jPk,eis bounded on Lp,2\neandQ≤jis bounded on Lp\ntL2\nxfor\n1≤p≤ ∞, with bound independent of j,k.\nProof.See the proof of Lemma 5.4 in [8]. /square\nLemma 4.2. Assumen≥3,k1,k2,k3∈Z. Then\n/ba∇dblPk1uPk2v/ba∇dblL2\nt,x/lessorsimilar2(n−1)k1/22−k2/2/ba∇dblPk1u/ba∇dblYk1+Fk1/ba∇dblPk2v/ba∇dblFk2,(4.1)\n/ba∇dblPk3(Pk1uPk2v)/ba∇dblL2\nt,x/lessorsimilar2(n−2)min( k1,k2,k3)\n2/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk2v/ba∇dblYk2. (4.2)\nProof.For the first inequality, we have\n/ba∇dblPk1uPk2v/ba∇dblL2\nt,x/lessorsimilarn/summationdisplay\nj=1/ba∇dblPk1uPk2,ejΘj\nk2v/ba∇dblL2\nt,x/lessorsimilarn/summationdisplay\nj=1/ba∇dblPk1u/ba∇dblL2,∞\nej/ba∇dblPk2,ejv/ba∇dblL∞,2\nej\n/lessorsimilar2(n−1)k1/22−k2/2/ba∇dblPk1u/ba∇dblYk1+Fk1/ba∇dblPk2v/ba∇dblFk2.\nFor the second inequality, if k3≤min(k1,k2), then\n/ba∇dblPk3(Pk1uPk2v)/ba∇dblL2\nt,x/lessorsimilar2k3(n−2)/2/ba∇dblPk1uPk2v/ba∇dbl\nL2\ntL2n\n2n−2\nx\n/lessorsimilar2k3(n−2)/2/ba∇dblPk1u/ba∇dbl\nL2\ntL2n\nn−2\nx/ba∇dblPk2v/ba∇dblL∞\ntL2x.\nIfk1≤min(k2,k3), then\n/ba∇dblPk3(Pk1uPk2v)/ba∇dblL2\nt,x/lessorsimilar/ba∇dblPk1u/ba∇dblL2\ntL∞x/ba∇dblPk2v/ba∇dblL∞\ntL2x\n/lessorsimilar2k1(n−2)/2/ba∇dblPk1u/ba∇dbl\nL2\ntL2n\nn−2\nx/ba∇dblPk2v/ba∇dblL∞\ntL2x.\nIfk2≤min(k1,k3), the proof is identical to the above case. /square10 Z. GUO AND C. HUANG\nLemma 4.3 (Algebra properties) .Ifs≥n/2, then we have\n/ba∇dbluv/ba∇dblYs/lessorsimilar/ba∇dblu/ba∇dblYs/ba∇dblv/ba∇dblYn/2+/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblYs.\nProof.We only show the case s=n/2. By the embedding ˙Bn/2\n2,1⊂L∞\nxwe get\n/ba∇dblu/ba∇dblL∞\nx,t≤ /ba∇dblu/ba∇dblL∞\nt˙Bn/2\n2,1/lessorsimilar/ba∇dblu/ba∇dblYn/2.\nThe Lebesgue component can be easily handled by para-product de composition and\nH¨ older’s inequality. Now we deal with Xs,b-type space. By (2.3) it suffices to show\n/summationdisplay\nk2nk/22−k/ba∇dblPk(fg)/ba∇dblX0,1+¯X0,1/lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2, (4.3)\nWe have\n/summationdisplay\nk2nk/22−k/ba∇dblPk(fg)/ba∇dblX0,1+¯X0,1\n/lessorsimilar/summationdisplay\nki2nk3/22−k3/ba∇dblPk3(Pk1fPk2g)/ba∇dblX0,1+¯X0,1\n/lessorsimilar(/summationdisplay\nki:k1≤k2+/summationdisplay\nki:k1>k2)2k3n/22−k3/ba∇dblPk3(Pk1fPk2g)/ba∇dblX0,1+¯X0,1\n:=I+II.\nBy symmetry, we only need to estimate the term I.\nAssumePk1f=Pk1f1+Pk1f2,Pk2g=Pk2g1+Pk2g2such that\n/ba∇dblPk1f1/ba∇dblX0,1+/ba∇dblPk1f2/ba∇dbl¯X0,1/lessorsimilar/ba∇dblPk1f/ba∇dblX0,1+¯X0,1,\n/ba∇dblPk2g1/ba∇dblX0,1+/ba∇dblPk2g2/ba∇dbl¯X0,1/lessorsimilar/ba∇dblPk2g/ba∇dblX0,1+¯X0,1.\nThen we have\nI/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3/ba∇dblPk3(Pk1fjPk2g1)/ba∇dblX0,1\n+/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3/ba∇dblPk3(Pk1fjPk2g2)/ba∇dbl¯X0,1\n:=I1+I2.\nWe only estimate the term I1since the term I2can be estimated in a similar way.\nWe have\nI1/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3/ba∇dblPk3(Pk1fjPk2g1)/ba∇dblX0,1. (4.4)\nFirst we assume k3≤k1+5 in the summation of (4.4). We have\nI1/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3(/ba∇dblPk3Q≤k1+k2+9(Pk1fjPk2g1)/ba∇dblX0,1\n+/ba∇dblPk3Q≥k1+k2+10(Pk1fjPk2g1)/ba∇dblX0,1)\n:=I11+I12.LANDAU-LIFSHITZ EQUATION 11\nFor the term I11we have\nI11/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12k3n/22−k32k1+k2/ba∇dblPk1fj/ba∇dblL∞\ntLnx/ba∇dblPk2g1/ba∇dbl\nL2\ntL2n\nn−2\nx\n/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12k3n/22−k32k22k1n/2/ba∇dblPk1fj/ba∇dblL∞\ntL2x/ba∇dblPk2g1/ba∇dbl\nL2\ntL2n\nn−2\nx\n/lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2.\nFor the term I12, we need to exploit the nonlinear interactions as in [8]. We have\nFPk3Q≥k1+k2+10(Pk1fjPk2g1)\n=χk3(ξ3)χ≥k1+k2+10(τ3+|ξ3|2)/integraldisplay\nξ3=ξ1+ξ2,τ3=τ1+τ2χk1(ξ1)/hatwidefj(τ1,ξ1)χk2(ξ2)/hatwideg1(τ2,ξ2).\nWe assume j= 1 since j= 2 is similar. On the plane {ξ3=ξ1+ξ2,τ3=τ1+τ2}we\nhave\nτ3+|ξ3|2=τ1+|ξ1|2+τ2+|ξ2|2−H(ξ1,ξ2) (4.5)\nwhereHis the resonance function in the product Pk3(Pk1fjPk2g1)\nH(ξ1,ξ2) =|ξ1|2+|ξ2|2−|ξ1+ξ2|2. (4.6)\nSince|H|/lessorsimilar2k1+k2, then one of Pk1fj,Pk2g1has modulation larger than the output\nmodulation, namely\nmax(|τ1+|ξ1|2|,|τ2+|ξ2|2|)/greaterorsimilar|τ3+|ξ3|2|.\nIfPk1fjhas larger modulation, then\nI12/lessorsimilar/summationdisplay\nki:k1≤k22nk3/22−k3/ba∇dbl2j3/ba∇dblPk3Qj3(Pk1fjPk2g1)/ba∇dblL2\nt,x/ba∇dbll2\nj3≥k1+k2+10\n/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22nk3/22−k3(/summationdisplay\nj3≥k1+k2+1022j3/ba∇dblQ≥j3Pk1fj/ba∇dbl2\nL2\nt,x)1/2/ba∇dblPk2g1/ba∇dblL∞\ntL2x\n/lessorsimilar/summationdisplay\nki:k1≤k22nk32−k3(/ba∇dblPk1f1/ba∇dblX0,1+/ba∇dblPk1f2/ba∇dbl¯X0,1)/ba∇dblPk2g1/ba∇dblYk2\n/lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2.\nIfPk2g1has larger modulation, then\nI12/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3/ba∇dblPk1fj/ba∇dblL∞\nt,x(/summationdisplay\nj3≥k1+k222j3/ba∇dblPk2Q≥j3g1/ba∇dbl2\nL2\ntL2x)1/2\n/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k32nk1/2/ba∇dblPk1fj/ba∇dblYk1/ba∇dblPk2g1/ba∇dblX0,1\n/lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2.12 Z. GUO AND C. HUANG\nNow we assume k3≥k1+6 in the summation of (4.4) and thus |k2−k3| ≤4. We\nhave\nI1/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3(/ba∇dblPk3Q≤k1+k2+9(Pk1fjPk2g1)/ba∇dblX0,1\n+/ba∇dblPk3Q≥k1+k2+10(Pk1fjPk2g1)/ba∇dblX0,1)\n:=˜I11+˜I12.\nBy Lemma 4.2 we get\n˜I11/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3(/ba∇dblPk3Q≤k1+k2+9(Pk1fjPk2g1)/ba∇dblX0,1\n/lessorsimilar/summationdisplay\nki:k1≤k22nk3/22k12(n−2)k1/2/ba∇dblPk1fj/ba∇dblYk1/ba∇dblPk2g1/ba∇dblYk2/lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2.\nFor the term ˜I12, similarly as the term I12, one ofPk1fj,Pk2g1has modulation larger\nthan the output modulation. If Pk1fjhas larger modulation, then\n˜I12/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3(/summationdisplay\nj322j3/ba∇dblPk1Q≥j3fj/ba∇dbl2\nL2\ntL∞x)1/2/ba∇dblPk2g1/ba∇dblL∞\ntL2x\n/lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2.\nIfPk2g1has larger modulation, then\n˜I12/lessorsimilar/summationdisplay\nki:k1≤k22/summationdisplay\nj=12nk3/22−k3/ba∇dblPk1fj/ba∇dblL∞\ntL∞x(/summationdisplay\nj322j3/ba∇dblPk2Q≥j3g1/ba∇dbl2\nL2\ntL2x)1/2\n/lessorsimilar/ba∇dblf/ba∇dblYn/2/ba∇dblg/ba∇dblYn/2.\nThus, we complete the proof. /square\nLemma 4.4. We have\n/summationdisplay\nkj2k3(n−2)/2/ba∇dblPk3[un/summationdisplay\ni=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2\nt,x/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.(4.7)\nProof.We have\nLHS of (4.7) /lessorsimilar/summationdisplay\nkj2k3(n−2)/2/ba∇dblPk3[P≥k3−10un/summationdisplay\ni=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2\nt,x\n+/summationdisplay\nkj2k3(n−2)/2/ba∇dblPk3[P≤k3−10un/summationdisplay\ni=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2\nt,x\n:=I+II.LANDAU-LIFSHITZ EQUATION 13\nBy symmetry, we may assume k1≤k2in the above summation. For the term II,\nsincen≥3, then we have\nII/lessorsimilar/ba∇dblu/ba∇dblYn/2/summationdisplay\nkj2k3(n−2)/2/ba∇dbl˜Pk3n/summationdisplay\ni=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2\nt,x\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/summationdisplay\nk1,k3≤k2+52k3(n−2)/22k1+k22[(n−1)k1−k2]/2/ba∇dblPk1v/ba∇dblFk1/ba∇dblPk2w/ba∇dblFk2\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nFor the term I, ifk3≤k2+20, then we get from Lemma 4.2 that\nI/lessorsimilar/summationdisplay\nkj2nk3/22k3(n−2)/2/ba∇dblP≥k3−10u/ba∇dblL∞\ntL2x/ba∇dbln/summationdisplay\ni=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2\nt,x\n/lessorsimilar/summationdisplay\nkj2nk3/22k3(n−2)/22(n+1)k1/22k2/2/ba∇dblP≥k3−10u/ba∇dblL∞\ntL2x/ba∇dblPk1v/ba∇dblFk1/ba∇dblPk2w/ba∇dblFk2\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nIfk3≥k2+20, then uhas frequency ∼2k3, and thus we get\nI/lessorsimilar/summationdisplay\nkj2k3(n−2)/2/ba∇dblPk3u/ba∇dblL∞\ntL2x/ba∇dbln/summationdisplay\ni=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2\ntL∞x\n/lessorsimilar/summationdisplay\nkj2k3(n−2)/2/ba∇dblPk3u/ba∇dblL∞\ntL2x2nk2/2/ba∇dbln/summationdisplay\ni=1(∂xiPk1v∂xiPk2w)]/ba∇dblL2\ntL2x\n/lessorsimilar/summationdisplay\nkj2k3(n−2)/22k1+k22(n−1)k1/22(n−1)k2/2/ba∇dblPk3u/ba∇dblL∞\ntL2x/ba∇dblPk1v/ba∇dblFk1/ba∇dblPk2w/ba∇dblFk2\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nTherefore we complete the proof. /square\nLemma 4.5. We have\n/ba∇dblun/summationdisplay\ni=1(∂xiv∂xiw)/ba∇dblNn/2/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2∩Zn/2/ba∇dblw/ba∇dblFn/2∩Zn/2. (4.8)\nProof.By the definition of Nn/2, theL2component was handled by the previous\nlemma. We only need to control\n/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1un/summationdisplay\ni=1(Pk2∂xiv∂xiPk3w)]/ba∇dblNk4. (4.9)14 Z. GUO AND C. HUANG\nBy symmetry we may assume k2≤k3in the above summation. If in the above\nsummation we assume k4≤k1+40, then\n(4.9)/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1uPk2∂xiv∂xiPk3w]/ba∇dblL1\ntL2x\n/lessorsimilar/summationdisplay\nki2k1n/2/ba∇dblPk1u/ba∇dblL∞\ntL2x2k2/ba∇dblPk2v/ba∇dblL2\ntL∞x2k3/ba∇dblPk3w/ba∇dblL2\ntL∞x\n/lessorsimilar/summationdisplay\nki2k1n/2/ba∇dblPk1u/ba∇dblL∞\ntL2x2k2n/2/ba∇dblPk2v/ba∇dbl\nL2\ntL2n\nn−2\nx2k3n/2/ba∇dblPk3w/ba∇dbl\nL2\ntL2n\nn−2\nx\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nThus from now on we assume k4≥k1+ 40 in the summation of (4.9). We bound\nthe summation case by case.\nCase 1: k2≤k1+20\nIn this case we have k4≥k2+20 and hence |k4−k3| ≤5. By Lemma 4.2 we get\n(4.9)/lessorsimilar/summationdisplay\nki2k4n/22−k4/2/ba∇dblPk4[Pk1uPk2∂xiv∂xiPk3w]/ba∇dblL1,2\ne\n/lessorsimilar/summationdisplay\nki2k4n/22−k4/2/ba∇dblPk1uPk3∂xiw/ba∇dblL2\nx,t/ba∇dblPk2∂xiv/ba∇dblL2,∞\ne\n/lessorsimilar/summationdisplay\nki2k4n/22−k4/22(n−1)k1/22−k3/22(n−1)k2/2/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk3∂xiw/ba∇dblFk3/ba∇dblPk2∂xiv/ba∇dblFk2\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nCase 2: k2≥k1+21\nIn this case we have k4≤k3+40. Let g=/summationtextn\ni=1(Pk2∂xiv·Pk3∂xiw). Then we have\n(4.9)/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1uQ≤k2+k3g]/ba∇dblNk4+/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1uQ≥k2+k3g]/ba∇dblNk4\n:=I+II.\nFirst we estimate the term II. We have\nII/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1Q≥k2+k3−10u·Q≥k2+k3g]/ba∇dblNk4\n+/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1Q≤k2+k3−10u·Q≥k2+k3g]/ba∇dblNk4\n:=II1+II2.LANDAU-LIFSHITZ EQUATION 15\nFor the term II1we have\nII1/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1Q≥k2+k3−10u·Q≥k2+k3g]/ba∇dblL1\ntL2x\n/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk1Q≥k2+k3−10u/ba∇dblL2\ntL∞x/ba∇dblQ≥k2+k3g]/ba∇dblL2\ntL2x\n/lessorsimilar/summationdisplay\nki2k4n/22k1n/2/ba∇dblPk1Q≥k2+k3−10u/ba∇dblL2\ntL2x/ba∇dblQ≥k2+k3g]/ba∇dblL2\ntL2x\n/lessorsimilar/summationdisplay\nki2k4n/22k1n/22−(k2+k3)/ba∇dblPk1Q≥2k1+10u/ba∇dblX0,1\n·2[(n−1)k2−k3]/22k2+k3/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nFor the term II2, sincek4≥k1+ 40, then we may assume ghas frequency of size\n2k4. The resonance function in the product Pk1u·Pk4gis of size /lessorsimilar2k1+k4. Thus the\noutput modulation is of size /greaterorsimilar2k2+k3. Then we get\nII2/lessorsimilar/summationdisplay\nki2k4n/22−(k2+k3)/2/ba∇dblPk4[Pk1Q≤k2+k3−10u·Q≥k2+k3g]/ba∇dblL2\nt,x\n/lessorsimilar/summationdisplay\nki2k4n/22−(k2+k3)/22k1n/2/ba∇dblPk1u/ba∇dblL∞\ntL2x·/ba∇dblg/ba∇dblL2\nt,x\n/lessorsimilar/summationdisplay\nki2k4n/22−(k2+k3)/22k1n/22[(n−1)k2−k3]/22k2+k3/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nNow we estimate the term I. We have\nI/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3n/summationdisplay\ni=1(Pk2∂xiQ≥k2+k3+40v·Pk3∂xiw)]/ba∇dblNk4\n+/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3n/summationdisplay\ni=1(Pk2∂xiQ≤k2+k3+39v·Pk3∂xiw)]/ba∇dblNk4\n:=I1+I2.\nFor the term I1, since the resonance function in the product Pk2v·Pk3wis of size\n/lessorsimilar2k2+k3, then we may assume Pk3whas modulation of size /greaterorsimilar2k2+k3. Then we get\nI1/lessorsimilar/summationdisplay\nki2k4n/22−k4/2/ba∇dblPk4[Pk1u·Q≤k2+k3n/summationdisplay\ni=1(Pk2∂xiQ≥k2+k3+40v·Pk3∂xiQ≥k2+k3−5w)]/ba∇dblL1,2\ne\n/lessorsimilar/summationdisplay\nki2k4n/22−k4/2/ba∇dblPk1u/ba∇dblL∞\nt,x2k2+k3/ba∇dblPk2v/ba∇dblL2,∞\ne/ba∇dblPk3Q≥k2+k3−5w/ba∇dblL2\nt,x\n/lessorsimilar/summationdisplay\nki2k4n/22−k4/22(k2+k3)/22(n−1)k2/22k1n/2/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.16 Z. GUO AND C. HUANG\nFinally, we estimate the term I2. For this term, we need to use the null structure\nobserved by Bejenaru [1]. We can rewrite\n−2∇u·∇v= (i∂t+∆)u·v+u·(i∂t+∆)v−(i∂t+∆)(u·v).(4.10)\nThen we have\nI2=/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3(Pk2LQ≤k2+k3+39v·Pk3w)]/ba∇dblNk4\n+/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3(Pk2Q≤k2+k3+39v·Pk3Lw)]/ba∇dblNk4\n+/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblNk4\n:=I21+I22+I23.\nFor the term I21, we have\nI21/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q≤k2+k3(Pk2LQ≤k2+k3+39v·Pk3w)]/ba∇dblL1\ntL2x\n/lessorsimilar/summationdisplay\nki2k4n/22k1n/2/ba∇dblPk1u/ba∇dblL∞\ntL2x2k2(n−2)/2/ba∇dblPk2Lv/ba∇dblL2\nt,x/ba∇dblPk3w/ba∇dbl\nL2\ntL2n\nn−2\nx\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblZn/2/ba∇dblw/ba∇dblFn/2.\nFor the term I22, we may assume whas modulation /lessorsimilar2k2+k3. Then we get\nI22/lessorsimilar/summationdisplay\nki2k4n/22−k4/2/ba∇dblPk4[Pk1u·Q≤k2+k3(Pk2Q≤k2+k3+39v·Pk3Q≤k2+k3+100Lw)]/ba∇dblL1,2\ne\n/lessorsimilar/summationdisplay\nki2k4n/22−k4/22k1n/2/ba∇dblPk1u/ba∇dblL∞\ntL2x/ba∇dblPk2v/ba∇dblL2,∞\ne/ba∇dblPk3Q≤k2+k3+100Lw)]/ba∇dblL2\nt,x\n/lessorsimilar/summationdisplay\nki2k4n/22−k4/22nk1/22(n−1)k2/22(k2+k3)/2/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblX0,1/2,∞\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nNext we estimate the term I23. We have\nI23/lessorsimilar/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q[k1+k4+100,k2+k3]L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblNk4\n+/summationdisplay\nki2k4n/2/ba∇dblPk4[Pk1u·Q≤k1+k4+99L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblNk4\n:=I231+I232.\nFor the term I232we have\nI232/lessorsimilar/summationdisplay\nki2k4n/22−k4/2/ba∇dblPk4[Pk1u·Q≤k1+k4+99L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblL1,2\ne\n/lessorsimilar/summationdisplay\nki2k4n/22−k4/2/ba∇dblPk1u/ba∇dblL2,∞\ne2k1+k42[(n−1)k2−k3]/2/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.LANDAU-LIFSHITZ EQUATION 17\nFor the term I231we have\nI231/lessorsimilar/summationdisplay\nkik2+k3/summationdisplay\nj2=k1+k4+1002k4n/2/ba∇dblPk4Q≤j2−10[Pk1u·Qj2L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblNk4\n+/summationdisplay\nkik2+k3/summationdisplay\nj2=k1+k4+1002k4n/2/ba∇dblPk4Q≥j2−9[Pk1u·Qj2L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblNk4\n:=I2311+I2312.\nFor the term I2312we have\nI2312/lessorsimilar/summationdisplay\nkik2+k3/summationdisplay\nj2=k1+k4+100/summationdisplay\nj3≥k2−92k4n/22−j3/2\n·/ba∇dblPk4Qj3[Pk1u·Qj2L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblL2\nt,x\n/lessorsimilar/summationdisplay\nki2k4n/22k1n/22(k2+k3)/22[(n−1)k2−k3]/2/ba∇dblPk1u/ba∇dblYk1/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nFor the term I2311we have\nI2311/lessorsimilar/summationdisplay\nkik2+k3/summationdisplay\nj2=k1+k4+1002k4n/2/ba∇dblPk4Q≤j2−10[Pk1˜Qj2u·Qj2L(Pk2Q≤k2+k3+39v·Pk3w)]/ba∇dblL1\ntL2x\n/lessorsimilar/summationdisplay\nkik2+k3/summationdisplay\nj2=k1+k4+1002k4n/22k1n/2/ba∇dblPk1˜Qj2u/ba∇dblL2\nt,x2j22[(n−1)k2−k3]/2/ba∇dblPk2v/ba∇dblFk2/ba∇dblPk3w/ba∇dblFk3\n/lessorsimilar/ba∇dblu/ba∇dblYn/2/ba∇dblv/ba∇dblFn/2/ba∇dblw/ba∇dblFn/2.\nTherefore, we complete the proof. /square\nCombining all the estimates above we get\nLemma 4.6 (Nonlinear estimates) .Assumeu∈Fn/2∩Zn/2with/ba∇dblu/ba∇dblYn/2≪1.\nThen/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble\nNn/2/lessorsimilar/ba∇dblu/ba∇dblYn/2\n1−/ba∇dblu/ba∇dbl2\nYn/2/ba∇dblu/ba∇dblFn/2∩Zn/2/ba∇dblu/ba∇dblFn/2∩Zn/2.\nProof.SinceYn/2⊂L∞, then\n¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2=∞/summationdisplay\nk=0¯u(−1)k|u|2kn/summationdisplay\nj=1(∂xju)2.\nThe lemma follows from Lemma 4.5, Lemma 4.4 and Lemma 4.3. /square\n5.The limit behaviour\nIn this section we prove Theorem 1.1. It is equivalent to prove\nTheorem 5.1. Assumen≥3,ε∈[0,1]. There exists 0< δ≪1such for any\nφ∈˙Bn/2\n2,1with/ba∇dblφ/ba∇dbl˙Bn/2\n2,1≤δ, there exists a unique global solution uεto(1.5)such that\n/ba∇dbluε/ba∇dblFn/2∩Zn/2/lessorsimilarδ,18 Z. GUO AND C. HUANG\nwhere the implicit constant is independent of ε. The map φ→uεis Lipshitz from\nBδ(˙Bn/2\n2,1)toC(R;˙Bn/2\n2,1)and the Lipshitz constant is independent of ε. Moreover,\nfor anyT >0,\nlim\nε→0+/ba∇dbluε−u0/ba∇dblC([0,T];˙Bn/2\n2,1)= 0.\nFortheuniformglobalwell-posedness, wecanproveitbystandard Picarditeration\nargument by using the linear and nonlinear estimates proved in the pr evious section.\nIndeed, define\nΦu0(u) :=eit∆+εt∆u0\n−i/integraldisplayt\n0ei(t−s)∆+ε(t−s)∆/bracketleftbigg2¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2−2iε¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2/bracketrightbigg\nds.\nThenusingtheLemma3.5andLemma4.6wecanshowΦ u0isacontractionmapping\nin the set\n{u:/ba∇dblu/ba∇dblFn/2∩Zn/2≤Cδ}\nif/ba∇dblu0/ba∇dbl˙Bn/2\n2,1≤δwithδ >0sufficiently small. Thus wehaveexistence anduniqueness.\nMoreover, by standard arguments we immediately have the persist ence of regularity,\nnamely if u0∈˙Bs\n2,1for some s > n/2, thenu∈Fs∩Zsand\n/ba∇dblu/ba∇dblFs∩Zs/lessorsimilar/ba∇dblu0/ba∇dbl˙Bs\n2,1(5.1)\nuniformly with respect to ε∈(0,1].\nNow we prove the limit behaviour. Assume uεis a solution to the Landau-Lifshitz\nequation with small initial data φ1∈˙Bn/2\n2,1, anduis a solution to the Schr¨ odinger\nmap with small initial data φ2∈˙Bn/2\n2,1. Letw=uε−u,φ=φ1−φ2, thenwsolves\n(i∂t+∆)w=iε∆uε+/bracketleftbigg2¯uε\n1+|uε|2n/summationdisplay\nj=1(∂xjuε)2−2¯u\n1+|u|2n/summationdisplay\nj=1(∂xju)2/bracketrightbigg\n−2iε¯uε\n1+|uε|2n/summationdisplay\nj=1(∂xjuε)2, (5.2)\nw(0) =φ.\nFirst we assume in addition φ1∈˙B(n+4)/2\n2,1. By the linear and nonlinear estimates,\nfor anyT >0 we get\n/ba∇dblw/ba∇dblFn/2∩Zn/2/lessorsimilar/ba∇dblφ/ba∇dbl˙Bn/2\n2,1+εT/ba∇dbluε/ba∇dblL∞\nt˙B(n+4)/2\n2,1+δ2/ba∇dblw/ba∇dblFn/2∩Zn/2+ε/ba∇dbluε/ba∇dbl3\nFn/2∩Zn/2.\nThen we get by (5.1)\n/ba∇dblw/ba∇dblFn/2∩Zn/2/lessorsimilar/ba∇dblφ/ba∇dbl˙Bn/2\n2,1+εT/ba∇dblφ1/ba∇dbl˙B(n+4)/2\n2,1+εδ3. (5.3)\nNow we assume φ1=φ2=ϕ∈˙Bn/2\n2,1with small norm. For fixed T >0, we need to\nprove that ∀η >0, there exists σ >0 such that if 0 < ε < σthen\n/ba∇dblSε\nT(ϕ)−ST(ϕ)/ba∇dblC([0,T];˙Bn/2\n2,1)< η (5.4)LANDAU-LIFSHITZ EQUATION 19\nwhereSε\nTis the solution map corresponding to (5.2) and ST=S0\nT. We denote\nϕK=P≤Kϕ. Then we get\n/ba∇dblSε\nT(ϕ)−ST(ϕ)/ba∇dblC([0,T];˙Bn/2\n2,1)\n≤/ba∇dblSε\nT(ϕ)−Sε\nT(ϕK)/ba∇dblC([0,T];˙Bn/2\n2,1)\n+/ba∇dblSε\nT(ϕK)−ST(ϕK)/ba∇dblC([0,T];˙Bn/2\n2,1)+/ba∇dblST(ϕK)−ST(ϕ)/ba∇dblC([0,T];˙Bn/2\n2,1).\nFrom the uniform global well-posedness and (5.3), we get\n/ba∇dblSǫ\nT(ϕ)−ST(ϕ)/ba∇dblC([0,T];˙Bn/2\n2,1)/lessorsimilar/ba∇dblϕK−ϕ/ba∇dbl˙Bn/2\n2,1+εC(T,K,/ba∇dblϕ/ba∇dbl˙Bn/2\n2,1).(5.5)\nWe first fix Klarge enough, then let εgo to zero, therefore (5.4) holds.\nAcknowledgment. Z. Guo is supported in part by NNSF of China (No.11371037),\nand C. Huang is supported in part by NNSF of China (No. 11201498).\nReferences\n[1] I. Bejenaru, On Schr¨ odinger maps, Amer. J. Math. 130 (2008 ), 1033-1065.\n[2] I. 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Small energy in two dim ensions, Commun. Math.\nPhys. 224 (2001), 443-544.\n[21] D. Tataru, Local andglobalresults forthe wavemaps I, Comm . PartialDifferential Equations,\n23 (1998), no. 9-10, 1781-1793.\n[22] B. Wang, Z. Huo, C. Hao and Z. Guo, Harmonic Analysis Method fo r Nonlinear Evolution\nEquations, I, World Scientific Press, 2011.\n[23] B. Wang, The limit behavior of solutions for the Cauchy problem of the Complex Ginzburg-\nLandau equation, Commu. Pure. Appl. Math., 55 (2002), 0481-050 8.\n[24] B. Wang and Y. Wang, The inviscid limit for the derivative Ginzburg- Landau equations, J.\nMath. Pures Appl., 83 (2004), 477-502.\nSchool of Mathematical Sciences, Monash University, VIC 38 00, Australia &\nLMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China\nE-mail address :zihua.guo@monash.edu\nSchool of Statistics and Mathematics, Central University o f Finance and Eco-\nnomics, Beijing 100081, China\nE-mail address :hcy@cufe.edu.cn"    },    {        "title": "1709.04911v2.Intrinsic_Damping_Phenomena_from_Quantum_to_Classical_Magnets_An_ab_initio_Study_of_Gilbert_Damping_in_Pt_Co_Bilayer.pdf",        "content": "Intrinsic Damping Phenomena from Quantum to Classical Magnets:\nAn ab-initio Study of Gilbert Damping in Pt/Co Bilayer\nFarzad Mahfouzi,1,\u0003Jinwoong Kim,1, 2and Nicholas Kioussis1,y\n1Department of Physics and Astronomy, California State University, Northridge, CA, USA\n2Department of Physics and Astronomy, Rutgers University, NJ, USA\nA fully quantum mechanical description of the precessional damping of Pt/Co bilayer is presented\nin the framework of the Keldysh Green function approach using ab initio electronic structure cal-\nculations. In contrast to previous calculations of classical Gilbert damping ( \u000bGD), we demonstrate\nthat\u000bGDin the quantum case does not diverge in the ballistic regime due to the \fnite size of\nthe total spin, S. In the limit of S!1 we show that the formalism recovers the torque correla-\ntion expression for \u000bGDwhich we decompose into spin-pumping and spin-orbital torque correlation\ncontributions. The formalism is generalized to take into account a self consistently determined de-\nphasing mechanism which preserves the conservation laws and allows the investigation of the e\u000bect\nof disorder. The dependence of \u000bGDon Pt thickness and disorder strength is calculated and the\nspin di\u000busion length of Pt and spin mixing conductance of the bilayer are determined and compared\nwith experiments.\nPACS numbers: 72.25.Mk, 75.70.Tj, 85.75.-d, 72.10.Bg\nI. INTRODUCTION\nMagnetic materials provide an intellectually rich arena\nfor fundamental scienti\fc discovery and for the invention\nof faster, smaller and more energy-e\u000ecient technologies.\nThe intimate relationship of charge transport and mag-\nnetic structure in metallic systems on one hand, and the\nrich physics occurring at the interface between di\u000berent\nmaterials in layered structures on the other hand, are the\nhallmark of the \rourishing research \feld of spintronics.1{5\nRecently, intense focus has been placed on the signi\f-\ncant role played by spin-orbit coupling (SOC) and the ef-\nfect of interfacial inversion symmetry breaking on the dy-\nnamics of the magnetization in ferromagnet (FM)-normal\nmetal (NM) bilayer systems. Of prime importance to this\n\feld is the (precessional) magnetization damping phe-\nnomena, usually treated phenomenologically by means\nof a parameter referred to as Gilbert damping constant,\n\u000bGD, in the LandauLifshitzGilbert (LLG) equation of\nmotiond~ m=dt =\r~ m\u0002~B+\u000bGD~ m\u0002d~ m=dt , which de-\nscribes the rate of the angular momentum loss of the\nFM.6Here,~ mis the unit vector along the magnetization\ndirection and ~Bis an e\u000bective magnetic \feld.\nIn FM/NM bilayer devices the e\u000bect of the NM on the\nGilbert damping of the FM is typically considered as an\nadditive e\u000bect, where the total Gilbert damping can be\nseparated into an intrinsic bulk contribution and an inter-\nfacial component due to the presence of the NM.7,8While\nthe interfacial Gilbert damping is usually attributed to\nthe loss of angular momentum due to pumped spin cur-\nrent into the NM,9,10in metallic bulk FMs the intrin-\nsic Gilbert damping constant is described by the cou-\npling between the conduction electrons and the (time-\ndependent) magnetization degree of freedom.11\nThe conventional approach to determine the Gilbert\ndamping constant involves calculating the imaginary part\nof the time-dependent susceptibility of the FM in thepresence of conduction electrons in the linear response\nregime.12{14In this case, the time-dependent magneti-\nzation term in the electronic Hamiltonian leads to the\nexcitation of electrons close to the Fermi surface trans-\nferring angular momentum to the conduction electrons.\nThe excited electrons in turn relax to the ground state\nby interacting with their environment, namely through\nphonons, photons and/or collective spin/charge excita-\ntions. These interactions are typically parameterized\nphenomenologically by the broadening of the energy lev-\nels,\u0011=~=2\u001c, where\u001cis the relaxation time of the elec-\ntrons close to the Fermi surface. The phenomenological\ntreatment of the electronic relaxation is valid when the\nenergy broadening is small which corresponds to clean\nsystems, i.e., \u0011D(EF)\u00141, whereD(EF) is the den-\nsity of states per atom at the Fermi energy. In the case\nof large\u0011[\u0011D(EF)&1)] however, this approach vio-\nlates the conservation laws and a more accurate descrip-\ntion of the relaxation mechanism that preserves the en-\nergy, charge and angular momentum conservation laws\nare required.15The importance of including the vertex\ncorrections has already been pointed out in the literature\nwhen the Gilbert damping is dominated by the interband\ncontribution,16{18i.e.,when there is a signi\fcant number\nof states available within the energy window of \u0011around\nthe Fermi energy.\nIn this paper we investigate the magnetic damping phe-\nnomena through a di\u000berent Lens in which the FM is as-\nsumed to be small and quantum mechanical. We show\nthat in the limit of large magnetic moments we recover\ndi\u000berent conventional expressions for the Gilbert damp-\ning of a classical FM. We calculate the Gilbert damping\nfor a Pt/Co bilayer system versus the energy broaden-\ning,\u0011and show that in the limit of clean systems and\nsmall magnetic moments the FM damping is governed\nby a coherent dynamics. We show that in the limit of\nlarge broadening \u0011 > 1meV which is typically the case\nat room temperature, the relaxation time approximationarXiv:1709.04911v2  [cond-mat.mes-hall]  14 Nov 20172\nfails. Hence, we employ a self consistent approach pre-\nserving the conservation laws. We calculate the Gilbert\ndamping versus the Pt and Co thicknesses and by \ftting\nthe results to spin di\u000busion model we calculate the spin\ndi\u000busion length and spin mixing conductance of Pt.\nII. THEORETICAL FORMALISM OF\nMAGNETIZATION DAMPING\nFor a metallic FM the magnetization degree of freedom\nis inherently coupled to the electronic degrees of freedom\nof the conduction electrons. It is usually convenient to\ntreat each degree of freedom separately with the corre-\nsponding time-dependent Hamiltonians that do not con-\nserve the energy. However, since the total energy of the\nsystem is conserved, it is possible to consider the total\nHamiltonian of the combined system and solve the corre-\nsponding stationary equations of motion. For an isolated\nmetallic FM the wave function of the coupled electron-\nmagnetic moment con\fguration system is of the form,\njm\u000b~ki=jS;mi\nj\u000b~ki, where the parameter Sdenotes\nthe total spin of the nano-FM ( S! 1 in the classi-\ncal limit),m=\u0000S:::; +S, are the eigenvalues of the\ntotal Szof the nano-FM,\nrefers to the Kronecker prod-\nuct, and\u000bdenotes the atomic orbitals and spin of the\nelectron Bloch states. The single-quasi-particle retarded\nGreen function and the corresponding density matrix can\nbe obtained from,19\n\u0012\nE\u0000i\u0011\u0000^H~k\u0000HM\u00001\n2S^\u0001~k^~ \u001b\u0001~S\u0013\n^Gr\n~k(E) =^1;(1)\nand\n^\u001a~k=ZdE\n\u0019^Gr\n~k(E)\u0011f(E\u0000HM)^Ga\n~k(E): (2)\nHere,HM=\r~B\u0001~S, is the Hamiltonian of the nano-\nFM in the presence of an external magnetic \feld ~Bwith\neigenstates,jS;mi,\ris the gyromagnetic ratio, f(E) is\nthe Fermi-Dirac distribution function, ^~ \u001bis the vector of\nthe Pauli matrices, ^H~kis the non-spin-polarized Hamilto-\nnian matrix in the presence of spin orbit coupling (SOC),\nand^\u0001~kis the~k-dependent exchange splitting matrix, dis-\ncussed in detail in Sec. III. We employ the notation that\nbold symbols operate on jS;mibasis set and symbols\nwith hat operate on the j\u000b~kis. Here, for simplicity we\nignore explicitly writing the identity matrices ^1 and 1as\nwell as the Kronecker product symbol in the expressions.\nA schematic description of the FM-Bloch electron en-\ntangled system and the damping process of the nano-FM\nis shown in Fig. 1. The presence of the magnetic Hamil-\ntonian in the Fermi distribution function in Eq. (2) act-\ning as a chemical potential leads to transition between\nmagnetic statesjS;mialong the direction in which the\nmagnetic energy is minimized19. The transition rate of\nthe FM from the excited states, jS;mi, to states with\nFIG. 1: (Color online) Schematic representation of the com-\nbined FM-Bloch electron system. The horizontal planes de-\nnote the eigenstates, jS;miof the total Szof the nano-FM\nwith eigenvalues m=\u0000S;\u0000S+ 1;:::; +S. For more details\nsee Fig. 2 in Ref.19\n.\nlower energy ( i.e.the damping rate) can be calculated\nfrom19,\nTm=1\n2=(T\u0000\nm\u0000T+\nm); (3)\nwhere,\nT\u0006\nm=1\n2SNX\n~kTrel[^\u0001~k^\u001b\u0007S\u0006\nm^\u001a~k;m;m\u00061]: (4)\nHere,Nis the number of ~k-points in the \frst Brillouin\nzone,Trel, is the trace over the Bloch electron degrees\nof freedom,S\u0006\nm=p\nS(S+ 1)\u0000m(m\u00061), and ^\u001b\u0007\u0011\n^\u001bx\u0007i^\u001by.\nThe precessional Gilbert damping constant can be de-\ntermined from conservation of the total angular mo-\nmentum by equating the change of angular momen-\ntum per unit cell for the Bloch electrons, Tm, and\nthe magnetic moment obtained from LLG equation,\n\u000bGDMtotsin2(\u0012)=2, which leads to,\n\u000bGD(m) =\u00002\nMtot!sin2(\u0012m)Tm\n\u0011\u0000S2\nMtot!(S(S+ 1)\u0000m2)Tm: (5)\nHere, cos(\u0012m) =mp\nS(S+1), is the cone angle of precession\nandMtotis the total magnetic moment per unit cell in\nunits of1\n2g\u0016Bwithgand\u0016Bbeing the Land\u0013 e factor and\nmagneton Bohr respectively. The Larmor frequency, !,\ncan be obtained from the e\u000bective magnetic \feld along\nthe precession axis, ~!=\rBz.\nThe exact treatment of the magnetic degree of freedom\nwithin the single domain dynamical regime o\u000bers a more\naccurate description of the damping phenomena that can\nbe used even when the classical equation of motion LLG\nis not applicable. However, since in most cases of in-\nterest the FM behaves as a classical magnetic moment,\nwhere the adiabatic approximation can be employed to\ndescribe the magnetization dynamics, in the following\ntwo sections we consider the S!1 limit and close to\nadiabatic regime for the FM dynamics.3\nA. Classical Regime: Relaxation Time\nApproximation\nThe dissipative component of the nonequilibrium elec-\ntronic density matrix, to lowest order in @=@t, can be\ndetermined by expanding the Fermi-Dirac distribution\nin Eq. (2) to lowest order in [ HM]mm0=\u000emm0m~!.\nPerforming a Fourier transformation with respect to the\ndiscrete Larmor frequency modes, m!\u0011i@=@t , we \fnd\nthat, ^\u001adis\nneq(t) =1\n\u0019~\u0011^Gri@^Ga=@t, where ^Gr=\u0002\nEF\u0000i\u0011\u0000\n^H(t)\u0003\u00001and ^Ga= (^Gr)yare the retarded and advanced\nGreen functions calculated at the Fermi energy, EF, and\na \fxed time t.\nThe energy absorption rate of the electrons can\nbe determined from the expectation value of the\ntime derivative of the electronic Hamiltonian, E0\ne=\n<(Tr(^\u001adis\nneq(t)@^H=@t )), where<() refers to the real part.\nCalculating the time-derivative of the Green function and\nusing the identity, \u0011^Gr^Ga=\u0011^Ga^Gr==(^Gr), where,=()\nrefers to the anti-Hermitian part of the matrix, the torque\ncorrelation (TC) expression for the energy excitation rate\nof the electrons is of the form,\nE0\ne=~\n\u0019NX\nkTrh\n=(^Gr)@^H\n@t=(^Gr)@^H\n@ti\n: (6)\nIn the case of semi-in\fnite NM leads attached to the FM,\nusing,=(^Gr) =^Gr^\u0000^Ga=^Ga^\u0000^Gr, Eq.(6) can be written\nas\nE0\ne=~\n\u0019NX\nkTrh\n^\u0000@^Gr\n@t^\u0000@^Ga\n@ti\n(7)\nwhere, ^\u0000 =\u0011^1 + ( ^\u0006r\u0000^\u0006a)=2i, with ^\u0006r=abeing the\nretarded=advanced self energy due to the NM lead at-\ntached to the FM which describes the escape rate of\nelectrons from/to the reservoir. It is useful to separate\nthe dissipation phenomena into local andnonlocal compo-\nnents as follows. Applying the unitary operator, ^U(t) =\nei!^\u001bzt=2ei\u0012^\u001bx=2e\u0000i!^\u001bzt=2= cos(\u0012\n2)^1 +isin(\u0012\n2)(^\u001b+ei!t+\n^\u001b\u0000e\u0000i!t), to \fx the magnetization orientation along z\nwe \fnd,\n@(^U^Gr\n0^Uy)\n@t\u0019!\n2sin(\u0012)\u0010\n^G0ei!t+^G0ye\u0000i!t\u0011\n;(8)\nwhere we have ignored higher order terms in \u0012and,\n^G0= [^Gr\n0;^\u001b+]\u0000^Gr\n0[^H0;^\u001b+]^Gr\n0: (9)\nHere, [;] refers to the commutation relation, ^H0is the\ntime independent terms of the Hamiltonian, and ^Gr=a\n0\nrefers to the Green function corresponding to magnetiza-\ntion alongz-axis. Using Eq. (7) for the average energyabsorption rate we obtain,\nE0\ne=~!2\n2\u0019Nsin2(\u0012)X\nkTr\u0010\n^\u0000^G0^\u0000^G0y\u0011\n=\u0000~!2\n2\u0019Nsin2(\u0012)X\nk<\u0010\nTr\u0010\n^\u0000[^Gr\n0;^\u001b+]^\u0000[^Ga\n0;^\u001b\u0000]\n+=(^Gr)[^H0;^\u001b+]=(^Gr)[^H0;^\u001b\u0000]\n\u00002 [=(^Gr\n0);^\u001b+]^\u0000^Ga\n0[^H0;^\u001b\u0000]\u0011\u0011\n: (10)\nIn the absence of the SOC, the \frst term in Eq.\n(10) is the only non-vanishing term which corresponds\nto the pumped spin current into the reservoir [i.e.\nISz=~Tr(^\u001bz^\u0000^\u001adis\nneq)=2] dissipated in the NM (no back\n\row). This spin pumping component is conventionally\nformulated in terms of the spin mixing conductance20,\nISz=~g\"#sin2(\u0012)=4\u0019, which acts as a nonlocal dissi-\npation mechanism. The second term, referred to as the\nspin-orbital torque correlation11,21(SOTC) expression for\ndamping, is commonly used to calculate the intrinsic con-\ntribution to the Gilbert damping constant for bulk metal-\nlic FMs. The third term arises when both SOC and the\nreservoir are present. It is important to note that the\nformalism presented above is valid only in the limit of\nsmall\u0011(ballistic regime). On the other hand, in the case\nof large\u0011, typical in experiments at room temperature,\nthe results may not be reliable due to the fact that in\nthe absence of metallic leads a \fnite \u0011acts as a \fctitious\nreservoir that yields a nonzero dissipation of spin cur-\nrent even in the absence of SOC. A simple approach to\nrectify the problem is to ignore the e\u000bect of \fnite \u0011in\nthe spin pumping term in calculating the Gilbert damp-\ning constant. A more accurate approach is to employ\na dephasing mechanism that preserves the conservation\nlaws, which we refer it to as conserving torque correlation\napproach discussed in the following subsection.\nB. Classical Regime: Conserving Dephasing\nMechanism\nRather than using the broadening parameter, \u0011, as a\nphenomenological parameter, we determine the self en-\nergy of the Bloch electrons interacting with a dephas-\ning bath associated with phonons, disorder, etc. using a\nself-consistent Green function approach22. Assuming a\nmomentum-relaxing self energy given by,\n^\u0006r=a\nint(E;t) =1\nNX\nk^\u0015k^Gr=a\nk(E;t)^\u0015y\nk; (11)\nwhere ^\u0015kis the interaction coupling matrix, the dressed\nGreen function, ^Gr=a\nk(E;t) , and corresponding self en-\nergy, ^\u0006r=a\nint(E;t), are calculated self-consistently. This\nwill in turn yield a renormalized broadening matrix,\n^\u0000int==(^\u0006r\nint), which is the vertex correction modi\f-\ncation of the in\fnitesimal initial broadening \u00110.4\nThe nonequilibrium density matrix is calculated from\n^\u001adis\nneq(k;t) =~\n\u0019^Gr\nk^\u0000int^Ga\nk\u0010@^Hk(t)\n@t+^Saa\nt\u0011\n^Ga\nk;(12)\nwhere the time derivative vertex correction term is\n^Saa\nt=1\nNX\nk^\u0015k^Ga\nk\u0010@^Hk(t)\n@t+^Saa\nt\u0011\n^Ga\nk^\u0015y\nk: (13)\nThe energy excitation rate for the Bloch electrons then\nreads,\nE0\ne=~\n\u0019NX\nk<h\nTr\u0010\u0010@^Hk(t)\n@t+^Sar\nt\u0011\n^\u001adis\nneq(k;t)\u0011i\n;(14)\nwhere\n^Sar\nt=1\nNX\nk^\u0015k^Ga\nk\u0010@^Hk(t)\n@t+^Sar\nt\u0011\n^Gr\nk^\u0015y\nk: (15)\nThe vertex correction Eqs. (13) and (15) can be solved\neither exactly by transforming them into a system of lin-\near equations or by solving them self consistently. Due to\nthe large number of orbitals and atoms per unit cell for\nthe Co/Pt bilayer the latter approach is computationally\nmore e\u000ecient. In the following numerical calculations we\nassume ^\u0015k=\u0015int^1 to be a constant independent of kand\nof orbitals, which can be viewed as the root mean square\nvalue of a random on-site potential, \u0015int=p\nhV2\nrandi,\nwhereh:::idenotes an ensemble averaging, where the self\nenergy in Eq. (11) corresponds to the self consistent Born\napproximation.\nC. Gilbert Damping Calculation\nHaving determined the energy absorption rate of the\nBloch electrons due to the precessing FM, from conserva-\ntion of energy one can deduce the energy dissipation rate\nof the FM from, E0\nM=\u0000E0\ne. Using the LLG equation\nof motion the energy dissipation rate per unit cell of the\nprecessing FM can be obtained from\nE0\nM=1\n2Mtot~!@mz\n@t=\u00001\n2\u000bGDMtot~!2sin2(\u0012);(16)\nwhere~ mis the unit vector along the magnetization of\nthe FM. The Gilbert damping parameter can then be\nobtained from\n\u000bGD=2E0\ne\nMtot~!2sin2(\u0012): (17)\nIII. COMPUTATIONAL SCHEME\nThe spin-polarized density functional theory calcula-\ntions for the hcp Co(0001)/fcc Pt(111) bilayer were car-\nried out using the Vienna ab initio simulation package(VASP)23,24. The pseudopotential and wave functions\nare treated within the projector-augmented wave (PAW)\nmethod25,26. Structural relaxations were carried using\nthe generalized gradient approximation as parameter-\nized by Perdew et al.27when the largest atomic force\nis smaller than 0.01 eV/ \u0017A. The plane wave cuto\u000b energy\nis 500 eV and a 14 \u000214\u00021kpoint mesh is used in the\n2D BZ sampling. The Pt( m)/Co(n) bilayer is modeled\nemploying the slab supercell approach along the [111]\nconsisting of mfcc Pt monolayers (MLs) (ABC stacking)\n(m=1, 2,:::, 6),nhcp Co MLs (AB stacking) Co ( n= 6),\nand a 25 \u0017A thick vacuum region separating the periodic\nslabs. The in-plane lattice constant of the hexagonal unit\ncell was set to the experimental value of 2.505 \u0017A for bulk\nCo.\nThe Gilbert damping constant was calculated us-\ning the tight-binding parameters obtained from VASP-\nWannier90 calculations28with a 250\u0002250k-mesh for\nthe bilayer and 250 \u0002250\u0002250k-mesh for bulk Co. The\nelectron Hamiltonian, ^H~k, and exchange splitting, ^\u0001~k,\nmatrices in Eq. (1) in the Wannier basis have the form\n^H~k=^HSOC+1\n2X\n~ n1\nD~ n(^H\"\"\n~ n+^H##\n~ n)e2i\u0019~ n\u0001~k(18)\n^\u0001~k=1\n2X\n~ n1\nD~ n(^H\"\"\n~ n\u0000^H##\n~ n)e2i\u0019~ n\u0001~k; (19)\nwhere ^HSOC is the SOC Hamiltonian matrix, ^H\"\"\n~ nand\n^H##\n~ nare the spin-majority and spin-minority matrices,\n~ n= (n1;n2;n3) are integers denoting the lattice vectors,\nD~ nis the degeneracy of the Wigner-Seitz grid point, and\nki2[0;1].\nThe ^H\"\"\n~ nand ^H##\n~ nare determined from spin-polarized\nVASP-Wannier90 calculations without SOC. On the\nother hand, ^HSOC, is determined from VASP-Wannier90\nnon-spin-polarized calculations with SOC as the follow-\ning.\nUsing the identity, Tr[^Li^Lj] =\u000eijl(l+1)(2l+1)\n3, where\n^Liis the angular momentum operator of orbital land\ni;j=x;y;z , the SOC strength of the Ith atom can be\ncalculated from\n\u0018I\nl=3Tr[^HP\nll;II^Li^\u001bi]\nl(l+ 1)(2l+ 1): (20)\nHere, the superscript Pdenotes the paramagnetic Hamil-\ntonian,IIare the on-site Hamiltonian matrix elements\nfor atomIat~ n= (0;0;0), andllis the block Hamiltonian\nmatrix corresponding to orbital l. The result is indepen-\ndent of the direction of the angular momentum operator.\nWe \fnd that \u0018Pt\nd=0.5 eV and \u0018Co\nd=70 meV for the d-\norbitals of Pt and Co, respectively, that are somewhat\nsmaller than the values considered in the literature7(i.e.\n\u0018Pt\nd=0.65 eV,\u0018Co\nd=85 meV). The SOC Hamiltonian can\nin turn be written as,\nhI;lmsj^HSOCjI0;l0m0s0i=1\n2\u000ell0\u000eII0\u0018I\nlX\nihlmj^Lijlm0i^\u001bi\nss0:\n(21)5\n0 45 90 135 1800.0280.02850.0290.02950.030.03050.0310.03150.032\nCone Angle (deg)Gilbert Damping\nStudent Version of MATLAB\nFIG. 2: (Color online). Gilbert damping versus precessional\ncone angle calculated from Eq. (5) for Pt(1 ML)/Co(6 ML)\nbilayer system for S=60 and\u0011=1 meV, respectively.\nIV. RESULTS AND DISCUSSION\nThe Gilbert damping constant (calculated from\nEq. (5))versus the precessional cone angle, \u0012m=\ncos\u00001\u0010\nmp\nS(S+1)\u0011\n, for the Pt(1 ML)/Co(6 ML) bilayer\nis shown in Fig. 2 with \u0011=1 meV and S=60. We \fnd\nthat the Gilbert damping is relatively independent of the\ncone angle. The small angular dependence of the Gilbert\ndamping is material dependent and it could increase or\ndecrease upon increasing the cone angle, depending on\nthe material.\nIn order to see the transition from quantum mechanical\nto classical dynamical regimes, in Fig. 3 we present the\nGilbert damping constant of the Pt(1 ML)/Co(6 ML)\nbilayer versus the broadening parameter, \u0011, for di\u000ber-\nent values of the total spin Sof the FM. For the case\nof \fniteSwe used Eq. (5) while for S=1we used\nthe TC expression Eq. (6). We \fnd that for \fnite S\nthe Gilbert damping value exhibits a peak in the small \u0011\nregime (clean system) where the peak value increases lin-\nearly withSand shifts to smaller broadening value with\nincreasingS. The underlying origin of the \u000bGD(\u0011) behav-\nior withSin coherent regime can be understood in terms\nof the coherent transport of quasi-particles along the aux-\niliary direction min Fig. 1, where the auxiliary current\n\row (damping rate) depends linearly on the chemical po-\ntential di\u000berence between the \frst ( m= +S) and last\nlayers (m=\u0000S) which is simply 2 S. This suggests that\nin the limit of in\fnite Sand ballistic regime \u0011!0 the\nintrinsic Gilbert damping diverges. It was shown that\nthe problem of in\fnite Gilbert damping in the ballistic\nregime can be removed by taking into account the collec-\ntive excitations.29,30\nAs we discussed in Sec. II A, the relaxation time ap-\nproximation is valid only in the small \u0011limit and it vi-\nolates the conservation law when \u0011is large. In order\n10−410−210010−310−210−1100\nBroadening, η, (eV)Gilbert Damping\n  \nS=10S=30S=60S=∞\nStudent Version of MATLABFIG. 3: (Color online) Gilbert damping versus broadening\nparameter for the Pt(1 ML)/Co(6 ML) bilayer for di\u000berent\nvalues of the total spin Sof the FM nanocluster. Eqs. (5)\nand (6) were used to calculate the Gilbert damping for \fnite\nand in\fnite S, respectively.\nto quantify the validity of the relaxation time approxi-\nmation, in Fig. 4 we display the Gilbert damping ver-\nsus broadening, \u0011;or interaction parameter, \u0015int, for\nthe Pt(1 ML)/Co(6 ML) bilayer with S=1, using\n(i)torque correlation(TC) expression (Eq. (6)); (ii)the\nspin-orbital torque correlation (SOTC) expression (sec-\nond term in Eq. (10);) and (iii)the conserving TC expres-\nsion (Eq. (14)). The upper horizontal-axis refers to the\ninteraction strength \u0015intof the conserving TC method\nand the lower one refers to the broadening parameter,\n\u0011. The calculations show that for \u0011 >20 meV the TC\nresults deviate substantially from those of the conserv-\ning TC method. Ignoring the spin pumping contribu-\ntion to the Gilbert damping in Eq. (10) and considering\nonly the SOTC component increases the range of the va-\nlidity of the relaxation time approximation. Therefore,\nthe overestimation of the Gilbert damping using the TC\nmethod can be attributed to the disappearance of elec-\ntrons (pumped spin current) in the presence of the \fnite\nnon-Hermitian term, i\u0011^1, in the Hamiltonian.\nWe have used the conserving TC approach to calculate\nthe e\u000bect of \u0015inton the Gilbert damping as a function of\nthe Pt layer thickness for the Pt( m)/Co(6 ML) bilayer.\nAs an example, we display in Fig. 5 the results of Gilbert\ndamping versus Pt thickness for \u0015int= 1eVwhich yields\na Gilbert damping value of 0.005 for bulk Co ( m= 0 ML)\nand is in the range of 0.00531,32to 0.01133{35reported\nexperimentally. Note that this large \u0015intvalue describes\nthe Gilbert damping in the resistivity-like regime which\nmight not be appropriate to experiment, where the bulk\nGilbert damping decreases with temperature, suggesting\nthat it is in the conductivity regime.36\nFor a given \u0015intwe \ftted the ab initio calculated\nGilbert damping versus Pt thickness to the spin di\u000bu-6\n10−410−210010−310−210−1100\nBroadening, η, (eV)Gilbert Damping10−1100Interaction Strength, λint, (eV)\nConserving TC MethodSOTC MethodTC Method\nStudent Version of MATLAB\nFIG. 4: (Color online). Gilbert damping of Pt(1 ML)/Co(6\nML) bilayer versus the broadening parameter \u0011(lower ab-\nscissa) and interaction strength, \u0015int, (upper abscissa), using\nthe torque correlation (TC), spin-orbital torque correlation\n(SOTC), and conserving TC expressions given by Eqs. (6),\n(10) and (14), respectively.\nsion model,37{39\n\u000bPt=Co =\u000bCo+ge\u000b\n\"#VCo\n2\u0019MCodCo(1\u0000e\u00002dPt=Lsf\nPt):(22)\nHere,ge\u000b\n\"#is the e\u000bective spin mixing conductance, dCo\n(dPt) is the thickness of Co (Pt), VCo= 10:5\u0017A3\n(MCo= 1:6\u0016B) is the volume (magnetic moment) per\natom in bulk Co, and Lsf\nPtis the spin di\u000busion length\nof Pt. The inset of Fig. 5 shows the variation of the ef-\nfective spin mixing conductance and spin di\u000busion length\nwith the interaction strength \u0015int. In the di\u000busive regime\n\u0015int>0:2eV,Lsf\nPtranges between 1 to 6 nm in agree-\nment with experiment \fndings which are between 0.5 and\n10 nm33,40. Moreover, the e\u000bective spin mixing conduc-\ntance is relatively independent of \u0015intoscillating around\n20 nm\u00002, which is approximately half of the experimen-\ntal value of\u001935 - 40 nm\u00002.33,41On the other hand,\nin the ballistic regime ( \u0015int<0.2 eV), although the er-\nrorbar in \ftting to the di\u000busion model is relatively large,\nthe value of Lsf\nPt\u00190.5 nm is in agreement with Ref.7and\nexperimental observation40.\nV. CONCLUDING REMARKS\nWe have developed an ab initio -based electronic struc-\nture framework to study the magnetization dynamics ofa nano-FM where its magnetization is treated quantum\nmechanically. The formalism was applied to investigate\nthe intrinsic Gilbert damping of a Co/Pt bilayer as a\n0 1 2 300.0050.010.0150.02\nPt Thickness, dPt (nm)Gilbert Damping\n  \n10−210−110002468\nInteraction Strength, λint (eV)Spin Diffusion Length (nm)100101102103\ng↑↓eff (nm−2)\nStudent Version of MATLAB\nFIG. 5: (Color online). Ab initio values (circles) of Gilbert\ndamping versus Pt thickness for Pt( mML)/Co(6 ML) bilayer\nwheremranges between 0 and 6 and \u0015int= 1eV. The dashed\ncurve is the \ft of the Gilbert damping values to Eq. (22).\nInset: spin di\u000busion length (left ordinate) and e\u000bective spin\nmixing conductance, ge\u000b\n\"#, (right ordinate) versus interaction\nstrength. The errorbar for ge\u000b\n\"#is equal to the root mean\nsquare deviation of the damping data from the \ftted curve.\nfunction of energy broadening. We showed that in the\nlimit of small Sand ballistic regime the FM damping is\ngoverned by coherent dynamics, where the Gilbert damp-\ning is proportional to S. In order to study the e\u000bect of\ndisorder on the Gilbert damping we used a relaxation\nscheme within the self-consistent Born approximation.\nTheab initio calculated Gilbert damping as a function of\nPt thickness were \ftted to the spin di\u000busion model for a\nwide range of disorder strength. In the limit of large dis-\norder strength the calculated spin di\u000busion length and\ne\u000bective spin mixing conductance are in relative agree-\nment with experimental observations.\nAcknowledgments\nThe work is supported by NSF ERC-Translational Ap-\nplications of Nanoscale Multiferroic Systems (TANMS)-\nGrant No. 1160504 and by NSF-Partnership in Research\nand Education in Materials (PREM) Grant No. DMR-\n1205734.7\n\u0003Electronic address: Farzad.Mahfouzi@gmail.com\nyElectronic address: nick.Kioussis@csun.edu\n1Ioan Mihai Miron, Gilles Gaudin, Stphane Au\u000bret,\nBernard Rodmacq, Alain Schuhl, Stefania Pizzini, Jan Vo-\ngel and Pietro Gambardella, Current-driven spin torque in-\nduced by the Rashba e\u000bect in a ferromagnetic metal layer,\nNat. Mater. 9, 230234 (2010).\n2Ioan Mihai Miron, Kevin Garello, Gilles Gaudin, Pierre-\nJean Zermatten, Marius V. 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B 93, 054402 (2016)."    },    {        "title": "2310.08807v1.Unified_framework_of_the_microscopic_Landau_Lifshitz_Gilbert_equation_and_its_application_to_Skyrmion_dynamics.pdf",        "content": "arXiv:2310.08807v1  [cond-mat.mes-hall]  13 Oct 2023Unified framework of the microscopic Landau-Lifshitz-Gilb ert equation\nand its application to Skyrmion dynamics\nFuming Xu§,1Gaoyang Li§,1Jian Chen,2Zhizhou Yu,3Lei Zhang,4, 5,∗Baigeng Wang,6and Jian Wang1, 2,†\n1College of Physics and Optoelectronic Engineering, Shenzh en University, Shenzhen 518060, China\n2Department of Physics, The University of Hong Kong, Pokfula m Road, Hong Kong, China\n3School of Physics and Technology, Nanjing Normal Universit y, Nanjing 210023, China\n4State Key Laboratory of Quantum Optics and Quantum Optics De vices,\nInstitute of Laser Spectroscopy, Shanxi University, Taiyu an 030006, China\n5Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China\n6National Laboratory of Solid State Microstructures and Dep artment of Physics, Nanjing University, Nanjing 210093, Ch ina\nThe Landau-Lifshitz-Gilbert (LLG) equation is widely used to describe magnetization dynamics. We develop\na unified framework of the microscopic LLG equation based on t he nonequilibrium Green’s function formalism.\nWe present a unified treatment for expressing the microscopi c LLG equation in several limiting cases, including\nthe adiabatic, inertial, and nonadiabatic limits with resp ect to the precession frequency for a magnetization\nwith fixed magnitude, as well as the spatial adiabatic limit f or the magnetization with slow variation in both\nits magnitude and direction. The coefficients of those terms in the microscopic LLG equation are explicitly\nexpressed in terms of nonequilibrium Green’s functions. As a concrete example, this microscopic theory is\napplied to simulate the dynamics of a magnetic Skyrmion driv en by quantum parametric pumping. Our work\nprovides a practical formalism of the microscopic LLG equat ion for exploring magnetization dynamics.\nI. INTRODUCTION\nSingle-molecule magnets (SMMs) are mesoscopic mag-\nnets with permanent magnetization, which show both classic al\nproperties and quantum properties.1–7SMMs are appealing\ndue to their potential applications as memory cells and pre-\ncessing units in spintronic devices.8,9Transport of SMMs cou-\npled with leads has been investigated both experimentally10–13\nand theoretically.7,14–20Transport measurements on magnetic\nmolecules such as Mn1210andFe811revealed interesting phe-\nnomena, including peaks in the differential conductance an d\nCoulomb blockades. Dc- and ac-driven magnetization switch -\ning and noise as well as the influence on I-V characteristics\nwere discussed in a normal metal/ferromagnet/normal metal\nstructure.15Current-induced switching of a SMM junction\nwas theoretically studied in the adiabatic regime within th e\nBorn-Oppenheimer approximation.16It was found that mag-\nnetic exchange interactions between molecular magnets can\nbe tuned by electric voltage or temperature bias.17Transient\nspin dynamics in a SMM was investigated with generalized\nspin equation of motion.21A microscopic formalism was re-\ncently proposed for consistent modeling of coupled atomic\nmagnetization and lattice dynamics.22\nFor a SMM with magnetization M, its magnetization\ndynamics can be semiclassically described by the Landau-\nLifshitz-Gilbert (LLG) equation of motion23–29\ndm\ndt=−γm×Heff+m×(αdm\ndt)+τSTT, (1)\nwherem=M/Mis the unit magnetization vector, γis the\ngyromagnetic ratio, and Heffis the effective magnetic field\naround which the magnet precesses. αis the Gilbert damping\ntensor describing the dissipation of the precession, and τSTT\nis the spin transfer torque due to the misalignment between\nthe magnetization and the transport electron spin.30–33\nThe LLG equation is widely adopted to describe magnetiza-\ntion dynamics in the adiabatic limit, where the magnetizati onprecesses slowly and the typical time scale is in the order of\nns. The Gilbert damping term is in general a 3×3tensor,32\nwhich can be deduced from experimental data, scattering ma-\ntrix theory,28,29or first-principles calculation.34–36Later, the\nLLG equation was generalized to study ultrafast dynamics in -\nduced bypselectrical pulse37,38orfslaser pulse39–42, which\nextends the magnetization switching time down to psor even\nsub-ps. This is refereed as the inertial regime43, where the\ntime scale involved is much shorter than that of the adiabati c\nlimit. In the inertial limit, a nonlinear inertial term was i ntro-\nduced into the LLG equation44–48, which was applied to sim-\nulate ultrafast spin dynamics.21,49,50Direct observation of in-\nertial spin dynamics was experimentally realized in ferrom ag-\nnetic thin films in the form of magnetization nutation at a fre -\nquency of 0.5THz .51When the magnetization varies in both\ntemporal and spatial domains, two adiabatic spin torques we re\nincorporated into the LLG equation52, which can describe the\ndynamics of magnetic textures such as Skymions.53\nMagnetic Skyrmions (Sk) stabilized by the Dzyaloshinskii-\nMoriya interaction (DMI) or competing interaction between\nfrustrated magnets are topologically nontrivial spin text ures\nshowing chiral particle-like nature. When an electron tra-\nverses the Sk, it acquires a Berry phase and experiences a\nLorentz-like force, leading to the topological Hall effect54.\nAt the same time, the Magnus force due to the back ac-\ntion on Sk gives rise to Skyrmion Hall effect55,56. The exis-\ntence of Sk has been verified in magnetic materials includ-\ning MnSi57and PdFe/Ir(111)58. The radius of an Sk can\nbe as small as a few nm59,60and is stable even at room\ntemperatures61,62. Sk can be operated at ultralow current\ndensity,63–65which makes it promising in spintronic appli-\ncations including the magnetic memory and logic gates.66,67\nVarious investigations show that Sk can be manipulated by\nspin torque due to the charge/spin current injection63,64, exter-\nnal electric field,68,69magnetic field gradient70, temperature\ngradient,71–74and strain,75etc. However, Sk driven by quan-\ntum parametric pumping has not been explored.2\nQuantum parametric pump refers to such a process: in an\nopen system without bias voltages, cyclic variation of syst em\nparameters can give rise to a net dc current per cycle76–84. In\nthe adiabatic limit, this quantum parametric pump requires at\nleast two pumping parameters with a phase difference and\nthe pumped current is proportional to the area enclosed by\nthe trajectory of pumping parameters in parameter space.76\nIt was found that the adiabatic pumped current is related\nto Berry phase.85Beyond the adiabatic limit, the cyclic fre-\nquency may serve as another dimension in parameter space\nand hence a single-parameter quantum pump is possible at\nfinite frequences.86,87In general, quantum parametric pump\ncan be formulated in terms of photon-assisted transport.88,89\nQuantum parametric pump can also generate heat current90,91,\nwhose lower bound is Joule heating during the pumping pro-\ncess. This defines an optimal quantum pump92,93that is noise-\nless and pumps out quantized charge per cycle94–96. Quantum\nparametric pumping theory has been extended to account for\nAndreev reflection in the presence of superconducting lead97,\ncorrelated charge pump98,99, and parametric spin pump100,101,\nproviding more physical insights. It is interesting to gene ral-\nize quantum parametric pump to Skyrmion transport, which\nmay offer new operating paradigms for spintronic devices.\nIn this work, we investigate the microscopic origin of the\nLLG equation and the Gilbert damping. We focus on sev-\neral limiting cases of the LLG equation. For a magnetization\nwith fixed magnitude, the adiabatic, inertial, and nonadiab atic\nlimits with respect to its precession frequency are discuss ed.\nWhen both the magnitude and direction of a magnetization\nvary slowly in space, which is referred as the adiabatic limi t in\nspatial domain, our formalism can also be extended to cover\nthis limit. We will provide a unified treatment of all these\ncases and explicitly express each term in the microscopic LL G\nequation in the language of nonequilibrium Green’s functio ns.\nAs an example, we apply the microscopic LLG equation to\nsimulate the dynamics of a Skyrmion driven by quantum para-\nmetric pumping in a two-dimensional (2D) system.\nThis paper is organized as follows. In Sec. II, a single-\nmolecule magnet (SMM) transport setup and corresponding\nHamiltonians are introduced. In Sec. III, a stochastic Langevin\nequation for magnetization dynamics is derived from the\nequation of motion by separating fast (electron) and slow\n(magnetization) degrees of freedom, forming a microscopic\nversion of the LLG equation. In Sec. IV, four limiting cases\nof the microscopic LLG equation are discussed. In Sec. V, we\nnumerically study Sk transport driven by quantum parametri c\npumping. Finally, a brief summary is given in Sec. VI.\nII. MODEL\nThe model system under investigation is shown in Fig. 1,\nwhere a noninteracting quantum dot (QD) representing a\nsingle-molecule magnet (SMM) with magnetization Mis\nconnected to two leads. A uniform magnetic field B=Bˆez\nis applied in the central region. In addition, we assume that\nthere is a dc bias or spin bias across the system providing a\nspin transfer torque or spin orbit torque.\nFIG. 1. Sketch of the model system. A single-molecule magnet\n(SMM) represented by the quantum dot (QD) is connected to the\nleft and right leads. A uniform magnetic field is applied in th e cen-\ntral region, around which the SMM magnetization precesses.\nThe Hamiltonian of this system is given by ( /planckover2pi1= 1)\nˆHtotal=ˆHL+ˆHR+ˆHD+ˆHT,\nwith the lead Hamiltonian ( α=L,R ),\nˆHα=/summationdisplay\nkσǫkασˆc†\nkασˆckασ, (2)\nand the Hamiltonian of the central region,\nˆHD=ˆH0+ˆH′+γˆM·B. (3)\nHereˆH0is the Hamiltonian of the QD with spin-orbit interac-\ntion (SOI)102\nˆH0=/summationdisplay\nnσǫnσˆd†\nnσˆdnσ+/summationdisplay\nmn(tSO\nnmd†\nm↑dn↓+H.c.),(4)\nwithtSO\nnm=−tSO\nmn.ˆH′is the interaction between the electron\nspin and the magnetization as well as the magnetic field,\nˆH′=J/summationdisplay\nnˆsn·ˆM+γe/summationdisplay\nnˆsn·B.\nWe can also add uniaxial anisotropy field to ˆH′. The coupling\nHamiltonian between the QD and the leads is\nˆHT=/summationdisplay\nkαn,σσ′[tσσ′\nkαnˆc†\nkασˆdnσ′+H.c.]. (5)\nIn the above equations, ˆd†\nnσ(ˆc†\nkασ) creates an electron with\nenergyǫnσ(ǫkασ) in the QD (lead α). In general, the leads\ncan be metallic or ferromagnetic. Here ˆsn=1\n2ψ†\nnσψnis the\nelectron spin in the central region, with ψ†\nn= (d†\nn↑,d†\nn↓). The\nPauli matrices satisfy [σx,σy] = 2iσz, and the magnetization\nMfollows the commutation relation [Mx,My] =i/planckover2pi1Mz.J\nis the exchange interaction between the magnetization and t he\nspin of conducting electrons. γ(γe) is the gyromagnetic ratio\nof the magnet (electron).\nIf we choose the magnetic field in the zdirection as the\nlaboratory frame, and (θ,φ)the polar and azimuthal angles3\nof the magnetization, the spin dependent coupling matrix is\ngiven by\ntσσ′\nkαn=/bracketleftbigˆRtkαn/bracketrightbig\nσσ′, (6)\nwithˆRthe rotational operator103\nˆR=e−iθ\n2ˆσye−iφ\n2ˆσz=/parenleftBigg\ne−iφ\n2cos(θ\n2)−eiφ\n2sin(θ\n2)\ne−iφ\n2sin(θ\n2)eiφ\n2cos(θ\n2)/parenrightBigg\n.(7)\nIII. MAGNETIZATION DYNAMICS\nFrom the Heisenberg equation of motion, the magnetization\ndynamics in the central region is governed by\n˙ˆM=−γˆM×B−JˆM׈sD, (8)\nwhereˆsD=/summationtext\nnˆsnis the total electron spin. In deriving the\nabove equation, the following relation is used:\n[ˆσ,ˆσ·A] =−2iˆσ×A. (9)\nNow we separate an operator into its quantum average and\nits fluctuation, then ˆsD=/an}b∇acketle{tˆsD/an}b∇acket∇i}ht+δˆsD, andˆM=/an}b∇acketle{tˆM/an}b∇acket∇i}ht+δˆM,\nwhereδˆsD(δˆM) is the fluctuation of the electron (magnet)\nspin. We can transform Eq. ( 8) into a Langevin equation. For\nthe expectation value M(t) =/an}b∇acketle{tˆM(t)/an}b∇acket∇i}ht,104\n˙M=M×[−γB−JsD+δˆB], (10)\nor\n˙M=−γM×[Heff−δˆB′],\nwhere\nsD=/an}b∇acketle{tˆsD/an}b∇acket∇i}ht=−i\n2Tr[σG<(t,t)]. (11)\nHereG<\nijσσ′(t′,t) =i/an}b∇acketle{td†\njσ′(t′)diσ(t)/an}b∇acket∇i}htis the lesser Green’s\nfunction of electrons, which will be discussed in detail be-\nlow. The effective magnetic field Heffis defined as the vari-\nation of the free energy of the system with respect to the\nmagnetization32,105,106\nHeff=1\nγδHtotal\nδM. (12)\nAndδˆB=γδˆB′contributes from the fluctuations\nM×δˆB=−δ˙ˆM−γδˆM×B−JM×δˆsD\n−JδˆM×sD−JδˆM×δˆsD.\nThese fluctuations can play an important role in determining\nthe motion of the magnetization, such as reducing or enhanc-\ning the threshold bias of magnetization switching.16\nTo transform Eq. ( 10) into the usual LLG equation, we fur-\nther separate sDin Eq. ( 11) into the time-reversal symmetricand antisymmetric components, ss\nDandsa\nD. ThenM×ss\nD\nandM×sa\nDcorrespond to the dissipative and dissipativeless\nterms, respectively. Thus, Eq. ( 10) is rewritten as\n˙M=−γM×B−JM×sa\nD−JM×ss\nD. (13)\nNote that the last term in Eq. ( 13),M×ss\nD, corresponds to the\ndamping of magnetization. As will be discussed below that in\nthe adiabatic approximation, it assumes the form M×(α˙M)\nwhereαis the Gilbert damping tensor which is expressed in\nterms of nonequilibrium Green’s function (see Eq. ( 19)). The\nsecond term in Eq. ( 13),M×sa\nD, corresponds to the spin\ntransfer torque. In the presence of SOI, M×sa\nDis the spin or-\nbit torque in collinear ferromagnetic systems, which has fie ld-\nlike and damping-like components, respectively, along the di-\nrectionsM×uandM×(M×u)withu·M= 0. Hereu\nis the unit vector of the spin current.31\nIV . MICROSCOPIC LLG EQUATION IN DIFFERENT\nLIMITS\nIn this section, we will drive the LLG equation and express\nthe Gilbert damping tensor in terms of the nonequilibrium\nGreen’s functions. We also discuss the fluctuation in the equ a-\ntion of motion and the spin continuity equation, showing tha t\nthe spin transfer torque is insufficient to describe magneti za-\ntion dynamics in general conditions.\nWe focus on several limiting cases of the microscopic LLG\nequation (Eq. ( 13)). These cases correspond to different lim-\nits: (1) Adiabatic limit in temporal domain where the preces s-\ning frequency of the magnet is low and sDcan be expanded\nup to the first order in frequency; (2) Inertial regime where t he\ntime scale is much shorter than that of the adiabatic limit, e .g.,\nmagnetization switching in psor even sub- psrange37–42; (3)\nNonadiabatic regime where adiabatic approximation in tem-\nporal domain is removed. We will work on the linear coupling\nbetween the magnetization and the environment,107and derive\nthe Gilbert damping coefficient as a function of the precess-\ning frequency; (4) In the above situations, we have assumed\nthat the magnetization has fixed magnitude and only its direc -\ntion varies in space. Our theory can be easily extended to ad-\ndress the motion of domain walls where the magnetization is\nnonuniform. In the simplest case, we assume that the magne-\ntization varies slowly in space so that adiabatic approxima tion\nin spatial domain can be taken. In this spatial adiabatic lim it,\ntwo additional toques are incorporated into the LLG equatio n\nwhich are naturally obtained in our theory.\nA. Adiabatic limit\nAs the magnetization precesses, the electron spin and hence\nspin-orbit energy of each state changes32,106, which drives the\nsystem out of equilibrium. In the language of frozen Green’s\nfunctions (Eqs. ( 41) and ( 44)), total spin of the QD (Eqs. ( 11))\ncan be expanded in terms of the precession frequency, which\nconsists of two parts: the quasi-static part s(0)\nD, and the adia-4\nbatic change s(1)\nDto the first order in frequency\nsD=s(0)\nD+s(1)\nD,\nwhere\ns(0)\nD=−i\n2/integraldisplaydE\n2πTr[σG<\nf], (14)\nand\ns(1)\nD=−1\n4/integraldisplaydE\n2πTr[G<\nfσGr\nfGr\nfσ−Ga\nfGa\nfσG<\nfσ]˙b,(15)\nwith˙b=JM˙m.\nConcerning the magnetization dynamics, the effective field\nHeff(t)(Eq. ( 12)) can be separated into two contributions: an\nanisotropy field and a damping field106,\nHeff(t) =Hani\neff(t)+Hdamp\neff(t), (16)\nwith\nHani\neff(t) =B+(J/γ)s(0)\nD,Hdamp\neff(t) = (J/γ)s(1)\nD.(17)\nSubstituting Eqs. ( 14) and ( 15) into Eq. ( 13), ignoring the\nfluctuation, and noting that ˙b=JM˙m, we obtain the deter-\nministic Landau-Lifshitz-Gilbert equation,\ndm\ndt=−γm×Hani\neff−m×(α˙m), (18)\nwhere\nm=M/M=/parenleftbig\nsinθcosφ,sinθsinφ,cosθ/parenrightbig\n,\nis the unit vector in the magnetization direction. αis the3×3\nGilbert damping tensor28,43, which is defined in terms of the\nfrozen Green’s functions:\nαij=(JM)2\n4/integraldisplaydE\n2πRe{Tr[G<\nfσiGr\nfGr\nfσj]}. (19)\nAs shown in Appendix D, this damping tensor recovers that\nobtained in Ref. [ 28] via the scattering matrix theory in the\nlimit of zero temperature and in the absence of external bias .\nIn general, the Gilbert damping tensor depends on m(t)and\nbias voltage through the frozen Green’s functions Gr\nfandG<\nf.\nThis agrees with the observation in Ref. [ 108] using the effec-\ntive field theory of breathing Fermi surface mode.\nB. Inertial regime\nIn this regime, the magnetization has both precessional and\nnutational motions. We focus on the linear coupling between\nthe magnetization and the environment so that an additional\n“inertial” term enters the LLG equation, which describes th e\nnutation of the magnet. In this case, the adiabatic approxim a-\ntion is not good enough. One has to expand the spin density\nsDat least to the second order in frequency. In the inertial\nregime, we assume that the magnitude of the magnetization isfixed while only its direction varies. Iterating Eq. ( 44) to the\nsecond order in frequency, we have\nGr=Gr\nf−iGr\nf˙Gr\nf+Gr\nf(˙Gr\nf)2+(Gr\nf)2¨Gr\nf,\nfrom which we find the contribution s(2)\nDin the inertial limit,\ns(2)\nD=−1\n8/integraldisplaydE\n2πIm{Tr[G<\nfσ(Gr\nf)3σ]}·∂2\ntb\n+1\n16/integraldisplaydE\n2πIm{Tr[G<\nfσ(Gr\nf)4(σ·∂tb)2]}\n−i\n8/integraldisplaydE\n2πTr[σ(Gr\nf)2G<\nf(Ga\nf)2(σ·∂tb)2].(20)\nTo make comparison with Ref. [ 43], we keep only the linear\nterm∂2\ntmand neglect other nonlinear terms such as (∂tm)2.\nWith this new term, the LLG equation in inertial regime is\nwritten as43–46,53\ndm\ndt=−γm×Hani\neff−m×[αdm\ndt]−m×[¯αd2m\ndt2],(21)\nwhere the inertial term is a 3×3tensor given by\n¯αij=(JM)2\n2/integraldisplaydE\n2πIm{Tr[G<\nfσi(Gr\nf)3σj]}. (22)\nThis additional inertial term has been obtained both\nphenomenologically44and semiclassically43. Ref. [ 35] pro-\nposed a first-principles method for calculating the inertia term\nin the semiclassical limit. Here we derive the quantum inert ial\ntensor in terms of the frozen Green’s function.\nC. Nonadiabatic regime\nNow we consider the magnetization dynamics at finite pre-\ncession frequency, whose time scale is still much larger tha n\nthat of electrons. Since no analytic solution exists in gene ral\nconditions, we only focus on the linear coupling in the ex-\nchange interaction J. In this nonadiabatic regime, we treat\nthe coupling Jas a small perturbation, and rewrite the equa-\ntion determining the Green’s function as\n/parenleftbig\ni∂\n∂t−˜H0−H′(t)−Σr/parenrightbig\nGr(t,t′) =δ(t−t′), (23)\nwhere˜H0=H0+γesD·Bis the unperturbed Hamilto-\nnian, including the bare Hamiltonian of the QD defined in\nEq. ( 4) and the Hamiltonian due to the constant external field.\nH′(t) =Jσ·M(t)is the perturbative term due to exchange\ncoupling between the magnetization and the electron spin.\nThe unperturbed retarded Green’s function satisfies\n(i∂\n∂t−˜H0−Σr)Gr\n0(t−t′) =δ(t−t′). (24)\nSinceGr\n0(t−t′)only depends on the time difference, it is\nconvenient to work in the energy representation,\nGr\n0(E) = [E−H0−γe\n2σ·B−Σr]−1, (25)5\nwhereGr\n0(t−t′)andGr(t,t′)are related through the Dyson\nequation\nGr(t,t′) =Gr\n0(t−t′)+/integraldisplay\ndt1Gr\n0(t−t1)H′(t1)Gr(t1,t′).\nIn the first order perturbation, we have\nGr=Gr\n0+Gr\n0H′Gr\n0,\nand\nG<=G<\n0+Gr\n0H′G<\n0+G<\n0H′Ga\n0.\nUsing\n/integraldisplay\ndt1Gr\n0(t−t1)H′(t1)G<\n0(t1−t)\n=/integraldisplaydE\n2π/integraldisplaydω\n2πe−iωtGr\n0(E+ω)H′(ω)G<\n0(E),\nand\n/integraldisplay\ndt1G<\n0(t−t1)H′(t1)Ga\n0(t1−t)\n=/integraldisplaydE\n2π/integraldisplaydω\n2πe−iωtG<\n0(E+ω)H′(ω)Ga\n0(E),\nwhereH′(ω) =J\n2σ·M(ω)withωthe precession frequency,\nthe spin density sDof the quantum dot can be evaluated\nsD=s(0)\nD+s(1)\nD. (26)\nHeres(0)\nDis independent of Mand timet:\ns(0)\nD=−i\n2/integraldisplaydE\n2πTr[σG<\n0(E)].\nAnds(1)\nDdepends linearly on M(t)\ns(1)\nD=−i\n2/integraldisplaydE\n2π/integraldisplaydω\n2πe−iωtTr[σGr\n0(E+ω)H′(ω)G<\n0(E)\n+σG<\n0(E+ω)H′(ω)Ga\n0(E)].\nUsing the anisotropic field Hani\neff(t)and the damping field\nHdamp\neff(t)expressed in Eq. ( 17) and ignoring the fluctuations,\nwe can obtain a deterministic dynamic equation from Eq. ( 13):\ndm\ndt=−γm×Hani\neff−γm×Hdamp\neff, (27)\nwhere\nHani\neff(t) =B−iJ\n2γ/integraldisplaydE\n2πTr[σG<\n0(E)], (28)\nand\nHdamp\neff(t) =−/integraldisplaydω\n2πe−iωtm(ω)˜α(ω). (29)Here˜α(ω)is the frequency dependent Gilbert damping tensor\ndefined as\n˜α(ω) =i\n4J2M2/integraldisplaydE\n2πTr/bracketleftbigg\nG<\n0(E)σGr\n0(E+ω)σ\n+Ga\n0(E)σG<\n0(E+ω)σ/bracketrightbigg\n. (30)\nIt is easy to confirm that when ωgoes to zero, we can recover\nthe results in the adiabatic and inertial limits.\nD. Adiabatic limit in spatial domain\nWhen both the magnitude and direction of the magnetiza-\ntion vary slowly in space, we refer to this situation as the ad ia-\nbatic limit in spatial domain. In this case, two additional t erms\nemerge in the LLG equation52,\ndm\ndt=−γm×Hani\neff−m×[αdm\ndt]\n+bJ(je·∇)m−cJm×(je·∇)m, (31)\nwherebJandcJare constants defined in Ref. [ 52]. Here\nthe term with coefficient bJis related to the adiabatic process\nof the nonequilibrium conducting electrons.52In contrast, the\nother term with coefficient cJcorresponds to the nonadiabatic\nprocess which changes sign upon time-reversal operation.\nIn this limit, the coupling between the magnetization and\nthe electron spin can be approximated as\nˆH′=Jˆsr·ˆM(r,t)+γeˆsr·B. (32)\nEqs. ( 13), (46), and ( 47) are still valid except that MandsD\nare local variables depending on position, where sD(x)is de-\nfined as\nsD(x) =−i\n2/integraldisplaydE\n2πTrs[σG<]xx, (33)\nwhere the trace is taken only in spin space.\nTo derive the adiabatic term in Eq. ( 31), we start from\nEq. ( 46) and then use Eq. ( 47). From Eq. ( 46), we have109\ndsD\ndt+∇·js=JM×sD, (34)\nwherejsis the spin current density and the term −γesD×B\nis neglected. Using the fact that js≈ −b0jem(whereb0=\nµBP/e andPis the polarization) and neglecting the second\norder terms such as ∂tδsD, we find from Eq. ( 34)110\nJM×δsD=−b0(je·∇)m, (35)\nwhereδsDdenotes the contribution due to the spatial varia-\ntion of the magnetization ∇m. The nonadiabatic term can be\ngenerated by iterating the following equation,\ndm\ndt=−γm×Hani\neff−m×[αdm\ndt]+bJ(je·∇)m,(36)6\nfrom which we arrive at105,\ndm\ndt=−γ\n1+α2m×Hani\neff−γα\n1+α2m×[m×Hani\neff]\n+bJ\n1+α2(je·∇)m+bJα\n1+α2m×(je·∇)m,(37)\nwhere we have assumed that the Gilbert damping tensor αis\ndiagonal, i.e., αij=αδij. The nonadiabatic term can also be\nderived explicitly, as shown in Appendix E.\nV . SKYRMION DYNAMICS DRIVEN BY QUANTUM\nPARAMETRIC PUMPING\nIn this section, we apply our microscopic theory to inves-\ntigate Skyrmion dynamics in a 2D system driven by quantum\nparametric pumping. Initially, an Sk is placed in the centra l re-\ngion of a two-lead system, as shown in Fig. 2. Then, we apply\ntwo time-dependent voltage gates with a phase difference in\nthe system to drive a dc electric current. The electron flow, i n\nturn, interacts with the Sk, which gives rise to quantum para -\nmetric pumping of the Sk. In the tight-binding representati on,\nthe Sk is described by the following Hamiltonian\nHSk=−Jex/summationdisplay\n/angbracketlefti,j/angbracketrightmi·mj+/summationdisplay\n/angbracketlefti,j/angbracketrightD·(mi×mj)\n−K/summationdisplay\ni(mi·ˆz)2−µ/summationdisplay\nimi·B. (38)\nHereJexis the Heisenberg exchange interaction. D=\nD(ri−rj)/|ri−rj|is the Dzyaloshinskii-Moriya interaction\n(DMI).Kis the perpendicular magnetic anisotropy constant,\nandµis the magnitude of the magnetic moment. To facilitate\nparametric pumping, we apply gate voltages in two different\nregions of the system with the following form,\nVp=V1cos(ωpt)+V2cos(ωpt+φ),\nwhereV1=Vδ(x−l1)andV2=Vδ(x−l2)are potential\nlandscapes with Vthe pumping amplitude, ωpis the pumping\nfrequency, and φis the phase difference. The central scatter-\ning region is discretized into a 40×40mesh. The positions\nof gate voltages are l1= 1 andl2= 5, which are displayed\nin Fig. 2. In the adiabatic pumping regime (small ωplimit),\nthe cyclic variation of two potentials V1andV2can pump out\na net current when φ/ne}ationslash=nπ76,82. Thus, the total Hamiltonian\nof the system consists of Hα,H0,HT,H′(given by Eqs. ( 2),\n(4), (5), and ( 32), respectively), HSk, andVp.\nSince the Sk has slow varying spin texture in space, its dy-\nnamics can be approximated by the adiabatic limit in spatial\ndomain. The following LLG equation describing the Sk dy-\nnamics driven by parametric pumping needs to be solved,\ndmi\ndt=−γmi×[Heff−(J/γ)sD]−mi×(α˙mi),(39)\nwhere the effective field Heffis defined as\nHeff=1\nγδHSk\nδmi.\n/s120/s121 \n/s109\n/s122/s49\n/s45/s49/s86\n/s49/s32/s32/s86\n/s50\n/s76/s101/s102/s116\n/s108/s101/s97/s100/s82/s105/s103/s104/s116\n/s108/s101/s97/s100\nFIG. 2. Schematic plot of a central region that hosts a Skyrmi on\nand is connected to two metallic leads. The central region co nsists\nof a square lattice of size 40 ×40. The arrows denote the in-plane\ncomponent of the magnetization texture of the Skyrmion. Pum ping\npotentials V1andV2are applied on the first and fifth column layers\nof the central region, which are labeled by dark gray bars.\nTABLE I. Unit conversion table for Jex= 1meV and a= 0.5nm.\nDistance x ˆx=a = 0.5nm\nTimet ˆt=/planckover2pi1/Jex ≈0.66ps\nCurrent density κˆκ= 2eJex/a2/planckover2pi1≈2×1012A/m2\nVelocityv ˆv=Jexa/(h) ≈7.59×102m/s\nHeresDis defined in terms of Green’s functions in Eq. ( 33).\nThe Gilbert damping tensor αis assumed to be a diagonal ma-\ntrix,αij=αδij. It is worth mentioning that Eq. ( 39) already\nincludes the (je· ∇)mandm×(je· ∇)mterms, which is\ndiscussed in Sec. IV D and Appendix E.\nInitial configuration of the Sk is generated by manually cre-\nating a topological unity charge at the center of the system a nd\nthen relaxing the spin texture numerically until the magnet ic\nenergy is stable. Note that mzat the Sk center is negative,\nwhile the outside values are positive. In numerical simula-\ntion, the central region is a 40a×40asquare lattice with a\nthe lattice spacing; the relaxed Sk radius is r0= 10 , which\nis the minimal distance between the Sk center mz\ni(0) =−1\nandmz\ni(r0) = 1 . Parameters are set as D= 0.2Jex26,\nK= 0.07Jex26,J= 2Jex,B= 0, andα= 0.4. The Heisen-\nberg exchange constant Jex=t= 1 is chosen as the energy\nunit, wheretis the hopping energy. We set /planckover2pi1=γ=a= 1,\nand then the coefficients to convert the time t, current den-\nsityκ, and velocity vto SI units are /planckover2pi1/Jex,2eJex/(a2/planckover2pi1),\nandJexa/h.111,112Table Ishows the expressions and partic-\nular values for Jex= 1meV anda= 0.5nm.\nOur numerical calculation proceeds as follows. First, with\nthe initial Sk configuration chosen at t=t0, we calculate the\ntotal Hamiltonian of the system and then the frozen Green’s\nfunction in Eq. ( 45) that determines sDin Eq. ( 33). Second,\nthe LLG equation in Eq. ( 39) is solved by using the fourth-7\n/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48\n/s48 /s50 /s52 /s54 /s56/s48/s52/s56/s49/s50/s49/s54/s48/s49/s48/s50/s48/s51/s48\n/s48 /s50 /s52 /s54 /s56/s48/s50/s52/s54/s56/s119/s105/s116/s104/s111/s117/s116/s32/s83/s107/s68/s79/s83/s47/s49/s48/s48/s84/s114/s97/s110/s115/s109/s105/s115/s115/s105/s111/s110\n/s32/s115/s112/s105/s110/s45 /s111/s114/s32/s40/s97/s41\n/s40/s98/s41/s32/s115/s112/s105/s110/s45\n/s32/s115/s112/s105/s110/s45/s40/s99/s41 /s119/s105/s116/s104/s32/s83/s107\n/s40/s100/s41\nFIG. 3. The transmission coefficient (a) and density of state s (b) as\na function of the electron energy in the absence of an Sk. (c) a nd (d)\nare the transmission and density of states for cases with an S k at the\ncenter. No pumping potential is added in the system ( V= 0).\norder Runge-Kutta method with a small time step dt. Then\nthe Sk Hamiltonian in Eq. ( 38) can be updated. We repeat the\nabove two-step calculation to simulate the Sk dynamics driv en\nby quantum parametric pumping, and monitor the pumped\ncurrent during the time evolution.\nFirst, we investigate the static transport properties of th e\nsystem without pumping. Fig. 3shows the transmission coef-\nficient and density of states (DOS) as a function of the electr on\nenergyEwith and without an Sk locating at the system center.\nWhen there is no Sk, Fig. 3(a) and (b) show spin-degenerate\ntransmission coefficients and DOS, which are standard trans -\nport properties for a metallic square lattice. However, in t he\npresence of the Sk, spin degeneracy of the system is lifted.\nIn Fig. 3(d), the whole energy range [0,8]can be typically\ndivided into the following three regions, irrespective to t he\nexchange strength J113,114.\n(i)0< E <|J|. The conduction electrons are fully spin-\npolarized. Since J =2 in our calculation, this region corre-\nsponds to 0< E <2. Only spin-down electrons can trans-\nmit in this energy region, and the largest spin polarization is\nreached near E= 2.\n(ii)|J|< E <8−|J|. Both spin-up and spin-down con-\nduction electrons exist in the system.\n(iii)8− |J|< E <8. The conduction electrons are fully\npolarized with spin up component.\nSecond, we study the parametric pumping effect on the dy-\nnamics of an Sk and the corresponding pumped current. Phys-\nically, the pumped current can drive the motion of Sk, while\nthe Sk’s motion can affect the pumped current in turn. The\npumped current at time tis defined as89\nIp(t) = Tr/bracketleftbigg\nΓRGr\nfdVp\ndtGa\nf/bracketrightbigg\n, (40)\nwhereΓR= Σr\nR−Σa\nRis the linewidth function of the\nright metallic lead. Σr,a\nRare the retarded and advanced self-/s45/s51/s48/s48/s51/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s45/s49/s48/s49/s50/s45/s51/s48/s48/s51/s48/s73\n/s112/s32/s119/s105/s116/s104/s32/s83/s107 /s32/s119/s105/s116/s104/s111/s117/s116/s32/s83/s107/s40/s97/s41/s83/s107/s32/s99/s101/s110/s116/s101/s114\n/s116/s47/s84/s32/s120\n/s32/s121 /s40/s99/s41/s73\n/s112/s32/s115/s112/s105/s110/s45 /s32/s115/s112/s105/s110/s45/s40/s98/s41\nFIG. 4. (a) The pumped current Ipversus time with or without the\nSk. The time is in unit of the pumping period T, withT= 2π/ωp.\n(b) The pumped spin-dependent current I↑/↓\npwith the Sk. (c) Time\nevolution of the Sk center position R= (x,y). Parameters: E=\n1.9,J= 2,V= 0.8,ωp= 1,φ=π/2.\nenergies. The Sk center R= (x,y)is defined as R=/summationtext\ni(mz\n0−mz\ni)ri//summationtext\ni(mz\n0−mz\ni)to characterize its motion,\nwhere index isums over sites with mz\ni<mz\n0=−0.1.\nAs shown in Fig. 4(a), in the absence of an Sk, the pumped\ncurrent is roughly a sine or cosine function in time. When\nan Sk is introduced at t=t0, the conduction electrons are\nscattered by the moving Sk. This results in the deviation of\nthe pump current from the smooth curve. Meanwhile, in the\npresence of the Sk, the pumped current is fully spin-polariz ed\nat the given energy, where only the spin-down component is\nnonzero in Fig. 4(b). At the same time, the Sk is driven by\nthe pumped current. Fig. 4(c) displays xandycoordinates of\nthe Sk center as a function of time. The remarkable charac-\nteristic is the quasi-periodic movement of the Sk along both\nˆxandˆydirections. Moreover, the motion of the Sk center\nhas the same period as the pumped current, but is delayed by\none quarter cycle in phase. In our system, the pumped current\n��ows in the ±xdirection, and hence the Sk moves faster in\nthis direction. Besides, the Sk acquires a velocity in the ±ˆy\ndirection. This indicates that the Sk Hall effect can also be\ndriven by parametric pumping.\nWe examine the influence of pumping parameters on the Sk\ndynamics. The pumping amplitude is first evaluated. Fig. 5\nshows the time evolution of the Sk center for different pump-\ning amplitudes V. We observe that the Sk’s speed along xdi-\nrection increases with the pumping amplitudes. For V= 0.4,\nthe Sk oscillates around its initial position and does not pr op-\nagate. As the pumping amplitude increases, the Sk moves\nfaster in+ˆxdirection and then saturates when the amplitude\nexceedsV= 0.8. The motion along the ˆydirection is always\nslower than that in the ˆxdirection.\nThe effect of the pumping frequency is also studied. When\nthere is no Sk, the pumped current in adiabatic pumping\nregime is independent of the pumping frequency.76,82In the\npresence of an Sk, We expect that the pumped current can8\n/s48/s49/s50\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s45/s50/s45/s49/s48/s49/s50/s83/s107/s32/s99/s101/s110/s116/s101/s114/s32/s120\n/s32/s48/s46/s52/s32 /s32/s48/s46/s54/s32 /s32/s48/s46/s56/s32 /s32/s49/s46/s48/s32 /s32/s49/s46/s50/s40/s97/s41/s83/s107/s32/s99/s101/s110/s116/s101/s114/s32/s121\n/s116/s47/s84/s40/s98/s41\n/s86/s58\nFIG. 5. (a) and (b): xandycoordinates of the Sk center Rversus\ntime for different pumping amplitudes V= 0.4,0.6,0.8,1,1.2. The\npumping frequency is fixed at ωp= 1. Other parameters: J=\n2,E= 1.9,l1= 1,l2= 5,φ=π/2.\nbe simply scaled by the pumping frequency. We examine the\npumped current for different pumping frequency when the Sk\nis fixed at its initial configuration, and numerical results a re\npresented in Fig. 6(a). Four periods are shown here. It is clear\nthat the pumped currents collapse precisely onto each other\nwhen scaled by the corresponding pumping frequencies. For\na free Sk, Fig. 6(b) and (c) show the xandycoordinates of the\nSk center under the pumping. At small frequencies ωp= 0.2\nand0.5, the Sk is driven along +ˆxdirection first, and then re-\nflected back periodically. Its motion in ydirection is similar.\nFor a larger frequency ωp= 1, there is no such oscillating be-\nhavior even for a time scale of 100 periods (not shown here).\nNotice that when the phase difference is reversed, both the\npumped current and the Sk motion change direction.\nVI. SUMMARY\nIn conclusion, we have developed a unified microscopic\ntheory of the LLG equation in terms of nonequilibrium\nGreen’s function. Four limiting cases of the microscopic LL G\nequation are discussed in detail, including the adiabatic, iner-\ntia, and nonadiabatic limits for the magnetization with fixe d\nmagnitude, as well as the adiabatic limit in spatial domain f or\nthe magnetization with slow varying magnitude and directio n\nin space. As a demonstration, the microscopic LLG equation\nis applied to investigate the motion of a Skyrmion state driv en\nby quantum parametric pumping. Our work not only provides\na unified microscopic theory of the LLG equation, but also of-\nfers a practical formalism to explore magnetization dynami cs\nwith nonequilibrium Green’s functions./s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48/s45/s56/s48/s48/s56/s48/s49/s54/s48\n/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53\n/s48 /s49/s48 /s50/s48 /s51/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s73\n/s112/s47\n/s112\n/s116/s47/s84/s32\n/s112/s61/s48/s46/s50\n/s32\n/s112/s61/s48/s46/s53\n/s32\n/s112/s61/s49/s40/s97/s41\n/s102/s105/s120/s101/s100/s32/s83/s107\n/s112/s61/s49\n/s112/s61/s48/s46/s53/s83/s107/s32/s99/s101/s110/s116/s101/s114/s32/s120/s40/s98/s41\n/s112/s61/s48/s46/s50/s83/s107/s32/s99/s101/s110/s116/s101/s114/s32/s121\n/s116/s47/s84/s40/s99/s41\n/s112/s61/s49\n/s112/s61/s48/s46/s53/s112/s61/s48/s46/s50\nFIG. 6. (a) The pumped current for different pumping frequen cies.\nThe Sk is fixed at its initial configuration during the time evo lution.\nThe pumped current Ipis scaled with ωp. (b) and (c): xandycoordi-\nnates of the Sk center for ωp= 0.2,0.5,1. The pumping amplitude\nis fixed at V= 0.8. Other parameters: J= 2,E= 1.9,l1=\n1,l2= 5,φ=π/2.\nACKNOWLEDGEMENT\nThis work was supported by the National Natural Science\nFoundation of China (Grants No. 12034014, No. 12174262,\nNo. 12074230, and No. 12074190). L. Zhang thanks the Fund\nfor Shanxi ”1331 Project”.\nAPPENDIX A\nIn this appendix, we express the charge and spin current in\nterms of the nonequilibirum Green’s functions.\nIn the presence of time-varying magnetization, the nonequi -\nlibrium Green’s function Gr(t,t′)depends on two time in-\ndicestandt′. If the magnetization changes slowly with time,\nwe can treat the time difference (t−t′)in energy space.115\nAfter taking the Fourier transformation, the Green’s funct ion\nin energy space only depends on one time variable t:\nGr(t,E) =/integraldisplay\ndτeiEτGr(t,t′),\nwhereτ=t−t′. The inverse Fourier transformation gives\nGr(t,t′) =/integraldisplaydE\n2πe−iEτGr(t,E).\nWith the above definition, it is easy to show that\nG<(t,t′) =/integraldisplaydE\n2πe−iEτGr(t,E)Σ<(E)Ga(E,t′).(41)9\nIn this representation, the particle current matrix is defin ed116\nIα\nop(t) =/integraldisplaydE\n2π[Gr(t,E)Σ<\nα(E)+G<(t,E)Σa\nα(E)+H.c.].\n(42)\nIn term of which, the charge current Icα(t)and spin current\nIsα(t)are expressed as\nIcα(t) =−qTr[Iα\nop],Isα(t) =−1\n2Tr[σIα\nop]. (43)\nAs shown in Appendix C, these time-dependent Green’s\nfunctions, such as Gr(t,E)(also called Floquet Green’s\nfunction115), can be expressed in terms of the instantaneous\nfrozen Green’s function Gr\nf, which satisfies the following re-\ncursive relation:\nGr(t,E) =Gr\nf(t,E)−iGr\nf(t,E)˙Gr(t,E), (44)\nwhere˙Gris the time derivative of Gr. The frozen Green’s\nfunction contains the effective magnetic field b(t)and is de-\nfined as\nGr\nf(t,E) = [E−H0−Σr−σ·b(t)/2]−1, (45)\nwhereb(t) =γeB+JM. The self energies (for ferromag-\nnetic leads) are given by\nΣγ\nmn(t) =ˆR†Σγ\n0,mnˆR,\nwithγ=r,a,< .ˆRis defined in Eq. ( 7).Σγ\n0,mnis the self-\nenergy when the magnetization is along z-axis:\nΣγ\n0,mn=/summationdisplay\nkαt∗\nkαmgγ\nkαtkαn=/parenleftbiggΣγ\nmn,↑0\n0 Σγ\nmn,↓/parenrightbigg\n,\nwheregγ\nkαis the surface Green’s function of lead α.\nAPPENDIX B\nIn this appendix, we express the microscopic LLG equation\nEq. ( 13) in terms of the spin current and spin operators.\nFor the spin transfer torque (STT) to occur in magnetic mul-\ntilayers, one requires a pair of FM layers in a noncollinear\nconfiguration. Then a spin-polarized current can be generat ed\nfrom the reference (fixed) layer, and the transverse spin can be\ntransferred to the switchable (free) layer. The current ind uced\nSTT can also be obtained from Eq. ( 13) in such a noncollinear\nmagnetic system. Denoting H1=Htotal−H′and defining\nthe spin operators for lead αand the QD:\nˆsα=1\n2/summationdisplay\nkσσ′ˆc†\nkσασσσ′ˆckσ′α,ˆsD=1\n2/summationdisplay\nnσσ′ˆd†\nnσσσσ′ˆdnσ′,\nwe have\ndˆsα\ndt=−i[ˆsα,H1],\ndˆsD\ndt=−i[ˆsD,ˆH1]−i[ˆsD,ˆH′],where\n−i[ˆsD,ˆH′] =−γeˆsD×B+JˆM׈sD.\nIt can be shown that117[/summationtext\nαˆsα+ˆsD,H1] = 0 , from which\nthe spin continuity equation of the system is expressed as118\n/summationdisplay\nαdˆsα\ndt+dˆsD\ndt=−γeˆsD×B+JˆM׈sD, (46)\nwhich indicates that the total spin is not conserved due to sp in\nprecession117. Substituting Eq. ( 46) into Eq. ( 8) and taking\nquantum average, we have\n˙M=−γM×B−/summationdisplay\nαIsα+A, (47)\nwhere the correction term Ais given by\nA=−dsD\ndt−γesD×B, (48)\nand/summationtext\nαIsα=/summationtext\nαdsα/dtis the total spin current. Notice\nthat Eq. ( 13) and Eq. ( 47) are equivalent. Form Eq. ( 48), it is\nfound that when sDis time-reversal symmetric, dsD/dtorA\nis time-reversal antisymmetric. If we decompose/summationtext\nαIsαand\nAinto the time-reversal symmetric (labeled with superscrip t\ns) and antisymmetric (labeled with superscript a) parts, we\nhave\n˙M=−γM×B−/summationdisplay\nαIs\nsα+As−JM×ss\nD. (49)\nHere we have used the fact that/summationtext\nαIsα−A=JM×sD.\nWhen the external magnetic field is strong enough, the elec-\ntron spin will approximately align with the direction of the\nfield and we may drop the term −γesa\nD×B.\nIn the adiabatic limit, the term −dsa\nD/dt in inAsof\nEq. ( 49) corresponds to −ds(0)\nD/dt. Expanding it to the first\norder in frequency, we find\n−ds(0)\nD\ndt=−iJM\n4/integraldisplaydE\n2πTr[G<\nfσGr\nfσ+Ga\nfσG<\nfσ]dm\ndt\n≡ −ηdm\ndt, (50)\nwhereηis a second rank tensor. This term can be absorbed by\nintroducing an effective gyromagnetic ratio γ′=γ(1+η)−1.\nTherefore, the driving force of magnetization precession o rig-\ninates from the total spin current/summationtext\nαIs\nsα, which corresponds\nto the current induced STT discussed in Ref. [ 119].\nAPPENDIX C\nIn this appendix, we derive the relation between Floquet\nGreen’s functions and frozen Green’s functions. We start wi th\nthe two-time retarded Green’s function defined as120\n[i∂\n∂t1−H(t1)]Gr(t1,t2)+/integraldisplay\ndt′Σr(t1,t′)Gr(t′,t2) =δ(t1,t2).10\nTo work in the Floquet Green’s function representation, we\nmake the Fourier transform with respect to the fast time scal e\nτ=t1−t2, and obtain\n/bracketleftBig\ni∂\n∂t+E−H(t)/bracketrightBig\nGr(t,E)\n−/integraldisplayt\n−∞dt′eiE(t−t′)Σr(t,t′)Gr(t′,E) =I.(51)\nThe last term on the left hand side can be written as\n/integraldisplayt\n−∞dt′eiE(t−t′)Σr(t,t′)Gr(t′,E)\n=/integraldisplaydE′\n2π/integraldisplayt\n−∞dt′ei(E′−E)t′ei(E−E′)tΣr(t,E′)Gr(t′,E)\n≈/integraldisplaydE′\n2π/integraldisplayt\n−∞dt′ei(E′−E)t′ei(E−E′)tΣr(t,E′)Gr(t,E)\n=/integraldisplaydE′\n2π2πδ(E′−E)ei(E−E′)tΣr(t,E′)Gr(t,E)\n= Σr(t,E)Gr(t,E),\nwhereGr(t′,E) =Gr(t−τ,E)≈Gr(t,E)is used and\nthe fast time variable τ=t−t′is neglected. Then we can\nintroduce the frozen Green’s function Gr\nf(t,E)satisfying\n[E−H(t)−Σr(t,E)]Gr\nf(t,E) =I. (52)\nSubstituting Eq. ( 52) into Eq. ( 51), we have\nGr(t,E) =Gr\nf(t,E)−iGr\nf(t,E)˙Gr(t,E). (53)\nSimilarly, the advanced Green’s function is\nGa(E,t) =Ga\nf(E,t)+i˙Ga(E,t)Ga\nf(E,t). (54)\nUp to the linear order in frequency, we find\nGr(t,E) =Gr\nf(t,E)−iGr\nf(t,E)˙Gr\nf(t,E),\nGa(E,t) =Ga\nf(E,t)+i˙Ga\nf(E,t)Ga\nf(E,t).(55)\nNotice that the frozen Green’s function is an instantaneous\nfunction, since it depends only on the present time.\nAPPENDIX D\nIn this appendix, we compare the Gilbert damping coeffi-\ncient derived in Ref. [ 28] using the scattering matrix theory\nwith that obtained in this work via nonequilibrium Green’s\nfunctions. According to Ref. [ 28], in the absence of exter-\nnal bias, the Gilbert tensor element at zero temperature is e x-\npressed in terms of the scattering matrix at the Fermi energy :\nGij(m) =γ2\n4πRe{Tr[∂S\n∂mi∂S\n∂mj]}. (56)To compare with Eq. ( 19), we calculate the dimensionless\nquantityα′\nij=Gij/γ2. The scattering matrix Sis connected\nto the Green’s function through the Fisher-Lee relation121,122\nSαβσσ′=−δαβσσ′+i/an}b∇acketle{tWασ|Gr|Wβσ′/an}b∇acket∇i}ht,\n(S†)βασ′σ=−δαβσσ′−i/an}b∇acketle{tWβσ′|Ga|Wασ/an}b∇acket∇i}ht,\nwhereΓαmnσσ′=|Wαmσ/an}b∇acket∇i}ht/an}b∇acketle{tWαnσ′|.|Wαm/an}b∇acket∇i}htis proportional\nto the eigenvector of Γα122. From Eq. ( 45), we see that\n∂Gr,a\nf\n∂mi=1\n2JMGr,a\nfσiGr,a\nf.\nThen we can express Eq. ( 56) with frozen Green’s functions\nTr[∂S\n∂mi∂S\n∂mj] =1\n4(JM)2Tr[Ga\nfΓGr\nfσiGr\nfΓGa\nfσj]\n= (JM)2Tr[Im(Gr\nf)σiIm(Gr\nf)σj].\nHence the dimensionless damping tensor element is\nα′\nij=1\n16π(JM)2Re{Tr[Ga\nfΓGr\nfσiGr\nfΓGa\nfσj]}.(57)\nNow we assume that there is no external bias and the temper-\nature is zero, which are the same conditions used in deriving\nEq. ( 56) in Ref. [ 28]. Note that Eq. ( 56) was derived from the\npumped energy current defined as ˙mα′˙mso that the resultant\ntensorα′\nijis always symmetric. Hence we symmetrize the\ndimensionless damping tensor in Eq. ( 19):\n˜αij=1\n2(αij+αji). (58)\nIn the absence of external bias, G<\nf= (Ga\nf−Gr\nf)f(E), and\nEq. ( 58) becomes\n˜αij=−/integraldisplay\ndETr[σiAσjGr\nfGr\nf−σjAσiGa\nfGa\nf\n+σjAσiGr\nfGr\nf−σiAσjGa\nfGa\nf]\n=−/integraldisplay\ndETr[σiAσj∂E(Gr\nf−Ga\nf)+σjAσi∂E(Gr\nf−Ga\nf)]\n=−/integraldisplay\ndEfTr∂E[σi(Gr\nf−Ga\nf)σj(Gr\nf−Ga\nf)]\n= Tr[σi(Gr\nf−Ga\nf)σj(Gr\nf−Ga\nf)], (59)\nwithA= (Gr\nf−Ga\nf)f. Apart from a constant factor\n(JM)2/16π, this expression is exactly the same as α′\nijshown\nin Eq. ( 57). Therefore, we confirm the equivalence of the\nGilbert damping tensor between our formalism (Eq. ( 19)) and\nthat obtained in Ref. [ 28] in the limiting case.\nAPPENDIX E\nIn this appendix, we derive the nonadiabatic term in\nEq. ( 31). Expanding m(r,t)up to the first order in ∂im≡\n∂m/∂xi, we have\nm(r,t) =m+δm=m+xi∂im,11\nwhere the Einstein summation convention is implied. The\nnonequilibrium Green’s functions have similar expansions\nGr=Gr\n0−(JM/4)Gr\n0σxi∂imGr\n0,\nG<=G<\n0−[(JM/4)Gr\n0σxi∂imG<\n0−H.c.].\nIt is straightforward to find the correction on s(0)\nDdue to∂im,\nδs(0)\nD=JMi\n8/integraldisplaydE\n2πTrs[σGrσxjG<]xx∂jm+H.c.,(60)\nwhere we have focused on the linear response regime52,123and\nkept only the linear term in ∂jm. HereTrs[...]xxdenotes\ntracing over spin space and then taking diagonal matrix el-\nement in real space and we have dropped the subscript 0in\nthe Green’s function. If we further neglect SOI, the Green’s\nfunction is diagonal in spin space in the linear regime, and\nTrs[σGrσxjG<]xx=1Trs[GrxjG<]xxis also diagonal in\nspin space. Using the relation\nxj=mc0[∇j,H−E],\nwhich is valid in the adiabatic approximation in spatial do-\nmain (c0is a constant having dimension T−2withTthe di-\nmension of time), Eq. 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Fol-\nlowing a Rayleigh dissipation functional approach within a Lagrangian classical \feld formulation,\nthe theory captures intra- as well as cross-sublattice terms in the Gilbert damping, parameterized\nby a 2\u00022 matrix. When spin-pumping into an adjacent conductor causes dissipation, we obtain\nthe corresponding Gilbert damping matrix in terms of the interfacial spin-mixing conductances.\nOur model reproduces the experimentally observed enhancement of the ferromagnetic resonance\nlinewidth in a ferrimagnet close to its compensation temperature without requiring an increased\nGilbert parameter. It also predicts new contributions to damping in an antiferromagnet and sug-\ngests the resonance linewidths as a direct probe of the sublattice asymmetry, which may stem from\nboundary or bulk.\n1arXiv:1808.04385v2  [cond-mat.mtrl-sci]  6 Nov 2018I. INTRODUCTION\nThe fundamental connection1between magnetic moment and spin angular momentum\nunderlies the important role for magnets in nearly all spin-based concepts. An applied mag-\nnetic \feld provides the means to manipulate the state of a ferromagnet (FM), and thus the\nassociated spin. Conversely, a spin-polarized current absorbed by the FM a\u000bects its mag-\nnetization2{5. Exploiting a related phenomenon, switching the state of an antiferromagnet\n(AFM) has also been achieved6. Emboldened by this newly gained control, there has been\nan upsurge of interest in AFMs7{10, which o\u000ber several advantages over FMs. These include\nthe absence of stray \felds and a larger anisotropy-induced gap in the magnon spectrum. The\ntwo-sublattice nature of the AFMs further lends itself to phenomena distinct from FMs11.\nConcurrently, ferrimagnets (FiMs) have been manifesting their niche in a wide range of\nphenomena such as ultrafast switching12{14and low-dissipation spin transport15{22. A class of\nFiMs exhibits the so-called compensation temperature23{28, at which the net magnetization\nvanishes, similar to the case of AFMs. Despite a vanishing magnetization in the compensated\nstate, most properties remain distinct from that of AFMs29. Thus, these materials can be\ntuned to mimic FMs and AFMs via the temperature. In conjunction with the possibility of a\nseparate angular-momentum compensation, when the magnetization does not vanish but the\ntotal spin does, FiMs provide a remarkably rich platform for physics and applications. An\nincreased complexity in the theoretical description29,30hence accompanies these structurally\ncomplicated materials, and may be held responsible for comparatively fewer theoretical\nstudies. Nevertheless, a two-sublattice model with distinct parameters for each sublattice\nqualitatively captures all the phenomena mentioned above.\nDissipation strongly in\ruences the response of a magnet to a stimulus and is thus cen-\ntral to the study of magnetic phenomena such as switching, domain wall motion and spin\ntransport. Nevertheless, magnetic damping has conventionally been investigated via the\nferromagnetic resonance (FMR) linewidth. It is accounted for phenomenologically in the\nLandau-Lifshitz description of the magnetization dynamics via the so-called Gilbert damp-\ning term31, which produces a good agreement with experiments for a wide range of systems.\nThe Gilbert damping represents the viscous contribution and may be `derived' within a\nLagrangian formulation of classical \feld theory by including the Rayleigh dissipation func-\ntional31. While the magnetic damping for FMs has been studied in great detail29,31{35,\n2from phenomenological descriptions to microscopic models, a systematic development of an\nanalogous description for ferri- and antiferromagnets has been lacking in literature. Further-\nmore, recent theoretical results on spin pumping in two-sublattice magnets36and damping\nin AFMs37suggest an important role for the previously disregarded29cross-sublattice terms\nin Gilbert damping, and thus set the stage for the present study. Yuan and co-workers have\nrecently presented a step in this direction focussing on spin torques in AFMs38.\nHere, we formulate the magnetization dynamics equations in a general two-sublattice\nmagnet following the classical Lagrangian approach that has previously been employed for\nFMs31. The Gilbert damping is included phenomenologically via a Rayleigh dissipation\nfunctional appropriately generalized to the two-sublattice system, which motivates intra-\nas well as cross-sublattice terms. The Gilbert damping parameter thus becomes a 2 \u00022\nmatrix, in contrast with its scalar form for a single-sublattice FM. Solving the system of\nequations for spatially homogeneous modes in a collinear ground state, we obtain the decay\nrates of the two eigenmodes \fnding direct pathways towards probing the dissipation mech-\nanism and asymmetries in the system. Consistent with recent experiments28,39, we \fnd an\nenhancement in the decay rates39close to the magnetization compensation in a FiM with\nan unaltered damping matrix28. The general description is found to be consistent with the\nspin pumping mediated damping in the magnet34{36, and allows for relating the Gilbert\ndamping matrix with the interfacial spin-mixing conductances. Focusing on AFMs, we ex-\npress the magnetization dynamics in terms of the Neel variable thus clarifying the origin\nof the di\u000berent damping terms in the corresponding dynamical equations38,40. Apart from\nthe usually considered terms, we \fnd additional contributions for the case when sublattice-\nsymmetry is broken in the AFM36,41{45. Thus, FMR linewidth measurements o\u000ber a direct,\nparameter-free means of probing the sublattice asymmetry in AFMs, complementary to the\nspin pumping shot noise36.\nThis paper is organized as follows. We derive the Landau-Lifshitz-Gilbert (LLG) equa-\ntions for the two-sublattice model in Sec. II. The ensuing equations are solved for the\nresonance frequencies and decay rates of the uniform modes in a collinear magnet in Sec.\nIII. Section IV presents the application of the phenomenology to describe a compensated\nferrimagnet and spin pumping mediated Gilbert damping. The case of AFMs is discussed in\nSec. V. We comment on the validity and possible generalizations of the theory in Sec. VI.\nThe paper is concluded with a summary in Sec. VII. The discussion of a generalized Rayleigh\n3dissipation functional and properties of the damping matrix is deferred to the appendix.\nII. MAGNETIZATION DYNAMICS AND GILBERT DAMPING\nWe consider a two-sublattice magnet described by classical magnetization \felds MMMA\u0011\nMMMA(rrr;t) andMMMB\u0011MMMB(rrr;t) corresponding to the sublattices AandB. The system is\ncharacterized by a magnetic free energy F[MMMA;MMMB] with the magnetization \felds assumed\nto be of constant magnitudes MA0andMB0. Here, the notation F[ ] is employed to emphasize\nthat the free energy is a functional over the magnetization \felds, i.e. an integration of the\nfree energy density over space.\nThe undamped magnetization dynamics is described by equating the time derivative of\nthe spin angular momentum associated with the magnetization to the torque experienced\nby it. The resulting Landau-Lifshitz equations for the two \felds may be written as:\nd\ndt\u0012MMMA;B\n\u0000j\rA;Bj\u0013\n=\u0000_MMMA;B\nj\rA;Bj=MMMA;B\u0002\u00160HHHA;B; (1)\nwhere\rA;B(<0) are the gyromagnetic ratios for the two sublattices, and HHHA;Bare the\ne\u000bective magnetic \felds experienced by the respective magnetizations. This expression of\nangular momentum \row may be derived systematically within the Lagrangian classical \feld\ntheory31. The same formalism also allows to account for a restricted form of damping via\nthe so-called dissipation functional R[_MMMA;_MMMB] in the generalized equations of motion:\nd\ndt\u000eL[\u0001]\n\u000e_MMMA;B\u0000\u000eL[\u0001]\n\u000eMMMA;B=\u0000\u000eR[_MMMA;_MMMB]\n\u000e_MMMA;B; (2)\nwhereL[\u0001]\u0011 L [MMMA;MMMB;_MMMA;_MMMB] is the Lagrangian of the magnetic system. Here,\n\u000eL[\u0001]=\u000eMMMArepresents the functional derivative of the Lagrangian with respect to the var-\nious components of MMMA, and so on. The left hand side of Eq. (2) above represents the\nconservative dynamics of the magnet and reproduces Eq. (1) with31\n\u00160HHHA;B=\u0000\u000eF[MMMA;MMMB]\n\u000eMMMA;B; (3)\nwhile the right hand side accounts for the damping.\nThe Gilbert damping is captured by a viscous Rayleigh dissipation functional parame-\nterized by a symmetric matrix \u0011ijwithfi;jg=fA;Bg:\nR[_MMMA;_MMMB] =Z\nVd3r\u0010\u0011AA\n2_MMMA\u0001_MMMA+\u0011BB\n2_MMMB\u0001_MMMB+\u0011AB_MMMA\u0001_MMMB\u0011\n; (4)\n4whereVis the volume of the magnet. The above form of the functional assumes the damping\nto be spatially homogeneous, isotropic, and independent of the equilibrium con\fguration.\nA more general form with a lower symmetry is discussed in appendix A. Including the\ndissipation functional via Eq. (2) leads to the following replacements in the equations of\nmotion (1):\n\u00160HHHA!\u00160HHHA\u0000\u0011AA_MMMA\u0000\u0011AB_MMMB; (5)\n\u00160HHHB!\u00160HHHB\u0000\u0011BB_MMMB\u0000\u0011AB_MMMA: (6)\nHence, the LLG equations for the two-sublattice magnet become:\n_MMMA=\u0000j\rAj(MMMA\u0002\u00160HHHA) +j\rAj\u0011AA\u0010\nMMMA\u0002_MMMA\u0011\n+j\rAj\u0011AB\u0010\nMMMA\u0002_MMMB\u0011\n; (7)\n_MMMB=\u0000j\rBj(MMMB\u0002\u00160HHHB) +j\rBj\u0011AB\u0010\nMMMB\u0002_MMMA\u0011\n+j\rBj\u0011BB\u0010\nMMMB\u0002_MMMB\u0011\n:(8)\nThese can further be expressed in terms of the unit vectors ^mmmA;B=MMMA;B=MA0;B0:\n_^mmmA=\u0000j\rAj(^mmmA\u0002\u00160HHHA) +\u000bAA\u0010\nmmmA\u0002_^mmmA\u0011\n+\u000bAB\u0010\n^mmmA\u0002_^mmmB\u0011\n; (9)\n_^mmmB=\u0000j\rBj(^mmmB\u0002\u00160HHHB) +\u000bBA\u0010\n^mmmB\u0002_^mmmA\u0011\n+\u000bBB\u0010\n^mmmB\u0002_^mmmB\u0011\n; (10)\nthereby introducing the Gilbert damping matrix ~ \u000bfor a two-sublattice system:\n~\u000b=0\n@\u000bAA\u000bAB\n\u000bBA\u000bBB1\nA=0\n@j\rAj\u0011AAMA0j\rAj\u0011ABMB0\nj\rBj\u0011ABMA0j\rBj\u0011BBMB01\nA; (11)\n\u000bAB\n\u000bBA=j\rAjMB0\nj\rBjMA0: (12)\nAs elaborated in appendix B, the positivity of the dissipation functional implies that the\neigenvalues and the determinant of ~ \u000bmust be non-negative, which is equivalent to the\nfollowing conditions:\n\u0011AA;\u0011BB\u00150; \u0011 AA\u0011BB\u0015\u00112\nAB=)\u000bAA;\u000bBB\u00150; \u000b AA\u000bBB\u0015\u000bAB\u000bBA: (13)\nThus, Eqs. (9) and (10) constitute the main result of this section, and introduce the damping\nmatrix [Eq. (11)] along with the constraints imposed on it [Eq. (12) and (13)] by the\nunderlying formalism.\n5III. UNIFORM MODES IN COLLINEAR GROUND STATE\nIn this section, we employ the phenomenology introduced above to evaluate the resonance\nfrequencies and the decay rates of the spatially homogeneous modes that can be probed in a\ntypical FMR experiment. We thus work in the macrospin approximation, i.e. magnetizations\nare assumed to be spatially invariant. Considering an antiferromagnetic coupling J(>0)\nbetween the two sublattices and parameterizing uniaxial easy-axis anisotropies via KA;B(>\n0), the free energy assumes the form:\nF[MMMA;MMMB] =Z\nVd3r\u0002\n\u0000\u00160H0(MAz+MBz)\u0000KAM2\nAz\u0000KBM2\nBz+JMMMA\u0001MMMB\u0003\n;(14)\nwhereH0^zzzis the applied magnetic \feld. The magnet is assumed to be in a collinear ground\nstate:MMMA=MA0^zzzandMMMB=\u0000MB0^zzzwithMA0>M B0. Employing Eq. (3) to evaluate the\ne\u000bective \felds, the magnetization dynamics is expressed via the LLG equations (9) and (10).\nConsidering MMMA=MAx^xxx+MAy^yyy+MA0^zzz,MMMB=MBx^xxx+MBy^yyy\u0000MB0^zzzwithjMAx;Ayj\u001c\nMA0,jMBx;Byj \u001cMB0, we linearize the resulting dynamical equations. Converting to\nFourier space via MAx=MAxexp (i!t) etc. and switching to circular basis via MA\u0006(B\u0006)=\nMAx(Bx)\u0006iMAy(By), we obtain two sets of coupled equations expressed succinctly as:\n0\n@\u0006!\u0000\nA\u0000i!\u000b AA\u0000\u0010\nj\rAjJMA0+i!\u000b ABMA0\nMB0\u0011\n\u0010\nj\rBjJMB0+i!\u000b BAMB0\nMA0\u0011\n\u0006!+ \n B+i!\u000b BB1\nA0\n@MA\u0006\nMB\u00061\nA=0\n@0\n01\nA; (15)\nwhere we de\fne \n A\u0011j\rAj(JMB0+ 2KAMA0+\u00160H0) and \n B\u0011j\rBj(JMA0+ 2KBMB0\u0000\n\u00160H0). Substituting !=!r\u0006+i!i\u0006into the ensuing secular equation, we obtain the\nresonance frequencies !r\u0006to the zeroth order and the corresponding decay rates !i\u0006to the\n\frst order in the damping matrix elements:\n!r\u0006=\u0006(\nA\u0000\nB) +p\n(\nA+ \n B)2\u00004J2j\rAjj\rBjMA0MB0\n2; (16)\n!i\u0006\n!r\u0006=\u0006!r\u0006(\u000bAA\u0000\u000bBB) +\u000bAA\nB+\u000bBB\nA\u00002Jj\rBjMA0\u000bAB\n!r++!r\u0000: (17)\nIn the expression above, Eq. (16) and Eq. (17), we have chosen the positive solutions of\nthe secular equations for the resonance frequencies. The negative solutions are equal in\nmagnitude to the positive ones and physically represent the same two modes. The positive-\npolarized mode in our notation corresponds to the typical ferromagnetic resonance mode,\nwhile the negative-polarized solution is sometimes termed `antiferromagnetic resonance'25.\n6020406080100120\n0 0.2 0.4 0.6 0.8 10.040.050.060.070.080.090.1FIG. 1. Resonance frequencies and normalized decay rates vs. the applied \feld for a quasi-\nferromagnet ( MA0= 5MB0).j\rAj=j\rBj= 1;1:5;0:5 correspond to solid, dashed and dash-dotted\nlines respectively. The curves in blue and red respectively depict the + and \u0000modes. The damping\nparameters employed are \u000bAA= 0:06,\u000bBB= 0:04 and\u000bAB= 0.\nIn order to avoid confusion with the ferromagnetic or antiferromagnetic nature of the un-\nderlying material, we call the two resonances as positive- and negative-polarized. The decay\nrates can further be expressed in the following form:\n!i\u0006\n!r\u0006=\u0016\u000b(\nA+ \n B)\u00002Jj\rBjMA0\u000bAB\n!r++!r\u0000\u0006\u0001\u0016\u000b; (18)\nwith \u0016\u000b\u0011(\u000bAA+\u000bBB)=2 and \u0001\u0016\u000b\u0011(\u000bAA\u0000\u000bBB)=2. Eq. (18) constitutes the main result\nof this section and demonstrates that (i) asymmetric damping in the two sublattices is\nmanifested directly in the normalized decay rates of the two modes (Figs. 1 and 2), and\n(ii) o\u000b-diagonal components of the damping matrix may reduce the decay rates (Fig. 2).\nFurthermore, it is consistent with and reproduces the mode-dependence of the decay rates\nobserved in the numerical studies of some metallic AFMs37.\nTo gain further insight into the results presented in Eqs. (16) and (18), we plot the\n7resonance frequencies and the normalized decay rates vs. the applied magnetic \feld for a\ntypical quasi-ferromagnet, such as yttrium iron garnet, in Fig. 1. The parameters employed\nin the plot arej\rBj= 1:8\u00021011,MB0= 105,KA=KB= 10\u00007, andJ= 10\u00005in SI units,\nand have been chosen to represent the typical order of magnitude without pertaining to a\nspeci\fc material. The plus-polarized mode is lower in energy and is raised with an increasing\napplied magnetic \feld. The reverse is true for the minus-polarized mode whose relatively\nlarge frequency makes it inaccessible to typical ferromagnetic resonance experiments. As\nanticipated from Eq. (18), the normalized decay rates for the two modes di\u000ber when \u000bAA6=\n\u000bBB. Furthermore, the normalized decay rates are independent of the applied \feld for\nsymmetric gyromagnetic ratios for the two sublattices. Alternately, a measurement of the\nnormalized decay rate for the plus-polarized mode is able to probe the sublattice asymmetry\nin the gyromagnetic ratios. Thus it provides essential information about the sublattices\nwithout requiring the measurement of the large frequency minus-polarized mode.\nIV. SPECIFIC APPLICATIONS\nWe now examine two cases of interest: (i) the mode decay rate in a ferrimagnet close to\nits compensation temperature, and (ii) the Gilbert damping matrix due to spin pumping\ninto an adjacent conductor.\nA. Compensated ferrimagnets\nFMR experiments carried out on gadolinium iron garnet23,39\fnd an enhancement in the\nlinewidth, and hence the mode decay rate, as the temperature approaches the compensation\ncondition, i.e. when the two e\u000bective46sublattices have equal saturation magnetizations.\nThese experiments have conventionally been interpreted in terms of an e\u000bective single-\nsublattice model thereby ascribing the enhancement in the decay rate to an increase in the\nscalar Gilbert damping constant allowed within the single-sublattice model24. In contrast,\nexperiments probing the Gilbert parameter in a di\u000berent FiM via domain wall velocity\n\fnd it to be essentially unchanged around compensation28. Here, we analyze FMR in a\ncompensated FiM using the two-sublattice phenomenology developed above and thus address\nthis apparent inconsistency.\n8020406080100120\n1 2 3 4 500.050.10.150.20.250.3FIG. 2. Resonance frequencies and normalized decay rates vs. relative saturation magnetizations\nof the sublattices. The curves which are not labeled as + or \u0000represent the common normalized\ndecay rates for both modes. The parameters employed are the same as for Fig. 1 with \rA=\rB.\nThe compensation behavior of a FiM may be captured within our model by allowing\nMA0to vary while keeping MB0\fxed. The mode frequencies and normalized decay rates\nare examined with respect to the saturation magnetization variation in Fig. 2. We \fnd an\nenhancement in the normalized decay rate, consistent with the FMR experiments23,39, for a\n\fxed Gilbert damping matrix. The single-sublattice interpretation ascribes this change to a\nmodi\fcation of the e\u000bective Gilbert damping parameter24, which is equal to the normalized\ndecay rate within that model. In contrast, the latter is given by Eq. (18) within the\ntwo-sublattice model and evolves with the magnetization without requiring a modi\fcation\nin the Gilbert damping matrix. Speci\fcally, the enhancement in decay rate observed at\nthe compensation point is analogous to the so-called exchange enhancement of damping in\nAFMs47. Close to compensation, the FiM mimics an AFM to some extent.\nWe note that while the spherical samples employed in Ref. 23 are captured well by our\nsimple free energy expression [Eq. (14)], the interfacial and shape anisotropies of the thin\n9\flm sample employed in Ref. 39 may result in additional contributions to decay rates. The\nsimilarity of the observed linewidth trends for the two kinds of samples suggests that these\nadditional anisotropy e\u000bects may not underlie the observed damping enhancement. Quan-\ntitatively accounting for these thin \flm e\u000bects requires a numerical analysis, as discussed\nin Sec VI below, and is beyond the scope of the present work. Furthermore, domain forma-\ntion may result in additional damping contributions not captured within our single-domain\nmodel.\nB. Spin pumping mediated Gilbert damping\nSpin pumping34from a FM into an adjacent conductor has been studied in great detail35\nand has emerged as a key method for injecting pure spin currents into conductors48. The\nangular momentum thus lost into the conductor results in a contribution to the magnetic\ndamping on top of the intrinsic dissipation in the bulk of the magnet. A variant of spin\npumping has also been found to be the dominant cause of dissipation in metallic magnets37.\nThus, we evaluate the Gilbert damping matrix arising due to spin pumping from a two-\nsublattice magnet36into an adjacent conductor acting as an ideal spin sink.\nWithin the macrospin approximation, the total spin contained by the magnet is given by:\nSSS=\u0000MA0V^mmmA\nj\rAj\u0000MB0V^mmmB\nj\rBj: (19)\nThe spin pumping current emitted by the two-sublattice magnet has the following general\nform36:\nIIIs=~\neX\ni;j=fA;BgGij\u0010\n^mmmi\u0002_^mmmj\u0011\n; (20)\nwithGAB=GBA, where the spin-mixing conductances Gijmay be evaluated within di\u000berent\nmicroscopic models36,49{51. Equating the spin pumping current to \u0000_SSSand employing Eqs.\n(9) and (10), the spin pumping contribution to the Gilbert damping matrix becomes:\n\u000b0\nij=~Gijj\rij\neMi0V; (21)\nwhich in turn implies\n\u00110\nij=~Gij\neMi0Mj0V; (22)\n10for the corresponding dissipation functional. The resulting Gilbert damping matrix is found\nto be consistent with its general form and constraints formulated in Sec. II. Thus, employing\nthe phenomenology developed above, we are able to directly relate the magnetic damping in\na two-sublattice magnet to the spin-mixing conductance of its interface with a conductor.\nV. ANTIFERROMAGNETS\nDue to their special place with high symmetry in the two-sublattice model as well as the\nrecent upsurge of interest7{10,52{54, we devote the present section to a focused discussion on\nAFMs in the context of the general results obtained above. It is often convenient to describe\nthe AFM in terms of a di\u000berent set of variables:\nmmm=^mmmA+^mmmB\n2; nnn=^mmmA\u0000^mmmB\n2: (23)\nIn contrast with ^mmmAand ^mmmB,mmmandnnnare not unit vectors in general. The dynamical\nequations for mmmandnnnmay be formulated by developing the entire \feld theory, starting with\nthe free energy functional, in terms of mmmandnnn. Such a formulation, including damping,\nhas been accomplished by Hals and coworkers40. Here, we circumvent such a repetition and\ndirectly express the corresponding dynamical equations by employing Eqs. (9) and (10) into\nEq. (23):\n_mmm=\u0000(mmm\u0002\rm\u00160HHHm)\u0000(nnn\u0002\rn\u00160HHHn) +X\np;q=fm;ng\u000bm\npq(ppp\u0002_qqq); (24)\n_nnn=\u0000(mmm\u0002\rn\u00160HHHn)\u0000(nnn\u0002\rm\u00160HHHm) +X\np;q=fm;ng\u000bn\npq(ppp\u0002_qqq); (25)\nwith\n\rm\u00160HHHm\u0011j\rAj\u00160HHHA+j\rBj\u00160HHHB\n2; (26)\n\rn\u00160HHHn\u0011j\rAj\u00160HHHA\u0000j\rBj\u00160HHHB\n2; (27)\n\u000bm\nmm=\u000bn\nnm=\u000bAA+\u000bBB+\u000bAB+\u000bBA\n2; (28)\n\u000bm\nmn=\u000bn\nnn=\u000bAA\u0000\u000bBB\u0000\u000bAB+\u000bBA\n2; (29)\n\u000bm\nnn=\u000bn\nmn=\u000bAA+\u000bBB\u0000\u000bAB\u0000\u000bBA\n2; (30)\n\u000bm\nnm=\u000bn\nmm=\u000bAA\u0000\u000bBB+\u000bAB\u0000\u000bBA\n2: (31)\n11A general physical signi\fcance, analogous to \rA;B, may not be associated with \rm;nwhich\nmerely serve the purpose of notation here. The equations obtained above manifest new\ndamping terms in addition to the ones that are typically considered in the description\nof AFMs. Accounting for the sublattice symmetry of the antiferromagnetic bulk while\nallowing for the damping to be asymmetric, we may assume \rA=\rBandMA0=MB0, with\n\u0016\u000b\u0011(\u000bAA+\u000bBB)=2, \u0001\u0016\u000b\u0011(\u000bAA\u0000\u000bBB)=2, and\u000bAB=\u000bBA\u0011\u000bod. Thus, the damping\nparameters simplify to\n\u000bm\nmm=\u000bn\nnm=\u0016\u000b+\u000bod; (32)\n\u000bm\nmn=\u000bn\nnn=\u0001\u0016\u000b; (33)\n\u000bm\nnn=\u000bn\nmn=\u0016\u000b\u0000\u000bod; (34)\n\u000bm\nnm=\u000bn\nmm=\u0001\u0016\u000b; (35)\nthereby eliminating the \\new\" terms in the damping when \u000bAA=\u000bBB. However, the sublat-\ntice symmetry may not be applicable to AFMs, such as FeMn, with non-identical sublattices.\nFurthermore, the sublattice symmetry of the AFM may be broken at the interface41{43via,\nfor example, spin mixing conductances36,45,55resulting in \u000bAA6=\u000bBB.\nThe resonance frequencies and normalized decay rates [Eqs. (16) and (18)] take a simpler\nform for AFMs. Substituting KA=KB\u0011K,\rA=\rB\u0011\r, andMA0=MB0\u0011M0:\n!r\u0006=\u0006j\rj\u00160H0+ 2j\rjM0p\n(J+K)K; (36)\n!i\u0006\n!r\u0006=J(\u0016\u000b\u0000\u000bod) + 2K\u0016\u000b\n2p\n(J+K)K\u0006\u0001\u0016\u000b\u0019(\u0016\u000b\u0000\u000bod)\n2r\nJ\nK+ \u0016\u000br\nK\nJ\u0006\u0001\u0016\u000b; (37)\nwhere we have employed J\u001dKin the \fnal simpli\fcation. The term /p\nK=J has typically\nbeen disregarded on the grounds K\u001cJ. However, recent numerical studies of damping in\nseveral AFMs37\fnd \u0016\u000b\u001d\u0016\u000b\u0000\u000bod>0 thus suggesting that this term should be comparable\nto the one proportional top\nJ=K and hence may not be disregarded. The expression above\nalso suggests measurement of the normalized decay rates as a means of detecting the sublat-\ntice asymmetry in damping. For AFMs symmetrical in the bulk, such an asymmetry may\narise due to the corresponding asymmetry in the interfacial spin-mixing conductance36,45,55.\nThus, decay rate measurements o\u000ber a method to detect and quantify such interfacial e\u000bects\ncomplementary to the spin pumping shot noise measurements suggested earlier36.\n12VI. DISCUSSION\nWe have presented a phenomenological description of Gilbert damping in two-sublattice\nmagnets and demonstrated how it can be exploited to describe and characterize the system\ne\u000bectively. We now comment on the limitations and possible generalizations of the formal-\nism presented herein. To begin with, the two-sublattice model is the simplest description of\nferri- and antiferromagnets. It has been successful in capturing a wide range of phenomenon.\nHowever, recent measurements of magnetization dynamics in nickel oxide could only be ex-\nplained using an eight-sublattice model56. The temperature dependence of the spin Seebeck\ne\u000bect in yttrium iron garnet also required accounting for more than two magnon bands57.\nA generalization of our formalism to a N-sublattice model is straightforward and can be\nachieved via a Rayleigh dissipation functional with N2terms, counting \u0011ijand\u0011jias sepa-\nrate terms. The ensuing Gilbert damping matrix will be N \u0002N while obeying the positive\ndeterminant constraint analogous to Eq. (13).\nIn our description of the collinear magnet [Eq. (14)], we have disregarded contributions\nto the free energy which break the uniaxial symmetry of the system about the z-axis. Such\nterms arise due to spin-nonconserving interactions58, such as dipolar \felds and magnetocrys-\ntalline anisotropies, and lead to a mixing between the plus- and minus-polarized modes30.\nIncluding these contributions converts the two uncoupled 2 \u00022 matrix equations [(15)] into\na single 4\u00024 matrix equation rendering the solution analytically intractable. A detailed\nanalysis of these contributions30shows that their e\u000bect is most prominent when the two\nmodes are quasi-degenerate, and may be disregarded in a \frst approximation.\nIn evaluating the resonance frequencies and the decay rates [Eqs. (16) and (18)], we\nhave assumed the elements of the damping matrix to be small. A precise statement of the\nassumption employed is !i\u001c!r, which simply translates to \u000b\u001c1 for a single-sublattice\nferromagnet. In contrast, the constraint imposed on the damping matrix within the two-\nsublattice model by the assumption of small normalized decay rate is more stringent [Eq.\n(18)]. For example, this assumption for an AFM with \u000bAB= \u0001\u0016\u000b= 0 requires \u0016 \u000b\u001c\np\nK=J\u001c1. This stringent constraint may not be satis\fed in most AFMs37, thereby\nbringing the simple Lorentzian shape description of the FMR into question. It can also be\nseen from Fig. 2 that the assumption of a small normalized decay rate is not very good for\nthe chosen parameters.\n13VII. SUMMARY\nWe have developed a systematic phenomenological description of the Gilbert damping\nin a two-sublattice magnet via inclusion of a Rayleigh dissipation functional within the La-\ngrangian formulation of the magnetization dynamics. Employing general expressions based\non symmetry, we \fnd cross-sublattice Gilbert damping terms in the LLG equations in con-\nsistence with other recent \fndings36{38. Exploiting the phenomenology, we explain the en-\nhancement of damping23,39in a compensated ferrimagnet without requiring an increase in\nthe damping parameters28. We also demonstrate approaches to probe the various forms\nof sublattice asymmetries. Our work provides a uni\fed description of ferro- via ferri- to\nantiferromagnets and allows for understanding a broad range of materials and experiments\nthat have emerged into focus in the recent years.\nACKNOWLEDGMENTS\nA. K. thanks Hannes Maier-Flaig and Kathrin Ganzhorn for valuable discussions. We\nacknowledge \fnancial support from the Alexander von Humboldt Foundation, the Research\nCouncil of Norway through its Centers of Excellence funding scheme, project 262633, \\QuS-\npin\", and the DFG through SFB 767 and SPP 1538.\nAppendix A: Generalized Rayleigh dissipation functional\nAs compared to the considerations in Sec. II, a more general approach to parameterizing\nthe dissipation functional is given by:\nR[_MMMA;_MMMB] =1\n2Z\nVZ\nVd3r0d3rX\np;q=fA;BgX\ni;j=fx;y;zg_Mpi(rrr)\u0011ij\npq(rrr;rrr0)_Mqj(rrr0): (A1)\nThis form allows to capture the damping in an environment with a reduced symmetry.\nHowever, the larger number of parameters also makes it di\u000ecult to extract them reliably\nvia typical experiments. The above general form reduces to the case considered in Sec. II\nwhen\u0011ij\npq(rrr;rrr0) =\u0011pq\u000eij\u000e(rrr\u0000rrr0) and\u0011pq=\u0011qp. Furthermore, the coe\u000ecients \u0011ij\npqmay depend\nuponMMMA(rrr) andMMMB(rrr) as has been found in recent numerical studies of Gilbert damping\nin AFMs37.\n14Appendix B: Damping matrix\nThe Rayleigh dissipation functional considered in the main text is given by:\nR[_MMMA;_MMMB] =Z\nVd3r\u0010\u0011AA\n2_MMMA\u0001_MMMA+\u0011BB\n2_MMMB\u0001_MMMB+\u0011AB_MMMA\u0001_MMMB\u0011\n; (B1)\nwhich may be brought into the following concise form with the notation~_MMM\u0011[_MMMA_MMMB]|:\nR[_MMMA;_MMMB] =1\n2Z\nVd3r~_MMM|~\u0011~_MMM; (B2)\nwhere ~\u0011is the appropriate matrix given by:\n~\u0011=0\n@\u0011AA\u0011AB\n\u0011AB\u0011BB1\nA: (B3)\nConsidering an orthogonal transformation~_MMM=~Q~_M, the dissipation functional can be\nbrought to a diagonal form\nR[_MMMA;_MMMB] =1\n2Z\nVd3r~_M|~Q|~\u0011~Q~_M; (B4)\nwhere ~Q|~\u0011~Qis assumed to be diagonal. 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We prove that if the damping is effectiv e, and the nonlinearity and other\nlower order terms can be regarded as perturbations, then the solution is approximated by the\nscaled Gaussian of the corresponding linear parabolic prob lem. The proof is based on the scaling\nvariables and energy estimates.\n1 Introduction\nWe consider the Cauchy problem of the semilinear damped wave equat ion with lower order perturba-\ntions\n/braceleftbigg\nutt+b(t)ut= ∆xu+c(t)·∇xu+d(t)u+N(u,∇xu,ut), t>0,x∈Rn,\nu(0,x) =εu0(x), ut(0,x) =εu1(x), x ∈Rn,(1.1)\nwhere the coefficients b,canddare smooth, bsatisfies\nb(t)∼(1+t)−β,−1≤β <1, (1.2)\nandc(t)·∇xu,d(t)u,N(u,∇xu,ut) can be regarded as perturbations (the precise assumption will be\ngiven in the next section). Also, εdenotes a small parameter.\nOur purpose is to give the asymptotic profile of global solutions to (1 .1) with small initial data as\ntime tends to infinity. By the assumption (1.2), the damping is effectiv e, and we can expect that the\nasymptotic profile of solutions is given by the scaled Gaussian (see (2 .7), (2.8) and (2.9)).\nTheexistenceofglobalsolutionsandtheasymptoticbehaviorofso lutionstodampedwaveequations\nhave been widely investigated for a long time. Matsumura [27] obtain ed decay estimates of solutions\nto the linear damped wave equation\nutt−∆u+ut= 0, (1.3)\nand applied them to nonlinear problems. After that, Yang and Milani [5 2] showed that the solution of\n(1.3) has the so-called diffusion phenomena , that is, the asymptotic profile of solutions to (1.3) is given\nby the Gaussian in the L∞-sense. Marcati and Nishihara [26] and Nishihara [31] gave more det ailed\ninformations about the asymptotic behavior of solutions. They fou nd that when n= 1,3, the solution\nof (1.3) is asymptotically decomposed into the Gaussian and a solution of the wave equation (with an\n1exponentially decaying coefficient) in the Lp–Lqsense (see Hosono and Ogawa [12] and Narazaki [30]\nforn= 2 andn≥4).\nFor the nonlinear problem\n/braceleftbigg\nutt−∆u+ut=N(u),\n(u,ut)(0,x) =ε(u0,u1)(x),(1.4)\nthere are many results about global existence and asymptotic beh avior of solutions (see for example,\n[13, 14,17, 19, 20, 22, 32]). Inparticular, TodorovaandYordano v[41] and Zhang[53] provedthat when\nN(u) =|u|p, the critical exponent of (1.4) is given by p= 1+2/n. More precisely, they showed that,\nfor initial data satisfying ( u0,u1)∈H1,0(Rn)×L2(Rn) and having compact support, if p>1+2/n,\nthen the global solution uniquely exists for small ε; ifp≤1+2/nand/integraltext\nRn(u0+u1)(x)dx >0, then\nthe local-in-time solution blows up in finite time for any ε >0. The number 1 + 2 /nis the same\nas the well-known Fujita exponent, which is the critical exponent of the semilinear heat equation\nvt−∆v=vp(see [7]), though the role of the critical exponent is different in the s emilinear heat\nequation and the semilinear damped wave equation. In fact, for the subcritical case 1 <p<1+2/n,\nthe solution of the semilinear damped wave equation blows up in finite tim e under the positive mass\ncondition/integraltext\nRn(u0+u1)(x)dx>0, while all positive solutions blow up in finite time for the semilinear\nheat equation.\nConcerning the asymptotic behavior of the global solution, Hayash i, Kaikina and Naumkin [10]\nproved that if Nsatisfies|N(u)| ≤C|u|pwithp>1+2/n, then the unique global solution exists for\nsuitably small data and the asymptotic profile of the solution is given b y a constant multiple of the\nGaussian. However, they used the explicit formula of the fundamen tal solution of the linear problem\nin the Fourier space, and hence, it seems to be difficult to apply their m ethod to variable coefficient\ncases.\nGallayandRaugel[8]consideredtheone-dimensionaldampedwave equationwithvariableprincipal\nterm and a constant damping\nutt−(a(x)ux)x+ut=N(u,ux,ut).\nThey used scaling variables\ns= log(t+t0), y=x√t+t0, (1.5)\nand showed that if a(x) is positive and has the positive limits lim x→±∞a(x) =a±, then the solution\ncan be asymptotically expanded in terms of the corresponding para bolic equation. Moreover, this\nexpansion can be determined up to the second order. Recently, Ta keda [39, 40] and Kawakami and\nTakeda [18] obtained the complete expansion for the linear and nonlin ear damped wave equation with\nconstant coefficients.\nThe wave equation with variable coefficient damping\nutt−∆u+b(t,x)ut= 0\nhas been also intensively studied. Yamazaki [50, 51] and Wirth [46, 47 , 48, 49] considered time-\ndependent damping b=b(t). Here we briefly explain their results by restricting the damping bto\nb(t) =µ(1+t)−βwithµ>0 andβ∈R, although they discussed more general b(t): (i) when β >1\n(scattering), the solution scatters to a solution of the free wave equation; (ii) when β= 1 (non-effective\nweak dissipation), the behavior of solutions depends on the consta ntµ, and the solution scatters with\nsome modification; (iii) when β∈[−1,1) (effective), the asymptotic profile of the solution is given by\nthe scaled Gaussian; (iv) when β <−1 (overdamping), the solution tends to some asymptotic state,\nwhich is nontrivial function for nontrivial data. Hence our assumpt ion (1.2) is reasonable because the\nasymptotic behavior of solutions to the linear problem completely cha nges whenβ <−1 orβ≥1.\nIn the space-dependent damping case b=b(x) = (1 + |x|2)−α/2, Mochizuki [28] (see also [29])\nproved that if α >1, then the energy of solution does not decay to zero in general an d solutions\n2with data satisfying certain condition scatter to free solutions. On the other hand, Todorova and\nYordanov [42] obtained energy decay of solutions when α∈[0,1) and the decay rates agree with\nthose of the corresponding parabolic equation. Moreover, the au thor of this paper [45] proved that\nthe solution actually has the diffusion phenomena when α∈[0,1). In the critical case α= 1, that\nis,b=µ(1+|x|2)−1/2, Ikehata, Todorova and Yordanov [16] obtained optimal decay es timates of the\nenergy of solutions and found that the decay rate depends on the constantµ. However, the precise\nasymptotic profile is still open. On the other hand, Radu, Todorova and Yordanov [37, 38] studied the\ndiffusion phenomena for solutions to the abstract damped wave equ ation\n(L∂2\nt+∂t+A)u= 0\nby the method of the diffusion approximation, where Ais a nonnegative self-adjoint operator, and\nLis a bounded positive self-adjoint operator. Recently, Nishiyama [36 ] studied the abstract damped\nwave equation having the form ( ∂2\nt+M∂t+A)u= 0, where Mis a bounded nonnegative self-adjoint\noperator. Moreover, as an application, he also determined the asy mptotic profile of solutions to the\ndamped wave equation with variable coefficients under a geometric co ntrol condition.\nFor the semilinear wave equation with space-dependent damping\nutt−∆u+b(x)ut=N(u),\nIkehata, Todorova and Yordanov [15] proved that when b(x)∼(1+|x|)−αwithα∈[0,1) andN(u) =\n|u|p, the critical exponent is p= 1+2/(n−α) (see also Nishihara [33] for the case N(u) =−|u|p−1u\nandb(x) = (1+ |x|2)−α/2withα∈[0,1)).\nRecently, the asymptoticbehaviorofsolutionsto the semilinearwav eequationwith time-dependent\ndamping\nutt−∆u+b(t)ut=N(u)\nwas also studied. When b(t) = (1 +t)−β(−1< β < 1) andN(u) =|u|p, Lin, Nishihara and\nZhai [25] determined the critical exponent as p= 1 + 2/n, provided that the initial data belong to\nH1,0(Rn)×L2(Rn) with compact support. D’Abbicco, Lucente and Reissig [5] (see also [4]) extended\nthis result to more general b(t) satisfying a monotonicity condition and a polynomial-like behavior.\nMoreover, they relaxed the assumption on the data to exponentia lly decaying condition. They also\ndealt with the initial data belong to the class ( L1(Rn)∩H1,0(Rn))×(L1(Rn)∩L2(Rn)) whenn≤4.\nWe also refer the reader to D’Abbicco [3] for the critical case β= 1. On the other hand, Nishihara\n[34] studied the asymptotic profile of solutions in the case n= 1,b= (1 +t)−β(−1< β <1),\n(u0,u1)∈H1,0(Rn)×L2(Rn) with compact support and N(u) =−|u|p−1u(see also [35]). He proved\nthattheasymptoticprofileisgivenbythescaledGaussian. However ,the asymptoticprofileofsolutions\nin higher dimensional cases n≥2 remains open. Furthermore, even for the small data global exist ence,\nthere are no results for non exponentially decaying initial data when n≥5. Here we also refer the\nreader to [21, 23, 24, 43, 44] for space and time dependent dampin g cases.\nIn this paper, we shall prove the existence of the global-in-time solu tion to the Cauchy problem\n(1.1) with suitably small εand determine the asymptotic profile. Our result extends that of [ 34] to\nhigher dimensional cases n≥2, more general damping b=b(t), non exponentially decaying initial\ndata and with lower order perturbations. Moreover, in the one-dim ensional case, we can treat more\ngeneral nonlinear terms N=N(u,ux,ut) including first order derivatives. For the proof, we basically\nfollow the method of Gallay and Raugel [8]. To extend their argument t o variable damping cases, we\nintroduce new scaling variables\ns= log(B(t)+1), y= (B(t)+1)−1/2x, B(t) =/integraldisplayt\n0dτ\nb(τ)\ninstead of (1.5). Then, we decompose the solution to the asymptot ic profile and the remainder term,\nand prove that remainder term decays to zero as time tends to infin ity by using the energy method.\nTo estimate the energy of the remainder term, in [8], they used the p rimitive of the remainder term\n3F(s,y) =/integraltexty\n−∞f(s,z)dz. However, this does not work in higher dimensional cases n≥2. To overcome\nthis difficulty, we employ the idea from Coulaud [2] in which asymptotic pr ofiles for the second grade\nfluids equationwerestudied in the three dimensionalspace. Namely, weshall usethe fractionalintegral\nof the form ˆF(ξ) =|ξ|−n/2−δˆf(ξ) with 0< δ <1, and apply the energy method to ˆFin the Fourier\nside. Since the remainder term fsatisfies ˆf(0) = 0, ˆFmakes sense and enables us to control the term\n/ba∇dblˆf/ba∇dblL2in energy estimates.\nThis paper is organized as follows. In the next section, we state the precise assumptions and our\nmain result. Section 3 is devoted to a proofof the main result. The pr oof of energy estimates is divided\ninto the one-dimensional case and the higher dimensional cases. Af ter that, we will unify both cases\nand complete the proof of our result except for the estimates of t he error terms. These error estimates\nwill be given in Section 4.\nWe end up this section with some notations used in this paper. For a co mplex number ζ, we denote\nby Reζits real part. The letter Cindicates a generic positive constant, which may change from line\nto line. In particular, we denote by C(∗,...,∗) constants depending on the quantities appearing in\nparenthesis. We use the symbol f∼g, which stands for C−1g≤f≤Cgwith some C≥1. For\na function u=u(t,x) : [0,∞)×Rn→R, we write ut=∂u\n∂t(t),∂xiu=∂u\n∂xi(i= 1,...,n),∇xu=\nt(∂x1u,...,∂ xnu) and ∆u(t,x) =/summationtextn\ni=1∂2\nxiu(t,x). Furthermore, we sometimes use /an}b∇acketle{tx/an}b∇acket∇i}ht:=/radicalbig\n1+|x|2.\nFor a function f=f(x) :Rn→R, we denote the Fourier transform of fbyˆf=ˆf(ξ), that is,\nˆf(ξ) = (2π)−n/2/integraldisplay\nRnf(x)e−ixξdx.\nLetLp(Rn) andHk,m(Rn) be usual Lebesgue and weighted Sobolev spaces, respectively, e quipped\nwith the norms defined by\n/ba∇dblf/ba∇dblLp=/parenleftbigg/integraldisplay\nRn|f(x)|pdx/parenrightbigg1/p\n(1≤p<∞),/ba∇dblf/ba∇dblL∞= esssupx∈Rn|f(x)|,\n/ba∇dblf/ba∇dblHk,m=/summationdisplay\n|α|≤k/ba∇dbl(1+|x|)m∂α\nxf/ba∇dblL2(k∈Z≥0,m≥0).\nFor an interval Iand a Banach space X, we define Cr(I;X) as the space of r-times continuously\ndifferentiable mapping from ItoXwith respect to the topology in X.\n2 Main result\nLet us introduce our main result. First, we put the following assumpt ions:\nAssumptions\n(i) The initial data ( u0,u1) belong to H1,m(Rn)×H0,m(Rn), wherem= 1 (n= 1) andm >\nn/2+1 (n≥2).\n(ii) The coefficient of the damping term b(t) satisfies\nC−1(1+t)−β≤b(t)≤C(1+t)−β,/vextendsingle/vextendsingle/vextendsingle/vextendsingledb\ndt(t)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(1+t)−1b(t) (2.1)\nwith someβ∈[−1,1).\n(iii) The functions c(t) andd(t) satisfy\n|c(t)| ≤C(1+t)−γ,|d(t)| ≤C(1+t)−ν(2.2)\nwith someγ >(1+β)/2 andν >1+β.\n4(iv)-(1) When n= 1, the nonlinearity Nis of the form\nN=k/summationdisplay\ni=1Ni(u,ux,ut)\nfor somek≥0 and each Ni=Ni(z) =Ni(z1,z2,z3) satisfies\n\n\n|Ni(z)| ≤C|z1|pi1|z2|pi2|z3|pi3, pij≥1 or = 0, pi1>1, pi2+pi3≤1,\npi1+2pi2+/parenleftbigg\n3−2β\n1+β/parenrightbigg\npi3>3,(2.3)\nwhere we note that when β=−1, the number −2β/(1+β) is interpreted as an arbitrary large\nnumber. Moreover, to ensure the existence of local-in-time solutio ns, we assume that, for any\nR>0, there exists a constant C(R)>0 such that\n|Ni(z)−Ni(w)| ≤C(R)[|z1−w1|(1+|z2|+|w2|+|z3|+|w3|)+|z2−w2|+|z3−w3|] (2.4)\nforzi,wi∈R(i= 1,2,3) satisfying |z1|,|w1| ≤R.\n(iv)-(2) When n≥2, the nonlinearity Nis of classC1and independent of ∇xu,ut, that is,N=N(u).\nMoreover,Nsatisfies\n/braceleftBigg|N(u)| ≤C|u|p,\n2<p<+∞(n= 2),1+2\nn<p≤n\nn−2(n≥3).(2.5)\nAlso, to ensure the existence of local-in-time solutions, we assume t hat\n|N(u)−N(v)| ≤C|u−v|(|u|+|v|)p−1. (2.6)\nRemark 2.1. (i) By the above assumptions, as we will see later, we can rega rd the terms c(t)·∇xu,\nd(t)uandN(u,∇xu,ut)as perturbations.\n(ii) We can treat the case where the coefficients b(t),c(t)andd(t)depend on both tandx. More\nprecisely, our result is also valid for b=b(t,x),c=c(t,x)andd=d(t,x)such thatb(t,x) =\nb0(t)+b1(t,x)withb0satisfying Assumption (ii), b1(t,x)fulfilling |b1(t,x)| ≤C(1+t)−µ(µ>β)\nandc(t,x),d(t,x)satisfying |c(t,x)| ≤C(1+t)−γ(γ >(1+β)/2),|d(t,x)| ≤C(1+t)−ν(ν >\n1+β).\n(iii) A typical example satisfying the assumptions (2.3)and(2.4)is\nN=|u|pu+|u|qux+|u|rut\nwithp>2,q>1andr>1.\n(iv) The assumption 1+2/n<pin(2.5)is sharp in the sense that, if N(u) =|u|p,1<p≤1+2/n\nand the initial data satisfies/integraltext\nRn(u0+b∗u1)(x)dx>0withb∗=/integraltext∞\n0exp(−/integraltextt\n0b(τ)dτ)dt, then the\nlocal-in-time solution blows up in finite time (see [15, 22, 2 4, 41, 53]).\n(v) Whenn= 1, we can also treat the principal term with variable coefficien t(a(x)ux)xsatisfying\ninf\nx∈Ra(x)>0,lim\nx→±∞a(x) =a±>0\ninstead ofuxx. However, the argument is the same as in Gallay and Raugel [8] and hence, we\ndo not pursue here for simplicity.\n5(vi) There are no mutual implication relations between the a ssumptions on the damping bin ours and\nWirth [48], D’Abbicco, Lucente and Reissig [5].\nTo state our result, we put\nB(t) =/integraldisplayt\n0dτ\nb(τ)(2.7)\nand\nG(t,x) = (4πt)−n/2exp/parenleftbigg\n−|x|2\n4t/parenrightbigg\n. (2.8)\nWe note that the assumption (2.1) implies that B(t) is strictly increasing, and lim t→∞B(t) = +∞.\nThe main result of this paper is the following:\nTheorem 2.1. Under the Assumptions (i)–(iv), there exists some ε0>0such that, for any ε∈(0,ε0],\nthere exists a unique solution\nu∈C([0,∞);H1,m(Rn))∩C1([0,∞);H0,m(Rn))\nfor the Cauchy problem (1.1). Moreover, there exists the limit\nα∗= lim\nt→∞/integraldisplay\nRnu(t,x)dx\nsuch that the solution usatisfies\n/ba∇dblu(t,·)−α∗G(B(t),·)/ba∇dblL2≤Cε(B(t)+1)−n/4−λ/ba∇dbl(u0,u1)/ba∇dblH1,m×H0,m (2.9)\nfort≥1. Hereλis defined by\nλ= min/braceleftbigg1\n2,m\n2−n\n4,λ0,λ1/bracerightbigg\n−η\nwith arbitrary small number η>0, andλ0andλ1are defined by\nλ0= min/braceleftbigg1−β\n1+β,γ\n1+β−1\n2,ν\n1+β−1/bracerightbigg\n,\nwhere we interpret 1/(1+β)as an arbitrary large number when β=−1, and\nλ1=\n\n1\n2min\ni=1,...,k/braceleftbigg\npi1+2pi2+/parenleftbigg\n3−2β\n1+β/parenrightbigg\npi3−3/bracerightbigg\n, n= 1,\nn\n2/parenleftbigg\np−1−2\nn/parenrightbigg\n, n ≥2.\nHere we interpret −2βpi3/(1+β)as an arbitrary large number when pi3/ne}ationslash= 0andβ=−1.\nRemark 2.2. IfN=c=d= 0, namely there are no perturbation terms, and if βis close to 1so that\nmin{1/2,m/2−n/4,(1−β)/(1+β)}= (1−β)/(1+β), thenλ= (1−β)/(1+β)−ηwith arbitrary\nsmallη>0, and we expect that the gain of the decay rate (1−β)/(1+β)is optimal, in other words, the\nsecond order approximation of udecays as (B(t) +1)−n/4−(1−β)/(1+β). The higher order asymptotic\nexpansion will be discussed in a forthcoming paper.\n63 Proof of the main theorem\n3.1 Scaling variables\nWe introduce the following scaling variables:\ns= log(B(t)+1), y= (B(t)+1)−1/2x (3.1)\nand\nv(s,y) =ens/2u(t(s),es/2y), w(s,y) =b(t(s))e(n+2)s/2ut(t(s),es/2y),\nor equivalently,\nu(t,x) = (B(t)+1)−n/2v(log(B(t)+1),(B(t)+1)−1/2x),\nut(t,x) =b(t)−1(B(t)+1)−n/2−1w(log(B(t)+1),(B(t)+1)−1/2x),(3.2)\nwhere we have used the notation t(s) =B−1(es−1). Then, the problem (1.1) is transformed as\n\n\nvs−y\n2·∇yv−n\n2v=w, s>0,y∈Rn,\ne−s\nb(t(s))2/parenleftBig\nws−y\n2·∇yw−/parenleftBign\n2+1/parenrightBig\nw/parenrightBig\n+w= ∆yv+r(s,y), s>0,y∈Rn,\nv(0,y) =εv0(y) =εu0(y), w(0,y) =εw0(y) =εb(0)u1(y), y∈Rn,(3.3)\nwhere\nr(s,y) =1\nb(t(s))2db\ndt(t(s))w+es/2c(t(s))·∇yv+esd(t(s))v\n+e(n+2)s/2N/parenleftBig\ne−ns/2v,e−(n+1)s/2∇yv,b(t(s))−1e−(n+2)s/2w/parenrightBig\n. (3.4)\n3.2 Preliminary lemmas\nFirst, we collect frequently used relations and estimates.\nLemma 3.1. We have\nd\ndsb(t(s)) =db\ndt(t(s))b(t(s))es,d\nds1\nb(t(s))2=−2\nb(t(s))2db\ndt(t(s))es. (3.5)\nProof.First, we note that the function σ=B(t) is strictly increasing, and hence, the inverse t=\nB−1(σ) exists and\nd\ndσB−1(σ) =/parenleftbiggdB\ndt(t)/parenrightbigg−1\n=b(t).\nCombining this with s= log(B(t)+1), we obtain\nd\ndsb(t(s)) =d\ndsb/parenleftbig\nB−1(es−1)/parenrightbig\n=db\ndt(t(s))d\ndsB−1(es−1)\n=db\ndt(t(s))/parenleftbiggdB\ndt(t(s))/parenrightbigg−1d\nds(es−1)\n=db\ndt(t(s))b(t(s))es.\n7This shows the first assertion of (3.5). Moreover, we have\nd\nds1\nb(t(s))2=−2\nb(t(s))3d\ndsb(t(s)) =−2\nb(t(s))2db\ndt(t(s))es,\nwhich shows the second assertion of (3.5).\nNext, the assumption (2.1) implies the following:\nLemma 3.2. Under the assumption (2.1), we have the following estimates.\n(i) Whenβ∈(−1,1), we have\nb(t(s))∼e−βs/(1+β),e−s\nb(t(s))2∼e−(1−β)s/(1+β),1\nb(t(s))2/vextendsingle/vextendsingle/vextendsingle/vextendsingledb\ndt(t(s))/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Ce−(1−β)s/(1+β).\n(ii) Whenβ=−1, we have\nb(t(s))∼exp(es),e−s\nb(t(s))2∼exp(−2es−s),1\nb(t(s))2/vextendsingle/vextendsingle/vextendsingle/vextendsingledb\ndt(t(s))/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤Cexp(−2es).\nProof.(i) Whenβ∈(−1,1), from (2.7) and (3.1) we compute as\nes=B(t(s))+1 =/integraldisplayt(s)\n0dτ\nb(τ)+1∼/integraldisplayt(s)\n0(1+τ)βdτ+1∼(1+t(s))1+β.\nTherefore, one has 1+ t(s)∼es/(1+β), and hence,\nb(t(s))∼(1+t(s))−β∼e−βs/(1+β).\nBy the assumption (2.1), the other estimates can be obtained in a sim ilar way.\n(ii) Whenβ=−1, we have\nes=B(t(s))+1∼/integraldisplayt(s)\n0(1+τ)−1dτ+1 = log(1+ t(s))+1,\nand hence, b(t(s))∼1+t(s)∼exp(es) holds. We can prove the other estimates in the same way, and\nthe proof is omitted.\nWe sometimes employ the Gagliardo-Nirenberg inequality:\nLemma 3.3 (Gagliardo-Nirenberg inequality) .Let1<p<∞(n= 1,2)and1<p≤n/(n−2) (n≥\n3). Then for any f∈H1,0(Rn), we have\n/ba∇dblf/ba∇dblL2p≤C/ba∇dbl∇f/ba∇dblσ\nL2/ba∇dblf/ba∇dbl1−σ\nL2,\nwhereσ=n(p−1)/(2p).\nFor the proof, see for example [6, 9].\n83.3 Local existence of solutions\nWe prove the local existence of solutions for the equation (1.1) and the system (3.3), respectively. To\nthis end, putting U(t,x) =/an}b∇acketle{tx/an}b∇acket∇i}htmu,U0(x) =/an}b∇acketle{tx/an}b∇acket∇i}htmu0andU1=/an}b∇acketle{tx/an}b∇acket∇i}htmu1, we change the problem (1.1) to\n/braceleftbigg\nUtt+b(t)Ut= ∆xU+˜c(t,x)·∇xU+˜d(t,x)U+˜N(U,∇xU,Ut), t>0,x∈Rn,\nU(0,x) =εU0(x), Ut(0,x) =εU1(x), x ∈Rn,(3.6)\nwhere ˜c=c−2m/an}b∇acketle{tx/an}b∇acket∇i}ht��2x,˜d=d−c·(m/an}b∇acketle{tx/an}b∇acket∇i}ht−2x)−m/an}b∇acketle{tx/an}b∇acket∇i}ht−4(n/an}b∇acketle{tx/an}b∇acket∇i}ht2−(m+2)|x|2) and\n˜N(U,∇xU,Ut) =/an}b∇acketle{tx/an}b∇acket∇i}htmN/parenleftbig\n/an}b∇acketle{tx/an}b∇acket∇i}ht−mU,/an}b∇acketle{tx/an}b∇acket∇i}ht−m∇xU−m/an}b∇acketle{tx/an}b∇acket∇i}ht−m−2xU,/an}b∇acketle{tx/an}b∇acket∇i}ht−mUt/parenrightbig\n.\nWe further put U=t(U,Ut) andU0=t(U0,U1). Then, the equation (3.6) is written as\n/braceleftbiggUt=AU+N(U),\nU(0) =εU0,(3.7)\nwhere\nA=/parenleftbigg\n0 1\n∆ 0/parenrightbigg\n,N(U) =/parenleftbigg0\n−bUt+˜c·∇xU+˜dU+˜N(U,∇xU,Ut)/parenrightbigg\n.\nThe operator AonH1,0(Rn)×L2(Rn) with the domain D(A) =H2,0(Rn)×H1,0(Rn) ism-dissipative\n(see [1, Proposition 2.6.9]) with dense domain, and hence, Agenerates a contraction semigroup etAon\nH1,0(Rn)×L2(Rn) (see [1, Theorem 3.4.4]). Thus, we consider the integral form\nU(t) =εetAU0+/integraldisplayt\n0e(t−τ)AN(U(τ))dτ (3.8)\nof the equation (3.7) in C([0,T);H1,0(Rn)×L2(Rn)).\nFirst, we define the mild and strong solutions and the lifespan of solut ions.\nDefinition 3.4. We say that uis a mild solution of the Cauchy problem (1.1)on the interval [0,T)\nifuhas the regularity\nu∈C([0,T);H1,m(Rn))∩C1([0,T);H0,m(Rn)). (3.9)\nand satisfies the integral equation (3.8)inC([0,T);H1,0(Rn)×L2(Rn)). We also call ua strong\nsolution of the Cauchy problem (1.1)on the interval [0,T)ifuhas the regularity\nu∈C([0,T);H2,m(Rn))∩C1([0,T);H1,m(Rn))∩C2([0,T);H0,m(Rn)) (3.10)\nand satisfies the equation (1.1)inC([0,T);H0,m(Rn)). Moreover, we say that (v,w)defined by (3.2)\nis a mild (resp. strong) solution of the Cauchy problem (3.3)on the interval [0,S)ifuis a mild (resp.\nstrong) solution of (1.1)on the interval [0,t(S)). We note that if (v,w)is a mild solution of (3.3)on\n[0,S), then(v,w)has the regularity\n(v,w)∈C([0,S);H1,m(Rn)×H0,m(Rn)),\nand if(v,w)is a strong solution of (3.3)on[0,S), then(v,w)has the regularity\n(v,w)∈C([0,S);H2,m(Rn)×H1,m(Rn))∩C1([0,S);H1,m(Rn)×H0,m(Rn)) (3.11)\nand satisfies the system (3.3)inC([0,S);H1,m(Rn)×H0,m(Rn)).\nWe also define the lifespan of the mild solutions uand(v,w)by\nT(ε) = sup{T∈(0,∞);there exists a unique mild solution uto(1.1)}\nand\nS(ε) = sup{S∈(0,∞);there exists a unique mild solution (v,w)to(3.3)},\nrespectively.\n9Proposition 3.5. Under the assumptions (i)–(iv) in the previous section, the re existsT >0depending\nonly onε/ba∇dbl(u0,u1)/ba∇dblH1,m×H0,m(the size of the initial data) such that the Cauchy problem (1.1)admits\na unique mild solution u. Also, if (u0,u1)∈H2,m(Rn)×H1,m(Rn)in addition to Assumption (i),\nthen the corresponding mild solution ubecomes a strong solution of (1.1). Moreover, if the lifespan\nT(ε)is finite, then usatisfies limt→T(ε)/ba∇dbl(u,ut)(t)/ba∇dblH1,m×H0,m=∞. Furthermore, for arbitrary fixed\ntimeT0>0, we can extend the solution to the interval [0,T0)by takingεsufficiently small.\nFrom this proposition, we easily have the following.\nProposition 3.6. Under the assumptions (i)–(iv) in the previous section, the re existsS >0depending\nonly onε/ba∇dbl(v0,w0)/ba∇dblH1,m×H0,m(the size of the initial data) such that the Cauchy problem (3.3)admits\na unique mild solution (v,w). Also, if (u0,u1)∈H2,m(Rn)×H1,m(Rn)in addition to Assumption (i),\nthen the corresponding mild solution (v,w)becomes a strong solution of (3.3). Moreover, if the lifespan\nS(ε)is finite, then (v,w)satisfies lims→S(ε)/ba∇dbl(v,w)(s)/ba∇dblH1,m×H0,m=∞. Furthermore, for arbitrary\nfixed timeS0>0, we can extend the solution to the interval [0,S0]by takingεsufficiently small.\nProof of Proposition 3.5. By using the assumption (iv), and the Sobolev inequality for n= 1, or the\nGagliardo-Nirenberginequality for n≥2 (see Lemma 3.3), we can see that N(U) is a locally Lipschitz\nmapping on H1,0(Rn)×L2(Rn). Therefore, by [1, Proposition 4.3.3], there exists a unique solution\nU ∈C([0,T);H1,0(Rn)×L2(Rn)) to the integral equation (3.8). This shows the existence of a uniq ue\nmild solution uto the Cauchy problem (1.1).\nIf (u0,u1)∈H2,m(Rn)×H1,m(Rn), then we have U0∈D(A), and hence, [1, Proposition 4.3.9]\nimplies that U ∈C([0,T);D(A))∩C1([0,T);H1,0(Rn)×L2(Rn)) andUbecomes the strong solution\nof the equation (3.7), namely, Usatisfies the equation (3.7) in C([0,T);H1,0(Rn)×L2(Rn)). Then, by\nthe definition of U, we conclude that uhas the regularity in (3.10) and satisfies the equation (1.1) in\nC([0,T);H0,m(Rn)). Moreover, employing [1, Theorem 4.3.4], we see that if the lifespan T(ε) is finite,\nthenusatisfies lim t→T(ε)/ba∇dbl(u,ut)(t)/ba∇dblH1,m×H0,m=∞.\nNext, we prove that for any fixed T0>0, the solution ucan be extended over the interval [0 ,T0]\nby takingεsufficiently small. To verify this, we reconsider the Cauchy problem (3 .6) and its inhomo-\ngeneous linear version\n/braceleftbigg\nUtt+b(t)Ut= ∆xU+˜c(t,x)·∇xU+˜d(t,x)U+˜N(t,x), t>0,x∈Rn,\nU(0,x) =εU0(x), Ut(0,x) =εU1(x), x ���Rn.(3.12)\nFor˜N∈L1(0,T0;L2(Rn)), the existence of a unique solution in the distribution sense is prov ed by\n[11, Theorem 23.2.2]. We also recall the standard energy estimate (s ee [11, Lemma 23.2.1])\nsup\n0<t<T 0/ba∇dbl(U,Ut)(t)/ba∇dblH1,0×L2≤C(T0)/parenleftBigg\nε/ba∇dbl(U0,U1)/ba∇dblH1,0×L2+/integraldisplayT0\n0/ba∇dbl˜N(t)/ba∇dblL2dt/parenrightBigg\n.(3.13)\nWe again construct the solution Uto (3.6) in\nK:=/braceleftbigg\nU∈C([0,T0];H1,0(Rn))∩C1([0,T0];L2(Rn)); sup\n0<t<T 0/ba∇dbl(U,Ut)(t)/ba∇dblH1,0×L2≤2C(T0)I0ε/bracerightbigg\n,\nwhereI0:=/ba∇dbl(U0,U1)/ba∇dblH1,0×L2. For each V∈K, we define the mapping by U=M(V), where\nUis the solution to (3.12) with ˜N=˜N(V,∇xV,Vt). Then, by using the Sobolev inequality or the\nGagliardo-Nirenberg inequality again with the estimate (3.13), we can see that\nsup\n0<t<T 0/ba∇dbl(U,Ut)(t)/ba∇dblH1,0×L2≤C(T0)I0ε+C(T0)(2C(T0)I0ε)pT0. (3.14)\nThus, noting p>1 and taking ε>0 sufficiently small, we deduce that MmapsKto itself. Further-\nmore, in the same manner, we easily obtain\nsup\n0<t<T 0/ba∇dbl(U1,U1\nt)−(U2,U2\nt)/ba∇dblH1,0×L1≤C(T0)(4C(T0)I0ε)p−1T0sup\n0<t<T 0/ba∇dbl(V1,V1\nt)−(V2,V2\nt)/ba∇dblH1,0×L2,\n10whereUj=M(Vj) (j= 1,2). Thus, noting p>1 again and taking εsufficiently small, we see that M\nis acontractionmappingon K. Therefore, by the contractionmappingprinciple, wefind aunique fi xed\npoint˜Uof the mapping Min the setK, and˜Usatisfies the equation (3.6) in the distribution sense.\nAlso, the uniqueness of the solution in the distribution sense to (3.6) in the class C([0,T0];H1,0(Rn))∩\nC1([0,T0];L2(Rn)) follows from (3.13). Since the mild solution Uconstructed before also satisfies the\nequation (3.6) in the distribution sense, we have U(t) =˜U(t) fort∈[0,min{T(ε),T0}). However,\nnoting that the estimate (3.14) implies sup0<t<T 0/ba∇dbl(˜U,˜Ut)(t)/ba∇dblH1,0×L2is finite, we have T0<T(ε) and\nthis completes the proof.\n3.4 A priori estimate implies the global existence\nIn what follows, to justify the energy method, we tacitly assume th at (u0,u1)∈H2,m(Rn)×H1,m(Rn),\nand the solution ( v,w) is in the class (3.11). Therefore, the following calculations make sen se. Once\nwe obtain the desired asymptotic estimate (2.9) for such a data, we can easily have the same estimate\nfor general ( u0,u1)∈H1,m(Rn)×H0,m(Rn) by applying the usual approximation argument.\nLet(v,w) bethelocal-in-timesolutionto(3.3)ontheinterval[0 ,S). Bythelocalexistencetheorem,\nit suffices to show an a priori estimate of solutions. The first goal of this section is the following a\npriori estimate:\nProposition 3.7. Under the assumptions (i)–(iv) in the previous section, the re exist constants s0>0,\nε1>0andC∗>0such that the following holds: if ε∈(0,ε1]and(v,w)is a mild solution of (1.1)on\nsome interval [0,S]withS >s0, then(v,w)satisfies\n/ba∇dblv(s)/ba∇dbl2\nH1,m+e−s\nb(t(s))2/ba∇dblw(s)/ba∇dbl2\nH0,m≤C∗ε2/ba∇dbl(v0,w0)/ba∇dbl2\nH1,m×H0,m. (3.15)\nBefore proving the above proposition, we show that Propositions 3 .6 and 3.7 imply the global\nexistence of solutions for small ε.\nProof of global existence part of Theorem 2.1. First,wenotethatProposition3.6guaranteesthatthere\nexitsε2>0 such that the mild solution ( v,w) uniquely exists on the interval [0 ,s0] forε∈(0,ε2],\nwheres0is the constant described in Proposition 3.7. In particular, we have S(ε)>s0forε∈(0,ε2].\nLetε0:= min{ε1,ε2}, whereε1is the constant described in Proposition 3.7. Then, we have S(ε) =∞\nforε∈(0,ε0]. Indeed, supposethat S(ε∗)<∞forsomeε∗∈(0,ε0] and let (v,w) be the corresponding\nmild solution of (3.3). Applying Proposition 3.7, we have the a priori est imate (3.15) with ε=ε∗. On\nthe other hand, Proposition 3.6 also implies\nlim\ns→S(ε∗)/ba∇dbl(v,w)(s)/ba∇dblH1,m×H0,m=∞.\nHowever, it contradicts the a priori estimate (3.15). Thus, we hav eS(ε) =∞forε∈(0,ε0].\n3.5 Spectral decomposition\nIn the following, we prove the a priori estimate (3.15) in Proposition 3 .7. At first, we decompose v\nandwinto the leading terms and the remainder terms, respectively.\nLetα(s) be\nα(s) =/integraldisplay\nRnv(s,y)dy. (3.16)\nSincev(s)∈H1,m(Rn) for eachs∈[0,S) andm>n/2,α(s) is well-defined. We also put\nϕ0(y) = (4π)−n/2exp/parenleftbigg\n−|y|2\n4/parenrightbigg\n.\n11Then, it is easily verified that\n/integraldisplay\nRnϕ0(y)dy= 1 (3.17)\nand\n∆ϕ0=−y\n2·∇yϕ0−n\n2ϕ0. (3.18)\nWe also put\nψ0(y) = ∆ϕ0(y).\nWe decompose v,was\nv(s,y) =α(s)ϕ0(y)+f(s,y),\nw(s,y) =dα\nds(s)ϕ0(y)+α(s)ψ0(y)+g(s,y).(3.19)\nWe shall prove that f,gcan be regarded as remainder terms.\nFirst, we note the following lemma.\nLemma 3.8. We have\ndα\nds(s) =/integraldisplay\nRnw(s,y)dy, (3.20)\ne−s\nb(t(s))2d2α\nds2(s) =e−s\nb(t(s))2dα\nds(s)−dα\nds(s)+/integraldisplay\nRnr(s,y)dy, (3.21)\nwhereris defined by (3.4).\nProof.Notingv∈C1([0,S);H1,m(Rn)),w∈C([0,S);H0,m(Rn)) andm > n/ 2, we immediately\nobtain (3.20) from\ndα\nds(s) =/integraldisplay\nRnvs(s,y)dy=/integraldisplay\nRn/parenleftBigy\n2·∇yv+n\n2v+w/parenrightBig\ndy=/integraldisplay\nRnw(s,y)dy.\nNext, by the regularity (3.11), we see thatdα\nds(s)∈C1([0,S);R). Differentiatingdα\nds(s) again and using\nthe second equation of (3.3), we have\ne−s\nb(t(s))2d2α\nds2(s) =e−s\nb(t(s))2/integraldisplay\nRnws(s,y)dy\n=e−s\nb(t(s))2/integraldisplay\nRn/parenleftBigy\n2·∇yw+/parenleftBign\n2+1/parenrightBig\nw/parenrightBig\ndy−/integraldisplay\nRnwdy+/integraldisplay\nRn∆yvdy+/integraldisplay\nRnrdy\n=e−s\nb(t(s))2/integraldisplay\nRnwdy−/integraldisplay\nRnwdy+/integraldisplay\nRnrdy.\nThus, we finish the proof.\nNext, we consider the remainder term ( f,g). Sincefandgare defined by (3.19), and we assumed\nthat (v,w) has the regularity in (3.11), so is ( f,g):\n(f,g)∈C([0,S);H2,m(Rn)×H1,m(Rn))∩C1([0,S);H1,m(Rn)×H0,m(Rn)).(3.22)\n12Therefore, from the system (3.3) and the equation (3.18), we see thatfandgsatisfy the following\nsystem:\n\n\nfs−y\n2·∇yf−n\n2f=g, s> 0,y∈Rn,\ne−s\nb(t(s))2/parenleftBig\ngs−y\n2·∇yg−/parenleftBign\n2+1/parenrightBig\ng/parenrightBig\n+g= ∆yf+h, s>0,y∈Rn,\nf(0,y) =v(0,y)−α(0)ϕ0(y), y ∈Rn,\ng(0,y) =w(0,y)−˙α(0)ϕ0(y)−α(0)ψ0(y), y ∈Rn,(3.23)\nwherehis given by\nh(s,y) =e−s\nb(t(s))2/parenleftbigg\n−2dα\nds(s)ψ0(y)+α(s)/parenleftBigy\n2·∇yψ0(y)+/parenleftBign\n2+1/parenrightBig\nψ0(y)/parenrightBig/parenrightbigg\n+r(s,y)−/parenleftbigg/integraldisplay\nRnr(s,y)dy/parenrightbigg\nϕ0(y). (3.24)\nMoreover, from (3.16), (3.17) and (3.20), it follows that\n/integraldisplay\nRnf(s,y)dy=/integraldisplay\nRng(s,y)dy= 0. (3.25)\nWe also notice that the condition (3.25) implies\n/integraldisplay\nRnh(s,y)dy= 0. (3.26)\nWe note that it suffices to show a priori estimates of f,g,αanddα\ndsfor the proof of global existence\nof solutions to the system (3.3). Therefore, hereafter, we cons ider the system (3.23) instead of (3.3).\n3.6 Energy estimates for n= 1\nTo obtain the decay estimates for f,g, we introduce\nF(s,y) =/integraldisplayy\n−∞f(s,z)dz, G(s,y) =/integraldisplayy\n−∞g(s,z)dz. (3.27)\nFrom the following lemma and the condition (3.25), we see that F,G∈C([0,S);L2(R)).\nLemma 3.9 (Hardy-type inequality) .Letf=f(y)belong toH0,1(R)and satisfy/integraltext\nRf(y)dy= 0, and\nletF(y) =/integraltexty\n−∞f(z)dz. Then it holds that\n/integraldisplay\nRF(y)2dy≤4/integraldisplay\nRy2f(y)2dy. (3.28)\nProof.First, we prove (3.28) when f∈C∞\n0(R). In this case/integraltext\nRf(y)dy= 0 leads to F∈C∞\n0(R).\nTherefore, we apply the integration by parts and have\n/integraldisplay\nRF(y)2dy=−2/integraldisplay\nRyF(y)f(y)dy≤2/integraldisplay\nRy2f(y)2dy+1\n2/integraldisplay\nRF(y)2dy.\nThus, we obtain (3.28). For general f∈H0,1(R) satisfying/integraltext\nRf(y)dy= 0, we can easily prove (3.28)\nby appropriately approximations.\n13Moreover, by the regularity assumption (3.22) on ( f,g), we see that\n(F,G)∈C([0,S);H3,0(R)×H2,0(R))∩C1([0,S);H2,0(R)×H1,0(R)). (3.29)\nSincefandgsatisfy the equation (3.23), we can show that FandGsatisfy the following system:\n\n\nFs−y\n2Fy=G, s> 0,y∈R,\ne−s\nb(t(s))2/parenleftBig\nGs−y\n2Gy−G/parenrightBig\n+G=Fyy+H, s> 0,y∈R,\nF(0,y) =/integraldisplayy\n−∞f(0,z)dz, G(0,y) =/integraldisplayy\n−∞g(0,z)dz, y∈R,(3.30)\nwhere\nH(s,y) =/integraldisplayy\n−∞h(s,z)dz. (3.31)\nWe define the following energy.\nE0(s) =/integraldisplay\nR/parenleftbigg1\n2/parenleftbigg\nF2\ny+e−s\nb(t(s))2G2/parenrightbigg\n+1\n2F2+e−s\nb(t(s))2FG/parenrightbigg\ndy,\nE1(s) =/integraldisplay\nR/parenleftbigg1\n2/parenleftbigg\nf2\ny+e−s\nb(t(s))2g2/parenrightbigg\n+f2+2e−s\nb(t(s))2fg/parenrightbigg\ndy,\nE2(s) =/integraldisplay\nRy2/bracketleftbigg1\n2/parenleftbigg\nf2\ny+e−s\nb(t(s))2g2/parenrightbigg\n+1\n2f2+e−s\nb(t(s))2fg/bracketrightbigg\ndy.\nBy using Lemma 3.2, the following equivalents are valid for s≥s1with sufficiently large s1>0.\nE0(s)∼/integraldisplay\nR/parenleftbigg\nF2\ny+e−s\nb(t(s))2G2+F2/parenrightbigg\ndy,\nE1(s)∼/integraldisplay\nR/parenleftbigg\nf2\ny+e−s\nb(t(s))2g2+f2/parenrightbigg\ndy, (3.32)\nE2(s)∼/integraldisplay\nRy2/bracketleftbigg\nf2\ny+e−s\nb(t(s))2g2+f2/bracketrightbigg\ndy.\nNext, we prove the following energy identity.\nLemma 3.10. We have\nd\ndsE0(s)+1\n2E0(s)+L0(s) =R0(s), (3.33)\nwhere\nL0(s) =/integraldisplay\nR/parenleftbigg1\n2F2\ny+G2/parenrightbigg\ndy,\nR0(s) =3\n2e−s\nb(t(s))2/integraldisplay\nRG2dy−1\nb(t(s))2db\ndt(t(s))/integraldisplay\nR/parenleftbig\nG2+2FG/parenrightbig\ndy+/integraldisplay\nR(F+G)Hdy.\nMoreover, we have\nd\ndsE1(s)+1\n2E1(s)+L1(s) =R1(s), (3.34)\n14where\nL1(s) =/integraldisplay\nR/parenleftbig\nf2\ny+g2/parenrightbig\ndy−/integraldisplay\nRf2dy,\nR1(s) = 3e−s\nb(t(s))2/integraldisplay\nRg2dy+2e−s\nb(t(s))2/integraldisplay\nRfgdy−1\nb(t(s))2db\ndt(t(s))/integraldisplay\nR(g2+4fg)dy+/integraldisplay\nR(2f+g)hdy.\nFurthermore, we have\nd\ndsE2(s)+1\n2E2(s)+L2(s) =R2(s), (3.35)\nwhere\nL2(s) =/integraldisplay\nRy2/parenleftbigg1\n2f2\ny+g2/parenrightbigg\ndy+2/integraldisplay\nRyfy(f+g)dy,\nR2(s) =3\n2e−s\nb(t(s))2/integraldisplay\nRy2g2dy−1\nb(t(s))2db\ndt(t(s))/integraldisplay\nRy2(2f+g)gdy+/integraldisplay\nRy2(f+g)hdy.\nProof.The proofs of (3.34) and (3.35) are the almost same as that of (3.33 ), and we only prove (3.33).\nWe calculate the derivatives of each term of E0(s). First, we have\nd\nds/bracketleftbigg1\n2/integraldisplay\nRF2dy/bracketrightbigg\n=/integraldisplay\nRFFsdy\n=/integraldisplay\nRF/parenleftBigy\n2Fy+G/parenrightBig\ndy\n=/integraldisplay\nR/parenleftbigg/parenleftBigy\n4F2/parenrightBig\ny−1\n4F2+FG/parenrightbigg\ndy\n=−1\n4/integraldisplay\nRF2dy+/integraldisplay\nRFGdy.\nHere we have used thaty\n2FFy∈L1(R), which enables us to justify the integration by parts. By\nLemma 3.1, we also have\nd\nds/bracketleftbigge−s\nb(t(s))2/integraldisplay\nRFGdy/bracketrightbigg\n=−2\nb(t(s))2db\ndt(t(s))/integraldisplay\nRFGdy−e−s\nb(t(s))2/integraldisplay\nRFGdy+e−s\nb(t(s))2/integraldisplay\nR(FsG+FGs)dy\n=−2\nb(t(s))2db\ndt(t(s))/integraldisplay\nRFGdy−e−s\nb(t(s))2/integraldisplay\nRFGdy+e−s\nb(t(s))2/integraldisplay\nR/parenleftBigy\n2Fy+G/parenrightBig\nGdy\n+e−s\nb(t(s))2/integraldisplay\nRF/parenleftBigy\n2Gy+G/parenrightBig\ndy−/integraldisplay\nRFGdy+/integraldisplay\nRFFyydy+/integraldisplay\nRFHdy\n=−1\n2e−s\nb(t(s))2/integraldisplay\nRFGdy−2\nb(t(s))2db\ndt(t(s))/integraldisplay\nRFGdy\n+e−s\nb(t(s))2/integraldisplay\nRG2dy−/integraldisplay\nRFGdy−/integraldisplay\nRF2\nydy+/integraldisplay\nRFHdy.\nAdding up the above identities, we conclude that\nd\nds/bracketleftbigg/integraldisplay\nR/parenleftbigg1\n2F2+e−s\nb(t(s))2FG/parenrightbigg\ndy/bracketrightbigg\n=−1\n4/integraldisplay\nRF2dy−1\n2e−s\nb(t(s))2/integraldisplay\nRFGdy−2\nb(t(s))2db\ndt(t(s))/integraldisplay\nRFGdy\n+e−s\nb(t(s))2/integraldisplay\nRG2dy−/integraldisplay\nRF2\nydy+/integraldisplay\nRFHdy. (3.36)\n15We also have\nd\nds/bracketleftbigg1\n2/integraldisplay\nRF2\nydy/bracketrightbigg\n=/integraldisplay\nRFyFysdy\n=/integraldisplay\nRFy/parenleftbiggy\n2Fyy+1\n2Fy+Gy/parenrightbigg\ndy\n=1\n4/integraldisplay\nRF2\nydy+/integraldisplay\nRFyGydy\nand\nd\nds/bracketleftbigg1\n2e−s\nb(t(s))2/integraldisplay\nRG2dy/bracketrightbigg\n=−1\nb(t(s))2db\ndt(t(s))/integraldisplay\nRG2dy−1\n2e−s\nb(t(s))2/integraldisplay\nRG2dy+e−s\nb(t(s))2/integraldisplay\nRGGsdy\n=−1\nb(t(s))2db\ndt(t(s))/integraldisplay\nRG2dy−1\n2e−s\nb(t(s))2/integraldisplay\nRG2dy\n+e−s\nb(t(s))2/integraldisplay\nRG/parenleftBigy\n2Gy+G/parenrightBig\ndy−/integraldisplay\nRG2dy+/integraldisplay\nRGFyydy+/integraldisplay\nRGHdy\n=−1\nb(t(s))2db\ndt(t(s))/integraldisplay\nRG2dy+1\n4e−s\nb(t(s))2/integraldisplay\nRG2dy\n−/integraldisplay\nRG2dy−/integraldisplay\nRFyGydy+/integraldisplay\nRGHdy.\nAdding up the above two identities, one has\nd\nds/bracketleftbigg1\n2/integraldisplay\nR/parenleftbigg\nF2\ny+e−s\nb(t(s))2G2/parenrightbigg\ndy/bracketrightbigg\n=1\n4/integraldisplay\nRF2\nydy+1\n4e−s\nb(t(s))2/integraldisplay\nRG2dy−1\nb(t(s))2db\ndt(t(s))/integraldisplay\nRG2dy−/integraldisplay\nRG2dy+/integraldisplay\nRGHdy. (3.37)\nFrom (3.36) and (3.37), we conclude that\nd\ndsE0(s)+1\n2E0(s)+/integraldisplay\nR/parenleftbigg1\n2F2\ny+G2/parenrightbigg\ndy=R0(s).\nThis completes the proof.\n3.7 Energy estimates for n≥2\nNext, we consider higher dimensional cases n≥2. In this case, we cannot use the primitives (3.27).\nTherefore, instead of (3.27), we define\nˆF(s,ξ) =|ξ|−n/2−δˆf(s,ξ),ˆG(s,ξ) =|ξ|−n/2−δˆg(s,ξ),ˆH(s,ξ) =|ξ|−n/2−δˆh(s,ξ),\nwhere 0< δ <1, andˆf(s,ξ) denotes the Fourier transform of f(s,y) with respect to the space\nvariable. First, to ensure that ˆF,ˆGandˆHmake sense as L2-functions, instead of Lemma 3.9, we\nprove the following lemma.\nLemma3.11. Letm>n/2+1andf(y)∈H0,m(Rn)be afunction satisfying ˆf(0) = (2π)−n/2/integraltext\nRnf(y)dy=\n0. LetˆF(ξ) =|ξ|−n/2−δˆf(ξ)with some 0<δ <1. Then, there exists a constant C(n,m,δ)>0such\nthat\n/ba∇dblF/ba∇dblL2≤C(n,m,δ)/ba∇dblf/ba∇dblH0,m (3.38)\nholds.\n16Proof.By the Plancherel theorem, it suffices to show that /ba∇dblˆF/ba∇dblL2≤C/ba∇dblf/ba∇dblH0,m. Using the definition of\nˆFand the condition ˆf(0) = 0 , we compute\n/integraldisplay\nRn|ˆF(ξ)|2dξ=/integraldisplay\nRn|ξ|−n−2δ|ˆf(ξ)|2dξ\n=/integraldisplay\n|ξ|≤1|ξ|−n−2δ|ˆf(ξ)|2dξ+/integraldisplay\n|ξ|>1|ξ|−n−2δ|ˆf(ξ)|2dξ\n=/integraldisplay\n|ξ|≤1|ξ|−n−2δ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay1\n0d\ndθˆf(θξ)dθ/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndξ+/integraldisplay\n|ξ|>1|ξ|−n−2δ|ˆf(ξ)|2dξ\n≤ /ba∇dbl∇ξˆf/ba∇dbl2\nL∞/integraldisplay\n|ξ|≤1|ξ|2−n−2δdξ+/ba∇dblˆf/ba∇dbl2\nL2\n≤C(n,δ)/parenleftBig\n/ba∇dbl∇ξˆf/ba∇dbl2\nL∞+/ba∇dblˆf/ba∇dbl2\nL2/parenrightBig\n.\nSincem>n/2+1, we have\n/ba∇dbl∇ξˆf/ba∇dblL∞=/ba∇dbl/hatwideryf/ba∇dblL∞≤C/ba∇dblyf/ba∇dblL1≤C(n,m)/ba∇dbl(1+|y|)mf/ba∇dblL2≤C(n,m)/ba∇dblf/ba∇dblH0,m.\nConsequently, we obtain\n/ba∇dblˆF/ba∇dblL2≤C(n,δ)/parenleftBig\n/ba∇dbl∇ξˆf/ba∇dblL∞+/ba∇dblˆf/ba∇dblL2/parenrightBig\n≤C(n,m,δ)/ba∇dblf/ba∇dblH0,m,\nwhich completes the proof.\nWe also notice that, for any small η>0, the inequality\n/integraldisplay\nRn|ˆf|2dξ=/integraldisplay\n|ξ|≥√η−1|ˆf|2dξ+/integraldisplay\n|ξ|<√η−1|ˆf|2dξ\n≤η/integraldisplay\n|ξ|≥√η−1|ξ|2|ˆf|2dξ+η(2−n−2δ)/2/integraldisplay\n|ξ|<√η−1|ξ|2−n−2δ|ˆf|2dξ\n≤η/integraldisplay\nRn|ξ|2|ˆf|2dξ+η(2−n−2δ)/2/integraldisplay\nRn|ξ|2|ˆF|2dξ (3.39)\nholds. This is provedby noting that 2 −n−2δ<0 (here we assumed that n≥2). The above inequality\nenables us to control /ba∇dblˆf/ba∇dblL2by/ba∇dbl|ξ|ˆf/ba∇dblL2and/ba∇dbl|ξ|ˆF/ba∇dblL2. Moreover, the coefficient in front of /ba∇dbl|ξ|ˆf/ba∇dblL2\ncan be taken arbitrarily small.\nBy applying the Fourier transform to (3.23), we obtain\n\n\nˆfs+1\n2∇ξ·/parenleftBig\nξˆf/parenrightBig\n−n\n2ˆf= ˆg, s> 0,ξ∈Rn,\ne−s\nb(t(s))2/parenleftbigg\nˆgs+1\n2∇ξ·(ξˆg)−/parenleftBign\n2+1/parenrightBig\nˆg/parenrightbigg\n+ ˆg=−|ξ|2ˆf+ˆh, s>0,ξ∈Rn.(3.40)\nBy noting that\n1\n2∇ξ·/parenleftBig\nξˆf/parenrightBig\n=ξ\n2·∇ξˆf+n\n2ˆf,\nwe rewrite (3.40) as\n\n\nˆfs+ξ\n2·∇ξˆf= ˆg, s> 0,ξ∈Rn,\ne−s\nb(t(s))2/parenleftbigg\nˆgs+ξ\n2·∇ξˆg−ˆg/parenrightbigg\n+ ˆg=−|ξ|2ˆf+ˆh, s>0,ξ∈Rn.\n17Making use of this, we calculate\nˆFs=|ξ|−n/2−δˆfs\n=|ξ|−n/2−δ/parenleftbigg\n−ξ\n2·∇ξˆf+ ˆg/parenrightbigg\n=|ξ|−n/2−δ/parenleftbigg\n−ξ\n2·∇ξ/parenleftBig\n|ξ|n/2+δˆF/parenrightBig\n+|ξ|n/2+δˆG/parenrightbigg\n=−ξ\n2·∇ξˆF−1\n2/parenleftBign\n2+δ/parenrightBig\nˆF+ˆG\nand\ne−s\nb(t(s))2ˆGs=e−s\nb(t(s))2|ξ|−n/2−δˆgs\n=|ξ|−n/2−δ/bracketleftbigge−s\nb(t(s))2/parenleftbigg\n−ξ\n2·∇ξˆg+ ˆg/parenrightbigg\n−ˆg−|ξ|2ˆf+ˆh/bracketrightbigg\n=|ξ|−n/2−δ/bracketleftbigge−s\nb(t(s))2/parenleftbigg\n−ξ\n2·∇ξ/parenleftBig\n|ξ|n/2+δˆG/parenrightBig\n+|ξ|n/2+δˆG/parenrightbigg\n−|ξ|n/2+δˆG−|ξ|2+n/2+δˆF+|ξ|n/2+δˆH/bracketrightBig\n=e−s\nb(t(s))2/parenleftbigg\n−ξ\n2·∇ξˆG−1\n2/parenleftBign\n2+δ−2/parenrightBig\nˆG/parenrightbigg\n−ˆG−|ξ|2ˆF+ˆH.\nHence,ˆFandˆGsatisfy the following system.\n\n\nˆFs+ξ\n2·∇ξˆF+1\n2/parenleftBign\n2+δ/parenrightBig\nˆF=ˆG, s> 0,ξ∈Rn,\ne−s\nb(t(s))2/parenleftbigg\nˆGs+ξ\n2·∇ξˆG+1\n2/parenleftBign\n2+δ−2/parenrightBig\nˆG/parenrightbigg\n+ˆG=−|ξ|2ˆF+ˆH, s> 0,ξ∈Rn.\nWe consider the following energy.\nE0(s) = Re/integraldisplay\nRn/parenleftbigg1\n2/parenleftbigg\n|ξ|2|ˆF|2+e−s\nb(t(s))2|ˆG|2/parenrightbigg\n+1\n2|ˆF|2+e−s\nb(t(s))2ˆF¯ˆG/parenrightbigg\ndξ,\nE1(s) =/integraldisplay\nRn/parenleftbigg1\n2/parenleftbigg\n|∇yf|2+e−s\nb(t(s))2g2/parenrightbigg\n+/parenleftBign\n4+1/parenrightBig/parenleftbigg1\n2f2+e−s\nb(t(s))2fg/parenrightbigg/parenrightbigg\ndy,\nE2(s) =/integraldisplay\nRn|y|2m/bracketleftbigg1\n2/parenleftbigg\n|∇yf|2+e−s\nb(t(s))2g2/parenrightbigg\n+1\n2f2+e−s\nb(t(s))2fg/bracketrightbigg\ndy.\nBy using Lemma 3.2 again, the following equivalents are valid for s≥s1with sufficiently large s1.\nE0(s)∼/integraldisplay\nRn/parenleftbigg\n|ξ|2|ˆF|2+e−s\nb(t(s))2|ˆG|2+|ˆF|2/parenrightbigg\ndξ,\nE1(s)∼/integraldisplay\nRn/parenleftbigg\n|∇yf|2+e−s\nb(t(s))2g2+f2/parenrightbigg\ndy, (3.41)\nE2(s)∼/integraldisplay\nRn|y|2m/bracketleftbigg\n|∇yf|2+e−s\nb(t(s))2g2+f2/bracketrightbigg\ndy.\nThen, in a similar way to the case n= 1, we obtain the following energy identities.\nLemma 3.12. We have\nd\ndsE0(s)+δE0(s)+L0(s) =R0(s), (3.42)\n18where\nL0(s) =1\n2/integraldisplay\nRn|ξ|2|ˆF|2dξ+/integraldisplay\nRn|ˆG|2dξ,\nR0(s) =3\n2e−s\nb(t(s))2/integraldisplay\nRn|ˆG|2dξ−1\nb(t(s))2db\ndt(t(s))Re/integraldisplay\nRn/parenleftBig\n2ˆF+ˆG/parenrightBig¯ˆGdξ+Re/integraldisplay\nRn/parenleftBig\nˆF+ˆG/parenrightBig¯ˆHdξ.\nMoreover, we have\nd\ndsE1(s)+δE1(s)+L1(s) =R1(s), (3.43)\nwhere\nL1(s) =1\n2(1−δ)/integraldisplay\nRn|∇yf|2dy+/integraldisplay\nRng2dy−/parenleftbiggn\n4+δ\n2/parenrightbigg/parenleftBign\n4+1/parenrightBig/integraldisplay\nRnf2dy,\nR1(s) =/parenleftBign\n2+δ/parenrightBig/parenleftBign\n4+1/parenrightBige−s\nb(t(s))2/integraldisplay\nRnfgdy+1\n2(n+3+δ)e−s\nb(t(s))2/integraldisplay\nRng2dy\n−1\nb(t(s))2db\ndt(t(s))/integraldisplay\nRn/parenleftBig\n2/parenleftBign\n4+1/parenrightBig\nf+g/parenrightBig\ngdy+/integraldisplay\nRn/parenleftBig/parenleftBign\n4+1/parenrightBig\nf+g/parenrightBig\nhdy.\nFurthermore, we have\nd\ndsE2(s)+(˜δ−η)E2(s)+L2(s) =R2(s), (3.44)\nwhere˜δ=m−n/2,η∈(0,˜δ)is an arbitrary number,\nL2(s) =η\n2/integraldisplay\nRn|y|2mf2dy+1\n2(η+1)/integraldisplay\nRn|y|2m|∇yf|2dy+/integraldisplay\nRn|y|2mg2dy\n+2m/integraldisplay\nRn|y|2m−2(y·∇yf)(f+g)dy,\nR2(s) =−ηe−s\nb(t(s))2/integraldisplay\nRn|y|2mfgdy−1\n2(η−3)e−s\nb(t(s))2/integraldisplay\nRn|y|2mg2dy\n−1\nb(t(s))2db\ndt(t(s))/integraldisplay\nRn|y|2m(2f+g)gdy+/integraldisplay\nRn|y|2m(f+g)hdy.\nProof.The proofs of (3.43) and (3.44) are the almost same as that of (3.42 ), and we only prove (3.42).\nFirst, we calculate\nd\nds/bracketleftbigg1\n2/integraldisplay\nRn|ˆF|2dξ/bracketrightbigg\n= Re/integraldisplay\nRn/parenleftbigg\n−ξ\n2·∇ξˆF−1\n2/parenleftBign\n2+δ/parenrightBig\nˆF+ˆG/parenrightbigg\n¯ˆFdξ\n=−δ\n2/integraldisplay\nRn|ˆF|2dξ+Re/integraldisplay\nRn¯ˆFˆGdξ\nand\nd\nds/bracketleftbigge−s\nb(t(s))2Re/integraldisplay\nRnˆF¯ˆGdξ/bracketrightbigg\n=−2\nb(t(s))2db\ndt(t(s))Re/integraldisplay\nRnˆF¯ˆGdξ−e−s\nb(t(s))2Re/integraldisplay\nRnˆF¯ˆGdξ\n+e−s\nb(t(s))2Re/integraldisplay\nRn/parenleftBig\nˆFs¯ˆG+¯ˆFˆGs/parenrightBig\ndξ\n=−2\nb(t(s))2db\ndt(t(s))Re/integraldisplay\nRnˆF¯ˆGdξ−δe−s\nb(t(s))2Re/integraldisplay\nRnˆF¯ˆGdξ\n+e−s\nb(t(s))2/integraldisplay\nRn|ˆG|2dξ−Re/integraldisplay\nRnˆF¯ˆGdξ\n−/integraldisplay\nRn|ξ|2|ˆF|2dξ+Re/integraldisplay\nRnˆF¯ˆHdξ.\n19Adding up these identities, we see that\nd\nds/bracketleftbigg\nRe/integraldisplay\nRn/parenleftbigg\n|ˆF|2+e−s\nb(t(s))2ˆF¯ˆG/parenrightbigg\ndξ/bracketrightbigg\n=−δ\n2/integraldisplay\nRn|ˆF|2dξ−2\nb(t(s))2db\ndt(t(s))Re/integraldisplay\nRnˆF¯ˆGdξ−δe−s\nb(t(s))2Re/integraldisplay\nRnˆF¯ˆGdξ\n+e−s\nb(t(s))2/integraldisplay\nRn|ˆG|2dξ−/integraldisplay\nRn|ξ|2|ˆF|2dξ+Re/integraldisplay\nRnˆF¯ˆHdξ (3.45)\nWe also have\nd\nds/bracketleftbigg1\n2/integraldisplay\nRn|ξ|2|ˆF|2dξ/bracketrightbigg\n= Re/integraldisplay\nRn|ξ|2/parenleftbigg\n−ξ\n2·∇ξˆF−1\n2/parenleftBign\n2+δ/parenrightBig\nˆF+ˆG/parenrightbigg\n¯ˆFdξ\n=1\n2(1−δ)/integraldisplay\nRn|ξ|2|ˆF|2dξ+Re/integraldisplay\nRn|ξ|2¯ˆFˆGdξ\nand\nd\nds/bracketleftbigg1\n2e−s\nb(t(s))2/integraldisplay\nRn|ˆG|2dξ/bracketrightbigg\n=−1\nb(t(s))2db\ndt(t(s))/integraldisplay\nRn|ˆG|2dξ−1\n2e−s\nb(t(s))2/integraldisplay\nRn|ˆG|2dξ\n+e−s\nb(t(s))2Re/integraldisplay\nRnˆGs¯ˆGdξ\n=−1\nb(t(s))2db\ndt(t(s))/integraldisplay\nRn|ˆG|2dξ+1\n2(1−δ)e−s\nb(t(s))2/integraldisplay\nRn|ˆG|2dξ\n−/integraldisplay\nRn|ˆG|2dξ−Re/integraldisplay\nRn|ξ|2ˆF¯ˆGdξ+Re/integraldisplay\nRnˆG¯ˆHdξ.\nSumming up the above identities, we have\nd\nds/bracketleftbigg1\n2e−s\nb(t(s))2/integraldisplay\nRn|ˆG|2dξ/bracketrightbigg\n=1\n2(1−δ)/integraldisplay\nRn|ξ|2|ˆF|2dξ−1\nb(t(s))2db\ndt(t(s))/integraldisplay\nRn|ˆG|2dξ\n+1\n2(1−δ)e−s\nb(t(s))2/integraldisplay\nRn|ˆG|2dξ−/integraldisplay\nRn|ˆG|2dξ+Re/integraldisplay\nRnˆG¯ˆHdξ. (3.46)\nFrom (3.45) and (3.46), we conclude (3.42).\n3.8 Proof of Proposition 3.7\nIn either case when n= 1 orn≥2, we have proved energy identities of Ej(s) with remainder terms\nRj(j= 0,1,2). Hereafter, we unify the both cases and complete the proof of Proposition 3.7. We\ndefine\nE3(s) =1\n2e−s\nb(t(s))2/parenleftbiggdα\nds(s)/parenrightbigg2\n+e−2λsα(s)2\nand\nE4(s) =C0E0(s)+C1E1(s)+E2(s)+E3(s),\nwhereλ >0 is determined later, and C0,C1are positive constants such that 1 ≪C1≪C0. By\nrecalling the equivalences (3.32) and (3.41), the following equivalence is valid for s≥s1:\nE4(s)∼ /ba∇dblf(s)/ba∇dbl2\nH1,m+e−s\nb(t(s))2/ba∇dblg(s)/ba∇dbl2\nH0,m+e−s\nb(t(s))2/parenleftbiggdα\nds(s)/parenrightbigg2\n+e−2λsα(s)2.(3.47)\nTo obtain the energy estimate of E4(s), we first notice the following lemma.\n20Lemma 3.13. We have\nd\ndsE3(s)+2λE3(s)+/parenleftbiggdα\nds(s)/parenrightbigg2\n=R3(s),\nwhere\nR3(s) =1\n2(2λ+1)e−s\nb(t(s))2/parenleftbiggdα\nds(s)/parenrightbigg2\n−1\nb(t(s))2db\ndt(t(s))/parenleftbiggdα\nds(s)/parenrightbigg2\n+dα\nds(s)/parenleftbigg/integraldisplay\nRnr(s,y)dy/parenrightbigg\n+2e−2λsα(s)dα\nds(s). (3.48)\nThen, we can also see the following energy estimate.\nLemma 3.14. We have\nd\ndsE4(s)+2λE4(s)+L4(s) =R4(s), (3.49)\nwhere\nL4(s) =/parenleftbigg1\n2−2λ/parenrightbigg\n(C0E0(s)+C1E1(s)+E2(s))+C0L0(s)+C1L1(s)+L2(s)+/parenleftbiggdα\nds(s)/parenrightbigg2\n,\nforn= 1,\nL4(s) =C0(δ−2λ)E0(s)+C1(δ−2λ)E1(s)+(˜δ−η−2λ)E2(s)\n+C0L0(s)+C1L1(s)+L2(s)+/parenleftbiggdα\nds(s)/parenrightbigg2\nforn≥2, and\nR4(s) =C0R0(s)+C1R1(s)+R2(s)+R3(s).\nHereR0,R1,R2andL0,L1,L2are defined in Lemmas 3.10 (n= 1)and 3.12 (n≥2), andR3is\ndefined by (3.48).\nThen, by the Schwarz inequality and the inequality (3.39), we obtain t he following lower estimate\nofL4. Here we recall that δ∈(0,1) is an arbitrary number, ˜δ=m−n/2 andη >0 is an arbitrary\nsmall number.\nLemma 3.15. If0<λ≤1/4 (n= 1),0<λ<min{1\n2,m\n2−n\n4}(n≥2), then\nL4(s)≥C/parenleftBigg\n/ba∇dblf(s)/ba∇dbl2\nH1,m+/ba∇dblg(s)/ba∇dbl2\nH0,m+/parenleftbiggdα\nds(s)/parenrightbigg2/parenrightBigg\nholds fors≥s1.\nProof.Letλsatisfy 0<λ≤1/4 (n= 1) and 0 <λ<min{1\n2,m\n2−n\n4}(n≥2). We take the parameter\nδso that 2λ<δ < 1. Then, recalling that ˜δ=m−n/2, we have 2 λ<min{δ,˜δ}. We also note that\nthe equivalences (3.32) and (3.41) of E0(s),E1(s),E2(s) yield the positivity of the first three terms of\nL4(s) fors≥s1. Therefore, it suffices to consider the terms L0(s),L1(s) andL2(s). Whenn= 1,\nnotingFy=fand applying the Schwarz inequality, we easily have\n/integraldisplay\nRf2dy=/integraldisplay\nRF2\nydy\n21and\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRyfy(f+g)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤1\n4/integraldisplay\nRy2f2\nydy+8/integraldisplay\nR(f2+g2)dy.\nHence, taking C1>8 andC0>2C1, we obtain the desired estimate.\nNext, when n≥2, we note that, for any small µ>0, we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRn|y|2m−2(y·∇yf)(f+g)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle\n≤µ/integraldisplay\nRn|y|2m|∇yf|2dy+8µ−1/integraldisplay\nRn|y|2m−2(f2+g2)dy\nand\nµ−1/integraldisplay\nRn|y|2m−2(f2+g2)dy=µ−1/integraldisplay\n|y|>µ−1|y|2m−2(f2+g2)dy+µ−1/integraldisplay\n|y|≤µ−1|y|2m−2(f2+g2)dy\n≤µ/integraldisplay\n|y|>µ−1|y|2m(f2+g2)dy+µ−2m+1/integraldisplay\n|y|≤µ−1(f2+g2)dy\n≤µ/integraldisplay\nRn|y|2m(f2+g2)dy+µ−2m+1/integraldisplay\nRn(f2+g2)dy.\nWe takeµsufficiently small so that µ≪ηand thenC0,C1sufficiently large so that µ−2m+1≪C1≪\nC0. Then, applying (3.39) to estimate the last term, we have the desire d estimate.\nFinally, we put\nE5(s) =E4(s)+1\n2α(s)2+e−s\nb(t(s))2α(s)dα\nds(s).\nThen, we easily obtain\nLemma 3.16. There exists s2≥s1such that we have\nE5(s)∼ /ba∇dblf(s)/ba∇dbl2\nH1,m+e−s\nb(t(s))2/ba∇dblg(s)/ba∇dbl2\nH0,m+α(s)2+e−s\nb(t(s))2/parenleftbiggdα\nds(s)/parenrightbigg2\n,\nE5(s)+/ba∇dblg(s)/ba∇dbl2\nH0,m+/parenleftbiggdα\nds(s)/parenrightbigg2\n∼ /ba∇dblf(s)/ba∇dbl2\nH1,m+/ba∇dblg(s)/ba∇dbl2\nH0,m+α(s)2+/parenleftbiggdα\nds(s)/parenrightbigg2\nfors≥s2.\nProof.By the Schwarz inequality, we have\n/vextendsingle/vextendsingle/vextendsingle/vextendsinglee−s\nb(t(s))2α(s)dα\nds(s)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C(˜η)e−s\nb(t(s))2α(s)2+ ˜ηe−s\nb(t(s))2/parenleftbiggdα\nds(s)/parenrightbigg2\n,\nwhere ˜η >0 is a small number determined later. By the equivalence (3.47) of E4(s) and taking ˜ η\nsufficiently small, we control the second term of the right-hand side and have\n˜ηe−s\nb(t(s))2/parenleftbiggdα\nds(s)/parenrightbigg2\n≤1\n2E4(s)\nfors≥s1. On the other hand, by Lemma 3.2 and taking s2≥s1sufficiently large, we estimate the\nfirst term as\nC(˜η)e−s\nb(t(s))2α(s)2≤1\n4α(s)2\n22fors≥s2. Combining them, we conclude that\nE5(s)≥1\n2E4(s)+1\n4α(s)2\nholds fors≥s2. Then, using the lower bound (3.47) again, we have the lower bound\n/ba∇dblf(s)/ba∇dbl2\nH1,m+e−s\nb(t(s))2/ba∇dblg(s)/ba∇dbl2\nH0,m+α(s)2+e−s\nb(t(s))2/parenleftbiggdα\nds(s)/parenrightbigg2\n≤CE5(s)\nfors≥s2. The upper bound of E5(s) immediately follows from the equivalence (3.47) of E4(s) and\nwe have the first assertion. The second assertion is also directly pr oved from the first one.\nBy using (3.21), we also have\nd\nds/bracketleftbigg1\n2α(s)2+e−s\nb(t(s))2α(s)dα\nds(s)/bracketrightbigg\n=e−s\nb(t(s))2/parenleftbiggdα\nds(s)/parenrightbigg2\n−2\nb(t(s))2α(s)db\ndt(t(s))dα\nds(s)\n+α(s)/parenleftbigg/integraldisplay\nRnr(s,y)dy/parenrightbigg\n=:˜R5(s).\nLettingR5(s) =R4(s)+˜R5(s), we obtain\nd\ndsE5(s)+2λE4(s)+L4(s) =R5(s). (3.50)\nWe give an estimate for the remainder term R5(s):\nLemma 3.17 (Estimate for the remainder terms) .Letλ0,λ1be\nλ0= min/braceleftbigg1−β\n1+β,γ\n1+β−1\n2,ν\n1+β−1/bracerightbigg\n(3.51)\n(where we interpret 1/(1+β)as an arbitrary large number when β=−1) and\nλ1=\n\n1\n2min\ni=1,...,k/braceleftbigg\npi1+2pi2+/parenleftbigg\n3−2β\n1+β/parenrightbigg\npi3−3/bracerightbigg\n, n= 1,\nn\n2/parenleftbigg\np−1−2\nn/parenrightbigg\n, n ≥2(3.52)\n(where we interpret −2βpi3/(1+β)as an arbitrary large number when pi3/ne}ationslash= 0andβ=−1). Then,\nthere exists s0≥s2such that we have the following estimates:\n(i)Whenn= 1,R4(s)andR5(s)satisfy\n|R4(s)| ≤˜ηL4(s)+C(˜η)e−2λ0sE5(s)+C(˜η)e−2λ1sk/summationdisplay\ni=1E5(s)pi1+pi2(E5(s)pi3+L4(s)pi3),\n|R5(s)| ≤˜ηL4(s)+C(˜η)e−λ0sE5(s)+C(˜η)e−2λ1sk/summationdisplay\ni=1E5(s)pi1+pi2(E5(s)pi3+L4(s)pi3)\n+C(˜η)e−λ1sk/summationdisplay\ni=1E5(s)(pi1+pi2+pi3+1)/2\nfors≥s0, where˜η>0is an arbitrary small number.\n23(ii)Whenn≥2,R4(s)andR5(s)satisfy\n|R4(s)| ≤˜ηL4(s)+C(˜η)e−2λ0sE5(s)+C(˜η)e−2λ1sE5(s)p,\n|R5(s)| ≤˜ηL4(s)+C(˜η)e−λ0sE5(s)+C(˜η)e−2λ1sE5(s)p+C(˜η)e−λ1sE5(s)(p+1)/2\nfors≥s0, where˜η>0is an arbitrary small number.\nWe postpone the proof of this lemma until the next section, and now we completes the proofs of\nProposition 3.7 and Theorem 2.1. We first consider the case n≥2. Taking ˜η= 1/2 in Lemma 3.17\nand using (3.50) and Lemmas 3.15, 3.16, we have\nd\ndsE5(s)≤Ce−λ0sE5(s)+Ce−2λ1sE5(s)p+Ce−λ1sE5(s)(p+1)/2(3.53)\nfors≥s0. Let\nΛ(s) := exp/parenleftbigg\n−C/integraldisplays\ns0e−λ0τdτ/parenrightbigg\n.\nWe note that e−Ce−λ0s0/λ0≤Λ(s)≤1 fors≥s0and Λ(s0) = 1. Multiplying (3.53) by Λ( s) and\nintegrating it over [ s0,s], we see that\nΛ(s)E5(s)≤E5(s0)+C/integraldisplays\ns0/bracketleftBig\nΛ(τ)e−2λ1τE5(τ)p+Λ(τ)e−λ1τE5(τ)(p+1)/2/bracketrightBig\ndτ\nholds fors≥s0. Putting\nM(s) := sup\ns0≤τ≤sE5(τ),\nwe further obtain\nM(s)≤CM(s0)+C(s0,λ0,λ1)/parenleftBig\nM(s)p+M(s)(p+1)/2/parenrightBig\n(3.54)\nfors≥s0. On the other hand, we easily estimate M(s0) as\nM(s0)≤C(s0)/parenleftbig\n/ba∇dbl(f(s0),g(s0))/ba∇dbl2\nH1,m×H0,m+α(s0)2+ ˙α(s0)2/parenrightbig\n≤C(s0)/ba∇dbl(v(s0),w(s0))/ba∇dbl2\nH1,m×H0,m\n≤C(s0)ε2/ba∇dbl(v0,w0)/ba∇dbl2\nH1,m×H0,m (3.55)\nby using the local existence result (see the proof of Proposition 3.5 ). Combining (3.54) with (3.55),\nwe have\nM(s)≤C2ε2/ba∇dbl(v0,w0)/ba∇dbl2\nH1,m×H0,m+C2/parenleftBig\nM(s)p+M(s)(p+1)/2/parenrightBig\nfors≥s0with some constant C2>0. Letε1be\nε1:=/parenleftBig/radicalbig\nC22(p+1)/4/parenrightBig−1\n.\nThen, a direct calculation implies\n2C2ε2I0>C2ε2I0+C2/bracketleftBig\n(2C2ε2I0)p+(2C2ε2I0)(p+1)/2/bracketrightBig\nholdsforε∈(0,ε1], whereI0=/ba∇dbl(v0,w0)/ba∇dbl2\nH1,m×H0,m. Combiningthiswith M(s0)≤C2ε2/ba∇dbl(v0,w0)/ba∇dbl2\nH1,m×H0,m\nand the continuity of M(s) with respect to s, we conclude that\nM(s)≤2C2ε2/ba∇dbl(v0,w0)/ba∇dbl2\nH1,m×H0,m (3.56)\n24holds fors≥s0andε∈(0,ε1]. Therefore, from Lemma 3.16, we obtain\n/ba∇dblf(s)/ba∇dbl2\nH1,m+e−s\nb(t(s))2/ba∇dblg(s)/ba∇dbl2\nH0,m+α(s)2+e−s\nb(t(s))2/parenleftbiggdα\nds(s)/parenrightbigg2\n≤Cε2/ba∇dbl(v0,w0)/ba∇dbl2\nH1,m×H0,m\nfors≥s0andε∈(0,ε1]. This implies\n/ba∇dblv(s)/ba∇dbl2\nH1,m+e−s\nb(t(s))2/ba∇dblw(s)/ba∇dbl2\nH0,m≤C∗ε2/ba∇dbl(v0,w0)/ba∇dbl2\nH1,m×H0,m (3.57)\nwith some constant C∗>0. This completes the proof of Proposition 3.7.\nWhenn= 1, to control the additional term E5(s)pi1L4(s) appearing in the estimate of R5(s), we\nuse (3.50) as\nd\ndsE5(s)+L4(s)≤Ce−λ0sE5(s)+C(˜η)e−2λ1sk/summationdisplay\ni=1E5(s)pi1+pi2(E5(s)pi3+L4(s)pi3)\n+C(˜η)e−λ1sk/summationdisplay\ni=1E5(s)(pi1+pi2+pi3+1)/2\ninstead of (3.53). In the same way as before, we multiply the both sid es by Λ(s) and integrate it over\n[s0,s] to obtain\nΛ(s)E5(s)+/integraldisplays\ns0Λ(τ)L4(τ)dτ\n≤E5(s0)+C/summationdisplay\ni=1,...,k\npi3=1/integraldisplays\ns0Λ(τ)e−2λ1τE5(τ)pi1+pi2L4(τ)pi3dτ\n+Ck/summationdisplay\ni=1/integraldisplays\ns0/bracketleftBig\nΛ(τ)e−2λ1τE5(τ)pi1+pi2+pi3+Λ(τ)e−λ1τE5(τ)(pi1+pi2+pi3+1)/2/bracketrightBig\ndτ.\nAs before, putting M(s) := sups0≤τ≤sE5(τ) and noting that Λ( s) is bounded by both above and\nbelow, we see that\nM(s)+/integraldisplays\ns0L4(τ)dτ≤C2ε2/ba∇dbl(v0,w0)/ba∇dbl2\nH1,m×H0,m+C2/summationdisplay\ni=1,...,k\npi3=1M(s)pi1+pi2/integraldisplays\ns0L4(τ)pi3dτ\n+C2k/summationdisplay\ni=1/parenleftBig\nM(s)pi1+pi2+pi3+M(s)(pi1+pi2+pi3+1)/2/parenrightBig\nfors≥s0with some constant C2>0. Takingε1sufficiently small so that\n2C2ε2I0+/integraldisplays\ns0L4(τ)dτ >C 2ε2I0+C2/summationdisplay\ni=1,...,k\npi3=1(2C2ε2I0)pi1+pi2/integraldisplays\ns0L4(τ)pi3dτ\n+C2k/summationdisplay\ni=1/parenleftBig\n(2C2ε2I0)pi1+pi2+pi3+(2C2ε2I0)(pi1+pi2+pi3+1)/2/parenrightBig\nholds forε∈(0,ε1], whereI0=/ba∇dbl(v0,w0)/ba∇dbl2\nH1,m×H0,m. Combining this with M(s0)≤C2ε2I0and the\ncontinuity of M(s) with respect to s, we conclude that\nM(s)≤2C2ε2/ba∇dbl(v0,w0)/ba∇dbl2\nH1,m×H0,m,\nwhich leads to (3.57) and completes the proof of Proposition 3.7 for n= 1.\n253.9 Proof of Theorem 2.1: asymptotic behavior\nNext, we prove the asymptotic behavior (2.9). For simplicity, we only consider the case n≥2, since\nthe proof of the one-dimensional case is similar. Putting\nλ= min/braceleftbigg1\n2,m\n2−n\n4,λ0,λ1/bracerightbigg\n−η, (3.58)\nwhereη >0 is an arbitrary small number, and λ0,λ1are defined by (3.51), (3.52), and turning back\nto (3.49) and using Lemma 3.17 with ˜ η=1\n2toR4(s), we have\nd\ndsE4(s)+2λE4(s)+1\n2L4(s)≤Ce−2λ0sE5(s)+Ce−2λ1sE5(s)p\n≤Ce−2λ2sε2/ba∇dbl(u0,u1)/ba∇dbl2\nH1,m×H0,m,\nwhereλ2= min{λ0,λ1}. Multiplying the above inequality by e2λs, we obtain\nd\nds/bracketleftbig\ne2λsE4(s)/bracketrightbig\n+e2λs\n2L4(s)≤Ce−2ηsε2/ba∇dbl(u0,u1)/ba∇dbl2\nH1,m×H0,m.\nIntegrating it over [ s0,s] and using Lemma 3.15, we have\nE4(s)+/integraldisplays\ns0e−2λ(s−τ)/parenleftBigg\n/ba∇dblf(s)/ba∇dbl2\nH1,m+/ba∇dblg(s)/ba∇dbl2\nH0,m+/parenleftbiggdα\ndτ(τ)/parenrightbigg2/parenrightBigg\ndτ≤Ce−2λsε2/ba∇dbl(u0,u1)/ba∇dbl2\nH1,m×H0,m.\nIn particular, for s0≤˜s≤s, one has\n|α(s)−α(˜s)|2=/parenleftBigg/integraldisplays\n˜s/parenleftbiggdα\ndτ(τ)/parenrightbigg2\ndτ/parenrightBigg2\n≤/parenleftbigg/integraldisplays\n˜se−2λτdτ/parenrightbigg/parenleftBigg/integraldisplays\n˜se2λτ/parenleftbiggdα\ndτ(τ)/parenrightbigg2\ndτ/parenrightBigg\n≤Ce−2λ˜sε2/ba∇dbl(u0,u1)/ba∇dbl2\nH1,m×H0,m,\nand hence, the limit α∗= lims→+∞α(s) exists and it follows that\n|α(s)−α∗|2≤Ce−2λsε2/ba∇dbl(u0,u1)/ba∇dbl2\nH1,m×H0,m.\nFinally, we have\n/ba∇dblv(s)−α∗ϕ0/ba∇dbl2\nH1,m≤ /ba∇dblf(s)/ba∇dbl2\nH1,m+|α(s)−α∗|2/ba∇dblϕ0/ba∇dbl2\nH1,m≤Ce−2λsε2/ba∇dbl(u0,u1)/ba∇dbl2\nH1,m×H0,m.\nRecalling the relation (3.2) and ( B(t) + 1)−n/2ϕ0((B(t) +1)−1/2x) =G(B(t) + 1,x), where Gis the\nGaussian defined by (2.8), we obtain\n/ba∇dblu(t,·)−α∗G(B(t)+1,·)/ba∇dbl2\nL2≤Cε2(B(t)+1)−n/2−2λ/ba∇dbl(u0,u1)/ba∇dbl2\nH1,m×H0,m,\nwhich completes the proof of Theorem 2.1.\n4 Estimates of the remainder terms\nIn this section, we give a proof to Lemma 3.17.\n26Lemma 4.1. Under the assumptions (2.1),(2.3)and(2.5), we have\n/vextenddouble/vextenddouble/vextenddoublee3s/2N/parenleftBig\ne−s/2v,e−svy,b(t(s))−1e−3s/2w/parenrightBig/vextenddouble/vextenddouble/vextenddouble2\nH0,1\n≤Ce−2λ1sk/summationdisplay\ni=1(/ba∇dblf(s)/ba∇dblH1,1+α(s))2(pi1+pi2)/parenleftbigg\n/ba∇dblg(s)/ba∇dblH0,1+α(s)+dα\nds(s)/parenrightbigg2pi3\n(4.1)\nforn= 1,s≥0and\n/vextenddouble/vextenddouble/vextenddoublee(n\n2+1)sN/parenleftbig\ne−n\n2sv/parenrightbig/vextenddouble/vextenddouble/vextenddouble2\nH0,m≤Ce−2λ1s(/ba∇dblf(s)/ba∇dblH1,m+α(s))2p(4.2)\nforn≥2,s≥0, whereλ1is given by (3.52).\nProof.Whenn= 1,β∈(−1,1), by the assumption (2.3) and Lemma 3.2, we compute\n(1+y2)e3sNi/parenleftBig\ne−s/2v,e−svy,b(t(s))−1e−3s/2w/parenrightBig2\n≤C(1+y2)e−2λ1s|v|2pi1|vy|2pi2|w|2pi3,\nwhereλ1is defined by (3.52). By the Sobolev inequality /ba∇dblv(s)/ba∇dblL∞≤C/ba∇dblv(s)/ba∇dblH1,0, we calculate\n(1+y2)e−2λ1s|v|2pi1|vy|2pi2|w|2pi3\n≤Ce−2λ1s|v2|pi1+pi2+pi3−1((1+y2)v2)1−pi2−pi3((1+y2)v2\ny)pi2((1+y2)w2)pi3\n≤Ce−2λ1s/ba∇dblv(s)/ba∇dbl2(pi1+pi2+pi3−1)\nH1,0 ((1+y2)v2)1−pi2−pi3((1+y2)v2\ny)pi2((1+y2)w2)pi3.\nTherefore, by the H¨ older inequality, we conclude\n/vextenddouble/vextenddouble/vextenddoublee3s/2Ni/parenleftBig\ne−s/2v,e−svy,b(t(s))−1e−3s/2w/parenrightBig/vextenddouble/vextenddouble/vextenddouble2\nH0,1\n≤Ce−2λ1s/ba∇dblv(s)/ba∇dbl2(pi1+pi2−1)\nH1,0/ba∇dblv(s)/ba∇dbl2(1−pi2)\nH1,1/ba∇dblv(s)/ba∇dbl2pi2\nH1,1/ba∇dblw(s)/ba∇dbl2pi3\nH0,1\n≤Ce−2λ1s(/ba∇dblf(s)/ba∇dblH1,1+α(s))2(pi1+pi2)/parenleftbigg\n/ba∇dblg(s)/ba∇dblH0,1+α(s)+dα\nds(s)/parenrightbigg2pi3\nWhenn= 1,pi3/ne}ationslash= 0,β=−1, we obtain\n(1+y2)e3sNi/parenleftBig\ne−s/2v,e−svy,b(t(s))−1e−3s/2w/parenrightBig2\n≤C(1+y2)e(3−pi1−2pi2−3pi3)sb(t(s))−pi3|v|2pi1|vy|2pi2|w|2pi3\n≤C(1+y2)e−λi1s|v|2pi1|vy|2pi2|w|2pi3,\nwhere we can take λi1as an arbitrary large number, since Lemma 3.2 shows b(t(s))−pi3∼exp(−pi3es).\nTherefore, by the same way, we obtain the desired estimate.\nNext, we consider the case n≥2. By the assumption (2.5) and Lemma 3.3, we have\n/vextenddouble/vextenddouble/vextenddoublee(n\n2+1)sN/parenleftbig\ne−n\n2sv/parenrightbig/vextenddouble/vextenddouble/vextenddouble2\nH0,m≤C/integraldisplay\nRne2(n\n2+1)s/an}b∇acketle{ty/an}b∇acket∇i}ht2m/vextendsingle/vextendsinglee−n\n2sv(s,y)/vextendsingle/vextendsingle2pdy\n≤Ce−2λ1s/integraldisplay\nRn/vextendsingle/vextendsingle/vextendsingle/an}b∇acketle{ty/an}b∇acket∇i}htm/pv(s,y)/vextendsingle/vextendsingle/vextendsingle2p\ndy\n≤Ce−2λ1s/vextenddouble/vextenddouble/vextenddouble∇/parenleftBig\n/an}b∇acketle{ty/an}b∇acket∇i}htm/pv(s)/parenrightBig/vextenddouble/vextenddouble/vextenddouble2pσ\nL2/vextenddouble/vextenddouble/vextenddouble/an}b∇acketle{ty/an}b∇acket∇i}htm/pv(s)/vextenddouble/vextenddouble/vextenddouble2p(1−σ)\nL2\n≤Ce−2λ1s/ba∇dblv(s)/ba∇dbl2p\nH1,m\n≤Ce−2λ1s(/ba∇dblf(s)/ba∇dblH1,m+α(s))2p,\nwhich completes the proof.\n27From Lemmas 3.2, 4.1 and the assumption (2.2), we immediately obtain t he following estimate:\nLemma 4.2. Letrbe defined by (3.4). Under the assumptions (2.1)–(2.5), we have\n/ba∇dblr(s)/ba∇dbl2\nH0,m≤Ce−2λ0s/parenleftBigg\n/ba∇dblf(s)/ba∇dbl2\nH1,m+/ba∇dblg(s)/ba∇dbl2\nH0,m+α(s)2+/parenleftbiggdα\nds(s)/parenrightbigg2/parenrightBigg\n+Ce−2λ1sk/summationdisplay\ni=1(/ba∇dblf(s)/ba∇dblH1,1+α(s))2(pi1+pi2)/parenleftbigg\n/ba∇dblg(s)/ba∇dblH0,1+α(s)+dα\nds(s)/parenrightbigg2pi3\nforn= 1,s≥0and\n/ba∇dblr(s)/ba∇dbl2\nH0,m≤Ce−2λ0s/parenleftBigg\n/ba∇dblf(s)/ba∇dbl2\nH1,m+/ba∇dblg(s)/ba∇dbl2\nH0,m+α(s)2+/parenleftbiggdα\nds(s)/parenrightbigg2/parenrightBigg\n+Ce−2λ1s(/ba∇dblf(s)/ba∇dblH1,m+α(s))2p\nforn≥2,s≥0, whereλ0,λ1are defined by (3.51),(3.52), respectively.\nProof.By Lemma 4.1, it suffices to estimate\n1\nb(t(s))2db\ndt(t(s))w+es/2c(t(s))·∇yv+esd(t(s))v.\nApplying Lemma 3.2, we have\n/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\nb(t(s))2db\ndt(t(s))w(s)/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nH0,m≤C/parenleftBigg\n/ba∇dblg(s)/ba∇dbl2\nH0,m+α(s)2+/parenleftbiggdα\nds(s)/parenrightbigg2/parenrightBigg\n×/braceleftBigg\ne−2(1−β)s/(1+β)β∈(−1,1),\nexp(−4es)β=−1.\nAlso, the assumption (2.2) implies\n/vextenddouble/vextenddouble/vextenddoublees/2c(t(s))·∇yv(s)/vextenddouble/vextenddouble/vextenddouble2\nH0,m≤C/parenleftbig\n/ba∇dblf(s)/ba∇dbl2\nH1,1+α(s)2/parenrightbig\n×/braceleftBigg\ne−((2γ)/(1+β)−1)sβ∈(−1,1),\nexp(−2γes+s)β=−1\nand\n/ba∇dblesd(t(s))v(s)/ba∇dbl2\nH0,m≤C/parenleftbig\n/ba∇dblf(s)/ba∇dbl2\nH1,1+α(s)2/parenrightbig\n×/braceleftBigg\ne−(2ν/(1+β)−2)β∈(−1,1),\nexp(−2νes+2s)β=−1.\nSumming up the above estimates and (4.1), (4.2), we obtain the desir ed estimate.\nNext, we estimate the term hgiven by (3.24). By Lemmas 3.2 and 4.2, we can easily have the\nfollowing estimate:\nLemma 4.3. Lethbe defined by (3.24). Under the assumption (2.1)–(2.5), we have\n/ba∇dblh(s)/ba∇dbl2\nH0,m≤Ce−2λ0s/parenleftBigg\n/ba∇dblf(s)/ba∇dbl2\nH1,m+/ba∇dblg(s)/ba∇dbl2\nH0,m+α(s)2+/parenleftbiggdα\nds(s)/parenrightbigg2/parenrightBigg\n+Ce−2λ1sk/summationdisplay\ni=1(/ba∇dblf(s)/ba∇dblH1,1+α(s))2(pi1+pi2)/parenleftbigg\n/ba∇dblg(s)/ba∇dblH0,1+α(s)+dα\nds(s)/parenrightbigg2pi3\nforn= 1,s≥0and\n/ba∇dblh(s)/ba∇dbl2\nH0,m≤Ce−2λ0s/parenleftBigg\n/ba∇dblf(s)/ba∇dbl2\nH1,m+/ba∇dblg(s)/ba∇dbl2\nH0,m+α(s)2+/parenleftbiggdα\nds(s)/parenrightbigg2/parenrightBigg\n+Ce−2λ1s(/ba∇dblf(s)/ba∇dblH1,m+α(s))2p\nforn≥2,s≥0, whereλ0,λ1are defined by (3.51),(3.52), respectively.\n28Proof.We easily estimate\n/vextenddouble/vextenddouble/vextenddouble/vextenddoublee−s\nb(t(s))2/parenleftbigg\n−2dα\nds(s)ψ0(y)+α(s)/parenleftBigy\n2·∇yψ0(y)+/parenleftBign\n2+1/parenrightBig\nψ0(y)/parenrightBig/parenrightbigg/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\nH0,m\n≤Ce−2λ0s/parenleftBigg\nα(s)2+/parenleftbiggdα\nds(s)/parenrightbigg2/parenrightBigg\n.\nFor the term r(s), we apply Lemma 4.2. Finally, for the term (/integraltext\nRnr(s,y)dy)ϕ0(y), we note that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRnr(s,y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤C/ba∇dblr(s)/ba∇dblH0,m, (4.3)\nholds due to m>n/2. Thus, we apply Lemma 4.2 again to obtain the conclusion.\nMoreover, combining (3.26) and the Hardy-type inequalities (3.28), (3.38), we also have\nLemma 4.4. LetHbe defined by (3.31). Under the assumption (2.1)–(2.5), we have\n/ba∇dblH(s)/ba∇dbl2\nH0,m≤Ce−2λ0s/parenleftBigg\n/ba∇dblf(s)/ba∇dbl2\nH1,m+/ba∇dblg(s)/ba∇dbl2\nH0,m+α(s)2+/parenleftbiggdα\nds(s)/parenrightbigg2/parenrightBigg\n+Ce−2λ1sk/summationdisplay\ni=1(/ba∇dblf(s)/ba∇dblH1,1+α(s))2(pi1+pi2)/parenleftbigg\n/ba∇dblg(s)/ba∇dblH0,1+α(s)+dα\nds(s)/parenrightbigg2pi3\nforn= 1,s≥0and\n/ba∇dblH(s)/ba∇dbl2\nH0,m≤Ce−2λ0s/parenleftBigg\n/ba∇dblf(s)/ba∇dbl2\nH1,m+/ba∇dblg(s)/ba∇dbl2\nH0,m+α(s)2+/parenleftbiggdα\nds(s)/parenrightbigg2/parenrightBigg\n+Ce−2λ1s(/ba∇dblf(s)/ba∇dblH1,m+α(s))2p\nforn≥2,s≥0, whereλ0,λ1are defined by (3.51),(3.52), respectively.\nNow we are at the position to prove Lemma 3.17.\nProof of Lemma 3.17. We first prove the estimate for R4(s). Let ˜η>0 be an arbitrary small number.\nThen, by the Schwarz inequality and Lemmas 3.15 and 3.16, there exis tss3≥s2such that the terms\nnot including the nonlinearity are easily bounded by ˜ ηL4(s)+C(˜η)e−2λ0sE5(s) fors≥s3. The terms\nincluding the nonlinearity consist of the following three terms:\n/integraldisplay\nRn(F+G)Hdy,/integraldisplay\nRn(1+|y|2m)(f+g)hdy,/parenleftbigg/integraldisplay\nRnr(s,y)dy/parenrightbiggdα\nds(s).\nBy the Schwarz inequality and Lemmas 3.9, 3.11, 3.15, 3.16, 4.3, 4.4, th ere existss4≥s2such that\nthe first two terms are easily bounded by\n˜ηL4(s)+C(˜η)e−2λ0sE5(s)+\n\nCe−2λ1sk/summationdisplay\ni=1E5(s)pi1+pi2(E5(s)pi3+L4(s)pi3) (n= 1),\nCe−2λ1s(/ba∇dblf(s)/ba∇dblH1,m+α(s))2p(n≥2)\nfors≥s4. For the third term, we apply the Schwarz inequality to obtain\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftbigg/integraldisplay\nRnr(s,y)dy/parenrightbiggdα\nds(s)/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤˜η/parenleftbiggdα\nds(s)/parenrightbigg2\n+C(˜η)/parenleftbigg/integraldisplay\nRnr(s,y)dy/parenrightbigg2\n.\n29Noting (4.3) and applying Lemma 4.2, and then Lemmas 3.15 and 3.16, we have the desired estimate.\nFinally, we prove the estimate for R5(s). Let ˜η >0 be an arbitrary small number. Recall that\nR5(s) =R4(s)+˜R5(s) with\n˜R5(s) =e−s\nb(t(s))2/parenleftbiggdα\nds(s)/parenrightbigg2\n−2\nb(t(s))2α(s)db\ndt(t(s))dα\nds(s)+α(s)/parenleftbigg/integraldisplay\nRnr(s,y)dy/parenrightbigg\n.\nWe have already estimated R4(s) and hence, it suffices to estimate ˜R5(s). By Lemmas 3.15 and 3.16,\nthere exists s5≥s2such that the first two terms are easily estimated by ˜ ηL4(s) +C(˜η)e−2λ0sE5(s)\nfors≥s5. Moreover, by (4.3), α(s)≤CE5(s)1/2and Lemma 4.2, there exists s6≥s2such that the\nthird term is estimated as\nα(s)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\nRnr(s,y)dy/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤α(s)/ba∇dblr(s)/ba∇dblH0,m\n≤˜ηL4(s)+Ce−λ0sE5(s)\n+\n\nCe−λ1sk/summationdisplay\ni=1E5(s)(pi1+pi2+1)/2/parenleftBig\nE5(s)pi3/2+L4(s)pi3/2/parenrightBig\n(n= 1),\nCe−λ1sE5(s)(p+1)/2(n≥2)\nfors≥s6. Whenn= 1, we further applythe Schwarzinequalityto the terms in the sum c orresponding\ntopi3= 1 and obtain\ne−λ1sE5(s)(pi1+pi2+1)/2L4(s)pi3/2≤˜ηL4(s)+C(˜η)e−2λ1sE5(s)(pi1+pi2+pi3).\nFinally,letting s0:= max{s3,s4,s5,s6}andcombiningtheaboveestimates,wehavetheconclusion.\nAcknowledgements\nThe author is deeply grateful to Professor Yoshiyuki Kagei for u seful suggestions and constructive\ncomments. 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Polishchuk,2Vladislav Korenivski,2and Arne Brataas1\n1Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, Trondheim, Norway\n2Nanostructure Physics, Royal Institute of Technology, Stockholm, Sweden\nSpin valves form a key building block in a wide range of spintronic concepts and devices from\nmagnetoresistive read heads to spin-transfer-torque oscillators. We elucidate the dependence of the\nmagnetic damping in the free layer on the angle its equilibrium magnetization makes with that in\nthe \fxed layer. The spin pumping-mediated damping is anisotropic and tensorial, with Gilbert- and\nBloch-like terms. Our investigation reveals a mechanism for tuning the free layer damping in-situ\nfrom negligible to a large value via the orientation of \fxed layer magnetization, especially when the\nmagnets are electrically insulating. Furthermore, we expect the Bloch contribution that emerges\nfrom the longitudinal spin accumulation in the non-magnetic spacer to play an important role in a\nwide range of other phenomena in spin valves.\nIntroduction. { The phenomenon of magnetoresistance\nis at the heart of contemporary data storage technolo-\ngies [1, 2]. The dependence of the resistance of a multi-\nlayered heterostructure comprising two or more magnets\non the angles between their respective magnetizations has\nbeen exploited to read magnetic bits with a high spatial\nresolution [3]. Furthermore, spin valves comprised of two\nmagnetic layers separated by a non-magnetic conductor\nhave been exploited in magnetoresistive random access\nmemories [2, 4, 5]. Typically, in such structures, one\n`free layer' is much thinner than the other `\fxed layer'\nallowing for magnetization dynamics and switching in\nthe former. The latter serves to spin-polarize the charge\ncurrents \rowing across the device and thus exert spin-\ntorques on the former [6{9]. Such structures exhibit a\nwide range of phenomena from magnetic switching [5] to\noscillations [10, 11] driven by applied electrical currents.\nWith the rapid progress in taming pure spin cur-\nrents [12{20], magnetoresistive phenomena have found\na new platform in hybrids involving magnetic insulators\n(MIs). The electrical resistance of a non-magnetic metal\n(N) was found to depend upon the magnetic con\fgura-\ntion of an adjacent insulating magnet [21{24]. This phe-\nnomenon, dubbed spin Hall magnetoresistance (SMR),\nrelies on the pure spin current generated via spin Hall\ne\u000bect (SHE) in N [25, 26]. The SHE spin current accu-\nmulates spin at the MI/N interface, which is absorbed\nby the MI depending on the angle between its magne-\ntization and the accumulated spin polarization. The\nnet spin current absorbed by the MI manifests as ad-\nditional magnetization-dependent contribution to resis-\ntance in N via the inverse SHE. The same principle of\nmagnetization-dependent spin absorption by MI has also\nbeen exploited in demonstrating spin Nernst e\u000bect [27],\ni.e. thermally generated pure spin current, in platinum.\nWhile the ideas presented above have largely been ex-\nploited in sensing magnetic \felds and magnetizations,\ntunability of the system dissipation is a valuable, un-\nderexploited consequence of magnetoresistance. Such\nan electrically controllable resistance of a magnetic wire\nFIG. 1. Schematic depiction of the device under investigation.\nThe blue arrows denote the magnetizations. The \fxed layer\nF2magnetization remains static. The free layer F 1magneti-\nzation precesses about the z-axis with an average cone angle\n\u0002\u001c1. The two layers interact dynamically via spin pumping\nand back\row currents.\nhosting a domain wall [28] has been suggested as a ba-\nsic circuit element [29] in a neuromorphic computing [30]\narchitecture. In addition to the electrical resistance or\ndissipation, the spin valves should allow for controlling\nthe magnetic damping in the constituent magnets [31].\nSuch an in-situ control can be valuable in, for example,\narchitectures where a magnet is desired to have a large\ndamping to attain low switching times and a low dissipa-\ntion for spin dynamics and transport [13, 16]. Further-\nmore, a detailed understanding of magnetic damping in\nspin valves is crucial for their operation as spin-transfer-\ntorque oscillators [10] and memory cells [5].\nInspired by these new discoveries [21, 27] and previous\nrelated ideas [31{34], we suggest new ways of tuning the\nmagnetic damping of the free layer F 1in a spin valve\n(Fig. 1) via controllable absorption by the \fxed layer\nF2of the spin accumulated in the spacer N due to spin\npumping [31, 35]. The principle for this control over spin\nabsorption is akin to the SMR e\u000bect discussed above and\ncapitalizes on altering the F 2magnetization direction.\nWhen spin relaxation in N is negligible, the spin lost by\nF1is equal to the spin absorbed by F 2. This lost spin\nappears as tensorial Gilbert [36] and Bloch [37] damp-arXiv:1811.00020v2  [cond-mat.mes-hall]  10 Apr 20192\ning in F 1magnetization dynamics. In its isotropic form,\nthe Gilbert contribution arises due to spin pumping and\nis well established [31{33, 35, 38{40]. We reveal that\nthe Bloch term results from back\row due to a \fnite dc\nlongitudinal spin accumulation in N. Our results for the\nangular and tensorial dependence of the Gilbert damping\nare also, to best of our knowledge, new.\nWe show that the dissipation in F 1, expressed in terms\nof ferromagnetic resonance (FMR) linewidth, varies with\nthe angle\u0012between the two magnetizations (Fig. 3).\nThe maximum dissipation is achieved in collinear or or-\nthogonal con\fgurations depending on the relative size\nof the spin-mixing g0\nrand longitudinal spin glconduc-\ntances of the NjF2subsystem. For very low gl, which\ncan be achieved employing insulating magnets, the spin\npumping mediated contribution to the linewidth vanishes\nfor collinear con\fgurations and attains a \u0012-independent\nvalue for a small non-collinearity. This can be used to\nstrongly modulate the magnetic dissipation in F 1electri-\ncally via, for example, an F 2comprised by a magneto-\nelectric material [41].\nFMR linewidth. { Disregarding intrinsic damping for\nconvenience, the magnetization dynamics of F 1including\na dissipative spin transfer torque arising from the spin\ncurrent lost IIIs1may be expressed as:\n_^mmm=\u0000j\rj(^mmm\u0002\u00160HHHe\u000b) +j\rj\nMsVIIIs1: (1)\nHere, ^mmmis the unit vector along the F 1magnetization\nMMMtreated within the macrospin approximation, \r(<0)\nis the gyromagnetic ratio, Msis the saturation magneti-\nzation,Vis the volume of F 1, andHHHe\u000bis the e\u000bective\nmagnetic \feld. Under certain assumptions of linearity\nas will be detailed later, Eq, (1) reduces to the Landau-\nLifshitz equation with Gilbert-Bloch damping [36, 37]:\n_^mmm=\u0000j\rj(^mmm\u0002\u00160HHHe\u000b) + ( ^mmm\u0002GGG)\u0000BBB: (2)\nConsidering the equilibrium orientation ^mmmeq=^zzz, Eq. (2)\nis restricted to the small transverse dynamics described\nbymx;y\u001c1, while the z-component is fully determined\nby the constraint ^mmm\u0001^mmm= 1. Parameterized by a diagonal\ndimensionless tensor \u0014 \u000b, the Gilbert damping has been in-\ncorporated via GGG=\u000bxx_mx^xxx+\u000byy_my^yyyin Eq. (2). The\nBloch damping is parametrized via a diagonal frequency\ntensor \u0014\n asBBB= \n xxmx^xxx+ \nyymy^yyy. A more familiar,\nalthough insu\u000ecient for the present considerations, form\nof Bloch damping can be obtained by assuming isotropy\nin the transverse plane: BBB= \n 0(^mmm\u0000^mmmeq). This form,\nrestricted to transverse dynamics, makes its e\u000bect as a\nrelaxation mechanism with characteristic time 1 =\n0ev-\nident. The Bloch damping, in general, captures the so-\ncalled inhomogeneous broadening and other, frequency\nindependent contributions to the magnetic damping.\nConsidering uniaxial easy-axis and easy-plane\nanisotropies, parametrized respectively by Kzand\n0 30 60 9000.10.20.30.40.5FIG. 2. Normalized damping parameters for F 1magneti-\nzation dynamics vs. spin valve con\fguration angle \u0012(Fig.\n1). ~\u000bxx6= ~\u000byysigni\fes the tensorial nature of the Gilbert\ndamping. The Bloch parameters ~\nxx\u0019~\nyyare largest for\nthe collinear con\fguration. The curves are mirror symmetric\nabout\u0012= 90\u000e. ~g0\nr= 1, ~gl= 0:01, \u0002 = 0:1,!0= 10\u00022\u0019\nGHz, and!ax= 1\u00022\u0019GHz.\nKx[42], the magnetic free energy density Fmis ex-\npressed as: Fm=\u0000\u00160MMM\u0001HHHext\u0000KzM2\nz+KxM2\nx;with\nHHHext=H0^zzz+hhhrfas the applied static plus microwave\n\feld. Employing the e\u000bective \feld \u00160HHHe\u000b=\u0000@Fm=@MMM\nin Eq. (2) and switching to Fourier space [ \u0018exp(i!t)],\nwe obtain the resonance frequency !r=p\n!0(!0+!ax).\nHere,!0\u0011j\rj(\u00160H0+ 2KzMs) and!ax\u0011j\rj2KxMs.\nThe FMR linewidth is evaluated as:\nj\rj\u00160\u0001H=(\u000bxx+\u000byy)\n2!+t(\nxx+ \nyy)\n2\n+t!ax\n4(\u000byy\u0000\u000bxx); (3)\nwhere!is the frequency of the applied microwave \feld\nhhhrfand is approximately !rclose to resonance, and t\u0011\n!=p\n!2+!2ax=4\u00191 for a weak easy-plane anisotropy.\nThus, in addition to the anisotropic Gilbert contribu-\ntions, the Bloch damping provides a nearly frequency-\nindependent o\u000bset in the linewidth.\nSpin \row. { We now examine spin transport in the\ndevice with the aim of obtaining the damping parame-\nters that determine the linewidth [Eq. (3)]. The N layer\nis considered thick enough to eliminate static exchange\ninteraction between the two magnetic layers [31, 40]. Fur-\nthermore, we neglect the imaginary part of the spin-\nmixing conductance, which is small in metallic systems\nand does not a\u000bect dissipation in any case. Disregarding\nlongitudinal spin transport and relaxation in the thin free\nlayer, the net spin current IIIs1lost by F 1is the di\u000berence\nbetween the spin pumping and back\row currents [31]:\nIIIs1=gr\n4\u0019\u0010\n~^mmm\u0002_^mmm\u0000^mmm\u0002\u0016\u0016\u0016s\u0002^mmm\u0011\n; (4)\nwheregris the real part of the F 1jN interfacial spin-\nmixing conductance, and \u0016\u0016\u0016sis the spatially homogeneous3\nspin accumulation in the thin N layer. The spin current\nabsorbed by F 2may be expressed as [31]:\nIIIs2=g0\nr\n4\u0019^mmm2\u0002\u0016\u0016\u0016s\u0002^mmm2+gl\n4\u0019(^mmm2\u0001\u0016\u0016\u0016s)^mmm2;\n\u0011X\ni;j=fx;y;zggij\n4\u0019\u0016sj^iii; (5)\nwhereglandg0\nrare respectively the longitudinal spin\nconductance and the real part of the interfacial spin-\nmixing conductance of the N jF2subsystem, ^mmm2denotes\nthe unit vector along F 2magnetization, and gij=gji\nare the components of the resulting total spin conduc-\ntance tensor. glquanti\fes the absorption of the spin\ncurrent along the direction of ^mmm2, the so-called longi-\ntudinal spin current. For metallic magnets, it is domi-\nnated by the di\u000busive spin current carried by the itin-\nerant electrons, which is dissipated over the spin re-\nlaxation length [31]. On the other hand, for insulat-\ning magnets, the longitudinal spin absorption is domi-\nnated by magnons [43, 44] and is typically much smaller\nthan for the metallic case, especially at low tempera-\ntures. Considering ^mmm2= sin\u0012^yyy+ cos\u0012^zzz(Fig. 1),\nEq. (5) yields gxx=g0\nr,gyy=g0\nrcos2\u0012+glsin2\u0012,\ngzz=g0\nrsin2\u0012+glcos2\u0012,gxy=gyx=gxz=gzx= 0,\nandgyz=gzy= (gl\u0000g0\nr) sin\u0012cos\u0012.\nRelegating the consideration of a small but \fnite spin\nrelaxation in the thin N layer to the supplemental ma-\nterial [45], we assume here that the spin current lost by\nF1is absorbed by F 2, i.e.,IIIs1=IIIs2. Imposing this spin\ncurrent conservation condition, the spin accumulation in\nN along with the currents themselves can be determined.\nWe are primarily interested in the transverse (x and y)\ncomponents of the spin current since these fully deter-\nmine the magnetization dynamics ( ^mmm\u0001^mmm= 1):\nIs1x=1\n4\u0019grgxx\ngr+gxx(\u0000~_my+mx\u0016sz);\nIs1y=1\n4\u0019\u0014grgyy\ngr+gyy(~_mx+my\u0016sz) +gyz\u0016sz(1\u0000ly)\u0015\n;\n\u0016sz=~gr(lxmx_my\u0000lymy_mx\u0000p_mx)\ngzz\u0000pgyz+gr\u0000\nlxm2x+lym2y+ 2pmy\u0001;\n(6)\nwherelx;y\u0011gxx;yy=(gr+gxx;yy) andp\u0011gyz=(gr+gyy).\nThe spin lost by F 1appears as damping in the magneti-\nzation dynamics [Eqs. (1) and (2)] [31, 35].\nWe pause to comment on the behavior of \u0016szthus ob-\ntained [Eq. (6)]. Typically, \u0016szis considered to be \frst\nor second order in the cone angle, and thus negligibly\nsmall. However, as discussed below, an essential new\n\fnding is that it becomes independent of the cone an-\ngle and large under certain conditions. For a collinear\ncon\fguration and vanishing gl,gzz=gyz= 0 results\nin ~\u0016sz\u0011\u0016sz=~!!1 [38]. Its \fnite dc value con-\ntributes to the Bloch damping [Eq. (6)] [38]. For a\nnon-collinear con\fguration, \u0016sz\u0019\u0000~grp_mx=(gzz\u0000pgyz)\n0 45 90 135 18000.10.20.30.40.50.6FIG. 3. Normalized ferromagnetic resonance (FMR)\nlinewidth of F 1for di\u000berent values of the longitudinal spin\nconductance ~ gl\u0011gl=grof NjF2bilayer. The various parame-\nters employed are ~ g0\nr\u0011g0\nr=gr= 1, \u0002 = 0:1 rad,!0= 10\u00022\u0019\nGHz, and!ax= 1\u00022\u0019GHz.grandg0\nrare the spin-mixing\nconductances of F 1jN and NjF2interfaces respectively. Only\nthe spin pumping-mediated contribution to the linewidth has\nbeen considered and is normalized to its value for the case of\nspin pumping into a perfect spin sink [31].\nand contributes to Gilbert damping via Is1y[Eq. (6)].\nThus, in general, we may express the spin accumulation\nas\u0016sz=\u0016sz0+\u0016sz1[46], where \u0016sz0is the dc value\nand\u0016sz1/_mxis the linear oscillating component. \u0016sz0\nand\u0016sz1contribute, respectively, to Bloch and Gilbert\ndamping.\nGilbert-Bloch dissipation. { Equations (1) and (6) com-\npletely determine the magnetic damping in F 1. However,\nthese equations are non-linear and cannot be captured\nwithin our linearized framework [Eqs. (2) and (3)]. The\nleading order e\u000bects, however, are linear in all but a nar-\nrow range of parameters. Evaluating these leading or-\nder terms within reasonable approximations detailed in\nthe supplemental material [45], we are able to obtain the\nGilbert and Bloch damping tensors \u0014 \u000band\u0014\n. Obtaining\nthe general result numerically [45], we present the ana-\nlytic expressions for two cases covering a large range of\nthe parameter space below.\nFirst, we consider the collinear con\fgurations in the\nlimit of ~gl\u0011gl=gr!0. As discussed above, we obtain\n~\u0016sz0\u0011\u0016sz0=~!!1 and ~\u0016sz1\u0011\u0016sz1=~!!0 [Eq. (6)].\nThus the components of the damping tensors can be di-\nrectly read from Eq. (6) as ~ \u000bxx;yy\u0011\u000bxx;yy=\u000bss=ly;x=\ng0\nr=(gr+g0\nr) = ~g0\nr=(1+~g0\nr);and~\nxx;yy\u0011\nxx;yy=(\u000bss!) =\n\u0000lx;y\u0016sz0=(~!) =\u0000g0\nr=(gr+g0\nr) =\u0000~g0\nr=(1 + ~g0\nr). Here,\nwe de\fned ~ g0\nr\u0011g0\nr=grand\u000bss\u0011~grj\rj=(4\u0019MsV) is the\nGilbert constant for the case of spin-pumping into an\nideal spin sink [31, 35]. Substituting these values in Eq.\n(3), we \fnd that the linewidth, or equivalently damping,\nvanishes. This is understandable since the system we\nhave considered is not able to relax the z component of\nthe spin at all. There can, thus, be no net contribution to4\nFIG. 4. Normalized FMR linewidth of F 1for very small ~ gl.\nThe squares and circles denote the evaluated points while the\nlines are guides to the eye. The linewidth increases from being\nnegligible to its saturation value as \u0012becomes comparable to\nthe average cone angle \u0002. ~ g0\nr= 1,!0= 10\u00022\u0019GHz, and\n!ax= 1\u00022\u0019GHz.\nmagnetic damping. \u0016sz0accumulated in N opposes the\nGilbert relaxation via a negative Bloch contribution [38].\nThe latter may also be understood as an anti-damping\nspin transfer torque due to the accumulated spin [6].\nNext, we assume the system to be in a non-collinear\ncon\fguration such that ~ \u0016sz0!0 and may be disre-\ngarded, while ~ \u0016sz1simpli\fes to:\n~\u0016sz1=\u0000_mx\n!(~gl\u0000~g0\nr) sin\u0012cos\u0012\n~g0r~gl+ ~glcos2\u0012+ ~g0rsin2\u0012; (7)\nwhere ~gl\u0011gl=grand ~g0\nr\u0011g0\nr=gras above. This in turn\nyields the following Gilbert parameters via Eq. (6), with\nthe Bloch tensor vanishing on account of ~ \u0016sz0!0:\n~\u000bxx=~g0\nr~gl\n~g0r~gl+ ~glcos2\u0012+ ~g0rsin2\u0012;~\u000byy=~g0\nr\n1 + ~g0r;(8)\nwhere ~\u000bxx;yy\u0011\u000bxx;yy=\u000bssas above. Thus, ~ \u000byyis\u0012-\nindependent since ^mmm2lies in the y-z plane and the x-\ncomponent of spin, the absorption of which is captured\nby ~\u000byy, is always orthogonal to ^mmm2. ~\u000bxx, on the other\nhand, strongly varies with \u0012and is generally not equal\nto ~\u000byyhighlighting the tensorial nature of the Gilbert\ndamping.\nFigure 2 depicts the con\fgurational dependence of nor-\nmalized damping parameters. The Bloch parameters are\nappreciable only close to the collinear con\fgurations on\naccount of their proportionality to \u0016sz0. The\u0012range over\nwhich they decrease to zero is proportional to the cone\nangle \u0002 [Eq. (6)]. The Gilbert parameters are described\nsu\u000eciently accurately by Eq. (8). The linewidth [Eq.\n(3)] normalized to its value for the case of spin pump-\ning into a perfect spin sink has been plotted in Fig. 3.\nFor low ~gl, the Bloch contribution partially cancels the\nGilbert dissipation, which results in a smaller linewidthclose to the collinear con\fgurations [38]. As ~ glincreases,\nthe relevance of Bloch contribution and \u0016sz0diminishes,\nand the results approach the limiting condition described\nanalytically by Eq. (8). In this regime, the linewidth\ndependence exhibits a maximum for either collinear or\northogonal con\fguration depending on whether ~ gl=~g0\nris\nsmaller or larger than unity. Physically, this change in\nthe angle with maximum linewidth is understood to re-\n\rect whether transverse or longitudinal spin absorption\nis stronger.\nWe focus now on the case of very low ~ glwhich can\nbe realized in structures with electrically-insulating mag-\nnets. Figure 4 depicts the linewidth dependence close to\nthe collinear con\fgurations. The evaluated points are\nmarked with stars and squares while the lines smoothly\nconnect the calculated points. The gap in data for very\nsmall angles re\rects the limited validity of our linear\ntheory, as discussed in the supplemental material [45].\nAs per the limiting case ~ gl!0 discussed above, the\nlinewidth should vanish in perfectly collinear states. A\nmore precise statement for the validity of this limit is\nre\rected in Fig. 4 and Eq. (6) as ~ gl=\u00022!0. For su\u000e-\nciently low ~ gl, the linewidth changes sharply from a neg-\nligible value to a large value over a \u0012range approximately\nequal to the cone angle \u0002. This shows that systems com-\nprised of magnetic insulators bearing a very low ~ glare\nhighly tunable as regards magnetic/spin damping by rel-\natively small deviation from the collinear con\fguration.\nThe latter may be accomplished electrically by employ-\ning magnetoelectric material [41] for F 2or via current\ndriven spin transfer torques [6, 9, 47].\nDiscussion. { Our identi\fcation of damping contribu-\ntions as Gilbert-like and Bloch-like [Eq. (6)] treats \u0016sz\nas an independent variable that may result from SHE,\nfor example. When it is caused by spin pumping cur-\nrent and\u0016sz/!, this Gilbert-Bloch distinction is less\nclear and becomes a matter of preference. Our results\ndemonstrate the possibility of tuning the magnetic damp-\ning in an active magnet via the magnetization of a passive\nmagnetic layer, especially for insulating magnets. In ad-\ndition to controlling the dynamics of the uniform mode,\nthis magnetic `gate' concept [48] can further be employed\nfor modulating the magnon-mediated spin transport in a\nmagnetic insulator [43, 44]. The anisotropy in the result-\ning Gilbert damping may also o\u000ber a pathway towards\ndissipative squeezing [49] of magnetic modes, comple-\nmentary to the internal anisotropy-mediated `reactive'\nsqueezing [50, 51]. We also found the longitudinal accu-\nmulated spin, which is often disregarded, to signi\fcantly\na\u000bect the dynamics. This contribution is expected to\nplay an important role in a wide range of other phenom-\nena such as spin valve oscillators.\nSummary. { We have investigated the angular modu-\nlation of the magnetic damping in a free layer via control\nof the static magnetization in the \fxed layer of a spin\nvalve device. 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Lett. 116, 146601 (2016).\n[51] Akashdeep Kamra, Utkarsh Agrawal, and Wolfgang\nBelzig, \\Noninteger-spin magnonic excitations in untex-\ntured magnets,\" Phys. Rev. B 96, 020411 (2017).\n[52] A.I. Akhiezer, V.G. Bar'iakhtar, and S.V. Peletminski,\nSpin waves (North-Holland Publishing Company, Ams-\nterdam, 1968).1\nSupplemental material with the manuscript Anisotropic and controllable\nGilbert-Bloch dissipation in spin valves by\nAkashdeep Kamra, Dmytro M. Polishchuk, Vladislav Korenivski and Arne Brataas\nCOLLINEAR CONFIGURATION WITHOUT LONGITUDINAL SPIN RELAXATION\nIn order to appreciate some of the subtleties, we \frst examine the collinear con\fguration in the limit of vanishing\nlongitudinal spin conductance. \u0012= 0;\u0019andgl= 0 imply the following values for the various parameters:\ngxx=gyy=g0\nr; g zz=gyz=p= 0; lx;y=g0\nr\ngr+g0r\u0011l; (S1)\nwhence we obtain:\n\u0016sz\n~=(mx_my\u0000my_mx)\nm2x+m2y; (S2)\n=!0+!ax\n1 +!ax\n2!0[1\u0000cos(2!t)]; (S3)\nwhere we have assumed magnetization dynamics as given by the Landau-Lifshitz equation without damping, and\nthe phase of mxis treated as the reference and set to zero. In order to obtain analytic expressions, we make the\nassumption !ax=!0\u001c1 such that we have:\n\u0016sz=\u0016sz0+\u0016sz2; with (S4)\n\u0016sz0=~\u0010\n!0+!ax\n2\u0011\n; (S5)\n\u0016sz2=~!ax\n4\u0000\ne\u0000i2!t+ei2!t\u0001\n: (S6)\nIn contrast with our assumptions in the main text, a term oscillating with 2 !appears. Furthermore, it yields\ncontributions to the Bloch damping via products such as my\u0016sz, which now have contributions oscillating at !due\nto the\u0016sz0as well as\u0016sz2. We obtain:\n~\u000bxx= ~\u000byy=l; (S7)\n~\nxx=\u0000l!0+3!ax\n4\n!0+!ax\n2and ~\nyy=\u0000l!0+!ax\n4\n!0+!ax\n2; (S8)\nsubstituting which into Eq. (3) from the main text yields a vanishing linewidth and damping. This is expected from\nthe general spin conservation argument that there can be no damping in the system if it is not able to dissipate the\nz-component of the spin. In fact, in the above considerations, \u0016sz2contributed with the opposite sign to ~\nxxand\n~\nyy, and thus dropped out of the linewidth altogether. This also justi\fes our ignoring this contribution in the main\ntext.\nFigure 1 depicts the dependence of the accumulated z-polarized spin and the normalized linewidth for small but\n\fniteglin the collinear con\fguration. The accumulated longitudinal (z-polarized) spin increases with the cone angle\nand the linewidth accordingly decreases to zero [38].\nNUMERICAL EVALUATION\nDespite the additional complexity in the previous section, we could treat the dynamics within our linearized frame-\nwork. However, in the general case, \u0016szhas contributions at all multiples of !and cannot be evaluated in a simple\nmanner. A general non-linear analysis must be employed which entails treating the magnetization dynamics numer-\nically altogether. Such an approach prevents us from any analytic description of the system, buries the underlying\nphysics, and is thus undesirable.\nFortunately, the e\u000bects of non-linear terms are small for all, but a narrow, range of parameters. Hence, we make\nsome simplifying assumptions here and continue treating our system within the linearized theory. We only show2\n10-310-210-100.10.20.30.40.5\nFIG. 1. Ferromagnetic resonance linewidth and the dc spin accumulation created in the spacer as a function of the average\ncone angle in the collinear con\fguration. Depending on ~ gl, there is a complementary transition of the two quantities between\nsmall and large values as the cone angle increases. ~ g0\nr= 1,!0= 10\u00022\u0019GHz, and!ax= 1\u00022\u0019GHz.\nresults in the parameter range where our linear analysis is adequate. Below, we describe the numerical routine for\nevaluating the various quantities. To be begin with the average cone angle \u0002 is de\fned as:\n\u00022=\nm2\nx+m2\ny\u000b\n; (S9)\nwhereh\u0001idenotes averaging over time. The spin accumulation is expressed as \u0016sz=\u0016sz0+\u0016sz1with:\n\u0016sz0=*\n~gr(lxmx_my\u0000lymy_mx\u0000p_mx)\ngzz\u0000pgyz+gr\u0000\nlxm2x+lym2y+ 2pmy\u0001+\n; (S10)\n\u0016sz1=\u0000*\ngrp\ngzz\u0000pgyz+gr\u0000\nlxm2x+lym2y+ 2pmy\u0001+\n~_mx: (S11)\nThe above expressions combined with the equations for the spin current \row (Eqs. (6) in the main text) directly yield\nthe Gilbert and Bloch damping tensors.\nVARIATION WITH ADDITIONAL PARAMETERS\nHere, we discuss the dependence of the FMR linewidth on the easy-plane anisotropy and the spin-mixing conduc-\ntanceg0\nrof the NjF2interface. The results are plotted in Fig. 2. A high easy-plane anisotropy is seen to diminish\nthe con\fguration dependence of the linewidth and is thus detrimental to the dissipation tunability. The easy-axis\nanisotropy, on the other hand, is absorbed in !0and does not need to be examined separately. We also see an increase\nin the con\fguration dependence of the damping with an increasing g0\nr. This is understood simply as an increased\ndamping when the spin is absorbed more e\u000eciently due to a larger g0\nr. The damping is expected to reach the case of\nspin pumping into a perfect spin sink in the limit of ~ g0\nr!1 and\u0012= 0;\u0019.\nEFFECT OF SPIN RELAXATION IN THE SPACER LAYER\nWe now address the role of the small but \fnite spin relaxation in the non-magnetic spacer layer. To this end, we\nconsider that a part of the spin current injected into N by F 1is lost as the \\spin-leakage current\" IIIsl, as depicted in\nFig. 3, such that IIIs1=IIIs2+IIIsl. In order to evaluate the leakage, we consider the spin di\u000busion equation in N which\nreads [31]:\nD@2\nx\u0016\u0016\u0016s=\u0016\u0016\u0016s\n\u001csf; (S12)3\n0 45 90 135 18000.10.20.30.40.50.6\n(a)\n0 45 90 135 18000.20.40.60.81 (b)\nFIG. 2. Normalized ferromagnetic resonance (FMR) linewidth of F 1. (a) Same as Fig. 3 in the main text with additional plots\nfor a large easy-plane anisotropy. (b) Linewidth dependence for di\u000berent spin-mixing conductances of N jF2interface. The\nparameters employed are the same as Fig. 2 in the main text.\nFIG. 3. Schematic depiction of the spin currents \rowing through the device, including the spin-leakage current IIIslthat is lost\non account of a \fnite spin relaxation in the spacer layer N.\nwhereDand\u001csfare di\u000busion constant and spin-\rip time, respectively. We now integrate the equation over the\nthickness of N:\nZ\nd(D@x\u0016\u0016\u0016s) =Zd\n0\u0016\u0016\u0016s\n\u001csfdx: (S13)\nSince the N-layer thickness dis typically much smaller than the spin di\u000busion length in N (e.g., a few nm versus a\nfew hundred nm for Cu), we treat \u0016\u0016\u0016son the right hand side as a constant. Furthermore, in simplifying the left hand\nside, we invoke the expression for the spin current [31]: IIIs= (\u0000~NSD=2)@x\u0016\u0016\u0016s, withNthe one-spin density of states\nper unit volume and Sthe interfacial area. Thus, we obtain\n2\n~NS(IIIs1\u0000IIIs2) =d\n\u001csf\u0016\u0016\u0016s; (S14)\nwhich simpli\fes to the desired relation IIIs1=IIIs2+IIIslwith\nIIIsl=~NVN\n2\u001csf\u0016\u0016\u0016s\u0011gsl\n4\u0019\u0016\u0016\u0016s; (S15)\nwhereVNis the volume of the spacer layer N.\nIt is easy to see that accounting for spin leakage, as derived in Eq. (S15), results in the following replacements to\nEqs. (6) of the main text:\ngxx!gxx+gsl; g yy!gyy+gsl; g zz!gzz+gsl: (S16)4\nSince all our speci\fc results are based on Eqs. (6) of the main text, this completes our assessment of the role played\nby spin relaxation in N. Physically, this new result means that the condition for no spin relaxation in the system,\nwhich was previously treated as gl!0, is now amended to gl+gsl!0. This, however, does not a\u000bect the generality\nand signi\fcance of the key results presented in the main text."    },    {        "title": "2102.10394v2.Fast_magnetization_reversal_of_a_magnetic_nanoparticle_induced_by_cosine_chirp_microwave_field_pulse.pdf",        "content": "Fast magnetization reversal of a magnetic nanoparticle induced by cosine\nchirp microwave field pulse\nM. T. Islam,1,a)M. A. S. Akanda,1M. A. J. Pikul,1and X. S. Wang2,b)\n1)Physics Discipline, Khulna University, Khulna 9208, Bangladesh\n2)School of Physics and Electronics, Hunan University, Changsha 410082, China\nWe investigate the magnetization reversal of single-domain magnetic nanoparticle driven by the circularly polarized\ncosine chirp microwave pulse (CCMP). The numerical findings, based on the Landau-Lifshitz-Gilbert equation, reveal\nthat the CCMP is by itself capable of driving fast and energy-efficient magnetization reversal. The microwave field am-\nplitude and initial frequency required by a CCMP are much smaller than that of the linear down-chirp microwave pulse.\nThis is achieved as the frequency change of the CCMP closely matches the frequency change of the magnetization pre-\ncession which leads to an efficient stimulated microwave energy absorption (emission) by (from) the magnetic particle\nbefore (after) it crosses over the energy barrier. We further find that the enhancement of easy-plane shape anisotropy\nsignificantly reduces the required microwave amplitude and the initial frequency of CCMP. We also find that there is an\noptimal Gilbert damping for fast magnetization reversal. These findings may provide a pathway to realize the fast and\nlow-cost memory device.\nI. INTRODUCTION\nAchieving fast and energy-efficient magnetization rever-\nsal of high anisotropy materials has drawn much attention\nsince it has potential application in non-volatile data stor-\nage devices1–3and fast data processing4. For high thermal\nstability and low error rate, high anisotropy materials are\nrequired5in device application. But one of the challenging\nissues is to find out the way which can induce the fastest mag-\nnetization reversal with minimal energy consumption. Over\nthe last two decades, many magnetization reversal methods\nhas been investigated, such as by constant magnetic fields6,7,\nby the microwave field of constant frequency, either with or\nwithout a polarized electric current8–10and by spin-transfer\ntorque (STT) or spin-orbit torque (SOT)11–28. However, all\nthe means are suffering from their own limitations. For in-\nstance, in the case of external magnetic field, reversal time\nis longer and has scalability and field localization issues6.\nIn case of the constant microwave field driven magnetiza-\ntion reversal, the large field amplitude and the long rever-\nsal time are emerged as limitations29–31. In the case of the\nSTT-MRAM, the threshold current density is a large and thus,\nJoule heat which may lead the device malfunction durability\nand reliability issues32–38. Moreover, there are several stud-\nies showing magnetization reversal induced by microwaves of\ntime-dependent frequency39–45. In the study39, the magnetiza-\ntion reversal, with the assistance of external field, is obtained\nby a radio-frequency microwave field pulse. Here, a dc ex-\nternal field acts as the main reversal force. In the study40,\nto obtain magnetization reversal, the applied microwave fre-\nquency needs to be the same as the resonance frequency, and\nin the studies42,43, optimal microwave forms are constructed.\nThese microwave forms are difficult to be realized in practice.\nThe study44reports magnetization reversal induced by the mi-\ncrowave pulse, but the pulse is applied such that the magne-\ntization just crosses over the energy barrier, i.e., only positive\na)Electronic mail: torikul@phy.ku.ac.bd\nb)Electronic mail: justicewxs@hnu.edu.cnfrequency range ( +f0to 0) is employed.\nA recent study45has demonstrated that the circularly polar-\nized linear down-chirp microwave pulse (DCMP) (whose fre-\nquency linearly decreases with time from the initial frequency\n+f0to\u0000f0) can drive fast magnetization reversal of uniax-\nial nanoparticles. The working principle of the above model\nis that the DCMP triggers stimulated microwave energy ab-\nsorption (emission) by (from) the magnetization before (after)\ncrossing the energy barrier. However, the efficiency of trig-\ngered microwave energy absorption or emission depends on\nhow closely the frequency of chirp microwave pulse matches\nthe magnetization precession frequency. In DCMP-driven\ncase, the frequency linearly decreases from f0to\u0000f0with\ntime but, in fact, the decrement of magnetization precession\nfrequency is not linear46,47during magnetization reversal . So\nthe frequency of DCMP only roughly matches the magneti-\nzation precession frequency. Thus, the DCMP triggers in-\nefficient energy absorption or emission and the required mi-\ncrowave amplitude is still large.\nTherefore, to achieve more efficient magnetization reversal,\nwe need to find a microwave pulse of proper time-dependent\nfrequency that matches the intrinsic magnetization precession\nfrequency better. In this study, we demonstrate that a cosine\nchirp microwave pulse (CCMP), defined as a microwave pulse\nwhose frequency sweeps in a cosine function with time from\n+f0to\u0000f0in first half-period of the microwave pulse, is ca-\npable of driving the fast and energy efficient magnetization\nreversal. This is because the frequency change of the CCMP\nmatches the nonlinear frequency change of magnetization pre-\ncession better than the DCMP. In addition, this study empha-\nsizes how the shape anisotropy influences the required param-\neters of CCMP and how the Gilbert damping affects the mag-\nnetization reversal. We find that the increase of easy-plane\nshape anisotropy makes the magnetization reversal easier. The\nmaterials with larger damping are better for fast magnetization\nreversal. These investigations might be useful in device appli-\ncations.arXiv:2102.10394v2  [cond-mat.mes-hall]  15 Sep 20212\nmf0\n0-f(b) (a)\nτ t\nFIG. 1. (a) Schematic diagram of the system in which mrepresents\na unit vector of the magnetization. A circularly polarized cosine\n(nonlinear) chirp microwave pulse is applied onto the single domain\nnanoparticle. (b) The frequency sweeping (from +f0to\u0000f0) of a\ncosine chirp microwave pulse.\nII. ANALYTICAL MODEL AND METHOD\nWe consider a square magnetic nanoparticle of area Sand\nthickness dwhose uniaxial easy-axis anisotropy directed in\nthez-axis as shown in FIG. 1(a). The size of the nanopar-\nticle is chosen so that the magnetization is considered as a\nmacrospin represented by the unit vector mwith the mag-\nnetic moment SdM s, where Msis the saturation magnetization\nof the material. The demagnetization field can be approxi-\nmated by a easy-plane shape anisotropy. The shape anisotropy\nfield coefficient is hshape=\u0000m0(Nz\u0000Nx)Ms, and the shape\nanisotropy field is hshape=hshapemzˆz, where NzandNxare\ndemagnetization factors48,49andm0=4p\u000210\u00007N=A2is the\nvacuum magnetic permeability. The strong uniaxial mag-\nnetocrystalline anisotropy hani=hanimzˆzdominates the total\nanisotropy so that the magnetization of the nanoparticle has\ntwo stable states, i.e., mparallel to ˆzand\u0000ˆz.\nThe magnetization dynamics min the presence of circularly\npolarized CCMP is governed by the Landau-Lifshitz-Gilbert\n(LLG) equation50\ndm\ndt=\u0000gm\u0002heff+am\u0002dm\ndt; (1)\nwhere aandgare the dimensionless Gilbert damping con-\nstant and the gyromagnetic ratio, respectively, and heffis the\ntotal effective field which includes the microwave magnetic\nfieldhmw, and the effective anisotropy field hkalong zdirec-\ntion (note that although we consider small nanoparticles that\ncan be treated as macrospins approximately, we still perform\nfull micromagnetic simulations with small meshes and full de-\nmagnetization field to be more accurate).\nThe effective anisotropy field can be expressed in terms of\nuniaxial anisotropy haniand shape anisotropy hshape ashk=\nhani+hshape= [hani\u0000m0(Nz\u0000Nx)Ms]mzˆz. Thus, the resonant\nfrequency of the nanoparticle is obtained from the well-known\nKittel formula\nf0=g\n2p[hani\u0000m0(Nz\u0000Nx)Ms]: (2).\nFor microwave field-driven magnetization reversal from the\nLLG equation, the rate of energy change is expressed as\ndE\ndt=\u0000agjm\u0002heffj2\u0000m\u0001dhmw\ndt: (3)\nThe first term is always negative since damping ais posi-\ntive. The second term can be either positive or negative for\ntime-dependent external microwave field. Therefore, the mi-\ncrowave field pulse can trigger the stimulated energy absorp-\ntion or emission, depending on the angle between the instan-\ntaneous magnetization manddhmw\ndt.\nInitially, because of easy-axis anisotropy, the magnetiza-\ntion prefers to stay in one of the two stable states, \u0006ˆz, cor-\nresponding to two energy minima. The objective of magne-\ntization reversal is to get the magnetization from one stable\nstate to the other. Along the reversal process, the magnetiza-\ntion requires to overcome an energy barrier at mz=0 which\nseparates two stable states. For fast magnetization reversal,\nthe external field is required to supply the necessary energy\nto the magnetization until crossing the energy barrier and, af-\nter crossing the energy barrier, the magnetization releases en-\nergy through damping and =or the external field is required\nto extract (by negative work done) energy from the magne-\ntization. It is mentioned that there is as intrinsic anisotropy\nfieldhanidue to the anisotropy which induces a magnetization\nnatural =resonant frequency proportional to mz. When mag-\nnetization goes from one stable state to another, the magneti-\nzation resonant frequency decreases while the magnetization\nclimbs up and becomes zero momentarily while crossing the\nenergy barrier and then increases with the opposite preces-\nsion direction while it goes down from the barrier. In princi-\nple, for fast and energy-efficient reversal, one requires a chirp\nmicrowave pulse whose frequency always matches the mag-\nnetization precession frequency to ensure the term m\u0001˙hmw\nto be maximal (minimal) before (after) crossing the energy\nbarrier. The study45, employs the DCMP (whose frequency\nlinearly decreases with time) to match the magnetization pre-\ncession frequency roughly. In fact, during the reversal from\nmz= +1 to mz=\u00001 , the decreasing of the resonant fre-\nquency ( while the spin climbs up the energy barrier) and in-\ncreasing of the resonant frequency (while it goes down from\nthe barrier) are not linear46,47. This leads us to consider a\ncosine chirp microwave pulse (CCMP) (a microwave pulse\nwhose frequency decreases non-linearly with time) in order\nto match the change of magnetization precession frequency\nclosely. Thus the CCMP might trigger more efficient stim-\nulated microwave absorptions (emissions) by (from) magne-\ntization before (after) the spin crosses the energy barrier to\ninduce fast and energy-efficient magnetization reversal.\nIn order to substantiate the above mentioned prediction,\nwe apply a circularly polarized cosine down-chirp microwave\npulse in the xyplane of the nanoparticle and solve the LLG\nequation numerically using the MUMAX3 package51. The\ncosine chirp microwave pulse (CCMP) takes the form hmw=\nhmw[cosf(t)ˆx+sinf(t)ˆy], where hmwis the amplitude of the\nmicrowave field and f(t)is the phase. Since the phase f(t)is\n2pf0cos(2pRt)t, where R(in units of GHz) is the controlling3\n/s45/s49\n/s48\n/s49/s45/s49/s48/s49\n/s45/s49/s48/s49\n/s109\n/s122\n/s109/s121\n/s109\n/s120\n/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54/s45/s49/s48/s49\n/s46\n/s116/s32/s40/s110/s115/s41/s109\n/s122\n/s45/s49/s48/s45/s53/s48/s53/s49/s48/s101 /s32/s40/s49/s48/s49/s53\n/s32/s74/s47/s115/s32/s109/s51\n/s41(a) (b)\n(c)\nFIG. 2. Model parameters of nanoparticle of Ms= 106A/m, Hk=\n0:75 T, g= 1:76\u00021011rad/(T\u0001s), and a=0:01. (a) Temporal evo-\nlutions of mzofV= (8\u00028\u00028)nm3driven by the CCMP (with the\nminimal hmw=0:035 T, f0=18:8 GHz and R= 0.32 GHz) (red line)\nand DCMP (with hmw=0:035 T, f0=18:8 GHz and R= 1.53 GHz\n) (blue line). For CCMP case, (b) the corresponding magnetization\nreversal trajectories and (c) temporal evolutions of mz(red lines) and\nthe energy changing rate ˙eof the magnetization against time (blue\nlines).\nparameter, the instantaneous frequency f(t)of CCMP is ob-\ntained as f(t) =1\n2pdf\ndt=f0[cos(2pRt)\u0000(2pRt)sin(2pRt)]\nwhich decreases with time from f0to final\u0000f0at a time\ndependent chirp rate h(t)(in units of ns\u00002) as shown\nin FIG. 1(b). The chirp rate takes the form h(t) =\n\u0000f0h\n(4pR)sin(2pRt)+(2pR)2tcos(2pRt)i\n.\nAccording to the applied CCMP, the second term of right\nhand side of Eq. (3), i.e., the energy changing rate can be\nexpressed as\n˙e=\u0000Hmwsinq(t)sinF(t)\u0014f(t)\nt\u0000d\ndt\u0012f(t)\nt\u0013\nt\u0015\n(4)\nwhereF(t)is the angle between mt(the in-plane component\nofmandhmw. Therefore, the microwave field pulse can trig-\nger the stimulated energy absorption (before crossing the en-\nergy barrier) with\u0000F(t)and emission (after crossing the en-\nergy barrier) with F(t).\nThe material parameters of this study are chosen from typ-\nical experiments on microwave-driven magnetization reversal\nasMs=106A=m,hani=0:75 T, g=1:76\u00021011rad=(T\u0001s),\nexchange constant A=13\u000210\u000012J=m and a=0:01. Al-\nthough, the strategy and other findings of this study would\nwork for other materials also. The cell size (2\u00022\u00022)nm3\nis used in this study. We consider the switching time win-\ndow 1 ns at which the magnetization switches/reverses to\nmz=\u00000:9.\nIII. NUMERICAL RESULTS\nWe first investigate the possibility of reversing the magne-\ntization of cubic sample (8\u00028\u00028)nm3by the cosine chirpmicrowave pulse (CCMP). Accordingly, we apply the CCMP\nwith the microwave amplitude hmw=0:045 T, initial fre-\nquency f0=21 GHz and R=1:6 ns\u00001which are same as\nestimated in the study45), to the sample and found that CCMP\ncan drive the fast magnetization reversal. Then, we search the\nminimal hmw,f0, and Rof the CCMP such that the fast re-\nversal is still valid. Interestingly, the CCMP with significantly\nsmaller parameters i.e., hmw=0:035 T, f0=18:8 GHz and\noptimal R=0:32 ns\u00001, is capable of reversing the magnetiza-\ntion efficiently shown by red line in FIG. 2(a).\nThen we intend to show how efficient the CCMP driven\nmagnetization reversal compare to DCMP driven case. For\nfair comparison, we choose the same pulse duration t(as\nshown in Fig. 1, tis the time at which the frequency changes\nfrom +f0to\u0000f0). For the CCMP, we solve cos (2pRt)\u0000\n(2pRt)sin(2pRt) =\u00001, which gives the relation t=1:307\n2pR.\nBut, in case of DCMP, we know that t=2f0\nhand for f0=\n18:8 GHz, the chirp rate becomes h=57:86 ns\u00002and hence\nfind the parameter R(=1=t)= 1.53 ns\u00001. Then we apply the\nDCMP with the hmw=0:035 T, f0=18:8 GHz (which are\nsame as CCMP-driven case), and R=1.53 ns\u00001and found that\nthe magnetization only precesses around the initial state, i.e.,\nthe DCMP is not able to reverse the magnetization as shown\nby blue line in FIG. 2(a). So, it is mentioned that the CCMP\ncan reverse the magnetization with lower energy consumption\nwhich is the desired in device application. To be more explicit,\nthe trajectories of magnetization reversal induced by CCMP\nis shown FIG. 2(b) which shows the magnetization reverses\nswiftly.For further justification of CCMP driven reversal, we\ncalculate the energy changing rate dE=dtrefers to (4) by de-\ntermining the angles F(t)andq(t)and plotted with time in\nFIG. 2(c). The stimulated energy absorption (emission) peaks\nare obtained before (after) crossing the energy barrier as ex-\npected for faster magnetization reversal. This is happened be-\ncause the frequency of the CCMP closely matches the fre-\nquency of magnetization precession frequency, i.e., before\ncrossing the energy barrier, the F(t)remains around\u000090\u000eand\nafter crossing the energy barrier F(t)around 90\u000eto maximize\nthe energy absorption and emission respectively.\nThen, this study emphasizes how shape-anisotropy field\nhshape affects the magnetization switching time, microwave\namplitude hmwand initial frequency f0of CCMP. Since de-\nmagnetization field or shape-anisotropy field hshape should\nhave significant effect on magnetization reversal process as\nit opposes the magnetocrystalline anisotropy field haniwhich\nstabilizes the magnetization along two stable states. Accord-\ningly, to induce the hshape in the sample, we choose square\ncuboid shape samples and, to increase the strength of hshape,\nthe cross-sectional area S=xyis enlarged gradually for the\nfixed thickness d(=z) =8 nm. Specifically, we focus on\nthe samples of S1=10\u000210,S2=12\u000212,S3=14\u000214,\nS4=16\u000216,S5=18\u000218,S6=20\u000220 and S7=22\u000222 nm2\nwith d=8 nm. For the samples of different S, by determining\nthe analytic demagnetization factors NzandNx48,52, the shape\nanisotropy field hshape=m0(Nz\u0000Nx)Msmzˆzare determined.\nThehshape actually opposes the anisotropy field haniand\nhence resonance frequencies f0=g\n2p[hani\u0000m0(Nz\u0000Nx)Ms]4\nTABLE I. Shape anisotropy coefficient, resonant frequency f0, simulated frequency, f0and frequency-band\nCross\u0000sectional area Shape anisotropy coefficient Resonant frequency, Simulated minimal frequency, Simulated frequency-band\nS(nm2) hshape (T) f0(GHz) f0(GHz) (GHz)\nS1 0.09606 18.3 17.8 17.8 \u000019.9\nS2 0.17718 16 14.9 14.9 \u000018.3\nS3 0.2465 14.1 13.7 13.7 \u000015.4\nS4 0.3064 12.4 12.2 12.2 \u000013.8\nS5 0.3588 11 10.7 10.7 \u000012.3\nS6 0.4049 9.6 8.8 8.8 \u000011.5\nS7 0.4459 8.5 7.7 7.4 \u00008.7\ndecreases as shown in the Table I. Then, for the samples of\ndifferent S, with the fixed hmw=0:035 T, the corresponding\nminimal f0and optimal Rof CCMP are determined through\nthe study of the magnetization reversal. The temporal evolu-\ntions of mzfor different Sare shown in FIG. 3(a) and found\nthat for S7\u001522\u000222 nm2, the magnetization smoothly reveres\nwith the shortest time. The switching time tsas a function of\nthe coefficient hshape (corresponding to S) is plotted in FIG.\n3(c). It is observed that, with the increase of hshape orS, the\ntsshows slightly increasing trend but for S7(22\u000222)nm2or\nhshape=0:4459 T , tsdrops to 0.43 ns. For further increment\nofSorhshape,tsremains constant around 0.43 ns which is\nclose to the theoretical limit (0.4 ns) refers to the study37.\nThis is because the hshape reduces the effective anisotropy\nand thus reduces the height of energy barrier (energy differ-\nence between the initial state and saddle point) which is shown\nin the FIG. 3(b). Therefore, after decreasing certain height\nof energy barrier, the magnetization reversal becomes fastest\neven with the same filed amplitude hmw=0:035 T. Due to the\nreduction of height of the energy barrier with the hshape, one\ncan expect that the hmwand f0should also decrease with the\nincrease of the anisotropy coefficient hshape and theses find-\nings are presented subsequently.\nHere we present the effect of the shape anisotropy hshape\non the microwave amplitude hmwand initial frequency f0of\nCCMP. Purposely, for each sample Sorhshape, we numerically\ndetermine (by tuning hmw,f0and optimal R) the minimally re-\nquired hmwandf0of CCMP for the the magnetization reversal\ntime window 1 ns. Interestingly, we find that the fast and effi-\ncient reversal is valid for a wide range of initial frequency f0.\nFIG. 4(a) shows the estimated frequency bands of f0by verti-\ncal dashed lines for different S( for example, f0=19:9\u001817:8\nforhshape=0:096 T) for time window 1 ns. So there is a great\nflexibility in choosing the initial frequency f0which is useful\nin device application. FIG. 3(d) shows how the minimal f0\n(red circles) and hmw(blue square) decrease with the increase\nofhshape. The decay of f0is expected as hshape reduces ef-\nfective anisotropy (refers to Eq.(2)). For more justification, in\nsame FIG. 3(d), theoretically resonant frequency (black solid\nline) and simulated minimal frequency (red circles) as func-\ntion of hshape indicate the agreement. The minimal frequency\nf0always smaller than the theoretical =resonant frequency.\nMoreover, the decreasing trend of hmwwith hshape can be\nattributed by the same reason as the height of the energy bar-\n1\n2\n3\n4\n5\n6\n72\n4\n6\n70\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s53/s49/s48/s49/s53/s50/s48/s32/s102\n/s111\n/s32/s102\n/s111/s32/s40/s84/s104/s101/s111/s114/s101/s116/s105/s99/s97/s108/s41\n/s32/s72\n/s109/s119\n/s104\n/s115/s104/s97/s112/s101/s32/s40/s84 /s41/s102\n/s111/s32/s40/s71/s72/s122/s41\n/s48/s46/s48/s50/s48/s48/s46/s48/s50/s53/s48/s46/s48/s51/s48/s104\n/s109/s119/s32/s40/s84/s41(a) (b)\n(c) (d)\n/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s52/s53/s54/s55/s56/s57/s116\n/s115/s32/s40/s110/s115/s41\n/s104\n/s115/s104/s97/s112/s101/s32/s40/s84 /s41FIG. 3. (a) Temporal evolution of mzinduced by CCMP (with hmw=\n0:035 T fixed) for different cross-sectional area, S. (b) The energy\nlandscape Ealong the line f=0. The symbols i and s represent the\ninitial state and saddle point. (c) tsas a function of hshape. (d) The\nminimal f0(red dotted) and hmw(blue square) as a function of hshape\nwhile switching time window 1 ns.\n0\n1\n32\n4\n5\n760\n1\n2\n3(a) (b)\nFIG. 4. (a) Minimal switching time tsas a function of estimated fo\nof CCMP with fixed hmwandRcorresponding to different S. (b)\nMinimal tsas a function of Gilbert damping afor different S.\nrier decreases with hshape (refers to FIG. 3(b)). For the larger\nhshape or lower height of energy barrier, the smaller microwave\nfield hmwcan induce the magnetization reversal swiftly.\nThe Gilbert damping parameter, ahas also a crucial effect\non the magnetization magnetization dynamics and hence re-\nversal process and reversal time53–55. In case of CCMP-driven5\n/s45/s49\n/s48\n/s49/s45/s49/s48/s49\n/s45/s49/s48/s49\n/s109\n/s122\n/s109/s121 /s109\n/s120/s45/s49\n/s48\n/s49/s45/s49/s48/s49\n/s45/s49/s48/s49\n/s109\n/s122\n/s109/s121\n/s109\n/s120(a) (b)\nFIG. 5. Magnetization reversal trajectories of biaxial shape (10\u0002\n10\u00028)nm3driven by CCMP for (a) a=0:010. (b) a=0:045.\nmagnetization reversal, smaller (larger) ais preferred while\nthe magnetization climbing the energy barrier (the magnetiza-\ntion goes down to stable states). Therefore, it is meaningful to\nfind the optimal afor samples of different Sat which the re-\nversal is fastest. For fixed hmw=0:035 T, using the optimal f0\nandRcorresponding to S0,S1,S2andS3, we study the CCMP-\ndriven magnetization reversal as a function of afor different\nS. FIG. 4(b) shows the dependence of switching time on the\nGilbert damping for different S. For each S, there is certain\nvalue or range of afor which the switching time is minimal.\nFor instance, the switching time is lowest at a=0:045 for\nthe sample of S1=10\u000210 nm2. To be more clear, one may\nlook at FIG. 5(a) and FIG. 5(b) which show the trajectories\nof magnetization reversal for a=0:01 and a=0:045 respec-\ntively and observed that for a=0:045, the reversal path is\nshorter. This is because, after crossing over the energy bar-\nrier, the larger damping dissipates the magnetization energy\npromptly and thus it leads to faster magnetization reversal.\nThis finding suggests that larger ashows faster magnetiza-\ntion reversal.\nIV. DISCUSSIONS AND CONCLUSIONS\nThis study investigates the CCMP-driven magnetization re-\nversal of a cubic sample at zero temperature limit and found\nthat the CCMP with significantly smaller hmw=0:035 T,\nf0=18:8 GHz and R=0:32 ns\u00001than that of DCMP (i.e.,\nhmw=0:045 T, f0=21 GHz and R= 1.6 ns\u00001) can drive fast\nmagnetization reversal. Since the frequency change of CCMP\nclosely matches the magnetization precession frequency and\nthus it leads fast magnetization reversal with lower energy-\ncost. Then we study the influence of demagnetization field =\nshape anisotropy field hshape, on magnetization reversal pro-\ncess and the optimal parameters of CCMP. Interestingly we\nfind that, with the increase of hshape, the parameters hmwand\nf0of CCMP decreases with increasing hshape.\nSo, we search a set of minimal parameters of CCMP for\nthe fast ( ts\u00181 ns) magnetization reversal of the sample 22 \u0002\n22\u00028 nm3, and estimated as hmw=0:03 T, f0=7:7 GHz\nandR=0:24 ns\u00001which are (significantly smaller than that\nof DCMP) useful for device application. This is happened\nbecause with increase of hshape, the effective anisotropy field\ndecreases and thus the energy barrier (which separates two\nstable states) decreases. In addition, it is observed that thematerials with the larger damping are better for fast magneti-\nzation reversal. There is a recent study56reported that ther-\nmal effect assists the magnetization reversal i.e., thermal ef-\nfect reduces the controlling parameters of chirp microwave\nfield pulse. Thus, it is expected that the parameters of CCMP\nalso might be reduced further at room temperature. To gener-\nate such a cosine down-chirp microwave pulse, several recent\ntechnologies46,47are available. Therefore, the strategy of the\ncosine chirp microwave chirp driven magnetization reversal\nand other findings may lead to realize the fast and low-cost\nmemory device.\nACKNOWLEDGMENTS\nThis work was supported by the Ministry of Education\n(BANBEIS, Grant No. SD2019972). X. S. W. acknowledges\nthe support from the Natural Science Foundation of China\n(NSFC) (Grant No. 11804045) and the Fundamental Research\nFunds for the Central Universities.\nAppendix A: Calculation of ˙e\nIn this Appendix, we show the details of the derivation of ˙e\nin Eq. 4. The rate of change of hmwis\n˙hmw=dhmw\ndt\n=d\ndt(hmw[cosf(t)ˆx+sinf(t)ˆy])\n=hmw[\u0000sinf(t)ˆx+cosf(t)ˆy]df\ndt\n=hmw[\u0000sinf(t)ˆx+cosf(t)ˆy]\u0014f(t)\nt\u0000d\ndt\u0012f(t)\nt\u0013\nt\u0015\nThe magnetization is given by\nm=mxˆx+myˆy\n=sinq(t)cosfm(t)ˆx+sinq(t)sinfm(t)ˆy\nwhere q(t)is the polar angle and fm(t)is the azimuthal angle\nof the magnetization m.\nSubstituting mxand˙hmwin Eq. 3, we get,\n˙e=\u0000m\u0001˙hmw\n=hmwsinq(t)[\u0000sinf(t)cosfm(t)+cosf(t)sinfm(t)]\u0001\u0014f(t)\nt\u0000d\ndt\u0012f(t)\nt\u0013\nt\u0015\n=hmwsinq(t)sin(f(t)\u0000fm(t))\u0014f(t)\nt\u0000d\ndt\u0012f(t)\nt\u0013\nt\u0015\nDefining F(t) =fm(t)\u0000f(t), we have finally\n˙e=hmwsinq(t)sinF(t)\u0014f(t)\nt\u0000d\ndt\u0012f(t)\nt\u0013\nt\u0015\nwhereh\nf(t)\nt\u0000d\ndt\u0010\nf(t)\nt\u0011\nti\nrepresents w(t).6\nREFERENCES\n1S. Sun, C. B. Murray, D. Weller, L. Folks, and A. 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Yuan,2,\u0003Ke Xia,1, 3and Zhe Yuan1,y\n1The Center for Advanced Quantum Studies and Department of Physics,\nBeijing Normal University, 100875 Beijing, China\n2Department of Physics, South University of Science and Technology of China, Shenzhen, Guangdong, 518055, China\n3Synergetic Innovation Center for Quantum E\u000bects and Applications (SICQEA),\nHunan Normal University, Changsha 410081, China\n(Dated: October 16, 2017)\nDamping in magnetization dynamics characterizes the dissipation of magnetic energy and is essen-\ntial for improving the performance of spintronics-based devices. While the damping of ferromagnets\nhas been well studied and can be arti\fcially controlled in practice, the damping parameters of an-\ntiferromagnetic materials are nevertheless little known for their physical mechanisms or numerical\nvalues. Here we calculate the damping parameters in antiferromagnetic dynamics using the gen-\neralized scattering theory of magnetization dissipation combined with the \frst-principles transport\ncomputation. For the PtMn, IrMn, PdMn and FeMn metallic antiferromagnets, the damping coef-\n\fcient associated with the motion of magnetization ( \u000bm) is one to three orders of magnitude larger\nthan the other damping coe\u000ecient associated with the variation of the N\u0013 eel order ( \u000bn), in sharp\ncontrast to the assumptions made in the literature.\nDamping describes the process of energy dissipation in\ndynamics and determines the time scale for a nonequi-\nlibrium system relaxing back to its equilibrium state.\nFor magnetization dynamics of ferromagnets (FMs), the\ndamping is characterized by a phenomenological dissipa-\ntive torque exerted on the precessing magnetization [1].\nThe magnitude of this torque that depends on material,\ntemperature and magnetic con\fgurations, has been well\nstudied in experiment [2{10] and theory [11{16].\nRecently, magnetization dynamics of antiferromagnets\n(AFMs) [17{20], especially that controlled by an electric\nor spin current [21{32], has attracted lots of attention in\nthe process of searching the high-performance spintronic\ndevices. However, the understanding of AFM dynamics,\nin particular the damping mechanism and magnitude in\nreal materials, is quite limited. Magnetization dynam-\nics of a collinear AFM can be described by two coupled\nLandau-Lifshitz-Gilbert (LLG) equations corresponding\nto the precessional motion of the two sublattices, respec-\ntively [33], i.e. ( i= 1;2)\n_mi=\u0000\rmi\u0002hi+\u000bimi\u0002_mi; (1)\nwhere\ris the gyromagnetic ratio, miis the magnetiza-\ntion direction on the i-th sublattice and _mi=@tmi.hi\nis the e\u000bective magnetic \feld on mi, which contains the\nanisotropy \feld, the external \feld and the exchange \feld\narising from the magnetization on the both sublattices.\nThe last contribution to himakes the dynamic equation\nof one sublattice coupled to the equation of the other one.\nSpeci\fcally, if the free energy of the AFM is given by the\nfollowing formF[m1;m2]\u0011\u00160MsVE[m1;m2] with the\npermeability of vacuum \u00160, the magnetization on each\nsublatticeMsand the volume of the AFM V, one has\nhi=\u0000\u000eE=\u000emi.\u000biin Eq. (1) is the damping param-\neter representing the dissipation rate of the magnetiza-tionmi. Due to the sublattice permutation symmetry,\nthe damping magnitudes of the two sublattices should\nbe equal. This approach has been used to investigate the\nAFM resonance [33, 34], temperature gradient induced\ndomain wall (DW) motion [35] and spin-transfer torques\nin an AFMjFM bilayer [36].\nAn alternative way to deal with the AFM dynamics is\nintroducing the net magnetization m\u0011m1+m2and the\nN\u0013 eel order n\u0011m1\u0000m2so that the precessional motion\nofmandncan be derived from the Lagrangian equa-\ntion [26]. The damping e\u000bect is then included arti\fcially\nwith two parameters \u000bmand\u000bnthat characterize the\ndissipation rate of mandn, respectively. This approach\nis widely used to investigate spin super\ruid in an AFM\ninsulator [37, 38], AFM nano-oscillator [39], and DW mo-\ntion induced by an electrical current [26, 40], spin waves\n[41] and spin-orbit torques [42, 43]. Using the above def-\ninitions of mandn, one can reformulate Eq. (1) and\nderive the following dynamic equations\n_n= (\rhm\u0000\u000bm_m)\u0002n+ (\rhn\u0000\u000bn_n)\u0002m;(2)\n_m= (\rhm\u0000\u000bm_m)\u0002m+ (\rhn\u0000\u000bn_n)\u0002n;(3)\nwhere hnandhmare the e\u000bective magnetic \felds exerted\nonnandm, respectively. They can also be written as\nthe functional derivative of the free energy [26, 41], i.e.\nhn=\u0000\u000eE=\u000enandhm=\u0000\u000eE=\u000em. The damping pa-\nrameters in Eqs. (1{3) have the relation \u000bn=\u000bm=\n\u000b1=2 =\u000b2=2 [36]. Indeed, the assumption \u000bm=\u000bnis\ncommonly adopted in the theoretical study of AFM dy-\nnamics with only a few exceptions, where \u000bmis ignored in\nthe current-induced skyrmion motion in AFM materials\n[44] and the magnon-driven DW motion [45]. However,\nthe underlying damping mechanism of an AFM and the\nrelation between \u000bmand\u000bnhave not been fully justi\fed\nyet [46, 47].arXiv:1710.04766v1  [cond-mat.mtrl-sci]  13 Oct 20172\nIn this paper, we generalize the scattering theory of\nmagnetization dissipation in FMs [48] to AFMs and cal-\nculate the damping parameters from \frst-principles for\nmetallic AFMs PtMn, IrMn, PdMn and FeMn. The\ndamping coe\u000ecients in an AFM are found to be strongly\nmode-dependent with \u000bmup to three orders of magni-\ntude larger than \u000bn. By analyzing the dependence of\ndamping on the disorder and spin-orbit coupling (SOC),\nwe demonstrate that \u000bnarises from SOC in analog to the\nGilbert damping in FMs, while \u000bmis dominated by the\nspin pumping e\u000bect between sublattices.\nTheory.| In analogue to the scattering theory of mag-\nnetization dissipation in FMs [48], the damping parame-\nters in AFMs, \u000bnand\u000bm, can be expressed in terms of\nthe scattering matrix. Following the previous de\fnition\nof the free energy, the energy dissipation rate of an AFM\nreads\n_E=\u0000\u00160MsV_E=\u00160MsV\u0012\n\u0000\u000eE\n\u000em\u0001_m\u0000\u000eE\n\u000en\u0001_n\u0013\n=\u00160MsV(hm\u0001_m+hn\u0001_n): (4)\nBy replacing the e\u000bective \felds hmandhnby the time\nderivative of magnetization order and N\u0013 eel order using\nEq. (2) and (3), one arrives at [49]\n_E=\u00160MsV\n\r\u0000\n\u000bn_n2+\u000bm_m2\u0001\n: (5)\nIf we place an AFM between two semi-in\fnite nonmag-\nnetic metals, the propagating electronic states coming\nfrom the metallic leads are partly re\rected and trans-\nmitted. The probability amplitudes of the re\rection and\ntransmission form the so-called scattering matrix S[50].\nFor such a scattering structure with only the order pa-\nrameter nof the AFM varying in time (see the insets of\nFig. 1), the energy loss that is pumped into the reservoir\nis given by\n_E=~\n4\u0019Tr\u0010\n_S_Sy\u0011\n=~\n4\u0019Tr\u0012@S\n@n@Sy\n@n\u0013\n_n2\u0011Dn_n2:(6)\nHere we de\fne Dn\u0011(~=4\u0019)Tr[(@S=@n)(@Sy=@n)]. Com-\nparing Eqs. (S7) and (6), we obtain\nDn=\u00160MsA\n\r\u000bnL; (7)\nwhere we replace the volume Vby the product of the\ncross-sectional area Aand the length Lof the AFM. We\ncan express \u000bmin the same manner,\nDm=\u00160MsA\n\r\u000bmL (8)\nwithDm\u0011(~=4\u0019)Tr[(@S=@m)(@Sy=@m)]. Using\nEqs. (7) and (8), we calculate the energy dissipation as a\nfunction of the length Land extract the damping param-\neters\u000bn(m)via a linear least squares \ftting. Note thatthe above formalism can be generalized to include non-\ncollinear AFM, such as DWs in AFMs, by introducing\nthe position-dependent order parameters n(r) and m(r).\nIt can also be extended for the AFMs containing more\nthan two sublattices, which may not be collinear with\none another [51]. For the latter case, one has to rede\fne\nthe proper order parameters instead of nandm[52].\nFirst-principles calculations.| The above formalism is\nimplemented using the \frst-principles scattering calcu-\nlation and is applied here in studying the damping of\nmetallic AFMs including PtMn, IrMn, PdMn and FeMn.\nThe lattice constants and magnetic con\fgurations are the\nsame as in the reported \frst-principles calculations [53].\nHere we take tetragonal PtMn as an example to illustrate\nthe computational details. A \fnite thickness ( L) of PtMn\nis connected to two semi-in\fnite Au leads along (001) di-\nrection. The lattice constant of Au is made to match\nthat of the aaxis of PtMn. The electronic structures are\nobtained self-consistently within the density functional\ntheory implemented with a minimal basis of the tight-\nbinding linear mu\u000en-tin orbitals (TB LMTOs) [54]. The\nmagnetic moment of every Mn atom is 3.65 \u0016Band Pt\natoms are not magnetized.\nTo evaluate \u000bnand\u000bm, we \frst construct a lateral\n10\u000210 supercell including 100 atoms per atomic layer in\nthe scattering region, where the atoms are randomly dis-\nplaced from their equilibrium lattice sites using a Gaus-\nsian distribution with the root-mean-square (RMS) dis-\nplacement \u0001 [15, 55]. The value of \u0001 is chosen to repro-\nduce typical experimental resistivity of the corresponding\nbulk AFM. The scattering matrix Sare obtained using a\n\frst-principles \\wave-function matching\" scheme that is\nalso implemented with TB LMTOs [56] and its derivative\nis obtained by \fnite-di\u000berence method [49].\nFigure 1(a) shows the calculated energy pumping rate\nDnof PtMn as a function of Lfornalong thecaxis\nwith \u0001=a= 0:049. The total pumping rate (solid sym-\nbols) increases linearly with increasing the volume of\nthe AFM. A linear least squares \ftting yields \u000bn=\n(0:67\u00060:02)\u000210\u00003, as plotted by the solid line. The \fnite\nintercept of the solid line corresponding to the interface-\nenhanced energy dissipation, which is essentially the spin\npumping e\u000bect at the AFM jAu interface [57, 58]. The\nN\u0013 eel order induced damping \u000bncompletely results from\nspin-orbit coupling (SOC). If we arti\fcially turn SOC o\u000b,\nthe calculated pumping rate is independent of the volume\nof the AFM indicating \u000bn= 0. This is because the spin\nspace is decoupled from the real space without SOC and\nthe energy is then invariant with respect to the direction\nofn. The spin pumping e\u000bect is nearly unchanged by\nthe SOC.\nThe energy pumping rate Dmof PtMn with nalong\nthecaxis is plotted in Fig. 1(b), where we \fnd three\nimportant features. (1) The extracted value of \u000bm=\n0:59\u00060:02, which is nearly 1000 times larger than \u000bn.\n(2) Turning SOC o\u000b only slightly increases the calculated3\n0 5 10 15 20 25 30L (nm)0102030γDm/(µ0Ms A) (nm)0 0.5 1SOC Factor00.40.8αm103αn\n0.020.030.04γDn/(µ0Ms A) (nm)PtMn1Mn2\nm1 m2 \nm1 m2 \n ξSO=0 ξSO=0(a)\n(b) ξSO≠0\n ξSO≠0abc\nFIG. 1. Calculated energy dissipation rate as a function of\nthe length of PtMn due to variation of the order parameters\nn(a) and m(b).Ais the cross-sectional area of the lat-\neral supercell. Arrows in each panels illustrate the dynamical\nmodes of the order parameters. The empty symbols are cal-\nculated without spin-orbit interaction. The inset of panel (a)\nshows atomic structure of PtMn with collinear AFM order.\nThe inset in (b) shows calculated \u000bnand\u000bmas a function\nof the scaled SOC strength. The factor 1 corresponds to the\nreal SOC strength that is determined by the derivative of the\nself-consistent potentials.\n\u000bmindicating that SOC is not the main dissipative mech-\nanism of\u000bm. The di\u000berence between the solid and empty\ncircles in Fig. 1(b) can be attributed to the SOC-induced\nvariation of electronic structure near the Fermi level. To\nsee more clearly the di\u000berent in\ruence of SOC on \u000bmand\n\u000bn, we plot in the inset of Fig. 1(b) the calculated damp-\ning parameters as a function of SOC strength. Indeed,\nas the SOC strength \u0018SOis arti\fcially tuned from its real\nvalue to zero, \u000bndecreases dramatically and tends to\nvanish at\u0018SO= 0, while\u000bmis less sensitive to \u0018SOthan\n\u000bn. (3) The intercepts of the solid and dashed lines are\nboth vanishingly small indicating that this speci\fc mode\ndoes not pump spin current into the nonmagnetic leads.\nThe pumped spin current from an AFM generally reads\nIpump\ns/n\u0002_n+m\u0002_m[58]. For the mode depicted in\nFig. 1(b), one has _n= 0 and _mkmsuch that Ipump\ns = 0.\nTo explore the disorder dependence of the damping pa-\nrameters\u000bnand\u000bm, we further perform the calculation\nby varying the RMS of atomic displacements \u0001. Fig-\nure 2(a) shows that the calculated resistivity increases\nmonotonically with increasing \u0001. The resistivity \u001acwith\nnalongcaxis is lower than \u001aawithnalongaaxis.\n0.51.01.52.0αn (10-3)(a)80160240ρ (µΩ cm)n//a(b)n//c4.6 5.4 6.2∆/a (10-2)01020AMR (%)\n4.6 5.0 5.4 5.8 6.2∆/a (10-2)0.20.40.60.8αm4 6 8 10 12σ (105Ω-1 m-1)0.20.50.8αm(b)(c)FIG. 2. Calculated resistivity (a) and damping parameters\n\u000bn(b) and\u000bm(c) of PtMn as a function of the RMS of\natomic displacements. The red squares and black circles are\ncalculated with nalongaaxis andcaxis, respectively. The\ninset of (a) shows the calculated AMR. \u000bmis replotted as a\nfunction of conductivity in the inset of (c). The blue dashed\nline illustrates the linear dependence.\nThe anisotropic magnetoresistance (AMR) de\fned by\n(\u001aa\u0000\u001ac)=\u001acis about 10%, which slightly decreases with\nincreasing \u0001, as plotted in the inset of Fig. 2(a). The\nlarge AMR in PtMn is useful for experimental detection\nof the N\u0013 eel order. The calculated AMR seems to be an\norder of magnitude larger than the reported values in lit-\nerature [59{61]. We may attribute the di\u000berence to the\nsurface scattering in thin-\flm samples and other types of\ndisorder that have been found to decrease the AMR of\nferromagnetic metals and alloys [62].\n\u000bnof PtMn plotted in Fig. 2(b) is of the order of 10\u00003,\nwhich is comparable with the magnitude of the Gilbert\ndamping of ferromagnetic transition metals [2{4, 15]. For\nnalongaaxis,\u000bnshows a weak nonmonotonic depen-\ndence on disorder, while \u000bnfornalongcaxis increases\nmonotonically. With the relativistic SOC, the electronic\nstructure of an AFM depends on the orientation of n.\nWhen nvaries in time, the occupied energy bands may\nbe lifted above the Fermi level. Then a longer relax-\nation time (weaker disorder) gives rise to a larger energy\ndissipation, corresponding to the increase in \u000bnwith de-\ncreasing \u0001 at small \u0001. It is analogous to the intraband\ntransitions accounting for the conductivity-like behavior\nof Gilbert damping at low temperature in the torque-\ncorrelation model [11, 12]. Su\u000eciently strong disorder4\nrenders the system isotropic and the variation of ndoes\nnot lead to electronic excitation but scattering of conduc-\ntion electrons by disorder still dissipates energy into the\nlattice through SOC. The higher the scattering rate, the\nlarger is the energy dissipation rate corresponding to the\ncontribution of the interband transitions [11, 12]. There-\nfore,\u000bnshares the same physical origin as the Gilbert\ndamping of metallic FMs.\nThe value of \u000bmis about three orders of magnitude\nlarger than \u000bnand it decreases monotonically with in-\ncreasing the structural disorder, as shown in Fig. 2(c).\nThis remarkable di\u000berence can be attributed to the en-\nergy involved in the dynamical motion of mandn. While\nthe precession of nonly changes the magnetic anisotropy\nenergy in an AFM, the variation of mchanges the ex-\nchange energy that is in magnitude much larger than the\nmagnetic anisotropy energy.\nPhysically, \u000bmcan be understood in terms of spin\npumping [63, 64] between the two sublattices of an AFM.\nThe sublattice m2pumps a spin current that can be ab-\nsorbed by m1resulting in a damping torque exerted on\nm1as\u000b0m1\u0002[m1\u0002(m2\u0002_m2)]. Here\u000b0is a dimen-\nsionless parameter to describe the strength of the spin\npumping. This torque can be simpli\fed to be \u000b0m1\u0002_m2\nby neglecting the high-order terms of the total magne-\ntization m. In addition, the spin pumping by m1also\ncontributes to the damping of the sublattice m1that is\nequivalent to a torque \u000b0m1\u0002_m1exerted on m1. Tak-\ning the inter-sublattice spin pumping into account, we are\nable to derive Eqs. (2) and (3) and obtain the damping\nparameters \u000bn=\u000b0=2 and\u000bm= (\u000b0+ 2\u000b0)=2 [49]. Here\n\u000b0is the intrinsic damping due to SOC for each sublat-\ntice. It is worth noting that the spin pumping strength\nwithin a metal is proportional to its conductivity [65{67].\nWe replot\u000bmas a function of conductivity in the inset\nof Fig. 2(c), where a general linear dependence is seen for\nbothnalongaaxis andcaxis.\nWe list in Table I the calculated \u001a,\u000bnand\u000bmfor\ntypical metallic AFMs including PtMn, IrMn, PdMn and\nFeMn. For IrMn, \u000bmis only 10 times larger than \u000bn,\nwhile\u000bmof the other three materials are about three\norders of magnitude larger than their \u000bn.\nTABLE I. Calculated resistivity and damping parameters for\nthe N\u0013 eel order nalongaaxis andcaxis.\nAFM n\u001a(\u0016\n cm)\u000bn(10\u00003)\u000bm\nPtMnaaxis 119 \u00065 1.60 \u00060.02 0.49 \u00060.02\ncaxis 108 \u00064 0.67 \u00060.02 0.59 \u00060.02\nIrMnaaxis 116 \u00062 10.5 \u00060.2 0.10 \u00060.01\ncaxis 116 \u00062 10.2 \u00060.3 0.10 \u00060.01\nPdMnaaxis 120 \u00068 0.16 \u00060.02 1.1 \u00060.10\ncaxis 121 \u00068 1.30 \u00060.10 1.30 \u00060.10\nFeMnaaxis 90 \u00061 0.76 \u00060.04 0.38 \u00060.01\ncaxis 91 \u00061 0.82 \u00060.03 0.38 \u00060.01\n0 10 20 30 40Hext (kOe)00.040.080.12∆ω (THz)2.0 2.4ω (THz)-1.6 -1.2-Im χ (arb.units)Hext=20 kOe\nαm=αnαm=103αnm1 m2 \nm1 m2 \nHextHext ωL\n//// ωR ωL ωRFIG. 3. Linewidth of AFMR as a function of the external\nmagnetic \feld. The black dashed lines and red solid lines\nare calculated with \u000bm=\u000bnand\u000bm= 103\u000bn, respectively.\nInset: the imaginary part of susceptibility as a function of\nthe frequency for the external magnetic \feld Hext= 20 kOe\nand\u000bm= 103\u000bn. The cartoons illustrate the corresponding\ndynamical modes. Here we use HE= 103kOe,HA= 5 kOe\nand\u000bn= 0:001.\nAntiferromagnetic resonance.| Ke\u000ber and Kittel for-\nmulated antiferromagnetic resonance (AFMR) with-\nout damping [33] and determined the resonant fre-\nquencies that depend on the external \feld Hext, ex-\nchange \feld HEand anisotropy \feld HA,!res=\n\rh\nHext\u0006p\nHA(2HE+HA)i\n. Here we follow their ap-\nproach, in which Hextis applied along the easy axis and\nthe transverse components of m1andm2are supposed\nto be small. Taking both the intrinsic damping due to\nSOC and spin pumping between the two sublattices into\naccount, we solve the dynamical equations of AFMR and\n\fnd the frequency-dependent susceptibility \u001f(!) that is\nde\fned by n?(!) =\u001f(!)\u0001h?(!). Here n?andh?\nare the transverse components of the N\u0013 eel order and mi-\ncrowave \feld, respectively. The imaginary part of the\ndiagonal element of \u001f(!) withHext= 20 kOe is plotted\nin the inset of Fig. 3, where two resonance modes can be\nidenti\fed. The precessional modes for the positive ( !R)\nand negative frequency ( !L) are schematically depicted\nin Fig. 3. The linewidth of the AFMR \u0001 !can be deter-\nmined from the imaginary part of the (complex) eigen-\nfrequency [68] by solving det j\u001f\u00001(!)j= 0 and is plotted\nin Fig. 3 as a function of Hext. Without Hext, the two\nmodes have the same linewidth. A \fnite external \feld\nincreases the linewidth of !Rand decreases that of !L,\nboth linearly. By including the spin pumping between\ntwo sublattices, both the linewidth at Hext= 0 and the\nslope of \u0001!as a function of Hextincrease by a factor\nof about 3.5. It indicates that the spin pumping e\u000bect\nbetween the two sublattices plays an important role in\nthe magnetization dynamics of metallic AFMs.5\nConclusions.| We have generalized the scattering the-\nory of magnetization dissipation in FMs to be applicable\nfor AFMs. Using \frst-principles scattering calculation,\nwe \fnd the damping parameter accompanying the mo-\ntion of magnetization ( \u000bm) is generally much larger than\nthat associated with the motion of the N\u0013 eel order ( \u000bn)\nin metallic AFMs PtMn, IrMn, PdMn and FeMn. While\n\u000bnarises from the spin-orbit interaction, \u000bmis mainly\ncontributed by the spin pumping between the two sublat-\ntices in an AFM via exchange interaction. 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Saib, Modeling and design of microwave devices based on ferromagnetic nanowires\n(Presses Universitaires du Louvain, 2004).1\nSupplementary Material for \\Mode-Dependent Damping in Metallic\nAntiferromagnets Due to Inter-Sublattice Spin Pumping\"\nQian Liu,1;\u0003H. Y. Yuan,2;\u0003Ke Xia,1;3and Zhe Yuan1;y\n1The Center for Advanced Quantum Studies and Department of Physics, Beijing Normal University, Beijing 100875, China\n2Department of Physics, Southern University of Science and Technology of China, Shenzhen, Guangdong, 518055, China\n3Synergetic Innovation Center for Quantum E\u000bects and Applications (SICQEA), Hunan Normal University, Changsha\n410081, China\nIn the Supplemental Material, we present the detailed derivation of the energy pumping arising from antiferromagnetic\ndynamics, the implementation of calculating the derivatives of scattering matrix and derivation of dynamic equations of nand\nmincluding the spin pumping between sublattices.\nDERIVATION OF ENERGY DISSIPATION IN ANTIFERROMAGNETIC DYNAMICS\nWe consider a collinear antiferromagnet (AFM) with two sublattices, both of which have the magnetization Ms.\nThe magnetization directions are denoted by the unit vectors m1andm2. Then we are able to de\fne the total\nmagnetization m=m1+m2and the N\u0013 eel order parameter n=m1\u0000m2. The dynamic equations of mandncan\nbe written as [S1, S2]\n_m=\u0000\r(m\u0002hm+n\u0002hn) +\u000bmm\u0002_m+\u000bnn\u0002_n; (S1)\n_n=\u0000\r(m\u0002hn+n\u0002hm) +\u000bmn\u0002_m+\u000bnm\u0002_n: (S2)\nHerehmandhnare the e\u000bective \felds acting on the total magnetization and the N\u0013 eel order. Speci\fcally, if the free\nenergy is written as F=\u00160MsVE, where\u00160is the vacuum permeability, Vis the volume of the AFM, and Eis a\nreduced free energy density, one has [S1]\nhm=\u0000\u000eE\n\u000em;andhn=\u0000\u000eE\n\u000en: (S3)\nIn Eqs. (S1) and (S2), \u000bmand\u000bnare used to characterize the damping due to the variation of the magnetization and\nthe N\u0013 eel order, respectively.\nIfmandnare the only time-varying parameters in the system, the energy dissipation can be represented by\n_E=\u0000_F=\u0000\u00160MsV_E\n=\u00160MsV\u0014\n_m\u0001\u0012\n\u0000\u000eE\n\u000em\u0013\n+_n\u0001\u0012\n\u0000\u000eE\n\u000en\u0013\u0015\n=\u00160MsV(_m\u0001hm+_n\u0001hn): (S4)\nWe then insert the dynamic Eqs. (S1) and (S2) into the above Eq. (S4) and obtain\n_E\n\u00160MsV= [\u0000\r(m\u0002hm+n\u0002hn) +\u000bmm\u0002_m+\u000bnn\u0002_n]\u0001hm\n+[\u0000\r(m\u0002hn+n\u0002hm) +\u000bmn\u0002_m+\u000bnm\u0002_n]\u0001hn\n=\u0000\rn\u0002hn\u0001hm+ (\u000bmm\u0002_m+\u000bnn\u0002_n)\u0001hm\u0000\rn\u0002hm\u0001hn+ (\u000bmn\u0002_m+\u000bnm\u0002_n)\u0001hn\n= (\u000bmm\u0002_m+\u000bnn\u0002_n)\u0001hm+ (\u000bmn\u0002_m+\u000bnm\u0002_n)\u0001hn\n=\u000bm(m\u0002_m\u0001hm+n\u0002_m\u0001hn) +\u000bn(n\u0002_n\u0001hm+m\u0002_n\u0001hn)\n=\u000bm(hm\u0002m\u0001_m+hn\u0002n\u0001_m) +\u000bn(hm\u0002n\u0001_n+hn\u0002m\u0001_n)\n=\u000bm(hm\u0002m+hn\u0002n)\u0001_m+\u000bn(hm\u0002n+hn\u0002m)\u0001_n\n=\u000bm1\n\r(_m\u0000\u000bmm\u0002_m\u0000\u000bnn\u0002_n)\u0001_m+\u000bn1\n\r(_n\u0000\u000bmn\u0002_m\u0000\u000bnm\u0002_n)\u0001_n\n=\u000bm\n\r_m2\u0000\u000bm\u000bn\n\rn\u0002_n\u0001_m+\u000bn\n\r_n2\u0000\u000bn\u000bm\n\rn\u0002_m\u0001_n\n=1\n\r\u0000\n\u000bm_m2+\u000bn_n2\u0001\n: (S5)2\nLeft Lead Right Lead Scattering Region \n!O!IOIm1 m2 \nFIG. S1. Schematic illustration of the scattering geometry that is used in the \frst-principles calculations. Since both the\nleft and right leads are semi-in\fnite with periodic crystalline structure, the propagating (incoming and outgoing) Bloch states\ncan be obtained by solving the Kohn-Sham equation self-consistently. Then the transmission and re\rection coe\u000ecients can be\nsolved using the numerical technique called \\wave function matching\" [S3].\nTherefore the energy dissipation during antiferromagnetic dynamics can be eventually obtained\n_E=\u00160MsV\n\r\u0000\n\u000bm_m2+\u000bn_n2\u0001\n: (S6)\nCALCULATING THE DERIVATIVE OF SCATTERING MATRIX\nNoting that the energy dissipation in a scattering geometry, i.e. the left lead{scattering region{the right lead (see\nFig. S1), can be written in terms of the parametric pumping [S4]\n_E=~\n4\u0019Tr\u0010\n_S_Sy\u0011\n: (S7)\nHereSis the scattering matrix. Supposing only the magnetic order \u0010(\u0010=mor\u0010=n) of the system is varying in\ntime, one can rewrite Eq. (S7) as\n_E=~\n4\u0019Tr\u0012@S\n@\u0010@Sy\n@\u0010\u0013\n_\u00102\u0011D\u0010_\u00102: (S8)\nThe quantity D\u0010is generally a positive-de\fnite and symmetric tensor [S5] with its elements de\fned by\nDij\n\u0010=~\n4\u0019Tr\u0012@S\n@\u0010i@Sy\n@\u0010j\u0013\n: (S9)\nNoting that @S=@\u0010i,@Sy=@\u0010jand their product are all matrices, so we rewrite Eq. (S9) in terms of the speci\fc matrix\nelements as\nDij\n\u0010=~\n4\u0019X\n\u0016\u0012@S\n@\u0010i@Sy\n@\u0010j\u0013\n\u0016\u0016=~\n4\u0019X\n\u0016X\n\u0017@S\u0016\u0017\n@\u0010i@\u0000\nSy\u0001\n\u0017\u0016\n@\u0010j=~\n4\u0019X\n\u0016X\n\u0017@S\u0016\u0017\n@\u0010i\u0012@S\u0016\u0017\n@\u0010j\u0013\u0003\n: (S10)\nIn particular, for i=j, we have the diagonal elements of D\u0010\nDii\n\u0010=~\n4\u0019X\n\u0016;\u0017\f\f\f\f@S\u0016\u0017\n@\u0010i\f\f\f\f2\n; (S11)\nwhich is a real number. All the remaining task is to numerically calculate the derivatives of the scattering matrix\nelements@S\u0016\u0017=@\u0010i.\nIn the following, we take \u0010=nas an example and illustrate the calculation of @S\u0016\u0017=@\u0010i. Considering the N\u0013 eel\norder along z-axis, i.e. m1=\u0000m2= ^zandn= 2^z, one can calculate the scattering matrix S(n). Then we add an\nin\fnitesimal transverse component \u0001 n=\u0011^xonto the N\u0013 eel order so that the new N\u0013 eel order becomes n0= 2^z+\u0011^x.\n(In practice, we \fnd that the calculated results are well converged with \u0011in the range of 10\u00003{10\u00005.) Under such a\nmagnetic con\fguration, we redo the scattering calculation to \fnd another scattering matrix S(n0). The derivatives of\nthe matrix element S\u0016\u0017can be obtained by\n@S\u0016\u0017\n@nx=S0\n\u0016\u0017\u0000S\u0016\u0017\n\u0011: (S12)3\nIn the same manner, we can \fnd another scattering matrix S00atn00= 2^z+\u0011^yand consequently we have\n@S\u0016\u0017\n@ny=S00\n\u0016\u0017\u0000S\u0016\u0017\n\u0011: (S13)\nFinally, we \fnd that the calculated o\u000b-diagonal elements Dxy\nn(m)andDyx\nn(m)are much smaller than the diagonal elements\nDxx\nn(m)andDyy\nn(m). The latter two are nearly the same. So we take their average in practice, i.e. Dn= (Dxx\nn+Dyy\nn)=2\nandDm= (Dxx\nm+Dyy\nm)=2.\nDYNAMICAL EQUATIONS WITH INTER-SUBLATTICE SPIN PUMPING\nWe start from the coupled dynamical equations of an AFM with the sublattice index i= 1;2,\n_mi=\u0000\rmi\u0002hi+\u000b0mi\u0002_mi: (S14)\nHerehiis the e\u000bective \feld exerted on mi, which can be calculated from the functional derivative of the free energy\nFas\nhi=\u00001\n\u00160MsV\u000eF\n\u000emi: (S15)\n\u000b0is the damping parameter, which must be equal for m1andm2because of the permutation symmetry. Now we\nconsider the spin pumping e\u000bect that discussed in the main text. The spin pumping by the sublattice m1contributes\na dissipative torque \u000b0m1\u0002_m1that is exerted on m1. Here\u000b0is a dimensionless parameter to quantify the magnitude\nof the inter-sublattice spin pumping. The pumped spin current by m1can be absorbed by m2resulting in a damping-\nlike torque m2\u0002[m2\u0002(\u000b0m1\u0002_m1)]\u0019\u000b0m2\u0002_m1, which is exerted on m2. In the same manner, we can identify\ntwo torques due to the spin pumping of m2:\u000b0m1\u0002_m2exerted on m1and\u000b0m2\u0002_m2exerted on m2. Eventually,\nwe obtain the coupled dynamical equations by including the inter-sublattice spin pumping as\n_m1=\u0000\rm1\u0002h1+ (\u000b0+\u000b0)m1\u0002_m1+\u000b0m1\u0002_m2;\n_m2=\u0000\rm2\u0002h2+ (\u000b0+\u000b0)m2\u0002_m2+\u000b0m2\u0002_m1: (S16)\nThe above form of the dynamical equations can be rigorously derived using the Rayleigh functional to describe the\ndissipation [S6].\nIn the following, we rewrite Eq. (S16) into the dynamical equations of the total magnetization m=m1+m2and\nthe N\u0013 eel order n=m1\u0000m2. The e\u000bective \feld hican be transformed as\nh1=\u00001\n\u00160MsV\u000eF\n\u000em1=\u00001\n\u00160MsV\u0012\u000eF\n\u000em@m\n@m1+\u000eF\n\u000en@n\n@m1\u0013\n=hm+hn;\nh2=\u00001\n\u00160MsV\u000eF\n\u000em2=\u00001\n\u00160MsV\u0012\u000eF\n\u000em@m\n@m2+\u000eF\n\u000en@n\n@m2\u0013\n=hm\u0000hn; (S17)\nwhere we have de\fned\nhm=\u00001\n\u00160MsV\u000eF\n\u000em;\nhn=\u00001\n\u00160MsV\u000eF\n\u000en: (S18)\nThen we \fnd\n_m=_m1+_m2=\u0000\r(m1\u0002h1+m2\u0002h2) + (\u000b0+\u000b0) (m1\u0002_m1+m2\u0002_m2) +\u000b0(m1\u0002_m2+m2\u0002_m1):(S19)\nUsing Eq. (S17), the \frst term in the right-hand side of Eq. (S19) can be simpli\fed as\n\u0000\r(m1\u0002h1+m2\u0002h2) =\u0000\r\u0014m+n\n2\u0002(hm+hn) +m\u0000n\n2\u0002(hm\u0000hn)\u0015\n=\u0000\r(m\u0002hm+n\u0002hn):(S20)4\nThe second and the third terms in the right-hand side of Eq. (S19) can be simpli\fed, respectively, as\n(\u000b0+\u000b0) (m1\u0002_m1+m2\u0002_m2) = (\u000b0+\u000b0)\u0012m+n\n2\u0002_m+_n\n2+m\u0000n\n2\u0002_m\u0000_n\n2\u0013\n=\u000b0+\u000b0\n2(m\u0002_m+n\u0002_n);\n(S21)\nand\n\u000b0(m1\u0002_m2+m2\u0002_m1) =\u000b0\u0012m+n\n2\u0002_m\u0000_n\n2+m\u0000n\n2\u0002_m+_n\n2\u0013\n=\u000b0(m\u0002_m\u0000n\u0002_n): (S22)\nFinally, Eq. (S19) is rewritten as\n_m=\u0000\r(m\u0002hm+n\u0002hn) +\u0010\u000b0\n2+\u000b0\u0011\nm\u0002_m+\u000b0\n2n\u0002_n: (S23)\nThe dynamical equation of the N\u0013 eel order ncan be obtained in the same way\n_n=\u0000\r(m\u0002hn+n\u0002hm) +\u0010\u000b0\n2+\u000b0\u0011\nn\u0002_m+\u000b0\n2m\u0002_n: (S24)\nComparing Eqs. (S23) and (S24) with Eqs. (S1) and (S2), we can identify the relations of the damping parameters,\ni.e.\n\u000bm=\u000b0\n2+\u000b0;and\u000bn=\u000b0\n2: (S25)\nThe above relations naturally show the spin pumping e\u000bect and is consistent with our \frst-principles calculations.\n\u0003These authors contributed equally to this work.\nyCorresponding author: zyuan@bnu.edu.cn\n[S1] K. M. D. Hals, Y. Tserkovnyak, and A. Brataas, \\Phenomenology of current-induced dynamics in antiferromagnets,\"\nPhys. Rev. Lett. 106, 107206 (2011).\n[S2] E. V. Gomonay and V. M. Loktev, \\Spintronics of antiferromagnetic systems (review article),\" Low Temp. Phys. 40, 17\n(2014).\n[S3] K. Xia, M. Zwierzycki, M. Talanana, P. J. Kelly, and G. E. W. Bauer, \\First-principles scattering matrices for spin\ntransport,\" Phys. Rev. B 73, 064420 (2006).\n[S4] J. E. Avron, A. Elgart, G. M. Graf, and L. Sadun, \\Optimal quantum pumps,\" Phys. Rev. Lett. 87, 236601 (2001).\n[S5] A. Brataas, Y. Tserkovnyak, and G. E. W. Bauer, \\Magnetization dissipation in ferromagnets from scattering theory,\"\nPhys. Rev. B 84, 054416 (2011).\n[S6] H. Y. Yuan, Qian Liu, Ke Xia, Zhe Yuan, and X. R. Wang, \\Proper dissipative torques in antiferromagnetic dynamics,\"\nunpublished (2017)."    },    {        "title": "1706.04670v2.Temperature_dependent_Gilbert_damping_of_Co2FeAl_thin_films_with_different_degree_of_atomic_order.pdf",        "content": "1 \n Temperature -dependent Gilbert damping of Co 2FeAl thin films with different degree of \natomic order  \nAnkit Kumar1*, Fan  Pan2,3, Sajid  Husain4, Serkan  Akansel1, Rimantas  Brucas1, Lars \nBergqvist2,3, Sujeet Chaudhary4, and Peter  Svedlindh1# \n \n1Department of Engineering Sciences, Uppsala University, Box 534, SE -751 21 Uppsala, \nSweden \n2Department of Applied Physics, School of Engineering Sciences, KTH Royal Institute of \nTechnology, Electrum 229, SE -16440 Kista, Sweden \n3Swedish e -Science Research Center, KTH Roy al Institute of Technology, SE -10044 \nStockholm, Sweden \n4Department of Physics, Indian Institute of Technology Delhi, New Delhi -110016, India  \n \nABSTRACT  \nHalf-metallicity and low magnetic damping are perpetually sought for in spintronics materials \nand full He usler alloys in this respect provide outstanding properties . However, it is \nchallenging to obtain the well -ordered half-metallic phase in as -deposited full Heusler alloys \nthin films and theory has struggled to establish  a fundamentals understanding of the \ntemperature dependent Gilbert damping in these systems. Here we present a study of the \ntemperature dependent Gilbert damping of differently ordered as -deposited Co 2FeAl full \nHeusler alloy thin films. The sum of inter - and intraband electron scattering in conjunction \nwith the finite electron lifetime in Bloch states govern the Gilbert damping for the well -\nordered phase in contrast to the damping of partially -ordered and disordered phases which is \ngoverned by interband electronic scat tering alone. These results, especially the ultralow room \ntemperature intrinsic damping observed for the well -ordered phase provide new fundamental \ninsight s to the physical origin of the Gilbert damping in full Heusler alloy thin films.   \n 2 \n INTRODUCTION \nThe Co-based full Heusler alloys have gained massive attention over the last decade due to \ntheir high Curie temperature and half-metallicity; 100% spin polarization of the density of \nstates at the Fermi level [1 -2]. The room temperature half- metallicity and lo w Gilbert \ndamping make them ideal candidates for magnetoresistive and thermoelectric spintronic \ndevices [3]. Co2FeAl (CFA), which is one of the most studied Co-based Heusler alloys , \nbelongs to the 𝐹𝐹𝐹𝐹 3𝐹𝐹 space group, exhibits half-metallicity  and a high C urie temperature \n(1000 K) [2, 4]. In CFA, half-metallicity is the result of hybridization between the d orbitals \nof Co and Fe. The d orbitals of Co hybridize resulting in bonding (2e g and 3t 2g) and non-\nbonding hybrids (2e u and 3t 1u). The bonding hybrids of Co further hybridise with the d \norbitals of Fe yielding bonding and anti -bonding hybrids. However, the non-bonding hybrids \nof Co cannot hybridise with the d orbitals of Fe. The half-metallic gap arises from the \nseparation of non-bonding states, i.e. the conduction band of e u hybrids and the valence band \nof t 1u hybrids [5, 6]. However, chemical or atomic disorder modifies the band hybridization \nand results in a reduc ed half-metallicity in CFA. The ordered phase of CFA is the L2 1 phase, \nwhich is half -metallic [7]. The partially ordered B2 phase forms when the Fe and Al atoms \nrandomly share their sites, while the disordered phase forms when Co, Fe, and Al atoms \nrandomly share all the sites [5-8]. These chemical disorders strongly influence the physical \nproperties and result in additional states at the Fermi level therefore reducing the half-\nmetallicity or spin polarization [7, 8]. It is challenging to obtain the ordered L2 1 phase of \nHeusler alloys in as-deposited films, which is expecte d to possess the lowest Gilbert damping \nas compared to the other phases [4, 9-11]. Therefore, in the last decade several attempts have \nbeen made to grow the ordered phase of CFA thin films employing different methods [4, 9-\n13]. The most successful attempts  used post -deposition annealing to reduce the anti -site \ndisorder by a thermal activation process [4]. The observed value of the Gilbert damping for \nordered thin films was found to lie in the range of 0.001-0.004 [7-13]. However, the \nrequirement of post -deposition annealing might not be compatible with the process constraints \nof spintronics and CMOS devices. The annealing treatment requirement for the formation of \nthe ordered phase can be circumvented by employing energy enhanced growth mechanisms \nsuch as io n beam sputtering where the sputtered species carry substantially larger  energy, ~20 \neV , compared to other deposition techniques [14, 15]. This higher energy of the sputtered \nspecies enhances the ad -atom mobility during coalescence of nuclei in the initial  stage of the \nthin film growth, therefore enabling the formation of the ordered phase. Recently we have 3 \n reported growth of the ordered CFA phase on potentially advantageous Si substrate  using ion \nbeam sputtering. The samples deposited in the range of 300°C to 500°C substrate temperature \nexhibited nearly equivalent I(002)/I(004) Bragg diffraction intensity peak ratio, which \nconfirms at least B2 order ed phase as it is difficult to identify  the formation of the L2 1 phase \nonly by X -ray diffraction analysis [16] .  \nDifferent theoretical approaches have been employed to calculate the Gilbert damping in Co -\nbased full Heusler alloys, including first principle calculations on the ba sis of (i) the torque \ncorrelation  model [17], (ii) the fully relativistic Korringa -Kohn-Rostoker model in \nconjunction with the coherent potential approximation and the linear response formalism [8] , \nand (iii) an approach considering different exchange correlation effects using both the local \nspin density approximation including the Hubbard  U and the local spin density approximation \nplus the dynamical mean field theory approximation [7]. However, very little is known about \nthe temperature dependence of the Gilbert damping in differently ordered Co-based Heusler \nalloys and a unifying conse nsus between theoretical and experimental results is still lacking. \nIn this study we report the growth of differently ordered  phases, varying from disordered to \nwell-ordered phases, of as -deposited CFA thin films grown on Si employing ion beam \nsputtering a nd subsequently the detailed temperature dependent measurements of the Gilbert \ndamping. The observed increase in intrinsic Gilbert damping with decreasing temperature in \nthe well -ordered sample is in contrast to the continuous decrease in intrinsic Gilbert  damping \nwith decreasing temperature observed for partially ordered and disordered phases. These \nresults are satisfactorily explained by employing spin polarized relativistic Korringa -Kohn-\nRostoker band structure calculations in combination with the local spin density \napproximation.  \nSAMPLES & METHODS  \nThin films of CFA were deposited on Si substrates at various growth temperatures using ion \nbeam sputtering system operating at 75W RF ion-source power ( 𝑃𝑃𝑖𝑖𝑖𝑖𝑖𝑖). Details of the \ndeposition process as well as st ructural and magnetic properties of the films have been \nreported elsewhere [16]. In the present work to study the temperature dependent Gilbert \ndamping of differently ordered phases (L2 1 and B2) we have chosen CFA thin films deposited \nat 573K, 673K and 773K substrate temperature ( 𝑇𝑇𝑆𝑆) and the corresponding samples are named \nas LP573K, LP673K and LP773K, respectively. The sample thickness was kept constant at 50 \nnm and the samples were capped with a 4 nm thick Al layer. The capping layer protects the \nfilms by forming a 1.5 nm thin protective layer of Al 2O3. To obtain the A2 disordered CFA 4 \n phase, the thin film was deposited at 300K on Si employing 100W ion-source power, this \nsample is referred to  as HP300K. Structural and magnetic properties of this film are presented \nin Ref. [18]. The absence of the (200) diffraction peak in the HP300K sample [18] reveals that \nthis sample exhibits the A2 disordered structure. The appearance of the (200) pea k in the LP \nseries samples clearly indicates at least formation of B2 order [16]. Employing the Webster model along with the analysis approach developed by Takakura et al. [19] we have calculated \nthe degree of B2 ordering in the samples, S\nB2= �I200 I220⁄\nI200full orderI220full order�� , where \nI200 I220⁄  is the experimentally obtained intensity ratio of the (200) and (220) diffractions and \nI200full orderI220full order⁄  is the theoretically calculated intensity ratio for fully ordered B2 structure \nin polycrystalline films  [20]. The estimated values of SB2 for the LP573, LP673, and LP773 \nsamples are found to be ∼  90 %, 90% and 100%, respectively , as presented in Ref.  [20]. The \nI200 I400⁄  ratio of the (200) and (400) diffraction peaks for all LP series samples is ∼  30 %, \nwhich compares well with  the theoretical value for perfect B2 order [21, 22]. Here it is \nimportant to note that the L21 ordering parameter, SL21, will take different values depending \non the degree of B2 ordering. S L21 can be calculated from the I111 I220⁄  peak ratio in \nconjunction with the SB2 ordering parameter [19]. However, in the recorded grazing incident \nXRD spectra on the polycrystalline LP samples (see Fig. 1 of Ref. [16]) we did not observe \nthe (111) peak. This could be attributed to the fact t hat theoretical intensity of this peak is only \naround two percent of the (220) principal peak. The appearance of this peak is typically \nobserved in textured/columnar thicker films [19, 23 ]. Therefore, here using the experimental \nresults of the Gilbert damping, Curie temperature and saturation magnetization, in particular employing the temperature dependence of the Gilbert damping that is very sensitive to the \namount of site disorder in CFA films, and comparing with corresponding results obtained \nfrom first principle calculations, we provide a novel method for determining the type of \ncrystallographic ordering in full Heusler alloy thin films.  \nThe observed values of the saturation magnetization ( µ0MS) and coercivity ( µ0Hci), taken \nfrom Refs. [16, 18] are presented in Table I. The temperature dependence of the magnetization \nwas recorded in the high temperature region (300–1000K) using a vibrating sample \nmagnetometer i n an external magnetic field of 𝜇𝜇 0𝐻𝐻=20 mT. An ELEXSYS EPR \nspectrometer from Bruker equipped with an X -band resonant cavity was used for angle \ndependent in-plane ferromagnetic resonance (FMR) measurements . For studying the 5 \n temperature dependent spin dynamics in the magnetic thin films, an in-house built out -of-\nplane FMR setup was used. The set up, using a Quantum Design Physical Properties \nMeasurement System covers the temperature range 4 – 350 K and the magnetic field range \n±9T. The system employs an Agilent N5227A PNA network analyser covering the frequency \nrange 1 – 67 GHz and an in-house made coplanar waveguide. The layout of the system is \nshown in Fig. 1. The complex transmission coefficient ( 𝑆𝑆21) was recorded as a function of \nmagnetic field for different frequencies in the range 9-20 GHz and different temperatures in \nthe range 50-300 K.  All FMR measurements were  recorded keeping constant 5 dB power.  \nTo calculate the Gilbert damping, we have the used the torque –torque correlation model [7, \n24], which includes both intra - and interband transitions. The electronic structure was \nobtained from the spin polarized relativistic Korringa -Kohn-Rostoker (SPR- KKR) band \nstructure method [24, 25] and the local spin density approximation (LSDA) [26] was used for \nthe exchange correlation potential. Relativistic effects were taken into account by solving the Dirac equation for the electronic states, and the atomic sphere approximation (ASA) was employed for the shape of potentials. The experimental bulk value of the lattice constant [27] \nwas used. The angular momentum cut -off of 𝑙𝑙\n𝑚𝑚𝑚𝑚𝑚𝑚 =4 was used in the mu ltiple -scattering \nexpansion.  A k-point grid consisting of ~1600 points in the irreducible Brillouin zone was \nemploye d in the self -consistent calculation while a substantially more dense grid of ~60000  \npoints was employe d for the Gilbert damping calculation.  The exchange parameters 𝐽𝐽 𝑖𝑖𝑖𝑖 \nbetween the atomic magnetic moments were calculated using the magnetic force theorem \nimplemented in the Liechtenstein -Katsnelson -Antropov-Gubanov (LKAG) formalism [28, 29] \nin order to construct a parametrized mod el Hamiltonian. For the B2 and L2 1 structures, the \ndominating exchange interactions were found to be between the Co and Fe atoms, while in A2 the Co-Fe and Fe -Fe interactions are of similar size. Finite temperature properties such as the \ntemperature dependent magnetization was obtained by performing Metropolis Monte Carlo \n(MC) simulations [30] as implemented in the UppASD software [31,  32] using the \nparametrized Hamiltoni an. The coherent potential approximation (CPA) [33, 34] was ap plied \nnot only for the treatment of the chemical disorder of the system, but also used to include the \neffects of quasi -static lattice displacement and spin fluctuations in the calculation of the \ntemperature dependent Gilbert damping [35–37] on the basis of linear response theory [38].  \nRESULTS & DISCUSSION  \nA. Magnetization vs. temperature measurements  6 \n Magnetization measurements were performed with the ambition to extract values for the \nCurie temperature ( 𝑇𝑇𝐶𝐶) of CFA films with different degree of atomic order; the results a re \nshown in Fig. 2(a). Defining 𝑇𝑇𝐶𝐶 as the inflection point in the magnetization vs. temperature \ncurve, the observed values are found to be 810  K, 890 K and 900 K for the LP573K, LP773K \nand LP673K samples, respectively. The 𝑇𝑇𝐶𝐶 value for the HP300K sample is similar to the \nvalue obtained for LP573. Using the theoretically calculated exchange interactions, 𝑇𝑇𝐶𝐶 for \ndifferent degree of atomic order in CFA varying from B2 to L2 1 can be calculated using MC \nsimul ations. The volume was kept fixed as the degree of order varied between B2 and L2 1 \nand the data presented here represent the effects of differently ordered CFA phases. To obtain \n𝑇𝑇𝐶𝐶 for the different phases, the occupancy of Fe atoms on the Heusler alloy 4a sites was varied \nfrom 50% to 100%, corresponding to changing the structure from B2 to L2 1. The estimated \n𝑇𝑇𝐶𝐶 values , cf. Fig. 2 (b), monotonously increases from 𝑇𝑇 𝐶𝐶=810 K (B2) to 𝑇𝑇𝐶𝐶=950 K \n(L2 1). A direct comparison between experimental and calculated  𝑇𝑇𝐶𝐶 values is hampered by the \nhigh temperature (beyond 800K) induced structural transition from well -ordered to partially -\nordered CFA phase which interferes with the magnetic transition [39, 40]. The irreversible \nnature of the recorded magnetization vs . temperature curve  indicates a distortion of structure \nfor the ordered phase during measurement , even though interface alloying at elevated \ntemperature cannot be ruled out . The experimentally observe d 𝑇𝑇𝐶𝐶 values are presented in \nTable I.  \nB. In-plane angle dependent FMR measurements  \nIn-plane angle dependent FMR measurements were performed at 9.8 GHz frequency for all \nsamples; the resonance field 𝐻𝐻𝑟𝑟 vs. in -plane angle  𝜙𝜙𝐻𝐻 of the applied magnetic field is plotted \nin Fig. 3. The  experimental results have been fitted using the expression [41],  \n𝑓𝑓=\n𝑔𝑔∥𝜇𝜇𝐵𝐵𝜇𝜇0\nℎ��𝐻𝐻𝑟𝑟cos(𝜙𝜙𝐻𝐻−𝜙𝜙𝑀𝑀)+2𝐾𝐾𝑐𝑐\n𝜇𝜇0𝑀𝑀𝑠𝑠cos4(𝜙𝜙𝑀𝑀−𝜙𝜙𝑐𝑐)+2𝐾𝐾𝑢𝑢\n𝜇𝜇0𝑀𝑀𝑠𝑠cos2(𝜙𝜙𝑀𝑀−𝜙𝜙𝑢𝑢)��𝐻𝐻𝑟𝑟cos(𝜙𝜙𝐻𝐻−\n𝜙𝜙𝑀𝑀)+ 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒+𝐾𝐾𝑐𝑐\n2𝜇𝜇0𝑀𝑀𝑠𝑠(3+cos4(𝜙𝜙𝑀𝑀−𝜙𝜙𝑐𝑐)+2𝐾𝐾𝑢𝑢\n𝜇𝜇0𝑀𝑀𝑠𝑠cos2 (𝜙𝜙𝑀𝑀−𝜙𝜙𝑢𝑢)��12�\n, (1) \nwhere 𝑓𝑓 is resonance frequency , 𝜇𝜇𝐵𝐵 is the Bohr magneton  and ℎ is Planck constant . 𝜙𝜙𝑀𝑀, 𝜙𝜙𝑢𝑢 \nand 𝜙𝜙𝑐𝑐 are the in -plane directions of the magnetization, uniaxial anisotropy and cubic \nanisotropy, respectively , with respect to the [100] direction of the Si substrate . 𝐻𝐻𝑢𝑢=2𝐾𝐾𝑢𝑢\n𝜇𝜇0𝑀𝑀𝑠𝑠 and \n𝐻𝐻𝑐𝑐=2𝐾𝐾𝑐𝑐\n𝜇𝜇0𝑀𝑀𝑠𝑠 are the in-plane uniaxial and cubic anisotropy fields , respectively, and 𝐾𝐾𝑢𝑢 and 𝐾𝐾𝑐𝑐 7 \n are the uniaxial and cubic magnetic anisotrop y constant s, respectively, 𝑀𝑀𝑠𝑠 is the saturation \nmagnetization and 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒 is the effective magnetization . By considering 𝜙𝜙 𝐻𝐻 ∼ 𝜙𝜙𝑀𝑀, 𝐻𝐻𝑢𝑢 and 𝐻𝐻𝑐𝑐 \n<<𝐻𝐻𝑟𝑟<< 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒, equation (1) can be simplified as:  \n𝐻𝐻𝑟𝑟=�ℎ𝑒𝑒\n𝜇𝜇0𝑔𝑔∥𝜇𝜇𝐵𝐵�21\n𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒−2𝐾𝐾𝑐𝑐\n𝜇𝜇0𝑀𝑀𝑠𝑠cos4(𝜙𝜙𝐻𝐻−𝜙𝜙𝑐𝑐)−2𝐾𝐾𝑢𝑢\n𝜇𝜇0𝑀𝑀𝑠𝑠cos2(𝜙𝜙𝐻𝐻−𝜙𝜙𝑢𝑢) . (2) \nThe extracted cubic anisotropy fields µ 0Hc ≤ 0.22mT are negligible for all the samples. The \nextracted in -plane Landé splitting factors g∥ and the uniaxial anisotropy fields  µ0Hu are \npresented in T able I. The purpose of the angle dependent FMR measurements was only to \ninvestigate the symmetry of the in -plane magnetic anisotropy. Therefore, care was not taken \nto have the same in -plane orientation of the samples during angle dependent FMR \nmeasurements, which explains why the maxima appear at diffe rent angles for the different \nsamples.  \nC. Out-of-plane FMR measurements  \nField -sweep out -of-plane FMR measurements were performed at different constant \ntemperatures in the range 50K – 300K and at different constant frequencies in the range of 9-\n20 GHz. Figure 1(b) shows the amplitude  of the complex transmission coefficient 𝑆𝑆21(10 \nGHz) vs. field measured for the LP673K thin film at different temperatures. The recorded \nFMR spectra were fitted using the equation [42],  \n𝑆𝑆21=𝑆𝑆�∆𝐻𝐻2��2\n(𝐻𝐻−𝐻𝐻𝑟𝑟)2+�∆𝐻𝐻2��2+𝐴𝐴�∆𝐻𝐻2��(𝐻𝐻−𝐻𝐻𝑟𝑟)\n(𝐻𝐻−𝐻𝐻𝑟𝑟)2+�∆𝐻𝐻2��2+𝐷𝐷∙𝑡𝑡,  (3) \nwhere 𝑆𝑆 represents the coefficient describing the transmitted microwave power, 𝐴𝐴 is used to \ndescribe a waveguide induced phase shift  contribution  which is, however, minute , 𝐻𝐻 is \napplied magnetic field, ∆𝐻𝐻 is the full-width of half maxim um, and 𝐷𝐷∙𝑡𝑡 describes the linear \ndrift in time  (𝑡𝑡) of the recorded signal. The extracted ∆ 𝐻𝐻 vs. frequency at different constant  \ntemperatures  are shown in Fig. 4 for all the samples. For brevity only data at a few \ntemperatures are plotted. The Gilbert damping was estimated using the equation [42 ],  \n∆𝐻𝐻=∆𝐻𝐻0+2ℎ𝛼𝛼𝑒𝑒\n𝑔𝑔⊥𝜇𝜇𝐵𝐵𝜇𝜇0   (4) \nwhere ∆𝐻𝐻0 is the inhomogeneous line -width broadening, 𝛼𝛼 is the experimental Gilbert \ndamping constant , and 𝑔𝑔⊥ is the Landé  splitting factor measured employing out -of-plane \nFMR. The insets in the figures show the temperature dependence of  𝛼𝛼. The effective 8 \n magnetization ( 𝜇𝜇0𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒) was estimated  from the 𝑓𝑓 vs. 𝐻𝐻𝑟𝑟 curves using out -of-plane Kittel’s \nequation [43],  \n𝑓𝑓=𝑔𝑔⊥𝜇𝜇0𝜇𝜇𝐵𝐵\nℎ�𝐻𝐻𝑟𝑟−𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒�,     (5) \nas shown in Fig. 5 . The temperature dependence of  𝜇𝜇0𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒 and 𝜇𝜇0∆𝐻𝐻0 are shown as  insets  in \neach figure . The observed room temperature values of  𝜇𝜇0𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒 are closely equal  to the  𝜇𝜇0𝑀𝑀𝑠𝑠 \nvalues obtained  from static magnetizat ion measurements, presented in T able I. The extracted \nvalues of g⊥ at different temperatures are within error limits  constant for all samples. \nHowever , the difference between estimated values of g ∥ and g⊥ is ≤ 3%. This difference c ould \nstem from the limited frequency range  used since  these values are quite sensitive to the value \nof 𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒, and even a minute uncertainty in this quantity can result in the observed small \ndifference between  the g∥ and g⊥ values.  \nTo obtain  the intrinsic Gilbert damping (𝛼𝛼 𝑖𝑖𝑖𝑖𝑖𝑖) all extrinsic contributions to the experimental 𝛼𝛼 \nvalue  need to be  subtracted. In metallic ferromagnets , the intrinsic Gilbert damping is mostly \ncaused by electron magnon scattering, but  several other extrinsic co ntributions can also \ncontribute to the experimental value of the damping constant. One  contribution is two -\nmagnon scattering which is however minimized for the  perpendicular geometry used in this \nstudy and therefore this contribution is disregarded [44].  Another contribution is spin-\npumping into the  capping layer as the LP573K, LP673K and LP773K samples are capped \nwith 4 nm of Al that naturally forms a thin top layer consisting of  Al2O3. Since spin pumping \nin low spin-orbit coupling materials with  thickness less than the spin-diffusion length is quite \nsmall this contribution is also disregarded in all samples. However, the HP300K sample is \ncapped with Ta and therefore a spin-pumping contribution have been subtracted from the \nexperimental 𝛼𝛼 value ; 𝛼𝛼𝑠𝑠𝑠𝑠= 𝛼𝛼𝐻𝐻𝐻𝐻300𝐾𝐾(with  Ta capping )−𝛼𝛼𝐻𝐻𝐻𝐻300𝐾𝐾(without capping )≈\n1×10−3. The third contribution arises  from the inductive coupling between  the precessi ng \nmagnetization and the CPW , a reciprocal phenomenon of FMR, known as radiative damping \n𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟 [45]. This damping is directly proportional to the magnetization and thickness of the thin \nfilms samples and therefore usually dominates in thicker and/or high magnetization samples. \nThe l ast contribution is eddy current damping ( αeddy) caused  by eddy current s in metallic \nferromagnetic  thin films [ 45, 46]. As per Faraday’s law the time varying magnetic flux density  \ngenerate s an AC voltage in the metallic ferromagnetic  layer and therefore result s in the eddy 9 \n current damping . Thi s damping is directly proportional to the square of the film thickness  and \ninversely proportional to the resistivity of the sample [ 45].  \nIn contrast to eddy -current damping, 𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟 is independent of the conductivity of the \nferromagnetic layer, hence this damping mechanism is also operati ve in ferromagnetic \ninsulators. Assuming a uniform magnetization of the sample the radiative damping can be \nexpressed as [45],  \n𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟= 𝜂𝜂𝜂𝜂𝜇𝜇02𝑀𝑀𝑆𝑆𝛿𝛿𝛿𝛿\n2 𝑍𝑍0𝑤𝑤 ,   (6) \nwhere 𝛾𝛾=𝑔𝑔𝜇𝜇𝐵𝐵ℏ� is the gyromagnetic ratio, 𝑍𝑍0 = 50 Ω is the waveguide impedance, 𝑤𝑤 = 240 \nµm is the width of the waveguide, 𝜂𝜂 is a dimensionless parameter which accounts for the \nFMR mode profile and depends on boundary conditions, and 𝛿𝛿 and 𝑙𝑙 are the thickness and \nlength of the sample on the waveguide, respectively. The strength of this inductive coupling \ndepends on the inductance of the FMR mode which is determined by the waveguide width, \nsample length over waveguide, sample saturation magnetization and sample thickness.  The \ndimensions of the LP573K, LP673K  and LP773K samples were 6.3×6.3 mm2, while the \ndimensions of the HP300K sample were  4×4 mm2. Th e 𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟 damping was estimated  \nexperimentally as explained by Schoen et al.  [45] by  placing a 200 µm thick glass spacer \nbetween  the waveguide and the sample , which decreases  the radiative damping by more than \none order magnitude as shown in Fig. 6(a). The measured radiative damping by placing the \nspacer  between the waveguide and the LP773 sample, \n𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟=𝛼𝛼𝑤𝑤𝑖𝑖𝑖𝑖ℎ𝑖𝑖𝑢𝑢𝑖𝑖  𝑠𝑠𝑠𝑠𝑚𝑚𝑐𝑐𝑒𝑒𝑟𝑟  −𝛼𝛼𝑤𝑤𝑖𝑖𝑖𝑖ℎ 𝑠𝑠𝑠𝑠𝑚𝑚𝑐𝑐𝑒𝑒𝑟𝑟≈ (2.36 ±0.10×10−3) − (1.57 ±0.20×10−3)=\n0.79±0.22×10−3. The estimated value matches  well with the calculated value using Eq. \n(6); 𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟= 0.78 ×10−3. Our results are also analogous to previously reported results  on \nradiative damping [45].  The estimated temperature dependent radiative damping values for all \nsamples are shown in Fig. 6(b).   \nSpin wave precession in ferromagnetic layers induces an AC current in the conducting \nferromagnetic layer which results in  eddy current damping. It can be  expressed as [45, 46], \n𝛼𝛼𝑒𝑒𝑟𝑟𝑟𝑟𝑒𝑒 = 𝐶𝐶𝜂𝜂𝜇𝜇02𝑀𝑀𝑆𝑆𝛿𝛿2\n16 𝜌𝜌 ,   (7) \nwhere 𝜌𝜌 is the resistivity of the sample and 𝐶𝐶 accounts for the eddy current distribution in the \nsample ; the smaller the value of 𝐶𝐶 the larger is the localization of eddy currents in the sample. \nThe measured  resistivity values between 300  K to 50 K temperature range fall in the ranges \n1.175 – 1.145  µΩ-m, 1 .055 – 1.034 µΩ -m, 1 .035 – 1.00 µΩ -m, and 1. 45 – 1.41 µΩ -m for the \nLP573K, LP673 , LP773 and HP300K samples, respect ively. The parameter 𝐶𝐶 was obtained 10 \n from thickness dependent experimental Gilbert damping constants measured for B2 ordered  \nfilms,  by line ar fitting of 𝛼𝛼−𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟≈𝛼𝛼𝑒𝑒𝑟𝑟𝑟𝑟𝑒𝑒 vs. 𝛿𝛿2 keeping other parameters  constant  (cf. Fig. \n6(c)). The fit to the data yield ed 𝐶𝐶 ≈ 0.5±0.1. These results are concurrent to those  \nobtained for permalloy  thin films [45]. Since the variations of the resistivity and \nmagnetization for the samples  are small , we have used the same 𝐶𝐶 value  for the estimation of \nthe eddy current damping in all the samples. The estimated temperature dependent values of \nthe eddy current damping are presented in Fig. 6(d).  \nAll these contributions have  been subtracted from the experimentally observed values of 𝛼𝛼. \nThe estimated intrinsic Gilbert damping 𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 values so obtained are plotted in Fig. 7(a) for all \nsamples.   \nD. Theoretical results: first principle calculations  \nThe calculated temperature dependent intrinsic Gilbert damping for Co 2FeAl phases with \ndifferent degree of atomic order are shown in Fig. 7(b). The temperature dependent Gilbert \ndamping indicates that the lattice displacements and spin fluctuations contribute differently in \nthe A2, B2 and L2 1 phases. The torque correlation model [47, 48] describes qualitatively two \ncontributions to the Gilbert damping. The first one is the intraband scattering where the band \nindex is always conserved. Since it has a linear dependence on the electron lifetime, in the \nlow temperature regime this term increases rapid ly, it is also known as the conductivity like \nscattering. The second mechanism is due to interband transitions where the scattering occurs \nbetween bands with different indices. Opposite to the intraband scattering, the resistivity like \ninterband scattering with an inverse depe ndence on the electron lifetime  increases with \nincreas ing temperature. The sum of the intra - and interband electron scattering contributions \ngives rise to a non-monotonic dependence of the Gilbert damping on temperature for the L2 1 \nstructure. In contrast to the case for L2 1, only interband scattering is present in the A2 and B2 \nphases, which results in a monotonic increase of the intrinsic Gi lbert damping with increas ing \ntemperature. This fact is also supported by a previous study [37 ] which showed that even a \nminute chemical disorder can inhibit the intraband scattering of the system. Our theoretical results manifest that the L2\n1 phase has the lowest Gilbert damping around 4.6 × 10−4 at 300 \nK, and that the value for the B2 phase is only slightly larger at room temperature. According \nto the torque correlation model, the two main contributions to damping are the spin orbit \ncoupling and the density of states (DOS) at the Fermi level [47 , 48]. Since the spin orbit \nstrength is the same for the different phases it is enough to focus the discussion on the DOS 11 \n that provide s a qualitative explanation  why damping is found lower in B2 and L2 1 structure s \ncompare d to A2 structure. The DOS at the Fermi level of the B2 phase (24.1 states/Ry/f.u;  f.u \n= formula unit ) is only slightly  larger to that of the L2 1 phase (20.2 states/Ry/f.u.) , but both \nare significantly smaller than for the A2 phase (59.6 states/Ry/f.u.) as shown in Fig. 8. The \ngap in the mi nority spin channel of the DOS for the B2 and L2 1 phases indicate half-\nmeta llicity, while the A2 phase is metallic. The atomically resolved spin polarized DOS \nindica tes that the Fermi -level states mostly have  contr ibutions from Co and Fe atoms. For \ntransition elements such as Fe and Ni, it has been reported that  the intrinsic Gilbert damping \nincreases significantly below 100K with decreas ing temperature [37]. The present electronic \nstructure calculations were performed using Green’s functions, which do rely on a \nphenomenological relaxation time parameter,  on the expense that the different contributions to \ndamping cannot  be separated eas ily. The reported results in Ref. [37] are by some means \nsimilar to our findings of the temperature dependent Gilbert damping in full Heusler alloy films with different degr ee of atomic order. The intermediate states of B2 and L2\n1 are more \nclose to the trend of B2 than L2 1, which indicates that even a tiny atomic orde r induced by the \nFe and Al site disorder will inhibit the conductivity -like channel in  the low temperature \nregion. The theoretically calculated Gilbert damping constants are matching qualitatively with \nthe experimentally observed  𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 values as shown in Fig. 7. However, the theoretically \ncalculated 𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 for the L2 1 phase increases rapidly below 100K, in co ntrast to the \nexperimental results for the well -ordered CFA thin film (LP673K ) indicating that  \n𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 saturates at low temperature. This discrepancy between the theoretical and experimental \nresults can be understood taking into account the low temperature behaviour of the life time τ \nof Bloch states. The present theoretical model assum ed that the Gilbert damping has a linear \ndependence on the electron lifetime in intraband transitions which is however correct only in \nthe limit of small lifetime, i.e., 𝑞𝑞𝑣𝑣𝐹𝐹𝜏𝜏≪1, where q is the magnon wave vector and 𝑣𝑣𝐹𝐹 is the \nelectron Fermi velocity. However, in the low temperature limit the lifetime  𝜏𝜏 increases and as \na result of the anomalous skin effect the intrinsic Gilbert damping saturates  \n𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖∝ tan−1𝑞𝑞𝑣𝑣𝐹𝐹𝜏𝜏𝑞𝑞𝑣𝑣𝐹𝐹�  at low temperature [37], which is evident from our experimental \nresults.  \nRemaining discrepancies between theoretical and experimental values of the intrinsic Gilbert \ndamping might stem from the fact that the samples used in the present study are 12 \n polycrystalline and because of sample imperfections these fil ms exhibit significant \ninhomogenous line -width broadening due to superposition of local resonance fields.  \nCONCLUSION  \nIn summary , we report temperature dependent FMR measurements on as -deposited Co 2FeAl \nthin films with different degree of atomic order. The degree of atomic ordering is established \nby comparing experimental and theoretical results for the temperature dependent intrinsic \nGilbert damping constant. It is evidenced that the experimentally observed intrinsic Gilbert \ndamping in samples with atomic  disorder (A2 and B2 phase samples) decreases with \ndecreasing temperature. In contrast, the atomically well -ordered sample, which we identify at \nleast partial L21 phase, exhibits an intrinsic Gilbert damping constant that increases with \ndecreasing temperat ure. These temperature dependent results are explained employing the \ntorque correction model including interband transitions and both interband as well as \nintraband transitions for samples with atomic disorder and atomically ordered phases, \nrespectively.  \nACKNOWLEDGEMENT \nThis work is supported by the Knut and Alice Wallenberg (KAW) Foundation,  Grant No.  \nKAW 2012.0031  and from Göran Gustafssons Foundation (GGS), Grant No.  GGS1403A. The \ncomputations were performed on resources provided by SNIC (Swedish National \nInfrastructure for Computing) at NSC (National Supercomputer Centre) in Linköping, \nSweden.  S. 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Lett. 99 , 027204 (2007) .  \n  15 \n Table I P arameters describing magnetic properties of the different CFA samples.  \nSample  𝜇𝜇0𝑀𝑀𝑆𝑆(𝜇𝜇0𝑀𝑀𝑒𝑒𝑒𝑒𝑒𝑒) \n(T) 𝜇𝜇0𝐻𝐻𝑐𝑐𝑖𝑖 \n(mT)  𝑔𝑔∥(𝑔𝑔⊥) \n 𝜇𝜇0𝐻𝐻𝑢𝑢 \n(mT)  𝑇𝑇𝐶𝐶 \n(K) 𝛼𝛼𝑖𝑖𝑖𝑖𝑖𝑖 \n(× 10-3) \nLP573K  1.2±0.1 (1.09 1±0.003)  0.75 2.06 (2.0)  1.56 810 2.56 \nLP673K  1.2±0.1 (1.11 0±0.002)  0.57 2.05 (2.0)  1.97 >900  0.76 \nLP773K  1.2±0.1 (1.081 ±0.002)  0.46 2.05 (2.0)  1.78 890 1.46 \nHP300K  0.9±0.1 (1.066 ±0.002)  1.32 2.01 (2.0)  3.12 -- 3.22  \n 16 \n Figure 1  \nFig. 1.  (a) Layout of the in -house made VNA-based out -of-plane ferromagnetic resonance  \nsetup . (b) Out -plane ferromagnetic resonance spectra recorded for the well -ordered LP673K \nsample at different temperatures  𝑓𝑓=10 GHz . \n  \n \n17 \n Figure 2  \nFig. 2. (a) Magnetization vs. temperature plots measured on the CFA films with different \ndegree of atomic order. (b) Theoretically calculated magnetization vs. temperature curves for \nCFA phases with different degree of atomic order, where 50 % (100 %) Fe atoms on Heusler \nalloy 4a sites indicate B2 (L2 1) ordered phase, and the rest are intermediate B2 & L2 1 mixed \nordered  phases.  \n  \n \n \n18 \n Figure 3  \nFig. 3.  Resonance field vs. in -plane orientation of the applied magnetic field of (a) 𝑇𝑇𝑆𝑆=\n300℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (b) 𝑇𝑇𝑆𝑆=400℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (c) 𝑇𝑇 𝑆𝑆=500℃ , \n𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, and (d) 𝑇𝑇 𝑆𝑆=27℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=100  𝑊𝑊 deposited films. Red lines \ncorrespond to fits to the data using Eq. (1).  \n    \n \n \n19 \n  \nFigure 4  \nFig. 4.   Line-width vs. frequency of (a) 𝑇𝑇𝑆𝑆=300℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (b) 𝑇𝑇𝑆𝑆=400℃ , \n𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (c) 𝑇𝑇 𝑆𝑆=500℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, and (d) 𝑇𝑇𝑆𝑆=27℃ , \n𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=100  𝑊𝑊 deposited samples. Red lines correspond to fits to the data to extract the \nexperimental Gilbert damping constant and inhomogeneous line -width. Respective insets \nshow the experimentally determined temperature dependent Gilbert damping constants.  \n  \n  \n \n \n20 \n Figure 5  \nFig. 5.  Frequency vs. applied field of (a) 𝑇𝑇𝑆𝑆=300℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (b) 𝑇𝑇𝑆𝑆=\n400℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, (c) 𝑇𝑇𝑆𝑆=500℃ , 𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=75 𝑊𝑊 deposited, and (d) 𝑇𝑇𝑆𝑆=27℃ , \n𝑃𝑃𝐼𝐼𝑖𝑖𝑖𝑖=100  𝑊𝑊 deposited samples. Red lines correspond to Kittel’s fits to the data. Respective \ninsets show the temperature dependent effective magnetization a nd inhomogeneous line -width \nbroadening values.  \n  \n  \n \n21 \n Figure 6  \nFig. 6.  (a) Linewidth vs. frequency with and without a glass spacer between the waveguide \nand the sample.  Red lines correspond to fits using Eq. (4).  (b) Temperature dependent values \nof the radiative damping using Eq.  (6). The lines are guide to the eye. (c) 𝛼𝛼−𝛼𝛼𝑟𝑟𝑚𝑚𝑟𝑟≈𝛼𝛼𝑒𝑒𝑟𝑟𝑟𝑟𝑒𝑒 \nvs 𝛿𝛿2. The red line corresponds to a fit  using Eq. (7) to extract the value of the correction \nfactor  𝐶𝐶. (d) Temperature dependent values of eddy current dampi ng using Eq. (7). The lines \nare guide to the eye.  \n  \n  \n \n \n22 \n Figure 7  \nFig. 7.  Experimental (a) and theoretical (b) results for the temperature dependent intrinsic \nGilbert damping constant for CFA samples with different degree of atomic order . The B2 & \nL21 mixed phase corresponds to the 75 % occupancy of Fe atoms on the Heusler alloy 4a \nsites. The lines are guide to the eye.  \n  \n \n \n23 \n Figure 8  \nFig. 8.  Total and atom -resolved spin polarized density of states plots for various \ncompositional CFA phases; (a) A2, (b) B2 and (c) L2 1. \n  \n \n \n"    },    {        "title": "1403.3914v2.Interpolating_local_constants_in_families.pdf",        "content": "arXiv:1403.3914v2  [math.NT]  20 Aug 2015INTERPOLATING LOCAL CONSTANTS IN FAMILIES\nGILBERT MOSS\n1.Introduction\nLetG=GLn(F), letkbe an algebraically closed field of characteristic ℓ, and\nW(k) its ring of Witt vectors. By an ℓ-adic family of representations we mean an\nA[G]-moduleVwhereAis a commutative W(k)-algebra with unit; then each point\npofAgives aκ(p)[G]-moduleV⊗Aκ(p) whereκ(p) denotes the residue field at p.\nIn [EH12], Emerton and Helm conjecture a local Langlands correspo ndence for ℓ-\nadicfamiliesofadmissiblerepresentations. ToanycontinuousGaloisr epresentation\nρ:GF→GLn(A), they conjecturally associate an admissible smooth A[G]-module\nπ(ρ), which interpolates the local Langlands correspondence for poin tsA→κwith\nκcharacteristic zero. They prove that any A[G]-module which is subject to this\ninterpolation property and a short list of representation-theore tic conditions (see\n[EH12, Thm 6.2.1]) must be unique.\nIn [Hel12b], Helm further investigates the structure of π(ρ) by taking the list of\nrepresentation-theoretic conditions in [EH12, Thm 6.2.1] as a start ing point for the\ntheory of “co-Whittaker” A[G]-modules (see Section 2.5 below for the definitions).\nUsing this theory, he is able to reformulate the conjecture in terms of the existence\nof a certain homomorphism between the integral Bernstein center and a universal\ndeformation ring ([Hel12b, Thm 7.8]).\nRoughly speaking, representations of GLn(F) overCare completely determined\nby data involving only local constants ([Hen93]), and in particular th e bijections\nof the classical local Langlands correspondence are uniquely dete rmined using L-\nand epsilon-factors (see, for example, [Jia13]). However, L- and epsilon-factors are\nabsent from the the local Langlands correspondence in families. Th us it is natural\nto ask whether it is possible to attach L- and epsilon-factors to an ℓ-adic family\nsuch asπ(ρ) as in [EH12], or more generally any co-Whittaker A[G]-module, in a\nway that interpolates the L- and epsilon factors at each point.\nOverC,L-factorsL(π,X) arise as the greatest common denominator of the zeta\nintegralsZ(W,X;j) of a representation πasWvaries over the space W(π,ψ) of\nWhittakerfunctions (seeSections2.2, 3.1fordefinitions). Epsilon- factorsǫ(π,X,ψ)\nare the constant of proportionality (i.e. not depending on W) in a functional\nequation relating the modified zeta integralZ(W,X)\nL(π,X)to its pre-composition with\na Fourier transform. Here, the formal variable Xreplaces the complex variable\nq−(s+n−1\n2)appearingin [JPSS79] and otherliterature, and weconsiderthese objects\nas formal series.\nIt appears difficult to construct L-factors in a way compatible with arbitrary\nchange of coefficients. To see this, consider the following simple exam ple: letq≡\n1 modℓ, and letχ1,χ2:F×→W(k)×be smooth characters such that χ1is\nDate: August 17, 2021.\n12 GILBERT MOSS\nunramifiedbut χ2isramified, andsuchthat χ1≡χ2modℓ. Followingtheclassical\nprocedure (see for example [BH06, 23.2]) for finding a generator of the fractional\nideal of zeta integrals, we get L(χi,X)∈W(k)(X) and find that L(χ1,X) =\n1\n1−χ1(̟F)X, andL(χ2,X) = 1. Now let Abe the Noetherian local ring {(a,b)∈\nW(k)×W(k) :a≡bmodℓ}, which has two characteristic zero points p1,p2and\na maximal ideal ℓA. Letπbe theA[F×]-moduleA, with the action of F×given\nbyx·(a,b) = (χ1(x)a,χ2(x)b). Interpolating L(χ1,X) andL(χ2,X) would mean\nfindinganelement L(π,X)inA[[X]][X−1] suchthatL(π,X)≡L(χi,X) modℓfor\ni= 1,2, but such a task is impossible because L(χ1,X) andL(χ2,X) are different\nmodℓ.\nOn the otherhand, zeta integralsthemselvesseem to be much more well-behaved\nwith respect to specialization. Classically, zeta integrals form elemen ts of the quo-\ntient field C(X) ofC[X,X−1]. Our first result is identifying, for more arbitrary\ncoefficient rings A, the correct fraction ring in which our naive generalization of\nzeta factors will live:\nTheorem 1.1. SupposeAis aNoetherian W(k)-algebra. Let Sbe the multiplicative\nsubset ofA[X,X−1]consisting of polynomials whose first and last coefficients ar e\nunits. Then if Vis a co-Whittaker A[G]-module,Z(W,X;j)lies in the fraction\nringS−1(A[X,X−1])for allW∈ W(V,ψ)and for0≤j≤n−2.\nThe proof of rationality in the setting of representations over a fie ld relies on\na useful decomposition of a Whittaker function into “finite” functio ns ([JPSS79,\nProp 2.2]). In the setting of rings, such a structure theorem is lack ing, but certain\nelements of its proof can be translated into a question about the fin iteness of the\n(n−1)st Bernstein-Zelevinsky derivative. This finiteness property, c ombined with\na simple translation property of the zeta integrals, yields Theorem 1 .1 (see§3.2).\nClassically, zeta integrals satisfy a functional equation which does n ot involve\ndividingbythe L-factor. Theconstantofproportionalityin thisfunctionalequat ion\nis called the gamma-factorand equals ǫ(π,X,ψ)L(πι,1\nqnX)\nL(π,X), when theL-factormakes\nsense. Our second main result is that gamma-factors interpolate in ℓ-adic families\n(see§4.1 for details on the notation):\nTheorem 1.2. SupposeAis a Noetherian W(k)-algebra and suppose Vis a prim-\nitive co-Whittaker A[G]-module. Then there exists a unique element γ(V,X,ψ)of\nS−1(A[X,X−1])such that\nZ(W,X;j)γ(V,X,ψ) =Z(/tildewidew′W,1\nqnX;n−2−j)\nfor anyW∈ W(V,ψ)and for any 0≤j≤n−2.\nToproveTheorem1.2weusethe theoryoftheintegralBernsteinc entertoreduce\nto the characteristic zero case of [JPSS79].\nThe question of interpolating local constants in ℓ-adic families has been inves-\ntigated in a simple case by Vigneras in [Vig00]. For supercuspidal repre sentations\nofGL2(F) overQℓ, Vigneras notes in [Vig00] that it is known that epsilon factors\ndefine elements of Zℓ, and proves that for two supercuspidal integral representa-\ntions to be congruent modulo ℓit is necessary and sufficient that they have epsilon\nfactors which are congruent modulo ℓ(we call a representation with coefficients\nin a local field Eintegral if it stabilizes an OE-lattice). The classical epsilon and\ngamma factors are equal in the supercuspidal case, so when the s pecialization of anINTERPOLATING LOCAL CONSTANTS IN FAMILIES 3\nℓ-adic family at a characteristic zero point is supercuspidal, the gamm a factor we\nconstruct in this paper specializes to the epsilon factor of [JPSS79, Vig00]. Since\ntwo representations V1,V2overOEwhich are congruent mod mEdefine a family\nV1×VV2over the connected W(k)-algebra OE×kEOE, Theorems 1.1 and 1.2 give\nthe following corollary (implying the “necessary” part of [Vig00]):\nCorollary 1.3. LetKdenote the fraction field of W(k). Ifπandπ′are absolutely\nirreducible integral representations of GLn(F)over a coefficient field Ewhich is a\nfinite extension of K, then:\n(1)γ(π,X,ψ)andγ(π′,X,ψ)have coefficients in the fraction ring\nS−1(OE[X,X−1]).\n(2) IfmEis the maximal ideal of OE, andπ≡π′modmE, thenγ(π,X,ψ)≡\nγ(π′,X,ψ) modmE.\nThe question of extending the theory of zeta integrals to the ℓ-modular setting\nhas been investigated in [M ´12], and very recently in [KM14] for the Rankin-Selberg\nintegrals. The question of deforming local constants over polynom ial rings over\nChas been investigated by Cogdell and Piatetski-Shapiro in [CPS10], an d the\ntechniques of this paper owe much to those in [CPS10].\nAnalogous to the results of Bernstein and Deligne in [BD84] for RepC(G), Helm\nshows in [Hel12a, Thm 10.8] that the category RepW(k)(G) has a decomposition\ninto full subcategories known as blocks. Our third main result is cons tructing for\neach block a gamma factor which is universal in the sense that it gives rise via\nspecialization to the gamma factor for any co-Whittaker module in th at block. We\nwill now state this result more precisely.\nEach block of the category RepW(k)(G) corresponds to a primitive idempotent\nin the Bernstein center Z, which is defined as the ring of endomorphisms of the\nidentity functor. It is a commutative ring whose elements consist of collections of\ncompatible endomorphisms of every object, each such endomorph ism commuting\nwith all morphisms. Choosing a primitive idempotent eofZ, the ringeZis the\ncenter of the subcategory e·RepW(k)(G) of representations satisfying eV=V. The\nringeZhas an interpretation as the ring of regular functions on an affine alg ebraic\nvariety over W(k), whosek-points are in bijection with the set of unramified twists\nof a fixed conjugacy class of supercuspidal supports in Repk(G). See [Hel12a]\nfor details. In [Hel12b], Helm determines a “universal co-Whittaker m odule” with\ncoefficientsin eZ, denoted hereby eW, whichgivesrisetoanyco-Whittakermodule\nvia specialization (see Proposition 2.31 below). By applying our theory of zeta\nintegrals to eWwe get a gamma factor which is universal in the same sense:\nTheorem 1.4. SupposeAis any Noetherian W(k)-algebra, and suppose Vis a\nprimitive co-Whittaker A[G]-module. Then there is a primitive idempotent e, a\nhomomorphism fV:eZ →A, and an element Γ(eW,X,ψ)∈S−1(eZ[X,X−1])\nsuch thatγ(V,X,ψ) =fV(Γ(eW,X,ψ)).\nInterpolating gamma factors of pairs may be the next step in obtain ing a local\nconversetheorem for ℓ-adic families. By capturing the interpolation property, fami-\nlies ofgammafactorsmight givean alternativecharacterizationoft he co-Whittaker\nmoduleπ(ρ) appearing in the local Langlands correspondence in families.\nThe author would like to thank his advisor David Helm for suggesting th is prob-\nlem and for his invaluable guidance, Keenan Kidwell for his helpful conv ersations,\nand Peter Scholze for his helpful questions and comments at the MS RI summer4 GILBERT MOSS\nschool on New Geometric Techniques in Number Theory in 2013. He wo uld also\nlike to thank the referee for her/his very helpful comments and su ggestions.\n1.1.Notation and Conventions. LetFbe a finite extension of Qp, letqbe the\norder of its residue field, and let kbe an algebraically closed field of characteristic\nℓ, whereℓ/ne}ationslash=pis an odd prime. Denote by W(k) the ring of Witt vectors over\nk. The assumption that ℓis odd is made so that W(k) contains a square root\nofq. Whenℓ= 2 all the arguments presented will remain valid, after possibly\nadjoining a square root of qtoW(k). The letter GorGnwill always denote the\ngroupGLn(F). Throughout the paper Awill be a Noetherian commutative ring\nwhich is aW(k)-algebra, with additional properties in various sections, and κ(p)\nwill denote the residue field of a prime ideal pofA. For any locally profinite group\nH, RepA(H) denotes the category of smooth representations of Hover the ring A,\ni.e.A[H]-modules for which every element is stabilized by an open subgroup of H.\nEven when this category is not mentioned, all representations are presumed to be\nsmooth. When Hisa closedsubgroupof G, wedefine the non-normalizedinduction\nfunctor IndG\nH(resp. c-IndG\nH) : RepA(H)→RepA(G) sendingτto the smooth part\noftheA[G]-module, underrighttranslation, offunctions(resp. functionsc ompactly\nsupported modulo H)f:G→τsuch thatf(hg) =τ(h)f(g),h∈H,g∈G.\nThe integral Bernstein center of [Hel12a] (see the discussion prec eding Theorem\n1.4) will alwaysbe denoted by Z. IfVis in RepA(G), then it is also in RepW(k)(G),\nand we frequently use the Bernstein decomposition of RepW(k)(G) to interpret\nproperties of V.\nIfAhas a nontrivial ideal I, thenI·Vis anA[H]-submodule of V, which shows\nthat most content would be missing if we developed the representat ion theory of\nRepA(H) around the notion of irreducible objects, or simple A[H]-modules. Thus\nconditions appear throughout the paper which in the traditional se tting are implied\nby irreducibility:\nDefinition 1.5. VinRepA(H)will be called\n(1)Schurif the natural map A→EndA[G](V)is an isomorphism;\n(2)G-finiteif it is finitely generated as an A[G]-module.\n(3)primitive if there exists a primitive idempotent ein the Bernstein center Z\nsuch thateV=V.\nWe say a ring is connected if it has connected spectrum or, equivalen tly, no\nnontrivial idempotents, for example any local ring or integral doma in. Note that if\nAis connected, Corollary2.32implies all co-Whittaker A[G]-modules are primitive.\nDenote by Nnthe subgroup of Gnconsisting of all unipotent upper-triangular\nmatrices. Let ψ:F→W(k)×be an additive character of Fwith kerψ=p. Then\nψdefines a character on any subgroup of Nn(F) by\n(u)i,j/mapsto→ψ(u1,2+···+un−1,n);\nwe abusively denote this character by ψas well.\nIfHis asubgroupnormalizedby anothersubgroupgroup K, andθis a character\nof the group H, denote by θkthe character given by θk(h) =θ(khk−1) forh∈H,\nk∈K. ForVin RepA(H), denote by VH,θthe quotient V/V(H,θ) whereV(H,θ)\nis the sub-A-module generated by elements of the form hv−θ(h)vforh∈Hand\nv∈V; it isK-stable ifθk=θ,k∈K. Given a standard Levi subgroup M⊂GnINTERPOLATING LOCAL CONSTANTS IN FAMILIES 5\nwith unipotent radical U, and1the trivial character, we denote by JMthe non-\nnormalized Jacquet functor RepA(G)→RepA(M) :V/mapsto→VU,1.\nFor eachm≤n, letGmdenoteGLm(F) and embed it in Gvia (Gm0\n0In−m).\nWe let{1}=P1⊂ ··· ⊂Pndenote the mirabolic subgroups of G1⊂ ··· ⊂Gn,\nwherePmis given by/braceleftbig\n(gm−1x\n0 1) :gm−1∈Gm−1, x∈Fm−1/bracerightbig\n. We also have the\nunipotent upper triangular subgroup UmofPmgiven by/braceleftbig\n(Im−1x\n0 1) :x∈Fm−1/bracerightbig\n.\nIn particular, Um≃Fm−1andPm=UmGm−1. Note that this is different from\nthe groups N(r) defined in Proposition 2.3.\nSinceGncontains a compact open subgroup whose pro-order is invertible in\nW(k), there exists a unique (for that choice of subgroup) normalized H aar mea-\nsure, defining integration on the space C∞\nc(G,A) of smooth compactly supported\nfunctionsG→A([Vig96, I.2.3]).\n2.Representation Theoretic Background\n2.1.Co-invariants and Derivatives. Asin[EH12,BZ77], wedefinethefollowing\nfunctors with respect to the character ψ.\nΦ+:RepA(Pn−1)→RepA(Pn) Ψ+:Rep(Gn−1)→Rep(Pn)\nV/mapsto→c-IndPn\nPn−1UnV(Unacts viaψ)V/mapsto→V(Unacts trivially)\nˆΦ+:Rep(Pn−1)→Rep(Pn) Ψ−:Rep(Pn)→Rep(Gn−1)\nV/mapsto→IndPn\nPn−1UnV V /mapsto→V/V(Un,1)\nΦ−:Rep(Pn)→Rep(Pn−1)\nV/mapsto→V/V(Un,ψ)\nNote that we give these functors the same names as the ones origin ally defined\nin [BZ76], but we use the non-normalized induction functors, as in [BZ7 7, EH12],\nbecause they are simpler for our purposes. As observed in [EH12], t hese functors\nretain the basic adjointness properties proved in [BZ77, §3.2]. This is because the\nmethods of proof in [BZ76, BZ77] use properties of l-sheaves which carry over to\nthe setting of smooth A[G]-modules where Ais a Noetherian W(k)-algebra.\nProposition 2.1 ([EH12],3.1.3) .(1) The functors Ψ−,Ψ+,Φ−,Φ+,ˆΦ+are ex-\nact.\n(2)Φ+is left adjoint to Φ−,Ψ−is left adjoint to Ψ+, andΦ−is left adjoint to\nˆΦ+.\n(3)Ψ−Φ+= Φ−Ψ+= 0\n(4)Ψ−Ψ+,Φ−ˆΦ+, andΦ−Φ+are naturally isomorphic to the identity functor.\n(5) For each VinRep(Pn)we have an exact sequence\n0→Φ+Φ−(V)→V→Ψ+Ψ−(V)→0.\n(6) (Commutativity with Tensor Product) If Mis anA-module and FisΨ−,Ψ+,\nΦ−,Φ+, orˆΦ+, we haveF(V⊗AM)∼=F(V)⊗AM.\nFor 1≤m≤nwe define the mth derivative functor\n(−)(m):= Ψ−(Φ−)m−1: Rep(Pn)→Rep(Gn−m).6 GILBERT MOSS\nThis gives a functor Rep( Gn)→Rep(Gn−m) by first restricting representations to\nPnand then applying ( −)(m); this functor is also denoted ( −)(m). The zero’th de-\nrivativefunctor ( −)(0)is the identity. We candescribe the derivativefunctor ( −)(m)\nmore explicitly by using the following lemma on the transitivity of coinvar iants:\nLemma 2.2 ([BZ76]§2.32).LetHbe a locally profinite group, θa character of\nH, andVa representation of H. SupposeH1,H2are subgroups of Hsuch that\nH1H2=HandH1normalizes H2. Then/parenleftig\nVH2,θ|H2/parenrightig\nH1,θ|H1=VH,θ.\nDefineN(r) to be the group of matrices whose first rcolumns are those of the\nidentity matrix, and whose last n−rcolumns are those of elements of Nn(recall\nthatNnis the group of unipotent upper triangular matrices). For 2 ≤r≤nwe\nhaveUrN(r) =N(r−1) andUrnormalizes N(r). AsN(r) is contained in Nn, we\ndefineψonN(r) via its superdiagonal entries. We can also define a character /tildewideψon\nN(r) slightly differently from the usual definition: /tildewideψwill be given as usual via ψ\non the last n−r−1 superdiagonal entries, but trivially on the ( r,r+1) entry, i.e.\n/tildewideψ(x) :=ψ(0+xr+1,r+2+···+xn−1,n) forx∈N(r).\nThe functors (Φ−)mand (−)(m), defined above, can be described more explicitly.\nLetm=n−r. By applying Lemma 2.2 repeatedly with H1=Ur, andH2=\nN(r−1), we get\nProposition 2.3 ([Vig96] III.1.8) .(1)(Φ−)mVequals the module of coinvariants\nV/V(N(n−m),ψ).\n(2)V(m)equals the module of coinvariants V/V(N(n−m),/tildewideψ).\nIn particular, if m=n, this gives V(n)=V/V(Nn,ψ). Note that V(n)is simply\nanA-module.\n2.2.Whittaker and Kirillov Functions. The character ψ:Nn→A×defines a\nrepresentationof NnintheA-moduleA, whichwealsodenoteby ψ. ByProposition\n2.3 we have Hom A(V(n),A) = Hom Nn(V,ψ).\nDefinition 2.4. ForVinRepA(Gn), we say that Vis of Whittaker type if V(n)\nis free of rank one as an A-module. As in [EH12, Def 3.1.8] , ifAis a field we refer\nto representations of Whittaker type as generic.\nIfVis of Whittaker type, Hom Nn(V,ψ) is free of rank one, so we may choose\na generator λ. The image of λunder the Frobenius reciprocity isomorphism\nHomNn(V,ψ)∼→HomGn(V,IndGn\nNnψ) is the map v/mapsto→WvwhereWv(g) =λ(gv).\nTheA[G]-module formed by the image of the map v/mapsto→Wvis independent of the\nchoice ofλ.\nDefinition 2.5. The image of the homomorphism V→IndGn\nNnψis called the space\nof Whittaker functions of Vand is denoted W(V,ψ)or justW.\nChoosing a generator of V(n)and allowing Nnto act via ψ, we get an iso-\nmorphismV(n)∼→ψ. Composing this with the natural quotient map V→V(n)\ngives anNn-equivariant map V→ψ, which is a generator λ. Note that the map\nV→ W(V,ψ) is surjective but not necessarily an isomorphism, unlike the setting\nofirreducible generic representations with coefficients in a field. Different A[G]-\nmodules of Whittaker type can have the same space of Whittaker fu nctions:INTERPOLATING LOCAL CONSTANTS IN FAMILIES 7\nLemma 2.6. SupposeV′,VinRepA(G)are of Whittaker type, and suppose there\nis aG-equivariant homomorphism α:V′→Vsuch thatα(n): (V′)(n)→V(n)\nis an isomorphism. Then W(V′,ψ)is the subrepresentation of W(V,ψ)given by\nW(α(V′),ψ).\nProof.Letq′:V′→V′/V′(Nn,ψ) andq:V→V/V(Nn,ψ) be the quotient maps.\nChoosing a generator for V(n)gives isomorphisms η,η′such that the following\ndiagram commutes.\nVq>V(n)η>A\nV′α∨\nq′\n>(V′)(n)α(n)\n∨ η′>\nGivenv′∈V′we get\nWα(v′)(g) =η(q(gαv′)) =η((q◦α)(gv′)) =η′(q′(gv′)) =Wv′(g), g∈G.\nThis shows W(V′,ψ) =W(α(V′),ψ)⊂ W(V,ψ). /square\nIfVin RepA(Gn) is Whittaker type and v∈V, we will denote by Wv|Pnthe\nrestriction of the function Wvto the subgroup Pn⊂Gn.\nDefinition 2.7. The image of the Pn-equivariant homomorphism V→IndPn\nNnψ:\nv/mapsto→Wv|Pnis called the Kirillov functions of Vand is denoted K(V,ψ)or justK.\nThe following properties of the Kirillov functions are well known for Re pC(G),\nbut we will need them for RepA(G):\nProposition 2.8. LetVbe of Whittaker type in RepA(Pn), and choose a generator\nofV(n)in order to identify V(n)withA. Then the following hold:\n(1)(Φ+)n−1V(n)= c-IndPn\nNnψand(ˆΦ+)n−1V(n)= IndPn\nNnψ.\n(2) The composition (Φ+)n−1V(n)→V→IndPn\nNnψdiffers from the inclusion\nc-IndPn\nNnψ֒→IndPn\nNnψby multiplication with an element of A×.\n(3) The Kirillov functions K(V,ψ)contains c-IndPn\nNnψas a sub-A[Pn]-module.\nProof.The proof in [BZ76] Proposition 5.12 (g) works to prove (1) in this con text.\nLetS= (Φ+)n−1V(n). There is an embedding S→Vby Proposition 2.1\n(5); denote by tthe composition S→V→Indψ. Thent(n):S(n)→Indψ(n)\nis a nonzero homomorphism between free rank one A-modules, hence given by\nmultiplication with an element aofA. By Proposition 2.1 (6), For any maximal\nidealmofA,t(n)⊗κ(m) must be an isomorphism because it is a nonzero element\nof\nHomκ(m)((S(V)⊗κ(m))(n),(Indψ⊗κ(m))(n)) =κ(m).\nThusais nonzero in κ(m) for allm, hence a unit, so t(n)is an isomorphism. On the\nother hand there is the natural embedding c-Ind ψ→Indψ, which we will denote\ns. Sinces(n)is an isomorphism by [BZ77, Prop 3.2 (f)], we have s(n)=ut(n)\nfor someu∈A×. Thus, if K:= ker(s−ut) thenK(n)=S(V)(n)=V(n),\nwhence Hom P(S(V)/K,Indψ)∼=HomA((S(V)/K)(n),A) = Hom A({0},A) = 0,\nwhich implies s−ut≡0.\nTo prove (3), note that since K(V,ψ) is defined to be the image of the map\nV→IndPn\nNnψ, this follows from (2). /square8 GILBERT MOSS\nDefinition 2.9 ([EH12],§3.1 ).IfVis inRep(Pn), the image of the natural inclu-\nsion(Φ+)n−1V(n)→Vis called the Schwartz functions of Vand is denoted S(V).\nForVinRep(Gn)we also denote by S(V)the Schwartz functions of Vrestricted\ntoPn.\nWe can ask how the functor Φ−is reflected in the Kirillov space of a represen-\ntation. First we observe that Φ−commutes with the functor K:\nLemma 2.10. For0≤m≤n, theA[Pm]-modules K((Φ−)n−mV,ψ)and\n(Φ−)n−mK(V,ψ)are identical.\nProof.The image of the Pn−m-submodule V(N(m),ψ) in the map V→ Kequals\nthe submodule K(N(m),ψ). The lemma then follows from Proposition 2.3 /square\nFollowing [CPS10], we can explicitly describe the effect of the functor Φ−on the\nKirillov functions K. Recall that K(Un,ψ) denotes the A-submodule generated by\n{uW−ψ(u)W:u∈Un, W∈ K}and Φ−K:=K/K(Un,ψ).\nProposition 2.11 ([CPS10] Prop 1.1) .\nK(Un,ψ) ={W∈ K:W≡0on the subgroup Pn−1⊂Pn}.\nProof.The proof of [CPS10, Prop 1.1] carries over in this setting. It utilizes the\nJacquet-Langlandscriterion for an element vof a representation Vto be in the sub-\nspaceV(Uni,ψ), which remains valid over more general coefficient rings Abecause\nall integrals are finite sums. /square\nThus Φ−has the same effect as restriction of functions to the subgroup Pn−1\ninsidePn:\nΦ−K∼=/braceleftbig\nW/parenleftbigp0\n0 1/parenrightbig\n:W∈ W(V,ψ), p∈Pn−1/bracerightbig\n.\nBy applying for each m= 1,...,n−2 the argument of [CPS10, Prop 1.1] to the\nPn−m+1representation\n/braceleftig\nW/parenleftig\np0\n0Im−1/parenrightig\n:W∈ W(V,ψ), p∈Pn−m+1/bracerightig\ninstead of to K, we can describe (Φ−)mK:\nCorollary 2.12. Form= 1,...,n−1,\n(Φ−)mK∼=/braceleftbig\nW/parenleftbigp0\n0Im/parenrightbig\n:W∈ W(V,ψ), p∈Pn−m/bracerightbig\n.\n2.3.Partial Derivatives. Given a product H1×H2of subgroups of G, and a\ncharacterψon the unipotent upper triangular elements of H2, we can define “par-\ntial” versionsof the functors Φ±, Ψ±as follows: given Vin RepA(H1×H2), restrict\nit to a representation of H1={1}×H2, then apply the functor Φ±or Ψ±, and\nobservethat H1×{1}acts naturally on the result, since it commutes with {1}×H2.INTERPOLATING LOCAL CONSTANTS IN FAMILIES 9\nMore precisely:\nΦ+,2:RepA(Gn−m×Pm−1)→RepA(Gn−m×Pm)\nV/mapsto→c-IndGn−m×Pm\nGn−m×Pm−1Um(V), with {1}×Umacting viaψ\nˆΦ+,2:Rep(Gn−m×Pm−1)→Rep(Gn−m×Pm)\nV/mapsto→c-IndGn−m×Pm\nGn−m×Pm−1Um(V)\nΦ−,2:Rep(Gn−m×Pm)→Rep(Gn−m×Pm−1)\nV/mapsto→V/V({1}×Um,ψ)\nΨ+,2:Rep(Gn−m×Gm−1)→Rep(Gn−m×Pm)\nV/mapsto→V({1}×Umacts trivially)\nΨ−,2:Rep(Gn−m×Pm)→Rep(Gn−m×Gm−1)\nV/mapsto→V/V({1}×Um,1)\nBecauseH1×{1}commutes with {1}×H2, we immediately get\nLemma 2.13. The analogue of Proposition 2.1 (1)-(6) holds for Φ+,2,ˆΦ+,2,Φ−,2,\nΨ+,2, andΨ−,2.\nDefinition 2.14. We define the functor (−)(0,m): RepA(Gn−m×Gm)→\nRepA(Gn−m)to be the composition Ψ−,2(Φ−,2)m−1.\nThe proof of the following Proposition holds for W(k)-algebrasA:\nProposition 2.15 ([Zel80] Prop 6.7, [Vig96] III.1.8) .LetM=Gn−m×Gm.\nFor0≤m≤nthem’th derivative functor (−)(m)is the composition of the\nJacquet functor JM: Rep(Gn)→Rep(Gn−m×Gm)with the functor (−)(0,m):\nRepA(Gn−m×Gm)→RepA(Gn−m).\nLemma 2.16. LetVbe inRepA(Gn−m×Gm). ThenVcontains an A-submodule\nisomorphic to V(0,m).\nProof.The image of the natural embedding (Φ+,2)m−1Ψ+,2(V(0,m))→V, which\nis given by Proposition 2.13 (5), will be denoted S0,2(V). By Proposition 2.13 (4),\nthe natural surjection V→V(0,m)restricts to a surjection S0,2(V)→V(0,m). By\nProposition 2.13 (6), the map of A-modules S0,2(V)→V(0,m)arises from the map\n(Φ+,2)m−1Ψ+,2(A)→Aby tensoring over AwithV(0,m). Take the A-submodule\ngenerated by any element of (Φ+,2)m−1Ψ+,2(A) that maps to the identity in A;\nthen tensor with V(0,m). /square\n2.4.Finiteness Results. In this subsection we gathercertain finiteness results in-\nvolving derivatives, most of which are well-known when Ais a field of characteristic\nzero.\nLetHbeanytopologicalgroupcontainingadecreasingsequence {Hi}i≥0ofopen\nsubgroups whose pro-order is invertible in A, and which forms a neighborhood base\nof the identity in H. IfVis a smooth A[H]-module we may define a projection\nπi:V→VHi:v/mapsto→/integraltext\nHihvfor a Haar measure on HiwhereHihas total measure\n1. TheA-submodules Vi:= ker(πi)∩VHi+1then satisfy/circleplustext\niVi=V.\nLemma 2.17 ([EH12] Lemma 2.1.5,2.1.6) .A smoothA[H]-moduleVis admissible\nif and only if each A-moduleViis finitely generated. In particular, quotients of\nadmissibleA[H]-modules by A[H]-submodules are admissible.10 GILBERT MOSS\nThus the following version of the Nakayama lemma applies to admissible A[H]-\nmodules:\nLemma2.18 ([EH12]Lemma3.1.9) .LetAbe a Noetherian local ring with maximal\nidealm, and suppose that Mis a submodule of a direct sum of finitely generated\nA-modules. If M/mMis finite dimensional then Mis finitely generated over A.\nIfVis admissible, then it is G-finite if and only if V/mVisG-finite. To see this,\ntakeS⊂V/mVan (A/m)[H]-generating set, let Wbe theA[H]-span of a lift to\nV. SinceV/Wis admissible, we can apply Nakayama to each factor ( V/W)ito\nconcludeV/W= 0.\nProposition 2.19 ([EH12] 3.1.7) .Letκbe aW(k)-algebra which is a field, and\nVan absolutely irreducible admissible representation of Gn. ThenV(n)is zero or\none-dimensional over κ, and is one-dimensional if and only if Vis cuspidal.\nProposition 2.20 ([Vig96] II.5.10(b)) .Letκbe aW(k)-algebra which is a field.\nIfVis aκ[G]-module, then Vis admissible and G-finite if and only if Vis finite\nlength over κ[G].\nProof.SupposeVis admissible and G-finite. Ifκwere algebraically closed of char-\nacteristic zero (resp. characteristic ℓ), this is [BZ77, 4.1] (resp. [Vig96, II.5.10(b)]).\nOtherwise, let κbe an algebraic closure, then V⊗κis finite length, so Vis finite\nlength.\nIfVis finite length, so is V⊗κκ. Overan algebraicallyclosed field ofcharacteris-\ntic different from p, irreducible representations are admissible ([BZ77, 3.25],[Vig96,\nII.2.8]). Since admissibility is preserved under taking extensions V⊗κbeing finite\nlength implies it is admissible, hence Vis admissible. Thus we can reduce prov-\ningG-finiteness to proving that, given any exact sequence of admissible objects,\n0→W0→V→W1→0 whereW0andW1areG-finite, then VisG-finite. But\nthere is a compact open subgroup Usuch thatW0andW1are generated by WU\n0\nandWU\n1, respectively. It follows that that Vis generated by VU. /square\nLemma 2.21. Letκbe aW(k)-algebra which is a field. If Vis an absolutely\nirreducible κ[Gn]-module, then for m≥0,V(m)is finite length as a κ[Gn−m]-\nmodule.\nProof.We follow [Vig96, III.1.10]. Given j,kpositive integers, let M=Gj×Gk\nand letP=MNbe the associated standard parabolic subgroup. Given τin\nRepκ(Gj) andσin Repκ(Gk), we define τ×σto be the normalized induction\nc-IndP(δ1/2\nN(σ⊗τ)) in Repκ(Gj+k), whereδNdenotes the modulus character of\nN(for the definition of δNsee [BZ77, 1.7]). There exists a multiset {π1,...,π r}\nof irreducible cuspidals such that V⊂π1× ··· ×πr. The Liebniz formula for\nderivatives says that ( π1×π2)(t)has a filtration whose successive quotients are\nπ(t−i)\n1×π(i)\n2. Its proof, given in [BZ77, §7], carries over in this generality. Then\nV(m)⊂(π1× ··· ×πr)(m), which is finite length by induction, using Proposition\n2.19 combined with the Liebniz formula. /square\nProposition 2.22 ([Hel12b] Prop 9.15) .LetMbe a standard Levi subgroup of G.\nIfVinRepA(G)is admissible and primitive, then JMVinRepA(M)is admissible.\nCorollary 2.23. IfAis a local Noetherian W(k)-algebra and Vis admissible and\nG-finite, then V(m)is admissible and G-finite for 0≤m≤n.INTERPOLATING LOCAL CONSTANTS IN FAMILIES 11\nProof.LetM=Gn−m×Gm. By Proposition 2.15, V(m)= (JMV)(0,m), so by\nLemma 2.16, there is an embedding V(m)→JMV. Admissibility and G-finiteness\nmeanVis generated over A[G] by vectors in VKfor some compact open subgroup\nK. SinceVKis finite over A,eVKis nonzero for only a finite set of primitive\nidempotents eofthe Bernstein center, so eV/ne}ationslash= 0for at most finitely many primitive\nidempotents eof the integral Bernstein center. Therefore, Proposition 2.22 ap plies,\nshowingV(m)embeds in an admissible module. Thus by Lemma 2.18, we are\nreduced to proving the result for V:=V/mV. SinceVis admissible and G-finite,\nandA/mischaracteristic ℓ, Lemma2.20shows Visfinite length, thereforeitfollows\nfrom Lemma 2.21 that V(m)is finite length. Applying Lemma 2.20 once more, we\nhave the result. /square\nLoosely speaking, the ( n−1)st derivative describes the restriction of a Gn-\nrepresentation to a G1-representation (see Corollary 2.12). The next result shows\nthat this restriction intertwines a finite set of characters:\nTheorem 2.24. IfAis a localW(k)-algebra and VinRepA(G)is admissible and\nG-finite, then V(n−1)is finitely generated as an A-module.\nProof.ByLemma2.18andCorollary2.23itissufficienttoshowthat V(n−1)isfinite\nover the residue field κ. We know V(n−1)isG-finite and admissible by Corollary\n2.23, hence finite length asa κ[G1]-module by Proposition2.20. Since G1is abelian,\nall composition factors are 1-dimensional, so V(n−1)being finite length implies it\nis finite dimensional over κ. /square\nSince the hypotheses of being admissible and G-finite are preserved under local-\nization by Proposition 2.1 (6), we can go beyond the local situation:\nCorollary 2.25. LetAbe a Noetherian W(k)-algebra and suppose that Vis ad-\nmissible and G-finite. Then for every pinSpecA,V(n−1)\npis finitely generated as\nanAp-module.\n2.5.Co-Whittaker A[G]-Modules. In this subsection we define co-Whittaker\nrepresentations and show that every admissible A[G]-moduleVof Whittaker type\ncontains a canonical co-Whittaker subrepresentation.\nDefinition 2.26 ([Hel12b] 3.3) .Letκbe a field of characteristic different from p.\nAn admissible smooth object UinRepκ(G)is said to have essentially AIG dual if it\nis finite length as a κ[G]-module, its cosocle cos(U)is absolutely irreducible generic,\nandcos(U)(n)=U(n)(the cosocle of a module is its largest semisimple quotient) .\nThis condition is equivalent to U(n)being one-dimensional over κand having\nthe property that W(n)/ne}ationslash= 0 for any nonzero quotient κ[G]-moduleW(see [EH12,\nLemma 6.3.5] for details).\nDefinition 2.27 ([Hel12b] 6.1) .An objectVinRepA(G)is said to be co-Whittaker\nif it is admissible, of Whittaker type, and V⊗Aκ(p)has essentially AIG dual for\neachp.\nProposition 2.28 ([Hel12b] Prop 6.2) .LetVbe a co-Whittaker A[G]-module.\nThen the natural map A→EndA[G](V)is an isomorphism.12 GILBERT MOSS\nLemma 2.29. SupposeVis admissible of Whittaker type and, for all primes p,\nany non-generic quotient of V⊗κ(p)equals zero. Then Vis generated over A[G]\nby a single element.\nProof.Letxbe a generator of V(n), and let ˜x∈Vbe a lift of x. IfV′is the\nA[G]-submodule of Vgenerated by ˜ x, then (V/V′)(n)= 0. Since any non-generic\nquotient of V⊗κ(p) equals zero, ( V/V′)⊗κ(p) = 0 for all p. SinceV/V′is\nadmissible, we can apply Lemma 2.18 over the local rings Apto conclude V/V′is\nfinitely generated, then apply ordinary Nakayama to conclude it is ze ro. /square\nThus every co-Whittaker module is admissible, Whittaker type, G-finite (in fact\nG-cyclic), and Schur, so satisfies the hypotheses of Theorem 3.5, b elow. Moreover,\nevery admissible Whittaker type representation contains a canonic al co-Whittaker\nsubmodule:\nProposition 2.30. LetVinRepA(G)be admissible of Whittaker type. Then the\nsub-A[G]-module\nT:= ker(V→/productdisplay\n{U⊂V: (V/U)(n)=0}V/U)\nis co-Whittaker.\nProof.(V/T)(n)= 0 soTis Whittaker type. Since Vis admissible so is T. Let\npbe a prime ideal and let T:=T⊗κ(p). We show that cos( T) is absolutely\nirreducible and generic. By its definition, cos( T) =/circleplustext\njWjwithWjan irre-\nducibleκ(p)[G]-module. Since the map T→/circleplustext\njWjis a surjection and ( −)(n)\nis exact and additive, the map ( T)(n)→/circleplustext\njW(n)\njis also a surjection. Hence\ndimκ(p)(/circleplustext\njW(n)\nj)≤dimκ(p)(T(n)). SinceTis Whittaker type and T(n)=T(n)is\nnonzero, there can only be one jsuch thatW(n)\njis potentially nonzero. On the\nother hand, suppose some W(n)\njwere zero, then Wjis a quotient appearing in the\ntarget of the map\nV→/productdisplay\n{U⊂V: (V /U)(n)=0}V/U,\nhence as a quotient of Tit would have to be zero, a contradiction. Hence precisely\noneWjis nonzero. Now applying [EH12, 6.3.4] with Abeingκ(p) andVbeing\ncos(T), we have that End G(cos(T))∼=κ(p) hence absolutely irreducible. It also\nshows that cos( T)(n)=W(n)\nj/ne}ationslash= 0. Hence T(n)= cos(T)(n). By Lemma 2.29, Tis\nκ(p)[G]-cyclic; since it is admissible, it is finite length by Lemma 2.20. /square\n2.6.The Integral Bernstein Center. IfAis a Noetherian W(k)-algebra and\nVis anA[G]-module, then in particular Vis aW(k)[G]-module, so we use the\nBernstein decomposition of RepW(k)(G) to studyV.\nLetWbe theW(k)[G]-module c-IndGn\nNnψ. Ifeis a primitive idempotent of Z,\nthe representation eWlies in the block eRepW(k)(G), and we may view it as an\nobject in the category RepeZ(G). With respect to extending scalars from eZtoA,\nthe module eWis “universal” in the following sense:\nProposition 2.31 ([Hel12b] Thm 6.3) .LetAbe a Noetherian eZ-algebra. Then\neW⊗eZAis a co-Whittaker A[G]-module. Conversely, if Vis a primitive co-\nWhittakerA[G]module in the block eRepW(k)(G), andAis aneZ-algebra viaINTERPOLATING LOCAL CONSTANTS IN FAMILIES 13\nfV:eZ →A, then there is a surjection α:W⊗A,fVA→Vsuch thatα(n):\n(W⊗A,fVA)(n)→V(n)is an isomorphism.\nIf we assume Ahas connected spectrum (i.e. no nontrivial idempotents), then\nthe mapfV:Z →Awould factor through a map eZ →Afor some primitive\nidempotent e, hence:\nCorollary 2.32. IfAis a connected Noetherian W(k)-algebra and Vis co-\nWhittaker, then Vmust be primitive for some primitive idempotent e.\nRemark 2.33. Theorems 1.1, 1.2, and 1.4 remain true if the hypothesis that Vis\nprimitive is replaced with the hypothesis that Ais connected.\n3.Zeta Integrals\nIn this section we use the representation theory of Section 2 to de fine zeta inte-\ngrals and investigate their properties.\n3.1.Definition of the Zeta Integrals. We first propose a definition of the zeta\nintegral which is the analog of that in [JPSS79], and then check that t he definition\nmakes sense.\nDefinition 3.1. ForW∈ W(V,ψ)and0≤j≤n−2, letXbe an indeterminate\nand define\nZ(W,X;j) =/summationdisplay\nm∈Z(qn−1X)m/integraldisplay\nx∈Fj/integraldisplay\na∈UFW/bracketleftbigg/parenleftbigg\n̟ma0 0\nx Ij0\n0 0In−j−1/parenrightbigg/bracketrightbigg\nd×adx,\nandZ(W,X) =Z(W,X;0)\nWe first show that Z(W,X;0) defines an element of A[[X]][X−1].\nLemma 3.2. LetWbe any element of IndG\nNnψ. Then there exists an integer\nN <0such thatW(a0\n0In−1)is zero for vF(a)< N. Moreover if Wis compactly\nsupported modulo Nn, then there exists an integer L >0such thatW(a0\n0In−1)is\nzero forvF(a)>L\nProof.There is an integer jsuch that/parenleftbigg\n1pj0\n0 1 0\n0 0In−2/parenrightbigg\nstabilizesW. Forxinpj, we\nhave\nW/parenleftiga0 0\n0 1 0\n0 0In−2/parenrightig\n=W/parenleftig/parenleftiga0 0\n0 1 0\n0 0In−2/parenrightig/parenleftig1x0\n0 1 0\n0 0In−2/parenrightig/parenrightig\n=ψ/parenleftig1ax0\n0 1 0\n0 0In−2/parenrightig\nW/parenleftiga0 0\n0 1 0\n0 0In−2/parenrightig\nWhenevervF(a) is negative enough that axlands outside of ker ψ=p, we get that\nψ/parenleftig1ax0\n0 1 0\n0 0In−2/parenrightig\nis a nontrivial p-power root of unity ζinW(k), hence 1 −ζis the\nlift of something nonzero in the residue field k, and defines an element of W(k)×.\nThis shows that W(a0\n0In−1) = 0. /square\nJust as in [JPSS79], the next two lemmas show that Z(W,X;j) defines an ele-\nment ofA[[X]][X−1] when 0<j <n−2 by reducing it to the case of Z(W,X;0).\nLemma 3.3 ([JPSS79] Lemma4.1.5) .LetHbe a function on G, locally fixed under\nright translation by G, and satisfying H(ng) =ψ(n)H(g)forg∈G,n∈Nn. Then\nthe support of the function on Fjgiven by\nx/mapsto→H/bracketleftbigg/parenleftbigga0 0\nx Ij0\n0 0In−j−1/parenrightbigg/bracketrightbigg14 GILBERT MOSS\nis contained in a compact set independent of a∈F×.\nCorollary 3.4. Ifρdenotes right translation (ρ(g)φ)(x) =φ(xg), andUis the\nunipotent radical of the standard parabolic subgroup of typ e(1,n−1), then there is\na finite set of elements u1,...,u rofUsuch that\nZ(W,X;j) =r/summationdisplay\ni=1Z(ρ(tui)W,X;0)\nfor anyW∈IndG\nNnψ.\nIn [JPSS79], the zeta integrals form elements of the field C((X)) and it is proved\nthat in fact they are elements of the subfield C(X) of rational functions. Whereas\nC((X)) (resp. C(X)) is the fraction field of the domain C[[X]] (resp. C[X,X−1]),\nour ringsA[[X]][X−1] andA[X,X−1] are not in general domains. The first main\nresult of this paper is determining the appropriate fraction ring of A[X,X−1] in\nwhich the zeta integrals Z(W,X;j) live:\nTheorem 3.5. SupposeAis aNoetherian W(k)-algebra. Let Sbe the multiplicative\nsubset ofA[X,X−1]consisting of polynomials whose first and last coefficients ar e\nunits. Then if Vis admissible, Whittaker type, and G-finite, then Z(W,X;j)lies\ninS−1A[X,X−1]for allWinW(V,ψ)for0≤j≤n−2.\nIn particular, the result holds if Vis co-Whittaker, as in Theorem 1.1. The proof\nof Theorem 3.5 will occupy the remainder of this section. The key idea is that the\nzeta integrals Z(W,X) are completely determined by the values W/parenleftbiga0\n0In−1/parenrightbig\nfor\na∈F×, and asWranges over W(V,ψ), the set of these values is equivalent to the\ndata of the P2-representation (Φ−)n−2K. Determining the rationality of Z(W,X)\nwill then reduce to a finiteness result for the quotient K(n−1), or more generally for\nV(n−1).\n3.2.Proof of Rationality. Denote byτthe righttranslationrepresentationof G1\nonK(n−1). LetBbe the commutative A-subalgebra of End A(K(n−1)) generated\nbyτ(̟) andτ(̟−1), where̟is a uniformizer of F. It follows from Corollary\n2.25 that K(n−1)\npis finitely generated over Ap. For every pof SpecA, the inclusion\nBp⊂End(K(n−1))p֒→End(K(n−1)\np), showsBpis finitely generated as an Ap-\nmodule.\nLemma 3.6. Bis finitely generated as an A-module.\nProof.Bis the image of the map A[X,X−1]→EndA(K(n−1)) sendingXtoτ(̟).\nBpis the image of the localized map Ap[X,X−1]→/parenleftbig\nEndA(K(n−1))/parenrightbig\np, which is\nfinitely generated. Thus for every p,τ(̟) andτ(̟−1) satisfy monic polynomials\nsp(X),tp(X) with coefficients in Ap. Sincespandtphave finitely many coefficients\nthere exists a global section fp/∈psuch thatsp(X),tp(X) lie inAfp[X]. The open\nsubsetsD(fp) cover Spec Aand we can take a finite subset {f1,...,f n} ⊂ {fp}\nsuch that (fi) = 1. Since τ(̟) andτ(̟−1) satisfy monic polynomials over Afi, we\nhave thatBfiis finitely generated over Afifor eachi. It follows that Bis finitely\ngenerated over A. /square\nSinceBis finitely generated over A,τ(̟) andτ(̟−1) satisfy monic polynomials\nc0+c1X+...cr−1Xr−1+Xrandb0+b1X+···+bs−1Xs−1+Xsrespectively.INTERPOLATING LOCAL CONSTANTS IN FAMILIES 15\nThe degrees randsare nonzero because τ(̟) andτ(̟−1) are units in B. Adding\nthese together we have\n0 =τ(̟)−s+bs−1τ(̟)−s+1+···+b0+c0+...cr−1τ(̟)r−1+τ(̟)r,\nhenceτ(̟) satisfies a Laurent polynomial whose first and last coefficients are units.\nThe final ingredient in proving rationality is the following transformat ion prop-\nerty.\nLemma 3.7. Z(̟nW,X) =X−nZ(W,X)for anyW∈ W(V,ψ), and any integer\nn.\nProof of Lemma. Denotebybmthecoefficient/integraltext\nUFW(̟mu)d×u. ThenZ(̟nW,X)\nis/summationtext\nm∈ZXmbm+n, which can be rewritten X−nZ(W,X). /square\nDeducing Theorem 3.5. The representation K(n−1)is formed by restricting the\nright translation representation on (Φ−)n−2KfromP2toG1, then taking the quo-\ntient by the G1-stable submodule (Φ−)n−2K(U2,1). By Corollary 2.12, the right\ntranslation representation on (Φ−)n−2Kis given by translations of the restricted\nKirillov functions W|(x0\n0I), denotedW(x) for short. As an endomorphism of the\nquotientmodule K(n−1),τ(̟) satisfiesapolynomial Xn−an−1Xn−1−···−a1X−a0\n(in fact we can take a0to be−1). Hence for any restricted Kirillov function W(x)\nwe have\n̟nW(x) =n−1/summationdisplay\ni=0ai̟iW(x)+W1(x),\nfor some element W1of ((Φ−)n−2K)(U2,1). Therefore we get a relation\nZ(̟nW,X) =n−1/summationdisplay\ni=0aiZ(̟iW,X)+Z1(X)\nwithZ1(X) being a Laurent polynomial by Lemma 3.2. Using Lemma 3.7, then\nmultiplying through by Xnand rearranging we get Z(W,X)(1−/summationtextn−1\ni=0aiXn−i) =\nXnZ1(X) which completes the proof since a0is a unit. /square\n4.Functional Equation and Gamma Factor\n4.1.Contragredient Whittaker Functions. There isan analogueofthe contra-\ngredient which is reflected on the level of Whittaker functions by a t ransform/tildewidest(−);\nthe functional equation will relate the zeta integral of Wto that of its transform.\nWe will need the following two matrices:\nw=\n0···0 1\n0··· −1 0\n...\n(−1)n−1···0 0\n, w′=\n(−1)n0···0\n0 0 ···(−1)n−2\n......\n0 (−1)0···0\n\nFor any element Wof IndG\nNnψ, define the transform /tildewiderWofWas/tildewiderW(g) :=W(wgι),\nwheregι:=tg−1.\nObservation 4.1. IfVis of Whittaker type, then for v∈V,/tildewiderWvis an element of\nIndG\nNψbecause\n/tildewiderW(ng) =W(w(ng)ι) =W(wnιw−1wgι) =ψ(wnιw−1)W(wgι) =ψ(n)/tildewiderW(g).16 GILBERT MOSS\nThusZ(/tildewidew′W,X;j) lands inA[[X]][X−1] by Lemma 3.2. In this section we state\nthe second main result and recover the rationality properties of Se ction 2.2 for\nZ(/tildewidew′W,X;j). The second main result is as follows:\nTheorem 4.2. SupposeAis a Noetherian W(k)-algebra, and suppose Vin\nRepA(G)is co-Whittaker and primitive. Let Sdenote the multiplicative subset of\nTheorem 3.5. Then there exists a unique element γ(V,X,ψ)ofS−1A[X,X−1]such\nthat for any W∈ W(V,ψ),\nZ(W,X;j)γ(V,X,ψ) =Z(/tildewidew′W,1\nqnX;n−2−j)\nfor0≤j≤n−2.\nThe proof of Theorem 4.2 is in Section 5. We now verify that Z(/tildewidew′W,1\nqnX;j)\nalways lives in S−1A[X,X−1].\nProposition 4.3. SupposeVinRepA(G)is admissible, Whittaker type, G-finite,\nSchur, and primitive. Let Vιdenote the smooth A[G]-module whose underlying A-\nmodule isVand whoseG-action is given by g·v=gιv. ThenVιis also admissible,\nWhittaker type, G-finite, Schur, and primitive.\nProof.Consider the map Hom Nn(V,ψ)→HomNn(Vι,ψ) given by λ/mapsto→/tildewideλ, where\n/tildewideλ:x/mapsto→λ(wx). We have /tildewideλ(n·v) =λ(wnιw−1wv) =ψ(n)/tildewideλ(v), which shows\n/tildewideλindeed defines an element of Hom Nn(Vι,ψ). Sincew2= (−1)n−1In, it is an\nisomorphism of A-modules. Admissibility, G-finiteness, and Schur-ness all hold for\nVιsinceg/mapsto→gιis a topological automorphism of the group G. SinceVis Schur,\nAmust be connected, hence Vmust be primitive since it is Schur. /square\nIn particular, ( Vι)(n)=Vι/Vι(Nn,ψ) is free of rank one and we may define\n(/tildewiderW)v(g) =/tildewideλ(gιv) and take W(Vι,ψ) to be the A-module {(/tildewiderW)v:v∈Vι}as\nbefore. Note that this is precisely the same as {/tildewide(Wv) :v∈V}. We record this\nsimple observation as a Lemma:\nLemma 4.4. Ifλis a generator of HomNn(V,ψ)then/tildewideλ:x/mapsto→λ(wx)is a generator\nofHomNn(Vι,ψ)and defines W(Vι,ψ). There is an isomorphism of G-modules\nW(V,ψ)→ W(Vι,ψ)sendingWto/tildewiderW.\nThus all the hypotheses for the results of the previous sections, in particular\nTheorem 3.5, apply to Vιwhenever they apply to V, so we get Z(/tildewidew′W,X;j)\nis inS−1A[X,X−1]. Now we can make the substition1\nqnXforXin the ratio of\npolynomials Z(/tildewidew′W,X;j)to getZ(/tildewidew′W,1\nqnX;j). It will againbe in S−1A[X,X−1]\nbecause this process swaps the first and last coefficients in the den ominator (and q\nis a unit in Asinceqis relatively prime to ℓ).\n4.2.Zeta Integrals and Tensor Products. The goal of this subsection is to\ncheck that the formation of zeta integrals commutes with change o f base ring A.\nFor anyf:A→B, denote by ψA⊗Bthe free rank one B-module with action\ngiven by the character f◦ψ. The group action on V⊗ABis given by acting in\nthe first factor. Let idenote the map V→V⊗AB. Proposition 2.1 (6), gives the\nfollowing lemma.\nLemma 4.5. (1) IfVis of Whittaker type, so is V⊗AB.INTERPOLATING LOCAL CONSTANTS IN FAMILIES 17\n(2) Letλgenerate HomA[N](V,ψ)as anA-module. Then λ⊗idis a generator of\nHomB[N](V⊗B,ψ⊗B).\n(3) LetWv⊗b(g) := (f◦λ)(gv)⊗bdefine elements of W(V⊗B,ψ⊗B). Then\nf◦Wv=Wi(v)for anyv∈V.\nFrom the definition of integration given in §1.1, it follows that if Φ kis the\ncharacteristic function of some Hk, then/integraltext\n(f◦Φk)d(f◦µ×) = (f◦µ×)(Hk) =\nf/parenleftbig/integraltext\nΦkd(µ×)/parenrightbig\n. It follows from the definitions that ( f◦/tildewiderW)(x) =/tildewiderf◦W(x).\nCorollary 4.6. LetFdenote the map of formal Laurentseries rings A[[X]][X−1]→\nB[[X]][X−1]induced by f, then we have\nF(Z(Wv,X;j)) =Z(f◦W,X;j) =Z(Wi(v),X;j) (1)\nF/parenleftig\nZ(/tildewidew′W,X;j)/parenrightig\n=Z(f◦/tildewidew′W,X;j) =Z(/tildewiderw′(f◦W),X;j) (2)\nfor anyWinW(V,ψ), and for 0≤j≤n−2.\nThe next proposition follows from the linearity of the zeta integrals a nd the\ntransform/tildewidest(−).\nProposition 4.7. Suppose there is an element γ(V,X,ψ)inA[[X]][X−1]satisfying\na functional equation as in Theorem 4.2 for all Wv∈ W(V,ψ). Then the element\nF(γ(V,X,ψ))∈B[[X]][X−1]satisfies the functional equation for all W∈ W(V⊗\nB,ψ⊗B).\n4.3.Construction of the Gamma Factor. We define the gamma factor to be\nwhat it must in order to satisfy the functional equation of Theorem 4.2 for a single,\nparticularly simple Whittaker function W0. We seek a W0such thatZ(W,X;0) is\na unit inS−1A[X,X−1].\nBy Proposition2.8 and Lemma 2.10, we have that c-IndP2\nU2ψ⊂(Φ−)n−2K. Since\nc-IndP2\nU2ψis isomorphic to C∞\nc(F×,A) via restriction to G1(recall that C∞\nc(F×,A)\ndenotes the locally constant compactly supported functions F×→A), we find the\nfollowing:\nProposition 4.8. SupposeVinRepA(G)is of Whittaker type. Then the charac-\nteristic function of U1\nFis realized as a restricted Whittaker function W0(g0\n0In−1)for\nsomeW0inW(V,ψ).\nFrom now on, the symbol W0will denote a choice of element in W(V,ψ) whose\nrestriction to (g0\n0In−1) is the characteristic function of U1\nF. ThenZ(W0,X) is/integraltext\nUFW1(a0\n0 1)d×a=µ×(U1\nF) = 1. Since we want our gamma factor to satisfy the\nfunctional equation for W0, we are left with no choice:\nDefinition 4.9 (The Gamma Factor) .LetAbe any Noetherian W(k)-algebra\nand suppose VinRepA(G)is of Whittaker type. We define the gamma factor\nofVwith respect to ψto be the element of A[[X]][X−1]given byγ(V,X,ψ) :=\nZ(/tildewiderw′W0,1\nqnX;n−2).\nWhenVis co-Whittaker and primitive the uniqueness of this gamma factor will\nfollow from the functional equation: if γandγ′both satisfy the functional equation\nfor all Whittaker functions, then γ=Z(/tildewiderw′W0,1\nqnX,n−2) =γ′. In particular for\nsuch representations our construction of the gamma factor doe s not depend on the\nchoice ofW0.18 GILBERT MOSS\n4.4.Functional Equation for Characteristic Zero Points. If the residue field\nκ(p) ofphas characteristic zero, the reduction modulo pofZ(W,X;j) forms an\nelement of κ(p)(X). Asκ(p) is an uncountable algebraically closed field of char-\nacteristic zero, we may fix an embedding C֒→κ(p). The proof of [JPSS83, Thm\n2.7(iii)(2)] (which occurs in [JPSS83, §2.11]) carries over verbatim to the setting\nwhereπandπ′are admissible, Whittaker type, G-finite representations over any\nfield containing C, hence for representations over κ(p). Thus the reduction modulo\npof Ψ(W,X;j) is precisely the integral Ψ( s,W;j) of [JPSS79, §4.1], after replac-\ning the complex variable q−(s+n−1\n2)with the indeterminate X, and there exists a\nunique element, which we will call γp(s,V⊗κ(p),ψ), inκ(p)(q−s) such that for all\nW∈ W(Vp⊗κ(p),ψp) and for all j≥0,\nΨ(1−s,/tildewidew′W;n−2−j) =γp(s,V⊗κ(p),ψp)Ψ(s,W,j).\nThe changeofvariable s/mapsto→1−scan be re-written as −(s+n−1\n2)/mapsto→(s+n−1\n2)−n, so\nwriting the functional equation in terms of Xwe have shown the following Lemma:\nLemma 4.10. SupposeVis admissible of Whittaker type, and G-finite. For each\nprimepofAwith residue characteristic zero, there exists a unique ele mentγp(V⊗\nκ(p),X,ψp)inκ(p)(X)such that for all WinW(V⊗κ(p),ψp)and for0≤j≤n−2\nwe have\nZ(/tildewidew′W,1\nqnX;n−2−j) =γp(X,V⊗κ(p),ψp)Z(W,X;j).\nMoreover,γp(V⊗κ(p),X,ψp) =γ(V,X,ψ) modpby uniqueness in [JPSS79] .\n4.5.Proof of Functional Equation When Ais Reduced and ℓ-torsion Free.\nIn the case that Ais reduced and ℓ-torsion free as a W(k)-algebra, we get a slightly\nstronger result than that of Theorem 4.2.\nTheorem 4.11. IfAis a Noetherian W(k)-algebra and Ais reduced and ℓ-torsion\nfree, then the conclusion of Theorem 4.2 holds for any VinRepA(G)which is\nG-finite, and admissible of Whittaker type.\nProof.Letpbe any characteristic zero prime, and let fp:A→κ(p) be reduction\nmodulop. Corollary 4.6 and Lemma 4.10 tell us that\nfp/parenleftbigg\nγ(V,X,ψ)Z(W,X)−Z(/tildewidew′W,1\nqnX;n−2)/parenrightbigg\n= 0\nforanyWinW(V,ψ), not justW0. This shows that the difference\nγ(V,X,ψ)Z(W,X)−Z(/tildewidew′W,1\nqnX;n−2)\nis in the intersection of all characteristic zero primes of A. WhenAis reduced its\nzero divisors are the union of its minimal primes, so it is ℓ-torsion free if and only\nif all minimal primes have residue characteristic zero. Thus when Ais reduced and\nℓ-torsion free, the intersection of all characteristic zero primes o fAequals zero, so\nthe functional equation holds for any WinW(V,ψ).\nWe now prove uniqueness. If there were another element γ′satisfying the same\nproperty, it would satisfy the functional equation in κ(p) for allWi(v)by reduction,\nso it satisfies the functional equation for all WinW(V⊗κ(p),ψp). But uniqueness\nin Lemma 4.10 then shows fp(γ(V,X,ψ)−γ′) = 0 for all characteristic zero primes\npofA. Again, this means γ′=γ(V,X,ψ).INTERPOLATING LOCAL CONSTANTS IN FAMILIES 19\nWe get rationality by observing that whenever Vis admissible of Whittaker\ntype, it has a canonical co-Whittaker submodule Tby Proposition 2.30, which is\nprimitive if Vis primitive. Since γ(T,X,ψ) satisfies the functional equation for all\nWinW(T,ψ), we must have γ(T,X,ψ) =γ(V,X,ψ) by the construction of the\ngamma factor. But γ(T,X,ψ) is inS−1A[X,X−1] by Theorem 3.5, which holds for\nprimitive co-Whittaker modules. /square\n5.Universal Gamma Factors\nWhenVis primitive and co-Whittaker, we can remove the hypothesis that A\nis reduced and ℓ-torsion free by specializing the gamma factor for the universal\nco-Whittaker module eW.\nTheorem 5.1 ([Hel12a] Thm 12.1) .Any block eZof the Bernstein center of\nRepW(k)(G)is a finitely generated (hence Noetherian), reduced, ℓ-torsion free\nW(k)-algebra.\nBy Proposition 2.31, eWis co-Whittaker, and since it is clearly primitive, all\nthe hypotheses of Proposition 4.11 are satisfied. Hence (Thm 4.11) there exists\na unique gamma factor in S−1(eZ[X,X−1]), which we will denote Γ( eW,X,ψ),\nsatisfying the functional equation for all WinW(eW,ψ).\nProof of Theorem 4.2. SinceVis primitive and co-Whittaker, there is a (unique)\nprimitive idempotent eofZand a ring homomorphism fV:eZ →EndG(V)∼→A,\nand a surjection of A[G]-moduleseW⊗fVA→Vpreserving the top derivative,\nso thatfV(Γ(eW,X,ψ)) =γ(eW⊗fVA,X,ψ). Since Γ( eW,X,ψ) satisfies the\nfunctional equation for all WinW(eW,ψ), we can apply Proposition 4.7 again to\nconcludethat γ(eW⊗A,X,ψ)satisfiesthe functionalequationforall WinW(eW⊗\nA, ψ). SinceeW⊗Ahas a surjection onto Vpreserving the top derivative, Lemma\n2.6 tells us that W(V,ψ) =W(eW⊗A,ψ). The functional equation shows that\nDefinition 4.9 gives a unique gamma factor, hence γ(V,X,ψ) =γ(eW⊗A,X,ψ); it\nsatisfies the functional equation for all WinW(V,ψ). Note that since Γ( eW,X,ψ)\nis inS−1(eZ[X,X−1]), its image in fVis in the corresponding fraction ring of\nA[X,X−1]. This proves Theorem 4.2. /square\nWe can extend the uniqueness and rationality result to a larger class of rep-\nresentations, though with a weaker functional equation coming on ly from the co-\nWhittaker case:\nCorollary 5.2. LetVbe admissible, primitive, of Whittaker type and let Tbe\nits canonical co-Whittaker submodule. Then there exists a u nique gamma factor\nγ(V,X,ψ)inS−1(A[X,X−1])which equals γ(T,X,ψ), and satisfies the functional\nequation for all WinW(T,ψ).\nProof.WhenVis admissible of Whittaker type it has a canonical co-Whittaker\nsub by Proposition 2.30. We have just shown that its gamma factor γ(T,X,ψ)\nsatisfies the functional equation for all WinW(T,ψ). Applying Proposition 2.6\nwithα:T→Vbeing the inclusion map, we conclude that W(T,ψ)⊂ W(V,ψ).\nThe coefficients of the series Z(/tildewiderw′W0,1\nqnX;n−2) in Definition 4.9 are determined\nbyG-translates of the Whittaker function W0, so occurs already in W(T,ψ), so by\ndefinitionγ(T,X,ψ) =γ(V,X,ψ). In particular γ(V,X,ψ) lies inS−1A[X,X−1]\nand satisfies the functional equation for all WinW(T,ψ). /square20 GILBERT MOSS\nWe can make precise the sense in which we have created a universal g amma\nfactor:\nCorollary 5.3. SupposeAis a Noetherian W(k)-algebra, and suppose Vis a\nco-Whittaker A[G]-module in the subcategory eRepW(k)(G)ofRepW(k)(G). Then\nthere is a homomorphism fV:eZ →AandfV(Γ(eW,X,ψ))equals the unique\nγ(V,X,ψ)satisfying a functional equation for all WinW(V,ψ).\nAgain,wecanbroadentheclassofrepresentationsatthecostof amorerestrictive\nfunctional equation:\nTheorem 5.4. SupposeAis any Noetherian W(k)-algebra, and suppose Vis an\nadmissibleA[G]-module of Whittaker type in the subcategory eRepW(k)(G). Then\nthere is a homomorphism fV:eZ →Aand the gamma factor of Corollary 5.2\nequalsfV(Γ(eW,X,ψ)).\nProof.We define fVto be the homomorphism eZ →EndG(T)∼→AwhereT\nis the canonical co-Whittaker submodule of Proposition 2.30. Since Tlies in\neRepW(k)(G),eW⊗fVAsurjectsonto T, andwehave fV(Γ(eW,X,ψ) =γ(T,X,ψ)\n(Prop2.6), andsince Tinjects into V(with topderivativepreserved),againbyProp\n2.6, we have\nfV(Γ(eW,X,ψ)) =γ(eW⊗eZ,fVA,X,ψ) =γ(T,X,ψ) =γ(V,X,ψ).\n/square\nReferences\n[BD84] J. Bernstein and P. Deligne. Le “centre” de Bernstein .Representations des groups re-\nductifs sur un corps local, Traveux en cours , 1984.\n[BH06] Colin Bushnell and Guy Henniart. The Local Langlands Conjecture for GL(2). Berlin\nHeidelberg: Springer-Verlag, 2006.\n[BZ76] I. N. Bernshtein and A. V. Zelevinskii. Representati ons of the group GL(n,F) where F\nis a nonarchimedean local field. Russian Math. Surveys 31:3 (1976), 1-68 , 1976.\n[BZ77] I. N. Bernshtein and A. V. Zelevinskii. Induced repre sentations of reductive p-adic\ngroups. I. Annales scientifiques de l’ ´E.N.S., 1977.\n[CPS10] James Cogdell and I.I. Piatetski-Shapiro. Derivat ives and L-functions for GL(n).The\nHeritage of B. Moishezon, IMCP , 2010.\n[EH12] Matthew Emerton and David Helm. The local Langlands c orrespondence for GL(n) in\nfamilies. Ann. Sci. E.N.S. , 2012.\n[Hel12a] David Helm. The Bernstein center of the category of smooth W(k)[GLn(F)]-modules.\narxiv:1201.1874 , 2012.\n[Hel12b] David Helm. Whittaker models and the integral Bern stein center for GL(n).\narXiv:1210.1789 , 2012.\n[Hen93] Guy Henniart. Caracterisation de la correspondanc e de langlands locale par les facteurs\nǫde paires. Inventiones Mathematicae , 1993.\n[Jia13] Dihua Jiang. On the local Langlands conjecture and r elated problems over p-adic local\nfields.Proceedings of the 6th International Congress of Chinese Ma thematicians , 2013.\n[JPSS79] Herve Jacquet, Ilja Iosifovitch Piatetski-Shapi ro, and Joseph Shalika. Automorphic\nforms on GL(3) I.The Annals of Mathematics , 1979.\n[JPSS83] Herve Jacquet, Ilja Iosifovitch Piatetski-Shapi ro, and Joseph Shalika. Rankin-Selberg\nconvolutions. American Journal of Mathematics , 1983.\n[KM14] Robert Kurinczuk and Nadir Matringe. Rankin-selber g local factors modulo ℓ.\narXiv:1408.5252v2 , 2014.\n[M´12] Alberto M´ ınguez. Fonctions Zˆ eta ℓ-modulaires. Nagoya Math. J. , 208, 2012.\n[Vig96] Marie-France Vigneras. Representations ℓ-modulaires d’un groupe reductif p-adique avec\nℓdifferent de p. Boston: Birkhauser, 1996.INTERPOLATING LOCAL CONSTANTS IN FAMILIES 21\n[Vig00] Marie-France Vigneras. Congruences modulo ℓbetween ǫfactors for cuspidal represen-\ntations of GL(2).Journal de theorie des nombres de Bordeaux , 2000.\n[Zel80] A. V. Zelevinsky. Induced representations of reduc tivep-adic groups. II. on irreducible\nrepresentations of GL(n).Annales scientifiques de l’ ´E.N.S., 1980."    },    {        "title": "1803.01280v2.Optimization_of_Time_Resolved_Magneto_optical_Kerr_Effect_Signals_for_Magnetization_Dynamics_Measurements.pdf",        "content": "Optimization of Time -Resolv ed Magneto -optical Kerr Effect S ignals for \nMagnetization Dynamics Measurements  \nDustin M. Lattery1, Delin Zhang2, Jie Zhu1, Paul Crowell3, Jian-Ping Wang2 and Xiaojia Wang1* \n1Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA  \n2Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN \n55455, USA   \n3School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455, USA  \n*Corresponding authors: wang4940@umn.edu \n \nAbstract: Recently magnetic storage and magnetic memory have shifted towards the use of \nmagnetic thin films with perpendicular magnetic anisotropy (PMA). Understanding the magnetic \ndamping in these ma terials is crucial, but normal Ferromagnetic Resonance (FMR) measurements \nface some limitations. The desire to quantify the damping in materials with PMA has resulted in \nthe adoption of Time -Resolved Magneto -optical Kerr Effect (TR -MOKE) measurements. In t his \npaper, we discuss the angle and field dependent signals in TR -MOKE, and utilize a numerical \nalgorithm based on the Landau -Lifshitz -Gilbert (LLG) equation to provide information on the \noptimal conditions to run TR -MOKE measu rements . \n \nI. INTRODUCTION  \nSpintronics utilizing perpendicular magneti c anisotropy (PMA) are very promising for the \nadvancement of computer memory, logic, and storage. Due to the time scale of magnetic switching \nin these devices (~ 1 ns), it is crucial to understand the ultrafast dy namic magnetization, which \nbehave according to the Landau -Lifshitz -Gilbe rt (LLG) equation. The application of this equation to understand magnetization dynamics requires knowledge of the magnetic anisotropy and the \nGilbert damping (α). While anisotropy can  be determined through magnetostatic measurements, \nextracting α requires measurements that can capture the dynamic magnetization at time scales \nfaster than magnetic switching. To date, the most common method to do this is through frequency \ndomain measureme nts of ferr omagnetic resonance ( FMR ). By measuring the resonance frequency \nand linewidth as a function of field, FMR can probe both the magnetic anisotropy a nd Gilbert \ndamping . As spintronic  applications  begin to use materials  with large PMA, the use  of another \ntechnique, time -resolved magneto -optical Kerr effect (TR -MOKE), has increased. This technique \n(which is essentially a time -domain FMR measurement technique) is able to measure at higher \nresonance frequencies and external fields, which allows ex tremely hard mag netic materials to be \nmeasured . \nThere are many papers discussing TR -MOKE measurements for measuring the Gilbert \nDamping. Most of these papers utilize similar polar MOKE measurement techniques, but there is \noften a large variation in both  the Hext range for measurements and in the angle of external field. \nWhile some papers utilize in -plane external field because of its well -understood frequency \ndependence, others choose to apply the field at a chosen angle away from the surface normal . It \nhas been theorized and shown in measurements that the process of applying the field at some angle \nbetween 0 and 90°  is beneficial to increase the TR -MOKE signal amplitude, but the explanations \nas to why this occurs are lacking . In this paper, we aim to discuss why the signal depends on the \nangle of external field and calculate the optimal angle for conducting TR -MOKE measurements \nof damping on magnetic materials with PMA.  \n \n II. FINITE DIFFERENCE METHOD LANDAU -LIFSHITZ -GILBERT EQUATIONS  \nSimulations in this  work utilize a finite difference approach to solve the LLG equation \n(Eq. 1) with an explicit solution for the magnetization vector ( M) as a function of time following \nthe forward Euler method.  \neff\nsdd\ndt M dt    MMM H M\n          (1) \nwhere M is the magneti zation vector with a magnitude of Ms (the saturation magnetization), γ is \nthe gyromagnetic ratio, Heff is the effective magnetic field, and α is the Gilbert damping parameter. \nThe vector Heff is determined by taking the gradient of the magnetic free energy  density ( F) with \nrespect to the magnetization direction (\neff FM H ). The scalar quantity F is the summation of \ncontributions from Zeeman energy (from the external magnetic field, Hext), perpendicular uniaxial \nmagnetic anisotropy ( Ku), and  the demagnetizing field (assuming the sample is a magnetic thin \nfilm).  \nWhile Eqn. 1 is often used to describe magneto -dynamics due to the use of α, it is not \nconducive to numerical solutions of this ordinary differential equation. To simplify the \ndevelopment of computational algorithms, it is preferential to utilize the Landau -Lifshitz equation \n(Eq. 2).  \n  eff eff 2\ns'd\ndt M    MM H M M H\n.             (2) \nThe coefficients in Eq. 2 can be related to the previously defined constants in Eqs. 3 and 4 [1]. \n2'\n1\n\n\n           (3) \ns'M \n           (4) In equilibrium, M is parallel to Heff, and so the magnetization does not precess . If the \nmagnetization is removed from the equilibrium direction, it will begin precessing around the \nequilibrium direction, finally damping towards equilibrium at a rate determined by the magnitude \nof α (shown in Fig. 1) .  \n \nFigure 1 . A three -dimensional representation of the magnetization vector ( M) precessing around the equilibrium \ndirection ( θ) displayed on the surface of a sphere of radius Ms. The equilibrium direction  is controlled by the magnitude \nand direction ( θH) of the external magnetic field vec tor (Hext). The change in the z-component of magnetization (Δ Mz) \nis proportional to the TR -MOKE signal.  \n \nTo initiate precession, a thermal  demagnetization process  is applied , emulating  TR-MOKE \nmeasurements. For TR -MOKE measurements, a “pump” laser pulse increases the temperature at \nan ultrafast time scale, causing a thermal demagnetization (a decrease in Ms caused by temperature) \n[2, 3] . This thermal demagnetization temporarily moves the equilibrium direction causing the \nmagnetization to begin precession, which is continued even when Ms has recovered to its original \nstate.  Here, the demagnetization process is treated as a step decrease in Ms that lasts for 2.5 ps \nbefore an instant recovery to the initial value.  All signal analysis discussed in this work  is following \nthe recovery of Ms. \nFor polar MOKE measurements, the projected magnetization in the z -direction ( Mz, \nthrough -plane magnetization ) is proportional to the Kerr rotation  [4]. The projection of Mz in time \nduring precession will appear is a decaying sinusoid  (\n sin exp /zM t t t      ), which is \nalso captured by TR -MOKE measurements. The amplitude of the precession will greatly depend \non the applied field magnitude and angle, which is also carried into TR -MOKE signal. By \nanalyzing the precession as a function of field and angle, the precession amplitude (delta Mz) can \nbe extracted. Figure 2 shows the process of extracting the amplitude as a function of angle for two \ndifferent regions of magnetic field. Tracking this signal amplitude as a function of θH, reveals  that \nthe precession (and thus the signal) will be maximized for a certain θH as shown in Fig. 2(b). \nMaximizing the oscillation implies that it will be beneficial to maximize the “magnetic torque” \nterm (M × Heff, which prefers a large angle between M and Heff), but it also important to factor in \nthat TR -MOKE measures the projection of the magnetization along the z-direction (which prefers \nθ = 90°). Because of this, the value of θH,MAX  requires weighing inputs from both the magnetic \ntorque and the z-direction projection of  magnetization.  \n \nFigure 2. For specific conditions, the LLG simulation will produce a time -dependent magnetization vector. The \ndifference between the maximum and mini mum of the z-component of magnetization in time (Δ Mz) provides \ninformation about the strength of the TR -MOKE signal. These simulations are conducted for a range of θH resulting \nin the curves in (b). The trend of signal with increasing θH also depends on th e magnitude of the external field relative \nto Hk,eff, as shown by the black ( Hk,eff < Hext) and red ( Hk,eff > Hext) lines.  \nDepending on whether the field ratio ( Hext/Hk,eff) the angular dependence on magnitude \nwill drastically change. For Hext<Hk,eff, the magnetization will be in equilibrium between the \nperpendicular direction and the in -plane direction (0  ≤ θ ≤ 90°). Maximizing the magnetic torque \nand projection in the z -direction in these cases will cause Hext applied in -plane (θH = 90°) to be the \noptimal setup [shown by the red line in Fig. 4(b)]. Once Hext exceeds Hkeff, the Stoner -Wolfarth \nminimum energy model predicts that the magnetization will approach the direction of external \nfield, but never align (except along θH = 0 or 90 °). Because these two directions will have no \nmagnetic torque, there should be no magnetic precession, and thus there will be amplitude minima \nat these extremes. Between these two angles, the two effects for optimizing signal will complete, \nleading to a n amplit ude maximum at an angle that depends on the size of the ratio.  \nFigure 3 shows a contour plot of the dependence on  signal amplitude as a function of both \nthe magnitude of Hext and θH. The highest amplitude of precession will occur near Hk,eff when the \nfield is applied in the film plane. If the external field is greater than Hk,eff, it is beneficial to conduct \nthe measurement at an angle that is out of the film plane. To reveal this trend, the dotted red line \nin Fig. 3 indicates the angle of maximum signal ( θH,MAX ) as a function at specified field ratios. \nNote that the curve does not follow the gradient of signal vs. θH and Hext. This is due to the \ndefinition of θH,MAX  as the value of θH that maximizes signal for a given Hext , instead of a \nmaximization of signal with both parameters.  Based on these results, measurement conditions can \nbe tuned to maximize the signal based on the field ratio. For example, if the maximum strength of \nthe magnetic field is only 2 Hk,eff, then it would be beneficial to set θH > 60°. Furthermore, \nmeasurements conducted at a constant field and a varied magnetic field angle, should not \nnecessarily conduct the measurement at the highest possible Hext if the goal is to maximize SNR.   \nFigure 3. A contour plot of the relative signal size  as a function of field ratio ( Hext/Hk,eff) and θH where a value of “1” \nindicates the maximum possible signal. The dotted line shows the θH where the signal is maximized at a specific field \nratio.  \n \nFor field -swept measurements, (where the angle is held constant and the field is swept) \nFig. 4 should provide a simple guide for maximizing signals  (a summary of θH,MAX  in Fig. 3). To \nfurther assist in the design of TR -MOKE signals to maximize SNR, we suggest a simplified \nestimation for the determination of th e amplitude of TR -MOKE signal. Equation 5 predicts the \nprecession amplitude based on the equilibrium direction ( θ, from Fig. 1) and the external field \nangle. The magnitude of Hext is integrated into Eq. 5 through the θ through Eq. 6 whic h provides \nthe mini mum energy condition.  \nH\nssin sinzM\nM  \n      (5) \n  ext H k,eff2 sin sin 2HH  \n         (6) \nThis simplified expression is based on the product of the two components for signal \nmaximization previously discussed: the projection of the magnetization in the z -direction , \n sin , \nand the magnetic torque, \nH sin . While the simplified expression presented in Eq. 2 cannot \ncapture all the details of a more complex LLG simulation, it is more than accurate enough for an \ninitial estimate of θH,MAX , as shown by the comparison in Fig. 4.  \n \nFigure 4. The trend of θH,MAX  at a given field ratio. The open circles indicate results from the LLG simulation discussed \nin Section I, while the red curve is the simplified model from Eq. 5.  \n \nIII. COMPARING SIMULATION RESULTS TO TR -MOKE MEASUREMENTS  \nTo verify the precited results for the m aximum TR -MOKE signal amplitude, a series of \nmeasurements were conducted on a 300 °C post -annealed W/CoFeB/MgO film (see our previous \npublication for more information ). After conducting measurements, the thermal background was \nsubtracted leaving purely the decaying sinusoidal term. The oscillation amplitude from \nmeasurement was calculated as shown in Fig. 2a. Results from four values of Hext and six value s \nof θH are summarized in Fig. 5.  \n \nFigure 5. Normalized TR -MOKE oscillation amplitudes directly for a W/CoFeB/MgO  when Hext is 4, 6, 8, and \n10 kOe. The open red circles show the measurement data (a line between points is provided to guide the eye) while \nthe black curves indicate the results from the LLG simulations for a material with Hk,eff ≈ 6 kOe.  \n  \nComparisons between the trends predicted simula tions and measurement results show \nremarkable agreement. As expected, the signal amplitude decreases with increasing angle for \nHext < Hk,eff (Hk,eff ≈ 6 kOe ) and decreases with increasing angle for Hext > Hk,eff. These \nmeasurements can even capture the predicted peak of amplitude at nearly the same θH for fields \nnear Hk,eff. For the 6 kOe measurements, there is a slight deviation in the amount of decay in signal \nstrength for decreasing θH (simulations predict a s lower decrease). This is most likely due to an \ninhomogeneous broadening effect (i.e. the Hk,eff in the sample has a distribution of values) leading \nto a deviation from theory near Hk,eff. While the θH in the setup used in this experiment was limited, \nthese  results verify that the excellent agreement between simulation and measurement.  \n \nIV.  CONCLUSION  \nIn conclusion, we utilized a numerical approach to calculate the dynamic  response of \nmagnetization to a demagnetization process. We find that the size of the magnetic precession, and \nthus the size of the TR -MOKE signal depends on the angle and amplitude of the external field \n(relative to Hk,eff). To verify the results of these  simulations, we conducted measurements on a \nW/CoFeB/MgO sample with perpendicular magnetic anisotropy. The results of the measurements \nshow that the magnitude of the TR -MOKE signal shows good agreement with our prediction. \nThese results should assist to m aximize the SNR in TR-MOKE measurements.  \n \nACKNOWLEDGEMENTS  \nThis work is supported by C -SPIN  (award #: 2013 -MA-2381) , one of six centers of STARnet, a \nSemiconductor Research Corporation progra m, sponsored by MARCO and DARPA.  \n \nREFERENCES  \n[1] Iida, S., 1963, \"The difference between gilbert's and landau -lifshitz's equations,\" Journal of \nPhysics and Chemistry of Solids, 24(5), pp. 625 -630. \n[2] van Kampen, M., Jozsa, C., Ko hlhepp, J. T., LeClair, P., Lagae, L., de Jonge, W. J. M., and \nKoopmans, B., 2002, \"All -Optical Probe of Coherent Spin Waves,\" Physical Review Letters, \n88(22), p. 227201.  \n[3] Zhu, J., Wu, X., Lattery, D. M., Zheng, W., and Wang, X., 2017, \"The Ultrafast Laser Pump -\nProbe Technique for Thermal Characterization of Materials With Micro/Nanostructures,\" \nNanoscale and Microscale Thermophysical Engineering, 21(3), pp. 177 -198. \n[4] You, C. -Y., and Shin, S. -C., 1998, \"Generalized analytic formulae for magneto -optical Kerr \neffects,\" Journal of Applied Physics, 84(1), pp. 541 -546. \n "    },    {        "title": "2401.12022v1.Damping_Enhanced_Magnon_Transmission.pdf",        "content": "Damping-Enhanced Magnon Transmission\nXiyin Ye,1Ke Xia,2Gerrit E. W. Bauer,3, 4and Tao Yu1,∗\n1School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China\n2School of Physics, Southeast University, Jiangsu 211189, China\n3WPI-AIMR and Institute for Materials Research and CSRN, Tohoku University, Sendai 980-8577, Japan\n4Kavli Institute for Theoretical Sciences, University of the Chinese Academy of Sciences, Beijing 100190, China\n(Dated: January 23, 2024)\nThe inevitable Gilbert damping in magnetization dynamics is usually regarded as detrimental\nto spin transport. Here we demonstrate in a ferromagnetic-insulator–normal-metal heterostructure\nthat the strong momentum dependence and chirality of the eddy-current-induced damping causes\nalso beneficial scattering properties. Here we show that a potential barrier that reflects magnon wave\npackets becomes transparent in the presence of a metallic cap layer, but only in one direction. We\nformulate the unidirectional transmission in terms of a generalized group velocity with an imaginary\ncomponent and the magnon skin effect. This trick to turn presumably harmful dissipation into useful\nfunctionalities should be useful for future quantum magnonic devices.\nIntroduction .—Magnonic devices save power by ex-\nploiting the collective excitations of the magnetic or-\nder, i.e., spin waves or their quanta, magnons, for non-\nreciprocal communication, reprogrammable logics, and\nnon-volatile memory functionalities [1–10]. The possibil-\nity to modulate magnon states and their transport in fer-\nromagnets by normal metals or superconductors brings\nfunctionalities to spintronics [11–14], quantum informa-\ntion [15–21], and topological materials [22, 23]. The pre-\ndiction of inductive magnon frequency shifts by supercon-\nducting gates on magnetic insulators [24–30] have been\nexperimentally confirmed [31]. Normal metals are not\nequally efficient in gating magnons [32–35], but the stray\nfields of magnetically driven “eddy currents” [36–43] sig-\nnificantly brake the magnetization dynamics [36].\nThe intrinsic Gilbert damping seems to be detrimental\nto transport since it suppresses the magnon propagation\nlength. However, in high-quality magnets such as yt-\ntrium iron garnet (YIG) films, this is not such an issue\nsince the magnon mobility is often limited by other scat-\ntering processes such as two-magnon scattering by disor-\nder, and measurements can be carried out in far smaller\nlength scales.\nNatural and artificial potential barriers are impor-\ntant instruments in electronics and magnonics by confin-\ning and controlling the information carriers. They may\nguide magnon transport [31, 44], act as magnonic logic\ngate [45], induce magnon entanglement [18, 46], and help\ndetecting exotic magnon properties [47–50]. In the lin-\near transport regime, the transmission of electrons and\nmagnons through an obstacle has always been assumed\nto be symmetric, i.e., the same for a wave or particle\ncoming from either side.\nIn this Letter, we address the counter-intuitive ef-\nfect that the strong momentum-dependent eddy-current-\ninduced damping by a normal metal overlayer as shown\nin Fig. 1 may help surmount obstacles such as mag-\nnetic inhomogeneities [51], artificial potential barriers\nformed by surface scratches [52], or dc-current carryingwires [46]. Here we focus on the band edges of magnetic\nfilms that are much thinner than the extinction length of\nthe Damon-Eshbach surface states in thick slabs and are\ntherefore not chiral. Instead, the effect therefore origi-\nnates from the Oersted fields generated by the eddy cur-\nrents in the overlayer that act in only half of the recip-\nrocal space [7] and causes magnon accumulations at the\nsample edges or magnon skin effect [8, 9]. The trans-\nmission through a barrier that is small and symmetric\nfor magnons with opposite wave numbers in an uncov-\nered sample becomes unidirectional with the assistance\nof dissipative eddy currents.\nFIG. 1. Ferromagnetic insulator-normal metal heterostruc-\nture. An in-plane external magnetic field H0orients the\nmagnetization at an angle θwith the ˆz-direction. The yellow\nsheet between the normal metal and ferromagnetic insulator\nindicates suppression of the exchange interaction and conven-\ntional spin pumping.\nModel and non-perturbation theory .—We consider the\nferromagnetic insulator (FI)-normal metal (NM) het-\nerostructure with thickness 2 dFanddMand an in-plane\nmagnetic field H0in Fig. 1. The saturated equilib-\nrium magnetization Msmakes an angle θwith the ˆz-\ndirection such that the torques exerted by the external\nand anisotropy fields cancel. For convenience, we set\nθ= 0 in the following discussion and defer results forarXiv:2401.12022v1  [cond-mat.mes-hall]  22 Jan 20242\nfinite θto the Supplemental Material (SM) [53]. We gen-\neralize a previous adiabatic theory [7, 36] to the full elec-\ntrodynamics of the system by self-consistently solving the\nMaxwell equations coupled with the linearized Landau-\nLifshitz (LL) equations and Ohm’s Law. This treatment\nbecomes exact in the limit of an instantaneous response\nof the metal electrons and high-quality ultrathin mag-\nnetic films.\nThe driving force is an externally generated spatiotem-\nporal magnetization dynamics M(r, t) =M(r, ω)e−iωtat\nfrequency ω. According to Maxwell’s theory, the electric\nfieldEobeys the wave equation ∇2E(r, ω)+k2\n0E(r, ω) =\n−iωµ0JM, where the wave number k0=ω√µ0ε0,µ0(ε0)\nis the vacuum permeability (permittivity), and JM=\n∇×Mis the “magnetization current” [54]. Disregarding\nthe intrinsic Gilbert damping, the LL equation\niωM=−µ0γM×Heff[M] (1)\ngoverns the magnetization dynamics in the FI, where γ\nis the gyromagnetic ratio. The effective magnetic field\nHeff[M] =−δF[M]/δM(r), where the free energy Fis\na functional of the magnetization. It includes the static\nfieldH0, the dipolar field Hd, and (in the FI) the ex-\nchange field Hex=αex∇2Mthat depends on the spin-\nwave stiffness αex. In the presence of the NM layer,\nHeff[M] also contains the Oersted magnetic fields gen-\nerated by the “eddy” currents J=σE, where the elec-\ntrical conductivity σis real. This defines a closed self-\nconsistency problem that we solve numerically.\nWe consider a thin FI film with constant Ms=\n(0,0, Ms). The transverse fluctuations M(r, ω) =\n(Mx(k, ω), My(k, ω),0)eik·rwith in-plane wave vectors\nk= (0, ky, kz) are small precessions with iMx(k, ω) =\nakMy(k, ω), where the complex ellipticity akbecomes\nunity for circular motion.\nThe electric-field modes outside the magnet are plane\nwaves with wave numbers km=p\nω2µ0ε0+iωµ0σ,\nwhere σ= 0 in the absence of an NM layer. The continu-\nity of electric and magnetic fields provides the interface\nboundary conditions. The field in the FI\nEη={x,y,z}(−dF⩽x⩽dF)\n=E(0)\nη(−dF⩽x⩽dF) +RkE(0)\nη(x=dF)e−iAk(x−dF)\nis now modified by the reflection coefficient\nRk=\u0000\nA2\nk−B2\nk\u0001\neiBkdM−\u0000\nA2\nk−B2\nk\u0001\ne−iBkdM\n(Ak−Bk)2eiBkdM−(Ak+Bk)2e−iBkdM,(2)\nwhere E(0)is the solution of Eq. (1) inside the FI without\nthe NM cap [53], Ak=p\nk2\n0−k2, and Bk=p\nk2m−k2.\nThe reflection is isotropic and strongly depends on the\nwave vector. Naturally, Rk= 0 when dM= 0. On the\nother hand, when |k|= 0, the electric field cannot escape\nthe FI, since the reflection is total with Rk=−1.A corollary of Maxwell’s equation—Faraday’s Law—\nreads in frequency space iωµ0[Hd(r, t) +M(r, t)] =∇ ×\nE(r, t). When the magnetization of sufficiently thin mag-\nnetic films is uniform, the Zeeman interaction is propor-\ntional to the spatial average Hdover the film thickness.\nReferring to SM for details [53], we find\nHd,x=\u0014\n−Rk\n4A2\nkdFak(e2iAkdF−1)2(−iAkak+ky)\n+i\n2AkdF(e2iAkdF−1)\u0015\nMx≡ζx(k)Mx,\nHd,y=\"\n−Rk\n4iAkdF(e2iAkdF−1)2 \n−ky\niAkak+k2\ny\nA2\nk+ 2!\n+k2\ny\nA2\nk−k2\ny\nA2\nk1\n2iAkdF(e2iAkdF−1)#\nMy≡ζy(k)My.\nBy substitution into the LL equation (1), the spin wave\neigenfrequencies and ellipticities become\nω(k) =µ0γq\n(˜H0−ζx(k)Ms)(˜H0−ζy(k)Ms),(3a)\nak=q\n(˜H0−ζy(k)Ms)/(˜H0−ζx(k)Ms), (3b)\nwhere ˜H0=H0+αexk2Ms. Imω(k)̸= 0 because of the\nJoule heating due to the eddy currents in the cap layer.\nChiral damping and frequency shifts .—The stray elec-\ntric fields of spin waves propagating perpendicular to the\nmagnetization are chiral, i.e., they depend on their prop-\nagation direction by a hand rule. When kz= 0,E=Ezˆz\nis along the equilibrium magnetization and Ez∝My\nis complex only for positive ky. We illustrate the re-\nsults of the self-consistent calculations for dF= 100 nm,\ndM= 500 nm, conductivity σ= 6.0×107(Ω·m)−1\nfor copper at room temperature [55], applied magnetic\nfield µ0H0= 0.02 T, µ0Ms= 0.178 T, the exchange\nstiffness αex= 3×10−16m2for YIG [56], and γ=\n1.77×1011(s·T)−1. The presence of the NM cap lay-\ners shifts the relative phases between the stray electric\nfields and that of the generating spin waves. We focus\nhere on the wave numbers ky=±1µm−1in Fig. 2(a)\n[Fig. 2(b)] at which the electric field is in-phase (out-\nof-phase) with the transverse magnetization Myˆy. The\nresponse to an in-phase (out-of-phase) electric field is dis-\nsipative (reactive). Both components decay in the FI and\nthe vacuum as ∝1/|k|. In the NM, the in-phase compo-\nnent is screened only in the metal region on the scale of\na skin depth λ=p\n2/(ωµ0σ)∼1.5µm at ω= 11 GHz.\nThe out-of-phase electric field, on the other hand, cre-\nates only a reactive response and is therefore symmetric\nabove and below the metallic film. Also in this case the\ndamping is modulated for constant Gilbert damping by\nthe associated spin wave frequency shift in Fig. 2(b), an\neffect that cannot be captured by the adiabatic approxi-\nmation [7, 36].3\nFIG. 2. The system responds strongly to a phase difference\nbetween the spin waves and their wave vector-dependent ac\nelectric stray fields E. ReEcauses damping [(a)] and Im E\na frequency shift [(b)]. Im Ezgoverns the spin wave vector\ndependence of the chiral damping [(c)]. (d) illustrates the\nstrong ky-dependence of the damping of the lowest standing\nspin wave for Cu thicknesses dM={50,100,200,500}nm. (e)\nshows the real and imaginary parts of the reflection coefficient\nRkthat causes the frequency shifts plotted in (f).\nThe chirality of the radiated electric field controls the\nbackaction of the NM layer that modifies the magnon\ndispersion in a chiral fashion. Figure 2(c) illustrates\nthe strong wave vector-dependent damping coefficient\nαeff(k) =|Imωk|/Reωk. Spin waves propagating in the\npositive ˆy-direction decay much faster than those along\nthe negative direction, while the damping for positive\nand negative kzis the same. According to Fig. 2(d), the\ncalculated damping for kz= 0 in Fig. 2(c) increases (de-\ncreases) with the thickness of the Cu (YIG) film. The\nenhancement of the damping saturates for NM thick-\nnesses dN>1/p\nk2+ 1/λ2, depending on the skin depth\n(λ∼1.5µm) and the wave number 1 /kof the electric\nfield. Moreover, the Kittel mode at k= 0 in Fig. 2(e)\nis not affected by the metal at all because the reflection\ncoefficient Rk=−1, which implies that the dynamics\nof the FI and metal fully decouple. Indeed, recent ex-\nperiments do not find a frequency shift of the FMR by a\nsuperconducting overlayer [57, 58]. The additional damp-\ning by eddy currents reported by Ref. [39] is caused bythe width of the exciting coplanar waveguide, a finite-size\neffect that we do not address here.\nThe real part of Rkin Fig. 2(e) causes an in-phase\nOersted magnetic field that chirally shifts the spin wave\nfrequencies by as much as ∼1 GHz, see Fig. 2(f). Refer-\nence [59] indeed reports a frequency shift of perpendicular\nstanding spin wave modes in Bi-YIG films in the presence\nof thin metallic overlayers.\nThe predicted effects differ strongly from those caused\nby spin pumping due to the interface exchange coupling\nαsp= (ℏγ/M sdF)Reg↑↓, where g↑↓is the interfacial spin\nmixing conductance [60]. αspdoes not depend on the\nthickness of the metal and vanishes like 1 /dF. The fre-\nquency shift scales like Im g↑↓/dFand is very small even\nfor very thin magnetic layers. In contrast, the eddy\ncurrent-induced damping is non-monotonic, scaling like\n∝dFwhen 2 kdF≪1, vanishing for much thicker mag-\nnetic layers, and reaching a maximum at dF∼2λ.\nUnidirectional transmission of wave packets through a\npotential barrier .—The transmission of a wave packet im-\npinging from the left or right at a conventional potential\nbarrier is the same [61]. In the presence of a metal cap,\nthis does not hold for magnons in thin magnetic films.\nBefore turning to the potential scattering in this\nmodel, we have to address the effect of the edges. When\nmagnons propagate in the negative direction without\ndamping but decay quickly when propagating in the op-\nposite one, those reflected at the left boundary of the\nsample accumulate, which is a non-Hermitian skin ef-\nfect [62–65]. We substantiate this conclusion by nu-\nmerical calculations for a two-dimensional square lat-\ntice model with ˆ mi= (1 /√\nN)P\nkˆmkeik·ri, where ˆ mk\nis the annihilation operator of magnons with frequency\nωkfrom Eq. (3a) and ilabels the sites and Nis the\nnumber of sites. The Hamiltonian in the real space\nˆH0=P\nijtjiˆm†\njˆmi, where tji= (1/N)P\nkℏωkeik·(rj−ri)is\na hopping amplitude between possibly distant sites iand\njand the summation is over the first Brillouin zone. With\na coarse-grained lattice constant of ay=az= 0.1µm the\nreciprocal lattice vector 2 π/ay,zis much larger than the\nmagnon modes of interest (refer to the SM [53] for de-\ntails). When the frequencies ωkare complex, the Hamil-\ntonian is non-Hermitian, i.e.,tji̸=t∗\nij.\nFigure 3(a) shows the winding path of the real and\nimaginary eigenfrequencies with wave number. In the\ninterval ky= [−25,25]µm−1and an applied magnetic\nfield parallel to the boundary with θ= 0, the complex\ncomponent is hysteretic, indicating localization of modes\nat opposite boundaries. Figure 3(b)-(c) show the average\nspatial distributions W(r) = (1 /Nm)PNm\nl=1|ϕl(r)|2ofNm\nlowest-frequency eigenstates ϕl(r) for ky∈[−1,1]µm−1\nandkz∈[−1,1]µm−1. When the static magnetic field\naligns with the sample boundary z-axis, i.e.θ= 0 in\nFig. 3(b), the magnons tend to accumulate at the left4\nedge. In the antiparallel configurations θ=π[Fig. 3(c)],\nthe magnons aggregate at the right. In the noncollinear\nconfiguration with θ=π/4 [Fig. 3(d)], the maxima shifts\nto the upper-left corner. While Wis an average, we\nalso illustrate the localization of individual low-frequency\nmodes in SM [53].\nFIG. 3. The magnon skin effect caused by chiral damping.\n(a) Complex spectral winding under periodic boundary con-\nditions when kyevolves from −25 to 25 µm−1forθ= 0. (b)-\n(d) corresponds to the edge or corner aggregations of magnon\neigenstates for other magnetic configurations θ∈ {0, π, π/ 4}.\nWe now illustrate the effect of square potential barri-\ners of width dand height u0,ˆV(y) =u0[Θ(y+d/2)−\nΘ(y−d/2)], where Θ( x) is the Heaviside step function,\non the magnon transmission along ˆy(⊥Ms). With in-\ncoming ⟨y|k0⟩=eik0y, the scattered states |ψs⟩obey the\nLippmann-Schwinger formula [66]\n|ψs⟩=|k0⟩+1\niℏ∂t−ˆH0+i0+ˆV|ψs⟩. (4)\nwhere ˆH0=P\nkℏωkˆm†\nkˆmkis the magnon Hamiltonian for\nan extended film. The transmitted waves read\n⟨y|ψs⟩=\u001aT+(k0)eik0y,{y, k0}>0\nT−(k0)eik0y,{y, k0}<0. (5)\nIn the weak scattering limit |u0d| ≪ | ℏvk0|,\nT±(k0) = 1±\u0012iℏvk0\nu0d−vk0\n2|vk0|\u0013−1\n≈1∓iu0d\nℏvk0,(6)\nwhere vk0=∂ωk/∂k|k=k0ˆyis a generalized group ve-\nlocity that dissipation renders complex. The imaginary\npart of the group velocity and transmission amplitudes\ndepend on the direction of the incoming wave:\nD±(k0) =|T±(k0)|2≈1±2Im\u0012u0d\nℏvk0\u0013\n. (7)For example, with u0/ℏ= 30.5 GHz, d= 0.1µm,k0=\n±0.8µm−1,vk0>0= (2.32 + 0 .52i) km/s and vk0<0=\n−(2.64 + 0 .16i) km/s lead to T+(k0>0)≈0.6 while\nT−(k0<0)≈0.9, so even in the weak scattering limit the\nNM cap layer significantly and asymmetrically reduces\nthe transmission probability.\nWe can assess the strong scattering regime with |u0d|≳\n|ℏvk0|by numerical calculations but find dramatic ef-\nfects on the time evolution of a real-space spin-wave\npacket as launched, e.g., by a current pulse in a mi-\ncrowave stripline. We adopt a Gaussian shape Ψ( r,0) =\ne−(r−r0)2/(2η2)eiq0·rcentered at r0with a width η≫ay,z\nthat envelopes a plane wave with wave vector q0and\nˆV(r) = u0f(r) with either f(|y−˜y0|< d) = 1 or\nf(|z−˜z0|< d) = 1, where ˜ y0and ˜z0are the center of the\nbarriers. According to Schr¨ odinger’s equation Ψ( r, t) =\neiˆHt/ℏΨ(r, t= 0) with ˆH=ˆH0+ˆV(r). Numerical results\nin Fig. 4(a) and (b) u0d≪ |ℏvk0|agree with perturbation\ntheory (7) in the weak scattering regime. However, when\n|ℏvk0|≲u0dand|Im(v−k0)| ≪ | Im(vk0)|≲|Re(v±k0)|\nthe transmission and unidirectionality becomes almost\nperfect. Figure 4(c) and (d) show a nearly unidirectional\ntransmission of the wave packet through the potential\nbarrier for the Damon-Eshbach configuration q0⊥Ms;\nit is transparent for spin waves impinging from the left,\nbut opaque for those from the right. In the calculations,\nq0=q(0)\nyˆywith q(0)\ny=±5µm−1andη= 3µm≫d.\nThe potential barrier is peaked with d=ay,z= 0.1µm\nand its height u0/ℏ= 15 GHz is relatively weak (the\nregular on-site energy ∼13 GHz). Also, dM= 50 nm\nanddF= 20 nm. The results are insensitive to the de-\ntailed parameter values (see SM [53]). The red and blue\ncurves are the incident and reflected wave packets, re-\nspectively. When q(0)\ny<0, the barrier does not affect the\nwave packet that propagates freely through the poten-\ntial barrier and accumulates on the left edge [Fig. 4(c)].\nWhen q(0)\ny>0, as shown in Fig. 4(d), the barrier reflects\nthe wave packet nearly completely, which we associate\nagain with the skin effect since these magnons cannot ac-\ncumulate on the right side. The unidirectional transmis-\nsion is therefore a non-local phase-coherent phenomenon\nthat involves the wave function of the entire sample.\nSince we find the skin effect to be crucial, its absence\nin waves propagating in the ˆz-direction must affect the\ntransport over the barrier. Indeed, our calculations in\nFig. 4(e) and (f) find strong reflection for both propaga-\ntion directions, even when reducing the barrier height by\nan order of magnitude to u0= 1.5 GHz (see SM [53]).\nDiscussion and conclusion .—In conclusion, we cal-\nculate the chiral damping, chiral frequency shift, and\nanomalous transport of magnonic modes in ferromag-\nnetic films with NM cap layers beyond the adiabatic ap-\nproximations. We predict anomalous unidirectional spin\ntransport over potential barriers. This effect is rooted\nin the non-Hermitian magnon skin effect and reflects the5\nFIG. 4. Calculated transmissions [(a) and (b)] and time evolu-\ntion of spin-wave packets in the presence of a potential barrier\nat the origin when q0⊥Ms[(c) and (d)] and q0∥Ms[(e)\nand (f)], where Msand the applied magnetic field are parallel\nto the sample edge with θ= 0. The red and blue curves rep-\nresent, respectively, the incident and scattered wave packets\nwith propagation directions indicated by arrows.\nglobal response of the entire system to a local perturba-\ntion. 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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": "1505.04087v2.Reliable_Damping_of_Free_Surface_Waves_in_Numerical_Simulations.pdf",        "content": "Reliable Damping of Free Surface Waves in Numerical\nSimulations\nRobinson Peri\u0013 c1, Moustafa Abdel-Maksoud\nHamburg University of Technology (TUHH), Institute for Fluid Dynamics and Ship\nTheory (M8), Hamburg, Germany\nAbstract\nThis paper generalizes existing approaches for free-surface wave damping via\nmomentum sinks for \row simulations based on the Navier-Stokes equations.\nIt is shown in 2D \row simulations that, to obtain reliable wave damping,\nthe coe\u000ecients in the damping functions must be adjusted to the wave pa-\nrameters. A scaling law for selecting these damping coe\u000ecients is presented,\nwhich enables similarity of the damping in model- and full-scale. The in-\n\ruence of the thickness of the damping layer, the wave steepness, the mesh\n\fneness and the choice of the damping coe\u000ecients are examined. An e\u000ecient\napproach for estimating the optimal damping setup is presented. Results of\n3D ship resistance computations show that the scaling laws apply to such\nsimulations as well, so the damping coe\u000ecients should be adjusted for every\nsimulation to ensure convergence of the solution in both model and full scale.\nFinally, practical recommendations for the setup of reliable damping in \row\nsimulations with regular and irregular free surface waves are given.\nKeywords: Damping of free surface waves, absorbing layer, volume of \ruid\n(VOF) method, damping coe\u000ecient, scaling law\n1. Introduction\nIn free surface \row simulations based on the Navier-Stokes equations, it\nis often required to model an in\fnite domain, which basically means min-\nimizing undesired wave re\rections at wave-maker and domain boundaries,\n1Corresponding author. Tel. +49 40 42878 6031.\nE-mail adress: robinson.peric@tuhh.de\nPreprint submitted to Ship Technology Research May 22, 2017arXiv:1505.04087v2  [physics.flu-dyn]  19 May 2017while choosing the solution domain as small as possible in order to lower\nthe computational e\u000bort. Since such simulations are among the computa-\ntionally most e\u000bortful techniques to solve numerical \row problems, they are\nemployed when simpler approaches (e.g. potential \row methods) cannot be\nused or when higher accuracy is required. As the schemes contain numer-\nical errors (discretization, iteration, modelling (in some cases)), to achieve\nthe required accuracy it is necessary to minimize avoidable uncertainties; of\nthese, the performance of the wave damping approach is often among the\nmost critical.\nThe elimination of wave re\rections at the domain boundaries is commonly\nachieved by\n(i)increasing the domain size\n(ii)beaches (e.g. Lal and Elangovan (2008)): a slope in the domain bottom\nleads to wave breaking and energy dissipation as in experiments\n(iii) grid damping (e.g. Kraskowski (2010), Peri\u0013 c and Abdel-Maksoud (2015)):\ncontinuously increasing the cell size towards the corresponding domain\nboundary increases numerical discretization and iteration errors, thus\ndamping the wave (this approach is also called numerical beach or grid\nextrusion)\n(iv) active wave absorption techniques (e.g. Cruz (2008); Higuera et al.\n(2013); Sch a\u000ber and Klopman (2000)): a boundary-based wave-maker\ngenerates waves which eliminate the incoming waves via destructive\ninterference\n(v)solution-forcing orsolver-coupling (e.g. Ferrant et al. (2008), Guignard\net al. (1999), Kim et al. (2012), Kim et al. (2013), W ockner-Kluwe\n(2013)): the \row is forced to a known solution in the vicinity of the\nboundary or the Navier-Stokes-based \row solver is coupled to another\n(e.g. potential \row based) solver\n(vi) damping layer approaches (e.g. Cao et al. (1993); Choi and Yoon\n(2009); Ha et al. (2011); Israeli and Orszag (1981); Park et al. (1999)):\nthe damping layer (also called sponge layer, absorbing layer, damping\nzone, porous media layer) is a zone set up next to the corresponding\nboundaries, in which momentum sinks are included in the governing\nequations to damp the waves propagating through the zone\n2Apart from the above methods, further approaches have been developed for\nother governing equations, like potential \row with boundary element method\nor Boussinesq-type equations (see e.g. Grilli and Horrillo (1997) and ref-\nerences therein). However, many of these approaches have not yet been\ntransferred to Navier-Stokes-type equations. From the above mentioned ap-\nproaches, (i) is the least feasible due to its (in many cases enormous) inherent\nincrease in computational e\u000bort. With (ii) it is di\u000ecult to minimize re\rec-\ntions to less then 10% of the incoming wave, which is a problem in experi-\nments as well. Approaches (iv) and (v) have recently attracted much interest\nand produced good results; however, since they are often used to simultane-\nously generate and damp waves, it is likely that their damping performance\nvaries depending on the kind of waves they are currently generating. At the\ntime of writing, (iii) and (vi) are the most widely-used wave damping ap-\nproaches in Navier-Stokes-type equation \row solvers, implemented in most\ncommercial and research codes. Although both approaches have been used\nwith success, the disadvantage of (iii) is that its performance depends on\ngrid, time step, temporal/spatial discretization schemes, etc., which for suf-\n\fciently \fne discretization is not the case for (vi), as the results in this work\nsuggest. Thus (iii) is less predictable than (vi) regarding the damping qual-\nity. This work focuses exclusively on damping layer approaches (vi) based\non momentum source terms, since a detailed investigation and comparison\nof all previously mentioned approaches was not possible in the scope of this\nstudy.\nThis paper discusses momentum source terms based on linear or quadratic\ndamping functions as de\fned in Sect. 3. The amount and character of\nthe damping is controlled by coe\u000ecients in the damping functions and the\nthickness of the damping layer. Such approaches are widely used and already\nimplemented in several open source as well as commercial computational \ruid\ndynamics (CFD) solvers. Section 3 shows how the existing implementations\ncan be generalized. Two of the most widely used implementations, the ones\nfrom CD-adapco STAR-CCM+, based on Choi and Yoon (2009), and ANSYS\nFluent, based on Park et al. (1999), are discussed in Sect. 4. Various other\nimplementations of linear or quadratic damping exist, see e.g. Cao et al.\n(1993) and Ha et al. (2011).\nThe nomenclature from Sect. 4 is used throughout the work. For all\nsimulations in this study, the damping function from Choi and Yoon (2009)\nis used, since it is rather generic and can be set up to act similar to the\nother approaches mentioned. Thus the results are easily applicable to all\n3damping approaches used in other CFD codes which can be generelized as\ndescribed in Sect. 3. Although widely used, much about the damping func-\ntions remains unknown at the time of writing. It has been observed that the\ncoe\u000ecients in the damping functions, and thus the damping performance, are\ncase-dependent. In the above mentioned codes, these damping coe\u000ecients\ncan be modi\fed by the user. However, no guidelines seem to exist for this.\nThus in practice, either the default settings or values from experience are\nused as coe\u000ecients in the damping functions. If during or after the simula-\ntion it is observed that the damping does not work satisfactorily, then the\ndamping coe\u000ecients are modi\fed by trial and error and the simulation is\nrestarted. This procedure is repeated until acceptable damping is obtained,\na process which requires human interaction and can cost considerable addi-\ntional time and computational e\u000bort. Especially in the light of the increased\nautomation of CFD computations, it would be preferable to be sure about the\ndamping quality before the simulation. Similar to wave damping in experi-\nments, where according to Lloyd (1989) seldom data on beach performance is\npublished, also with CFD simulations, the performance of the wave damping\nis often not su\u000eciently accounted for. Thus with most CFD publications,\nit is di\u000ecult to judge on the damping quality even if the damping setup is\ngiven.\nThe aim of the present work is to clarify how the damping functions\nwork, on which factors the damping quality depends and how reliable wave\ndamping can be set up case-independently, so that no more \fne-tuning of\nthe damping coe\u000ecients is necessary.\nIn Sect. 3, the generalized forms of linear and quadratic damping func-\ntions are given. In the following Sect. 4, the damping approach used in\nthis work is described. Moreover, it is shown exemplarily for two widely\nused damping functions how these can be generalized to the equations given\nin Sect. 3 and how results from one implementation can be transferred to\nanother.\nStarting from the analogy of the damped harmonic oscillator, Sect. 5\nattempts to show similarities to damping phenomena outside the hydrody-\nnamics \feld and constructs scaling laws for wave damping based on this\ninformation. These scaling laws are fully formulated and veri\fed in Sects.\n13 and 14. Furthermore, recommendations for selecting the damping coe\u000e-\ncients are given.\nSections 8 to 12 investigate via 2D \row simulations how the damping qual-\nity and behavior is in\ruenced by the choice of the coe\u000ecients in the damping\n4functions, the thickness of the damping layer, the computational grid and\nthe wave steepness. As the investigations so far have been mainly concerned\nwith regular, monochromatic waves, recommendations for the damping for\nirregular waves are given in Sect. 15 and illustrated with simulation results.\nSince wave damping is also widely used to speed up convergence for ex-\nample in ship resistance computations, it is veri\fed in Sect. 16 via 3D \row\nsimulations that the presented scaling laws hold for such cases as well.\n2. Governing Equations and Solution Method\nThe governing equations for the simulations are the Navier-Stokes equa-\ntions, which consist of the equation for mass conservation and the three\nequations for momentum conservation:\nd\ndtZ\nV\u001adV+Z\nS\u001av\u0001ndS= 0; (1)\nd\ndtZ\nV\u001auidV+Z\nS\u001auiv\u0001ndS=\nZ\nS(\u001cijij\u0000pii)\u0001ndS+Z\nV\u001agiidV+Z\nVqidV : (2)\nHereVis the control volume (CV) bounded by the closed surface S, vis the\nvelocity vector of the \ruid with the Cartesian components ui,nis the unit\nvector normal to Sand pointing outwards, tis time,pis the pressure, \u001aand\u0016\nare \ruid density and dynamic viscosity, \u001cijare the components of the viscous\nstress tensor, ijis the unit vector in direction xj,gcomprises the body forces\nandqiis an optional momentum source term. For the present simulations,\nthe only body force considered was the gravitational acceleration, i.e. g=\n(0;0;\u00009:81m\ns2)T. Only incompressible Newtonian \ruids are considered in this\nstudy. Thus \u001cijtakes the form\n\u001cij=\u0016\u0012@ui\n@xj+@uj\n@xi\u0013\n:\nNo turbulence modeling was applied to the above equations since, unless\nwaves break, the \row inside them can be considered practically laminar and\nall structures of interest can be resolved with acceptable computational ef-\nfort. For all simulations, the software STAR-CCM+ 8 :02:008 was used. The\n5volume of \ruid (VOF) method implemented in the STAR-CCM+ software\nis used to account for the two phases (air and water). Further details on the\nmethod can be found in Muzaferija and Peri\u0013 c (1999). The governing equa-\ntions are applied to each cell and discretized according to the Finite Volume\nMethod (FVM). All integrals are approximated by the midpoint rule. The\ninterpolation of variables from cell center to face center and the numerical\ndi\u000berentiation are performed using linear shape functions, leading to ap-\nproximations of second order. The integration in time is based on assumed\nquadratic variation of variables in time, which is also a second-order approx-\nimation. Each algebraic equation contains the unknown value from the cell\ncenter and the centers of all neighboring cells with which it shares common\nfaces. The resulting coupled equation system is then linearized and solved\nby the iterative STAR-CCM+ implicit unsteady segregated solver, using an\nalgebraic multigrid method with Gauss-Seidl relaxation scheme, V-cycles for\npressure and volume fraction of water, and \rexible cycles for velocity cal-\nculation. For each time step, one iteration consists of solving the governing\nequations for the velocity components, the pressure-correction equation (us-\ning the SIMPLE method for collocated grids to obtain the pressure values\nand to correct the velocities) and the transport equation for the volume frac-\ntion of water. For further information on the discretization of and solvers for\nthe governing equations, the reader is referred to Ferziger and Peri\u0013 c (2002)\nor the STAR-CCM+ software manual.\n3. General Formulation for Linear and Quadratic Damping\nLinear wave damping is obtained by inserting the momentum source term\nqd;lin\niforqiin Eq. (2):\nqd;lin\ni=\u001aCi;linui; (3)\nwith coe\u000ecient Ci;lin, which usually depends on the spatial location. Ci;linreg-\nulates the strength of the damping and is used to provide a smooth blending-\nin of the damping, by increasing the amount of damping from weak damping\nwhere the waves enter the damping layer to strong damping at the end of\nthe damping layer. This is to prevent undesired re\rections at the entrance\nto the damping layer as shown in Sects. 8 and 9. Commonly, the blending-in\nis realized in an exponential or quadratic fashion.\nQuadratic wave damping takes the form\nqd;quad\ni =\u001aCi;quadjuijui; (4)\n6with coe\u000ecient Ci;quad, which regulates strength and blending-in of the damp-\ning. In contrast to Eq. (3), \ruid particles with higher velocities experience\ndisproportionately high damping.\nIn most CFD codes, Ci;linandCi;quadcan be adjusted by the user. Usually,\nbut not necessarily, the momentum source terms are only applied to the\nequation for the vertical velocity component.\n4. Common Implementations of Linear and Quadratic Wave Damp-\ning\nA widely used approach is the one in Choi and Yoon (2009), which is e.g.\nimplemented in the commercial software code STAR-CCM+ by CD-adapco.\nIt features a combination of linear and quadratic damping, which allows the\nuse of either one or a combination of both approaches. Taking zas the\nvertical direction (i.e. perpenticular to the free surface) and was the vertical\nvelocity component, then the following source term appears for qiin Eq. (2):\nqd\nz=\u001a(f1+f2jwj)e\u0014\u00001\ne1\u00001w ; (5)\n\u0014=\u0012x\u0000xsd\nxed\u0000xsd\u0013n\n: (6)\nHerexstands for the wave propagation direction with xsdbeing the start\nandxedthe endx-coordinate of the damping layer, thus the thickness of the\ndamping layer xd=jxed\u0000xsdj.f1is the damping constant for the linear\npart andf2is the damping constant for the quadratic part of the damping\nterm. The default values are f1= 10:0 s\u00001andf2= 10:0 m\u00001according to\nthe STAR-CCM+ manual (release 8 :02:008). The terme\u0014\u00001\ne1\u00001blends in the\ndamping term, i.e. it is zero at xsd, and it equals one at xed.\u0014describes\nvianthe character of the blending functions, i.e. for n= 1 the blending\nis nearly linear. When increasing n, the blending becomes smoother at the\nentrance to the damping layer and at the same time more abrupt at xed, see\nFig. 1.\n7Figure 1: The functione\u0014(x)\u00001\ne1\u00001evaluated for n= 1,n= 2 andn= 4 over the dimensionless\ndistancex\u0000xsd\nxed\u0000xsd; a wave enters the damping layer atx\u0000xsd\nxed\u0000xsd= 0 and is damped during\npropagation towardsx\u0000xsd\nxed\u0000xsd= 1, where the damping layer ends\nAnother widely used approach was presented by Park et al. (1999), which\nis implemented in ANSYS Fluent.:\nqd\nz=\u001a(0:5f3jwj)\u0014\u0012\n1\u0000z\u0000zfs\nzb\u0000zfs\u0013\nw ; (7)\nwith damping constant f3,\u0014as given in Eq. (6) with n= 2, vertical coordi-\nnatezandz-coordinates of the domain bottom zband the free water surface\nzfs. The default value for f3is 10:0 m\u00001.\nThis approach can be generalized to a quadratic damping function accord-\ning to Eq. (4). It corresponds to the quadratic part of Eq. (5), except for a\nquadratic instead of exponential blending in x-direction and an added verti-\ncal fade-in. The latter term is a linear fade between domain bottom, where\nno damping is applied, to free surface level, where full damping is applied;\nfor practical deep water conditions of several wavelengths water depth, the\nin\ruence of the z-fading term on the applied damping becomes small, since\nmost wave energy is concentrated in the vicinity of the free surface. Thus for\nsuch cases, this damping approach can be modeled using Eqs. (5) and (6)\n8with accordingly adjusted coe\u000ecients. In a similar manner, also the other\ndamping layer approaches from Sect. 1 (vi) can be modeled. Therefore in the\nfollowing, only the approach by Choi and Yoon (2009) will be considered to\nstudy wave damping. Thus the \fndings in this work can easily be transferred\nto other damping approaches.\n5. Dependence of Damping Coe\u000ecients\nFor gravity waves, all scaling laws should be consistent with Froude scal-\ning. Thus to obtain similar wave damping for waves of di\u000berent scale, the\nblending part of Ci;linandCi;quadfrom Eqs. (3) and (4) must be geomet-\nrically similar. Therefore, the thickness xdof the damping layer must be\ndirectly proportional to the wavelength \u0015. For the remaining part of Ci;lin\nandCi;quad, which regulates the magnitude of the damping, the scaling laws\ncan be obtained by dimensional analysis. Due to its dimension of [ s\u00001],Ci;lin\nis directly proportional to the wave frequency !, whereasCi;quadhas the di-\nmension [m\u00001] and is thus directly proportional to \u0015\u00001(or!2in deep water).\nThus to achieve similar damping when changing scale and/or wave, it is nec-\nessary to:\n9Scaling Laws\n1. Set the damping thickness xdso that it is geometrically similar to the\ndamping layer thickness xd;refof the reference case (i.e. scale xdwith\nthe percentual change in wavelength)\nxd=xd;ref\u0001\u0015\n\u0015ref; (8)\n2. For linear damping , scalef1with the change of wave frequency !\nf1=f1;ref\u0001!\n!ref; (9)\nwith wave frequency !refof the reference case. This holds for all damp-\ning formulations that can be generalized by Eq. (3).\n3. For quadratic damping , scalef2with the change of wavelength \u0015to the\npower of minus one\nf2=f2;ref\u0001\u0015ref\n\u0015; (10)\nwith wavelength \u0015refof the reference case. This holds for all damping\nformulations that can be generalized by Eq. (4).\n6. Numerical Simulation Setup\nFigure 2 shows the solution domain, a 2D deep water wave tank with\nlengthLx= 6\u0015and height Lz= 4:5\u0015, given in relation to wavelength\n\u0015. It is \flled with water to a depth of d= 4\u0015, the rest of the tank is\n\flled with air. The origin of the coordinate system is set in the bottom left\nfront corner of the tank in Fig. 2. The top boundary (i.e. z=Lz) is a\npressure outlet boundary, i.e. the pressure there is set constant and equal\nto atmospheric pressure. This corresponds to an open water tank, where air\ncan \row in and out through the tank top. The x=Lxboundary is a pressure\noutlet, where volume fraction and hydrostatic pressure are prescribed for an\nundisturbed free surface. The waves are generated by prescribing velocities\nand volume fraction of a 5th-order Stokes waves according to Fenton (1985)\nat thex= 0 boundary. The wave damping layer extends over a distance of\nxd= 2\u0015in boundary-normal direction from the x=Lxboundary. Thus the\n10waves are created at x= 0, propagate in x-direction, enter the damping layer\natx=Lx\u0000xd, are subjected to damping until x=Lxis reached, where\nthe remaining waves are re\rected and, while further subjected to damping,\npropagate back to x=Lx\u0000xd. If the damping was set up correctly, then\nthe re\rected waves are either completely damped or their height is decreased\nso much, that their in\ruence on the simulation results can be neglected.\nOtherwise they will be evident in the simulation results as additional error,\nwhich can be substantial as will be shown in the following sections.\nFigure 2: 2D wave tank \flled with water (light gray) and air (white). The damping layer\nis shown in dark grey\nLocal mesh re\fnement is used, so that the grid is \fnest in the vicinity\nof the free surface, where the waves are discretized by roughly 100 cells per\nwavelength and 16 cells per wave height, as seen in Fig. 3. The temporal\ndiscretization involved more than 500 time steps \u0001 tper wave period T, thus\nin all investigated cases the Courant number C= (ui\u0001t)=\u0001xiremains well\nbelow 0:5 for every cell with size \u0001 xiand corresponding \ruid velocity ui.\nWaves were generated for over 12 wave periods with 8 iterations per time\nstep and under-relaxation of 0 :4 for pressure and 0 :9 for all other variables.\nThe discretization is based on the grid convergence study conducted by Peri\u0013 c\n(2013).\n11Figure 3: Computational grid for the whole domain (right), and a close up of the dis-\ncretization around free surface level (left)\nThe numerical setup is the same for all simulations in this study, only\nthat the scale, wave and damping parameters are varied. Variations of the\nsetup are mentioned where they occur.\n7. Assessing the Damping Performance for Regular, Monochro-\nmatic Waves\nWhen wave re\rections occur within a damping zone, the waves will prop-\nagate back into the solution domain with a diminished height. Due to the\nresulting superposition of the partly re\rected wave and the original wave,\nthe original wave will seem to grow and shrink in a time-periodic fashion.\nFor regular, monochromatic waves, this occurs uniformly in all parts of the\ndomain where the two wave systems are superposed as seen later e.g. in\nSect. 11 from Fig. 9. In the present work, this phenomenon is called a\npartial standing wave , since the more the damping deviates from the optimal\nvalue, the more does the phenomenon resemble a standing wave, until for the\nbounding cases (i.e. zero damping or in\fnitely large damping) there is 100%\nre\rection, and a standing wave occurs.\nAssessing the amount of re\rections for regular monochromatic waves is\nan intricate issue. Since only a single wave period occurs, a wave spectrum\n12cannot be constructed. As seen from Fig. 9, the wave height envelope changes\nover space. However, this cannot be predicted since it is not known in advance\nat which point in the damping layer the re\rection mainly occurs. Thus\nfrom the recordings of a single wave probe, the height of the re\rected wave\ncannot be determined. However, the surface elevation in close vicinity to\nthe boundary to which the damping zone is attached shows how much of\nthe incident wave reaches the domain boundary despite the damping, since\na maximum ampli\fcation occurs at the domain boundary when the wave is\nre\rected.\nThus in this study, \frst the free surface elevation is recorded at x= 5:75\u0015,\ni.e. next to the domain boundary. The average wave height at this location is\nobtained and plottet in relation to the average wave height of the undamped\nwave, to show how much the wave height is reduced after propagating from\nthe entrance to the outer boundary of the damping layer. As shown in Sects.\n8 and 9, this is a good criterion for insu\u000ecient to optimal wave damping,\nwhere wave re\rections occur mainly at the outer boundary of the damping\nlayer, atx=Lx; however, this criterion will not detect re\rections if the\ndamping is too strong, since then the waves will be re\rected at the entrance\nto the damping layer.\nTherefore, secondly a re\rection coe\u000ecient CRis computed as proposed\nby Ursell et al. (1960). CRcorresponds to the ratio of the heights of the\nincident wave to the re\rected wave. It can be written as\nCR= (Hmax\u0000Hmin)=(Hmax+Hmin); (11)\nwhereHmaxis the maximum and Hminthe minimum value of the wave height\nenvelope. It holds 0 \u0014CR\u00141, so thatCR= 1 for perfect wave re\rection\nandCR= 0 for no wave re\rection.\nAs explained above, the wave height for a partial standing wave is not\nconstant, but oscillates around a mean value. This occurs uniformly in all\ndomain parts where the partial standing wave has fully developed. Thus\nHmaxandHminare obtained in the following fashion in this work: The free\nsurface elevation in the whole tank is recorded at 40 evenly timed instances\nper wave period starting at 10 periods simulation time for 2 wave periods.\nFor each recording j, the average wave height \u0016Hjis calculated for the inter-\nval 2\u0015\u0014x\u00144\u0015, which is adjacent to the damping zone but not subject to\nwave damping and su\u000eciently away from the inlet; furthermore, the partial\nstanding wave at this location is fully developed during this time interval,\n13while wave re-re\rections at the wave-maker have not yet developed signi\f-\ncantly. Therefore, the maximum \u0016Hmaxand minimum \u0016Hminof all \u0016Hjvalues\ncan be taken as HmaxandHmin, respectively. This approach detects all wave\nre\rections that propagate back into the solution domain. The accuracy of\nthe scheme can be increased if the surface elevation in the tank is recorded in\nsmaller time intervals, however this also increases the computational e\u000bort\nsigni\fcantly. Furthermore, the above procedure requires undisturbed regular\nmonochromatic waves. Yet in many applications, \ruid-structure interactions\ncreate \row disturbances, so the above scheme cannot be applied directly in\nthe simulations, but an additional 2D simulation of undisturbed wave prop-\nagation and damping for otherwise the same setup is required to obtain CR.\nAs long as this process is not fully automized, running and post-processing\nthe 2D simulation requires additional human e\u000bort. Overall, this procedure\nis rather e\u000bortful, which explains why CRis rarely computed in practice.\nWith the two presented approaches, it is possible to distinguish between\nthose wave re\rections, which occur mainly at the entrance to the damping\nzone, and those re\rections which occur mostly at the boundary to which the\ndamping zone is attached. Furthermore, the in\ruence of wave re\rections on\nthe solution can be quanti\fed.\n8. Variation of Damping Coe\u000ecient for Linear Damping\nTo investigate which range for the linear damping coe\u000ecient according\nto Eq. (5) produces satisfactory wave damping, waves with wavelength \u0015=\n4 m and height 0 :16 m are investigated. Only linear damping is considered,\nsof2= 0. Simulations are performed for damping coe\u000ecient f1between\nf12[0:625 s\u00001;1000 s\u00001]. The recorded surface elevations near the end of the\ndamping layer in Fig. 4 show that the damping is strongly in\ruenced by the\nchoice off1, while the wave phase and period remain nearly unchanged.\n14Figure 4: Surface elevation scaled by height Hof the undamped wave over time scaled by\nthe wave period T; recorded at x= 5:75\u0015for simulations with f12[0:625 s\u00001;80 s\u00001] and\nf2= 0\nFigure 5 shows how a variation of f1a\u000bects the re\rection coe\u000ecient CR,\nwhich describes the amount of re\rected waves that are present in the solution\ndomain outside of the damping zone, and the mean measured wave height\nHmeanrecorded in close vicinity to the boundary to which the damping layer\nis attached.\n\u000fFor smaller damping coe\u000ecients ( f1.20 s\u00001), wave re\rections occur\nmainly at the domain boundary to which the damping zone is attached.\nHere, the both show the same trend, with CR<H mean=(2H) since the\nre\rected waves loose further height until they leave the damping zone\natx= 4\u0015. Forf1<1:25 s\u00001, the in\ruence of the damping is so small\nthat a partial standing wave (as described in Sect. 7) higher than the\ninitial wave appears even inside the damping layer.\n\u000fFor stronger damping ( f1>20 s\u00001), wave re\rection occurs mainly at\nthe entrance to the damping zone. Thus Hmean continually decreases\nwhen increasing f1, since less of the incoming wave can pass through\nthe damping zone, so the water surface will be virtually \rat near the\ndomain boundary. This is also visible later in Figs. 13 from Sect.\n13. The aforementioned increase in the amount of wave re\rections\ndetectable in the solution domain can be seen in the curve for CR. This\nshows that, for a given fade-in function and damping layer thickness,\n15there is an optimum for f1so that the wave re\rections propagating\nback into the solution domain are minimized. The best damping was\nachieved for f1= 10 s\u00001, in which case the e\u000bects of wave re\rections\non the wave height within the solution domain are less than 0 :7%.\nFigure 5: Mean wave height Hmean recorded at x= 5:75\u0015scaled by twice the height Hof\nthe undamped wave and re\rection coe\u000ecient CRover damping coe\u000ecient f1, whilef2= 0\nThis provides the following conclusions: Hmean is not a suitable indicator\nfor damping quality if the damping is stronger than optimal, however it is\nuseful to characterize where the re\rections originate from: if CRshows that\nnoticeable re\rections are present within the solution domain, while at the\nsame timeHmeanis negligibly small, then the re\rections cannot occur at the\ndomain boundary, but must occur closer to the entrance to the damping layer.\nSigni\fcant re\rections ( CR>2%) in form of partial standing waves appear for\nroughlyf1<5 s\u00001andf1>80 s\u00001. The height of the partial standing wave\nincreases the more f1deviates from the regime where satisfactory damping\nis observed; however, the increase is slower if the damping is stronger than\noptimal instead of weaker.\n9. Variation of Damping Coe\u000ecient for Quadratic Damping\nTo investigate which range for the quadratic damping coe\u000ecient accord-\ning to Eq. (5) produces satisfactory wave damping, waves with wavelength\n16\u0015= 4 m and height 0 :16 m are investigated. Only quadratic damping is con-\nsidered, so f1= 0. Simulations are performed for damping coe\u000ecient f2\nbetweenf22[0:625 m\u00001;10240 m\u00001]. The recorded surface elevations near\nthe end of the damping layer in Fig. 6 show that the damping is strongly\nin\ruenced by the choice of f2, while the wave phase and period remain nearly\nunchanged.\nFigure 6: Surface elevation scaled by height Hof the undamped wave over time scaled by\nthe wave period T; recorded at x= 5:75\u0015for simulations with f22[0:625 m\u00001;10240 m\u00001]\nandf1= 0\nFigure 7 shows how a variation of f2a\u000bects the re\rection coe\u000ecient CR\nand the mean wave height Hmeanrecorded in close vicinity to the boundary\nto which the damping layer is attached.\nThe results show the same trends as the ones in Sect. 8. For the given\nfade-in function and damping layer thickness, there is an optimum for f2\nso that the wave re\rections propagating back into the solution domain are\nminimized. The best damping was achieved for f2= 160 m\u00001, in which case\nthe e\u000bects of wave re\rections on the wave height within the solution domain\nare less than 0 :7%. The e\u000bects of wave re\rections increase the further f2\ndeviates from the optimum. For smaller f2values, the waves are re\rected\nmainly at the domain boundary, for larger f2values the re\rection occurs\nmainly at the entrance to the damping zone. Signi\fcant re\rections ( CR>\n2%) in form of partial standing waves appear for roughly f2<80 m\u00001and\nf2>640 m\u00001.\n17Figure 7: Mean wave height Hmean recorded at x= 5:75\u0015scaled by twice the height Hof\nthe undamped wave and re\rection coe\u000ecient CRover damping coe\u000ecient f2, whilef1= 0\nCompared to the linear damping functions from the previous section, the\nuse of quadratic damping functions does not o\u000ber a signi\fcant improvement\nin damping quality. With an optimal setup, both approaches provide roughly\nthe same damping quality if the setup is optimized. However, the range of\nwave frequencies that are damped satisfactorily is narrower for quadratic\ndamping.\n10. In\ruence of Computational Mesh on Achieved Damping\nThe simulations from Sect. 8 are rerun with same setup except for a grid\ncoarsened by factor 2. Thus whereas the \fne mesh simulations discretize the\nwave with 100 cells per wavelength and 16 cells per wave height, the coarse\nmesh simulations have 50 cells per wavelength and 8 cells per wave height.\nAll coarse grid re\rection coe\u000ecients di\u000ber from their corresponding \fne grid\nre\rection coe\u000ecient by CR;coarse =CR;\fne\u00060:8%. This is also visible in Fig.\n8. Thus for su\u000ecient resolution, the damping e\u000bectiveness can be considered\ngrid-independent. This is expected since the damping-related terms in Eqs.\n(2) to (4) do not depend on cell volume. The used grids in this study are\nthus adequate.\n18Figure 8: Re\rection coe\u000ecient CRover damping coe\u000ecient f1for coarse and \fne mesh\nsimulations, with f2= 0\n11. In\ruence of the Thickness of the Damping Layer\nThe simulation for \u0015= 4:0 m,H= 0:16 m andf1= 10 s\u00001from Sect.\n8 is repeated for xd= 0\u0015;0:25\u0015;0:5\u0015;0:75\u0015;1:0\u0015;1:25\u0015;1:5\u0015;2:0\u0015;2:5\u0015with\notherwise the same setup. The evolution of the free surface elevation in the\ntank over time in Fig. 9 shows that xdhas a strong in\ruence on the achieved\ndamping. Setting xd= 0\u0015deactivates the damping and produces at \frst\na nearly perfect standing wave, which then degenerates due to the in\ru-\nence of the pressure outlet boundary, since prescribing hydrostatic pressure\nestablishes an oscillatory in-/out\row of water through this boundary which\ndisturbs the standing wave. For xd= 0:5\u0015, a strong partial standing wave oc-\ncurs, and for xd= 1:0\u0015only slight re\rections are still observable CR\u00191:6%.\nFor largerxd, the in\ruence of wave re\rections continues to decrease. This is\nevident from the plot of CRfor the simulations shown in Fig. 10.\n19Figure 9: Free surface elevation over x-location in tank shown for 40 equally spaced\ntime instances over one period starting at t= 16 s; from top to bottom xd=\n0\u0015;0:5\u0015;1:0\u0015;1:5\u0015;2:0\u0015;2:5\u0015; the damping zone is depicted as shaded gray\n20Figure 10: Re\rection coe\u000ecient CRover damping zone thickness xd\nSubsequently, the simulations from Sect. 8 have been rerun with the\ndamping thickness set to xd= 1\u0015. Comparing the resulting curves for CR\noverf1shows that increasing xdnot only improves the damping quality; it\nalso widens the range of damping coe\u000ecients for which satisfactory damping\nis obtained; thus the wave damping then becomes less sensitive to !the more\nxdincreases. However, this also increases the computational e\u000bort.\nFigure 11: Re\rection coe\u000ecient CRover damping coe\u000ecient f1forxd= 1\u0015andxd= 2\u0015,\nwhilef2= 0\n21The results show that, if the damping coe\u000ecients are set up close to the\noptimum and it is desired that CR<2%, thenxd= 1\u0015su\u000eces. This knowl-\nedge is useful, since by reducing xdthe computational domain can be kept\nsmaller and thus the computational e\u000bort can be reduced. However, if better\ndamping is desired or when complex \row phenomena are considered, espe-\ncially when irregular waves or wave re\rections from bodies are present, then\nthe damping layer thickness should be increased to damp all wave compo-\nnents successfully. The present study suggests that a damping layer thickness\nof 1:5\u0015\u0014xd\u00142\u0015can be recommended.\n12. In\ruence of Wave Steepness on Achieved Damping\nThe simulations in this section are based on those from Sect. 8, i.e.\n\u0015= 4:0 m,H= 0:16 m and varying f1in the range [0 :625 s\u00001;1000 s\u00001]. The\nsimulations were rerun with the same setup except for two modi\fcations: The\nwave height was changed to H= 0:4 m, resulting in a steepness of H=\u0015 = 0:1\ninstead of the previous H=\u0015 = 0:04. Furthermore, the grid was adjusted to\nmaintain the same number of cells per wave height as well as per wavelength,\nso that both results are comparable.\nAs can be seen from Figure 12, the in\ruence of the increased wave steep-\nness is comparatively small, except for the cases with signi\fcantly smaller\nthan optimum damping ( f1\u00142:5 s\u00001). For the rest of the range, i.e. 5 \u0014f1\u0014\n1000 s\u00001, the di\u000berence in re\rection coe\u000ecients is only CR(H=\u0015 = 0:1) =\nCR(H=\u0015 = 0:04)\u00061:7%. Therefore, although the damping performs slightly\nbetter for waves of smaller steepness, the in\ruence of wave steepness can be\nassumed negligible for most practical cases. If stronger wave steepness is con-\nsidered and less uncertainty is required, then the thickness of the damping\nlayer can be further increased to decrease CR.\n22Figure 12: Re\rection coe\u000ecient CRover damping coe\u000ecient f1for waves of same period\nbut di\u000bering steepness H=\u0015 = 0:04 andH=\u0015 = 0:1, whilef2= 0\n13. The Scaling Law for Linear Damping\nIn order to verify the assumed scaling law for linear damping from Sect.\n5, the simulation setup with the best damping performance ( f1= 10:0 s\u00001)\nfrom Sect. 8 was scaled geometrically and kinematically so that the generated\nwaves are completely similar, for wavelengths \u0015= 0:04 m;4 m;400 m and thus\ncorresponding heights of H= 0:0016 m;0:16 m;16 m. This corresponds to a\nrealistic scaling, since geometrically scaling by 1 : 100 is common in both\nexperimental and computational model- and full scale investigations. As\nnecessary requirement to obtain similar damping (i.e. similarity of CRand\nsurface elevation), the damping length xdis scaled directly proportional to\nthe wavelength, according to Eq. (8).\nAt \frst, the simulations are run with no scaling of f1, so thatf1= 10 s\u00001in\nall cases. Figure 13 shows the free surface in the tank after the simulations. In\nthe vicinity of the inlet (0 <x=\u0015< 1) the e\u000bects of wave re\rections have not\nyet fully established, thus the di\u000berences between the curves are small. This\nshows that the wave-maker generated similar waves in the three simulations.\nAs the plot shows an arbitrarily selected time instant, the e\u000bects of wave\nre\rections cannot be judged from it, since the partial standing waves were\nnot captured at their maximum ampli\fcation. Therefore, for \u0015= 400 m\nno noticeable di\u000berence to the reference case \u0015= 4 m is seen outside the\ndamping zone (0 < x=\u0015 < 4), whereas obvious re\rections are visible for\n23\u0015= 0:04 m. However regarding the surface elevation within the damping\nzone (4<x=\u0015< 6), the three curves di\u000ber considerably, and thus the wave\ndamping also does not act in a similar fashion. A close look at the surface\nelevation in the damping zones shows that wave re\rections will mostly occur\nat domain boundary x=\u0015= 6 for\u0015= 0:04 m, in contrast to \u0015= 400 m, where\nthey will mostly occur close to the entrance to the damping zone at x=\u0015= 4.\nFigure 13: Surface elevation in the whole domain after \u001912:6 periods; for similar waves\nwith wavelengths \u0015= 0:04 m;4 m;400 m; no scaling of f1, thusf1= 10 s\u00001andf2= 0 for\nall cases\nSubsequently, based on the \fndings in Sect. 5, the simulations are rerun\nwithf1scaled with wave frequency !according to Eq. (9), based on reference\ncase\u0015ref= 4 m. It is apparent from Fig. 14, that the surface elevations are\nsimilar everywhere in the tank and that the proposed scaling law is correct.\n24Figure 14: Surface elevation in the whole domain after \u001912:6 periods; for similar waves\nwith wavelengths \u0015= 0:04 m;4 m;400 m; scaling off1according to Eq. (9) , thusf1=\n10 s\u00001;3:16 s\u00001;1 s\u00001andf2= 0\nTable 1 compares the re\rection coe\u000ecients CRfor both unscaled and\ncorrectly scaled f1. Whenf1is held constant, an up- or down-scaling of\nthe wave with factor 100 will lead to a signi\fcant increase in re\rections.\nTherefore without adjusting f1, it is not possible to obtain similar damping\nin model- and full scale.\nIf insteadf1is scaled according to Eqs. (8) and (9), CRstays nearly\nthe same; slight \ructuations of CRare due to the scheme for obtaining CR.\nTherefore, the damping quality can be reliably reproduced when the scaling\nlaw is applied.\nThis is even evident in the CRvalues for \fxed f1: For the scaled waves\nconsidered, f1is either roughly 10 times larger ( \u0015= 400 m) or smaller ( \u0015=\n0:04 m) than the optimum value; compared with the 10 times larger or smaller\nthan optimal f1values from Fig. 5, the CRvalues coincide.\n25Table 1: Re\rection coe\u000ecient CRfor unscaled and correctly scaled simulations\n\u0015CR(\fxedf1)CR(scaledf1)\n0:04 m 39:8% 1 :1%\n4 m 0:7% 0 :7%\n400 m 2:4% 0 :6%\nPractical Recommendation\nWhen using linear damping according to Eqs. (5) and (6), the following\ndamping setup can be recommended based on the results from Sects. 8 to\n15:\nf1= \t 1! ; (12)\nwith \t 1=\u0019, wave frequency !,f2= 0,xd= 2\u0015andn= 2.\n14. The Scaling Law for Quadratic Damping\nIn order to verify the assumed scaling law for quadratic damping from\nSect. 5, the simulation setup with the best damping performance ( f2=\n160:0 m\u00001) from Sect. 9 was scaled geometrically and kinematically so that\nthe generated waves are completely similar, for wavelengths \u0015= 0:04 m;4 m;400 m\nand thus corresponding heights of H= 0:0016 m;0:16 m;16 m. As in the pre-\nvious section, the damping length xdis scaled directly proportional to the\nwavelength, according to Eq. (8).\nAt \frst the simulations are run with no scaling of f2, so thatf2= 160 m\u00001\nin all cases. Subsequently, based on the \fndings in Sect. 5, the simulations\nare rerun with f2scaled with wave frequency !squared according to Eq.\n(10), based on reference case \u0015= 4 m. It is apparent from Fig. 15, that the\nsurface elevations are similar everywhere in the tank and that the proposed\nscaling law is correct.\n26Figure 15: Surface elevation in the whole domain after \u001912:6 periods; for similar waves\nwith wavelengths \u0015= 0:04 m;4 m;400 m; scaling off2according to Eq. (10) , thusf2=\n160 m\u00001;16 m\u00001;1:6 m\u00001andf1= 0\nTable 2 compares the re\rection coe\u000ecients CRfor both unscaled and\ncorrectly scaled f2. Whenf2is held constant, an up- or down-scaling of\nthe wave with factor 100 will lead to a signi\fcant increase in re\rections. If\ninsteadf2is scaled according to Eqs. (8) and (9), CRstays nearly the same;\nslight \ructuations of CRare due to the scheme for obtaining CR. This shows\nthat the damping quality can be reliably reproduced when the scaling law is\napplied.\nTable 2: Re\rection coe\u000ecient CRfor unscaled and correctly scaled simulations\n\u0015CR(\fxedf2)CR(scaledf2)\n0:04 m 65:9% 1 :1%\n4 m 0:6% 0 :6%\n400 m 17:7% 0 :5%\nThe present results show that if the damping parameters are not ad-\njusted when performing model- and full-scale simulations, then unsatisfac-\ntory damping can be expected.\n27Compared to linear wave damping, quadratic damping has a narrower\nrange of wave frequencies for which satisfactory damping is obtained. There-\nfore, since the damping performance is more sensitive to changes in wave\nfrequency for quadratic (or higher order) damping functions, and since with\noptimum set up the damping performance is the same as with linear damp-\ning, it is recommended to use linear damping functions. For this reasons,\nonly linear damping was examined in the following Sects. 15 and 16.\nPractical Recommendation\nWhen using quadratic damping according to Eqs. (5) and (6), the following\ndamping setup can be recommended based on the results from Sect. 9:\nf2= \t 2\u0015\u00001; (13)\nwith \t 2= 2\u0019\u0001102, wavelength \u0015,f1= 0,xd= 2\u0015andn= 2.\n15. Damping of Irregular Free-Surface Waves\nThe previous results indicate that, to simulate irregular waves, the damp-\ning setup should be based on the wave component with the longest wave-\nlength. If the damping setup is optimal for the longest wave component, then\non the one hand the damping coe\u000ecient for all shorter wave components will\nbe smaller then optimal, which has a negative e\u000bect on the damping perfor-\nmance for the corresponding wave component as seen in Sect. 8. On the\nother hand, the damping layer thickness relative to the wavelength will be\nlarger for these wave components, which has a positive e\u000bect on the damping\nperformance as shown in Sect. 11. From the previous results, the latter e\u000bect\nis expected be the prevailing in\ruence on the damping quality. Furthermore,\nthe shorter wave components are discretized with less cells per wavelength,\nso the discretization errors are stronger for these wave components, which\nprovides a faster wave energy dissipation for these components which is also\nbene\fcial in this respect.\nExemplarily, linear damping of three superposed wave components with\nsame steepnessHi\n\u0015i= 0:03 and wavelengths di\u000bering by 400% is demonstrated\nin the following. At the wave-maker, the following surface elevation is pre-\nscribed\n\u0011(t) =3X\ni=1Hi\n2cos(\u0000!i(t+ 1:19 s) +\u001ei); (14)\n28with wave height Hi, wave frequencies !i, phase shifts \u001eiand timet. The\nactual parameters are shown in Table 3.\nTable 3: Parameters of the wave components\ni\u0015i(m)Hi(m)!i(rad\ns)\u001ei(rad)\n1 4:0 0:12 3:926 0:0\n2 2:0 0:06 5:551\u00000:9176\n3 1:0 0:03 7:851 3:543\nVelocity and volume fraction corresponding to Eq. (14) are taken from\nlinear wave theory and are applied at the inlet boundary x= 0 as linear\nsuperposition. The domain length is Lx= 3\u00151. The grid dimensions in free\nsurface zone are \u0001 x= 0:0195 m and \u0001 z= 0:0024 m with a time step of\n\u0001t= 0:001 s.\nThe damping setup is based on the wave component with the longest\nwavelength according to Eq. (12): The damping layer thickness is xd= 2\u00151\nand the damping coe\u000ecients are f1= 20 s\u00001andf2= 0. With this setup, all\nwave components were damped satisfactorily without noticeable re\rections.\nAs can be seen from Fig. 16, the waves enter the damping layer at x= 4 m\nand after propagating a distance of 2 \u00151into the damping layer, the wave\nheight is reduced by two orders of magnitude. Thus linear damping seems\nwell suited for the damping of irregular waves. Although the approach from\nSect. 5 to determine CRcannot be applied in this case, the results from\nSect. 8 show that for all wave components re\rections will mostly occur at the\ndomain boundary, and not at the entrance to the damping layer. Thus in this\ncase, measuring the surface elevation in close vicinity to the corresponding\ndomain boundary is enough to evaluate the damping performance.\n29Figure 16: Free surface elevation in the solution domain after 20 s; top: whole domain;\nbottom: close up of the last part of the damping layer\nHowever, for realistic wave spectra the above approach is not feasible, as\nthese consist of wave components over a broad range of wavelengths. Thus\nselecting the component with the longest wavelength for the damping zone\nsetup would not be practical, since this would require a very large damping\nzone and increase the total amount of grid cells and computational e\u000bort\nsubstantially (possibly one or more orders of magnitude if the mesh size is\nnot changed).\nFor practical purposes, an e\u000ecient approach is outlined in the following.\nAccording to Lloyd (1989), most realistic sea spectra are narrow banded.\nThis means that most wave energy is concentrated in a narrow band of wave\nfrequencies around a peak frequency. Although larger wavelengths occur as\nwell, these carry a comparatively small amount of energy. Thus we propose\nto set up the wave damping based on the peak wave frequency. As seen in\nSect. 11 the range of frequencies that are damped can be modi\fed with the\n30damping zone thickness, so the broader-banded the spectrum is, the larger\nhasxdto be.\nExemplarily, this approach is investigated for the widely used JONSWAP\nspectrum, which was originally developed by Hasselmann et al. (1973). Ir-\nregular waves are prescribed at the inlet boundary, with parameters peak\nwave period Tp= 1:6 s, signi\fcant wave height Hs= 0:12 m, peak shape pa-\nrameter\r= 3:3, spectral width parameter \u001b= 0:07 for!\u0014!pand\u001b= 0:09\nfor!>! p. The wave spectrum was discretized into 100 components.\nThe choice of xd= 2\u0015peak, with wavelength \u0015peakcorresponding to the\npeak wave frequency, seems adequate for such cases. No visible disturbance\ne\u000bects were noted in the simulation. The wave heights over time are shown\nrecorded close to the wave-maker in Fig. 17 and recorded close to the bound-\nary to which the damping zone is attached in Fig. 18. This shows that the\naverage wave height is reduced by roughly two orders of magnitude.\nFigure 17: Surface elevation over time recorded directly before the wave-maker\n31Figure 18: Surface elevation over time recorded in close vicinity to the boundary, to which\nthe damping layer is attached\nFrom the surface elevation in the tank in Fig. 19 no undesired wave\nre\rections can be noticed. Although a more detailed study of the error of\nthis approximation regarding di\u000berent parameters of the spectrum was not\npossible in the scope of this study, the proposed approach seems to work\nreasonably well for practical purposes. Further research in this respect is\nrecommended to reduce uncertainties regarding the re\rections.\nFigure 19: Surface elevation in the whole domain at t\u001915:6 s\n16. Application of Scaling Law to Ship Resistance Prediction\nThis section compares results from model and full scale resistance compu-\ntations in 3D of the Kriso Container Ship (KCS) at Froude number 0 :26. The\n32simulations in this section are based on the simulations reported in detail in\nEnger et al. (2010). For detailed information on the setup and discretiza-\ntion, the reader referred to Enger et al. (2010). In the following, only a brief\noverview of the setup and di\u000berences to the original simulations are given for\nthe sake of brevity. The present model scale simulation di\u000bers only in the\ndamping setup (and slight modi\fcations of the used grid) from the \fne grid\nsimulation in Enger et al. (2010). The KCS is \fxed in its \roating position\nat zero speed. To simulate the hull being towed at speed U, this velocity\nis applied at the inlet domain boundaries and the hydrostatic pressure of\nthe undisturbed water surface is applied at the outlet boundary behind the\nship. The domain is initialized with a \rat water surface and \row velocity U\nfor all cells. As time accuracy is not in the focus here, \frst-order implicit\nEuler scheme is used for time integration. Apart from the use of the k-\u000f\nturbulence model by Launder and Spalding (1974), the computational setup\nis similar to the one used in the rest of this work. The computational grid\nconsists of roughly 3 million cells. For comparability, we compare only the\npressure components of the drag and vertical forces, which are obtained by\nintegrating the x-component for the drag and z-component for the vertical\ncomponent of the pressure forces over the ship hull. The simulation starts\nat 0 s and is stopped at tmax= 90 s simulation time. The Kelvin wake of the\nship can be decomposed into a transversal and a divergent wave component.\nThe wave damping setup is based on the ship-evoked transversal wave (wave-\nlength\u0015t\u00193:1 m), the phase velocity of which equals the service speed U.\nWave damping according to Eqs. (5) and (6) has been applied to inlet, side\nand outlet boundaries with parameters xd= 2:3\u0015t,f1= 22:5 s\u00001,f2= 0,\nandn= 2. This setup provided satisfactory convergence of drag and vertical\nforces in model scale. The wake pattern for the \fnished simulation is shown\nin Fig. 20 and the results are in agreement with the \fndings from Enger et\nal. (2010). The obtained resistance coe\u000ecient CT;sim= 3:533\u000110\u00003compares\nwell with the experimental data ( CT;exp= 3:557\u000110\u00003, 0:68% di\u000berence to\nCT;sim) by Kim et al. (2001) and to the simulation results by Enger et al.\n(2001) (CT;Enger = 3:561\u000110\u00003, 0:11% di\u000berence to CT;exp).\n33Figure 20: Wave pro\fle for model scale ship with f1= 22:5 s\u00001att= 90 s\nAdditionally, full scale simulations are performed with Froude similarity.\nThe scaled velocity and ship dimensions are shown in Table 4. The grid is\nsimilar to the one for the model scale simulation except scaled with factor\n31:6. Assuming similar damping can be obtained with the presented scaling\nlaws,xdwas scaled according to Eq. 8 by factor 31 :6 as well. Otherwise the\nsetup corresponds to the one from the model scale simulation.\n34Table 4: KCS parameters\nscale waterline length L(m) service speed U(m=s)\nmodel 7:357 2 :196\nfull 232:5 12 :347\nFigure 21 shows drag and vertical forces over time when both model and\nfull scale simulations are run with the same value for damping coe\u000ecient\nf1. In contrast to the model scale forces, the full scale forces oscillate in a\ncomplicated fashion. Therefore without a proper scaling of f1, no converged\nsolution can be obtained for the full scale case.\n35Figure 21: Drag (top image) and vertical (bottom image) pressure forces on ship over time\nfor model (red) and full scale (grey); no scaling of f1, thusf1= 22:5 s\u00001is the same in\nboth simulations\nFinally, the full scale simulation is rerun with f1scaled according to Eq. 9\nto obtain similar damping as in the model scale simulation with f1= 22:5 s\u00001.\nThe correctly scaled value for the full scale simulation is thus f1;full=f1;model\u0001\n!full=!model = 22:5 s\u00001\u00011=p\n31:6\u00194 s\u00001. The resulting drag and vertical\nforces in Fig. 22 show that indeed similar damping is obtained, since in\nboth cases the forces converge in a qualitatively similar fashion. Note that a\nperfect match of the curves in Fig. 22 is not expected, since Froude-similarity\nis given, but not Reynolds-similarity. Thus although the pressure components\nof the forces on the ship will converge to the same values (if scaled by L3), the\nway they converge (amplitude and frequency of the oscillation) may not be\n36exactly similar, since this depends on the solution of all equations. However,\nthe tendency of the convergence will be qualitatively similar as shown in the\nplots.\nFigure 22: Drag (top image) and vertical (bottom image) pressure forces on ship over time\nfor model and full scale; the damping setup for the full scale simulation is obtained by\nscaling the model scale setup according to Eqs. 8 and 9\n17. Discussion and Conclusion\nIn order to obtain reliable wave damping with damping layer approaches,\nthe damping coe\u000ecients must be adjusted according to the wave parameters,\nas shown in Sects. 8, 9, 13 and 14. It is described in Sect. 7 that the\nprocedure to quantify the damping quality is quite e\u000bortful, and thus it\n37is seldom carried out in practice. This underlines the importance of the\npresent \fndings for practical applications, since unless the damping quality\ncan be reliably set to ensure that the in\ruence of undesired wave re\rections\nare small enough to be neglected, a large uncertainty will remain in the\nsimulation results. The optimum values for damping coe\u000ecients f1andf2\ncan be assumed not to depend on computational grid, wave steepness and\nthicknessxdof the damping layer as shown in Sects. 10, 11 and 12. As shown\nin Sect. 11, the damping layer thickness has the strongest in\ruence on the\ndamping quality. If it increases, the range of waves that will be damped\nsatisfactorily broadens and the re\rection coe\u000ecient for the optimum setup\nshrinks; unfortunately, the computational e\u000bort increases at the same time as\nwell, thus optimizing the damping setup is important. In contrast, Sect. 12\nshows that the wave steepness has a smaller e\u000bect on the damping, with the\ntendency towards better damping for smaller wave steepness. For su\u000eciently\n\fne discretizations, Sect. 10 shows that the damping can be considered not\na\u000bected by the grid. A practical approach for e\u000ecient damping of irregular\nwaves has been presented in Sect. 15. The scaling laws in Sect. 5 and\nrecommendations given in Sects. 13 and 14 provide a reliable way to set\nup optimum wave damping for any regular wave. Moreover, similarity of\nthe wave damping can be guaranteed in model- and full-scale simulations as\nshown in Sects. 13, 14 and 16. The \fndings can easily be applied to any\nimplementation of wave damping which accords to Sect. 3.\nReferences\n[1] Cao, Y., Beck, R. F. and Schultz, W. W. 1993. An absorbing beach for\nnumerical simulations of nonlinear waves in a wave tank, Proc. 8th Intl.\nWorkshop Water Waves and Floating Bodies , 17-20.\n[2] Choi, J. and Yoon, S. B. 2009. Numerical simulations using momentum\nsource wave-maker applied to RANS equation model, Coastal Engineer-\ning, 56, (10), 1043-1060.\n[3] Cruz, J. 2008. Ocean wave energy, Springer Series in Green Energy and\nTechnology , UK, 147-159.\n[4] Enger, S., Peri\u0013 c, M. and Peri\u0013 c, R. 2010. Simulation of Flow Around KCS-\nHull, Gothenburg 2010: A Workshop on CFD in Ship Hydrodynamics,\nGothenburg.\n38[5] Fenton, J. D. 1985. A \ffth-order Stokes theory for steady waves, J.\nWaterway, Port, Coastal and Ocean Eng. , 111, (2), 216-234.\n[6] Ferrant, P., Gentaz, L., Monroy, C., Luquet, R., Ducrozet, G., Alessan-\ndrini, B., Jacquin, E., Drouet, A. 2008. Recent advances towards the\nviscous \row simulation of ships manoeuvring in waves, Proc. 23rd Int.\nWorkshop on Water Waves and Floating Bodies , Jeju, Korea.\n[7] Ferziger, J. and Peri\u0013 c, M. 2002. Computational Methods for Fluid Dy-\nnamics, Springer.\n[8] Grilli, S. T., Horrillo, J. 1997. Numerical generation and absorption of\nfully nonlinear periodic waves. J. Eng. Mech. , 123, (10), 1060-1069.\n[9] Guignard, S., Grilli, S. T., Marcer, R., Rey, V. 1999. Computation of\nshoaling and breaking waves in nearshore areas by the coupling of BEM\nand VOF methods, Proc. ISOPE1999 , Brest, France.\n[10] Ha, T., Lee, J.W. and Cho, Y.S. 2011. Internal wave-maker for Navier-\nStokes equations in a three-dimensional numerical model, J. Coastal\nResearch , SI 64, 511-515.\n[11] Hasselmann, K., Barnett, T. P., Bouws, E., Carlson, H., Cartwright,\nD. E., Enke, K., Ewing, J. A., Gienapp, H., Hasselmann, D. E., Kruse-\nman, P., Meerburg, A., M uller, P., Olbers, D. J., Richter, K., Sell, W.\nand Walden, H. 1973. Measurements of wind-wave growth and swell de-\ncay during the Joint North Sea Wave Project (JONSWAP), Deutsche\nHydrographische Zeitschrift , (8), Reihe A.\n[12] Higuera, P., Lara, J. L., Losada, I. J. 2013. Realistic wave generation\nand active wave absorption for Navier-Stokes models Application to\nOpenFOAM r,Coastal Eng. , 71, 102-118.\n[13] Jha, D. K. 2005. Simple Harmonic Motion and Wave Theory, Discovery\nPublishing House.\n[14] Kim, J., O'Sullivan, J., Read, A. 2012. Ringing analysis of a vertical\ncylinder by Euler overlay method, Proc. OMAE2012 , Rio de Janeiro,\nBrazil.\n39[15] Kim, J., Tan, J. H. C., Magee, A., Wu, G., Paulson, S., Davies, B.\n2013. Analysis of ringing ringing response of a gravity based structure\nin extreme sea states, Proc. OMAE2013 , Nantes, France.\n[16] Kim, W. J., Van, S. H., Kim, D. H. 2001. Measurement of \rows around\nmodern commercial ship models, Experiments in Fluids , 31, Springer-\nVerlag, 567-578.\n[17] Kraskowski, M. 2010. Simulating hull dynamics in waves using a RANSE\ncode, Ship technology research , 57, 2, 120-127.\n[18] Lal, A., Elangovan, M. 2008. CFD simulation and validation of \rap type\nwave-maker, WASET , 46, 76-82.\n[19] Launder, B.E., and Spalding, D.B. 1974. The numerical computation of\nturbulent \rows, Comput. Meth. Appl. Mech. Eng. , 3, 269-289.\n[20] Lloyd, A. R. J. M. 1989. Seakeeping: Ship Behaviour in Rough Weather,\nEllis Horwood Limited.\n[21] Muzaferija, S. and Peri\u0013 c, M. 1999. Computation of free surface \rows us-\ning interface-tracking and interface-capturing methods, Nonlinear Water\nWave Interaction, Chap. 2, 59-100, WIT Press, Southampton.\n[22] Park, J. C., Kim, M. H. and Miyata, H. 1999. Fully non-linear free-\nsurface simulations by a 3D viscous numerical wave tank, International\nJournal for Numerical Methods in Fluids , 29, 685-703.\n[23] Peri\u0013 c, R. 2013. Internal generation of free surface waves and application\nto bodies in cross sea, MSc Thesis, Schriftenreihe Schi\u000bbau, Hamburg\nUniversity of Technology, Hamburg, Germany.\n[24] Peri\u0013 c, R., Abdel-Maksoud, M., 2015. Assessment of uncertainty due to\nwave re\rections in experiments via numerical \row simulations, Proc.\nISOPE2015 , Hawaii, USA.\n[25] Sch a\u000ber, H.A., Klopman, G. 2000. Review of multidirectional active\nwave absorption methods, Journal of Waterway, Port, Coastal, and\nOcean Engineering , 88-97.\n40[26] Ursell, F., Dean, R. G. and Yu, Y. S. 1960. Forced small-amplitude\nwater waves: a comparison of theory and experiment, Journal of Fluid\nMechanics , 7, (01), 33-52.\n[27] W ockner-Kluwe, K. 2013. Evaluation of the unsteady propeller perfor-\nmance behind ships in waves, PhD thesis at Hamburg University of\nTechnology, Schriftenreihe Schi\u000bbau , 667, Hamburg.\n41"    },    {        "title": "1602.07325v1.Experimental_Investigation_of_Temperature_Dependent_Gilbert_Damping_in_Permalloy_Thin_Films.pdf",        "content": "1    Experimental Investigation of Temperature-Dependent Gilbert \nDamping in Permalloy Thin Films  \nYuelei Zhao1,2†, Qi Song1,2†, See-Hun Yang3, Tang Su1,2, Wei Yuan1,2, Stuart S. P. Parkin3,4, Jing \nShi5*, and Wei Han1,2*   \n1International Center for Quantum Materials,  Peking University, Beijing, 100871, P. R. China \n2Collaborative Innovation Center of Quantum Matter, Beijing 100871, P. R. China \n3IBM Almaden Research Center, San Jose, California 95120, USA \n4Max Planck Institute for Microstructu re Physics, 06120 Halle (Saale), Germany \n5Department of Physics and Astronomy, Univers ity of California, Riverside, California 92521, \nUSA \n†These authors contributed equally to the work \n*Correspondence to be addressed to: jing.shi @ucr.edu (J.S.) and weihan@pku.edu.cn (W.H.) \n \n \nAbstract \nThe Gilbert damping of ferromagnetic materials is arguably the most important but least \nunderstood phenomenological parameter that dictates real-time magnetization dynamics. \nUnderstanding the physical origin of  the Gilbert damping is highly relevant to developing future \nfast switching spintronics devices such as magnetic sensors and magnetic random access memory. Here, we report an experimental stud y of temperature-dependent Gilbert damping in \npermalloy (Py) thin films of varying thicknesses  by ferromagnetic resonance. From the thickness \ndependence, two independent cont ributions to the Gilbert damping are identified, namely bulk \ndamping and surface damping. Of particular inte rest, bulk damping decreases monotonically as \nthe temperature decreases, while surface da mping shows an enhancement peak at the 2 temperature of ~50 K. These results provide an important insight to the physical origin of the \nGilbert damping in ultr athin magnetic films.  \n \nIntroduction \nIt is well known that the magnetization dynamics  is described by the Landau-Lifshitz-Gilbert \nequation with a phenomenological parameter called the Gilbert damping ( α),1,2:   \n eff\nSdM dMMH Mdt M dtαγ=− × + × \n   (1) \nwhere M\nis the magnetization vector,  γis the gyromagne tic ratio, and SM M=\n is the saturation \nmagnetization. Despite intense theore tical and experimental efforts3-15, the microscopic origin of \nthe damping in ferromagnetic (FM) metallic ma terials is still not well understood. Using FM \nmetals as an example, vanadium doping decreases the Gilbert damping of Fe3 while many other \nrare-earth metals doping increase s the damping of permalloy (Py)4-6,16. Theoretically, several \nmodels have been developed to explain some  key characteristics. For example, spin-orbit \ncoupling is proposed to be the intrinsic or igin for homogenous time-varying magnetization9. The \ns-d exchange scattering model assumes that damp ing results from scattering of the conducting \nspin polarized electrons with the magnetization10. Besides, there is the Fermi surface breathing \nmodel taking account of the spin scattering with  the lattice defects ba sed on the Fermi golden \nrule11,12. Furthermore, other damping mechanisms in clude electron-electron scattering, electron-\nimpurity scattering13 and spin pumping into the adjacent nonmagnetic layers14, as well as the two \nmagnon scattering model, which refers to that pa irs of magnon are scatte red by defects, and the \nferromagnetic resonance (FMR) mode  moves into short wavelength spin waves, leading to a 3 dephasing contribution to the linewidth15. In magnetic nanostructu res, the magnetization \ndynamics is dictated by the Gilbert damping of the FM materials which can be simulated by \nmicromagnetics given the boundaries and dimens ions of the nanostructures. Therefore, \nunderstanding the Gilbert damping in FM materials is particularly important for characterizing \nand controlling ultrafast responses in magnetic  nanostructures that ar e highly relevant to \nspintronic applications such as  magne tic sensors and magnetic random access memory17.  \nIn this letter, we report an expe rimental investigation of the G ilbert damping in Py thin films \nvia variable temperature FMR in a modified  multi-functional insert  of physical property \nmeasurement system with a coplanar waveguide (see methods for details). We choose Py thin \nfilms since it is an interesting FM metallic material for spintronics due to its high permeability, nearly zero magnetostriction, low coercivity, a nd very large anisotropi c magnetoresistance. In \nour study, Py thin films are gr own on top of ~25 nm SiO\n2/Si substrates with a thickness ( d) range \nof 3-50 nm by magnetron sputtering (see methods  for details). A capping layer of TaN or Al 2O3 \nis used to prevent oxidation of the Py during m easurement. Interestingly, we observe that the \nGilbert damping of the thin Py films ( d <= 10 nm) shows an enhanced peak at ~ 50 K, while \nthicker films ( d >= 20 nm) decreases monotonically as the temperature decreases. The distinct \nlow-temperature behavior in the Gilbert dampi ng in different thickness regimes indicates a \npronounced surface contribution in the thin limit. In  fact, from the linear  relationship of the \nGilbert damping as a function of the 1/ d, we identify two contribu tions, namely bulk damping \nand surface damping. Interestingl y, these two contributions show  very different temperature \ndependent behaviors, in whic h the bulk damping decreases m onotonically as the temperature \ndecreases, while the surface damping indicates an  enhancement peak at ~ 50 K. We also notice \nthat the effective magnetization sh ows an increase at the same temperature of ~50 K for 3 and 5 4 nm Py films.  These observations could be all related to the magnetization reorientation on the \nPy surface at a certain temperatur e. Our results are important for theoretical investigation of the \nphysical origins of Gilbert damping and also us eful for the purpose of designing fast switching \nspintronics devices.  \nResults and Discussion \nFigure 1a shows five representative curves  of the forward amplitude of the complex \ntransmission coefficients (S 21) vs. in plane magnetic field meas ured on the 30 nm Py film with \nTaN capping at the frequencies of 4, 6, 8, 10 an d 12 GHz and at 300 K after renormalization by \nsubtracting a constant background. These experiment al results could be fitted using the Lorentz \nequation18: \n 2\n21 0 22()\n() ( )resHSSHH HΔ∝Δ+ −  (2) \nwhere S0 is the constant describing the coefficient for the transmitted microwave power, H is the \nexternal magnetic field, Hres is the magnetic field under the resonance condition, and ΔH is the \nhalf linewidth. The extracted ΔH vs. the excitation frequency ( f) is summarized in Figures 1b and \n1c for the temperature of 300 K and 5 K respect ively. The Gilbert damp ing could be obtained \nfrom the linearly fitted curves (red lin es), based on the following equation:  \n 02()H fHπαγΔ= + Δ   (3) \nin which γ is the geomagnetic ratio and ΔH0 is related to the inhom ogeneous properties of the \nPy films. The Gilbert damping at 300 K and 5 K is calculated to be 0.0064 ± 0.0001 and 0.0055 \n± 0.0001 respectively.  5 The temperature dependence of the Gilbert damp ing for 3-50 nm Py films with TaN capping \nlayer is summarized in Figure 2a. As d decreases, the Gilbert damping increases, indicative of \nthe increasing importance of the film surfaces. Interestingly, fo r thicker Py films (e.g. 30 nm), \nthe damping decreases monotonically as the temper ature decreases, which is expected for bulk \nmaterials due to suppressed sca ttering at low temperature. As d decreases down to 10 nm, an \nenhanced peak of the damping is obser ved at the temperature of ~ 50 K. As d decreases further, \nthe peak of the damping becomes more pronounce d. For the 3 nm Py film, the damping shows a \nslight decrease first from 0.0126 ± 0.0001 at 3 00 K to 0.0121 ± 0.0001 at 175 K, and a giant \nenhancement up to 0.0142 ± 0.0001 at 50 K, and then a sharp decrease back down to 0.0114 ± \n0.0003 at 5 K.  \nThe Gilbert damping as a function of the Py th icknesses at each temperature is also studied. \nFigure 2b shows the thickness dependence of the Py damping at 300 K. As d increases, the \nGilbert damping decreases, which indicates a surface/interface enhanced damping for thin Py \nfilms19. To separate the damping due to the bul k and the surface/interface contribution, the \ndamping is plotted as a function of 1/ d, as shown in Figure 2c, and it follows this equation as \nsuggested by theories19-21. \n 1()BSdαα α=+   (4) \nin which the Bα and Sα represent the bulk and surface da mping, respectively. From these \nlinearly fitted curves, we are able to separate  the bulk damping term and the surface damping \nterm out. In Figure 2b, the best fitted parameters for Bα and Sα are 0.0055 ± 0.0003 and 0.020 ± \n0.002 nm. To be noted, there are two insulating mate rials adjacent to the Py films in our studies. 6 This is very different from previous studies on Py/Pt bilayer systems, where the spin pumping \ninto Pt leads to an enhanced magnetic dampi ng in Py. Hence, the enhanced damping in our \nstudies is very unlikely resulti ng from spin pumping into SiO 2 or TaN. To our knowledge, this \nsurface damping could be related to  interfacial spin f lip scattering at the interface between Py \nand the insulating layers, which ha s been included in a generalized  spin-pumping theory reported \nrecently21. \nThe temperature dependence of the bulk damp ing and the surface damping are summarized \nin Figures 3a and 3b. The bulk damping of Py is  ~0.0055 at 300 K. As the temperature decreases, \nit shows a monotonic decrea se and is down to ~0.0049 at 5 K. Th ese values are consistent with \ntheoretical first principle calculations21-23 and the experimental valu es (0.004-0.008) reported for \nPy films with d ≥ 30 nm24-27. The temperature dependence of the bulk damping could be \nattributed to the magnetization rela xation due to the spin-lattice scattering in the Py films, which \ndecreases as the temperature decreases. \nOf particular interest, the surface damping sh ows a completely different characteristic, \nindicating a totally different mechanism from th e bulk damping. A strong enhancement peak is \nobserved at ~ 50 K for the surface damping. Could this enhancement of this surface/interface \ndamping be due to the strong spin-orbit coupli ng in atomic Ta of Ta N capping layer? To \ninvestigate this, we measure the damping of the 5 nm and 30 nm Py films with Al 2O3 capping \nlayer, which is expected to exhibit much lo wer spin-orbit coupling compared to TaN. The \ntemperature dependence of the Py damping is su mmarized in Figures 4a and 4b. Interestingly, \nthe similar enhancement of the damping at ~ 50 K is observed for 5 nm Py film with either Al 2O3 \ncapping layer or TaN layer, whic h excludes that the origin of the feature of the enhanced 7 damping at ~50 K results from th e strong spin-orbit coupling in TaN layer. These results also \nindicate that the mechanism of this feature is most likely related to the common properties of Py \nwith TaN and Al 2O3 capping layers, such as the crysta lline grain boundary and roughness of the \nPy films, etc. \nOne possible mechanism for the observed peak of  the damping at ~50 K could be related to a \nthermally induced spin reorientation transition  on the Py surface at that temperature. For \nexample, it has been show n that the spin reorientation of Py  in magnetic tunnel junction structure \nhappens due to the competition of different magne tic anisotropies, which c ould give rise to the \npeak of the FMR linewidth around the temperature of ~60 K28. Furthermore, we measure the \neffective magnetization ( Meff) as a function of temperature. Meff is obtained from the resonance \nfrequencies ( fres) vs. the external magnetic field via the Kittel formula29:  \n 12() [ ( 4 ) ]2res res res efffH H Mγππ=+   (4) \nin which Hres is the magnetic field at the resonance condition, and Meff is the effective \nmagnetization which contains the saturation ma gnetization and other anisotropy contributions. \nAs shown in Figures 5a and 5b, the 4π*M eff for 30 nm Py films w ith TaN capping layer are \nobtained to be ~10.4 and ~10.9 kG at 300 K and 5 K respectively. The temperature dependences \nof the 4π*M eff for 3nm, 5 nm, and 30 nm Py films are s hown in Figures 6a-6c. Around ~50 K, an \nanomaly in the effective magnetization for thin Py films (3 and 5 nm) is observed. Since we do \nnot expect any steep change in Py’s saturation magnetization at this temperature, the anomaly in \n4π*M eff should be caused by an anisot ropy change which coul d be related to a sp in reorientation. \nHowever, to fully understand the underlying mechan isms of the peak of the surface damping at ~ \n50 K, further theoretical and e xperimental studies are needed. 8 Conclusion  \nIn summary, the thickness and temperature dependences of the Gilbert damping in Py thin \nfilms are investigated, from which the contributio n due to the bulk damping and surface damping \nare clearly identified. Of particular interest, the bulk damping decreases monotonically as the \ntemperature decreases, while the surface damping develops an enhancement peak at ~ 50 K, \nwhich could be related to a thermally induced spin reorientation for the surface magnetization of \nthe Py thin films. This model is also consistent  with the observation of an enhancement of the \neffective magnetization below ~50 K. Our expe rimental results will contribute to the \nunderstanding of the intrinsic and ex trinsic mechanisms of the Gilber t damping in FM thin films.  \n \nMethods   \nMaterials growth. The Py thin films are deposited on ~25 nm SiO 2/Si substrates at room \ntemperature in 3×10- 3 Torr argon in a magnetron sputtering sy stem with a base pressure of ~ \n1×10-8 Torr. The growth rate of the Py is ~ 1  Å/s. To prevent ex situ  oxidation of the Py film \nduring the measurement,  a ~ 20 Å TaN or Al 2O3 capping layer is grown in situ environment. The \nTaN layer is grown by reactive sputtering of a Ta target in an argon-nitrogen gas mixture (ratio: \n90/10). For Al 2O3 capping layer, a thin Al (3 Å) layer is deposited first, and the Al 2O3 is \ndeposited by reactive spu ttering of an Al target in an ar gon-oxygen gas mixture (ratio: 93/7).  \nFMR measurement.  The FMR is measured using the vector network analyzer (VNA, Agilent \nE5071C) connected with a coplanar wave guide30 in the variable temperature insert of a \nQuantum Design Physical Properties Measuremen t System (PPMS) in the temperature range \nfrom 300 to 2 K. The Py sample is cut to be 1 × 0.4 cm and attached to the coplanar wave guide 9 with insulating silicon paste. For each temper ature from 300 K to 2 K, the forward complex \ntransmission coefficients (S 21) for the frequencies between 1 - 15 GHz are recorded as a function \nof the magnetic field sweeping from ~2500 Oe to 0 Oe.  \n \nContributions  \nJ.S. and W.H. proposed and supervised the studies. Y.Z. and Q.S. performed the FMR \nmeasurement and analyzed the data. T.S. and W.Y.  helped the measurement. S.H.Y. and S.S.P.P. \ngrew the films. Y.Z., J.S. and W.H. wrote the manuscript. All authors commented on the \nmanuscript and contributed to its final version. \n \nAcknowledgements   \nWe acknowledge the fruitful discussions with  Ryuichi Shindou, Ke Xia, Ziqiang Qiu, Qian \nNiu, Xincheng Xie and Ji Feng and the support of National Basic Research Programs of China \n(973 Grants 2013CB921903, 2014CB920902 and 2015 CB921104). Wei Han also acknowledges \nthe support by the 1000 Talents Program for Young Scientists of China. \n \nCompeting financial interests \nThe authors declare no compe ting financial interests. \n \n \nReferences: \n \n1 Landau, L. & Lifshitz, E. On the theory of the dispersion of magnetic permeability in \nferromagnetic bodies. Phys. Z. Sowjetunion  8, 153 (1935). \n2 Gilbert, T. L. A phenomenological theory  of damping in ferromagnetic materials. \nMagnetics, IEEE Transactions on  40, 3443-3449, doi:10.1109/TMAG.2004.836740 \n(2004). 10 3 Scheck, C., Cheng, L., Barsukov, I., Frait, Z.  & Bailey, W. E. 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B  78, 020404 (2008). \n14 Brataas, A., Tserkovnyak, Y. & Bauer, G. E. W. Scattering Theory of Gilbert Damping. \nPhys. Rev. Lett.  101, 037207 (2008). \n15 Arias, R. & Mills, D. L. Extrinsic contri butions to the ferromagne tic resonance response \nof ultrathin films. Phys. Rev. B  60, 7395-7409 (1999). \n16 Walowski, J., Müller, G., Djordjevic, M., M ünzenberg, M., Kläui, M., Vaz, C. A. F. & \nBland, J. A. C. Energy Equilibration Pro cesses of Electrons, Magnons, and Phonons at \nthe Femtosecond Time Scale. Phys. Rev. Lett.  101, 237401 (2008). \n17 Stiles, M. D. & Miltat, J. in Spin Dynamics in Confined Magnetic Structures III  Vol. 101 \nTopics in Applied Physics  (eds Burkard Hillebrands & André Thiaville) Ch. 7, 225-308 \n(Springer Berlin Heidelberg, 2006). \n18 Celinski, Z., Urquhart, K. B. & Heinrich, B. Using ferromagnetic resonance to measure \nthe magnetic moments of ultrathin films. J. Magn. Magn. Mater.  166, 6-26 (1997). \n19 Barati, E., Cinal, M., Edwards, D. M. & Umerski, A. Gilbert damping in magnetic \nlayered systems. Phys. Rev. B  90, 014420 (2014). \n20 Tserkovnyak, Y., Brataas, A., Bauer, G. E. W.  & Halperin, B. I. Nonlocal magnetization \ndynamics in ferromagnetic heterostructures. Rev. Mod. Phys.  77, 1375-1421 (2005). \n21 Liu, Y., Yuan, Z., Wesselink, R. J. H., St arikov, A. A. & Kelly, P. J. Interface \nEnhancement of Gilbert Damping from First Principles. Phys. Rev. Lett.  113, 207202 \n(2014). 11 22 Starikov, A. A., Kelly, P. J ., Brataas, A., Tserkovnyak, Y. & Bauer, G. E. W. Unified \nFirst-Principles Study of Gilbert Dampi ng, Spin-Flip Diffusion, and Resistivity in \nTransition Metal Alloys. Phys. Rev. Lett.  105, 236601 (2010). \n23 Mankovsky, S., Ködderitzsch, D., Woltersdo rf, G. & Ebert, H. First-principles \ncalculation of the Gilbert damping paramete r via the linear response formalism with \napplication to magnetic transition metals and alloys. Physical Review B  87, 014430 \n(2013). \n24 Bailey, W., Kabos, P., Mancoff, F. & Russe k, S. Control of magnetization dynamics in \nNi81Fe19 thin films through the us e of rare-earth dopants. Magnetics, IEEE Transactions \non 37, 1749-1754 (2001). \n25 Rantschler, J. O., Maranville, B. B., Malle tt, J. J., Chen, P., McMichael, R. D. & \nEgelhoff, W. F. Damping at no rmal metal/permalloy interfaces. Magnetics, IEEE \nTransactions on  41, 3523-3525 (2005). \n26 Luo, C., Feng, Z., Fu, Y., Zha ng, W., Wong, P. K. J., Kou, Z. X., Zhai, Y., Ding, H. F., \nFarle, M., Du, J. & Zhai, H. R. Enhancem ent of magnetization damping coefficient of \npermalloy thin films with dilute Nd dopants. Phys. Rev. B  89, 184412 (2014). \n27 Ghosh, A., Sierra, J. F., Auffret, S., Ebels,  U. & Bailey, W. E. Dependence of nonlocal \nGilbert damping on the ferromagnetic layer t ype in ferromagnet/Cu/Pt heterostructures. \nAppl. Phys. Lett.  98, 052508 (2011). \n28 Sierra, J. F., Pryadun, V. V., Russek, S. E ., García-Hernández, M., Mompean, F., Rozada, \nR., Chubykalo-Fesenko, O., Snoeck, E., Miao, G. X., Moodera, J. S. & Aliev, F. G. \nInterface and Temperature Dependent Magnetic  Properties in Permalloy Thin Films and \nTunnel Junction Structures. Journal of Nanoscience and Nanotechnology  11, 7653-7664 \n(2011). \n29 Kittel, C. On the Theory of Ferromagnetic Resonance Absorption. Phys. Rev.  73, 155 \n(1948). \n30 Kalarickal, S. S., Krivosik, P., Wu, M., Patt on, C. E., Schneider, M. L., Kabos, P., Silva, \nT. J. & Nibarger, J. P. Ferromagnetic reso nance linewidth in metallic thin films: \nComparison of measurement methods. J. Appl. Phys.  99, 093909 (2006). \n 12  \nFigure Captions  \n \nFigure 1.  Measurement of Gilbert damping in Py thin films via ferromagnetic resonance \n(Py thickness = 30 nm).   a, Ferromagnetic resonance spectra of the absorption for 30 nm Py thin \nfilms with TaN capping layer at gigahertz frequencies of 4, 6, 8, 10 and 12 GHz at 300 K after \nnormalization by background subtraction. b, c, The half linewidths as a function of the resonance \nfrequencies at 300 K and 5 K respectively. The red solid lines indicate the fitted lines based on \nequation (3), where the Gilbert damp ing constants could be obtained.  \n \nFigure 2. Temperature dependence of the Gilber t damping of Py thin films with TaN \ncapping.  a, The temperature dependence of the Gilbert damping fo r 3, 5, 10, 15, 20, 30, and 50 \nnm Py films. b, The Gilbert damping as a function of the Py thickness, d, measured at 300 K. c, \nThe Gilbert damping as a function of 1/ d measured at 300 K. The linear fitting corresponds to \nequation (4), in which the slope and the intercep t are related to the surf ace contribution and bulk \ncontribution to the total Gilber t damping. Error bars correspond to one standard deviation. \n Figure 3. Bulk and surface damping of Py  thin films with TaN capping layer.   a, b,  The \ntemperature dependence of the bulk damping an d surface damping, respectively. The inset table \nsummarizes the experimental values reported in early studies. Error bars correspond to one \nstandard deviation.  \nFigure 4. Comparison of the Gilbert damping of Py films with different capping layers.  a, \nb, Temperature dependence of the Gilbert dampi ng of Py thin films with TaN capping layer 13 (blue) and Al 2O3 capping layer (green) for 5 nm Py a nd 30 nm Py, respectively. Error bars \ncorrespond to one standard deviation.  \nFigure 5. Measurement of effective magnetizat ion in Py thin films via ferromagnetic \nresonance (Py thickness = 30 nm).   a, b, The resonance frequencies vs. the resonance magnetic \nfield at 300 K and 5 K, respectively. The fitted li nes (red curves) are obtained using the Kittel \nformula.   \nFigure 6. Effective magnetization of Py fi lms as a function of the temperature.  a, b, c, \nTemperature dependence of the effective magnetizati on of Py thin films of a thickness of 3 nm,  \n5 nm and 30 nm Py respectively. In b, c, the blue/green symbols correspond to the Py with \nTaN/Al\n2O3 capping layer.  \n \n 0\n500\n1000\n1500\n2000\n-0.3\n-0.2\n-0.1\n0.0\n0.1\n   4 \n   6 \n   8\n 10 \n 12 \n \nS\n21\n (dB) \n \nH (Oe)\nT=300 K\nf\n (GHz)\n0\n4\n8\n12\n16\n0\n10\n20\n30\n \n\nH (Oe)\n \nf (GHz)\nT=300 K\n0\n4\n8\n12\n16\n0\n10\n20\n30\n \n\nH (Oe)\n \nf (GHz)\nT=5 K\nb\nc\na\nFigure 10\n50\n100\n150\n200\n250\n300\n0.006\n0.008\n0.010\n0.012\n0.014\nd\n (nm)\n 3      \n 15        \n 5      \n 20\n 10    \n 30    \n               \n 50 \n \na\n \nTemperature (K)\n0.0\n0.1\n0.2\n0.3\n0.004\n0.006\n0.008\n0.010\n0.012\n0.014\n \na\n \n \n1/\nd\n (nm\n-1\n)\n0\n10\n20\n30\n0.006\n0.008\n0.010\n0.012\n0.014\n \n \nd\n (nm)\n \na\na\nb\nc\nFigure \n20\n50\n100\n150\n200\n250\n300\n0.0040\n0.0045\n0.0050\n0.0055\n0.0060\nTheory\n Ref. 21, 22\n Ref. 23\n Temperature (K)\n \na\nB\n \na\na\nExp.\n0.006\nRef. 24\n0.004\n-\n0.008\nRef.\n25\n0.007\nRef.\n26\n0.0067\nRef. 27\n0\n50\n100\n150\n200\n250\n300\n0.016\n0.018\n0.020\n0.022\n0.024\n0.026\n0.028\n0.030\n Temperature (K)\na\nS\n (nm)\n \nb\nFigure \n30\n50\n100\n150\n200\n250\n300\n0.004\n0.006\n0.008\n0.010\n 5 nm Py/TaN\n 5 nm Py/Al\n2\nO\n3\n Temperature (K)\na\n \n \n0\n50\n100\n150\n200\n250\n300\n0.004\n0.005\n0.006\n0.007\n 30 nm Py/TaN\n 30 nm Py/Al\n2\nO\n3\n Temperature (K)\na\n \n \na\nb\nFigure \n4a\nb\n0\n500\n1000\n1500\n2000\n0\n4\n8\n12\n16\n \nf\n (GHz)\n \nH\n (Oe)\nT=300 K\n0\n500\n1000\n1500\n2000\n0\n4\n8\n12\n16\n \nf\n (GHz)\n \nH (Oe)\nT=5 K\nFigure \n58.6\n8.8\n9.0\n9.2\n9.4\n9.6\n4\n\nM\neff\n (kG) \n 5 nm Py/TaN\n 5 nm Py/Al\n2\nO\n3\n \n6.2\n6.3\n6.4\n6.5\n6.6\n6.7\n6.8\n6.9\n4\n\nM\neff\n (kG) \n 3 nm Py/TaN\n \n0\n50\n100\n150\n10.6\n10.7\n10.8\n10.9\n11.0\n 30 nm Py/TaN\n 30 nm Py/Al\n2\nO\n3\n4\n\nM\neff\n (kG) \n \na\nb\nc\nTemperature (K) \nFigure \n6"    },    {        "title": "1704.01559v1.Relativistic_theory_of_magnetic_inertia_in_ultrafast_spin_dynamics.pdf",        "content": "Relativistic theory of magnetic inertia in ultrafast spin dynamics\nRitwik Mondal,\u0003Marco Berritta, Ashis K. Nandy, and Peter M. Oppeneer\nDepartment of Physics and Astronomy, Uppsala University, P. O. Box 516, SE-75120 Uppsala, Sweden\n(Dated: November 12, 2018)\nThe influence of possible magnetic inertia effects has recently drawn attention in ultrafast mag-\nnetization dynamics and switching. Here we derive rigorously a description of inertia in the\nLandau-Lifshitz-Gilbert equation on the basis of the Dirac-Kohn-Sham framework. Using the Foldy-\nWouthuysen transformation up to the order of 1=c4gives the intrinsic inertia of a pure system\nthrough the 2ndorder time-derivative of magnetization in the dynamical equation of motion. Thus,\nthe inertial damping Iis a higher order spin-orbit coupling effect, \u00181=c4, as compared to the\nGilbert damping \u0000that is of order 1=c2. Inertia is therefore expected to play a role only on ultra-\nshort timescales (sub-picoseconds). We also show that the Gilbert damping and inertial damping\nare related to one another through the imaginary and real parts of the magnetic susceptibility tensor\nrespectively.\nPACS numbers: 71.15.Rf, 75.78.-n, 75.40.Gb\nI. INTRODUCTION\nThe foundation of contemporary magnetization dy-\nnamics is the Landau-Lifshitz-Gilbert (LLG) equation\nwhich describes the precession of spin moment and a\ntransverse damping of it, while keeping the modulus of\nmagnetization vector fixed [1–3]. The LLG equation of\nmotion was originally derived phenomenologically and\nthe damping of spin motion has been attributed to rela-\ntivistic effects such as the spin-orbit interaction [1, 4–6].\nIn recent years there has been a flood of proposals for the\nfundamental microscopic mechanism behind the Gilbert\ndamping: the breathing Fermi surface model of Kamber-\nský, where the damping is due to magnetization preces-\nsion and the effect of spin-orbit interaction at the Fermi\nsurface [4], the extension of the breathing Fermi surface\nmodel to the torque-torque correlation model [5, 7], scat-\ntering theory description [8], effective field theories [9],\nlinear response formalism within relativistic electronic\nstructure theory [10], and the Dirac Hamiltonian theory\nformulation [11].\nFor practical reasons it was needed to extend the orig-\ninal LLG equation to include several other mechanisms\n[12, 13]. To describe e.g. current induced spin-transfer\ntorques, the effects of spin currents have been taken\ninto account [14–16], as well as spin-orbit torques [17],\nand the effect of spin diffusion [18]. A different kind of\nspin relaxation due to the exchange field has been intro-\nduced by Bar’yakhtar et al.[19]. In the Landau-Lifshitz-\nBar’yakhtar equation spin dissipations originate from the\nspatial dispersion of exchange effects through the second\norder space derivative of the effective field [20, 21]. A\nfurther recent work predicts the existence of extension\nterms that contain spatial as well temporal derivatives of\nthe local magnetization [22].\nAnother term, not discussed in the above investiga-\ntions, is the magnetic inertial damping that has recently\n\u0003ritwik.mondal@physics.uu.sedrawnattention[23–25]. Originally, magneticinertiawas\ndiscussed following the discovery of earth’s magnetism\n[26]. Within the LLG framework, inertia is introduced\nas an additional term [24, 27–29] leading to a modified\nLLG equation,\n@M\n@t=\u0000\rM\u0002He\u000b+M\u0002\u0012\n\u0000@M\n@t+I@2M\n@2t\u0013\n;(1)\nwhere \u0000is the Gilbert damping constant [1–3], \rthe gy-\nromagnetic ratio, He\u000bthe effective magnetic field, and I\nis the inertia of the magnetization dynamics, similar to\nthe mass in Newton’s equation. This type of motion has\nthe same classical analogue as the nutation of a spinning\nsymmetric top. The potential importance of inertia is il-\nlustrated in Fig. 1. While Gilbert damping slowly aligns\nthe precessing magnetization to the effective magnetic\nfield, inertial dynamics causes a trembling or nutation of\nthe magnetization vector [24, 30, 31]. Nutation could\nconsequently pull the magnetization toward the equa-\ntor and cause its switching to the antiparallel direction\n[32, 33], whilst depending crucially on the strength of\nthe magnetic inertia. The parameter Ithat character-\nizes the nutation motion is in the most general case a\ntensor and has been associated with the magnetic suscep-\ntibility [29, 31, 33]. Along a different line of reasoning,\nFähnle et al.extended the breathing Fermi surface model\nto include the effect of magnetic inertia [27, 34]. The\ntechnological importance of nutation dynamics is thus\nits potential to steer magnetization switching in memory\ndevices [23–25, 32] and also in skyrmionic spin textures\n[35]. Magnetization dynamics involving inertial dynam-\nics has been investigated recently and it was suggested\nthat its dynamics belongs to smaller time-scales i.e., the\nfemtosecond regime [24]. However, the origin of inertial\ndamping from a fundamental framework is still missing,\nand, moreover, although it is possible to vary the size of\nthe inertia in spin-dynamics simulations, it is unknown\nwhat the typical size of the inertial damping is.\nNaturally the question arises whether it is possible\nto derive the extended LLG equation including iner-arXiv:1704.01559v1  [cond-mat.other]  20 Mar 20172\nMHeff\nPrecession\nNutation\nFigure 1. (Color online) Schematic illustration of magnetiza-\ntion dynamics. The precessional motion of Maround He\u000bis\ndepicted by the blue solid-dashed curve and the nutation is\nshown by the red curve.\ntia while starting from the fully relativistic Dirac equa-\ntion. Hickey and Moodera [36] started from a Dirac\nHamiltonian and obtained an intrinsic Gilbert damping\nterm which originated from spin-orbit coupling. How-\never they started from only a part of the spin-orbit cou-\npling Hamiltonian which was anti-hermitian [37, 38]. A\nrecent derivation based on Dirac Hamiltonian theory for-\nmulation [11] showed that the Gilbert damping depends\nstrongly on both interband and intraband transitions\n(consistent with Ref. [39]) as well as the magnetic sus-\nceptibility response function, \u001fm. This derivation used\nthe relativistic expansion to the lowest order 1=c2of the\nhermitian Dirac-Kohn-Sham (DKS) Hamiltonian includ-\ning the effect of exchange field [40].\nIn this article we follow an approach similar to that of\nRef. [11] but we consider higher order expansion terms\nof the DKS Hamiltonian up to the order of 1=c4. This is\nshown to lead to the intrinsic inertia term in the modi-\nfied LLG equation and demonstrates that it stems from\na higher-order spin-orbit coupling term. A relativistic\norigin of the spin nutation angle, caused by Rashba-like\nspin-orbit coupling, was previously concluded, too, in the\ncontext of semiconductor nanostructures [41, 42].\nIn the following, we derive in Sec. II the relativistic\ncorrection terms to the extended Pauli Hamiltonian up\nto the order of 1=c4, which includes the spin-orbit inter-\naction and an additional term. Then the corresponding\nmagnetization dynamics is computed from the obtained\nspin Hamiltonian in Sec. III, which is shown to contain\nthe Gilbert damping and the magnetic inertial damping.\nFinally, we discuss the size of the magnetic inertia in re-\nlation to other earlier studies.II. RELATIVISTIC HAMILTONIAN\nFORMULATION\nWe start our derivation with a fully relativistic par-\nticle, a Dirac particle [43] inside a material and in the\npresence of an external field, for which we write the DKS\nHamiltonian:\nH=c\u000b\u0001(p\u0000eA) + (\f\u0000 1)mc2+V 1\n=O+ (\f\u0000 1)mc2+E; (2)\nwhereVis the effective crystal potential created by the\nion-ion, ion-electron and electron-electron interactions,\nA(r;t)is the magnetic vector potential from the external\nfield,cis the speed of light, mis particle’s mass and 1\nis the 4\u00024unit matrix. \u000band\fare the Dirac matrices\nthat have the form\n\u000b=\u0012\n0\u001b\n\u001b0\u0013\n; \f =\u0012\n10\n0\u00001\u0013\n;\nwhere\u001bis the Pauli spin matrix vector and 1is2\u00022unit\nmatrix. TheDiracequationisthenwrittenas i~@ (r;t)\n@t=\nH (r;t)for a Dirac bi-spinor  . The quantityO=c\u000b\u0001\n(p\u0000eA)defines the off-diagonal, or odd terms in the\nmatrix formalism and E=V 1are the diagonal, i.e., even\nterms. The latter have to be multiplied by a 2\u00022block\ndiagonal unit matrix in order to bring them in a matrix\nform. To obtain the nonrelativistic Hamiltonian and the\nrelativistic corrections one can write down the Dirac bi-\nspinor in double two component form as\n (r;t) =\u0012\n\u001e(r;t)\n\u0011(r;t)\u0013\n;\nand substitute those into the Dirac equation. The up-\nper two components represent the particle and the lower\ntwo components represent the anti-particle. However the\nquestion of separating the particle’s and anti-particle’s\nwave functions is not clear for any given momentum. As\nthe part\u000b\u0001pis off-diagonal in the matrix formalism, it\nretains the odd components and thus links the particle-\nantiparticle wave function. One way to eliminate the an-\ntiparticle’s wave function is by an exact transformation\n[44] which gives terms that require a further expansion in\npowers of 1=c2. Another way is to search for a represen-\ntation where the odd terms become smaller and smaller\nand one can ignore those with respect to the even terms\nand retain only the latter [45]. The Foldy-Wouthuysen\n(FW) transformation [46, 47] was the very successful at-\ntempt to find such a representation.\nIt is an unitary transformation obtained by suitably\nchoosing the FW operator,\nUFW=\u0000i\n2mc2\fO: (3)\nThe minus sign in front of the operator is because \fand\nOanti-commute with each other. The transformation of\nthewavefunctionadoptstheform  0(r;t) =eiUFW (r;t)3\nsuch that the probability density remains the same,\nj j2=j 0j2. The time-dependent FW transformation\ncan be expressed as [45, 48]\nHFW=eiUFW\u0012\nH\u0000i~@\n@t\u0013\ne\u0000iUFW+i~@\n@t:(4)\nThe first term can be expanded in a series as\neiUFWHe\u0000iUFW=H+i[UFW;H] +i2\n2![UFW;[UFW;H]]\n+::::+in\nn![UFW;[UFW;:::[UFW;H]:::]] +::: :(5)\nThe time dependency enters through the second term of\nEq. (4) and for a time-independent transformation one\nworks with@UFW\n@t= 0. It is instructive to note that the\naim of the whole procedure is to make the odd termssmaller and one can notice that as it goes higher and\nhigher in the expansion, the corresponding coefficients\ndecrease of the order 1=c2due to the choice of the unitary\noperator. After a first transformation, the new Hamilto-\nnian will contain new even terms, E0, as well as new odd\nterms,O0of1=c2or higher. The latter terms can be used\nto perform a next transformation having the new unitary\noperator as U0\nFW=\u0000i\n2mc2\fO0. After a second transfor-\nmation the new Hamiltonian, H0\nFWis achieved that has\nthe odd terms of the order 1=c4or higher. The trans-\nformation is a repetitive process and it continues until\nthe separation of positive and negative energy states are\nguaranteed.\nAfter a fourth transformation we derive the new trans-\nformed Hamiltonian with all the even terms that are cor-\nrect up to the order of1\nm3c6as [48–50]\nH000\nFW= (\f\u0000 1)mc2+\f\u0012O2\n2mc2\u0000O4\n8m3c6\u0013\n+E\u00001\n8m2c4h\nO;[O;E] +i~_Oi\n+\f\n16m3c6fO;[[O;E];E]g+\f\n8m3c6n\nO;h\ni~_O;Eio\n+\f\n16m3c6n\nO;(i~)2Oo\n: (6)\nNote that [A;B]defines the commutator, while fA;Bgrepresents the anti-commutator for any two operators Aand\nB. A similar Foldy-Wouthuysen transformation Hamiltonian up to an order of 1=m3c6was derived by Hinschberger\nand Hervieux in their recent work [51], however there are some differences, for example, the first and second terms in\nthe second line of Eq. (6) were not given. Once we have the transformed Hamiltonian as a function of odd and even\nterms, the final form is achieved by substituting the correct form of odd terms Oand calculating term by term.\nEvaluating all the terms separately, we derive the Hamiltonian for only the positive energy solutions i.e. the upper\ncomponents of the Dirac bi-spinor as a 2\u00022matrix formalism [40, 51, 52]:\nH000\nFW=(p\u0000eA)2\n2m+V\u0000e~\n2m\u001b\u0001B\u0000(p\u0000eA)4\n8m3c2\u0000e~2\n8m2c2r\u0001Etot+e~\n8m3c2n\n(p\u0000eA)2;\u001b\u0001Bo\n\u0000e~\n8m2c2\u001b\u0001[Etot\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002Etot]\n\u0000e~2\n16m3c4f(p\u0000eA);@tEtotg\u0000ie~2\n16m3c4\u001b\u0001[@tEtot\u0002(p\u0000eA) + (p\u0000eA)\u0002@tEtot]; (7)\nwhere@t\u0011@=@tdefines the first-order time derivative.\nThehigherorderterms( 1=c6ormore)willinvolvesimilar\nformulations and more and more time derivatives of the\nmagnetic and electric fields will appear that stem from\nthe time derivative of the odd operator O[48, 51].\nThe fields in the last Hamiltonian (7) are defined as\nB=r\u0002A, the external magnetic field, Etot=Eint+\nEextare the electric fields where Eint=\u00001\nerVis the\ninternal field that exists even without any perturbation\nandEext=\u0000@A\n@tis the external field (only the temporal\npart is retained here because of the Coulomb gauge).\nThe spin Hamiltonian\nThe aim of this work is to formulate the magnetiza-\ntion dynamics on the basis of this Hamiltonian. Thus,we split the Hamiltonian into spin-independent and spin-\ndependentpartsandconsiderfromnowonelectrons. The\nspin Hamiltonian is straightforwardly given as\nHS(t) =\u0000e\nmS\u0001B+e\n4m3c2n\n(p\u0000eA)2;S\u0001Bo\n\u0000e\n4m2c2S\u0001[Etot\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002Etot]\n\u0000ie~\n8m3c4S\u0001[@tEtot\u0002(p\u0000eA) + (p\u0000eA)\u0002@tEtot];\n(8)\nwhere the spin operator S= (~=2)\u001bhas been used. Let\nus briefly explain the physical meaning behind each term\nthat appears inHS(t). The first term defines the Zee-\nman coupling of the electron’s spin with the externally\napplied magnetic field. The second term defines an indi-\nrect coupling of light to the Zeeman interaction of spin\nand the optical B-field, which can be be shown to have4\nthe form of a relativistic Zeeman-like term. The third\nterm implies a general form of the spin-orbit coupling\nthat is gauge invariant [53], and it includes the effect of\nthe electric field from an internal as well as an external\nfield. Thelasttermisthenewtermofrelevanceherethat\nhas only been considered once in the literature by Hin-\nschberger et al.[51]. Note that, although the last term\nin Eq. (8) contains the total electric field, only the time-\nderivative of the external field plays a role here, because\nthe time derivative of internal field is zero as the ionic\npotential is time independent. In general if one assumes\na plane-wave solution of the electric field in Maxwell’s\nequation asE=E0ei!t, the last term can be written as\ne~!\n8m3c4S\u0001(E\u0002p)and thus adopts the form of a higher-\norder spin-orbit coupling for a general E-field.\nThe spin-dependent part can be easily rewritten in a\nshorter format using the identities:\nA\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002A= 2A\u0002(p\u0000eA)\n+i~r\u0002A (9)\nA\u0002(p\u0000eA) + (p\u0000eA)\u0002A=\u0000i~r\u0002A(10)\nfor any operator A. This allows us to write the spin\nHamiltonian as\nHS=\u0000e\nmS\u0001B+e\n2m3c2S\u0001B\u0014\np2\u00002eA\u0001p+3e2\n2A2\u0015\n\u0000e\n2m2c2S\u0001\u0002\nEtot\u0002(p\u0000eA)\u0003\n+ie~\n4m2c2S\u0001@tB\n+e~2\n8m3c4S\u0001@ttB: (11)\nHere, the Maxwell’s equations have been used to derive\nthe final form that the spatial derivative of the electric\nfield will generate a time derivative of a magnetic field\nsuch that r\u0002Eext=\u0000@B\n@t, whilst the curl of a internal\nfield results in zero as the curl of a gradient function is\nalways zero. The final spin Hamiltonian (11) bears much\nimportance for the strong laser field-matter interaction\nas it takes into account all the field-spin coupling terms.\nIt is thus the appropriate fundamental Hamiltonian to\nunderstand the effects of those interactions on the mag-\nnetization dynamics described in the next section.\nIII. MAGNETIZATION DYNAMICS\nIn general, magnetization is given by the magnetic mo-\nment per unit volume in a magnetic solid. The magnetic\nmomentisgivenby g\u0016BhSi,wheregistheLandég-factor\nand\u0016Bis the unit of Bohr magneton. The magnetization\nis then written\nM(r;t) =X\njg\u0016B\n\nhSji; (12)\nwhere \nis the suitably chosen volume element, the sum\njgoes over all electrons in the volume element, and h::i\nis the expectation value. To derive the dynamics, wetake the time derivative in both the sides of Eq. (12)\nand, withintheadiabaticapproximation, wearriveatthe\nequation of motion for the magnetization as [36, 54, 55]\n@M\n@t=X\njg\u0016B\n\n1\ni~h\u0002\nSj;HS(t)\u0003\ni:(13)\nNow the task looks simple, one needs to substitute the\nspin Hamiltonian (11) and calculate the commutators in\norder to find the equation of motion. Note that the dy-\nnamics only considers the local dynamics as we have not\ntaken into account the time derivative of particle density\noperator (for details, see [11]). Incorporating the latter\nwould give rise the local as well as non-local processes\n(i.e., spin currents) within the same footing.\nThe first term in the spin Hamiltonian produces the\ndynamics as\n@M(1)\n@t=\u0000\rM\u0002B; (14)\nwith\r=gjej=2mdefines the gyromagnetic ratio and\nthe Landé g-factor g\u00192for spins, the electronic charge\ne < 0. Using the linear relationship of magnetization\nwith the magnetic field B=\u00160(H+M), the latter is\nreplaced in Eq. (14) to get the usual form in the Landau-\nLifshitz equations, \u0000\r0M\u0002H, where\r0=\u00160\ris the\neffective gyromagnetic ratio. This gives the Larmor pre-\ncessionofmagnetizationaroundaneffectivefield H. The\neffective field will always have a contribution from a ex-\nchange field and the relativistic corrections to it, which\nhas not been explicitly taken into account in this article,\nas they are not in the focus here. For detailed calcula-\ntions yet including the exchange field see Ref. [11].\nThe second term in the spin Hamiltonian Eq. (11) will\nresult in a relativistic correction to the magnetization\nprecession. Within an uniform field approximation (A=\nB\u0002r=2), the corresponding dynamics will take the form\n@M(2)\n@t=\r\n2m2c2M\u0002BD\np2\u0000eB\u0001L+3e2\n8(B\u0002r)2E\n;\n(15)\nwithL=r\u0002pthe orbital angular momentum. The\npresence of \r=2m2c2implies that the contribution of this\ndynamics to the precession is relatively small, while the\nleading precession dynamics is given by Eq. (14). For\nsake of completeness we note that a relativistic correc-\ntion to the precession term of similar order 1=m2c2was\nobtained previously for the exchange field [11].\nThe next term in the Hamiltonian is a bit tricky to\nhandle as the third term in Eq. (11) is not hermitian, not\neven the fourth term which is anti-hermitian. However\ntogether they form a hermitian Hamiltonian [11, 37, 38].\nTherefore one has to work together with those terms and\ncannot only perform the dynamics with an individual\nterm. In an earlier work [11] we have shown that taking\nanuniformmagneticfieldalongwiththegauge A=B\u0002r\n2\nwill preserve the hermiticity. The essence of the uniform\nfield lies in the assumption that the skin depth of the5\nelectromagnetic field is longer than the thickness of the\nthin-film samples used in experiments. The dynamical\nequation of spin motion with the second and third terms\nthus thus be written in a compact form for harmonic ap-\nplied fields as [11]\n@M(3;4)\n@t=M\u0002\u0012\nA\u0001@M\n@t\u0013\n; (16)\nwith the intrinsic Gilbert damping parameter Athat is\na tensor defined by\nAij=\r\u00160\n4mc2X\nn;kh\nhripk+pkrii\u0000hrnpn+pnrni\u000eiki\n\u0002\u0010\n1+\u001f\u00001\nm\u0011\nkj:(17)\nHere\u001fmis the magnetic susceptibility tensor of rank 2\n(a3\u00023matrix) and 1is the 3\u00023unit matrix. Note that\nfor diagonal terms i.e., i=kthe contributions from the\nexpectation values of rkpicancel each other. The damp-\ning tensor can be decomposed to have contributions from\nan isotropic Heisenberg-like, anisotropic Ising-like and\nDzyaloshinskii-Moriya-like tensors. The anti-symmetric\nDzyaloshinskii-Moriya contribution has been shown to\nlead to a chiral damping of the form M\u0002(D\u0002@M=@t)\n[11]. Experimental observations of chiral damping have\nbeen reported recently [56]. The other cross term having\nthe formE\u0002Ain Eq. (11) is related to the angular mo-\nmentum of the electromagnetic field and thus provides\na torque on the spin that has been at the heart of an-\ngular magneto-electric coupling [53]. A possible effect in\nspin dynamics including the light’s angular momentum\nhas been investigated in the strong field regime and it\nhas been shown that one has to include this cross term\nin the dynamics in order to explain the qualitative and\nquantitative strong field dynamics [57].\nFor the last term in the spin Hamiltonian (11) it is\nrather easy to formulate the spin dynamics because it is\nevidently hermitian. Working out the commutator with\nthe spins gives a contribution to the dynamics as\n@M(5)\n@t=\u000eM\u0002@2B\n@t2; (18)\nwith the constant \u000e=\r~2\n8m2c4.\nLet us work explicitly with the second-order time\nderivative of the magnetic induction by the relation B=\n\u00160(H+M), using a chain rule for the derivative:\n@2B\n@t2=@\n@t\u0010@B\n@t\u0011\n=\u00160@\n@t\u0010@H\n@t+@M\n@t\u0011\n=\u00160\u0010@2H\n@t2+@2M\n@t2\u0011\n: (19)\nThis is a generalized equation for the time-derivative of\nthe magnetic induction which can be used even for non-\nharmonic fields. The magnetization dynamics is then\ngiven by\n@M(5)\n@t=\u00160\u000eM\u0002\u0010@2H\n@t2+@2M\n@t2\u0011\n:(20)Thus the extended LLG equation of motion will have\nthese two additional terms: (1) a field-derivative torque\nand (2) magnetization-derivative torque, and they ap-\npear with their 2ndorder time derivative. It deserves to\nbe noted that, in a previous theory we also obtained a\nsimilar term–a field-derivative torque in 1storder-time\nderivative appearing in the generalized Gilbert damping.\nSpecifically, the extended LLG equation for a general\ntime-dependent field H(t)becomes\n@M\n@t=\u0000\r0M\u0002H+M\u0002h\n\u0016A\u0001\u0010@H\n@t+@M\n@t\u0011i\n+\u00160\u000eM\u0002\u0010@2H\n@t2+@2M\n@t2\u0011\n; (21)\nwhere \u0016AisamodifiedGilbertdampingtensor(fordetails,\nsee [11]).\nHowever for harmonic fields, the response of the ferro-\nmagnetic materials is measured through the differential\nsusceptibility, \u001fm=@M=@H, because there exists a net\nmagnetization even in the absence of any applied field.\nWith this, the time derivative of the harmonic magnetic\nfield can be further written as:\n@2H\n@t2=@\n@t\u0010@H\n@M@M\n@t\u0011\n=@\n@t\u0010\n\u001f\u00001\nm\u0001@M\n@t\u0011\n=@\u001f\u00001\nm\n@t\u0001@M\n@t+\u001f\u00001\nm\u0001@2M\n@t2: (22)\nIn general the magnetic susceptibility is a spin-spin re-\nsponse function that is wave-vector and frequency depen-\ndent. Thus, Eq. (18) assumes the form with the first and\nsecond order time derivatives as\n@M(5)\n@t=M\u0002\u0012\nK\u0001@M\n@t+I\u0001@2M\n@t2\u0013\n;(23)\nwhere the parameters Iij=\u00160\u000e\u0000\n1+\u001f\u00001\nm\u0001\nijandKij=\n\u00160\u000e@t(\u001f\u00001\nm)ijare tensors. The dynamics of the second\nterm is that of the magnetic inertia that operates on\nshorter time scales [25].\nHaving all the required dynamical terms, finally the\nfull magnetization dynamics can be written by joining\ntogether all the individual parts. Thus the full magneti-\nzation dynamics becomes, for harmonic fields,\n@M\n@t=M\u0002\u0012\n\u0000\r0H+ \u0000\u0001@M\n@t+I\u0001@2M\n@t2\u0013\n:(24)\nNote that the Gilbert damping parameter \u0000has two con-\ntributions, one from the susceptibility itself, Aij, which\nis of order 1=c2and an other from the time derivative of\nit,Kijof order 1=c4. Thus, \u0000ij=Aij+Kij. However we\nwill focus on the first one only as it will obviously be the\ndominant contribution, i.e., \u0000ij\u0019Aij. Even though we\nconsider only the Gilbert damping term of order 1=c2in\nthe discussions, we shall explicitly analyze the other term\nof the order 1=c4. For an ac susceptibility i.e., \u001f\u00001\nm/ei!t\nwe find thatKij/\u00160\u000e@t(\u001f\u00001\nm)ij/i\u00160!\u000e\u001f\u00001\nm, which\nsuggests again that the Gilbert damping parameter of6\nthe order 1=c4will be given by the imaginary part of the\nsusceptibility,Kij/\u0000\u00160!\u000eIm\u0000\n\u001f\u00001\nm\u0001\n.\nThe last equation (24) is the central result of this\nwork, as it establishes a rigorous expression for the in-\ntrinsic magnetic inertia. Magnetization dynamics in-\ncluding inertia has been discussed in few earlier articles\n[24, 30, 31, 58]. The very last term in Eq. (24) has been\nassociated previously with the inertia magnetization dy-\nnamics [32, 59, 60]. As mentioned, it implies a magne-\ntization nutation i.e., a changing of the precession angle\nas time progresses. Without the inertia term we obtain\nthe well-known LLG equation of motion that has already\nbeen used extensively in magnetization dynamics simu-\nlations (see, e.g., [61–65]).\nIV. DISCUSSIONS\nMagnetic inertia was discussed first in relation to the\nearth’s magnetism [26]. From a dimensional analysis,\nthe magnetic inertia of a uniformly magnetized sphere\nundergoing uniform acceleration was estimated to be of\nthe order of 1=c2[26], which is consistent with the here-\nobtained relativistic nature of magnetic inertia.\nOur derivation based on the fundamental Dirac-Kohn-\nSham Hamiltonian provides explicit expressions for both\nthe Gilbert and inertial dampings. Thus, a comparison\ncan be made between the Gilbert damping parameter\nand the magnetic inertia parameter of a pure system.\nAs noticed above, both the parameters are given by the\nmagnetization susceptibility tensor, however it should be\nnoted that the quantiy hr\u000bp\fiis imaginary itself, because\n[11],\nhr\u000bp\fi=\u0000i~\n2mX\nn;n0;kf(Enk)\u0000f(En0k)\nEnk\u0000En0kp\u000b\nnn0p\f\nn0n:(25)\nThus the Gilbert damping parameter should be given by\nthe imaginary part of the susceptibility tensor [36, 66].\nOn the other hand the magnetic inertia tensor must be\ngiven by the real part of the susceptibility [31]. This is\nin agreement with a recent article where the authors also\nfound the same dependence of real and imaginary parts\nof susceptibility to the nutation and Gilbert damping re-\nspectively [33]. In our calculation, the Gilbert damping\nand inertia parameters adopt the following forms respec-tively,\n\u0000ij=i\r\u00160\n4mc2X\nn;k[hripk+pkrii\u0000hrnpn+pnrni\u000eik]\n\u0002Im\u0000\n\u001f\u00001\nm\u0001\nkj\n=\u0000\u00160\r~\n4mc2X\nn;k\u0014hripk+pkrii\u0000hrnpn+pnrni\u000eik\ni~\u0015\n\u0002Im\u0000\n\u001f\u00001\nm\u0001\nkj\n=\u0000\u0010X\nn;k\u0014hripk+pkrii\u0000hrnpn+pnrni\u000eik\ni~\u0015\n\u0002Im\u0000\n\u001f\u00001\nm\u0001\nkj;(26)\nIij=\u00160\r~2\n8m2c4h\n1+Re\u0000\n\u001f\u00001\nm\u0001\nkji\n=\u0010~\n2mc2h\n1+Re\u0000\n\u001f\u00001\nm\u0001\nkji\n; (27)\nwith\u0010\u0011\u00160\r~\n4mc2. Note that the change of sign from damp-\ningtensortotheinertiatensorthatisalsoconsistentwith\nRef. [33], and also a factor of 2 present in inertia. How-\never, most importantly, the inertia tensor is ~=mc2times\nsmallerthan the damping tensor as is revealed in our\ncalculations. Considering atomic units we can evaluate\n\u0010\u0018\u00160\n4c2\u00180:00066\n4\u00021372\u00188:8\u000210\u00009;\n\u0010~\n2mc2\u0018\u0010\n2c2\u00188:8\u000210\u00009\n2\u00021372\u00182:34\u000210\u000013:\nThis implies that the intrinsic inertial damping is typi-\ncally 4\u0002104times smaller than the Gilbert damping and\nit is not an independently variable parameter. Also, be-\ncause of its smallness magnetic inertial dynamics will be\nmore significant on shorter timescales [24].\nA further analysis of the two parameters can be made.\nOnecanusetheKramers-Kronigtransformationtorelate\nthe real and imaginary parts of a susceptibility tensor\nwith one another. This suggests a relation between the\ntwo parameters that has been found by Fähnle et al.[34],\nnamelyI=\u0000\u0000\u001c,where\u001cisarelaxationtime. Weobtain\nhere a similar relation, I/\u0000 \u0000\u0016\u001c, where \u0016\u001c=~=mc2has\ntime dimension.\nEven though the Gilbert damping is c2times larger\nthan the inertial damping, the relative strength of the\ntwo parameters also depends on the real and imaginary\nparts of the susceptibility tensor. In special cases, when\nthe real part of the susceptibility is much higher than\nthe imaginary part, their strength could be comparable\nto each other. We note in this context that there exist\nmaterials where the real part of the susceptibility is 102\u0000\n103times larger than the imaginary part.\nFinally, we emphasize that our derivation provides the\nintrinsic inertial damping of a pure, isolated system. For\nthe Gilbert damping it is already well known that en-\nvironmental effects, such as interfaces or grain bound-\naries, impurities, film thickness, and even interactions of7\nthe spins with quasi-particles, for example, phonons, can\nmodify the extrinsic damping (see, e.g., [67–69]). Simi-\nlarly, it can be expected that the inertial damping will\nbecome modified through environmental influences. An\nexample of environmental effects that can lead to mag-\nnetic inertia have been considered previously, for the case\nof a local spin moment surrounded by conduction elec-\ntrons, whose spins couple to the local spin moment and\naffect its dynamics [31, 32].\nV. CONCLUSIONS\nIn conclusion, we have rigorously derived the magne-\ntization dynamics from the fundamental Dirac Hamilto-\nnian and have provided a solid theoretical framework for,\nand established the origin of, magnetic inertia in pure\nsystems. We have derived expressions for the Gilbert\ndamping and the magnetic inertial damping on the same\nfooting and have shown that both of them have a rela-\ntivistic origin. The Gilbert damping stems from a gen-\neralized spin-orbit interaction involving external fields,\nwhiletheinertialdampingisduetohigher-order(in 1=c2)\nspin-orbit contributions in the external fields. Both have\nbeen shown to be tensorial quantities. For general time\ndependent external fields, a field-derivative torque with\na 1storder time derivative appears in the Gilbert-type\ndamping, and a 2ndorder time-derivative field torque ap-\npears in the inertial damping.\nIn the case of harmonic external fields, the expressions\nof the magnetic inertia and the Gilbert damping scalewith the real part and the imaginary part, respectively,\nof the magnetic susceptibility tensor, and they are op-\nposite in sign. Alike the Gilbert damping, the magnetic\ninertia tensor is also temperature dependent through the\nmagnetic response function and also magnetic moment\ndependent. Importantly, we find that the intrinsic iner-\ntial damping is much smaller than the Gilbert damping,\nwhich corroborates the fact that magnetic inertia was\nneglected in the early work on magnetization dynamics\n[1–3, 19]. This suggests, too, that the influence of mag-\nnetic inertia will be quite restricted, unless the real part\nof the susceptibility is much larger than the imaginary\npart. Another possibility to enhance the magnetic iner-\ntia would be to use environmental influences to increase\nits extrinsic contribution. 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Lett.\n102, 102401 (2013)."    },    {        "title": "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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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": "2308.03353v1.__textit_In_situ___electric_field_control_of_ferromagnetic_resonance_in_the_low_loss_organic_based_ferrimagnet_V_TCNE____x_sim_2__.pdf",        "content": "1 In situ electric-field control of ferromagnetic resonance in the low-\nloss organic-based ferrimagnet V[TCNE] x∼2 \nSeth W. Kurfman1, Andrew Franson1, Piyush Shah2, Yueguang Shi3, Hil Fung Harry  \nCheung4, Katherine E. Nygren5, Mitchell Swyt5, Kristen S. Buchanan5, Gregory D. Fuchs4, \nMichael E. Flatté3,6, Gopalan Srinivasan2, Michael Page2, and Ezekiel Johnston-Halperin†1 \n1Department of Physics, The Ohio State University \n2Materials and Manufacturing Directorate, Air Force Research Laboratory \n3Department of Physics and Astronomy, University of Iowa \n4Department of Physics, Cornell University \n5Department of Physics, Colorado State University \n6Department of Applied Physics, Eindhoven University of Technology \n†Corresponding author email: johnston-halperin.1@osu.edu  \nWe demonstrate indirect electric-field control of ferromagnetic resonance (FMR) in devices that \nintegrate the low-loss, molecule-based, room-temperature ferrimagnet vanadium \ntetracyanoethylene (V[TCNE] x∼2) mechanically coupled to PMN-PT piezoelectric transducers. \nUpon straining the V[TCNE] x films, the FMR frequency is tuned by more than 6 times the resonant \nlinewidth with no change in Gilbert damping for samples with α = 6.5 × 10−5. We show this tuning \neffect is due to a strain-dependent magnetic anisotropy in the films and find the magnetoelastic \ncoefficient | λs| ∼ (1 − 4.4) ppm, backed by theoretical predictions from DFT calculations and \nmagnetoelastic theory. Noting the rapidly expanding application space for strain-tuned FMR, we \ndefine a new metric for magnetostrictive materials, magnetostrictive agility,  given by the ratio of the \nmagnetoelastic coefficient to the FMR linewidth. This agility allows for a direct comparison \nbetween magnetostrictive materials in terms of their comparative efficacy for magnetoelectric \napplications requiring ultra-low loss magnetic resonance modulated by strain. With this metric, we \nshow V[TCNE] x is competitive with other magnetostrictive materials including YIG and Terfenol-\nD. This combination of ultra-narrow linewidth and magnetostriction in a system that can be \ndirectly integrated into functional devices without requiring heterogeneous integration in a thin-\nfilm geometry promises unprecedented functionality for electric-field tuned microwave devices \nranging from low-power, compact filters and circulators to emerging applications in quantum \ninformation science and technology. \nKeywords : magnetostriction, magnonics, molecule-based magnets  2 Introduction \nThe use of electric fields for control of magnetism has been a long-term goal of magnetoelectronics in \nits many manifestations ranging from metal and semiconductor spintronics [1, 2], to microwave electronics \n[3, 4], to emerging applications in quantum information [5]. This interest arises from the potential for clear \nimprovements in scaling, high-speed control, and multifunctional integration. However, while the promise \nof this approach is well established its realization has proven challenging due to the strong materials \nconstraints imposed by the existing library of magnetic materials [3, 4]. The most common approach to \nachieving this local control is through linking piezoelectricity with magnetostriction to achieve electric-\nfield control of magnetic anisotropy, either intrinsically through inherent coupling in multiferroic materials \nor extrinsically through piezoelectric/magnetic heterostructures [3]. Ideally, magnetic materials chosen for \nsuch applications should exhibit large magnetostriction, low magnetic damping and narrow linewidth (high-\nQ) ferromagnetic resonance (FMR), and robust mechanical stability upon strain cycling [4]. While the use \nof multiferroic materials promises relative simplicity in device design, they typically suffer from poor \nmagnetic properties and minimal tunability. The alternative approach of employing heterostructures of \nmagnetic thin films and piezoelectric substrates effectively creates a synthetic multiferroic exploiting the \nconverse magnetoelectric effect (CME) and in principle allows independent optimization of piezoelectric \nand magnetic properties [6, 7, 8, 9, 10, 11]. \nFurther, for applications that rely on magnetic resonance ( e.g., microwave electronics and magnon-based \nquantum information systems), the traditional metrics of magnetostriction, λs, or the CME coefficient, A, \ndo not capture the critical parameters governing damping and loss in this regime ( e.g., linewidth or Gilbert \ndamping coefficient). To date, materials with large magnetoelastic constants and CME coefficients ( e.g., \nTerfenol-D) suffer from high damping, broad magnetic resonance features, and are particularly fragile and \nbrittle [4, 12]. Ferrites such as yttrium iron garnet (YIG), on the other hand, are attractive due to their low-\nloss magnetic resonance properties but typically exhibit minimal to moderate magnetoelastic coefficients. \nFurther, these low-loss ferrites require high growth temperatures (800-900 ◦C) and lattice-matched \nsubstrates to produce high-quality material, which makes integrating these materials on-chip while \nmaintaining low-loss properties challenging [13, 14, 15], and limits their applicability for magnetic \nmicroelectronic integrated circuits (MMIC). Accordingly, alternative low-loss, magnetostrictive materials \nwith facile integration capabilities are desired for applications in electrically-controlled devices. Recently, \na complementary material to YIG, vanadium tetracyanoethylene (V[TCNE] x, x ~ 2), has gained significant \ninterest from the spintronics and quantum information science and engineering (QISE) communities due to \nits ultra-low damping under magnetic resonance and benign deposition characteristics [16, 17, 18, 19, 20, \n21, 22, 23, 24, 25, 26, 27].  3 Here we present the first systematic experimental study of the magnetostrictive properties of V[TCNE] x. \nComposite heterostructures of V[TCNE] x films and piezoelectric substrates demonstrate shifts in the FMR \nfrequency by 35 – 45.5 MHz, or more than 6 linewidths upon application of compressive strains up to 𝜀=\n−2.4×10ିସ . Further, a systematic analysis shows that the Gilbert damping, α, and inhomogeneous \nbroadening linewidth, 0, are insensitive to strain in this regime and robust to repeated cycling. Density-\nfunctional theory (DFT) calculations provide insight into the elastic and magnetoelastic properties of \nV[TCNE] x and predict a magnetoelastic coefficient 𝜆ሚଵ  = −2.52 ppm. Experimental measures of the \neffective magnetoelastic constant, λs, determined by combining optical measurements of the distortion with \nthe corresponding FMR frequency shifts, yield values of  λs = −(1 − 4 .4) ppm, which is in good agreement \nwith these DFT predictions. Finally, we define a new figure of merit, magnetostrictive agility, ζ , as the ratio \nof the magnitude of the magnetoelastic coefficient to the FMR linewidth ζ  = |λs|/Γ that more closely aligns \nwith the performance requirements for emerging applications of magnetostrictive materials. These results \nestablish a foundation for utilizing strain or acoustic (phononic) excitations for highly efficient strain-\nmodulated magnetoelectronic devices based on V[TCNE] x and other next-generation organic-molecule-\nbased magnetically-ordered materials for coherent information processing and straintronic applications \n[28].  \n \nResults \nV[TCNE] x is a room-temperature, organic-molecule-based ferrimagnet ( Tc ∼ 600 K) that exhibits superb \nlow-damping properties ( α = (3.98 ± 0.22) × 10−5) and high-quality factor FMR ( Q = fR/Γ > 3,000) [17, 18, \n19, 20, 21, 22, 23, 24, 25]. V[TCNE] x thin films are deposited via chemical vapor deposition (CVD) at \nrelatively low temperature and high pressure (50 ◦C and 35 mTorr, respectively) and is largely insensitive \nto substrate lattice constant or surface termination [17, 20]. Further, V[TCNE] x can be patterned via e-beam \nlithography techniques without increase in its damping [19]. The highly coherent and ultra-low loss \nmagnonic properties of V[TCNE] x have driven interest in applications in microwave electronics [26, 29] \nand magnon-based quantum information science and engineering (QISE) [21, 30, 31]. These benign \ndeposition conditions, combined with patterning that does not degrade performance, highlight the versatility \nof V[TCNE] x for facile on-chip integration with pre-patterned microwave circuits and devices [26, 27, 29, \n32, 33, 34]. These excellent magnetic properties are even more surprising given that V[TCNE] x lacks long-\nrange structural order [25]. Early studies indicated that V[TCNE] x films do not exhibit magnetic anisotropy \nbeyond shape effects due to this lack of long-range ordering [19, 20]. However, recent FMR studies on \nV[TCNE] x nanowires, microstructures, and thin films [19, 20, 21], coupled with combined DFT and \nelectron energy loss spectroscopy (EELS) of the crystal structure [25], suggest there is a residual nematic 4 ordering of the c-axis of the V[TCNE] x unit cell, giving rise to an averaged crystal field anisotropy that is \nsensitive to structural and thermally induced strain. However, dynamic measurements of these crystal fields \nand their dependence on strain are currently lacking, preventing a quantitative analysis of the \nmagnetostrictive properties of this material.  \nFor this work, PMN-PT/epoxy/V[TCNE] x/glass heterostructure devices are fabricated such that upon \nelectrically biasing the PMN-PT[001] substrate, the piezoelectric effect produces a lateral in-plane strain in \nthe V[TCNE] x thin film, schematically shown in Fig. 1a. PMN-PT is selected for its strong piezoelectric \neffects and high strain coefficients ( d31 ∼ −(500 to 1 ,000) pm/V) to maximize the strain in the devices [3], \nand the epoxy encapsulation layer is selected to allow for device operation under ambient conditions [22]. \nThis device structure allows investigation of the magnetoelastic properties of V[TCNE] x via standard FMR \ncharacterization and analysis. In the main text of this work, measurements on three devices denoted Samples \n1 - 3 are presented. Sample 1 is measured via broadband FMR (BFMR) techniques. Sample 2 is studied via \nX-band (∼9.8 GHz) cavity FMR techniques. Sample 3 is used to directly measure and calibrate strain in the \ndevices via optical techniques. Additional devices characterized via BFMR techniques are presented in the \nSupplemental Information and their characteristics summarized in Table 1.  \nThe BFMR response of Sample 1 is described in Fig. 1(b), where individual scans (inset) are fit to a \nLorentzian lineshape to extract the resonance frequency as a function of applied field, HR vs fR. This data \ncan be modeled by considering the V[TCNE] x thin film as an infinite sheet with attendant shape anisotropy \nand with a uniaxial crystal field anisotropy oriented in the out-of-plane direction (as described above). \nAccordingly, the Kittel equation for ferromagnetic resonance reduces to [19, 20, 21]  \n  (1) \nwhere fR = ω/2π is the FMR resonance frequency, γ is the gyromagnetic ratio, HR is the applied magnetic \nfield at resonance, Heff = 4πMeff =4πMs−H⊥ is the effective magnetization of the V[TCNE] x film with \nsaturation magnetization 4 πMs and uniaxial strain-dependent anisotropy field H⊥, and θ describes the \norientation of the external field as defined in Fig. 1(a). This equation is valid for films where HR ≫ 4πMeff, \nand is appropriate here as the effective magnetization for V[TCNE] x is typically ∼100 G [19] while the \nresonance field is typically between 3500 – 3650 G at X-band frequencies (9.86 GHz) for all magnetic field \norientations. As-grown films exhibit no in-plane anisotropy, consistent with the literature [19, 20, 21], and \nso ϕ-dependences are neglected.  When the external magnetic field is held out-of-plane ( θ = 0◦), Eq. (1) \nreduces to  \n  (2) \n5 Further information about the magnetic damping in thin films can be revealed by comparing the FMR \nfull-width-half-max (FWHM) frequency linewidth, Γ, to the FMR resonance frequency, fR, via (for \n4𝜋𝑀≪𝐻ோ and Γ ≪𝑓ோ [19]) \n Γ = 2αfR + Γ0 (3) \nwhere α is the (dimensionless) Gilbert damping constant and Γ 0 is the inhomogeneous broadening. It \nshould be noted this form for the Gilbert damping utilizing the frequency-swept linewidth is appropriate \ndirectly for out-of-plane magnetized thin films due to symmetry conditions resulting in the linear \nrelationship between 𝐻ோ  and 𝑓ோ  [19]. Accordingly, Eqs. (1 - 3) show that performing FMR at various \nfrequencies, fields, and magnetization orientations with and without applied strains in V[TCNE] x thin films \nshould provide information regarding the magnetoelastic properties of V[TCNE] x. \nFitting the data from Sample 1 to Eq. (2) reveals an effective magnetization 4 πMeff = 106.2 G and \ngyromagnetic ratio | γ|/2π = 2.756 MHz/Oe, consistent with literature [17, 18, 19, 20, 21, 22]. The Lorentzian \nfits of the FMR response also reveal the linewidth Γ as a function of the resonant frequency, seen in Fig. \n1(c), where fitting to Eq. (3) yields α = 1.02 ± 0.52 × 10−4 and Γ0 = 8.48 ± 1.22 MHz in Sample 1. These \ndamping characteristics are also consistent with literature values for V[TCNE] x [19, 27], and show that the \ndevices incorporate high-quality magnetic films exhibiting superb low-damping properties [32, 33, 34]. \nMoving beyond measurements of the as-grown strain-free sample, the FMR response of the device is \nnow measured while straining the V[TCNE] x film (Fig. 1(d)). Comparing the FMR response of Sample 1 \nwith no applied strain ( EB = 0 kV/cm) and maximum-applied strain ( EB = 13.3 kV/cm) yields a shift in the \nresonance frequency of 45.5 MHz at a resonance frequency fR = 9.8 GHz, corresponding to a CME \ncoefficient A = 3.38 MHz cm/kV (1.23 Oe cm/kV). It is worth noting that while this absolute shift in \nfrequency, and consequent value for CME, is modest when compared to other magnetostrictive materials \n[4], it represents a shift of over 6 magnetic resonance linewidths due to the ultra-low damping and narrow \nFMR linewidths of the V[TCNE] x thin film. This ability to shift cleanly on and off resonance with an applied \nelectric field is central to the functionality of many dynamically tuned MMIC devices, motivating a more \nin depth and systematic investigation of this phenomenon. \nThe magnetostriction in this composite device is explored by biasing the piezoelectric transducer \nbetween 0 kV/cm and 13.3 kV/cm, and the shift in the resonance frequency (for 𝜃= 0∘) tracks the linear \nstrain produced by the transducer [35], as seen in Fig. 2(a). For maximally strained films, fitting to Eq. (2) \nnow reveals 4 πMeff = 122.9 G, a difference of +16 .7 G between EB = 0 kV/cm and EB = 13.3 kV/cm (14% \nchange). Panels (b-d) of Fig. 2 show the FMR linewidth Γ, inhomogeneous broadening Γ 0, and Gilbert \ndamping α, for Sample 1 as a function of applied electric field (strain). These parameters do not vary over \nthe entire tuning range and are robust to repeated cycling ( >300 cycles - see Supplemental Information). 6 This stability indicates that the shift in resonance frequency is due to a true magnetoelastic effect under \nlinear deformation rather than some fatigue induced structure or morphology change in the film, and further \ndemonstrates the potential for device applications. Finally, it is noteworthy that the linewidths and damping \ncoefficients observed in these proof of principle devices are much narrower than typical magnetostrictive \nmaterials, but are roughly twice the value observed in optimized bare V[TCNE] x films (Γ is typically ~3 \nMHz [17, 19, 25]). This suggests that the tuning ratio of 6 times the linewidth may be further extended to \nmore than 10 times the linewidth in fully optimized devices [19, 25]. \nTo confirm that these shifts in the resonance position are due to strain-dependent crystal-field anisotropy \nin V[TCNE] x as prior studies suggest [20, 21], angular-dependent measurements on unstrained and \nmaximally strained films are performed. Sample 2 is mounted in an X-band (∼9.8 GHz) microwave cavity \nso that the structure can be rotated to vary the polar angle, θ. In-plane ( 𝜃= 90∘) and out-of-plane ( 𝜃= 0∘) \nFMR spectra are shown in Supplemental Fig. S1, with FWHM linewidths of 2.17 Oe (5.97 MHz) and 2.70 \nOe (7.45 MHz), respectively. By tracking the resonance field as a function of rotation and fitting to Eq. (1) \nthe effective magnetization Heff = 4πMeff = 74.0 G is extracted for Sample 2, as seen in Fig. 3. The difference \nin 4πMeff between Samples 1 and 2 can be attributed to sample-to-sample variation and remains consistent \nwith literature values [19]. Repeating the measurement with an applied bias of 13.3 kV/cm to the PMN-PT \nreveals an increase of 4 πMeff to 79.4 G, an increase of 5.4 Oe (8% change), which is like the change observed \nin Sample 1. This confirms that strain is modulating the magnetic anisotropy in V[TCNE] x through the \ncrystal field term H⊥ where 4πMeff = 4πMs−H⊥. This strain-dependent crystal field H⊥ is consistent with and \nsupports previous measurements of V[TCNE] x with both thermally and structurally induced strain [19, 20, \n21].  \nAn approximate upper bound to the strain in these devices can be simply calculated through the relation \nε = d31EB = (d31VB)/t ∼ −(6 − 12) × 10−4 for typical PMN-PT d31 piezo coefficients [4, 35]. However, the \naddition of epoxy, V[TCNE] x, and the glass substrate affect the overall stiffness of the device, thereby the \npiezo coefficient changes from d31 of the bare piezo to an effective coefficient deff of the entire stack. This \ndeff is directly measured by exploiting the color change of V[TCNE] x upon laser heating [23] to pattern \nfiducial marks on the samples and monitor their positions under strain using optical microscopy (see \nSupplemental Information). This approach yields an effective piezoelectric coefficient of deff ∼ −180 pm/V \nor strain of ε ∼ −2.4×10−4, reasonable for the PMN-PT heterostructures used here [4, 35].  \nDensity functional theory  calculations on the relaxed and strained V[TCNE] x unit cell provide further \ninsight into the elastic and magnetoelastic properties of V[TCNE] x. These properties are calculated using \nthe Vienna ab initio Simulation Package (VASP) (version 5.4.4) with a plane-wave basis, projector-\naugmented-wave pseudopotentials [36, 37, 38, 39], and hybrid functional treatment of Heyd-Scuseria-\nErnzerhof (HSE06) [40, 41]. The experimentally verified [25] local structure of the V[TCNE] x unit cell is 7 found by arranging the central V atom and octahedrally-coordinated TCNE ligands according to \nexperimental indications [42, 43, 44, 45, 46], and subsequently allowing the structure to relax by \nminimizing the energy. These DFT results previously produced detailed predictions of the structural \nordering of V[TCNE] x, along with the optoelectronic and inter-atomic vibrational properties of V[TCNE] x \nverified directly by EELS [25] and Raman spectroscopy [23], respectively. This robust and verified model \ntherefore promises reliable insight into the elastic and magnetoelastic properties of V[TCNE] x. \nThe magnetoelastic energy density for a cubic lattice f = fel + fme = E/V is a combination of the elastic \nenergy density  \n  (4) \nwhere Cij are the elements of the elasticity tensor and εij are the strains applied to the cubic lattice, and \nthe magnetoelastic coupling energy density  \n \nwhere Bi are the magnetoelastic coupling constants and αi where i ∈ {x,y,z} represent the cosines of the \nmagnetization vector [47].  \nThe elastic tensor C = Cij for V[TCNE] x is found by applying various strains to the unit cell and observing \nthe change in the energy. The calculated Cij tensor results in a predicted Young’s modulus for V[TCNE] x YV \n= 59.92 GPa. By directly applying compressive and tensile in-plane strains to the DFT unit cell (i.e. in the \nequatorial TCNE ligand plane [25]), one may calculate the overall change in the total energy density, both \nparallel and perpendicular to the easy axis. The difference between these two, Δ𝐸, is the magnetic energy \ndensity change, which is proportional to the magnetoelastic coupling constant B1 [47] \n ∆E/V = −(ν2D + 1)B1ε|| (6) \nwhere ν2D is the 2-dimension in-plane Poisson ratio and ε|| is the applied in-plane (equatorial TCNE \nplane) epitaxial strain while allowing out-of-plane (apical TCNE direction) relaxation. The elastic and \nmagnetoelastic coefficients are related via the magnetostriction constant λ100 = λs via \n . (7) \nAs a result, the calculated changes of the magnetoelastic energy density with strain provide direct \npredictions of the elasticity tensor ( Cij) and magnetoelastic coefficients ( Bi) for V[TCNE] x. For \npolycrystalline samples of cubic materials, the overall (averaged) magnetoelastic coefficient λs also \nconsiders the off-axis contribution from λ111 such that λs = (2/5)λ100 + (3/5)λ111. However, the off-axis \ncomponent is not considered here for two reasons: (i) the apical TCNE ligands are assumed to align along \n8 the out-of-plane direction ( z-axis, θ = 0◦), and (ii) difficulties in calculating the magnetoelastic energy \ndensity changes upon applying a shear strain that provides the estimate of B2 needed to calculate λ111. The \nformer argument is reasonable as previous experimental results indicate the magnetocrystalline anisotropy \nfrom strain is out-of-plane [21], consistent with the ligand crystal field splitting between the equatorial and \napical TCNE ligands [25]. Further, the lack of in-plane ( ϕ-dependent) anisotropy suggests the distribution \nin the plane averages out to zero. Therefore, the magnetoelastic coefficient calculated here considers an \naverage of the in-plane Cii components in determining λ100. That is, the DFT predicts a magnetoelastic \ncoefficient for V[TCNE] x of \n  (8) \nwhere 𝐶𝐼𝑃 = (1/2)(𝐶11 + 𝐶22)  = 60.56 GPa and C12 = 37.84 GPa. Accordingly, utilizing the \ncalculated value of B1 = 85.85 kPa (see Supplemental Information) predicts a theoretically calculated 𝜆ሚଵ = \n−2.52 ppm for V[TCNE] x magnetized along the apical TCNE ligand (i.e. θ = 0◦). \nCombining these ferromagnetic resonance, direct strain measurements, and DFT calculations provides \nthe information necessary to determine the magnetoelastic properties of V[TCNE] x. Here, we follow the \nconvention in the literature using the magnetoelastic free energy form from the applied stress σ = Y ε to the \nmagnetostrictive material, Fme = (3/2)λsσ [2, 3, 4]. Accordingly, this free energy yields an expression for the \nstrain-dependent perpendicular (out-of-plane) crystal field [3]  \n  (9) \nwhere λS is the magnetoelastic coefficient, Y is the Young’s modulus of the magnetic material, d31 is the \npiezoelectric coefficient of the (multiferroic) crystal, and EB is the electric field bias. Here, ε = d31EB is the \nstrain in the magnetic layer obtained based on the assumption that the electrically induced strain is perfectly \ntransferred to the magnetic film. For this study, the direct optical measurement of the strain in the V[TCNE] x \nfilms allows the modification of Eq. 9 by replacing d31EB by the measured ε = deffEB = −2.4 × 10−4 to account \nfor the mechanical complexity of the multilayered device. The magnetoelastic coefficient of V[TCNE] x can \nthen be calculated from Eq. 9 using the values of 4 πMS and H⊥ from FMR characterization, the direct \nmeasurement of ε from optical measurements, and the calculated value of YV = 59.92 GPa from DFT. \nAccordingly, inserting the corresponding values into Eq. 9 yields a magnetoelastic constant for V[TCNE] x \nof λs ∼ −1 ppm to  λs ∼ − 4 ppm for the devices measured here. This range shows excellent agreement with \nthe DFT calculations of the magnetoelastic coefficient 𝜆ሚଵ = −2.52 ppm from Eq. 8. This agreement \nprovides additional support for the robustness of the DFT model developed in previous work [23, 25]. \n9 Further, comparison with past studies of the temperature dependence of Heff [21] allows for the extraction \nof the thermal expansion coefficient of V[TCNE]  x, 𝛼௧ = 11 ppm/K, at room temperature. \n \nDiscussion \nThe results above compare V[TCNE] x thin films to other candidate magnetostrictive materials using the \nestablished metrics of CME and λs. However, while these parameters are effective in capturing the impact \nof magnetoelastic tuning on the DC magnetic properties of magnetic thin films and magnetoelectric devices, \nthey fail to capture the critical functionality for dynamic (AC) magnetoelectric applications: the ability to \ncleanly tune on and off magnetic resonance with an applied electric field. For example, Terfenol-D is \nconsidered a gold standard magnetostrictive material due to its record large magnetoelastic coefficient λs up \nto 2,000 and CME coefficients 𝐴 as large as 590 Oe cm/kV [3]. However, due to its broad ∼1 GHz FMR \nlinewidths, large 4 πMeff > 9,000 G, high Gilbert damping α = 6×10ିଶ, and brittle mechanical nature, it is \nnot practical for many applications in MMIC. As a result, we propose a new metric that appropriately \nquantifies the capability of magnetostrictive materials for applications in microwave magnonic systems [28, \n33, 34] that takes into account both the magnetostrictive characteristics and the linewidth (loss) under \nmagnetic resonance of a magnetically-ordered material. Accordingly, a magnetostrictive agility ζ is \nproposed here, which is the ratio of the magnetoelastic coefficient λs to the FMR linewidth (in MHz) ζ(fR) \n= |λs|/Γ. For the V[TCNE] x films studied here the magnetostrictive agility at X-band frequencies (9.8 GHz) \nis in the range ζ = {0.164 – 0.660}, comparable to YIG ζ = {0.139 − 0.455} and Terfenol-D ζ = {0.301 – \n0.662} as shown in Table 1. Further, we note that the growth conditions under which high-quality V[TCNE] x \nfilms can be obtained make on-chip integration with microwave devices significantly more practical than \nfor YIG, and that the narrow linewidth (low loss) is more attractive for applications such as filters and \nmicrowave multiplexers than Terfenol-D. \nConclusion \nWe have systematically explored indirect electric-field control of ferromagnetic resonance in the low-\nloss organic-based ferrimagnet V[TCNE] x in V[TCNE] x/PMN-PT heterostructures. These devices \ndemonstrate the ability to shift the magnetic resonance frequency of V[TCNE] x by more than 6 linewidths \nupon application of compressive in-plane strains 𝜀 ~ 10ିସ . Further, we find there is no change in the \nmagnetic damping of the films with strain and that the samples are robust to repeated cycling (> 300 cycles), \ndemonstrating the potential for applications in MMIC without sacrificing the ultra-low damping of \nmagnetic resonance in V[TCNE] x. The changes in the FMR characteristics along with direct optical 10 measurements of strain provide an experimentally determined range for the magnetoelastic coefficient, λS \n= −(1 − 4 .3) ppm, showing excellent agreement to DFT calculations of the elastic and magnetoelastic \nproperties of V[TCNE] x. Finally, we present a discussion on the metrics used in the magnetostriction \ncommunity wherein we point out the shortcomings on the commonly used metrics of the magnetostriction \nand CME coefficients. In this context, we propose a new metric, the magnetostrictive agility, ζ, for use of \nmagnetoelastic materials for coherent magnonics applications.  \nThese results develop the framework necessary for extended studies into strain-modulated magnonics \nin V[TCNE] x. Additionally, these magnetoelastic properties in V[TCNE] x suggest that large phonon-\nmagnon coupling in V[TCNE] x might be achieved, necessary and useful for applications in acoustically-\ndriven FMR (ADFMR) or, in conjunction with high-Q phonons, for quantum information applications [28]. \nOther recent work has identified V[TCNE] x as a promising candidate for QISE applications utilizing \nsuperconducting resonators [31] and NV centers in diamond ranging from enhanced electric-field sensing \n[5] to coupling NV centers over micron length scales [30]. These findings lay a potential framework for \ninvestigating the utilization of V[TCNE] x in quantum systems based on magnons and phonons. \n \nAcknowledgments \nS. W. K. developed the project idea, and S. W. K., E. J.-H., P. S., and M. P. developed the project plan for \nexperimental analysis. S. W. K.  fabricated the V[TCNE] x heterostructure devices, performed FMR \ncharacterization and analysis, and wrote the manuscript. A. F. developed the analysis software used for \nfitting FMR linewidths and extracting parameter fits. P. S., G. S., and M. P. provided PMN-PT substrates. \nY. S. performed and analyzed DFT calculations of the elasticity tensor and the magnetoelastic coefficients \nfor V[TCNE] x. H. F. H. C. performed and analyzed optical measurements of strain in the devices. K. E. \nN. and M. S. performed BLS measurements on V[TCNE] x/Epoxy devices to extract elastic properties. All \nauthors discussed the results and revised the manuscript. S. W. K., A. F., and E. J.-H. were supported by \nNSF DMR-1808704. P. S., G. S, and M. P. were supported by the Air Force Office of Scientific Research \n(AFOSR) Award No. FA955023RXCOR001. 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Appl. \nPhys. 99, 093909 (2006). \nMethods \nSynthesis of V[TCNE] x and Device Fabrication.  \nV[TCNE] x films are deposited via ambient-condition chemical vapor deposition (CVD) in a custom CVD reactor \ninside an argon glovebox (O 2 < 1 ppm, H 2O < 1 ppm) in accordance with literature [17, 18, 19, 20, 21, 22, 23, 24, \n25, 26, 27, 29, 31]. Argon gas flows over TCNE and V(CO) 6 precursors that react to form a V[TCNE] x thin film on \nthe substrates. The pressure inside the CVD reactor for all growths was 35 mmHg, and TCNE, V(CO) 6, and the \nsubstrates are held at 65◦C, 10◦C, and 50◦C, respectively. All substrates were cleaned via solvent chain (acetone, \nmethanol, isopropanol, and deionized (DI) water (×2)) and dried with N 2, followed by a 10 minute UV/Ozone clean \nin a UVOCS T10x 10/OES to remove any residual organic contaminants. \nNominally 400 nm V[TCNE] x films are deposited onto microscope cover glass substrates ( t = 100µm). These \nV[TCNE] x/glass substrates are then mechanically fixed to a PMN-PT transducer (4 mm×10 mm ×0.15 mm) with an \nOLED epoxy (Ossila E130) to create a PMN-PT/Epoxy/V[TCNE] x/Glass heterostructure. The epoxy here not only \nprotects V[TCNE] x from oxidation [22], but also propagates lateral strain into V[TCNE] x film from the piezo \ntransducer upon biasing. While the primary deformation in the piezo transducer is along the poling direction of the 16 PMN-PT ( z), the distortion of the PMN-PT in the thickness direction also produces a lateral in-plane strain in the \nPMN-PT through the Poisson effect (i.e. one must consider here the d31 piezo coefficient of PMN-PT). Therefore, the \nprimary strain experienced by the V[TCNE] x film is in-plane. The PMN-PT electrodes are connected to a Keithley \n2400 voltage source so that electric fields up to EB = VB/tPMN−PT = 13.3 kV/cm can be applied across the PMN-PT layer. \nFerromagnetic Resonance Characterization \nBroadband FMR (BFMR) measurements on Sample 1 and Supplemental Devices A-C were taken using a \ncommercial microstrip (Southwest Microwave B4003-8M-50) and Agilent N5222A vector network analyzer \n(VNA). The devices are mounted so that the magnetic field is normal to the V[TCNE] x film (𝜃 = 0∘ ). S21 \nmeasurements (P = −20 dBm) show the FMR peak upon matched magnetic field and frequency conditions \nin accordance with Eq. 2. A Keithley 2400 Sourcemeter is used to apply up to 200 V to the piezoelectric \ntransducers – accordingly, the maximum-applied strain in the 150 μm PMN-PT corresponds to an electric \nfield 𝐸 = 13.3 kV/cm as mentioned in the main text. \nAll angular-dependent FMR measurements (Sample 2) were performed in a Bruker X-band (~9.6 GHz) \nEPR (Elexsys 500) spectrometer. The frequency of the microwave source is tuned to match the resonant \nfrequency of the cavity before each scan to ensure optimal cavity tuning. All scans had a 0.03 G modulation \nfield at 100 kHz modulation frequency and were performed at the lowest possible microwave power (0.2 \nμW) to prevent sample heating and non-linear effects distorting the FMR lineshape. The V[TCNE] x/PMN-PT \ndevices are mounted on a sapphire wafer and loaded into glass \ntubes for FMR measurements such that the samples can be rotated in-plane (IP: 𝜃 = 90∘ ) to out-of-plane \n(OOP: 𝜃 = 0∘ ) for FMR measurements in 10 degree increments, where resonance occurs upon matched field \nand frequency conditions according to Eq. 1. \nDensity Functional Theory Calculations \nThe pseudopotentials used are default options from VASP’s official PAW potential set, with five valence electrons \nper vanadium, four per carbon and five per nitrogen [36, 37, 38, 39]. For the rest of the calculation we used 400 eV \nfor the energy cutoff and a Γ centered 5x5x3 k-mesh sampling. From these results, the elastic tensor Cij for V[TCNE] x \nis calculated. Using the elastic tensor, the Young’s modulus for V[TCNE] x is averaged over the C11, C22, and C33 \ncomponents to yield YV = 59.92 GPa. From the DFT calculations, the full elastic matrix from the Cij is given by (in \nunits of GPa) \n𝐶=\n⎣⎢⎢⎢⎢⎡66.4437.847.96\n37.8454.683.79\n7.96 3.79 58.641.38−0.200.53\n0.09−1.55−0.37\n−0.69 0.76 0.31\n1.38 0.09 −0.69\n−0.20 −1.55 0.76\n0.53 −0.37 0.3135.16 0.25 −0.95\n0.25 6.65 −0.17\n−0.95 −0.17 9.94 ⎦⎥⎥⎥⎥⎤\n \n      \n 17 Optical Measurements of Strain in V[TCNE] x \nV[TCNE] x films can be patterned via laser heating techniques, whereupon the material changes color when heated \nabove its thermal degradation temperature ( ∼ 370 K) [16, 23]. To more appropriately calibrate strain in the \nV[TCNE] x films versus applied bias, we directly measure the deformation in the films by exploiting the color change \nof V[TCNE] x upon laser heating [23] and optical microscopy techniques. Fresh V[TCNE] x/PMN-PT devices are \nexposed to a focused laser spot to create ad hoc fiducial marks on the film in Sample 3 (Supplementary Fig. S5) in \na 50 µm × 50 µm square. By measuring the distance between these laser-written structures with and without applied \nstrain, we can precisely and directly measure the strain in the V[TCNE] x films upon electric bias thus allowing a \nmore precise calculation of magnetoelastic coefficients. Using these methods, we apply a bias of 13.3 kV/cm on \nSample 3 and find a strain ε ∼ 2.4×10−4 which is in reasonable agreement with estimated values of strain using the \nthickness of the PMN-PT (150 µm) and typical piezo coefficient d31 ∼ 500 − 1000 pm/V).  18  \n \n \n \nFigure 1: (a) Effective device schematic, coordinate system, and wiring diagram for V[TCNE] x/PMN-PT \nheterostructures. (b) Ferromagnetic resonance frequency fR vs external field Hext with the field held OOP ( θ \n= 0◦) measured via BFMR. The external field is held constant as the microwave frequency is swept. (Inset) \nRepresentative BFMR scan at f0 = 9.8 GHz, Hext = 3,660 Oe. (c) FMR linewidth Γ versus FMR frequency \nfor OOP field ( θ = 0◦). A linear fit (red line) extracts the dimensionless Gilbert damping parameter α = 1.02 \n± 0.52 × 10−4 and the inhomogeneous broadening Γ 0 = 8.48 ± 1.22 MHz. (d) BFMR scans for unstrained (0 \nkV/cm – black) and maximally strained (13.3 kV/cm – red). The shift in the FMR frequency is ∼45 MHz, \na shift ∼4 linewidths.  \n 19   \nFigure 2:  V[TCNE] x damping analysis with applied strain: (a) Plot showing differential (shifted) \nFMR resonance position f R−f0 where f 0 is the resonance at 9.8 GHz, (b) FWHM linewidth Γ, (c) \ninhomogeneous broadening Γ 0, and (d) Gilbert damping α versus applied electric field bias E B. \nWhile the resonance position shifts by multiple linewidths, there is negligible effect in the \nlinewidth or damping of the material.  20  \n  \n \n \n \n \n \nFigure 3:  Cavity X-band FMR measurements on V[TCNE] x/PMN-PT devices. Angular dependence of \nFMR resonant field H R at 𝑓ோ ∼ 9.6 GHz is measured without (black squares) and with (red circles) strain. \nIn-plane and out-of-plane peak-to-peak (FWHM) linewidths are 1.25 (2.16) Oe and 1.56 (2.70) Oe, \nrespectively.  \n 21  \n \n \n \n \n \n \n \n \nTable 1: Extracted parameters from V[TCNE] x strain devices compared to YIG, Terfenol-D, and \nother magnetostrictive materials. Asterisk indicates the frequency-equivalent linewidth calculated \nfrom the field-swept FMR linewidth and accounts for the ellipticity of FMR precession for in-plane \nmagnetized materials following the method in Ref. [56].  \n22 Supplemental Information:  In situ electric-field control of \nferromagnetic resonance in the low-loss organic-based ferrimagnet \nV[TCNE] x∼2 \n \n \nCavity X-band FMR of Sample 2  \n \n \n \n \n \n \n \n \n \n \n \n \n \nFigure S1:  X-band (9.8 GHz) cavity FMR scans of Sample 2 for DC magnetic field in-plane ( 𝜃=90°) and out-\nof-plane ( 𝜃 = 0° ). Peak-to-peak (p2p) linewidths and resonance positions determined from a fit to a \nLorentzian derivative, from which the full-width-half-max (FWHM) linewidth Γ is found by multiplying by \n√3. 23 V[TCNE] x Resonance Frequency with Piezo Switching Strain  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \nFigure S2: V[TCNE] x out-of-plane magnetized differential resonance frequency 𝛿𝑓ோ=9.8 𝐺𝐻𝑧−𝑓ோ \nas a function of applied bias voltage to 150 µm PMN-PT . The “butterfly” hysteresis arises from the \nhysteretic behavior of the piezo strain upon the polarization direction switching. The inset shows \nfrequency-swept FMR spectra and fits at maximum and minimum frequency shift.  24  \n \n \n \n \nFigure S3: Differential resonance frequency ( 𝑓ோ,=9.8 GHz) in the same device from Fig. S2 switching \nbetween 𝑉= ±25 V (𝐸= ±1.67 kV/cm) as a function of the number of the number of switches \nbetween positive and negative applied strain. The FWHM linewidth of the V[TCNE] x remains \neffectively constant for over 300 positive/negative (tensile/compressive) strain applications, and the \nresonant frequency for the respective compressive and tensile additionally remains effectively constant. \nThese conditions were selected based on the linewidth and resonance frequency tuning such that the \nresonance features do not overlap, thereby demonstrating a means to electrically-bias a device on and \noff resonance.  \n25 V[TCNE] x Density Functional Theory Calculations of Strain-Dependent \nMagnetoelastic Energy  \n \nV[TCNE] x Optical Strain Characterization   \nFigure S5: Positions of fiducial marks “burned” onto the V[TCNE] x film measured via optical techniques \nupon biasing a 150 µm PMN-PT piezo transducer. The extracted strain in x and y is averaged to 𝜀 =\n2.4×10ିସ and is used to calculate the magnetoelastic coefficients 𝜆ௌ presented in the text.  \nFigure S4: DFT-calculated magnetic energy difference of the V[TCNE] x unit cell upon manipulating \nthe applied strain. The orange line is a tangential linear fit at 𝜀 = 0 to solve for 𝛥𝐸/𝑉 in the main \ntext that provides the magnetoelastic coupling 𝐵ଵ. 26  \nAdditional V[TCNE] x Device Strain Characterizations  \n \n \n   \nFigure S6:  Supplemental devices measured via BFMR techniques ( 𝜃=0°). Supplemental device C varies from the \nothers only by the thickness of the PMN-PT piezo ( 𝑡= 500 𝜇𝑚 ), so that the electric field across the device \n(hence the strain) is adjusted accordingly.  27 Additional Linewidth and Damping Analysis: Supplemental Device C ( 𝒕𝑷=\n𝟓𝟎𝟎 𝝁𝒎)  \n \n \n \n \nFigure S7:  Gilbert analysis of Supplemental Device C as a function of applied electric-field bias. (a) Resonance \nfrequency for an out-of-plane magnetization orientation and applied external field 𝐻ோ= 3,658.8  G (𝑓ோ,=\n9.83  GHz). (b) FWHM linewidth corresponding to the resonance frequencies in panel (a). (c) \nInhomogeneous broadening and (d) Gilbert damping parameters.  Note there is negligible change in \nlinewidth, inhomogeneous broadening, and Gilbert damping for positive and negative bias up to the piezo \nswitching fields at ±3.2 kV/cm. The error bars in (a) and (b) are smaller than the markers used for the data \npoints. "    },    {        "title": "0908.0600v1.Persistence_effects_in_deterministic_diffusion.pdf",        "content": "Persistence effects in deterministic diffusion\nThomas Gilbert1,\u0003and David P. Sanders2, †\n1Center for Nonlinear Phenomena and Complex Systems,\nUniversit ´e Libre de Bruxelles, C. P . 231, Campus Plaine, B-1050 Brussels, Belgium\n2Departamento de F ´ısica, Facultad de Ciencias, Universidad Nacional\nAut´onoma de M ´exico, Ciudad Universitaria, 04510 M ´exico D.F ., Mexico\nIn systems which exhibit deterministic diffusion, the gross parameter dependence of the diffusion coefficient\ncan often be understood in terms of random walk models. Provided the decay of correlations is fast enough,\none can ignore memory effects and approximate the diffusion coefficient according to dimensional arguments.\nBy successively including the effects of one and two steps of memory on this approximation, we examine the\neffects of “persistence” on the diffusion coefficients of extended two-dimensional billiard tables and show how\nto properly account for these effects, using walks in which a particle undergoes jumps in different directions\nwith probabilities that depend on where they came from.\nPACS numbers: 05.60.Cd, 05.45.-a, 05.10.-a, 02.50.-r\nI. INTRODUCTION\nDiffusion is a fundamental macroscopic phenomenon in\nphysical systems, which, for instance, characterizes the\nspreading of tracer particles in a solvent. At a mescoscopic\nscale, it can be traced to the cumulative effect of many “ran-\ndom” displacements, as in Brownian motion [1]. At the under-\nlying microscopic scale, however, the dynamics of a system\nare deterministic. Deterministic diffusion concerns the study\nof microscopic models whose deterministic dynamics also ex-\nhibit diffusive behavior at a macroscopic scale [2, 3, 4].\nA particularly appealing, physically motivated model\nwhich does exhibit this phenomenon is the periodic Lorentz\ngas [5]. Here, independent point particles in free motion un-\ndergo elastic collisions with fixed hard disks in a periodic ar-\nray. The diffusive motion can then be considered to be a result\nof the chaotic nature of the microscopic dynamics, accord-\ning to which nearby initial conditions separate exponentially\nfast due to the convex nature of the obstacles. Thus a cloud\nof (non-interacting) particles in this Lorentz gas spreads out\nover time in a way similar to that of solutions of the diffusion\nequation,\n\n[x(t)\u0000x(0)]2\u000b\n\u00184Dt; (1)\nwhere x(t)denotes the position of a tracer at time t, with ini-\ntial position x(0), and the mean squared displacement is com-\nputed as an average h\u0001iover many realizations of this process.\nThe diffusion coefficient, D, is a constant which depends on\nthe geometrical parameters of the system, i.e., the underlying\nmicroscopic dynamics.\nThe diffusion coefficient summarizes the macroscopic be-\nhavior of the system while capturing the microscopic prop-\nerties of the dynamics that lead to it. A central question in\ndeterministic diffusion is to understand how this dependence\n\u0003Electronic address: thomas.gilbert@ulb.ac.be\n†Electronic address: dps@fciencias.unam.mx; URL: http://sistemas.\nfciencias.unam.mx/ ~dsanderson the geometrical parameters comes about. This has been ad-\ndressed in particular by Machta and Zwanzig [6], who showed\nthat in the limit where the obstacles are close together, the mo-\ntion reduces to a stochastic Bernoulli-type hopping process—\narandom walk —between “traps”. By calculating the diffu-\nsion coefficient of this random walk, they were able to obtain\na reasonable agreement with the numerically-measured value\nof the diffusion coefficient.\nThe approach of Machta and Zwanzig was extended heuris-\ntically by Klages and Dellago [7], by including important\nphysical effects not taken into account in the simple random-\nwalk picture, namely a possibly non-isotropic probability of\nchanging directions, and of crossing two traps at once. Klages\nand Korabel [8] then provided an alternate approach, in which\nthey employed a Green-Kubo expansion of the diffusion co-\nefficient to obtain a series of increasingly accurate approx-\nimations, based on numerically-calculated multi-step transi-\ntion probabilities. In one particular Lorentz gas model, they\nshowed that their results are in good agreement with this ex-\npansion, see also [4]. Nonetheless, as we emphasize below,\nthe physical motivation, and indeed the physical meaning, of\nthis approach, are not clear.\nThe purpose of this paper is to show that in fact the correct\nexpansion beyond the Machta-Zwanzig approximation is to\nincorporate this type of correction in the framework of persis-\ntent random walks . In other words, to be consistent, memory\neffects of a given length, whether one or several steps, must\nbe accounted for through their contribution at all orders in the\nGreen-Kubo formula relating the diffusion coefficient to the\nvelocity auto-correlations. This is physically strongly moti-\nvated, and provides the correct way of incorporating correla-\ntion effects, in principle, of any finite order.\nIn the usual periodic Lorentz gas on a triangular lattice con-\nsidered in [8], the model is sufficiently isotropic that higher or-\nder contributions are small and can be safely neglected. How-\never, when the correlation effects are very strong, this is no\nlonger the case. We introduce a billiard model with this prop-\nerty and show that, whereas a Green-Kubo–type expansion\nfails except very close to the Machta-Zwanzig limiting case,\nthe approximation based on a first-order persistent random\nwalk is in reasonable agreement with the data. By further con-arXiv:0908.0600v1  [nlin.CD]  5 Aug 20092\nFIG. 1: Periodic Lorentz gas on a triangular lattice. A typical tra-\njectory is shown, which starts at the upper right disk in the initial\ncell—marked by the highlighted triangle—and moves across the ta-\nble, performing a diffusive motion.\nsidering memory effects up to two successive steps, we find\nthat the agreement between the numerically-measured diffu-\nsion coefficient of the billiard table and that of the second-\norder persistent random walk extends to an appreciably larger\nrange of parameters.\nII. PERIODIC LORENTZ GAS ON A TRIANGULAR\nLATTICE\nConsider the periodic Lorentz gas on a triangular lattice,\nshown in Fig. 1. The centers of three nearby disks are iden-\nti���ed as the vertices of equilateral triangles which, in our no-\ntation, will be taken to be of unit side length. Denoting by r\nthe radius of the disks, we let d\u00111\u00002rdenote the spacing\nbetween disks. When d=0, the triangles form closed traps\nfrom which the tracer particles cannot escape. If, however,\n0<d\u001c1, then we expect the particles to remain inside the\ncell for a long time before they can escape to another cell.\nThis argument can be made precise using ergodic theory [6].\nGiven a tracer of unit velocity, the mean trapping time tis\ngiven in terms of the ratio of the cell area to the lengths of the\nholes by [9]\nt=pp\n3=4\u0000pr2=2\n3d=pp\n3=4\u0000p(1\u0000d)2=8\n3d:(2)\nAssuming that this time is longer than the typical decay of\ncorrelations [18], Machta and Zwanzig [6] argued that the dif-\nfusion coefficient of the Lorentz gas can be approximated by a\nmemory-less random walk (called the “short-memory approx-\nimation” in [10]),\nDMZ=`2\n4t; (3)where, in our notation, the lattice spacing `=1=p\n3.\nTaking further account of memory effects, Klages & Ko-\nrabel [8] noted that the Machta-Zwanzig approximation (3)\nis the zeroth order expansion of a series given by the Green-\nKubo formula for the diffusion coefficient:\nD=DMZ \n1+2¥\nå\nk=1hv0\u0001vki!\n; (4)\nwhere vkis the jump vector (“velocity”) between traps on the\nkth step, andhv0\u0001vkiareauto-correlations of the velocity at\nsteps 0 and k.\nThe approach taken in [8] was to truncate the expression\n(4) at a finite value of k, assuming that all higher-order cor-\nrelations are 0. They showed that by numerically calculating\nthe terms appearing in this equation, the results evaluated by\nthis truncation converged to the numerically-obtained diffu-\nsion coefficient.\nHowever, this ad hoc truncation has no physical meaning:\nifhv0\u0001v1i6=0, it is not true that higher-order correlations\nhv0\u0001vkivanish. Rather, assuming that the process has mem-\nory of the previous step alone—which we will refer to as the\nsingle-step memory approximation—one must compute the\ncorrelationshv0\u0001vkiin a consistent way, and evaluate the dif-\nfusion coefficient (4) by taking all the k’s into consideration.\nA. Single-step memory approximation\nThe process for which the motion of a particle at step k+1\ndepends explicitly on the state at step kis known as a persis-\ntentorcorrelated random walk [11, 12, 13, 14]. The tech-\nnique for studying such walks is well developed, and consists\nof treating it as a random walk with internal states, which, in\nthis case, describe the direction with which the walker arrives\nat a site.\nConsidering a persistent random walk on a honeycomb lat-\ntice, we denote by Pbthe conditional probability to return in\nthe direction opposite to the current one, Prto turn right and\nPlto turn left. In terms of these quantities, the velocity auto-\ncorrelation is found to be [15]\nhv0\u0001vki=(\u00001)k\n2(\"\nPb\u0000Pr+Pl\n2\u0000ip\n3\n2(Pr\u0000Pl)#k\n+\"\nPb\u0000Pr+Pl\n2+ip\n3\n2(Pr\u0000Pl)#k)\n; (5)\nwhere i=p\u00001. In the case of a symmetric walk for which\nPr=Pl\u0011Ps= (1\u0000Pb)=2, the diffusion coefficient (4) is\nD1SMA =DMZ3(1\u0000Pb)\n1+3Pb: (6)\nIn comparison, the first order approximation made in [8] is to\nwrite\nD(1)\nKK=1+2hv0\u0001v1i=DMZ(2\u00003Pb); (7)3\nwhich corresponds to the first-order approximation of Eq. (6)\nwhen expanding D1SMA about the isotropic process for which\nPb=1=3.\nB. Two-step memory approximation\nFor a persistent process with two-step memory approxima-\ntion, there are nine conditional transition probabilities, whichwe denote by Pbb,Pbr,Pbl,Prb,Prr,Prl,Plb,PlrandPll, where,\nfor instance, Pbrdenotes the probability that the tracer first\nmoves backwards and then turns right, and similarly for the\nother symbols. Although the corresponding Markov chain in-\nvolves a 9\u00029 stochastic matrix, there are symmetries in the\nsystem which can be exploited to reduce the computations\nto 3\u00023 matrices involving the transition probabilities listed\nabove. The computation of the velocity auto-correlation then\nyields [15]\nhv0\u0001vki= (\u00001)k3\n2\u0010\n1 1 1\u00112\n640\n@Pbb Plb Prb\nfPblfPllfPrl\nf2Pbrf2Plrf2Prr1\nAk\u000010\n@1 0 0\n0f0\n0 0 f21\nA+0\n@Pbb Plb Prb\nf2Pblf2Pllf2Prl\nfPbrfPlrfPrr1\nAk\u000010\n@1 0 0\n0f20\n0 0 f1\nA3\n750\n@p1\np2\np31\nA;\n(8)\nwhere f=exp(2ip=3)and(p1;p2;p3)are the first three components of the stationary distribution of the Markov chain, which,\nfor a left-right symmetric process for which Prr=Pll\u0011Pss,Prb=Plb\u0011PsbandPrl=Plr=1\u0000Pss\u0000Psb, read\np1=Psb\n3\u00003Pbb+3Psb;p2=p3=1\u0000Pbb\n6\u00006Pbb+6Psb: (9)\nThe corresponding diffusion coefficient (4) is\nD2SMA =DMZ3(1\u0000Pbb)(1+Pbb\u0000Psb)(2\u0000Psb\u00002Pss)\n(1\u0000Pbb+Psb)[Psb(7+Pbb\u00008Pss)+2(1+Pbb)Pss\u00004P2\nsb]: (10)\nThis compares to the second-order approximation following\nthe truncation scheme in [8],\nD(2)\nKK=D(1)\nKK+2hv0\u0001v2i;\n=DMZ5\u00005Pbb\u00007Psb+9PbbPsb\n2\u00002Pbb+2Psb: (11)\nWe note that this expression is actually different from that\ngiven in [8], where the stationary distribution (9) was erro-\nneously assumed to be uniform.\nC. Numerical results\nThe transition probabilities of the single- and two-step\nmemory approximation random walks can be computed for\nthe Lorentz gas by estimating the relative frequencies of the\ncorresponding events and taking into consideration the left-\nright symmetry of these transitions. Plugging their values into\nEqs. (6) and (10), we obtain the corresponding coefficients\nand compare them to the diffusion coefficient of the billiard\ncalculated from direct numerical simulations of the billiard\ndynamics. These results are shown in Fig. 2, including the\nresults of the truncations (7) and (11).\nIn the limit d!0, we see that the Machta-Zwanzig ap-\nproximation (3) is recovered. Looking at a broader range\nof parameter values, whereas the single-step approximation\nyields a good estimate of the actual diffusion coefficient ofthe Lorentz gas only for values of d<\u001810\u00003, the extent of the\nrange of validity of the two-step approximation is much larger,\nd<\u00180:1.\n00.060.121.1.21.4\nd\n-10-8-6-4-20.91.01.11.21.31.4\nlogHdL4Dtl-2\nFIG. 2: Numerical computation of the diffusion coefficient of the pe-\nriodic Lorentz gas on a triangular lattice, here divided by the dimen-\nsional factor, Eq. (3). The lines correspond to the different approx-\nimate results discussed above: (long-dashed, black line) single-step\nmemory approximation (6); (dot-dashed, red line) first-order trunca-\ntion (7); (solid, magenta line) two-step memory approximation (10);\n(dashed green) second-order truncation (11).\nAs seen in the figure, the successive approximations (6) and\n(10) are slightly better than their respective counterparts (7)\nand (11). The differences between the results of Eqs. (6) and4\n(7) on the one hand, and Eqs. (10) and (11) on the other hand\nare, however, quite small and not everywhere easy to appre-\nciate. The reason is that the transition probabilities are nearly\nisotropic for all values of din the range of allowed values, viz.p\n3=4<d<1=2 (the lower bound corresponds to the finite-\nhorizon condition). This fact, which we illustrate in Fig. 3 for\nthe transition probabilities of the single-step memory approxi-\nmation, explains the validity of the Klages–Korabel truncation\nscheme in this case.\nàààààààààààà\nàààààààààààààààààààæææææææææææææææææææææææææææææææ\n0.000.020.040.060.080.100.120.260.280.300.320.340.360.380.40\ndPb,Ps\nFIG. 3: (Color online) Numerical computation of the transition prob-\nabilities Pb(squares) and Ps(circles) of the single-step memory ap-\nproximation associated to the periodic Lorentz gas on a triangular\nlattice. Similar results were reported in [7].\nIII. BILLIARD TABLE ON A SQUARE LATTICE\nIn order to better emphasize the validity of our approxima-\ntion scheme over simpler truncation methods, we turn to a\nclass of billiard tables such as shown in Fig. 4, which alter-\nnates small and large disks on a square lattice. By inserting\nrigid horizontal and vertical barriers between the small disks,\nwith gaps of size din their centers, we introduce a control pa-\nrameter of relevance to the dynamical properties of the model\nwhich is independent of the other system parameters. How-\never large the gaps, the trajectories in these billiards can be\nreconstructed from the trajectories of the same billiard with a\nsingle unit cell on the torus.\nThe geometry of the lattice is such that the distance between\nneighboring disks is equal to the lattice spacing, which we\ntake to be unity, `=1. Given the radii riof the large disks\nandroof the small disks, the trapping time is\nt=p1\u0000p(r2\ni+r2\no)\n4d: (12)\nGiven these two parameters, the Machta-Zwanzig approxima-\ntion is here again given by Eq. (3). Note that in the present\nmodel, DMZdepends linearly on d, contrary to the Lorentz\ngas discussed in the previous section, for which the area of\nthe cells varied with d.\nFIG. 4: Periodic Lorentz table on a square lattice. A typical trajectory\nis shown, with the initial position marked by a thick dot. We take the\ninner and outer radii to be respectively ri=0:36`andro=0:15`\nand vary only the size of the gaps.\nA. Single-step memory approximation\nConsider a persistent random walk with single-step mem-\nory on a square lattice. We denote by Pf,Pr,PbandPl, respec-\ntively, the conditional probabilities that the particle proceeds\nin the same direction, turns right, reverses direction, or turns\nleft. The velocity auto-correlation is\nhv0\u0001vki=1\n2f[Pf\u0000Pb\u0000i(Pr\u0000Pl)]k+[Pf\u0000Pb+i(Pr\u0000Pl)]kg:\n(13)\nPlugging this into (4) and assuming a symmetric process such\nthatPr=Pl\u0011Ps= (1\u0000Pf\u0000Pb)=2, we obtain the diffusion\ncoefficient\nD1SMA =DMZ1+Pf\u0000Pb\n1\u0000Pf+Pb: (14)\nThis expression compares to\nD(1)\nKK=1+2hv0\u0001v1i=DMZ(1+2Pf\u00002Pb); (15)\nwhich is the first-order approximation of Eq. (14) when ex-\npanding D1SMA about the isotropic process for which Pf=\nPb=1=4.\nB. Two-step memory approximation\nApplying the same procedure to the persistent random walk\nwith two-step memory, we have 16 conditional transition\nprobabilities, in terms of which it is possible to write the ve-\nlocity auto-correlation in the compact form [15]5\nhv0\u0001vki=2\u0010\n1 1 1 1\u00112\n6640\nB@Pff Plf Pbf Prf\niPfliPlliPbl iPrl\n\u0000Pfb\u0000Plb\u0000Pbb\u0000Prb\n\u0000iPfr\u0000iPlr\u0000iPbr\u0000iPrr1\nCAk\u000010\nB@1 0 0 0\n0i0 0\n0 0\u00001 0\n0 0 0\u0000i1\nCA\n+0\nB@Pff Plf Pbf Prf\n\u0000iPfl\u0000iPll\u0000iPbl\u0000iPrl\n\u0000Pfb\u0000Plb\u0000Pbb\u0000Prb\niPfriPlriPbr iPrr1\nCAk\u000010\nB@1 0 0 0\n0\u0000i0 0\n0 0\u00001 0\n0 0 0 i1\nCA3\n7750\nB@p1\np2\np3\np41\nCA; (16)\nwhere the pi’s are the first four components of the stationary distribution of this process, which, assuming a left-right symmetric\nprocess, have the expressions\np1=PbfPsb+Psf\u0000PbbPsf\n4(1\u0000Pff+Psb\u0000PffPsb+Pbf(\u0000Pfb+Psb)+Pbb(\u00001+Pff\u0000Psf)+Psf+PfbPsf);\np2=1\u0000Pbb\u0000PbfPfb\u0000Pff+PbbPff\n8(1\u0000Pff+Psb\u0000PffPsb+Pbf(\u0000Pfb+Psb)+Pbb(\u00001+Pff\u0000Psf)+Psf+PfbPsf);\np3=Psb\u0000PffPsb+PfbPsf\n4(1\u0000Pff+Psb\u0000PffPsb+Pbf(\u0000Pfb+Psb)+Pbb(\u00001+Pff\u0000Psf)+Psf+PfbPsf);\np4=1\u0000Pbb\u0000PbfPfb\u0000Pff+PbbPff\n8(1\u0000Pff+Psb\u0000PffPsb+Pbf(\u0000Pfb+Psb)+Pbb(\u00001+Pff\u0000Psf)+Psf+PfbPsf):(17)\nThe diffusion coefficient of this process may then be ob-\ntained by substituting Eqs. (16)-(17) into Eq. (4) and summing\nover k. We will however not write down its lengthy explicit\nexpression, referring the reader instead to [15].\nLet us also notice that Eqs. (16)-(17) can be used to write\ndown the corresponding second-order approximation in [8],\nhere properly taking into account the stationary distribution,\nD(2)\nKK=1+2hv0\u0001v1i+2hv0\u0001v2i=D(1)\nKK+2hv0\u0001v2i:(18)\nC. Numerical results\nThe transition probabilities of the random walks with\nsingle- and two-step memory approximation can be computed\nnumerically for the billiard table by estimating the relative fre-\nquencies of the corresponding events and taking into account\nthe left-right symmetry of these transitions. as above. Plug-\nging their values into Eqs. (14) and (16), we obtain the cor-\nresponding coefficients and compare them to the numerically-\ncomputed diffusion coefficient of the billiard. These results\nare reported in Fig. 5, including the results of the truncations\n(15) and (18).\nIn the limit d!0, we again see that the Machta-Zwanzig\napproximation (3) is recovered. Zooming into the lower range\nof the parameter d, we see that the single-step approximation\nyields a good estimate of the actual diffusion coefficient of the\nLorentz gas only for values of d<\u00182\u000210\u00003. The extent of the\nrange of validity of the two-step approximation, on the other\nhand, is again much larger, d<\u00180:5.\nThe other main observation is that the truncated estimates\n00.20.40.600.51\nd\n-6-5-4-3-2-100.00.20.40.60.81.0\nlogHdL4Dtl-2FIG. 5: Numerical computation of the diffusion coefficient of the bil-\nliard table on a square lattice as shown in Fig. 4, here divided by the\ndimensional factor, Eq. (3). The lines correspond to the different ap-\nproximate results discussed above: (long-dashed, black line) single-\nstep memory approximation (14); (dot-dashed, red line) first-order\ntruncation (15); (solid, magenta line) two-step memory approxima-\ntion, obtained from Eqs. (4) and (16); (dashed, green line) second-\norder truncation (18).\n(15) and (18) are inaccurate as soon as d>\u00180:05. The rea-\nson can be traced to the anisotropy of the hopping processes.\nFigure 6 shows the transition rates of the single-step mem-\nory random walk. Although the probability of a right or left\nturn remains close to 1 =4 throughout the parameter range,\nthe backscattering probability starts growing linearly above\n1=4 with small d’s and saturates near 1 =2 at around d=0:5.\nCorrespondingly, the forward-scattering probability decreases6\nand is close to zero at around d=0:5.\nèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèè ìììììììììììììììììììììììììììììììììì\n0.00.10.20.30.40.50.60.70.00.10.20.30.40.50.6\ndPf,Ps,Pb\nFIG. 6: (Color online) Numerical computation of the transition prob-\nabilities Pb(squares), Pf(circles) and Ps(diamonds) of the single-step\nmemory approximation associated to the billiard table on a square\nlattice. These rates reflect the anisotropy of the process.\nLooking at the transition probabilities of the two-step mem-\nory process, shown in Fig. 7, we notice the differences among\nthese probabilities, for instance comparing Pbb,PfbandPsb, as\nwell as between these probabilities and that of the single-step\nmemory process, in our example Pb. These differences jus-\ntify the necessity of resorting to the two-step memory process\nover the single-step process.\nIV . CONCLUSIONS\nThe diffusion coefficients of deterministic systems with\nrapid decay of correlations can be well approximated by that\nof correlated walks, where a walker’s transition probabilities\nare determined according to its motion over the last few steps.\nBilliards provide good, physically motivated, examples of\nsuch systems. The Machta-Zwanzig dimensional prediction\n[6], according to which the diffusion coefficient is approxi-\nmated by the ratio between the distance between traps squaredand the trapping time, provides a gross estimate of this quan-\ntity. However, the importance of memory effects in the de-\nterministic diffusion of tracer particles is apparent as soon as\none moves away from the limit where the trapping times are\ninfinite.\nTruncation schemes based on the Green-Kubo formula,\nsuch as considered in [4, 8], may provide accurate results for\nmodels with little anisotropy, but they are physically inconsis-\ntent: given a hopping process with finite memory effects, ve-\nlocity auto-correlations of all orders yield non-vanishing con-\ntributions to the Green-Kubo formula.\nThis is particularly clear where anisotropies come into play.\nEstimates of the diffusion coefficients based on persistent ran-\ndom walks, however, do provide accurate results where the\ntruncation schemes breakdown.\nIt will be interesting to find out how these results trans-\npose to models where disorder is present, such as with tagged-\nparticle diffusion in interacting particle systems. Dimensional\npredictions similar to the Machta-Zwanzig one also appeared\nrecently in the context of models of heat conduction [16, 17].\nEstimating the deviations of the heat conductivities of these\nmodels from dimensional predictions remains an open prob-\nlem.\nAcknowledgments\nThe authors thank Felipe Barra, Mark Demers, Hern ´an Lar-\nralde and Carlangelo Liverani for helpful discussions. This\nresearch benefitted from the joint support of FNRS (Belgium)\nand CONACYT (Mexico) through a bilateral collaboration\nproject. The work of TG is financially supported by the Bel-\ngian Federal Government under the Inter-university Attrac-\ntion Pole project NOSY P06/02. TG is financially supported\nby the Fonds de la Recherche Scientifique F.R.S.-FNRS. DPS\nacknowledges financial support from DGAPA-UNAM project\nIN105209, and the hospitality of the Universit ´e Libre de Brux-\nelles, where most of this work was carried out. TG acknowl-\nedges the hospitality of the Weizmann Institute of Science,\nwhere part of this work was completed.\n[1] S. Chandrasekar, Stochastic problems in physics and astronomy\nRev. Mod. Phys. 15, 1(1943).\n[2] P. Gaspard, Chaos, Scattering and Statistical Mechanics (Cam-\nbridge University Press, Cambridge, UK, 1998).\n[3] J. R. Dorfman, An Introduction to Chaos in Nonequilib-\nrium Statistical Mechanics (Cambridge University Press, Cam-\nbridge, UK, 1999).\n[4] R. Klages, Microscopic chaos, fractals and transport in\nnonequilibrium statistical mechanics (World Scientific, Singa-\npore, 2007).\n[5] L. A. Bunimovich and Ya. Sinai, Markov Partition for Dis-\npersed Billiard Comm. Math. Phys. 78, 247 (1980). Statistical\nproperties of Lorentz gas with periodic configuration of scatter-\nersComm. Math. Phys. 78, 479 (1980).\n[6] J. Machta and R Zwanzig, Diffusion in a periodic Lorentz gas\nPhys. Rev. Lett. 501959 (1983).[7] R. Klages and C. Dellago, Density-dependent diffusion in the\nperiodic Lorentz gas J. Stat. Phys. 101145 (2000).\n[8] R. Klages and N. Korabel, Understanding deterministic diffu-\nsion by correlated random walks J. Phys A math. gen. 354823\n(2002).\n[9] N. Chernov and R. Markarian, Chaotic billiards Math. Surveys\nand Monographs 127(AMS, Providence, RI, 2006).\n[10] R. Zwanzig, From classical dynamics to continuous time ran-\ndom walks J. Stat. Phys. 30255 (1983).\n[11] B. D. Hughes, Random Walks and Random Environments. Vol-\nume 1: Random Walks (Clarendon Press, Oxford, 1995)\n[12] G. H. Weiss and R. J. Rubin Random Walks: Theory and\nSelected Applications Advances in Chemical Physics 52563\n(1983)\n[13] J. W. Haus and K.W. Kehr, Diffusion in regular and disordered\nlattices Phys. Rep. 150, 263 (1987).7\nèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèììììììììììììììììììììììììììììììììì\n0.00.10.20.30.40.50.60.70.00.10.20.30.40.50.6\ndPff,Pfb,Pfs\nèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèìììììììììììììììììììììììììììììììììì\n0.00.10.20.30.40.50.60.70.00.10.20.30.40.50.6\ndPbf,Pbb,Pbs\nèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèè\nìììììììììììììììììììììììììììììììììì òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò\n0.00.10.20.30.40.50.60.70.00.10.20.30.40.50.6\ndPsf,Pss,Psb\nFIG. 7: (Color online) Numerical computation of the transition probabilities Pxb(squares), Pxf(circles) and Pxs(diamonds) of the two-step\nmemory approximation associated to the billiard table on a square lattice, where x stands respectively for f (left), b (middle) and s (right). In\nthe right figure, the triangles are the probabilities of turning right after turning left and, the other way around, turning left after turning right.\nThe differences between these three figures and the single-step transition probabilities shown in Fig. 6 justify resorting to a two-step memory\nprocess.\n[14] G. H. Weiss, Aspects and Applications of the Random Walk\n(North-Holland, Amsterdam, 1994).\n[15] T. Gilbert and D. P. Sanders, Diffusion coefficients for multi-\nstep persistent random walks on lattices unpublished.\n[16] P. Gaspard and T. Gilbert, Heat conduction and Fourier’s law\nby consecutive local mixing and thermalization Phys. Rev. Lett.\n101020601 (2008).\n[17] T. Gilbert and R. Lefevere, Heat Conductivity from MolecularChaos Hypothesis in Locally Confined Billiard Systems Phys.\nRev. Lett. 101200601 (2008).\n[18] The exponential decay of correlations has been proved in the\nperiodic Lorentz gas, see N. Chernov and L.-S. Young, Decay\nof correlations for Lorentz gases and hard balls , in: Hard Ball\nSystems and the Lorentz Gas , ed. by D. Szasz, Encyclopaedia\nof Mathematical Sciences 101, 89 (Springer, 2000)."    },    {        "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. 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Melkov, Magnetization Oscillation and Waves ( CRC, Boca\nRaton, 1996)\n8"    },    {        "title": "1910.11200v1.Spin_waves_in_ferromagnetic_thin_films.pdf",        "content": "arXiv:1910.11200v1  [cond-mat.mes-hall]  24 Oct 2019Spin waves in ferromagnetic thin films\nZhiwei Sun\nSchool of Mathematical Sciences, Soochow University, Suzh ou, China\nJingrun Chen∗\nSchool of Mathematical Sciences, Soochow University, Suzh ou, China and\nMathematical Center for Interdisciplinary Research, Sooc how University, Suzhou, China\n(Dated: October 25, 2019)\nA spin wave is the disturbance of intrinsic spin order in magn etic materials. In this paper, a spin\nwave in the Landau-Lifshitz-Gilbert equation is obtained b ased on the assumption that the spin\nwave maintains its shape while it propagates at a constant ve locity. Our main findings include:\n(1) in the absence of Gilbert damping, the spin wave propagat es at a constant velocity with the\nincrement proportional tothe strength of the magnetic field ; (2) in the absence of magnetic field, at a\ngiven time the spin wave converges exponentially fast to its initial profile as the damping parameter\ngoes to zero and in the long time the relaxation dynamics of th e spin wave converges exponentially\nfast to the easy-axis direction with the exponent proportio nal to the damping parameter; (3) in\nthe presence of both Gilbert damping and magnetic field, the s pin wave converges to the easy-axis\ndirection exponentially fast at a small timescale while pro pagates at a constant velocity beyond\nthat. These provides a comprehensive understanding of spin waves in ferromagnetic materials.\nPACS numbers: 05.45.Yv, 75.70.-i, 75.78.-n\nI. INTRODUCTION\nA spin wave is the disturbance of intrinsic spin order in magnetic mater ials. It is usually excited using magnetic\nfields and offers unique properties such as charge-less propagatio n and high group velocities, which are important for\nsignal transformations and magnetic logic applications [1–6].\nThepropagationofspinwavesisdescribedbytheLandau-Lifshitz- Gilbert(LLG) equation[7,8]in thedimensionless\nform\nmt=−m×h−αm×(m×h), (1)\nwhere the magnetization m= (m1,m2,m3)Tis a three dimensional vector with unit length, αis the Gilbert damping\nparameter. The effective field hincludes the exchange term, the anisotropy term with easy axis alon g the x-axis and\nthe anisotropy constant q, and the external field\nh= ∆m+qm1e1+hexte1. (2)\nhextis the strength of the external field applied along the x-axis with e1the unit vector. This model is often used to\ndescribe the magnetization dynamics in ferromagnetic thin films.\nFrom a theoretical perspective, a spin wave is known as a solitotary wave, which appears as the solution of a weakly\nnonlinear dispersive partial differential equation. In LLG equation ( 1)-(2), a soliton is caused by the cancellation of\nnonlinear and dispersive effects in the magnetic material. Solitons are of interests for quite a long time [9–13]. Most\nof works consider the one dimensional case and drop the damping te rm [9, 10, 13]. In [11], using the stereographic\nprojection, the authors found that the presence of Gilbert damp ing was merely a rescaling of time by a complex\nconstant. However, this was found to be valid only for a single spin in a constant magnetic field [12].\nIn this work, we give a comprehensive study of an explicit spin wave in t he LLG equation. Our starting point is\nthat the spin wave maintains its shape while it propagates at a consta nt velocity and the derivation is based on the\ngeneralization of the method of characteristics. The main findings a re: (1) in the absence of Gilbert damping, the\nspin wave propagates at a constant velocity with the increment pro portional to the strength of the magnetic field; (2)\nin the absence of magnetic field, at a given time the spin wave converg es exponentially fast to its initial profile as the\ndamping parameter goesto zeroand in the long time the relaxationdy namics of the spin waveconvergesexponentially\n∗Electronic address: jingrunchen@suda.edu.cn2\nfast to the easy-axis direction with the exponent proportional to the damping parameter; (3) in the presence of both\nGilbert damping and magnetic field, the spin wave converges to the ea sy-axis direction exponentially fast at a small\ntimescale while propagates at a constant velocity beyond that.\nII. DERIVATION AND RESULTS\nAs mentioned above, we start with the assumption that a spin wave m aintains its shape while it propagates at a\nconstant velocity. This can be seen from the method characterist ics in simple situations.\nIn 1D when α=q=hext= 0, one can check that\nm(x,t) =\ncosθ0\nsinθ0cos/parenleftig\nc\ncosθ0(x+ct)/parenrightig\nsinθ0sin/parenleftig\nc\ncosθ0(x+ct)/parenrightig\n(3)\nsolvesmt=−m×mxx. Hereθ0is determined by the initial condition and u=x+ctis the characteristic line. (3)\nprovides a solitary solution with the traveling speed c. A detailed derivation of (3) can be found in Chapter 2 of [13].\nA generalization of the method of characteristics yields a spin wave t omt=−m×∆m\nm(x,t) =\ncosθ0\nsinθ0cosv\ncosθ0\nsinθ0sinv\ncosθ0\n, (4)\nwherev=c1x+c2y+c3z+(c2\n1+c2\n2+c2\n3)t=c·x+(c·c)twithc= (c1,c2,c3)T. The speed field is cwith magnitude\n|c|. Actually, both (3) and (4) can be rewritten as\nm(x,t) =\ncosθ0\nsinθ0cos(w0·x+ϕ(t))\nsinθ0sin(w0·x+ϕ(t))\n, (5)\nwherew0=c/cosθ0andϕ(t) =/parenleftbig\n|c|2/cosθ0/parenrightbig\nt.\n(3)-(5) are obtained in the absence of Gilbert damping. In order to take the Gilbert damping and the other terms\nin (2) into account, we make an ansatz for the spin wave profile in the following form\nm(x,t) =\ncosθ(t)\nsinθ(t)cos(w0·x+ϕ(t))\nsinθ(t)sin(w0·x+ϕ(t))\n, (6)\nwhereθandϕare independent of xand only depend on t.\nSubstituting (6) into (2) and (1) and denoting w0·x+ϕ(t) byu(x,t), we have\nh=\n0\n−|w0|2sinθcosu(x,t)\n−|w0|2sinθsinu(x,t)\n+q\ncosθ\n0\n0\n+hext\n1\n0\n0\n,\nm×h=\n0\n|w0|2sinθcosθsinu(x,t)\n−|w0|2sinθcosθcosu(x,t)\n+q\n0\nsinθcosθsinu(x,t)\n−sinθcosθcosu(x,t)\n+hext\n0\nsinθsinu(x,t)\n−sinθcosu(x,t)\n,\nm×(m×h) =\n−|w0|2sin2θcosθ\n|w0|2sinθcos2θcosu(x,t)\n|w0|2sinθcos2θsinu(x,t)\n+q\n−sin2θcosθ\nsinθcos2θcosu(x,t)\nsinθcos2θsinu(x,t)\n+hext\n−sin2θ\nsinθcosθcosu(x,t)\nsinθcosθsinu(x,t)\n,\nand\nmt=\n−θtsinθ\nθtcosθcosu(x,t)−ϕtsinθsinu(x,t)\nθtcosθsinu(x,t)+ϕtsinθcosu(x,t)\n.\nAfter algebraic simplifications, we arrive at\n/braceleftbigg\nθt=−α(|w0|2+q)sinθcosθ−αhextsinθ\nϕt= (|w0|2+q)cosθ+hext. (7)3\nA. The absence of Gilbert damping\nWhenα= 0, we have θ=θ0andϕ=/parenleftbig\n(|w0|2+q)cosθ0+hext/parenrightbig\nt. Therefore we have the solution\nm=\ncosθ0\nsinθ0cos/parenleftbig\nw0·x+t/parenleftbig\n|w0|2cosθ0+qcosθ0+hext/parenrightbig/parenrightbig\nsinθ0sin/parenleftbig\nw0·x+t/parenleftbig\n|w0|2cosθ0+qcosθ0+hext/parenrightbig/parenrightbig\n. (8)\nNote that this recovers (5) when q= 0 and hext= 0. It is easy to see that the spin wave (8) propagates at a consta nt\nvelocity. The increment of the velocity field is qcosθ0w0\n|w0|2with magnitude|qcosθ0|\n|w0|, due to the magnetic anisotropy.\nThe increment of the velocity field is hextw0\n|w0|2with magnitude|hext|\n|w0|, due to the magnetic field.\nB. The absence of magnetic field\nWhenhext= 0, (7) reduces to\n/braceleftbigg\nθt=−α(|w0|2+q)sinθcosθ\nϕt= (|w0|2+q)cosθ. (9)\nFor the first equation in (9), assuming 0 ≤θ0< π/2, by separation of variables, we have\nα(|w0|2+q)t+C1= lncotθ,\nwhereC1is a constant determined by the initial condition.\nDenote˜t=α(|w0|2+q)t+C1. It follows that\ntanθ=e−˜t, (10)\nfrom which one has\ncosθ=1/radicalbig\n1+e−2˜t, (11)\nsinθ=1/radicalbig\n1+e2˜t. (12)\nWhent= 0, (11) turns to\ncosθ0=1√\n1+e−2C1, (13)\nfrom which we can determine C1by the initial condition θ0.\nAs forϕ, from the second equation in (9), one has that\ndϕ\ndθ=dϕ\ndt·dt\ndθ=−1\nαsinθ.\nTherefore\nαϕ=−/integraldisplaydθ\nsinθ=1\n2ln/parenleftbigg1+cosθ\n1−cosθ/parenrightbigg\n+C2= lncot1\n2θ+C2, (14)\nwhere\nC2=−1\n2ln/parenleftbigg1+cosθ0\n1−cosθ0/parenrightbigg\n.\nSubstituting (11) and (13) into (14) yields\nϕ=1\nαln/parenleftigg\ne˜t+/radicalbig\ne2˜t+1\neC1+√\ne2C1+1/parenrightigg\n=1\nα/parenleftbigg\nlncot1\n2θ−lncot1\n2θ0/parenrightbigg\n. (15)4\nIn short summary, the spin wave when α/negationslash= 0 takes the form\nm=1/radicalbig\n1+e2˜t\ne˜t\ncos(w0·x+ϕ)\nsin(w0·x+ϕ)\n. (16)\nThe above derivation is valid when 0 ≤θ0< π/2. Ifπ/2< θ0≤π, we choose the other solution of (11)\ncosθ=−1/radicalbig\n1+e−2˜t, (17)\nand\nϕ=−1\nαln/parenleftigg\ne˜t+/radicalbig\ne2˜t+1\neC1+√\ne2C1+1/parenrightigg\n.\n(16) remains unchanged.\nWhenα→0, we have ˜t→C1and\nlim\nα→0m=1√\n1+e2C1\neC1\ncos/parenleftig\nw0·x+ lim\nα→0ϕ/parenrightig\nsin/parenleftig\nw0·x+ lim\nα→0ϕ/parenrightig\n.\nBy L’Hospital’s rule, one has that\nlim\nα→0ϕ= lim\nα→0d\ndα/parenleftigg\nln/parenleftigg\ne˜t+/radicalbig\ne2˜t+1\neC1+√\ne2C1+1/parenrightigg/parenrightigg\n= lim\nα→0e˜t\n/radicalbig\ne2˜t+1(|w0|2+q)t=eC1\n√\neC1+1(|w0|2+q)t.(18)\nTherefore it follows that\nlim\nα→0m=\ncosθ0\nsinθ0cos/parenleftbig\nw·x+cosθ0/parenleftbig\n|w0|2+q/parenrightbig\nt/parenrightbig\nsinθ0sin/parenleftbig\nw·x+cosθ0/parenleftbig\n|w0|2+q/parenrightbig\nt/parenrightbig\n. (19)\nThis is exactly the solution (8) when hext= 0.\nIn addition, when α→0, bothθandϕconverges exponentially fast to initial conditions; see equations (1 1), (12),\n(13), and (18). Therefore, at a giventime t, (19) convergesexponentiallyfast to the initial spin wave(8) when hext= 0\nwith the exponent proportionalto the damping parameter α. Moreover,in the long time, i.e., when t→+∞,˜t→+∞\nandθ→0, (16) converges to (1 ,0,0)T(the easy-axis direction) exponentially fast with the rate proport ional to the\ndamping parameter α. Whenπ/2< θ0≤π, from (17), we have that (16) converges to ( −1,0,0)T(again the easy-axis\ndirection) exponentially fast with the rate proportional to the dam ping parameter α.\nIt is easy to check that the right-hand side of (19) is the solution of (1) when hext= 0 with the initial condition\nθ0=π/2. Therefore, Gilbert damping does not have any influence on magne tization dynamics in this case.\nIn [11], the authors used the stereographic projection and obser ved that the effect of Gilbert damping was only a\nrescaling of time by a complex constant. However, this was latter fo und to be valid only for a single spin in a constant\nmagnetic field [12]. Our result provides an explicit characterization of magnetization dynamics in the presence of\nGilbert damping.\nC. The presence of both Gilbert damping and magnetic field\nIt is difficult to get the explicit solution of (7) in general. To understan d the magnetization dynamics, we use the\nmethod of asymptotic expansion. For small external magnetic field ,θandϕadmit the following expansions\nθ(t,hext) =θ0(t)+θ1(t)hext+θ2(t)h2\next+···,\nϕ(t,hext) =ϕ0(t)+ϕ1(t)hext+ϕ2(t)h2\next+···.5\nTherefore one has that\nθt(t,hext) =θ0\nt(t)+θ1\nt(t)hext+θ2\nt(t)h2\next+···, (20)\nϕt(t,hext) =ϕ0\nt(t)+ϕ1\nt(t)hext+ϕ2\nt(t)h2\next+···. (21)\nOn the other hand, from (7), it follows that\nθt=−α(|w0|2+q)sinθ0cosθ0−/parenleftbig\nα(|w0|2+q)θ1cos2θ0+αsinθ0/parenrightbig\nhext+···, (22)\nϕt= (|w0|2+q)cosθ0+/parenleftbig\n−(|w0|2+q)θ1sinθ0+1/parenrightbig\nhext+···. (23)\nCombining (20) and (21) with (22) and (23), for the zero-order te rm, one has\n/braceleftbigg\nθ0\nt=−α(|w0|2+q)sinθ0cosθ0\nϕ0\nt= (|w0|2+q)cosθ0 , (24)\nwhich recovers (9) with solution (10) and (15).\nAs for the first-order term, one has that\n/braceleftbigg\nθ1\nt=−α(|w0|2+q)θ1cos2θ0−αsinθ0\nϕ1\nt=−(|w0|2+q)θ1sinθ0+1. (25)\nUsing variation of parameters, one can assume θ1=C(t)\ne˜t+e−˜tand it follows that\nC′(t) =−α(e˜t+e−˜t)sinθ0=−α(tanθ0+tan−1θ0)sinθ0.\nSince\n/integraldisplay\n−αtanθ0sinθ0dt=/integraldisplay\n−α(|w0|2+q)sinθ0cosθ0∗(|w0|2+q)−1sinθ0\ncos2θ0dt\n=/integraldisplay\n(|w0|2+q)−1sinθ0\ncos2θ0dθ0\n=(|w0|2+q)−11\ncosθ0,\nand\n/integraldisplay\n−αtan−1θ0sinθ0dt=/integraldisplay\n−αcosθ0dt\n=−α(|w0|2+q)−1ϕ0,\none can get C(t) = (|w0|2+q)−1(1\ncosθ0−αϕ0), and it follows that\nθ1= (|w0|2+q)−1(sinθ0−αsinθ0cosθ0ϕ0). (26)\nSubstituting the first equation in (24) into the second equation in (2 5), one has\n−/integraldisplay\n(|w0|2+q)θ1sinθ0dt\n=1\nα/integraldisplayθ1\ncosθ0dθ0\n=1\nα(|w0|2+q)/integraldisplay\ntanθ0−sinθ0(lncot1\n2θ0+C2)dθ0(using (26))\n=−t+1\n|w0|2+q(ϕ0cosθ0+α−1C1),\nand thus\nϕ1=1\n|w0|2+q(ϕ0cosθ0+α−1C1). (27)6\nTherefore, when hextis small, it has the approximate solution\n/braceleftbigg\nθ∗=θ0+(|w0|2+q)−1(sinθ0−αsinθ0cosθ0ϕ0)hext\nϕ∗=ϕ0+(|w0|2+q)−1(ϕ0cosθ0+α−1C1)hext(28)\nwithθ0andϕ0satisfying (9).\nFrom (10) and (15), θ0converges exponentially fast to the easy-axis direction, while ϕ0grows linearly. Therefore,\nfrom (26), θ1converges exponentially fast to 0 as well with a larger exponent. Th is relaxation dynamics happens at\na small timescale.\nMeanwhile, from (25), the difference between ϕ∗andϕ0satisfies\nϕ∗\nt−ϕ0\nt=ϕ1\nthext=−(|w0|2+q)θ1sinθ0hext+hext. (29)\nSinceθ1sinθ0convergesto 0 at a small timescale, the dynamics of ϕ∗−ϕ0is determined by the external field at longer\ntimescales. As a consequence, the increment of the velocity field is hextw0\n|w0|2with magnitude|hext|\n|w0|. This validates the\nWalker’s ansatz [14] for a spin wave.\nWhenπ/2< θ0≤πand the magnetic field is applied along the negative x-axis, and if ( |w|2+q)|cosθ0| ≤hext, the\nresult above will be correct.\nNote that θ0=π/2 does not fall into the above two cases since the magnetization dyn amics will change the spin\nwave profile. In fact, as t→+∞,θ→0 if the magnetic field is applied along the positive x-axis direction and θ→π\nif the magnetic field is applied along the negative x-axis direction.\nIII. CONCLUDING REMARKS\nIn this work, we study the magnetization dynamics in Landau-Lifshit z-Gilbert equation. By generalizing the\nmethod of characteristics, we are able to have an explicit characte rization of spin wave dynamics in the presence of\nboth Gilbert damping and magnetic field. Gilbert damping drives the spin wave converge exponentially fast to the\neasy-axis direction with the exponent proportional to the damping parameter at a small timescale and the magnetic\nfield drives the spin wave propagate at a constant velocity at longer timescales.\nIt will be of interests whether the technique developed here applies to the antiferromagnetic case [15, 16] and how\nrigorous the results obtained here can be proved from a mathemat ical perspective.\nIV. ACKNOWLEDGEMENTS\nWe thank Professor Yun Wang for helpful discussions. This work wa s partially supported by National Natural\nScience Foundation of China via grant 21602149 and 11971021.\n[1] M. P. Kostylev, A. A. Serga, T. Schneider, B. Leven, and B. Hillebrands, Applied Physics Letters 87, 153501 (2005).\n[2] Y. Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H. Umezawa, H. Kawai, K. Ando, K. Takanashi,\net al., Nature 464, 262 (2010).\n[3] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and B. Hillebra nds, Nature Physics 11, 453 (2015).\n[4] S. Woo, T. Delaney, and G. S. D. Beach, Nature Physics (201 7).\n[5] A. V. Chumak and H. Schultheiss, Journal of Physics D: App lied Physics 50, 300201 (2017).\n[6] M. Langer, R. A. Gallardo, T. Schneider, S. Stienen, A. Ro ld´ an-Molina, Y. Yuan, K. Lenz, J. Lindner, P. Landeros, and\nJ. Fassbender, Physical Review B 99(2019).\n[7] L. Landau and E. Lifshitz, Physikalische Zeitschrift de r Sowjetunion 8, 153 (1935).\n[8] T. Gilbert, Physical Review 100, 1243 (1955).\n[9] K. Nakamura and T. Sasada, Physics Letters A 48, 321 (1974).\n[10] H. J. Mikeska, Journal of Physics C: Solid State Physics 11, L29 (1977).\n[11] M. Lakshmanan and K. Nakamura, Physical Review Letters 53, 2497 (1984).\n[12] E. Magyari, H. Thomas, and R. Weber, Physical Review Let ters56, 1756 (1986).\n[13] B. Guo and S. Ding, Landau-Lifshitz Equation (World Scientific, 2007).\n[14] N. L. Schryer and L. R. Walker, Journal of Applied Physic s45, 5406 (1974).\n[15] H. J. Mikeska, Journal of Physics C: Solid State Physics 13, 2913 (1980).\n[16] V. Baltz, A. Manchon, M. Tsoi, T. Moriyama, T. Ono, and Y. Tserkovnyak, Reviews of Modern Physics 90, 015005 (2018)."    },    {        "title": "1501.05216v1.Lévy_walks_on_lattices_as_multi_state_processes.pdf",        "content": "Lévy walks on lattices as multi-state processes\nGiampaolo Cristadoro y, Thomas Gilbert z, Marco Lenci y§ and\nDavid P. Sanders k\nyDipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5,\n40126 Bologna, Italy\nzCenter for Nonlinear Phenomena and Complex Systems, Université Libre de\nBruxelles, C. P. 231, Campus Plaine, B-1050 Brussels, Belgium\n§ Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Via Irnerio 46, 40126\nBologna, Italy\nkDepartamento de Física, Facultad de Ciencias, Universidad Nacional Autónoma de\nMéxico, Ciudad Universitaria, 04510 México D.F., Mexico\nAbstract. Continuous-time random walks combining diffusive scattering and\nballistic propagation on lattices model a class of Lévy walks. The assumption that\ntransitions in the scattering phase occur with exponentially-distributed waiting times\nleads to a description of the process in terms of multiple states, whose distributions\nevolveaccordingtoasetofdelaydifferentialequations,amenabletoanalytictreatment.\nWe obtain an exact expression of the mean squared displacement associated with such\nprocesses and discuss the emergence of asymptotic scaling laws in regimes of diffusive\nand superdiffusive (subballistic) transport, emphasizing, in the latter case, the effect of\ninitial conditions on the transport coefficients. Of particular interest is the case of rare\nballistic propagation, in which case a regime of superdiffusion may lurk underneath\none of normal diffusion.\nSubmitted to: J. Stat. Mech. Theor. Exp.\nPACS numbers: 05.40.Fb, 05.60.-k, 02.50.-r, 02.30.Ks, 02.70.-c\nE-mail: giampaolo.cristadoro@unibo.it, thomas.gilbert@ulb.ac.be,\nmarco.lenci@unibo.it, dpsanders@ciencias.unam.mx\n1. Introduction\nStochastic processes in which independent particles scatter randomly at finite speed\nand may occasionally propagate over a long distance in single bouts are known as Lévy\nwalks. Use of these models has become ubiquitous in the study of complex diffusive\nprocesses [1–5]. They are particularly relevant to situations such that the probability\nof a long jump decays slowly with its length [6]. The scale-free superdiffusive motion of\nLévy walkers [7] has been identified as an efficient foraging strategy [8], spurring a large\ninterest in these models, including with regards to human mobility patterns [9,10].arXiv:1501.05216v1  [cond-mat.stat-mech]  21 Jan 2015Lévy walks on lattices as multi-state processes 2\nA central quantity in this formalism is the distribution of free path lengths,\nwhich gives the probability that a particle propagates over a distance xbetween two\nsuccessive scattering events. Its asymptotic scaling determines how the moments of the\ndisplacement asymptotically scale with time. Assuming the probability of a free path\nof lengthxscales asymptotically as x\u0000\u000b\u00001, one finds, in the so-called velocity picture\nof Lévy walks, the following scaling laws for the mean squared displacement after time\nt[11–15]:\nhr2it\u00188\n>>>>>>><\n>>>>>>>:t2; 0<\u000b< 1;\nt2=logt; \u000b = 1;\nt3\u0000\u000b; 1<\u000b< 2;\ntlogt; \u000b = 2;\nt; \u000b> 2 ;(1)\nsee below for a precise definition of this quantity. A scaling parameter value 0<\u000b\u00142is\nsuchthatthevarianceofthedistributionoffreepathsdiverges, inwhichcasetheprocess\nis often called scale-free [9]. Correspondingly, the asymptotic divergence of the mean\nsquared displacement gives rise to anomalous transport in the form of superdiffusion.\nIn a recent paper [16], we considered Lévy walks on lattices and generalized\nthe standard description of the velocity picture of Lévy walks, according to which\na new jump event takes place as soon as the previous one is completed, to include\nan exponentially-distributed waiting time separating successive jumps. As emphasised\nin reference [16], this additional phase induces a differentiation between the states of\nparticles which are in the process of completing a jump and those that are waiting to\nstart a new one. We call the former states propagating and the latter scattering. The\ndistinction between these states leads to a new theoretical framework of Lévy walks in\nterms of multistate processes [17], whereby the generalized master equation approach to\ncontinuous time random walks [18,19] translates into a set of delay differential equations\nfor the corresponding distributions.\nIt is the purpose of this paper to show that a complete characterization of\nthe solutions of such multistate processes can be obtained, which yields exact time-\ndependent analytic expressions of their mean squared displacement. These expressions\ncan, on the one hand, be compared with the asymptotic solutions already reported\nin [16], and, on the other, also prove useful when the asymptotic regime is not reached,\nwhich is often the case with studies dealing with observational data.\nSuch a situation arises when the probability of a transition from scattering to\npropagating states is small. A particle will then spend most of its time undergoing\ntransitions among scattering states, performing a seemingly standard continuous-time\nrandom walk, only seldom undergoing a transition to a propagating state, during which\nit moves ballistically over a distance distributed according to the scaling parameter\n\u000b. If\u000bis small enough ( \u000b\u00142), these occurrences, although rare, have a dramatic\neffect on the asymptotic scaling properties of the mean squared displacement, such thatLévy walks on lattices as multi-state processes 3\na crossover from normal to anomalous diffusive transport is observed. This crossover\ntime may in some situations, however, be much larger than the times accessible to\nmeasurements. Here we provide analytic results which give a precise characterisation of\nthe emergence of an anomalous contribution out of a normally scaling one, showing how\nthe asymptotic scaling (1) comes about. This is of particular relevance to the regime\n\u000b= 2, for which the logarithmic divergence in time of the mean squared displacement\nis indeed very slow. Our results are tested and confirmed by numerical simulations of\nthe processes under consideration.\nThe paper is organized as follows. In section 2, we introduce the multi-state\ndescription of Lévy walks with respect to scattering and propagating states, and define\nthe transition probabilities between them in terms of two parameters, one relating to\nthe probability of a transition from a scattering to a propagating state and the other\ncharacterizing the asymptotic scaling of the free path distribution. The fraction of\nparticles in a scattering state evolves according to a delay differential equation which is\nderived and solved in section 3. These results are exploited in section 4 to obtain the\nmean squared displacement of the processes. A comparison with numerical simulations\nis provided in section 5. Conclusions are drawn in section 6.\n2. Lévy walks as multi-state processes\nWe consider a continuous-time random walk on a square lattice which generalizes the\nstandard model of Montroll & Weiss [18] in that it assumes displacements to non-\nneighbouring sites occur in a time span given by the ratio of the distance traveled to\nthe walker’s speed v, which itself remains fixed throughout. In this sense, the model\nis similar to the so-called velocity picture of Lévy walks [20], the difference being that\nsuccessive propagations, irrespective of their lengths, are separated by random waiting\ntimes, which we assume have exponential distributions. The model itself is not new and,\nin some sense, is a simplification of other models of Lévy walks interrupted by rests;\nsee reference [21]. As we show below, however, the combination of a discretized spatial\nstructure and exponentially-distributed waiting times yields a novel description of the\nprocess in terms of delay differential equations amenable to analytic treatment.\nA natural distinction arises between the states of walkers which are moving across\nthe lattice structure and those at rest. We call propagating the state of a particle which\nis undergoing a displacement phase and scattering the state of a particle at rest, waiting\nto start a new displacement. In the framework of intermittent random walks [22], the\nformer state is usually referred to as ballistic and the latter as diffusive or reactive,\ndepending on context. A scattering state may therefore be thought of as one associated\nwith a local diffusive process bound to the scale `of the distance between neighbouring\nlattice sites.\nThe interplay between the two states is as follows. A particle in a propagating\nstate switches to a scattering state upon completing a displacement. A particle in\na scattering state, on the other hand, can make transitions to both scattering andLévy walks on lattices as multi-state processes 4\npropagating states; as soon as its randomly-distributed waiting time has elapsed, it\nmoves on to a neighbouring site and, in doing so, may switch to a propagating state\nand continue its motion to the next site, or start anew in a scattering state.\nConsider a d-dimensional cubic lattice of individual cells n2(`Z)d. The state of\na walker at position n\u0011(n1;:::;nd)`and timetcan take on a countable number of\ndifferent values, (k;j), specified by an integer k2N, and direction j2f1;:::;zg, where\nz\u00112ddenotes the coordination number of the lattice, that is the number of different\nlattice directions. States (0;j)are associated with a scattering state, irrespective of\ndirectionj, while states (k;j)withk\u00151refer to propagating states, with kbeing the\nremaining number of lattice sites the walker will travel in direction jto complete its\ndisplacement.\nTime evolution proceeds in steps, characterized by a waiting-time density function\nand a transition probability. A particle at position nin a scattering state will wait for\na random time t, exponentially distributed with mean z\u001cR, before updating its state\nto(k;j)with probability \u001ak=z, simultaneously changing its location to n+ej, where\nejdenotes the lattice vector (of length `) associated with direction j. In contrast, a\nparticle in a propagating state (k;j),k\u00151, will change its state to (k\u00001;j)after\na time\u001cB\u0011`=v, simultaneously moving from site nton+ej. The waiting times\nassociated with scattering states are drawn from a standard Poisson process, while the\nrenewal process generated by the combination of scattering and propagating states has\narbitrary holding times, whose distribution is determined by the transition probabilities\n\u001ak.\nThe waiting-time density of the process is thus the function\n k(t) =(\n\u001c\u00001\nRe\u0000t=\u001cR; k = 0;\n\u000eD(t\u0000\u001cB); k6= 0;(2)\nwhere\u000eD(:)denotes the Dirac delta function. When a step takes place, the transition\nprobability to go from state (k;j)to state (k0;j0)is\np(k;j);(k0;j0)=(\nz\u00001\u001ak0; k = 0;\n\u000ek\u00001;k0\u000ej;j0k6= 0;(3)\nwhere\u000e\u0001;\u0001is the Kronecker symbol.\nA particle which makes a transition from the scattering state to a propagating state\n(k;j)willthereforetraveladistance (k+1)`awayindirection j, untiliteventuallycomes\nback to the scattering state and may change directions at the next transition.\nWe now introduce a characterization of the transition probabilities \u001akin terms of\ntwo parameters. The first, which we refer to as the scattering parameter , is denoted by\n\u000f,0\u0014\u000f\u00141, and gives the total probability of a transition from the scattering state,\nzThe subscript R in \u001cRstands for residence as in “residence time.” In contrast, B in \u001cBstands for\nballistic as in “ballistic propagation time.”Lévy walks on lattices as multi-state processes 5\nk= 0, into a propagating state, k\u00151:\n1X\nk=1\u001ak=\u000f; (4)\nThe remaining transition probability,\n\u001a0\u00111\u0000\u000f; (5)\nis that of a transition from a scattering state into another scattering state. The value\n\u000f= 0thus corresponds to the absence of propagating states: the process is then a simple\ncontinuous-time random walk with transitions to nearest neighbouring sites only and\nexhibits normal diffusion with coefficient `2=(z\u001cR). The opposite extreme, \u000f= 1, assigns\nzero probability to transitions from scattering to scattering states, which means that\nevery transition involves a displacement over a distance of at least two sites. Scattering\nstates remain populated, however, due to the decay of propagating states.\nThesecondparameter, \u000b>0, isthe scaling parameter ofthetransitionprobabilities\n\u001ak, which controls their asymptotic behaviour,\n\u001ak/k\u0000\u000b\u00001(k\u001d1); (6)\nand determines the scaling law of the mean squared displacement (1). The specific form\nof\u001akhas no effect on this scaling law, but is does affect the time-dependent properties of\nthe mean squared displacement. To be specific, we consider in this paper the following\ndouble-telescopic form for the transition probabilities \u001ak:\n\u001ak=\u000f(1\u000021\u0000\u000b)\u00001\u0002\nk1\u0000\u000b\u00002(k+ 1)1\u0000\u000b+ (k+ 2)1\u0000\u000b\u0003\n(k\u00151); (7)\nits structure is particularly helpful for some of the computations presented below and\nis motivated by our study of anomalous transport in the infinite-horizon periodic\nLorentz gas [23]. In this respect, the model is slightly different from that presented\nin reference [16], where a simple telescopic structure of the transition probabilities was\nused.\nFor future reference, we define\n\u0017k=1X\nj=k\u001aj; (8)\nto be the probability of a transition to a state larger than or equal to k, such that, in\nparticular, \u00170= 1and\u00171=\u000f, and, for the choice of transition probabilities (7),\n\u0017k=\u000f(1\u000021\u0000\u000b)\u00001\u0002\nk1\u0000\u000b\u0000(k+ 1)1\u0000\u000b\u0003\n: (9)Lévy walks on lattices as multi-state processes 6\nWe also note the following two identities:\nkX\nj=1\u001aj=\u000f\b\n1\u0000(1\u000021\u0000\u000b)\u00001[(k+ 1)1\u0000\u000b\u0000(k+ 2)1\u0000\u000b]\t\n;\nkX\nj=1j\u001aj=\u000f(1\u000021\u0000\u000b)\u00001[1\u0000(k+ 1)2\u0000\u000b+ (k+ 2)2\u0000\u000b\u00002(k+ 2)1\u0000\u000b]:(10)\n3. Fraction of particles in the scattering state\nA quantity which plays a central role in the analysis of the process generated by the\nwaiting-time density (2) and transition probabilities (3) is the average return time to\nthe scattering state,\n\u001c\b=1X\nk=0\u001ak(\u001cR+k\u001cB); (11)\nThe average return time is finite only when x\u000b > 1. The process is then said to be\npositive recurrent . In the remainder, we restrict our attention to this range of parameter\nvalues. The null-recurrent case, when 0<\u000b\u00141, for which the average return time to\nthe scattering state diverges, is considered in reference [16].\nThe occupation probability of particles at site nand timet,P(n;t), is a sum of\nthe probabilities over the different states, Pk;j(n;t):\nP(n;t) =zX\nj=11X\nk=0Pk;j(n;t): (12)\nIn reference [16], we obtained the following set of delay differential equations for the\ntime-evolution of these occupation probabilities:\n@tP0;j(n;t) =1\nz\u001cRzX\nj0=11X\nk=0\u001akP0;j0(n\u0000(k+ 1)ej;t\u0000k\u001cB)\u00001\n\u001cRP0;j(n;t)\n+1X\nk=0\u001bk;j(n\u0000kej;t\u0000k\u001cB); (13)\n@tPk;j(n;t) =1\nz\u001cRzX\nj0=11X\nk0=1\u001ak+k0\u00001h\nP0;j0(n\u0000k0ej;t\u0000(k0\u00001)\u001cB)\n\u0000P0;j0(n\u0000k0ej;t\u0000k0\u001cB)i\n+1X\nk0=0h\n\u001bk+k0;j(n\u0000k0ej;t\u0000k0\u001cB)\nxGenerally speaking, \u001akmust decay asymptotically faster than k\u00002for the average return time to be\nfinite.Lévy walks on lattices as multi-state processes 7\n\u0000\u001bk+k0;j(n\u0000k0ej;t\u0000(k0+ 1)\u001cB)i\n; (14)\nwhere the inclusion of terms \u001bk;jaccounts for the possibility of external sources, such\nthat\u001bk;j(n;t)\u00150is the rate of injection of particles at site nand timetin the state\n(k;j).\nTo simplify matters, we assume that these source terms are independent of the\nlattice direction and write \u001bk;j(n;t)\u0011z\u00001\u001bk(n;t). Furthermore, we will usually let\n\u001bk;j(n;t) = 0fork\u00151, which amounts to assuming that particles are injected only in\na scattering state. The injection of particles in a propagating state is easily treatable\nas well, and will be considered explicitly in order to initiate the process in a stationary\nstate of equations (13)–(14). We will, however, limit such considerations to this specific\nchoice and avoid the possibility that source terms interfere with the asymptotic scaling\nof the process generated by particles initially injected in a scattering state, as might\narise from alternative choices of injection rates of propagating states.\nOf particular interest is the fraction of particles in the scattering state,\nS0(t)\u0011X\nn2ZdzX\nj=1P0;j(n;t); (15)\nwhose time-evolution is described by the following delay differential equation:\n_S0(t) =\u001c\u00001\nR1X\nk=1\u001akS0(t\u0000k\u001cB)\u0000\u000f\u001c\u00001\nRS0(t) +1X\nk=0\u001bk(t\u0000k\u001cB): (16)\nHere,\u001bk(t\u0000k\u001cB)denotes a source term for the rate of injection of particles in state k,\nirrespective of their positions, that is, \u001bk(t\u0000k\u001cB) =P\nn\u001bk(n;t\u0000k\u001cB). The first two\nterms on the right-hand side of this equation have the typical gain and loss structure\nof jump processes. Indeed, the second term corresponds to particles lost by scattering\nstates due to transitions to propagating states, which occur at rate \u000f=\u001cR. Each such\ntransition is gained back after a delay given by the length of the ballistic segment in a\npropagating state. The sum of those terms yields the first term on the right-hand side\nof equation (16).\nBefore turning to the integration of this differential equation, we note that the\nasymptotic value of S0(t),t!1, has a simple expression. By ergodicity of the process,\nthe equilibrium ratio of particles in the scattering state is given by the ratio between\nthe average time spent in the scattering state, \u001cR, and the average return time to it, \u001c\b,\nequation (11),\nlim\nt!1S0(t) =\u001cR\n\u001c\b: (17)\nAssuming a positive recurrent process, such as when the transition probabilities \u001akare\nspecified by equation (7) with \u000b>1, the average return time \u001c\b=\u001cR+\u000f(1\u000021\u0000\u000b)\u00001\u001cB\nis finite, so that the asymptotic fraction of particles in a scattering state is strictlyLévy walks on lattices as multi-state processes 8\npositive. For the particular choice of the parameters \u001ak, equation (7), the return time\nis given by equation (11) and equation (17) becomes\nlim\nt!1S0(t) =\u001cR\n\u001cR+\u000f(1\u000021\u0000\u000b)\u00001\u001cB: (18)\nNote, however, that the convergence to this asymptotic value follows a power law whose\nexponent goes to zero as \u000b!1; see figure 1a. Furthermore, S0(t)converges to 0when\nthe scaling parameter falls into the null recurrent regime, 0<\u000b\u00141.\nCorrespondingly, the fraction of particles in the propagating states,\nSk(t)\u0011X\nn2ZdzX\nj=1Pk;j(n;t); (19)\ntends to\nlim\nt!1Sk(t) =\u0017k\u001cB\n\u001cR+\u000f(1\u000021\u0000\u000b)\u00001\u001cB; (20)\nwhich follows from the observation that a particle that makes a transition to state k\nspends an equal amount of time in all states jsuch that 1\u0014j\u0014k.\n3.1. Time-dependent fraction of particles in the scattering state\nThe integration of equation (16) relies on the specification of initial conditions. For\ninstance, the choice k\nS0(t) =(\n0; t< 0;\n1; t= 0;(21)\ncorresponds, when all particles are injected at the lattice origin, to the injection rate\n\u001bk(n;t) =\u000eD(t)\u000ek;0\u000en;0; (22)\nand will be henceforth referred to as the all-scattering initial condition .\nThe time-dependent fraction of particles in the scattering state, S0(t),t>0, may\nthenbeobtainedbythemethodofsteps[24],whichconsistsofintegratingthedifferential\nequation (16) successively over the intervals k\u001cB\u0014t\u0014(k+ 1)\u001cB,k2N, matching the\nsolutions at the upper and lower endpoints of successive intervals. For the all-scattering\ninitial condition (21), the fraction of particles in the scattering state is, for times t\u00150,\nS0(t) =e\u0000\u000ft=\u001cR+bt=\u001cBcX\nk=1e\u0000\u000f(t\u0000k\u001cB)=\u001cRkX\nn=1a(njk)\u001c\u0000n\nR(t\u0000k\u001cB)n; (23)\nwherebt=\u001cBc(resp.dt=\u001cBe, used below) denotes the largest (resp. smallest) integer\nsmaller (resp. larger) than or equal to t=\u001cB, and each coefficient a(njk)is the sum of all\nkThe discussion below can be easily generalized, by linear superposition, to processes where particles\nin a scattering state are continuously injected into the process.Lévy walks on lattices as multi-state processes 9\npossible combinations of products \u001ai1:::\u001ain, where the sequences fijgn\nj=1are such that\ni1+\u0001\u0001\u0001+in=k,\na(njk)=1\nn!X\n1\u0014i1;:::;in\u0014k\ni1+\u0001\u0001\u0001+in=knY\nj=1\u001aij: (24)\nThecontributionsto a(njk)consistofalldistinctwaysoftravellingadistance kinnsteps,\ndivided by their number of permutations; the derivation of equation (23) is provided\nin Appendix A. A comparison between the time-dependent fraction of particles in the\nscattering state (23) and the asymptotic value (17) is illustrated in figure 1a, where\nthe difference between the two expressions is plotted vs. time for different values of\nthe scaling parameter \u000band a specific choice of the scattering parameter \u000f, with the\ntransition probabilities \u001akspecified by equation (7).\n��-������������-���-���-�\n���(�)-τ�/[τ�+ϵ τ�/(�-��-α)]\n(a)\u000f(1\u000021\u0000\u000b)\u00001= 1\n��-����������������������������������\n���(�) (b)\u000f(1\u000021\u0000\u000b)\u00001= 5\u000210\u00002\nFigure 1: Convergence in time of the fraction of particles in a scattering state, S0(t),\ncomputed from equation (23), to the asymptotic value (18). The transition probabilities\nare taken according to equation (7), with \u001cB= 1and\u001cR= 1 +\u000f(1\u000021\u0000\u000b)\u00001. The value\nof the scattering parameter is chosen such that \u000f=(1\u000021\u0000\u000b) = 1(left panel) and 5\u000210\u00002\n(rightpanel). Theleftpanelshowsthedifferencebetween S0(t)anditsasymptoticvalue.\nIn both panels, the scaling parameter takes the values \u000b= 3=2(green curve), 2(cyan\ncurve), and 5=2(red curve). The right panel corresponds to a perturbative regime, such\nthatS0(t)can be approximated by a first-order polynomial in \u000f(darker curves) whose\nfirst-order coefficient varies with time; see equation (29). The agreement between the\nexact and the approximate solutions improves as the value of \u000fdecreases.\n3.2. Small-parameter expansion\nSuppose that the scattering parameter is small, \u000f\u001c1, so that the overwhelming\nmajority of transitions occur between scattering states, and only rarely does a particle\nmake an excursion into a propagating state, which may, however, last long, depending\non the value of the scaling parameter \u000b>1.Lévy walks on lattices as multi-state processes 10\nThe first-order expansion of equation (23) yields the following approximation of S0:\nS0(t)'1\u0000\u000ft\n\u001cR+bt=\u001cBcX\nk=1\u001akt\u0000k\u001cB\n\u001cR;\n=S0(bt=\u001cBc\u001cB)\u0000t\u0000bt=\u001cBc\u001cB\n\u001cR1X\nk=dt=\u001cBe\u001ak: (25)\nThat is to say, for \u000f\u001c1,S0(t)is a sequence of straight line segments joining the values\nthe function takes at integer multiples of the propagation time \u001cB,\nS0(k\u001cB)'1\u0000\u000f\u001cB\n\u001cRh\nk\u0000kX\nj=1\u001aj\n\u000f(k\u0000j)i\n: (26)\nIn this regime, S0(t)converges asymptotically to the constant value (17), which, for the\nchoice of parameters (7), is given by equation (18), i.e., with all other parameters being\nfixed,\nlim\nt!1S0(t)'1\u0000\u000f(1\u000021\u0000\u000b)\u00001\u001cB\n\u001cR: (27)\nFor these parameters, we make use of identities (10) to evaluate equation (26),\nS0(k\u001cB)'1\u0000\u000f(1\u000021\u0000\u000b)\u00001[1\u0000(k+ 1)1\u0000\u000b]\u001cB\n\u001cR; (28)\nor, for general time values,\nS0(t)'1\u0000\u000f(1\u000021\u0000\u000b)\u00001nt\n\u001cRh\u0000\nbt=\u001cBc+ 1\u00011\u0000\u000b\u0000\u0000\nbt=\u001cBc+ 2\u00011\u0000\u000bi\n+\u001cB\n\u001cRh\n1\u0000\u0000\nbt=\u001cBc+ 1\u00012\u0000\u000b+\u0000\nbt=\u001cBc+ 2\u00012\u0000\u000b\n\u00002\u0000\nbt=\u001cBc+ 2\u00011\u0000\u000bio\n: (29)\nA comparison between this approximate value and the exact one, equation (23), is\ndisplayed in figure 1b, where the value of \u000fwas taken to be large enough that the curves\nremain distinguishable.\n4. Mean squared displacement\nAssuming initial injection of particles at the origin, the mean squared displacement of\nparticles as a function of time, given by the second moment of the displacement vector\nr\u0011`n,hr2it=`2hn2it=`2P\nn2Zdn2P(n;t), wherer2=r\u0001randn2=n\u0001n, evolves\naccording to the differential equationLévy walks on lattices as multi-state processes 11\nd\ndthn2it=\u001c\u00001\nR1X\nk=0(2k+ 1)\u0017kS0(t\u0000k\u001cB) +1X\nk=1(2k\u00001)1X\nk0=0\u001bk+k0(0;t\u0000k\u001cB);(30)\nwhere, incontrasttotheexpressionderivedinreference[16], wehavehereaddedpossible\ncontributions from source terms \u001bk;j(n;t),k\u00151, which we assume to be concentrated\nat site n=0only. To conform with stationarity of the process, propagating states\nshould be uniformly injected in the time interval \u0000\u001cB<t\u00140so they decay uniformly\nin the time interval 0<t\u0014\u001cB. We thus let ( k\u00151)\n\u001bk(n;t) =\u001c\u00001\nB\u0016k\u000en;0\u0002(\u0000t)\u0002(t+\u001cB); (31)\nwheref\u0016kgk\u00151is a sequence of positive numbers such thatP\nk\u0016k= 1and\u0002(:)is the\nHeaviside step function, such that the product \u0002(\u0000t)\u0002(t+\u001cB)guarantees the uniform\ndistribution of propagating source terms in the desired time interval. Equation (30)\ntransforms to\nd\ndthn2it=\u001c\u00001\nR1X\nk=0(2k+ 1)\u0017kS0(t\u0000k\u001cB) +\u001c\u00001\nB(2bt=\u001cBc+ 1)1X\nk=1\u0016bt=\u001cBc+k:(32)\nIntegrating over time, we obtain the mean squared displacement:\nhn2it=\u001cB\n\u001cRbt=\u001cBcX\nk=0(2k+ 1)\u0017kZt=\u001cB\u0000k\n0dxS0(x\u001cB) +1X\nk=1Zt=\u001cB\n0dx(2bxc+ 1)\u0016bxc+k:(33)\nThere are two contributions to this expression; the first arises from the fraction of\nparticles in a scattering state, and the second from particles initially distributed among\npropagating states. This term transforms to\n1X\nk=1Zt=\u001cB\n0dx(2bxc+ 1)\u0016bxc+k=1X\nk=0bt=\u001cBcX\nj=1(2j\u00001)\u0016j+k\n+ 2(t=\u001cB\u0000bt=\u001cBc)(bt=\u001cBc+ 1=2)1X\nk=1\u0016bt=\u001cBc+k:(34)\n4.1. Stationary regimes\nThe stationary initial condition , i.e. such that the initial fractions of particles in\nscattering and propagating states are stationary solutions of equations (13)-(14), are\nrealized if, in equation (31), we identify \u0016kwith the stationary fraction of particles in\nstatek,\n\u0016k=\u0017k\u001cB\n\u001cR+\u000f(1\u000021\u0000\u000b)\u00001\u001cB; (35)Lévy walks on lattices as multi-state processes 12\ncf. equation (20), and let\n\u001b0(n;t) =\u00160\u000eD(t)\u000en;0; \u0016 0=\u001cR\n\u001cR+\u000f(1\u000021\u0000\u000b)\u00001\u001cB: (36)\nWith this choice, the mean squared displacement (33) yields the exact expression\nhn2it=t\n(1\u000021\u0000\u000b)\u001cR+\u000f\u001cB\u001a\n1\u000021\u0000\u000b+\u000f\ntbt=\u001cBcX\nk=1(t\u0000k\u001cB)(2k+ 1)[k1\u0000\u000b\u0000(k+ 1)1\u0000\u000b]\u001b\n+\u000f\u001cB\n(1\u000021\u0000\u000b)\u001cR+\u000f\u001cB\u001abt=\u001cBcX\nk=12k\u00001\nk\u000b\u00001+ 2(t=\u001cB\u0000bt=\u001cBc)bt=\u001cBc+ 1=2\n(bt=\u001cBc+ 1)\u000b\u00001\u001b\n:\n(37)\nTreatingn\u0011t=\u001cB\u001d1asanintegerandletting H(\f)\nn\u0011Pn\nk=1k\u0000\fdenotethegeneralized\nharmonic numbers, the two terms with sums over kin the above expression evaluate\nrespectively to\nnX\nk=1n\u00001(n\u0000k)(2k+ 1)[k1\u0000\u000b\u0000(k+ 1)1\u0000\u000b]'1\u00004n\u00001H(\u000b\u00002)\nn + 2H(\u000b\u00001)\nn;(38)\nfor the term arising from the fraction of particles in a scattering state, and to\nnX\nk=1(2k\u00001)k1\u0000\u000b= 2H(\u000b\u00002)\nn\u0000H(\u000b\u00001)\nn; (39)\nfor the term arising from particles initially distributed among the propagating states.\nDepending on the value of the scaling parameter \u000b, we have the following three\nasymptotic regimes.\n4.1.1. Normal diffusion For parameter values \u000b > 2, the asymptotic properties\nof the harmonic numbers,\nlim\nn!1n\u00001H(\u000b\u00002)\nn = 0;\nlim\nn!1H(\u000b\u00001)\nn =\u0010(\u000b\u00001);(40)\nare such that only the terms arising from the fraction of particles in a scattering\nasymptotically contribute to equation (37):\nlim\nt!11\nthn2it=1\u000021\u0000\u000b+\u000f[1 + 2\u0010(\u000b\u00001)]\n(1\u000021\u0000\u000b)\u001cR+\u000f\u001cB; (41)\nwhich, up to a factor z\u00001, is the diffusion coefficient of the process.Lévy walks on lattices as multi-state processes 13\n4.1.2. Weak superdiffusion For the marginal parameter value \u000b= 2, the\nharmonic numbers in equation (38) evaluate to\nH(0)\nn=n;\nH(1)\nn= logn+\r+ O(n\u00001);(42)\nwhere\r'0:577216is Euler’s constant.\nWe therefore have the asymptotic limit of equation (37),\nlim\nt!11\ntlog(t=\u001cB)hn2it=4\u000f\n\u001cR+ 2\u000f\u001cB; (43)\nwhich is due to the fraction of particles in a scattering state alone. This result\nis, however, of limited use because the asymptotic regime only emerges provided\n\u000flog(t=\u001cB)\u001d1(where 1corresponds to the order of the sub-leading term), which may\nnot be attainable, especially when the scattering parameter is small. For this reason, it\nis preferable to retain also the next-order term in equation (37),\n1\nthn2it'1\n\u001cR+ 2\u000f\u001cBn\n1 + 4\u000f\u0002\nlog(t=\u001cB) +\r\u00001=2 + O(t=\u001cB)\u00001\u0003o\n;(44)\nwhere a fraction 4\u000f=(\u001cR+ 2\u000f\u001cB)is contributed by particles initially distributed among\npropagating states.\n4.1.3. Superdiffusion In the range of parameters 1< \u000b < 2, we substitute the\nscaling properties of the harmonic numbers,\nlim\nn!1n\u000b\u00003H(\u000b\u00002)\nn = (3\u0000\u000b)\u00001;\nlim\nn!1n\u000b\u00002H(\u000b\u00001)\nn = (2\u0000\u000b)\u00001:(45)\nand obtain from equation (37) the limit\nlim\nt!11\n(t=\u001cB)3\u0000\u000bhn2it=\u0014\u000b\u00001\n(2\u0000\u000b)(3\u0000\u000b)+1\n3\u0000\u000b\u00152\u000f\u001cB\n(1\u000021\u0000\u000b)\u001cR+\u000f\u001cB;\n=1\n(2\u0000\u000b)(3\u0000\u000b)2\u000f\u001cB\n(1\u000021\u0000\u000b)\u001cR+\u000f\u001cB: (46)\nAs emphasized by the first line of this equation, the leading-order coefficient is the sum\nof two distinct non-trivial contributions from the two terms on the right-hand side of\nequation (37): one from the fraction of particles in a scattering state and the other\nfrom particles initially injected in propagating states. As a consequence, the transport\ncoefficient corresponding to the all-scattering initial condition differs by a factor \u000b\u00001\nfrom that above, corresponding to the stationary initial condition. This observation is\nconsistent with the results of reference [25], where non-recurrent regimes of the scaling\nparameter\u000bwere also investigated.Lévy walks on lattices as multi-state processes 14\n4.2. Exact time-dependent solution\nThe following exact expression of the mean squared displacement (33) is obtained after\nsubstitution of the fraction of particles in a scattering state, equation (23), and applies\nto the all-scattering initial condition (22):\nhn2it=bt=\u001cBcX\nk=0(2k+ 1)1X\nl=k\u000f\u00001\u001al(\n1\u0000e\u0000\u000f(t\u0000k\u001cB)=\u001cR\n+bt=\u001cBc\u0000k\u00001X\nj=1jX\nm=1mX\nn=1\u000f\u0000na(njm)e\u0000\u000f(j\u0000m)\u001cB=\u001cRnX\ni=0n!\ni!\u000fi\u0012\u001cB\n\u001cR\u0013i\n\u0002h\n(j\u0000m)i\u0000(j\u0000m+ 1)ie\u0000\u000f\u001cB=\u001cRi\n+bt=\u001cBc\u0000kX\nm=1mX\nn=1\u000f\u0000na(njm)e\u0000\u000f(bt=\u001cBc\u0000k\u0000m)\u001cB=\u001cRnX\ni=0n!\ni!\u000fi\u0012\u001cB\n\u001cR\u0013i\n\u0002h\n(bt=\u001cBc\u0000k\u0000m)i\u0000(t=\u001cB\u0000k\u0000m)ie\u0000\u000f(t\u0000bt=\u001cBc\u001cB)=\u001cRi)\n;(47)\nwhere, for consistency, one must interpret terms such as (j\u0000m)0as unity, even when\nj=m.\n��-����������������������������\n�〈��〉/�\n(a)\u000f(1\u000021\u0000\u000b)\u00001= 1\n��-���������������������������������������\n�〈��〉/� (b)\u000f(1\u000021\u0000\u000b)\u00001= 5\u000210\u00002\nFigure 2: Growth in time of the rescaled mean squared displacement, hn2it=t, computed\nfrom equation (47). The transition probabilities were set according to equations (7),\nwith\u001cB= 1and\u001cR= 1 +\u000f(1\u000021\u0000\u000b)\u00001. The value of the scattering parameter was\nchosen such that \u000f=(1\u000021\u0000\u000b) = 1(left panel) and 5\u000210\u00002(right panel). In both figures,\nthe scaling parameter takes the values \u000b= 3=2(green curve), 2(cyan curve), and 5=2\n(red curve). For comparison, in the left panel, the asymptotic scaling values predicted\nby equations (41), (43) and (46) are displayed in dashed lines with matching colors.\nThe right panel shows a comparison between the exact solution and an approximated\nsolution, exact to first order in \u000f(darker curves); see equation (53).Lévy walks on lattices as multi-state processes 15\nFigure 2 displays the results of explicit evaluations of equation (47) for three\ndifferent values of the scaling parameter, \u000b > 1, with\u000f= 1\u000021\u0000\u000b. According to\nthe asymptotic results (41), (43) and (46), the scaling parameter value \u000b= 2gives\nrise to a logarithmic divergence of hn2it=t, which separates the superdiffusive regime,\nfor1< \u000b < 2, from that of normal diffusion, for \u000b > 2. A comparison between the\nexact solution and the asymptotic ones is displayed in figure 2a. On the time scale of\nthe figure, convergence to the asymptotic regimes is convincingly observed only in the\nsuperdiffusive case. Note that while the computation of equation (47) up to t= 50\u001cB\ntakes less than one half hour of CPU time on a reasonably fast computer, doubling the\ntime range would increase the required CPU time to over ten days.\nAlthough equation (47) gives the exact time-dependence of the mean squared\ndisplacement of the process for the initial condition (21), it does not give much insight\ninto its time development. In particular, how to extract the large-time behaviour of\nthe mean squared displacement and connect this result to the asymptotic scalings (41),\n(43) and (46) is not transparent. As reflected by figure 2a, a numerical evaluation of\nequation (47) is necessarily limited to moderately large times, due to the number of\nterms involved.\nAregimeofspecificinterestwhichallowsustoinfertheemergenceoftheasymptotic\nscalings (41), (43) and (46) from the solution (47) is, however, that of small values of the\nscattering parameter \u000f, i.e., such that transitions from scattering to propagating states\noccur only rarely. This is discussed below. We will otherwise have to resort to numerical\nsimulations of the underlying stochastic processes to observe this convergence. Those\nresults are presented in section 5.\n4.3. Small-parameter expansion\nFor small values of the scattering parameter \u000f, when transitions from scattering\nto propagating states are much rarer than transitions between scattering states, an\nexpansion of equation (47) in powers of this parameter provides an approximate\nexpression of the mean squared displacement from which the different asymptotic\nregimes discussed in section 4.1 can be inferred. In the anomalous regime of the scaling\nparameter, \u000b\u00152, terms constant in time, that carry a normal contribution to the mean\nsquared displacement, may bring about contributions which, even for large times, may\nbe much larger than that of terms diverging in time; an anomalous contribution to the\nmean squared displacement may then be masked by a normal one.\nTo proceed, we substitute equation (28) into equation (33) and let t\u0011k\u001cB. Having\ndropped all terms of order \u000f2and higher, we obtain\nhn2ik\u001cB=\u001cB\n\u001cRZk\n0dxn\n1\u0000\u000f(1\u000021\u0000\u000b)\u00001\u001cB\n\u001cR[1\u0000(bxc+ 1)1\u0000\u000b]\n\u0000\u001cB\n\u001cR(x\u0000bxc)1X\ni=dxe\u001aio\n+\u001cB\n\u001cRkX\nj=1(2j+ 1)(k\u0000j)1X\ni=j\u001aiLévy walks on lattices as multi-state processes 16\n=k\u001cB\n\u001cRn\n1\u0000\u001cB\n\u001cR\u000f(1\u000021\u0000\u000b)\u00001\u0002\n1\u0000k\u00001(H(\u000b\u00001)\nk\u00001=2 + 1=2(k+ 1)1\u0000\u000b)\u0003\n+\u000f(1\u000021\u0000\u000b)\u00001h\n1\u0000(k+ 1)1\u0000\u000b+ (2\u0000k\u00001)(H(\u000b\u00001)\nk+1\u0000(k+ 1)2\u0000\u000b)\n\u00002k\u00001(2H(\u000b\u00002)\nk+1\u0000H(\u000b\u00001)\nk+1\u0000(k+ 1)3\u0000\u000b)io\n;\n'k\u001cB\n\u001cRn\n1 +\u000f(1\u000021\u0000\u000b)\u00001h\n1 + 2H(\u000b\u00001)\nk\u00004k\u00001H(\u000b\u00002)\nk\u0000\u001cB\n\u001cRio\n; (48)\nwhere, in the last line, we have omitted terms that decay to zero as kgrows large.\nPlugging into this expression the asymptotic forms of the generalized harmonic numbers\n(40), (42) and (45),\nH(\u000b\u00001)\nk\u00002k\u00001H(\u000b\u00002)\nk'8\n>><\n>>:\u0010(\u000b\u00001); \u000b> 2;\nlogk+\r\u00002; \u000b = 2;\n\u000b\u00001\n(2\u0000\u000b)(3\u0000\u000b)k2\u0000\u000b+\u0010(\u000b\u00001);1<\u000b< 2;(49)\nwe obtain approximations of the mean squared displacement (47), which can be\ncompared with the asymptotic expressions (41), (43) and (46).\nIn the regime of normal diffusion, \u000b > 2, equation (48) reduces to the first order\nexpansion in \u000fof equation (41). For 1< \u000b\u00142, the mean squared displacement (48)\ndisplays normal diffusion at short times and anomalous diffusion at large times. Thus\nin the weak superdiffusive regime, \u000b= 2, equation (48) reduces to\n1\nthn2it'1\n\u001cRn\n1 + 2\u000fh\n2 logt\n\u001cB+ 2\r\u00003\u0000\u001cB\n\u001cRio\n; (50)\nwhich differs from the stationary expression (44) by a factor 4\u000f=\u001cR, subleading with\nrespect to the logarithmically diverging term, identical in both expression. This\ndifference stems from the absence of particles populating propagating states in our\nchoice of initial condition. In the superdiffusive regime, 1< \u000b < 2, the divergence of\nthe harmonic numbers, equation (49), dominates the large time behaviour,\n1\nthn2it'1\n\u001cRn\n1 +\u000f(1\u000021\u0000\u000b)\u00001h2(\u000b\u00001)\n(2\u0000\u000b)(3\u0000\u000b)\u0010t\n\u001cB\u00112\u0000\u000b\n+ 1 + 2\u0010(\u000b\u00001)\u0000\u001cB\n\u001cRio\n:(51)\nHere again we note a difference between this expression and equation (46), obtained in\nthe stationary regime. At variance with the case \u000b= 2, however, the difference occurs\nin the coefficient of the leading term on the right-hand side of equation (51), whose\nnumerator is 1\u0000\u000binstead of 1in the stationary regime. The transport coefficient on\nthe right-hand side of equation (51) therefore changes, depending on the choice of initial\nconditions.\nAt large times, after the mean squared displacement crosses over from normal\ndiffusion to superdiffusion, the above expressions are equivalent to O(\u000f)expansions of\nthe asymptotic values (43) and (46). The value of the crossover time, tc, that separatesLévy walks on lattices as multi-state processes 17\nthe short time normal diffusion from the large time anomalous diffusion can be inferred\nfrom the above expressions:\ntc\u0019\u001cB\u00028\n<\n:h\n(2\u0000\u000b)(3\u0000\u000b)\n2(\u000b\u00001)1\u000021\u0000\u000b\n\u000fi1\n2\u0000\u000b;1<\u000b< 2;\nexp\u00001\n4\u000f\u0001\n; \u000b = 2;(52)\nwhich can be large, in particular when \u000b= 2.\nFor short times, equation (48) can be improved by removing the assumption that t\nis an integer multiple of \u001cB. One then finds\nhn2it=t\n\u001cR+\u000f(1\u000021\u0000\u000b)\u00001\u001a\nt\n\u001cRh\n1\u00002(bt=\u001cBc+ 1)2\u0000\u000b\u0000(bt=\u001cBc+ 1)1\u0000\u000b+ 2H(\u000b\u00001)\nbt=\u001cBc+1i\n+\u001cB\n\u001cRh\n2(bt=\u001cBc+ 1)3\u0000\u000b+ (bt=\u001cBc+ 1)2\u0000\u000b\u00004H(\u000b\u00002)\nbt=\u001cBc+1+H(\u000b\u00001)\nbt=\u001cBc+1i\n\u0000\u0012\u001cB\n\u001cR\u00132n\nbt=\u001cBc\u0000H(\u000b\u00001)\nbt=\u001cBc+ 1=2\u00001=2(bt=\u001cBc+ 1)1\u0000\u000b\n+ 1=2\u0002\n(t=\u001cB)2\u0000bt=\u001cBc2\u0003\u0002\n(bt=\u001cBc+ 1)1\u0000\u000b\u0000(bt=\u001cBc+ 2)2\u0000\u000b\u0003\n+\u0000\nt=\u001cB\u0000bt=\u001cBc\u0001\u0002\n1\u0000(bt=\u001cBc+ 1)2\u0000\u000b+ (bt=\u001cBc+ 2)2\u0000\u000b\n\u00002(bt=\u001cBc+ 2)1\u0000\u000b\u0003o\u001b\n: (53)\nA comparison between this approximate solution and the exact one, equation (47), is\nshown in figure 2b for different values of the scaling parameter, \u000b.\n5. Numerical computations\nNumerical simulations of the process with rates (2) and probabilities (3) are based on a\nclassic kinetic Monte Carlo algorithm [26], taking into account the possibility of ballistic\nmotion of particles in a propagating phase.\nA collection of independent walkers are initialized at time t= 0at the origin of\nthe two-dimensional square lattice Z2, either in the scattering state, k0= 0, for the\nall-scattering initial condition, or in state k0\u00150with relative weights \u0016k0specified by\nequations (35)–(36), for the stationary initial condition. For each walker, we generate\na sequencef(kn;jn);tngn2Nof successive states (kn;jn),kn2N,jn= 1;:::;z\u00114, and\ncorresponding times tnas follows.\nWheninastate kn\u00001= 0, thenexttransitionisdeterminedbydrawingthefollowing\nthree random numbers. This is referred to as a scattering step .\n(i) The first random number, \u00112[0;1], is drawn from a uniform distribution, and\nyields the state kn, such thatPkn\u00001\na=0\u001aa\u0014\u0011<Pkn\na=0\u001aa.Lévy walks on lattices as multi-state processes 18\n(ii) The corresponding waiting time, i.e., the time tnthe particle waits in the state\nkn\u00001= 0before the transition to state kntakes place, is obtained by drawing a\nsecond random number, whose distribution is exponential with mean \u001cR.\n(iii) Athirdrandominteger,withuniformdistributionamongthesetoflatticedirections\nf1;:::; 4g, identifies the direction jnof the corresponding displacement by one\nlattice unit.\nDepending on the new state kn, another scattering step is taken if kn= 0or, if\nkn\u00151, a sequence of knpropagating steps f(km;jm);tmgn+kn\nm=n+1takes place, such that:\n(i) the state decreases by one unit, km=km\u00001\u00001,\n(ii) the direction remains unchanged jm=jm\u00001, and\n(iii) the time step takes value tm=\u001cB, corresponding to the propagation time over a\nsingle lattice cell.\nThe above loop repeats itself until m=n+kn, that is until km= 0, at which point\nanother scattering step is taken.\nKeeping track of the positions of all walkers as functions of time, which we measure\nat regular intervals on a logarithmic timescale, measurements of the mean squared\ndisplacementhn2itrescaled by 1=tare performed by taking averages of this quantity over\nthe set of all walkers, which typically consists of 108trajectories. Such measurements\nare reported below for the different scaling regimes in the positive recurrent range of\nthe scaling parameter values, \u000b>1.\nThroughout this section, we set the transition probabilities according to equation\n(7) and, for convenience, change the scattering parameter \u000fto\n\u000e\u00112\u000f\n1\u000021\u0000\u000b+\u000f: (54)\nWe further let\n\u001cB\u00111;\n\u001cR\u00112\n2\u0000\u000e;(55)\nsuch that the asymptotic fraction of particles in the scattering state (18) becomes\nlim\nt!1S0(t) =2\n2 +\u000e: (56)Lévy walks on lattices as multi-state processes 19\n5.1. All-scattering vs. stationary initial conditions\nWith the choice of parameters (54)-(55), The mean squared displacement, for tlarge\nand with the all-scattering initial condition, is expected to scale as\nhn2it\nt'2\n2 +\u000e\u00028\n>><\n>>:1 +\u000e\u0010(\u000b\u00001); \u000b> 2;\n1 +\u000e(logt+\r\u00002); \u000b = 2;\n1 +\u000eh\n\u000b\u00001\n(2\u0000\u000b)(3\u0000\u000b)t2\u0000\u000b+\u0010(\u000b\u00001)i\n;1<\u000b< 2;(all-scattering i. c.)\n(57)\nwhere, in the two anomalous cases, we kept terms constant in time to reflect the\npossibility that, when \u000eis small, the normal term may not be negligible with respect\nto the anomalous one over a large range of times. This is particularly relevant for the\nmarginal case, \u000b= 2, where logtand\r\u00002remain of comparable sizes throughout\nthe time range accessible to numerical computations. In comparison, for the stationary\ninitial condition, the above expression remains unchanged in the normal diffusive case,\nbut is modified in the two anomalous cases,\nhn2it\nt'2\n2 +\u000e\u00028\n>><\n>>:1 +\u000e\u0010(\u000b\u00001); \u000b> 2;\n1 +\u000e(logt+\r\u00001); \u000b = 2;\n1 +\u000eh\n1\n(2\u0000\u000b)(3\u0000\u000b)t2\u0000\u000b+\u0010(\u000b\u00001)i\n;1<\u000b< 2:(stationary i. c.)\n(58)\nThe first order expansion in \u000eof the two anomalous regimes in equation (57) (in\nall-scattering initial condition) yields results equivalent to equations (50) and (51)\nrespectively.\nA computation of the time evolution of the mean squared displacement in regimes\nof normal ( \u000b>2) and anomalous ( 1<\u000b\u00142) diffusion is shown in figure 3, providing\na comparison between the two initial conditions analyzed in section 4, with the two\ncorresponding sets of asymptotic regimes given by equations (57) and (58). The scaling\nparameter values are set to \u000b= 5=2(figure 3a), 2(figure 3b), and 3=2(figure 3c). The\nscattering parameter value is set to \u000f= 1throughout, such that transitions between\nscattering states are forbidden. As expected, the numerically computed mean squared\ndisplacement obtained from the stationary initial condition (magenta curves in figures\n3a-3c) follow precisely the analytic result (37) in all three regimes.\nWe further note that, in the regime of normal diffusion ( \u000b= 5=2), both data sets\nin figure 3a display consistent convergence to the same asymptotic regime, given by the\nfirst lines of equations (57) and (58), limt!1hn2it=t'2:597. A similar result is observed\nin figure 3b, where the convergence of the two data sets to the common leading value\nof equations (57) and (58), limt!1hn2it=tlogt= 4=5, is apparent. The effect of the\ndiffering subleading terms, the one in (57) negative, the other in (58) positive (the\nlatter value is about one half the absolute value of the former for \u000e= 4=3), is, however,Lévy walks on lattices as multi-state processes 20\n���������������������\n�〈�〉�/�\n�����������������������������(�)\n(a)\u000b= 5=2,\u000e'1:215(\u000f= 1)\n������������������������\n�〈�〉�/[����(�)]\n��������������������������(�)\n(b)\u000b= 2,\u000e= 4=3(\u000f= 1)\n������������������������������������\n�〈�〉�/����\n��������������������������(�)\n(c)\u000b= 3=2,\u000e'1:547(\u000f= 1)\nFigure 3: Numerical measurement of the time-evolution of the normally or anomalously\nrescaled mean squared displacement in the three distinct regimes of the scaling\nparameter. Time evolutions obtained from the stationary initial condition (magenta\ncurves) are compared with those generated by the all-scattering initial condition (cyan\ncurves). The dashed black curves correspond to the exact result (37) and the dotted\nlines to the asymptotic regimes (57) and (58), identical for regimes of normal diffusion,\nbut otherwise differing according to the type of initial conditions. Insets: time evolution\nof the fraction of particles in the scattering state compared with the stationary value\n(56) (black dotted line).Lévy walks on lattices as multi-state processes 21\nmanifest. Finally, in the superdiffusive regime, \u000b= 3=2, figure 3c exhibits the two\nasymptotic regimes of the anomalously rescaled mean squared displacement given by\nequations (57) and (58), limt!1hn2it=t3\u0000\u000b'0:582(all-scattering initial condition) or\n1:163(stationary initial condition), whose values differ by a 1 : 2ratio for this value of\nthe scaling parameter.\n5.2. Perturbative regimes of the scattering parameter\nThe time evolution of the mean squared displacement in the perturbative regime of the\nscattering parameter, \u000e\u001c1, is analyzed in figure 4 for particles initially distributed\nin the scattering state at the origin, where the rescaled mean squared displacement\nis compared with the O(\u000f)solution, equation (53). Excellent agreement between\nthe analytic and numerical results is observed throughout the time range when the\nparameter is small enough ( \u000e= 10\u00002figures 4b, 4d, 4f). Numerical computations are\nalso consistent with the asymptotic regime (57) for all cases.\nThe scattering parameter value \u000e= 10\u00001, shown in figures 4a ( \u000b= 5=2), 4c (\u000b= 2),\nand 4e (\u000b= 3=2), is on the one hand small enough that, for short times, the computed\nmean squared displacement is barely distinguishable from the perturbative expansion\n(53). The effect of next order corrections is, on the other hand, apparent for larger times\n(t&102), beyond which a convergence to the asymptotic result (57) is observed.\nWe note that the crossover time (52) is tc'1:3\u0002104for\u000e= 10\u00001(figure 4c)\nand much larger yet for \u000e= 10\u00002. For\u000b= 2, subleading terms in equation (57)\ntherefore dominate the logarithmically divergent term throughout the time range of\nmeasurements. Similarly, in the superdiffusive case, \u000b= 3=2, the respective crossover\ntimes (52) are tc'2\u0002102(figure 4e) and tc'2:2\u0002104(figure 4f), such that subleading\nterms in equation (57) remain important throughout the time range of measurements\nin both cases.\nWe conclude this section by pointing out that the time range of numerical\nmeasurements such as presented in figure 4 is limited by the finite precision of the\nrandom number generator used to draw the transition probabilities \u001akand sets an\neffective maximal scale of free flights, kmax. This observation is analogous to the effect\nof machine-dependent limitations recently reported in reference [27] and is particularly\nrelevanttotheregimeofweaksuperdiffusion. Theeffectivemaximalscale kmaxinducesa\nsaturation of the logarithmic growth of the second moment for times larger than kmax\u001cB.\nThe process would thus become diffusive for times sufficiently large. Although this effect\ncan be pushed upward to larger times by increasing the precision of the random number\ngenerator, it cannot be eliminated altogether.\n6. Conclusions\nThe inclusion of exponentially-distributed waiting times separating the successive jump\nevents of Lévy walkers leads to a natural distinction between propagating and scatteringLévy walks on lattices as multi-state processes 22\n������������������������������������\n�〈�〉�/�\n�����������������������������(�)\n(a)\u000b= 5=2,\u000e= 10\u00001(\u000f'3:4\u000210\u00002)\n������������������������������������������\n�〈�〉�/�\n�����������������������������(�) (b)\u000b= 5=2,\u000e= 10\u00002(\u000f'3:25\u000210\u00003)\n������������������������������\n�〈�〉�/�\n�����������������������������(�)\n(c)\u000b= 2,\u000e= 10\u00001(\u000f'2:63\u000210\u00002)\n������������������������������������\n�〈�〉�/�\n�����������������������������(�) (d)\u000b= 2,\u000e= 10\u00002(\u000f'2:51\u000210\u00003)\n������������������������\n�〈�〉�/�\n�����������������������������(�)\n(e)\u000b= 3=2,\u000e= 10\u00001(\u000f'1:54\u000210\u00002)\n��������������������������������\n�〈�〉�/�\n�����������������������������(�) (f)\u000b= 3=2,\u000e= 10\u00002(\u000f'1:47\u000210\u00003)\nFigure 4: Numerical computations (cyan curves) of the normally rescaled mean squared\ndisplacement in the three different regimes of the scaling parameter \u000bfor the all-\nscattering initial condition. The scattering parameter takes values in or near the\nperturbative regime: (a), (c), (e) \u000e= 10\u00001, and (b), (d), (f) \u000e= 10\u00002. The data\nare compared with the asymptotic solution (57) (black dotted curves) and the O(\u000f)\nsolution (53) (dashed curves). Insets: time evolution of the fraction of particles in the\nscattering state with the stationary value (18) (dotted lines) and the the O(\u000f)solution\n(29) (dashed curves).Lévy walks on lattices as multi-state processes 23\nstates, whose respective concentrations evolve in time according to a set of generalized\nmaster equations. In particular, the fraction of walkers in a scattering state obeys a\nlinear delay differential equation with a countable hierarchy of delays whose analytic\nsolutions were obtained.\nAsopposedtoclassicmethodsbasedontheFourier–Laplacetransformofanintegral\nkernel and the use of Tauberian theorems [11,13], the approach presented in this paper\nis based solely on the solutions of such delay differential equations. Our method thus\nyields a simple expression of the mean squared displacement of Lévy walkers in terms\nof the distribution of free paths and the time integral of the fraction of particles in the\nscattering state.\nBothexactandasymptoticexpressionsofthemeansquareddisplacementofwalkers\nwere obtained in regimes ranging from normal to superdiffusive subballistic transport.\nIn these regimes, the mechanism through which a walker can change directions between\nsuccessive propagation phases plays an important role in determining the values of\nthe transport coefficients, whether normal or anomalous. The transition through a\nscattering phase brings about a description in terms of two parameters, the first\nspecifying the typical timescale of scattering events as opposed to the timescale of\npropagationacrossanelementarycell, thesecondweightingtheprobabilityoftransitions\nbetween scattering and propagating states.\nRelying on the stationary fractions of particles in scattering and propagating states,\na detailed derivation of the transport coefficients, similar to that reported in [16], was\ngiven, exhibiting the precise effect of the two scattering parameters on these coefficients,\nas well as the influence of the initial distribution of walkers [25]. Our formalism also\nyieldstheexacttimeevolutionofthemeansquareddisplacement,whichwasinvestigated\nwhen particles are initially found in a scattering state.\nThe comparison between the exact and asymptotic solutions is generally not\nimmediate, but is fairly straightforward in perturbative regimes such that the likelihood\nof a transition from scattering to propagating states is small. A case in point – though\nby far not the only application of the present results – is given by the infinite-horizon\nLorentz gas [28], for which the scaling parameter of the distribution of free paths takes\nonthemarginalvalue \u000b= 2. Asonemightexpect, measuringthelogarithmicdivergence\nin time of the rescaled mean squared displacement is typically hindered by dominant\nconstant terms. This is particularly so in the regime of narrow corridors, which yields\na stochastic description in terms of multistate Lévy walks such that the scattering\nparameter is small, \u000f\u001c1[23]. The parameter \u000finduces a further separation between\nthe time scales of the scattering and propagating phases, \u001cR\u001d\u001cB. To accurately\nmodel the scattering phase is therefore of paramount importance to understanding the\ndynamics of the infinite-horizon Lorentz gas in terms of a Lévy walk.\nTheresultsobtainedinthispaperprovidetheframeworktotransposingsuchresults\nto a range of applications exhibiting other regimes of transport, such as have been\nstudied in the context of optimal intermittent search strategies [22]. Our theoretical\nframework brings about new perspectives to extend the study of intermittent walks toLévy walks on lattices as multi-state processes 24\npower-law distributed ballistic phases, which may be relevant to the increasing body of\nliterature on optimal search strategies [8,29,30].\nAcknowledgments\nThis work was partially supported by FIRB-Project No. RBFR08UH60 (MIUR, Italy),\nby SEP-CONACYT Grant No. CB-101246 and DGAPA-UNAM PAPIIT Grant No.\nIN117214 (Mexico), and by FRFC convention 2,4592.11 (Belgium). T.G. is financially\nsupported by the (Belgian) FRS-FNRS.\nAppendix A. Time-dependent fraction of particles in the scattering state\nTo derive equation (23), we note that a(1jk)=\u001akand, for 2\u0014n\u0014k,\na(njk)=1\nnk\u0000n+1X\nj=1\u001aja(n\u00001jk\u0000j): (A.1)\nBy recursive application of this formula, we obtain\na(njk)=1\nnk\u0000n+1X\nj1=1\u001aj1a(n\u00001jk\u0000j1);\n=1\nn(n\u00001)k\u0000n+1X\nj1=1k\u0000j1\u0000n+2X\nj2=1\u001aj1\u001aj2a(n\u00002jk\u0000j1\u0000j2);\n=1\nn!k\u0000n+1X\nj1=1\u0001\u0001\u0001k\u00001\u0000j1\u0000\u0001\u0001\u0001\u0000jn\u00002X\njn\u00001=1\u001aji:::\u001ajn\u00001\u001ak\u0000j1\u0000\u0001\u0001\u0001\u0000jn\u00001; (A.2)\nwhich is equivalent to equation (24).\nTaking the derivative of equation (23), we verify equation (16):\n_S0(t) =\u0000\u000f\u001c\u00001\nRS0(t) +bt=\u001cBcX\nk=1e\u0000\u000f(t\u0000k\u001cB)=\u001cRkX\nn=1na(njk)\u001c\u0000n\nR(t\u0000k\u001cB)n\u00001;\n=\u001c\u00001\nRbt=\u001cBcX\nk=1\u001akS0(t\u0000k\u001cB)\u0000\u000f\u001c\u00001\nRS0(t)\n+bt=\u001cBcX\nk=1e\u0000\u000f(t\u0000k\u001cB)=\u001cR\"kX\nn=2na(njk)\u001c\u0000n\nR(t\u0000k\u001cB)n\u00001\n\u0000\u001akbt=\u001cBc\u0000kX\nj=1e\u000fj\u001cB=\u001cRjX\nn=1a(njj)\u001c\u0000n\u00001\nR(t\u0000k\u001cB\u0000j\u001cB)n#\n: (A.3)Lévy walks on lattices as multi-state processes 25\nWe have to show that the last term on the RHS vanishes, which amounts to proving the\nidentity\nbt=\u001cBcX\nk=1e\u000fk\u001cB=\u001cR\"kX\nn=2na(njk)\u001c\u0000n+1\nR(t\u0000k\u001cB)n\u00001\n\u0000\u001akbt=\u001cBc\u0000k\u001cBX\nj=1e\u000fj\u001cB=\u001cRjX\nn=1a(njj)\u001c\u0000n\nR(t\u0000k\u001cB\u0000j\u001cB)n#\n= 0;(A.4)\nor, after rearrangement,\nbt=\u001cBcX\nk=2e\u000fk\u001cB=\u001cRk\u00001X\nn=1(n+ 1)a(n+1jk)\u001c\u0000n\nR(t\u0000k\u001cB)n\n=bt=\u001cBcX\nk=1\u001akbt=\u001cBc\u0000kX\nj=1e\u000f(k+j)\u001cB=\u001cRjX\nn=1a(njj)\u001c\u0000n\nR(t\u0000k\u001cB\u0000j\u001cB)n:(A.5)\nThe second term in this expression, by successively changing the index jtoj\u0000k, then\nswapping the sums over kandjand exchanging the indices jandk, transforms to\nbt=\u001cBcX\nk=1\u001akbt=\u001cBc\u0000kX\nj=1e\u000f(k+j)\u001cB=\u001cRjX\nn=1a(njj)\u001c\u0000n\nR(t\u0000k\u001cB\u0000j\u001cB)n\n=bt=\u001cBcX\nk=1\u001akbt=\u001cBcX\nj=k+1e\u000fj\u001cB=\u001cRj\u0000kX\nn=1a(njj\u0000k)\u001c\u0000n\nR(t\u0000j\u001cB)n;\n=bt=\u001cBcX\nk=2e\u000fk\u001cB=\u001cRk\u00001X\nj=1\u001ajk\u0000jX\nn=1a(njk\u0000j)\u001c\u0000n\nR(t\u0000k\u001cB)n: (A.6)\nNow swapping the sums over jandnin the last line and plugging this expression back\ninto equation (A.5), we obtain, after factorization of the common factors, the condition\nbt=\u001cBcX\nk=2e\u000fk\u001cB=\u001cRk\u00001X\nn=1\"\n(n+ 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   {        "title": "2201.11498v3.Effect_of_vertex_corrections_on_the_enhancement_of_Gilbert_damping_in_spin_pumping_into_a_two_dimensional_electron_gas.pdf",        "content": "E\u000bect of vertex corrections on the enhancement of Gilbert damping in spin pumping\ninto a two-dimensional electron gas\nM. Yama,1M. Matsuo,2;3;4;5T. Kato1,\n1Institute for Solid State Physics,\nThe University of Tokyo, Kashiwa, Japan\n2Kavli Institute for Theoretical Sciences,\nUniversity of Chinese Academy of Sciences, Beijing, China\n3CAS Center for Excellence in Topological Quantum Computation,\nUniversity of Chinese Academy of Sciences, Beijing, China\n4Advanced Science Research Center,\nJapan Atomic Energy Agency, Tokai, Japan\n5RIKEN Center for Emergent Matter Science (CEMS),\nWako, Saitama, Japan\n(Dated: May 15, 2023)\nWe theoretically consider the e\u000bect of vertex correction on spin pumping from a ferromagnetic\ninsulator (FI) into a two-dimensional electron gas (2DEG) in which the Rashba and Dresselhaus\nspin-orbit interactions coexist. The Gilbert damping in the FI is enhanced by elastic spin-\ripping\nor magnon absorption. We show that the Gilbert damping due to elastic spin-\ripping is strongly\nenhanced by the vertex correction when the ratio of the two spin-orbit interactions is near a special\nvalue at which the spin relaxation time diverges while that due to magnon absorption shows only\nsmall modi\fcation. We also show that the shift in the resonant frequency due to elastic spin-\ripping\nis strongly enhanced in a similar way as the Gilbert damping.\nI. INTRODUCTION\nIn the \feld of spintronics1,2, spin pumping has long\nbeen used as a method of injecting spins into various\nmaterials3{5. Spin pumping was \frst employed to inject\nspins from a ferromagnetic metal into an adjacent normal\nmetal (NM)6{9. Subsequently, it was used on ferromag-\nnetic insulator (FI)/NM junctions10. Because spin injec-\ntion is generally related to the loss of the magnetization\nin ferromagnets, it a\u000bects the Gilbert damping measured\nin ferromagnetic resonance (FMR) experiments11. When\nwe employ spin injection from the FI, the modulation of\nthe Gilbert damping re\rects the properties of the spin ex-\ncitation in the adjacent materials, such as magnetic thin\n\flms12, magnetic impurities on metal surfaces13, and su-\nperconductors14{17. This is in clear contrast with the\nGilbert damping of a bulk FI, which re\rects properties\nof electrons and phonons18{20.\nAn attractive strategy is to combine spin pumping\nwith spin-related transport phenomena in semiconduc-\ntor microstructures1,21. A two-dimensional electron gas\n(2DEG) in a semiconductor heterostructure is an easily\ncontrolled physical system that has been used in spin-\ntronics devices22{25. A 2DEG system has two types of\nspin-orbit interaction, i.e., Rashba26,27and Dresselhaus\nspin-orbit interactions28,29.\nIn our previous work30, we theoretically studied spin\npumping into a 2DEG in semiconductor heterostructures\nwith both Rashba and Dresselhaus spin-orbit interac-\ntions, which can be regarded as a prototype for a 2DEG\nwith a complex spin-texture near the Fermi surface [see\nFig. 1 (a)]. In that study, we formulated the modu-\nlation of the Gilbert damping in the FI by using the\n2DEG\nmicrowave\nFI\n(a) (b)\ntotFIG. 1. (a) Schematic picture of junction composed of a fer-\nromagnetic insulator (FI) and a two-dimensional electron gas\n(2DEG) realized in a semiconductor heterostructure. Stotin-\ndicates the total spin of the FI. We consider a uniform spin\nprecession of the FI induced by microwave irradiation. (b)\nLaboratory coordinates ( x;y;z ) and the magnetization-\fxed\ncoordinates ( x0;y0;z0). The red arrow indicates the expec-\ntation value of the spontaneous spin polarization of the FI,\nhSi.\nsecond-order perturbation with respect to the interfa-\ncial coupling15,31{35and related it to the dynamic spin\nsusceptibility of the 2DEG. We further calculated the\nspin susceptibility and obtained characteristic features\nof the Gilbert damping modulation. This modulation\ncontains two contributions: elastic spin-\ripping, which\ndominates at low resonant frequencies, and magnon ab-\nsorption, which dominates at high resonant frequencies.\nIn addition, we clari\fed that these contributions have\ndi\u000berent dependence on the in-plane azimuth angle \u0012of\nthe ordered spin in the FI [see Fig. 1 (b)].\nWhen the Rashba and Dresselhaus spin-orbit interac-\ntions have almost equal magnitudes, spin relaxation byarXiv:2201.11498v3  [cond-mat.mes-hall]  12 May 20232\nnonmagnetic impurity scattering is strongly suppressed\nbecause the direction of the e\u000bective Zeeman \feld gen-\nerated by the spin-orbit interactions is unchanged along\nthe Fermi surface. Due to this substantial suppression of\nspin relaxation, there emerge characteristic physical phe-\nnomena such as the persistent spin helix state36{39. In\ngeneral, the vertex corrections have to be taken into ac-\ncount to treat various conservation laws, i.e., the charge,\nspin, momentum, and energy conservation laws in cal-\nculation of the response functions40{43. Therefore, for\nbetter description of realistic systems, we need to con-\nsider vertex correction, which captures e\u000bect of impurity\nmore accurately by re\recting conservation laws. How-\never, the vertex corrections were neglected in our pre-\nvious work30. This means that our previous calculation\nshould fail when the Rashba and Dresselhaus spin-orbit\ninteractions compete.\nIn this study, we consider the same setting, i.e., a\njunction composed of an FI and a 2DEG as shown in\nFig. 1 (a), and examine e\u000bect of the spin conservation law\nby taking the vertex correction into account. We theo-\nretically calculate the modulation of the Gilbert damping\nand the shift in the FMR frequency by solving the Bethe-\nSalpeter equation within the ladder approximation. We\nshow that the vertex correction substantially changes the\nresults, in particular, when the strengths of the Rashba-\nand Dresselhaus-type spin-orbit interactions are chosen\nto be almost equal but slightly di\u000berent; Speci\fcally,\nboth the Gilbert damping and the FMR frequency shift\nare largely enhanced at low resonant frequencies re\rect-\ning strong suppression of spin relaxation. This remark-\nable feature should be able to be observed experimen-\ntally. In contrast, the vertex correction changes their\nmagnitude only slightly at high resonant frequencies.\nBefore describing our calculation, we brie\ry comment\non study of the vertex corrections in a di\u000berent context.\nIn early studies of the spin Hall e\u000bect, there was a de-\nbate on the existence of intrinsic spin Hall e\u000bect44{46. By\nconsidering the vertex corrections, the spin Hall conduc-\ntivity, which is calculated from the correlation function\nbetween the current and spin current, vanishes in the\npresence of short-range disorder for simple models even if\nits strength is in\fnitesimally small47{49. This seemingly\ncontradictory result stimulated theoretical researches on\nrealistic modi\fed models50,51as well as de\fnition of the\nspin current52{56. However, we stress that the vertex\ncorrections for the dynamic spin susceptibility, which is\ncalculated from the spin-spin correlation function, have\nno such subtle problem57because it does not include the\nspin current.\nThe rest of this work is organized as follows. In Sec. II,\nwe brie\ry summarize our model of the FI/2DEG junc-\ntion and describe a general formulation for the magnon\nself-energy following Ref. 30. In Sec. III, we formulate\nthe vertex correction that corresponds to the self-energy\nin the Born approximation. We show the modulation of\nthe Gilbert damping and the shift in the FMR frequency\nin Secs. IV and V, respectively, and discuss the e\u000bect\nky\nkx\n(a) (b)\nFIG. 2. Schematic picture of the spin-splitting energy bands\nof 2DEG for (a) \f=\u000b= 0 and (b) \f=\u000b= 1. The red and blue\narrows represent spin polarization of each band. In the case\nof (b), the spin component in the direction of the azimuth\nangle 3\u0019=4 is conserved.\nof the vertex correction in detail. Finally, we summa-\nrize our results in Sec. VI. The six Appendices detail the\ncalculation in Sec. III.\nII. FORMULATION\nHere, we describe a model for the FI/2DEG junction\nshown in Fig. 1 (a) and formulate the spin relaxation\nrate in an FMR experiment. Because we have already\ngiven a detailed formulation on this model in our previous\npaper30, we will brie\ry summarize it here.\nA. Two-dimensional electron gas\nWe consider a 2DEG whose Hamiltonian is given as\nHNM=Hkin+Himp, whereHkinandHimpdescribe the\nkinetic energy and the impurity, respectively. The kinetic\nenergy is given as\nHkin=X\nk(cy\nk\"cy\nk#)^hk\u0010ck\"\nck#\u0011\n; (1)\n^hk=\u0018k^I\u0000he\u000b(k)\u0001\u001b; (2)\nwhereck\u001bis the annihilation operator of conduction elec-\ntrons with wave number k= (kx;ky) andzcomponent\nof the spin, \u001b(=\";#),^Iis a 2\u00022 identity matrix, \u001ba\n(a=x;y;z ) are the Pauli matrices, \u0018k=~2k2=2m\u0003\u0000\u0016\nis the kinetic energy measured from the chemical poten-\ntial, andm\u0003is an e\u000bective mass. Hereafter, we assume\nthat the Fermi energy is much larger than the other en-\nergy scales such as the spin-orbit interactions, the tem-\nperature, and the ferromagnetic resonance energy. Then,\nthe low-energy part of the spin susceptibility depends on\nthe chemical potential \u0016and the e\u000bective mass m\u0003only\nthrough the density of states at the Fermi energy, D(\u000fF).\nThe spin-orbit interaction is described by the e\u000bective3\nZeeman \feld,\nhe\u000b(k) =jkj(\u0000\u000bsin'\u0000\fcos';\u000bcos'+\fsin';0)\n'kF(\u0000\u000bsin'\u0000\fcos';\u000bcos'+\fsin';0); (3)\nwhere\u000band\frespectively denote the amplitudes of the\nRashba- and Dresselhaus-type spin-orbit interactions and\nthe electron wave number is expressed by polar coordi-\nnates as (kx;ky) = (jkjcos';jkjsin'). In the second\nequation of Eq. (3), we have approximated jkjwith the\nFermi wave number kFassuming that the spin-orbit in-\nteraction energies, kF\u000bandkF\f, are much smaller than\nthe Fermi energy58. When only the Rashba spin-orbit in-\nteraction exists ( \f= 0), the energy band is spin-splitted\nas shown in Fig. 2 (a). The spin polarization of each band\ndepends on the azimuth angle 'because it is determined\nby the e\u000bective Zeeman \feld he\u000bwhich is a function of\n'as seen in Eq. (3). In the special case of \f=\u000b= 1, the\nspin polarization always becomes parallel to the direction\nof the azimuth angle 3 \u0019=4 in thexyplane as shown in\nFig. 2 (b). Then, the spin component in this direction is\nconserved. This observation indicates that e\u000bect of the\nspin conservation may become important when the two\nspin-orbit interactions compete ( \u000b'\f).\nThe Hamiltonian of the impurity potential is given as\nHimp=uX\ni2impX\n\u001b\ty\n\u001b(ri)\t\u001b(ri); (4)\nwhere \t\u001b(r) =A\u00001=2P\nkck\u001beik\u0001r,Ais the area of the\njunction,uis the strength of the impurity potential, and\nriis the position of the impurity site.\nThe \fnite-temperature Green's function for the con-\nduction electrons is de\fned by a 2 \u00022 matrix ^g(k;i!m)\nwhose elements are\ng\u001b\u001b0(k;i!m) =Z~\f\n0d\u001cei!m\u001cg\u001b\u001b0(k;\u001c); (5)\ng\u001b\u001b0(k;\u001c) =\u0000~\u00001hck\u001b(\u001c)cy\nk\u001b0i; (6)\nwhereck\u001b(\u001c) =eHNM\u001c=~ck\u001be\u0000HNM\u001c=~,HNM=Hkin+\nHimp,!m=\u0019(2m+ 1)=~\fis the fermionic Matsubara\nfrequency, and \fis the inverse temperature. By em-\nploying the Born approximation, the \fnite-temperature\nGreen's function can be expressed as\n^g(k;i!m) =(i~!m\u0000\u0018k+i\u0000sgn(!m)=2)^I\u0000he\u000b\u0001\u001bQ\n\u0017=\u0006(i~!m\u0000E\u0017\nk+i\u0000sgn(!m)=2);\n(7)\nwhereE\u0006\nk=\u0018k\u0006jhe\u000b(')jis the spin-dependent electron\ndispersion,\n\u0000 = 2\u0019niu2D(\u000fF) (8)\nis level broadening, and niis the impurity concentration\n(see Appendix A and Ref. 30 for detailed derivation).\nAs already mentioned, the case of \f=\u000b = 1 is special\nbecause the spin component parallel to the direction ofthe azimuth angle 3 \u0019=4 in thexyplane is conserved (see\nFig. 2 (b)). By de\fning the spin component in this di-\nrection as\ns3\u0019=4\ntot\u00111\n2X\nk(cy\nk+ck+\u0000cy\nk\u0000ck\u0000); (9)\n\u0012\nck+\nck\u0000\u0013\n=\u0012\n1=p\n2e\u0000i3\u0019=4=p\n2\n\u0000ei3\u0019=4=p\n2 1=p\n2\u0013\u0012\nck\"\nck#\u0013\n;(10)\nwe can prove [ Hkin+Himp;s3\u0019=4\ntot] = 0. When the value of\n\f=\u000b is slightly shifted from 1, the spin conservation law\nis broken slightly and this leads to a slow spin relaxation.\nAs will be discussed in Secs. IV and V, this slow spin re-\nlaxation, which is a remnant of the spin conservation at\n\f=\u000b = 1, strongly a\u000bects the spin injection from the FI\ninto the 2DEG. To describe this feature, we need to con-\nsider the vertex correction to take the conservation law\ninto account in our calculation as explained in Sec. III.\nB. Ferromagnetic insulator\nWe consider the quantum Heisenberg model for the FI\nand employ the spin-wave approximation assuming that\nthe temperature is much lower than the magnetic tran-\nsition temperature and the magnitude of the localized\nspins,S0, is su\u000eciently large. We write the expectation\nvalue of the localized spins in the FI as hSi, whose direc-\ntion is (cos \u0012;sin\u0012;0) as shown in the Fig. 1 (b). Using\nthe Holstein-Primakov transformation, the Hamiltonian\nin the spin-wave approximation is obtained as\nHFI=X\nk~!kby\nkbk; (11)\nwherebkis the magnon annihilation operator with wave\nnumberk,~!k=Dk2+~\rhdcis the energy dispersion of\na magnon,Dis the spin sti\u000bness, \ris the gyromagnetic\nratio, andhdcis the externally applied DC magnetic \feld.\nWe note that the external DC magnetic \feld controls the\ndirection of the ordered spins. We introduce new coor-\ndinates (x0;y0;z0) \fxed on the ordered spins by rotating\nthe original coordinates ( x;y;z ) as shown in Fig. 1 (b).\nThen, the magnon annihilation operator is related to the\nspin ladder operator by the Holstein-Primakov transfor-\nmation asSx0+\nk\u0011Sy0\nk+iSz0\nk= (2S0)1=2bk. The spin\ncorrelation function is de\fned as\nG0(k;i!n) =Z~\f\n0d\u001cei!n\u001cG0(k;\u001c); (12)\nG0(k;\u001c) =\u00001\n~hSx0+\nk(\u001c)Sx0\u0000\nk(0)i; (13)\nwhere!n= 2n\u0019=~\fis the bosonic Matsubara fre-\nquency. The spin correlation function is calculated from\nthe Hamiltonian (11), as\nG0(k;i!n) =2S0=~\ni!n\u0000!k\u0000\u000bGj!nj; (14)4\nwhere\u000bG>0 is a phenomenological dimensionless pa-\nrameter that describes the strength of the Gilbert damp-\ning in the bulk FI.\nC. E\u000bect of the FI/2DEG interface\nThe coupling between the FI and 2DEG can be ac-\ncounted for by the Hamiltonian,\nHint=X\nk(TkSx0+\nksx0\u0000\nk+T\u0003\nksx0+\nkSx0\u0000\nk); (15)\nwhereTkis an exchange interaction at a clean interface,\nfor which the momentum of spin excitation is conserved.\nThe spin ladder operators for conduction electrons, sx0\u0006\nk,\nare obtained using a coordinate rotation as30\nsx0\u0006\nk=1\n2X\n\u001b;\u001b0X\nk0cy\nk0\u001b(^\u001bx0\u0006)\u001b\u001b0ck0\u0006k\u001b0; (16)\n^\u001bx0\u0006=\u0000sin\u0012\u001bx+ cos\u0012\u001by\u0006i\u001bz; (17)\nwhere ^\u001bx0\u0006\u0011^\u001by0\u0006i^\u001bz0and\n0\n@^\u001bx0\n^\u001by0\n^\u001bz01\nA=0\n@cos\u0012sin\u00120\n\u0000sin\u0012cos\u00120\n0 0 11\nA0\n@\u001bx\n\u001by\n\u001bz1\nA:\nAssuming that the interfacial exchange interaction is\nmuch smaller than the spin-orbit interactions, kF\u000band\nkF\f59,60, we perform a second-order perturbation theory\nwith respect to the interfacial exchange interaction Hint.\nAccordingly, the spin correlation function of the FI is\ncalculated as\nG(k;i!n) =1\n(G0(k;i!n))\u00001\u0000\u0006(k;i!n); (18)\n\u0006(k;i!n) =jTkj2A\u001f(k;i!n); (19)\nwhere \u0006(k;i!n) is the self-energy due to the interfacial\nexchange coupling and \u001f(k;i!n) is the spin susceptibility\nfor conduction electrons per unit area, de\fned as\n\u001f(k;i!n) =Z~\f\n0d\u001cei!n\u001c\u001f(k;\u001c); (20)\n\u001f(k;\u001c) =\u00001\n~Ahsx0+\nk(\u001c)sx0\u0000\nk(0)i; (21)\nwheresx0\u0006\nk(\u001c) =eHNM\u001c=~sx0\u0006\nke\u0000HNM\u001c=~. Within the\nsecond-order perturbation, we only need to calculate the\nspin susceptibility for pure 2DEG without considering\nthe junction because the interfacial coupling is already\ntaken into account in the prefactor of the self-energy in\nEq. (19). The uniform component of the retarded spin\ncorrelation function is obtained by analytic continuation\n(a)(b)\nFIG. 3. Feynman diagrams of (a) the uniform spin susceptibil-\nity and (b) the Bethe-Salpeter equation for the ladder-type\nvertex function derived from the Born approximation. The\ncross with two dashed lines indicates interaction between an\nelectron and an impurity.\ni!n!!+i\u000e, as\nGR(0;!) =2S0=~\n!\u0000(!0+\u000e!0) +i(\u000bG+\u000e\u000bG)!;(22)\n\u000e!0\n!0'2S0jT0j2A\n~!0Re\u001fR(0;!0); (23)\n\u000e\u000bG'\u00002S0jT0j2A\n~!0Im\u001fR(0;!0); (24)\nwhere the superscript Rindicates the retarded compo-\nnent,!0=!k=0(=\rhdc) is the FMR frequency, and\n\u000e!0and\u000e\u000bGare respectively the changes in the FMR\nfrequency and Gilbert damping due to the FI/2DEG in-\nterface. We note that in contrast with the bulk Gilbert\ndamping\u000bG, the increase of the Gilbert damping, \u000e\u000bG,\ncan be related directly to the spin susceptibility of 2DEG\nas shown by Eq. (24). In fact, measurement of \u000e\u000bG\nhas been utilized as a qualitative indicator of spin cur-\nrent through a junction61,62. In Eqs. (23) and (24), we\nmade an approximation by replacing !with the FMR\nfrequency!0by assuming that the FMR peak is su\u000e-\nciently sharp ( \u000bG+\u000e\u000bG\u001c1). Thus, both the FMR\nfrequency shift and the modulation of the Gilbert damp-\ning are determined by the uniform spin susceptibility of\nthe conduction electrons, \u001f(0;!). In what follows, we\ninclude the vertex correction for calculation of \u001f(0;!),\nwhich was not taken into account in our previous work30.\nIII. VERTEX CORRECTION\nWe calculate the spin susceptibility in the ladder\napproximation42,43that obeys the Ward-Takahashi re-\nlation with the self-energy in the Born approximation57.\nThe Feynman diagrams for the corresponding spin sus-\nceptibility and the Bethe-Salpeter equation for the vertex\nfunction are shown in Figs. 3 (a) and 3 (b), respectively.5\nThe spin susceptibility of 2DEG is written as\n\u001f(0;i!n) =1\n4\fAX\nk;i!mTrh\n^g(k;i!m)^\u0000(k;i!m;i!n)\n^g(k;i!m+i!n)^\u001bx0\u0000i\n; (25)\nwhere the vertex function ^\u0000(k;i!m;i!n) is a 2\u00022 matrix\nwhose components are determined by the Bethe-Salpeter\nequation [see Fig. 3 (b)],\n\u0000\u001b0\u001b(k;i!m;i!n)\n= (^\u001bx0+)\u001b0\u001b+u2ni\nAX\nqX\n\u001b1\u001b2g\u001b0\u001b2(q;i!m)\n\u0002\u0000\u001b2\u001b1(q;i!m;i!n)g\u001b1\u001b(q;i!m+i!n):(26)\nSince the right-hand side of this equation is indepen-\ndent ofk, the vertex function can simply be described\nas^\u0000(i!m;i!n). We express the vertex function with the\nPauli matrices as\n^\u0000(i!m;i!n)\u0011E^I+X^\u001bx0+Y^\u001by0+Z^\u001bz0; (27)\nwhereE,X,Y, andZwill be determined self-\nconsistently later. The Green's function for the conduc-\ntion electrons can be rewritten as\n^g(q;i!m) =A^I+B^\u001bx0+C^\u001by0\nD; (28)\nA(i!m) =i~!m\u0000\u0018q+i\u0000\n2sgn(!m); (29)\nB=\u0000he\u000bcos(\u001e\u0000\u0012); (30)\nC=\u0000he\u000bsin(\u001e\u0000\u0012); (31)\nD(i!m) =Y\n\u0017=\u0006[i~!m\u0000E\u0017\nq+i\u0000\n2sgn(!m)]; (32)\nwhere\u001eis the azimuth angle by which the e\u000bective Zee-\nman \feld is written as he\u000b= (he\u000bcos\u001e;he\u000bsin\u001e;0).\nThishe\u000bis written as he\u000b'kFp\n\u000b2+\f2+ 2\u000b\fsin 2'\nusing the Fermi wave number kF. By substituting\nEqs. (27) and (28) into the second term of Eq. (26) and\nby the algebra of Pauli matrices, we obtain\nu2ni\nAX\nq^g(q;i!m)^\u0000(q;i!m;i!n)^g(q;i!m+i!n)\n=E0^I+X0^\u001bx0+Y0^\u001by0+Z0^\u001bz0; (33)\nwhere\n0\nB@E0\nX0\nY0\nZ01\nCA=0\nB@\u00030+ \u00031 0 0 0\n0 \u0003 0+ \u00032 \u00033 0\n0 \u0003 3 \u00030\u0000\u00032 0\n0 0 0 \u0003 0\u0000\u000311\nCA0\nB@E\nX\nY\nZ1\nCA;\n(34)and \u0003j(i!m;i!n) (j= 0;1;2;3) are expressed as\n\u00030(i!m;i!n) =u2ni\nAX\nqAA0\nDD0; (35)\n\u00031(i!m;i!n) =u2ni\nAX\nqh2\ne\u000b\nDD0; (36)\n\u00032(i!m;i!n) =u2ni\nAX\nqh2\ne\u000bcos 2(\u001e\u0000\u0012)\nDD0; (37)\n\u00033(i!m;i!n) =u2ni\nAX\nqh2\ne\u000bsin 2(\u001e\u0000\u0012)\nDD0; (38)\nusing the abbreviated symbols, A=A(i!m),A0=\nA(i!m+i!n),D=D(i!m), andD0=D(i!m+i!n).\nHere, we have used the fact that the contributions of the\n\frst-order terms of BandCbecome zero after replacing\nthe sum with the integral with respect to qand perform-\ning the azimuth integration. We can solve for E,X,\nY, andZby combining Eq. (34) and the Bethe-Salpeter\nequation (26), which we rewrite as\nE^I+X^\u001bx0+Y^\u001by0+Z^\u001bz0\n= ^\u001bx0++E0^I+X0^\u001bx0+Y0^\u001by0+Z0^\u001bz0; (39)\nwith ^\u001bx0+= ^\u001by0+i^\u001bz0. The solution is\nE= 0; (40)\nX=\u00033\n(1\u0000\u00030)2\u0000\u00032\n2\u0000\u00032\n3; (41)\nY=1\u0000\u00030\u0000\u00032\n(1\u0000\u00030)2\u0000\u00032\n2\u0000\u00032\n3; (42)\nZ=i\n1\u0000\u00030+ \u00031: (43)\nBy replacing the sum with an integral as \u0018\u0011\u0018q,\n1\nAX\nq(\u0001\u0001\u0001)'D(\u000fF)Z1\n\u00001d\u0018Z2\u0019\n0d'\n2\u0019(\u0001\u0001\u0001); (44)\nEqs. (35)-(38) can be rewritten as\n\u0003j(i!m;i!n) =\u0012(\u0000!m)\u0012(!m+!n)~\u0003j(i!n); (45)\n~\u0003j(i!n) =i\u0000\n4Z2\u0019\n0d'\n2\u0019\n\u0002X\n\u0017;\u00170=\u0006fj(\u0017;\u00170;')\ni~!n+ (\u0017\u0000\u00170)he\u000b(') +i\u0000;\n(46)\nwhere we have used Eq. (8), \u0012(x) is a step function, and\nf0(\u0017;\u00170;') = 1; (47)\nf1(\u0017;\u00170;') =\u0017\u00170; (48)\nf2(\u0017;\u00170;') =\u0017\u00170cos 2(\u001e(')\u0000\u0012); (49)\nf3(\u0017;\u00170;') =\u0017\u00170sin 2(\u001e(')\u0000\u0012): (50)6\nFor detailed derivation, see Appendix B. Substituting the\nGreen's function and the vertex function into Eq. (25),\nwe obtain\n\u001f(0;i!n) =1\n4\fAX\nk;i!m2\nDD0h\n2BCX\n+ (AA0\u0000B2+C2)Y\u0000i(AA0\u0000B2\u0000C2)Zi\n:(51)\nBy summing over kand!mand by analytical continu-\nation,i!n!!+i\u000e, the retarded spin susceptibility is\nobtained as63\n\u001fR(0;!)\n=D(\u000fF)~!\n2i\u0000\u0014~\u0003R\n0(1\u0000~\u0003R\n0)\u0000~\u0003R\n2(1\u0000~\u0003R\n2) + ( ~\u0003R\n3)2\n(1\u0000~\u0003R\n0)2\u0000(~\u0003R\n2)2\u0000(~\u0003R\n3)2\n+~\u0003R\n0\u0000~\u0003R\n1\n1\u0000~\u0003R\n0+~\u0003R\n1\u0015\n\u0000D(\u000fF); (52)\nwhere\n~\u0003R\nj=~\u0003R\nj(!) =~\u0003j(i!n!!+i\u000e)\n=i\u0000\n4\u00010Z2\u0019\n0d'\n2\u0019\n\u0002X\n\u0017\u00170fj(\u0017;\u00170;')\n~!=\u00010+ (\u0017\u0000\u00170)he\u000b=\u00010+i\u0000=\u00010:(53)\nA detailed derivation is given in Appendix C. Here, we\nhave introduced a unit of energy, \u0001 0=kF\u000b, for the con-\nvenience of making the physical quantities dimensionless.\nUsing Eqs. (23) and (24), we \fnally obtain the shift in\nthe FMR frequency and the modulation of the Gilbert\ndamping as\n\u000e!0\n!0=\u000bG;0ReF(!0); (54)\n\u000e\u000bG=\u0000\u000bG;0ImF(!0); (55)\nF(!) =\u00010\n2\u0019i\u0000\u0014~\u0003R\n0(1\u0000~\u0003R\n0)\u0000~\u0003R\n2(1\u0000~\u0003R\n2) + ( ~\u0003R\n3)2\n(1\u0000~\u0003R\n0)2\u0000(~\u0003R\n2)2\u0000(~\u0003R\n3)2\n+~\u0003R\n0\u0000~\u0003R\n1\n1\u0000~\u0003R\n0+~\u0003R\n1\u0015\n\u0000\u00010\n\u0019~!; (56)\nwhere\u000bG;0= 2\u0019S0jT0j2AD(\u000fF)=\u00010is a dimensionless\nparameter that describes the coupling strength at the\ninterface. This is our main result.\nThe spin susceptibility without the vertex correction\ncan be obtained by taking the \frst-order term with re-spect to ~\u0003R\nj:\n\u001fR(0;!)'~!D(\u000fF)\n2i\u0000\u0002\n2~\u0003R\n0\u0000~\u0003R\n1\u0000~\u0003R\n2\u0003\n\u0000D(\u000fF)\n=~!D(\u000fF)Zd'\n2\u0019h1\n~!+i\u00001\u0000cos2(\u001e(')\u0000\u0012)\n2\n+1\n~!\u00002he\u000b(') +i\u00001 + cos2(\u001e(')\u0000\u0012)\n4\n+1\n~!+ 2he\u000b(') +i\u00001 + cos2(\u001e(')\u0000\u0012)\n4i\n\u0000D(\u000fF):\n(57)\nThe imaginary part of \u001fR(0;!) reproduces the result of\nRef. 30. Using this expression, the shift in the FMR\nfrequency and the modulation of the Gilbert damping\nwithout the vertex correction are obtained as\n\u000e!nv\n0\n!0=\u000bG;0ReFnv(!0); (58)\n\u000e\u000bnv\nG=\u0000\u000bG;0ImFnv(!0); (59)\nFnv(!) =\u00010\n2\u0019i\u0000\u0014\n2~\u0003R\n0\u0000~\u0003R\n1\u0000~\u0003R\n2\u0015\n\u0000\u00010\n\u0019~!; (60)\nIV. MODULATION OF THE GILBERT\nDAMPING\nFirst, we show the result for the modulation of the\nGilbert damping, \u000e\u000bG, for\f=\u000b = 0, 1, and 3 and dis-\ncuss the e\u000bect of the vertex correction by comparing it\nwith the result without the vertex correction in Sec. IV A.\nNext, we discuss the strong enhancement of the Gilbert\ndamping near \f=\u000b= 1 in Sec. IV B.\nA. E\u000bect of vertex corrections\nFirst, let us discuss the case of \f=\u000b = 0, i.e., the\ncase when only the Rashba spin-orbit interaction exists64.\nFigure 4 (a) shows the e\u000bective Zeeman \feld he\u000balong\nthe Fermi surface. Figures 4 (b) and 4 (c) show the\nmodulations of the Gilbert damping without and with\nthe vertex correction. The horizontal axes of Figs. 4 (b)\nand 4 (c) denote the resonant frequency !0=\rhdcin\nthe FMR experiment. Note that the modulation of the\nGilbert damping, \u000e\u000bG, is independent of \u0012, i.e., the az-\nimuth angle ofhSi. The four curves in Figs. 4 (b) and\n4 (c) correspond to \u0000 =\u00010= 0:1, 0:2, 0:5, and 1:065. We\n\fnd that these two graphs have a common qualitative\nfeature; the modulation of the Gilbert damping has two\npeaks at!0= 0 and!0= 2\u0001 0and their widths become\nlarger as \u0000 increases. The peak at !0= 0 corresponds\nto elastic spin-\ripping of conduction electrons induced\nby the transverse magnetic \feld via the exchange bias\nof the FI, while the peak at ~!0= 2\u0001 0is induced by\nspin excitation of conduction electrons due to magnon7\nWithout vertex corrections\nWithout vertex corrections\nWithout vertex corrections With vertex correctionsWith vertex correctionsWith vertex corrections\n\u0001\nWithout vertex corrections With vertex corrections With vertex corrections\nFIG. 4. (Left panels) E\u000bective Zeeman \feld he\u000bon the Fermi surface. (Middle panels) Modulation of the Gilbert damping,\n\u000e\u000bnv\nG, without vertex correction. (Right panels) Modulation of the Gilbert damping with vertex correction, \u000e\u000bG. In the middle\nand right panels, the modulation of the Gilbert damping is plotted as a function of the FMR frequency, !0=\rhdc. The\nspin-orbit interactions are as follows. (a), (b), (c): \f=\u000b= 0. (d), (e), (f): \f=\u000b= 1. (g), (h), (i): \f=\u000b= 3. We note that (b),\n(e), (h) are essentially the same result as Ref. 30.\nabsorption30. In the case of \f=\u000b = 0, the vertex cor-\nrection changes the modulation of the Gilbert damping\nmoderately [compare Figs. 4 (c) with 4 (b)]. The widths\nof the two peaks at !0= 0 and!0= 2\u0001 0become nar-\nrower when the vertex correction is taken into account\n(see Appendix D for the analytic expressions).\nThe case of \f=\u000b = 1 is special because the e\u000bective\nZeeman \feld he\u000balways points in the direction of ( \u00001;1)\nor (1;\u00001), as shown in Fig. 4 (d). The amplitude of\nhe\u000bdepends on the angle of the wave number of the\nconduction electrons, ',\nhe\u000b(') = 2\u0001 0jsin('+\u0019=4)j; (61)\nand varies in the range of 0 \u00142he\u000b\u00144\u00010. Figures 4 (e)\nand 4 (f) show the modulation of the Gilbert dampingwithout and with the vertex correction for \u0000 =\u00010= 0:5.\nThe \fve curves correspond to \fve di\u000berent angles of hSi,\n\u0012=\u0000\u0019=4;\u0000\u0019=8;0;\u0019=8, and\u0019=4. The most remarkable\nfeature revealed by comparing Figs. 4 (f) with 4 (e) is that\nthe peak at !0= 0 disappears if the vertex correction\nis taken into account (see Appendix E for the analytic\nexpressions). In the subsequent section, we will show\nthat\u000e\u000bG(!0) has a\u000e-function-like singularity at !0= 0\nfor\f=\u000b = 1 due to the spin conservation law along the\ndirection ofhe\u000b.\nIn the case of \f=\u000b = 3, the direction of the e\u000bec-\ntive Zeeman \feld he\u000bvaries along the Fermi surface\n[Fig. 4 (g)]. Figures 4 (h) and 4 (i) show the modula-\ntion of the Gilbert damping without and with the ver-\ntex correction for \u0000 =\u00010= 0:5. For\f=\u000b = 3, a peak8\nWithout vertex corrections\nWithout vertex corrections\nWithout vertex corrections With vertex correctionsWith vertex correctionsWith vertex corrections\n\u0001\nWithout vertex corrections With vertex corrections With vertex corrections\nFIG. 5. Modulation of the Gilbert damping calculated for \f=\u000b= 1:1 (a) without the vertex correction and (b) with the vertex\ncorrection. The horizontal axis is the FMR frequency !0and the \fve curves correspond to \fve di\u000berent angles of hSi, i.e.,\n\u0012=\u0000\u0019=4;\u0000\u0019=8;0;\u0019=8, and\u0019=4. (c) Enlarged plot of the modulations of the Gilbert damping as a function of the FMR\nfrequency!0. The angle ofhSiis \fxed as\u0012=\u0019=4 and the three curves correspond to \f=\u000b = 1:03, 1:05, and 1:1. In all the\nplots, we have chosen \u0000 =\u00010= 0:5.\nat!0= 0 appears even when the vertex correction is\ntaken into account. The broad structure in the range of\n4\u00010\u0014~!0\u00148\u00010is caused by the magnon absorption\nprocess where its range re\rects the distribution of the\nspin-splitting energy 2 he\u000balong the Fermi surface. By\ncomparing Figs. 4 (h) and 4 (i), we \fnd that the vertex\ncorrection changes the result only moderately as in the\ncase of\f=\u000b = 0; the peak structure at !0= 0 becomes\nsharper when the vertex correction is taken into account\nwhile the broad structure is slightly enhanced.\nB. Strong enhancement of the Gilbert damping\nHere, we examine the strong enhancement of the\nGilbert damping for \f=\u000b'1. As explained in Sec. II A,\nthe spin component in the direction of the azimuth angle\n3\u0019=4 in thexyplane is exactly conserved at \f=\u000b = 1\n[see also Fig. 4 (d)]. When the value of \f=\u000b is shifted\nslightly from 1, the spin conservation law is broken but\nthe spin relaxation becomes remarkably slow. To see this\ne\u000bect, we show the modulation of the Gilbert damping\nwithout and with the vertex correction for \f=\u000b = 1:1\nin Figs. 5 (a) and 5 (b), respectively. The \fve curves\ncorrespond to \fve di\u000berent azimuth angles of hSi, and\nthe energy broadening is set as \u0000 =\u00010= 0:5. Figs. 5 (a)\nand 5 (b) indicate that the Gilbert damping is strongly\nenhanced at !0= 0 only when the vertex correction is\ntaken into account. This is the main result of our work.\nFigure 5 (c) plots the modulation of the Gilbert damp-\ning with the vertex correction for \u0000 =\u00010= 0:5 and\n\u0012=\u0019=4, the latter of which corresponds to the case of\nthe strongest enhancement at !0= 0. The three curves\ncorrespond to \f=\u000b= 1:03, 1:05, and 1:1. As the ratio of\n\f=\u000bapproaches 1, the peak height at !0= 0 gets larger.\nWithout vertex corrections\nWithout vertex corrections\nWithout vertex corrections With vertex correctionsWith vertex correctionsWith vertex corrections\nWithout vertex corrections With vertex corrections With vertex corrections\n\u0001\n\u0001\n\u0001\n\u0001\n\u0001\n\u0001FIG. 6. Modulation of the Gilbert damping as a function\nof\f=\u000b. The \fve curves correspond to ~!0=\u00010= 0;0:005,\n0:01, 0:02, and 0:05. We have taken the vertex correction into\naccount and have chosen \u0000 =\u00010= 0:5. The inset illustrates\nmaximum values of the modulation of the Gilbert damping,\n\u000e\u000bG;max, in varying \f=\u000b for a \fxed value of ~!0=\u00010.\nFor\f=\u000b'1,\u000e\u000bGis calculated approximately as\n\u000e\u000bG\n\u000bG;0'\u00010\n2\u0019\u0000s\n(~!0)2+ \u00002ssin2\u0010\n\u0012+\u0019\n4\u0011\n; (62)\n\u0000s\u00112\n\u0000Z2\u0019\n0d'\n2\u0019(hx+hy)2\n1 + (2he\u000b=\u0000)2; (63)\nwhere \u0000sgives the peak width in Figs. 5 (b) and 5 (c) (see\nAppendix F for a detailed derivation). For \f=\u000b= 1 +\u000e\n(\u000e\u001c1), \u0000sis proportional to \u000e2and approaches zero in\nthe limit of \u000e!0. This indicates that \u0000 scorresponds\nto the spin relaxation rate due to a small breakdown of\nthe spin conservation law away from the special point of\n\f=\u000b = 1. Note that the peak height of \u000e\u000bGat!0= 0\ndiverges at \f=\u000b = 1. This indicates that for \f=\u000b =\n1,\u000e\u000bG(!0) has a\u000e-function-like singularity at !0= 0,\nwhich is not drawn in Fig. 4 (f).9\nFIG. 7. (Upper panels) Modulations of the Gilbert damping, \u000e\u000bG=\u000bG;0for (a)\f=\u000b= 0, (b)\f=\u000b= 1, and (c) \f=\u000b= 3. (Lower\npanels) Shifts in the FMR frequency, \u000e!0=(\u000bG;0!0), for (d)\f=\u000b = 0, (e)\f=\u000b = 1, and (f) \f=\u000b = 3. The horizontal axes are\nthe FMR frequency, !0=\rhdc, while the vertical axes show the azimuth angle of the spontaneous spin polarization, \u0012, in the\nFI. In all the plots, we have considered vertex corrections and have chosen \u0000 =\u00010= 0:5. In (a), (c), and (e) there are regions\nin which the values exceed the upper limits of the color bar located in the right side of each plot; the maximum value is about\n0:45 in (a), 0 :65 in (c), and about 10 in (e) (see also Fig. 8). In addition, (b) cannot express a \u000e-function-like singularity at\n!0= 0 (see the main text).\nFigure 6 plots the modulation of the Gilbert damping\nfor \u0000=\u00010= 0:5 and\u0012=\u0019=4 as a function of \f=\u000b. The\n\fve curves correspond to ~!0=\u00010= 0;0:005;0:01;0:02,\nand 0:05, respectively. This \fgure indicates that when\nwe \fx the resonant frequency !0and vary the ratio of\n\f=\u000b, the Gilbert damping is strongly enhanced when \f=\u000b\nis slightly smaller or larger than 1. We expect that this\nenhancement of the Gilbert damping is strong enough to\nbe observed experimentally. We note that \u000e\u000bG=\u000bG;0ap-\nproaches 0:378 (0:318) for\f=\u000b!0 (\f=\u000b!1 ). The\ninset in Fig. 6 plots maximum values of \u000e\u000bG=\u000bG;0when\n\f=\u000bis varied for a \fxed value of ~!0=\u00010. In other words,\nthe vertical axis of the inset corresponds to the peak\nheight in the main panel for each value of ~!0=\u00010. We\n\fnd that the maximum value of \u000e\u000bG=\u000bG;0diverges as !0\napproaches zero.\nV. SHIFT IN THE FMR FREQUENCY\nNext, we discuss the shift in the FMR frequency when\nthe vertex correction is taken into account. The den-\nsity plots in Figs. 7 (a), 7 (b), and 7 (c) for \f=\u000b= 0, 1,\nand 3 summarize the modulation of the Gilbert damping,\n\u000e\u000bG. These plots have the same features as in Figs. 4 (c),\n4 (f), and 4 (i). Figures. 7 (d), 7 (e), and 7 (f) plot the\nshift in the FMR frequency \u000e!0=!0with density plots\nfor\f=\u000b = 0, 1, and 3. By comparing Figs. 7 (a), 7 (b),\nWithout vertex corrections\nWithout vertex corrections\nWithout vertex corrections With vertex correctionsWith vertex correctionsWith vertex corrections\nWithout vertex corrections With vertex corrections With vertex corrections\n\u0001\n\u0001\n\u0001\n\u0001\n\u0001\n\u0001FIG. 8. Shift in FMR frequency, \u000e!0=(\u000bG;0!0), as a func-\ntion of the resonance frequency !0for\f=\u000b = 1:1. The in-\nset shows the same quantities in the low-frequency range of\n0\u0014~!0=\u00010\u00140:05 with a larger scale on the vertical axis.\nWe have taken the vertex correction into account and have\nchosen \u0000=\u00010= 0:5.\nand 7 (c) with 7 (d), 7 (e), and 7 (f), we \fnd that some\nof the qualitative features of the FMR frequency shift\nare common to those of the modulation of the Gilbert\ndamping,\u000e\u000bG; (i) they depend on \u0012for\f=\u000b > 0, while\nthey do not depend on \u0012for\f=\u000b = 0, (ii) the structure10\nat!0= 0 due to elastic spin-\ripping appears, and (iii)\nthe structure within a \fnite range of frequencies due to\nmagnon absorption appears. We can also see a few dif-\nferences between \u000e\u000bGand\u000e!0=!0. For example, \u000e!0=!0\nhas a dip-and-peak structure at ~!0=\u00010= 2 where\u000e\u000bG\nhas only a peak. Related to this feature, \u000e!0=!0has a\ntail that decays more slowly than that for \u000e\u000bG. The most\nremarkable di\u000berence is that \u000e!0=!0diverges at !0= 0\nfor\f=\u000b = 1 except for \u0012= 3\u0019=4;7\u0019=4, re\recting the \u000e-\nfunction-like singularity of \u000e\u000bGat!0= 0. These features\nare reasonable because \u000e!0=!0and\u000e\u000bG, which are de-\ntermined by the real and imaginary parts of the retarded\nspin susceptibility, are related to each other through the\nKramers-Kronig conversion.\nThe main panel of Fig. 8 shows the frequency shift\n\u000e!0=!0for\f=\u000b = 1:1 as a function of the resonant\nfrequency!0. The \fve curves correspond to \u0012=\n\u0000\u0019=4;\u0000\u0019=8;0;\u0019=8, and\u0019=4. Although the frequency\nshift appears to diverge in the limit of !0!0 in the\nscale of the main panel, it actually grows to a \fnite value\nand then goes to zero as !0approaches zero (see the inset\nof Fig. 8). For \f=\u000b= 1 +\u000e(\u000e\u001c1), the frequency shift\nis calculated approximately as\n\u000e!0\n\u000bG;0!0'\u00010\n2\u0019~!0\n(~!0)2+ \u00002ssin2\u0010\n\u0012+\u0019\n4\u0011\n; (64)\nwhere \u0000sis the spin relaxation rate de\fned in Eq. (63)\n(see Appendix F for the detailed derivation). We expect\nthat this strong enhancement of the frequency shift near\n\f=\u000b= 1 can be observed experimentally.\nVI. SUMMARY\nWe theoretically investigated spin pumping into a two-\ndimensional electron gas (2DEG) with a textured e\u000bec-\ntive Zeeman \feld caused by Rashba- and Dresselhaus-\ntype spin-orbit interactions. We expressed the change\nin the peak position and the linewidth in a ferromag-\nnetic resonance (FMR) experiment that is induced by\nthe 2DEG within a second-order perturbation with re-\nspect to the interfacial exchange coupling by taking the\nvertex correction into account. The FMR frequency\nand linewidth are modulated by elastic spin-\ripping or\nmagnon absorption. We found that, for almost all of the\nparameters, the vertex correction modi\fes the modula-\ntion of the Gilbert damping only moderately and does not\nchange the qualitative features obtained in our previous\npaper30. However, we found that the Gilbert damping at\nlow frequencies, which is caused by elastic spin-\ripping,\nis strongly enhanced when the Rashba- and Dresselhaus-\ntype spin-orbit interactions are chosen to be almost equal\nbut slightly di\u000berent. Even in this situation, the Gilbert\ndamping at high frequencies, which is caused by magnon\nabsorption, shows small modi\fcation. This strong en-\nhancement of the Gilbert damping at low frequencies ap-\npears only when the vertex correction is taken into ac-\ncount and is considered to originate from the slow spinrelaxation related to the spin conservation law that holds\nwhen the two spin-orbit interactions completely match.\nA similar enhancement was found for the frequency shift\nof the FMR due to elastic spin-\ripping. We expect that\nthis remarkable enhancement can be observed experi-\nmentally.\nOur work provides a theoretical foundation for spin\npumping into two-dimensional electrons with a spin-\ntextured Zeeman \feld on the Fermi surface. Although\nwe have treated a speci\fc model for two-dimensional\nelectron systems with both the Rashba and Dresselhaus\nspin-orbit interactions, our formulation and results will\nbe helpful for describing spin pumping into general two-\ndimensional electron systems such as surface/interface\nstates66{68and atomic layer compounds69,70.\nACKNOWLEDGEMENTS\nThe authors thank Y. Suzuki, Y. Kato, and A. Shi-\ntade for helpful discussion. T. K. acknowledges sup-\nport from the Japan Society for the Promotion of Sci-\nence (JSPS KAKENHI Grant No. JP20K03831). M. M.\nis \fnancially supported by a Grant-in-Aid for Scienti\fc\nResearch B (Grants No. JP20H01863, No. JP21H04565,\nand No. JP21H01800) from MEXT, Japan. M. Y. is sup-\nported by JST SPRING (Grant No. JPMJSP2108).\nAppendix A: Calculation of Green's function\nIn our work, Green's function of conduction electrons\nis calculated by taking e\u000bect of impurity scattering into\naccount. In general, the \fnite-temperature Green's func-\ntion ^g(k;i!m) after the impurity average is described\nby the Dyson equation with the impurity self-energy\n^\u0000(k;!m) as\n^g(k;i!m) =1\n^g0(k;i!m)\u00001\u0000^\u0000(k;i!m); (A1)\nwhere ^g0(k;i!m)\u00001is Green's function of electrons in the\nabsence of impurities. In our work, we employ the Born\napproximation in which the self-energy is approximated\nby second-order perturbation with respect to an impurity\npotential. In the Born approximation, the self-energy is\ngiven as\n^\u0000(k;i!m) =niu2Zd2k\n(2\u0019)2^g0(k;i!m); (A2)\nwhereniis the impurity concentration. The correspond-\ning Feynman diagram of the Dyson equation is shown in\nFig. 9. By straightforward calculation, Eq. (7) can be\nderived. For a detailed derivation, see Ref. 30.11\nFIG. 9. The Feynman diagram for Green's function within\nthe Born approximation.\nAppendix B: Derivation of Equations. (45)-(50)\nEqs. (35)-(38) can be rewritten with \u0000 = 2 \u0019niu2D(\u000fF)\nas\n\u00030(i!m;i!n) =i\u0000\n4Z2\u0019\n0d'\n2\u0019X\n\u0017;\u00170I\u0017\u00170; (B1)\n\u00031(i!m;i!n) =i\u0000\n4Z2\u0019\n0d'\n2\u0019X\n\u0017;\u00170\u0017\u00170I\u0017\u00170; (B2)\n\u00032(i!m;i!n) =i\u0000\n4Z2\u0019\n0d'\n2\u0019cos 2('\u0000\u0012)X\n\u0017;\u00170\u0017\u00170I\u0017\u00170;\n(B3)\n\u00033(i!m;i!n) =i\u0000\n4Z2\u0019\n0d'\n2\u0019sin 2('\u0000\u0012)X\n\u0017;\u00170\u0017\u00170I\u0017\u00170;\n(B4)\nwhere\nI\u0017\u00170=Z1\n\u00001d\u0018\n2\u0019i1\ni~!m\u0000\u0018\u0000\u0017he\u000b+i(\u0000=2)sgn(!m)\n\u00021\ni~(!m+!n)\u0000\u0018\u0000\u00170he\u000b+i(\u0000=2)sgn(!m+!n):\n(B5)\nWe note that one needs to calculate this integral only for\n!n>0 to obtain the retarded component by analytic\ncontinuation. Then, we can easily prove by the residue\nintegral that I\u0017\u00170= 0 for!m>0 and!m+!n>0\n(!m<0 and!m+!n<0) because both of the two poles\nin the integrand are located only in the upper (lower) half\nof the complex plane of \u0018. For!m<0 and!m+!n>0,\nthe integral is evaluated by the residue integral as\nI\u0017\u00170=1\ni~!n+ (\u0017\u0000\u00170)he\u000b+i\u0000: (B6)\nBy combining these results, Eqs. (45)-(50) can be de-\nrived.\n(a) (b)\nFIG. 10. Schematic picture of the change in the contour in-\ntegral. (a) The original contour. (b) The modi\fed contour.\nAppendix C: Derivation of Eq. (52)\nIn this Appendix, we give a detailed derivation of\nEq. (52) from Eq. (51). First, we modify Eq. (51) as\n\u001f(0;i!n) =1\n8AX\nkX\n\u0017;\u00170\"\n\u0017\u00170sin 2(\u001e\u0000\u0012)I\u0017\u00170;1\n+n\n1\u0000\u0017\u00170cos 2(\u001e\u0000\u0012)o\nI\u0017\u00170;2\n\u0000i(1\u0000\u0017\u00170)I\u0017\u00170;3#\n; (C1)\nwhere\nI\u0017\u00170;j\u00111\n\fX\ni!mXj\ni~!m\u0000E\u0017\nk+i\u0000=2 sgn(!m)\n\u00021\ni~!m+i~!n\u0000E\u00170\nk+i\u0000=2 sgn(!m+!n);(C2)\nand (X1;X2;X3) = (X;Y;Z ). A standard procedure\nbased on the residue integral enables us to express the\nsumI\u0017\u00170;jfor!n>0 as a complex integral on the con-\ntour C shown in Fig. 10 (a). This contour can be modi\fed\ninto a sum of the four contours, C l(l= 1;2;3;4), shown\nin Fig. 10 (b). Accordingly, I\u0017\u00170;jis written as\nI\u0017\u00170;j=4X\nl=1ICl\n\u0017\u00170;j; (C3)\nICl\n\u0017\u00170;j=\u0000Z\nCldz\n2\u0019if(z)Xj(z;i!n)\nz\u0000E\u0017\nk+i\u0000=2 sgn(Imz)\n\u00021\nz+i~!n\u0000E\u00170\nk+i\u0000=2 sgn(Imz+!n);(C4)\nwheref(z) = 1=(e\fz+ 1) is the Fermi distribution func-\ntion. The sum of the contributions from the two contours,12\nC2and C 3, is calculated as\nIC2\n\u0017\u00170;j+IC3\n\u0017\u00170;j\n=\u0000~Xj(i!n)ZdE\n2\u0019if(E)\n\u0002\"\n\u00001\nE\u0000E\u0017\nk\u0000i\u0000=21\nE+i~!n\u0000E\u00170\nk+i\u0000=2\n+1\nE\u0000i~!n\u0000E\u0017\nk\u0000i\u0000=21\nE\u0000E\u00170\nk+i\u0000=2#\n:(C5)\nHere, we have used the fact that Xj(z;i!n) is indepen-\ndent ofzfor 0<Imz < !nfrom Eq. (45) and have\nde\fned its value as ~Xj(i!n) (j= 1;2;3). From Eqs. (41)-\n(43), ~Xj(i!n) are calculated as\n~X1(i!n) =~\u00033(i!n)\n(1\u0000~\u00030(i!n))2\u0000~\u00032(i!n)2\u0000~\u00033(i!n)2;(C6)\n~X2(i!n) =1\u0000~\u00030(i!n)\u0000~\u00032(i!n)\n(1\u0000~\u00030(i!n))2\u0000~\u00032(i!n)2\u0000~\u00033(i!n)2;(C7)\n~X3(i!n) =i\n1\u0000~\u00030(i!n) +~\u00031(i!n): (C8)\nBy changing the integral variable to E0=E\u0000E\u0017\nkin the\n\frst term and to E0=\u0000(E\u0000E\u00170\nk) in the second term in\nEq. (C5), we obtain\nIC2\n\u0017\u00170;j+IC3\n\u0017\u00170;j=\u0000~Xj(i!n)ZdE0\n2\u0019i1\nE0\u0000i\u0000=2\n\u0002\"\nf(\u0000E0+E\u00170\nk)\u0000f(E0+E\u0017\nk)\nE0+i~!n+E\u0017\nk\u0000E\u00170\nk+i\u0000=2#\n:(C9)\nUsing formula (44), we replace the sum over kin Eq. (C1)\nby the integral with respect to \u0018and'. We can perform\nthe\u0018-integral by using\n\u0000Z1\n\u00001d\u0018[f(E0+E\u0017\nk)\u0000f(\u0000E0+E\u00170\nk)]\n= 2E0+E\u0017\nk\u0000E\u00170\nk: (C10)\nThen, by performing the E0-integral, we obtain\nZ1\n\u00001d\u0018(IC2\n\u0017\u00170;j+IC3\n\u0017\u00170;j) =i~!n~Xj(i!n)\nE\u0017\nk\u0000E\u00170\nk+i~!n+i\u0000:\n(C11)\nNext, let us consider the contribution from C 1and C 4.\nOn these two contours, Xj(z;i!n) is independent of z\nand its value is de\fned by ~X0\nj(i!n) (j= 1;2;3). Because\n\u0003j(z;i!n) (j= 0;1;2;3) becomes zero for Im z < 0 or\n!n<Imzfrom Eq. (45), ~X0\nj(i!n) are given as\n~X0\n1(i!n) = 0;~X0\n2(i!n) = 1;~X0\n3(i!n) =i:(C12)A similar calculation to that of C 2and C 3yields\nZ1\n\u00001d\u0018(IC1\n\u0017\u00170;j+IC4\n\u0017\u00170;j) =\u0000~X0\nj(i!n): (C13)\nBy substituting these results into Eq. (C1), we obtain\n\u001f(0;i!n) =D(\u000fF)\n8X\n\u0017;\u00170Z2\u0019\n0d'\n2\u0019\"\n\u0017\u00170sin 2(\u001e\u0000\u0012) ~x1(i!n)\n+ [1\u0000\u0017\u00170cos 2(\u001e\u0000\u0012)][\u00001 + ~x2(i!n)]\n\u0000i(1\u0000\u0017\u00170)[\u0000i+ ~x3(i!n)]#\n: (C14)\nwhere\n~xj(i!n) =i~!n~Xj(i!n)\nE\u0017\nk\u0000E\u00170\nk+i~!n+i\u0000: (C15)\nFinally, Eq. (52) is derived by substituting the expres-\nsions for ~Xj(i!n) and by analytic continuation i!n!\n!+i\u000e.\nAppendix D: Analytic Expression for \f=\u000b= 0\nIn this appendix, we derive analytic expressions of the\nmodulation of the Gilbert damping when \f=\u000b= 0, only\nthe Rashba spin-orbit interaction exists, to see the quan-\ntitative e\u000bect of taking the vertex correction into account.\nFor\f=\u000b= 0, the spin-splitting energy 2 he\u000b= 2\u0001 0is con-\nstant along the Fermi surface, and ~\u0003R\nj(!) (j= 0;1;2;3)\nis simpli\fed as\n~\u0003R\n0(!) =i\u0000\n4\u00010X\n\u0017\u001701\n~!=\u00010+ (\u0017\u0000\u00170) +i\u0000=\u00010;(D1)\n~\u0003R\n1(!) =i\u0000\n4\u00010X\n\u0017\u00170\u0017\u00170\n~!=\u00010+ (\u0017\u0000\u00170) +i\u0000=\u00010;(D2)\n~\u0003R\n2(!) =~\u0003R\n3(!) = 0: (D3)\nThen, we obtain the modulation of the Gilbert damping\nwith the vertex corrections,\n\u000e\u000bG\n\u000bG;0'\u00010\n2\u0019\u0000Re\"~\u0003R\n0(!0)\n1\u0000~\u0003R\n0(!0)+~\u0003R\n0(!0)\u0000~\u0003R\n1(!0)\n1\u0000~\u0003R\n0(!0) +~\u0003R\n1(!0)#\n:\n(D4)\nThe modulation of the Gilbert damping without the ver-\ntex correction is obtained by considering only the \frst-\norder term with respect to ~\u0003R\nj(!0),\n\u000e\u000bnv\nG\n\u000bG;0'\u00010\n2\u0019\u0000Re\"\n2~\u0003R\n0(!0)\u0000~\u0003R\n1(!0)#\n: (D5)\nWhen \u0000\u001c\u00010, the contribution of \u0017=\u00170is dominant\nfor the peak at !0= 0 and the modulation of the Gilbert13\ndamping can be analytically calculated as\n\u000e\u000bG\n\u000bG;0'\u00010\n4\u0019\u0001\u0000=2\n(~!0)2+ (\u0000=2)2; (D6)\n\u000e\u000bnv\nG\n\u000bG;0'\u00010\n4\u0019\u0001\u0000\n(~!0)2+ \u00002: (D7)\nThis indicates that the peak width is halved by taking\nthe vertex correction into account, which is consistent\nwith the results shown in Figs. 4 (b) and 4 (c).\nIn a similar way, we can evaluate the modulation of\nthe Gilbert damping near the peak at !0= 2\u0001 0=~as\n\u000e\u000bG\n\u000bG;0'\u00010\n4\u0019\u0001\u00141\n23\u0000=4\n(~!0\u00002\u00010)2+ (3\u0000=4)2\n+\u0000=2\n(~!0\u00002\u00010)2+ (\u0000=2)2\u0015\n; (D8)\n\u000e\u000bnv\nG\n\u000bG;0'\u00010\n4\u0019\u00013\u0000=2\n(~!0\u00002\u00010)2+ \u00002: (D9)\nAs well, for the peak at !0= 2\u0001 0=~, the peak width\nbecomes smaller when the vertex correction is taken into\naccount. This observation is consistent with the results\nshown in Figs. 4 (b) and 4 (c). For a \fnite value of \u0000, a\nsum of Eqs. (D6) and (D8) [Eqs. (D7) and (D9)] gives a\nbetter analytic form which \fts the numerical result with\n(without) the vertex correction. Note that \u000e\u000bGand\u000e\u000bnv\nG\ndepend on the impurity potential strength, u, and im-\npurity concentration, ni, through \u0000 = 2 \u0019niu2D(\u000fF) [see\nEq. (8)]. As shown in Eqs. (D6)-(D9), the peak widths\nof the Lorentzian functions in \u000e\u000bGand\u000e\u000bnv\nGare deter-\nmined by \u0000 [see Figs. 4 (b) and 4 (c)]. It is remarkable\nthat the peak width in \u000e\u000bGis reduced from \u0000 to \u0000 =2 by\ntaking the vertex correction into account. To summarize\nthe e\u000bect of the vertex correction, we show \u000e\u000bG\u0000\u000e\u000bnv\nG\nand\u000e!0\u0000\u000e!nv\n0in Figs. 11 (a) and 11 (d), respectively.\nWe \fnd that the vertex correction modi\fes mainly the\npeak width around ~!0=\u00010= 0 and 2, in consistent with\nthe above analytic expressions.\nFinally, we note that the same analytical expressions\nfor\u000e\u000bGand\u000e\u000bnv\nGcan be obtained for the case of \u000b=\f=\n0, i.e., when only the Dresselhaus spin-orbit interaction\nexists. We also note that for general values of \f=\u000b,\u000e\u000bG\nand\u000e\u000bnv\nGdepend on \u0000 in a more complicated way.\nAppendix E: Analytic Expression for \f=\u000b= 1\nIn this Appendix, we derive analytic expressions of the\nmodulation of the Gilbert damping when \f=\u000b = 1. In\nthis case, the e\u000bective Zeeman \feld is parallel to the\n(\u00001;1;0) direction and its amplitude is given as\nhe\u000b(') = 2\u0001 0jsin('+\u0019=4)j: (E1)Then, ~\u0003R\nj(!) (j= 0;1;2;3) becomes\n~\u0003R\n0(!) =i\u0000\n4\u00010X\n\u0017\u00170J\u0017\u00170 (E2)\n~\u0003R\n1(!) =i\u0000\n4\u00010X\n\u0017\u00170\u0017\u00170J\u0017\u00170; (E3)\n~\u0003R\n2(!) =\u0000sin 2\u0012~\u0003R\n1(!); (E4)\n~\u0003R\n3(!) =\u0000cos 2\u0012~\u0003R\n1(!) (E5)\nwhere\nJ\u0017\u00170(!)\u0011Z2\u0019\n0d'\n2\u0019\u00010\n~!+ (\u0017\u0000\u00170)he\u000b(') +i\u0000:(E6)\nIn the case of \u0012=\u0019=4, the modulation of the Gilbert\ndamping with the vertex correction is expressed as\n\u000e\u000bG\n\u000bG;0=\u00010\n2\u0019\u0000Re\"\n\u00002 +1\n1\u0000~\u0003R\n0(!0) +~\u0003R\n1(!0)\n+1\n1\u0000~\u0003R\n0(!0)\u0000~\u0003R\n1(!0)#\n: (E7)\nThe third term of the above equation is calculated as\n1\n1\u0000~\u0003R\n0(!0)\u0000~\u0003R\n1(!0)=1\n1\u0000i\u0000\n~!0+i\u0000=~!0+i\u0000\n~!0:(E8)\nThis indicates that the expansion with respect to ~\u0003R\njcan\nnot be allowed for !0\u001c\u0000. This is why the modula-\ntion without the vertex correction, which is obtained by\ntaking from the \frst-order term of ~\u0003R\njin Eq. (E7) as\n\u000e\u000bnv\nG\n\u000bG;0=\u00010\n2\u0019\u0000Re\u0014\n2~\u0003R\n0(!0)\u0015\n; (E9)\ngives a di\u000berent result near !0'0. Actually, for \u0012=\u0019=4,\n\u000e\u000bGand\u000e\u000bnv\nGare calculated as\n\u000e\u000bG\n\u000bG;0=\u00010\n2\u0019\u0000Re\"\ni\u0000\n2\u00010(J+\u0000+J\u0000+)\n1\u0000i\u0000\n2\u00010(J+\u0000+J\u0000+)#\n; (E10)\n\u000e\u000bnv\nG\n\u000bG;0=1\n4\u0019Re\"\ni\u0012\nJ+\u0000+J\u0000++J+++J\u0000\u0000\u0013#\n:(E11)\nNote that Eq.(E10) is not valid for !0= 0. As indicated\nfrom the absence of J++andJ\u0000\u0000, the graph of \u000e\u000bG(!0)\nhas no peak at zero frequency even though \u000e\u000bnv\nG(!0) has a\npeak there. This observation is consistent with Figs. 4 (e)\nand 4 (f).\nIn the case of \u0012=\u0000\u0019=4, the modulations of the Gilbert\ndamping with and without the vertex correction are\n\u000e\u000bG\n\u000bG;0=\u00010\n\u0019\u0000Re\"\ni\u0000\n2\u00010(J+\u0000+J\u0000+)\n1\u0000i\u0000\n2\u00010(J+\u0000+J\u0000+)#\n; (E12)\n\u000e\u000bnv\nG\n\u000bG;0=1\n2\u0019Re\"\ni(J+\u0000+J\u0000+)#\n: (E13)14\nFIG. 11. (Upper panels) Change of the Gilbert damping due to the vertex correction, \u000e\u000bG\u0000\u000e\u000bnv\nG, for (a)\f=\u000b= 0, (b)\f=\u000b= 1,\nand (c)\f=\u000b= 3. (Lower panels) Change of the FMR frequency due to the vertex correction, \u000e!0\u0000\u000e!nv\n0, for (d)\f=\u000b= 0, (e)\n\f=\u000b= 1, and (f) \f=\u000b= 3. The horizontal axes are the FMR frequency, !0=\rhdc, whereas the vertical axes are the azimuth\nangle of the spontaneous spin polarization, \u0012, in the FI. In all the plots, we have set \u0000 =\u00010= 0:5.\nNote that\u000e\u000bnv\nGis obtained by taking the \frst-order term\nin Eq. (E12). As indicated by the absence of the terms,\nJ++andJ\u0000\u0000, neither\u000e\u000bGnor\u000e\u000bnv\nGhas any structure\naround!0= 0. It can be checked that these two expres-\nsions give almost the same result when \u0000 .\u00010, which is\nconsistent with Figs. 4 (e) and 4 (f). Note as well that\n\u000e\u000bGis just doubled compared with the result for \u0012=\u0019=4\nin Eq. (E10).\nTo summarize the e\u000bect of the vertex correction, we\nshow\u000e\u000bG\u0000\u000e\u000bnv\nGand\u000e!0\u0000\u000e!nv\n0in Figs. 11 (b) and\n11 (e), respectively. We \fnd that the vertex correction\nmodi\fes mainly the peak width around ~!0=\u00010= 0.\nIn addition, the broad peak in the range of 0 <~!0<\n2\u00010is enhanced or suppressed depending on the azimuth\nangle of the ordered spin. These features are consistent\nwith the above analytic expressions. We note that similar\nfeatures are observed for \f=\u000b= 3 as seen in Figs. 11 (c)\nand 11 (f).\nAppendix F: Approximate Expressions near \f=\u000b= 1\nIn this Appendix, we derive the approximate expres-\nsions Eqs. (62) and (64) for \f=\u000b = 1 +\u000e(\u000e\u001c1) and\n!'0. For\f=\u000b= 1 +\u000e(\u000e\u001c1), we can use the approx-\nimation,\ncos 2(\u001e\u0000\u0012)'sin 2\u0012\u0010\n\u00001 +(hx+hy)2\nh2\ne\u000b\u0011\n; (F1)\nsin 2(\u001e\u0000\u0012)'cos 2\u0012\u0010\n\u00001 +(hx+hy)2\nh2\ne\u000b\u0011\n: (F2)Then, we obtain\n~\u0003R\n2'Xsin 2\u0012; (F3)\n~\u0003R\n3'Xcos 2\u0012; (F4)\nX\u0011i\u0000\n4Z2\u0019\n0d'\n2\u0019X\n\u0017\u00170\u0017\u00170\u0010\n\u00001 +(hx+hy)2\nh2\neff\u0011\n~!+ (\u0017\u0000\u00170)he\u000b+i\u0000(F5)\nin the low-frequency region. Here, the contribution of\nthe second term of the bracket in Eq. (56) does not have\na singularity at !0= 0 because ~\u0003R\n0and ~\u0003R\n1do not de-\npend on the e\u000bective Zeeman \feld he\u000b. Therefore, the\nsingularity comes from the \frst term of the bracket in\nEq. (56) and we can approximate F(!) as\nF(!)'\u00010\n2\u0019i\u0000~\u0003R\n0(1\u0000~\u0003R\n0)\u0000~\u0003R\n2(1\u0000~\u0003R\n2) + ( ~\u0003R\n3)2\n(1\u0000~\u0003R\n0)2\u0000(~\u0003R\n2)2\u0000(~\u0003R\n3)2\n=\u00010\n2\u0019i\u0000\u0014\n\u00001 +(1\u0000sin 2\u0012)=2\n1\u0000~\u0003R\n0\u0000X+(1 + sin 2\u0012)=2\n1\u0000~\u0003R\n0+X\u0015\n:\n(F6)\nFinally, using the equation,\n1\u0000~\u0003R\n0+X=\u0000s\n\u0000\u0000i~!\n\u0000+O(!2); (F7)\nwe \fnd that the third term in the bracket in Eq. (F6) is\ndivergent at != 0 in the limit of \u000e!0 since the denom-\ninator vanishes. By substituting Eq. (F7) into Eq. (F6),\nthe most singular part is calculated as\nF(!)'\u00010\n2\u0019isin2(\u0012+\u0019=4)\n\u0000s+i~!: (F8)15\nUsing Eqs. (54) and (55), it is straightforward to obtain\nEqs.(62) and (64).\n1I.\u0014Zuti\u0013 c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys.\n76, 323 (2004).\n2E. Y. Tsymbal and I. 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An\neffective trapping mechanism emerges according to which the process can be modeled by a L ´evy walk com-\nbining exponentially-distributed trapping times with free propagation along paths whose precise probabilities\nwe compute. This description yields an approximation of the mean squared displacement of infinite-horizon\nbilliards in terms of two transport coefficients which generalizes to this anomalous regime the Machta-Zwanzig\napproximation of normal diffusion in finite-horizon billiards [Phys. Rev. Lett. 50, 1959 (1983)].\nPACS numbers: 05.60.-k ,05.40.Fb,05.45.-a,02.50.-r\nThe study of stochastic processes exhibiting anomalous\ntransport has attracted considerable attention in the recent\npast, in particular in the context of continuous-time random\nwalks [1–6]. It has since evolved into a highly interdisci-\nplinary field, and includes a large array of applications [7].\nFrom a fundamental perspective, a problem of specific in-\nterest concerns the emergence of such stochastic processes\nfrom deterministic dynamics [8, 9], and the derivation of their\nlimiting distributions [10], which typically differ from Gaus-\nsian, as, for instance, L ´evy stable distributions [11].\nInfinite-horizon periodic billiard tables, which are the fo-\ncus of this Letter, provide an example of mechanical systems\nwhose transport properties exhibit a weak form of anomalous\nbehavior, in the sense that there is a logarithmic correction to\nthe linear growth in time of the mean squared displacement\n[12–14]; this is in fact a marginal case, separating diffusive\nand superdiffusive regimes [15]. It is known, in particular,\nthat the distribution of the anomalously rescaled displacement\nvector, [r(t)\u0000r(0)]=ptlogt, converges weakly to a normal\ndistribution, whose covariance matrix is given in terms of a\nscalar expression of the model’s parameter, the so-called Ble-\nher variance [16–18].\nAs opposed to finite-horizon billiards, which are such that\nthe displacement between two successive collisions of a point\nparticle with obstacles remains bounded, infinite-horizon bil-\nliards admit, among their solutions, collisionless trajectories;\npoint particles can therefore move arbitrarily far through in-\nfinite corridors devoid of obstacles. When the widths of\nthese corridors is taken to be vanishingly small, a separa-\ntion between two timescales occurs, the first associated with\nthe propagation phases of a trajectory across the billiard table\nalong the corridors and the second with an effective trapping\nmechanism which consists of a scattering phase that takes\nplace within a single billiard cell, separating successive prop-\nagation phases. While the former timescale is independent of\nthe widths of the corridors, the latter grows unbounded in the\nnarrow-corridor regime.\nThe purpose of this Letter is to show that, in the limit ofnarrow corridors, a novel regime emerges, such that a normal\ncontribution to the finite-time diffusion coefficient may be-\ncome arbitrarily larger than the anomalous component, even\nthough the latter diverges logarithmically as time increases\nwhile the former remains asymptotically constant. Due to\nits connection with the Machta-Zwanzig approximation of\nthe diffusion coefficient of finite-horizon billiard tables [19],\nwhich is obtained in a limit similar to that of narrow corridors,\nwe refer to this limit as a Machta-Zwanzig regime of anoma-\nlous diffusion in infinite-horizon billiards.\nTo analyze this regime, we model the process on the\ninfinite-horizon billiard table by a L ´evy walk [20, 21]. Our\napproach is based on the framework of continuous-time ran-\ndom walks and relies on the distinction between the states of\nparticles in propagating and scattering phases and the deriva-\ntion of a multistate generalized master equation [22, 23]. The\nprocess whereby a walker on a lattice remains in a scatter-\ning phase during an exponentially-distributed waiting time\nand then propagates in a random direction over a free path\nof random length is such a multistate process and it can be\nsolved analytically [24]. When applied to the infinite-horizon\nbilliard, the probability distribution of free paths decays al-\ngebraically with the third power of their lengths [13, 16].\nThe mean squared displacement then has two relevant compo-\nnents: The first associated with the anomalous rescaling of the\ndisplacement vector [16–18], and the second with a normal\ndiffusion component, similar to the Machta-Zwanzig approx-\nimation obtained in the finite-horizon regime [19]. Whereas\nthe former contribution is second order in the widths of the\ncorridors, the latter is first order and, for finite time, becomes\nmuch larger than the former.\nWe report numerical measurements of the mean square dis-\nplacement of infinite-horizon periodic billiard tables which\nexhibit those two contributions, in agreement with the results\npredicted by the L ´evy walk model.\nModel. We consider the simplest kind of billiard in the\nplane, which is defined by a periodic array of square cells\nof sides `with identical circular scatterers of radii r, 0<arXiv:1408.0349v1  [cond-mat.stat-mech]  2 Aug 20142\nHm-3,nLHm-2,nLHm-1,nLHm,nLHm+1,nLHm+2,nL\n{d\nFIG. 1. (Color online) Infinite-horizon periodic billiard table of\nsquare cells of sides `. The discs centered at the cells’ corners have\nradii r=0:45`. Infinite corridors of widths d=`\u00002rspan along\nhorizontal and vertical axes, through the centers of each cell. The\nbroken red line shows a trajectory.\nr< `= 2, whose centers are placed at the corners of the cells.\nA point-particle moves freely on the exterior of these obsta-\ncles, performing specular collisions upon their boundaries;\nsee Fig. 1.\nThis is an infinite-horizon configuration, which refers to the\nexistence of collisionless trajectories along the vertical and\nhorizontal corridors that separate the obstacles. In contrast,\nafinite-horizon configuration would occur if, for instance, an\nadditional circular obstacle of radius r\f, with`=2\u0000r<r\f<\n`=p\n2\u0000r, were placed at the center of each cell [25], such\nthat a particle would have to perform at least one collision in\neach visited cell. The widths of the corridors is d=`\u00002r,\nwhich we take to be a control parameter.\nTransport of particles on such billiard tables can be studied\nin terms of the time-evolution of the coarsegrained distribu-\ntion of particles in cell n2Z2, whose changes are determined\nby the transitions particles make as they go from one cell to\nanother. Generally speaking, this is a complicated process\nwhich is affected by correlations between successive collision\nevents. The situation however simplifies when these correla-\ntions become negligible, which occurs when d\u001c`.\nInfinite vs finite horizon. In finite-horizon tables, where\ndiffusion is always normal [26], this regime yields an approx-\nimation of the process by a continuous-time random walk for\ndisplacements on the lattice structure [27]. The associated\ntimescale is given in terms of the residence time,\ntR=pA\n4p0d; (1)\nwhere Adenotes the area of the elementary cell, viz.A=\n`2\u0000p(r2+r2\n\f), and p0is the speed of point particles [28].\nWhen tRis large with respect to the intercollisional time,\na particle typically spends a long time rattling about the same\ncell, making many collisions with its obstacles, before reach-\ning one of the four slits of widths d. Under these conditions,\nthe velocity vectors of a particle entering and subsequently\nexiting a cell will effectively be uncorrelated. The motion of\na particle on the billiard table can thus be approximated by\na succession of scattering phases with random waiting times,\nexponentially-distributed with scale tR, punctuated by instan-\ntaneous random hops from one cell to a neighboring one.In this regime, the mean squared displacement grows lin-\nearly in time, with coefficient approximated by 4 DMZ, where\nDMZ=`2\n4tR(2)\nis a dimensional expression, known as the Machta-Zwanzig\napproximation to the diffusion coefficient [19]. To leading\norder, the actual diffusion coefficient differs from Eq. (2) by\na correction which is expected to be linear in d=`[29]. The\nvalidity of the Machta-Zwanzig approximation thus relies on\nthe separation d\u001c`.\nIn the case of infinite horizon, the presence of corridors ren-\nders the Machta-Zwanzig regime more complex. In addition\nto the residence time tR, Eq. (1), one has to take into consid-\neration the timescale of propagation across a cell, tF, which,\nin the narrow-corridor regime is small with respect to tR[30],\ntF\u0011`\np0\u001ctR: (3)\nWhen the phase-space coordinates of a particle are such that it\ncrosses the boundary between two cells with velocity almost\nperpendicular to it, the next scattering phase may take place\nat a remote distance, after a time approximately equal to tF\nmultiplied by the number of cells traveled free of collisions.\nThe assumption that correlations between successive scat-\ntering phases become negligible amounts to approximating\nthe transport process on the infinite-horizon billiard table by\na L´evy walk, whose distribution of free paths, i.e. the dis-\ntance a particle propagates free of collisions, matches that of\nthe infinite-horizon billiard, taking into account the timede-\nlay induced by the propagation along free paths. The general\nframework for analyzing L ´evy walks where a distinction oc-\ncurs between the states of particles in scattering and propa-\ngating phases was described in Ref. [24]. Whereas particles\nin a scattering state make transitions at random times, expo-\nnentially distributed with scale tR, and move in a random di-\nrection to a neighboring cell, the transitions of particles in a\npropagating state are deterministic: They take place after ex-\nactly time tFand are accompanied by displacements along the\nsame direction as the previous transition.\nParameter values. We label the internal state of a walker\nbyk2Z, with k=0 denoting the scattering state and k\u00151\na propagating state, where kcorresponds to the total number\nof steps of duration tFuntil the propagating state returns to\na scattering one. The direction of propagation is fixed in the\npropagating state. When the particle is in the scattering state,\nwe let mkdenote the probability of a transition to either a scat-\ntering state, k=0, or a propagating state, k\u00151, for any of the\nfour lattice directions the walker can move in.\nIn the narrow-corridor regime, the overwhelming major-\nity of transitions are from scattering to scattering states; only\nrarely do particles perform transitions from scattering to prop-\nagating states. When they do, however, as mentioned ear-\nlier, the probability of a long excursion to a propagating state,\nk\u001d1, decays with the third power of their lengths, mk\u0018k\u00003.3\nGjH0LGjH0L GjH1L\nGjH1L\nGjH2LGjH2L\n-d\n{-d\n2{0d\n2{d\n{-d\n2-d\n40d\n4d\n2\np°p0s\nFIG. 2. (Color online) Graph showing two-dimensional phase-space\nsets according to the distance separating phase points at the boundary\nbetween two cells from their next collision on a disc. The darkest\ncolor encodes sets of points whose next collision takes place in the\ncell they move into; the brighter the colors the more distant the next\ncollision. The two axes correspond, respectively, to pk=p0, which\ndenotes the fraction of the velocity component directed along the\nboundary,\u00001<pk=p0<1 (here restricted to \u0000d=` < pk=p0<d=`)\nand to the position salong the boundary, \u0000d=2<s<d=2, where\nd=`\u00002r(r=0:45`), is the width of the corridor. With these\ncoordinates, the area corresponds to the natural invariant measure,\nwhich is normalized if divided by 2 d.\nIn fact, to first order in d=`, we can write\nmk=(\n1\u0000d\n4`; k=0;\nd\nk(k+1)(k+2)`;k\u00151:(4)\nTo obtain this result, consider the phase-space sets that lie\nat the boundary between two neighboring cells nandn+ej,\nwhere ejis a unit vector in one of the four lattice directions,\nj2f1;:::; 4g. A particle in cell nwhich crosses over to cell\nn+ejis mapped at this boundary to a phase point with coor-\ndinatejsj<d=2, along the direction of the separation between\nthe two cells, with velocity psuch that p\u0001ej\u0011p?>0.\nLetGj(k)denote the set of such phase points which are\nmapped by the flow to the next collision event on an obstacle\nin cell n+kej. Since the points of these sets can be mapped\nback to their preceding collision on an obstacle in cell n\u0000lej,\nfor some l\u00150, they are associated with point particles in a\npropagation phase of length at least k. We must measure these\npoints by means of the natural invariant measure, the one in-\nduced by the Liouville measure on the cross-sections defined\nby collisions or cell crossings, which is known to be the area\nin phase space as parameterized in Fig. 2. Therefore, up to a\nnormalization factor, the area of Gj(k)is given by å¥\ni=kmi. By\ngeometric arguments, and as can also be seen from Fig. 2, the\nmeasure of[¥\ni=kGj(i)is, to leading order in d=`, proportionalto the area of a right triangle of base d=(k`). We therefore\nhave, for k\u00151,\n¥\nå\ni=k¥\nå\nj=imj=d\n2k`; (5)\nwhich implies, and thus justifies, Eq. (4).\nAnomalous diffusion. We let r=n`denote the displace-\nment of L ´evy walkers on the two-dimensional square lattice,\nmeasured from the origin where they are initially located, and\nobtain an expression of their mean squared displacement as\na function of time, hr2it, in terms of the time-integral of the\noverall fraction of particles in a scattering state, s(t),\nhr2it\n`2=1\ntRZt\n0dss(s)+d\n2`tRbt=tFc\nå\nk=12k+1\nk(k+1)Zt\u0000ktF\n0dss(s);\n(6)\nsee Ref. [24] for further details. In the stationary state, the\nfraction of particles in a scattering state is simply given by the\nratio of the average time spent in the scattering state, tR, to the\naverage return time to this state, tR+å¥\nk=1kmktF. Substituting\nthe transition probabilities, Eq. (4), yields\nlim\nt!¥s(t) =tR\ntR+d\n2`tF'1\u0000d\n2`tF\ntR: (7)\nPlugging this expression into Eq. (6), we obtain an expres-\nsion of the mean square displacement of walkers in terms of\nharmonic numbers, which, in the long-time limit, reduces to\nhr2it\n4t=`2\n4tR+d`\n4tR[logt+O(1)]: (8)\nThis is our main result. It generalizes to infinite-horizon\nbilliard tables in the narrow-corridor regime the Machta-\nZwanzig approximation of the diffusion coefficient for finite-\nhorizon tables, Eq. (2).\nFor short times, the right-hand side of Eq. (8) is dominated\nby the constant term, which, to leading order in d=`, corre-\nsponds to a regime of normal diffusion, with the coefficient\n(2), consistent with the Machta-Zwanzig approximation of the\n(normal) diffusion coefficient.\nThe term carrying the logarithmic correction has coefficient\nd`=(4tR), which is identical to the Bleher limiting variance\nof the anomalously rescaled limiting distribution [16–18],\ni.e. such that the displacement vector rescaled by the square\nroot of tlogtconverges in distribution to a centered normal\ndistribution whose covariance matrix reduces to a scalar given\nby this coefficient [31].\nWhereas the coefficient of the constant term on the right-\nhand side of Eq. (8) is first order in the small parameter d=`,\nthe coefficient of the logarithmic term is second order. Such\na contribution thus becomes significant only for times such\nthat log t\u0019`=d. In the narrow-corridor regime, however, the\nconstraints on the integration times are such that log tremains\nsmall with respect `=d.4\n0.000.020.040.060.080.100.121.01.11.21.31.41.51.6\nd4tRa,4tRbdr=0.485\n1011021035253545556\ntXrHtL2\\H4xflow tL\nFIG. 3. (Color online) Computed slopes, b(blue, lower curve), and\nintercepts, a(red, upper curve), of the mean squared displacement\nof point particles on the infinite-horizon Lorentz gas, fitted according\nto Eq. (9) for different values of the parameter d. The values found\nare here normalized by those predicted by Eqs. (10) and (11). To\nillustrate the fitting procedure, the inset shows the mean squared dis-\nplacement (blue curve, including error bars) measured, as a function\nof time, for d=0:03. The red dashed line is the result of a linear\nfit performed in the interval marked by the two vertical lines; see\nRef. [32] for further details on this procedure. The units are chosen\nsuch that `\u0011p0\u00111.\nNumerical results. Following the results presented in\nRef. [32], we perform numerical measurements of the mean\nsquared displacement of infinite-horizon billiard tables such\nas shown in Fig. 1 and determine a range of time values such\nthat the distribution of free paths is accurately sampled, which,\naccording to Eq. (4), scales like the square root of the to-\ntal number of initial conditions taken (typically 109). In that\nrange, we seek an asymptotically affine fitting function of log t\nfor the normally rescaled mean squared displacement,\nhr2it\n4t\u0018a+blogt; (9)\nwhere aandbare implicit functions of time, and are expected\nto converge to the values predicted by Eq. (8) as t!¥, i.e.\nlim\nd!0lim\nt!¥4tRa\n`2=1; (10)\nlim\nt!¥4tRb\nd`=1; (11)\nwhere, in the first line, the narrow-corridor limit takes care of\nd-dependent corrections to aour theory does not account for.\nValues found for these fitting parameters are reported in\nFig. 3 for different values of d. For the parameter b, on the\none hand, one expects Eq. (11) to hold for all values of d\nin the range shown and, in view of the difficulties presented\nby such measurements [32], the agreement is indeed rather\ngood, especially given the prevalence of finite-time effects\nwhen d!0. There is, on the other hand, no analytic pre-\ndiction for the value of the parameter a, other than that given\nin the narrow-corridor limit, Eq. (10). Nevertheless, our data\nprovides clear evidence in support of this result.We should note that, in contrast to the approximating L ´evy\nwalk, for which corrections to the zeroth order result (8) are\nfound to be negative, the corrections to the zeroth order result\nfor measurements performed for billiards appear to be positive\nat first order in d=`. This is to be expected, since memory ef-\nfects should indeed bring about corrections of the same order,\nas is the case with finite-horizon billiards [29]; such correc-\ntions may well predominate.\nConclusions Infinite-horizon billiard tables with narrow\ncorridors display anomalous transport properties such that the\nlogarithmic divergence in time of the mean squared displace-\nment must effectively be treated as a subleading contribution\nwith respect to a normally diffusive one.\nOur stochastic analysis of the process in terms of a L ´evy\nwalk with both scattering phases, characterized by random\nwaiting times with exponential distributions, and propagat-\ning phases along the table’s corridors provides two quanti-\ntative predictions which match the Machta-Zwanzig dimen-\nsional prediction of the (normal) diffusion coefficient, on the\none hand, and the Bleher variance of the anomalously rescaled\nprocess, on the other hand. Their physical interpretations is,\nmoreover, transparent: (i) the overwhelming majority of tran-\nsitions taking place on the billiard table are similar to those\nobserved in finite-horizon billiard tables, giving rise to the pre-\ndominant normal contribution to the mean squared displace-\nment, and (ii) rare scattering events allow propagation along\nthe billiard’s corridors over long distrances whose lengths fol-\nlow a precise distribution, at the origin of the anomalous con-\ntribution to the mean squared displacement. As our numerical\nresults make clear, ignoring the first of these two contributions\nwould obstruct the accurate measurement of the second.\nWe conclude by observing that the scaling properties of the\ntransition probabilities, Eq. (4), can be generalized to other\nvalues, extending the relevance of our approach well beyond\nthe regime studied in this Letter. As discussed in Ref. [24],\ntuning the parameter values allows the description of both\nnormal and anomalous transport regimes, including ballistic\ntransport. Further applications will be reported elsewhere.\nThis work was partially supported by FIRB-Project No.\nRBFR08UH60 (MIUR, Italy), by SEP-CONACYT Grant\nNo. CB-101246 and DGAPA-UNAM PAPIIT Grant No.\nIN117214 (Mexico), and by FRFC convention 2,4592.11\n(Belgium). T.G. is financially supported by the (Belgian)\nFRS-FNRS.\n[1] J. W. Haus and K. W. Kehr, Physics Reports 150, 263 (1987).\n[2] J.-P. Bouchaud and A. Georges, Physics Reports 195, 127\n(1990).\n[3] G. H. Weiss, Aspects and Applications of the Random Walk\n(North-Holland, Amsterdam, 1994).\n[4] M. F. Shlesinger, G. M. Zaslavsky, and U. Frisch, eds., L´evy\nFlights and Related Topics in Physics , Lecture Notes in Physics,\nV ol. 450 (Springer, Berlin, Heidelberg, 1995).\n[5] S. Benkadda and G. M. Zaslavsky, eds., Chaos, Kinetics and5\nNonlinear Dynamics in Fluids and Plasmas , Lecture Notes in\nPhysics, V ol. 511 (Springer, Berlin Heidelberg, 1998).\n[6] G. M. Zaslavsky, Physics Reports 371, 461 (2002).\n[7] R. Klages, G. Radons, and I. M. Sokolov, Anomalous trans-\nport: Foundations and applications (Wiley-VCH Verlag, Wein-\nheim, 2008).\n[8] M. F. Shlesinger, G. M. Zaslavsky, and J. Klafter, Nature 363,\n31 (1993).\n[9] G. M. Zaslavsky, Physics Today 52, 39 (1999).\n[10] J. Klafter, M. F. Shlesinger, and G. Zumofen, Physics Today\n49, 33 (1996).\n[11] B. V . Gnedenko and A. N. Kolmogorov, Limit Distributions for\nSums of Independent Random Variables , revised ed. (Addison-\nWesley, New York, 1968).\n[12] B. Friedman and R. F. Martin, Jr., Physics Letters A 105, 23\n(1984).\n[13] J.-P. Bouchaud and P. Le Doussal, Journal of Statistical Physics\n41, 225 (1985).\n[14] A. Zacherl, T. Geisel, J. Nierwetberg, and G. Radons, Physics\nLetters A 114, 317 (1986).\n[15] T. Geisel, J. Nierwetberg, and A. Zacherl, Physical Review\nLetters 54, 616 (1985).\n[16] P. M. Bleher, Journal of Statistical Physics 66, 315 (1992).\n[17] D. Sz ´asz and T. Varj ´u, Journal of Statistical Physics 129, 59\n(2007).\n[18] D. I. Dolgopyat and N. I. Chernov, Russian Mathematical Sur-\nveys 64, 651 (2009).\n[19] J. Machta and R. Zwanzig, Physical Review Letters 50, 1959\n(1983).\n[20] J. Klafter, A. Blumen, and M. F. Shlesinger, Physical ReviewA35, 3081 (1987).\n[21] G. Zumofen and J. Klafter, Physical Review E 47, 851 (1993).\n[22] V . M. Kenkre, E. W. Montroll, and M. F. Shlesinger, Journal of\nStatistical Physics 9, 45 (1973).\n[23] U. Landman, E. W. Montroll, and M. F. Shlesinger, Proceed-\nings of the National Academy of Sciences of the United States\nof America 74, 430 (1977).\n[24] G. Cristadoro, T. Gilbert, M. Lenci, and D. P. Sanders,\narXiv:1407.0227 (2014).\n[25] Without loss of generality, we assume r\u0015r\f. For definiteness\nof a unique relevant timescale, Eq. (1), the separation between\nthe central disc and the outer ones should also be substantially\nlarger than the separation between two outer discs.\n[26] L. A. Bunimovich, Y . G. Sinai, and N. Chernov, Russian Math-\nematical Surveys 46, 47 (1991).\n[27] P. Gaspard and T. Gilbert, Chaos 22, 026117 (2012).\n[28] N. Chernov, Journal of Statistical Physics 88, 1 (1997).\n[29] T. Gilbert and D. P. Sanders, Physical Review E 80, 041121\n(2009).\n[30] Strictly speaking, the residence time, Eq. (1), accounts for the\npossibility of collisionless motion inside a cell. However in the\nnarrow-corridor regime, the difference between tRand the ac-\ntual residence time conditioned on particles performing colli-\nsions inside the cell is next order in the small parameter d=`\nand will thus be neglected.\n[31] The second moment actually scales with coefficient given by\ntwice the Gaussian value. See discussion in Ref. [32].\n[32] G. Cristadoro, T. Gilbert, M. Lenci, and D. P. Sanders, Physical\nReview E, in press; arXiv:1405.0975 (2014)."    },    {        "title": "2106.14858v3.Stability_of_a_Magnetically_Levitated_Nanomagnet_in_Vacuum__Effects_of_Gas_and_Magnetization_Damping.pdf",        "content": "Stability of a Magnetically Levitated Nanomagnet in Vacuum: E\u000bects of Gas and\nMagnetization Damping\nKatja Kustura,1, 2Vanessa Wachter,3, 4Adri\u0013 an E. Rubio L\u0013 opez,1, 2and Cosimo C. Rusconi5, 6\n1Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria.\n2Institute for Theoretical Physics, University of Innsbruck, A-6020 Innsbruck, Austria.\n3Max Planck Institute for the Science of Light, Staudtstra\u0019e 2, 91058 Erlangen, Germany\n4Department of Physics, University of Erlangen-N urnberg, Staudtstra\u0019e 7, 91058 Erlangen, Germany\n5Max-Planck-Institut f ur Quantenoptik, Hans-Kopfermann-Strasse 1, 85748 Garching, Germany.\n6Munich Center for Quantum Science and Technology,\nSchellingstrasse 4, D-80799 M unchen, Germany.\n(Dated: June 1, 2022)\nIn the absence of dissipation a non-rotating magnetic nanoparticle can be stably levitated in a\nstatic magnetic \feld as a consequence of the spin origin of its magnetization. Here we study the\ne\u000bects of dissipation on the stability of the system, considering the interaction with the background\ngas and the intrinsic Gilbert damping of magnetization dynamics. At large applied magnetic \felds\nwe identify magnetization switching induced by Gilbert damping as the key limiting factor for\nstable levitation. At low applied magnetic \felds and for small particle dimensions magnetization\nswitching is prevented due to the strong coupling of rotation and magnetization dynamics, and\nthe stability is mainly limited by the gas-induced dissipation. In the latter case, high vacuum\nshould be su\u000ecient to extend stable levitation over experimentally relevant timescales. Our results\ndemonstrate the possibility to experimentally observe the phenomenon of quantum spin stabilized\nmagnetic levitation.\nI. INTRODUCTION\nThe Einstein{de Haas [1, 2] and Barnett e\u000bects [3] are\nmacroscopic manifestations of the internal angular mo-\nmentum origin of magnetization: a change in the mag-\nnetization causes a change in the mechanical rotation\nand conversely. Because of the reduced moment of in-\nertia of levitated nano- to microscale particles, these ef-\nfects play a dominant role in the dynamics of such sys-\ntems [4{10]. This o\u000bers the possibility to harness these\ne\u000bects for a variety of applications such as precise magne-\ntometry [11{16], inertial sensing [17, 18], coherent spin-\nmechanical control [19, 20], and spin-mechanical cool-\ning [21, 22] among others. Notable in this context is\nthe possibility to stably levitate a ferromagnetic parti-\ncle in a static magnetic \feld in vacuum [23, 24]. Stable\nlevitation is enabled by the internal angular momentum\norigin of the magnetization which, even in the absence of\nmechanical rotation, provides the required angular mo-\nmentum to gyroscopically stabilize the system. Such a\nphenomenon, which we refer to as quantum spin stabi-\nlized levitation to distinguish it from the rotational stabi-\nlization of magnetic tops [25{27], relies on the conserva-\ntive interchange between internal and mechanical angular\nmomentum. Omnipresent dissipation, however, exerts\nadditional non-conservative torques on the system which\nmight alter the delicate gyroscopic stability [26, 28]. It\nthus remains to be determined if stable levitation can\nbe observed under realistic conditions, where dissipative\ne\u000bects cannot be neglected.\nIn this article, we address this question. Speci\f-\ncally, we consider the dynamics of a levitated magnetic\nnanoparticle (nanomagnet hereafter) in a static magnetic\n\feld in the presence of dissipation originating both fromthe collisions with the background gas and from the\nintrinsic damping of magnetization dynamics (Gilbert\ndamping) [29, 30], which are generally considered to be\nthe dominant sources of dissipation for levitated nano-\nmagnets [8, 13, 31{33]. Con\fned dynamics can be ob-\nserved only when the time over which the nanomagnet is\nlevitated is longer than the period of center-of-mass os-\ncillations in the magnetic trap. When this is the case, we\nde\fne the system to be metastable . We demonstrate that\nthe system can be metastable in experimentally feasible\nconditions, with the levitation time and the mechanism\nbehind the instability depending on the parameter regime\nof the system. In particular, we show that at weak ap-\nplied magnetic \felds and for small particle dimensions\n(to be precisely de\fned below) levitation time can be\nsigni\fcantly extended in high vacuum (i.e. pressures be-\nlow 10\u00003mbar). Our results evidence the potential of\nunambiguous experimental observation of quantum spin\nstabilized magnetic levitation.\nWe emphasize that our analysis is particularly timely.\nPresently there is a growing interest in levitating and con-\ntrolling magnetic systems in vacuum [9, 34, 35]. Current\nexperimental e\u000borts focus on levitation of charged para-\nmagnetic ensembles in a Paul trap [19, 36, 37], diamag-\nnetic particles in magneto-gravitational traps [38{40], or\nferromagnets above a superconductor [14, 20, 41]. Lev-\nitating ferromagnetic particles in a static magnetic trap\no\u000bers a viable alternative, with the possibility of reaching\nlarger mechanical trapping frequencies.\nThe article is organized as follows. In Sec. II we in-\ntroduce the model of the nanomagnet, and we de\fne\ntwo relevant regimes for metastability, namely the atom\nphase and the Einstein{de Haas phase. In Sec. III and\nIV we analyze the dynamics in the atom phase and thearXiv:2106.14858v3  [cond-mat.mes-hall]  31 May 20222\nFigure 1. (a) Illustration of a spheroidal nanomagnet levi-\ntated in an external \feld B(r) and surrounded by a gas at the\ntemperature Tand the pressure P. (b) Linear stability dia-\ngram of a non-rotating nanomagnet in the absence of dissipa-\ntion, assuming a= 2b. Blue and red regions denote the stable\natom and Einstein{de Haas phase, respectively; hatched area\nis the unstable region. Dashed lines show the critical values\nof the bias \feld which de\fne the two phases. In particular,\nBEdH,1\u00115\u0016=[4\r2\n0(a2+b2)M],BEdH,2\u00113 [\u0016B02=(4\r0M)]1=3,\nandBatom = 2kaV=\u0016. Numerical values of physical parame-\nters used to generate panel (b) are given in Table I.\nEinstein{de Haas phase, respectively. We discuss our re-\nsults in Sec. V. Conclusions and outlook are provided in\nSec. VI. Our work is complemented by three appendices\nwhere we de\fne the transformation between the body-\n\fxed and laboratory reference frames (App. A), analyze\nthe e\u000bect of thermal \ructuations (App. B), and provide\nadditional \fgures (App. C).\nII. DESCRIPTION OF THE SYSTEM\nWe consider a single domain nanomagnet levitated in\na static1magnetic \feld B(r) as shown schematically in\nFig. 1(a). We model the nanomagnet as a spheroidal\nrigid body of mass density \u001aMand semi-axes lengths a;b\n(a > b ), having uniaxial magnetocrystalline anisotropy,\nwith the anisotropy axis assumed to be along the major\nsemi-axisa[42]. Additionally, we assume that the mag-\nnetic response of the nanomagnet is approximated by a\npoint dipole with magnetic moment \u0016of constant mag-\nnitude\u0016\u0011j\u0016j, as it is often justi\fed for single domain\nparticles [42, 43]. Let us remark that such a simpli\fed\nmodel has been considered before to study the classical\ndynamics of nanomagnets in a viscous medium [31, 44{\n49], as well as to study the quantum dynamics of mag-\nnetic nanoparticles in vacuum [5, 13, 50, 51]. Since the\nmodel has been successful in describing the dynamics of\nsingle-domain nanomagnets, we adopt it here to inves-\ntigate the stability in a magnetic trap. In particular,\nour study has three main di\u000berences as compared with\n1We denote a \feld static if it does not have explicit time depen-\ndence, namely if @B(r)=@t= 0.Table I. Physical parameters of the model and the values used\nthroughout the article. We calculate the magnitude of the\nmagnetic moment as \u0016=\u001a\u0016V, where\u001a\u0016=\u001aM\u0016B=(50amu),\nwith\u0016Bthe Bohr magneton and amu the atomic mass unit.\nParameter Description Value [units]\n\u001aM mass density 104[kg m\u00003]\na;b semi-axes see main text [m]\n\u001a\u0016 magnetization 2 :2\u0002106[J T\u00001m\u00003]\nka anisotropy constant 105[J m\u00003]\n\r0 gyromagnetic ratio 1 :76\u00021011[rad s\u00001T\u00001]\nB0 \feld bias see main text [T]\nB0\feld gradient 104[T m\u00001]\nB00\feld curvature 106[T m\u00002]\n\u0011 Gilbert damping 10\u00002[n. u.]\nT temperature 10\u00001[K]\nP pressure 10\u00002[mbar]\nM molar mass 29 [g mol\u00001]\n\u000bc re\rection coe\u000ecient 1 [n. u.]\nprevious work. (i) We consider a particle levitated in\nhigh vacuum, where the mean free path of the gas parti-\ncles is larger than the nanomagnet dimensions (Knudsen\nregime [52]). This leads to gas damping which is gen-\nerally di\u000berent from the case of dense viscous medium\nmostly considered in the literature. (ii) We consider\ncenter-of-mass motion and its coupling to the rotational\nand magnetic degrees of freedom, while previous work\nmostly focuses on coupling between rotation and mag-\nnetization only (with the notable exception of [48]). (iii)\nWe are primarily interested in the center-of-mass con\fne-\nment of the particle, and not in its magnetic response.\nWithin this model the relevant degrees of freedom of\nthe system are the center-of-mass position r, the linear\nmomentum p, the mechanical angular momentum L, the\norientation of the nanomagnet in space \n, and the mag-\nnetic moment \u0016. The orientation of the nanomagnet\nis speci\fed by the body-\fxed reference frame Oe1e2e3,\nwhich is obtained from the laboratory frame Oexeyez\naccording to ( e1;e2;e3)T=R(\n)(ex;ey;ez)T, where\n\n= (\u000b;\f;\r )Tare the Euler angles and R(\n) is the\nrotational matrix. We provide the expression for R(\n)\nin App. A. The body-\fxed reference frame is chosen such\nthate3coincides with the anisotropy axis. The magnetic\nmoment\u0016is related to the internal angular momentum F\naccording to the gyromagnetic relation \u0016=\r0F, where\n\r0is the gyromagnetic ratio of the material2.\n2The total internal angular momentum Fis a sum of the individ-\nual atomic angular momenta (spin and orbital), which contribute\nto the atomic magnetic moment. For a single domain magnetic\nparticle, it is customary to assume that Fcan be described as\na vector of constant magnitude, jFj=\u0016=\r0(macrospin approxi-\nmation) [43].3\nA. Equations of Motion\nWe describe the dynamics of the nanomagnet in the\nmagnetic trap with a set of stochastic di\u000berential equa-\ntions which model both the deterministic dissipative evo-\nlution of the system and the random \ructuations due to\nthe environment. In the following it is convenient to de-\n\fne dimensionless variables: the center-of-mass variables\n~r\u0011r=a,~p\u0011\r0ap=\u0016, the mechanical angular momen-\ntum`\u0011\r0L=\u0016, the magnetic moment m\u0011\u0016=\u0016, and\nthe magnetic \feld b(~r)\u0011B(a~r)=B0, whereB0denotes\nthe minimum of the \feld intensity in a magnetic trap,\nwhich we hereafter refer to as the bias \feld. Note that\nwe choose to normalize the position r, the magnetic mo-\nment\u0016and the magnetic \feld B(r) with respect to the\nparticle size a, the magnetic moment magnitude \u0016, and\nthe bias \feld B0, respectively. The scaling factor for an-\ngular momentum, \u0016=\r0, and linear momentum, \u0016=(a\r0),\nfollow as a consequence of the gyromagnetic relation.\nThe dynamics of the nanomagnet in the laboratory\nframe are given by the equations of motion\n_~r=!I~p; (1)\n_e3=!\u0002e3; (2)\n_~p=!Lr~r[m\u0001b(~r)]\u0000\u0000cm~p+\u001ep(t); (3)\n_`=!Lm\u0002b(~r)\u0000_m\u0000\u0000rot`+\u0018l(t); (4)\n_m=m\n1 +\u00112\u0002[!e\u000b\u0000\u0011m\u0002(!+!e\u000b+\u0011!\u0002m)\n+\u0010b(t)]: (5)\nHere we de\fne the relevant system frequencies: !I\u0011\n\u0016=(\r0Ma2) is the Einstein{de Haas frequency, with M\nthe mass of the nanomagnet, !L\u0011\r0B0is the Larmor\nfrequency,!A\u0011kaV\r0=\u0016is the anisotropy frequency,\nwithVthe volume of the nanomagnet and kathe ma-\nterial dependent anisotropy constant [43], !\u0011I\u00001L\nis the angular velocity, with Ithe tensor of inertia,\nand!e\u000b\u00112!A(m\u0001e3)e3+!Lb(~r). Dissipation is\nparametrized by the dimensionless Gilbert damping pa-\nrameter\u0011[29, 53], and the center-of-mass and rotational\nfriction tensors \u0000 cmand \u0000 rot, respectively [32]. The e\u000bect\nof stochastic thermal \ructuations is represented by the\nrandom variables \u001ep(t) and\u0018l(t) which describe, respec-\ntively, the \ructuating force and torque exerted by the\nsurrounding gas, and by \u0010b(t) which describes the ran-\ndom magnetic \feld accounting for thermal \ructuations\nin magnetization dynamics [54]. We assume Gaussian\nwhite noise, namely, for X(t)\u0011(\u001ep(t);\u0018l(t);\u0010b(t))Twe\nhavehXi(t)i= 0 andhXi(t)Xj(t0)i\u0018\u000eij\u000e(t\u0000t0).\nEquations (1-4) describe the center-of-mass and rota-\ntional dynamics of a rigid body in the presence of dis-\nsipation and noise induced by the background gas [32].\nThe expressions for \u0000 cmand \u0000 rotdepend on the parti-\ncle shape { here we take the expressions derived in [32]for a cylindrical particle3{, and on the ratio of the sur-\nface and the bulk temperature of the particle, which\nwe assume to be equal to the gas temperature T. Fur-\nthermore, they account for two di\u000berent scattering pro-\ncesses, namely the specular and the di\u000busive re\rection\nof the gas from the particle, which is described by a\nphenomenological interpolation coe\u000ecient \u000bc. The or-\nder of magnitude of the di\u000berent components of \u0000 cmand\n\u0000rotis generally well approximated by the dissipation\nrate \u0000\u0011(2Pab=M )[2\u0019M=(NAkBT)]1=2, wherePand\nMare, respectively, the gas pressure and molar mass,\nkBis the Boltzmann constant and NAis the Avogadro\nnumber. The magnetization dynamics given by Eq. (5)\nis the Landau-Lifshitz-Gilbert equation in the laboratory\nframe [8, 57], with the e\u000bective magnetic \feld !e\u000b=\r0.\nWe remark that Eqs. (1-5) describe the classical dynam-\nics of a levitated nanomagnet where the e\u000bect of the\nquantum spin origin of magnetization, namely the gy-\nromagnetic relation, is taken into account phenomeno-\nlogically by Eq. (5). This is equivalent to the equations\nof motion obtained from a quantum Hamiltonian in the\nmean-\feld approximation [24].\nLet us discuss the e\u000bect of thermal \ructuations on\nthe dynamics of the nanomagnet at subkelvin temper-\natures and in high vacuum. These conditions are com-\nmon in recent experiments with levitated particles [58{\n60]. The thermal \ructuations of magnetization dy-\nnamics, captured by the last term in Eq. (5), lead\nto thermally activated transition of the magnetic mo-\nment between the two stable orientations along the\nanisotropy axis [54, 61]. Such process can be quan-\nti\fed by the N\u0013 eel relaxation time, which is given by\n\u001cN\u0019(\u0019=! A)p\nkBT=(kaV)ekaV=(kBT). Thermal acti-\nvation can be neglected when \u001cNis larger than other\ntimescales of magnetization dynamics, namely the pre-\ncession timescale given by \u001cL\u00111=j!e\u000bj, and the Gilbert\ndamping timescale given by \u001cG\u00111=(\u0011j!e\u000bj). Con-\nsidering for simplicity j!e\u000bj\u00182!A, for a particle size\na= 2b= 1 nm and temperature T= 1 K, and the\nvalues of the remaining parameters as in Table I, the ra-\ntio of the timescales is of the order \u001cN=\u001cL\u0018103, and\nit is signi\fcantly increased for larger particle sizes and\nat smaller temperatures. We remark that, for the val-\nues considered in this article, \u001cNis much larger than the\nlongest dynamical timescale in Eqs. (1-5) which is associ-\nated with the motion along ex. Thermal activation of the\nmagnetic moment can therefore be safely neglected. The\nstochastic e\u000bects ascribed to the background gas, cap-\ntured by the last terms in Eqs. (3-4), are expected to be\nimportant at high temperatures (namely, a regime where\nMkBT\r2\n0a2=\u00162&1 [32]). At subkelvin temperatures and\nin high vacuum these \ructuations are weak and, con-\nsequently, they do not destroy the deterministic e\u000bects\n3The expressions for \u0000 cmand \u0000 rotfor a cylindrical particle\ncapture the order of magnitude of the dissipation rates for a\nspheroidal particle [55, 56].4\ncaptured by the remaining terms in Eqs. (1-5) [33]. In-\ndeed, for the values of parameters given in Table I and\nfora= 2b,MkBT\r2\n0a2=\u00162\u00190:8T=(a[nm]). For sub-\nkelvin temperatures and particle sizes a>1 nm, thermal\n\ructuations due to the background gas can therefore be\nsafely neglected.\nIn the following we thus neglect stochastic e\u000bects by\nsetting\u001ep=\u0018l=\u0010b= 0, and we consider only the de-\nterministic part of Eqs. (1-5) as an appropriate model\nfor the dynamics [8, 33, 54]. In App. B we carry out\nthe analysis of the dynamics including the e\u000bects of gas\n\ructuations in equations (1-5), and we show that the\nresults presented in the main text remain qualitatively\nvalid even in the presence of thermal noise. For the mag-\nnetic \feld B(r) we hereafter consider a Io\u000be-Pritchard\nmagnetic trap, given by\nB(r) =ex\u0014\nB0+B00\n2\u0012\nx2\u0000y2+z2\n2\u0013\u0015\n\u0000ey\u0012\nB0y+B00\n2xy\u0013\n+ez\u0012\nB0z\u0000B00\n2xz\u0013\n;(6)\nwhereB0;B0andB00are, respectively, the \feld bias, gra-\ndient and curvature [62]. We remark that this is not a\nfundamental choice, and di\u000berent magnetic traps, pro-\nvided they have a non-zero bias \feld, should result in\nsimilar qualitative behavior.\nB. Initial conditions\nThe initial conditions for the dynamics in Eqs. (1-5),\nnamely at time t= 0, depend on the initial state of the\nsystem, which is determined by the preparation of the\nnanomagnet in the magnetic trap. In our analysis, we\nconsider the nanomagnet to be prepared in the thermal\nstate of an auxiliary loading potential at the temperature\nT. Subsequently, we assume to switch o\u000b the loading\npotential at t= 0, while at the same time switching\non the Io\u000be-Pritchard magnetic trap. The choice of the\nauxiliary potential is determined by two features: (i) it\nallows us to simply parametrize the initial conditions by a\nsingle parameter, namely the temperature T, and (ii) it is\nan adequate approximation of general trapping schemes\nused to trap magnetic particles.\nRegarding point (i), we assume that the particle is lev-\nitated in a harmonic trap, in the presence of an external\nmagnetic \feld applied along ex. This loading scheme\nprovides, on the one hand, trapping of the center-of-mass\ndegrees of freedom, with trapping frequencies denoted by\n!i(i=x;y;z ). On the other hand, the magnetic moment\nin this case is polarized along ex. The Hamiltonian of the\nsystem in such a con\fguration reads Haux=p2=(2M) +P\ni=x;y;zM!2\nir2\ni=2+LI\u00001L=2\u0000kaVe2\n3;x\u0000\u0016xBaux, where\nBauxdenotes the magnitude of the external magnetic\n\feld, which we for simplicity set to Baux=B0in all our\nsimulations. At t= 0 the particle is released in the mag-\nnetic trap given by Eq. (6). For the degrees of freedomx\u0011(~r;~p;`;mx)T, we take as the initial displacement\nfrom the equilibrium the corresponding standard devia-\ntion in a thermal state of Haux. More precisely, xi(0) =\nxi;e+ (hx2\nii\u0000hxii2)1=2, wherexi;edenotes the equilib-\nrium value, andhxk\nii\u0011Z\u00001R\ndxxk\niexp[\u0000Haux=(kBT)],\nwithk= 1;2 and the partition function Z. For the Eu-\nler angles \nwe use \n 1(0)\u0011cos\u00001[\u0000p\nhcos2\n1i] and\n\ni(0)\u0011cos\u00001[p\nhcos2\nii] (i= 2;3). The initial condi-\ntions for e3follow from \nusing the transformation given\nin App. A.\nRegarding point (ii), the initial conditions obtained in\nthis way describe a trapped particle prepared in a ther-\nmal equilibrium in the presence of an external loading\npotential where the center of mass is decoupled from the\nmagnetization and the rotational dynamics. It is outside\nthe scope of this article to study in detail a particular\nloading scheme. However, we point out that an auxil-\niary potential given by Hauxcan be obtained, for exam-\nple, by trapping the nanomagnet using a Paul trap as\ndemonstrated in recent experiments [19, 21, 37, 63{70].\nIn particular, trapping of a ferromagnetic particle has\nbeen demonstrated in a Paul trap at P= 10\u00002mbar,\nwith center-of-mass trapping frequency of up to 1 MHz,\nand alignment of the particle along the direction of an\napplied \feld [19]. We note that particles are shown to\nremain trapped even when the magnetic \feld is varied\nover many orders of magnitudes or switched o\u000b. We re-\nmark further that alignment of elongated particles can\nbe achieved using a quadrupole Paul trap even in the\nabsence of magnetic \feld [55, 71].\nC. Linear stability\nIn the absence of thermal \ructuations, an equilibrium\nsolution of Eqs. (1-5) is given by ~re=~pe=`e= 0 and\ne3;e=me=\u0000ex. This corresponds to the con\fguration\nin which the nanomagnet is \fxed at the trap center, with\nthe magnetic moment along the anisotropy axis and anti-\naligned to the bias \feld B0. Linear stability analysis of\nEqs. (1-5) shows that the system is unstable, as expected\nfor a gyroscopic system in the presence of dissipation [28].\nHowever, when the nanomagnet is metastable, it is still\npossible for it to levitate for an extended time before\nbeing eventually lost from the trap, as in the case of a\nclassical magnetic top [25{27]. As we show in the fol-\nlowing sections, the dynamics of the system, and thus its\nmetastability, strongly depend on the applied bias \feld\nB0. We identify two relevant regimes: (i) strong-\feld\nregime, de\fned by bias \feld values B0> B atom, and\n(ii) weak-\feld regime, de\fned by B0< B atom, where\nBatom\u00112kaV=\u0016. This di\u000berence is reminiscent of the\ntwo di\u000berent stable regions which arise as a function of\nB0in the linear stability diagram in the absence of dis-\nsipation [see Fig. 1(b)] [23, 24]. In Sec. III and Sec. IV\nwe investigate the possibility of metastable levitation by\nsolving numerically Eqs. (1-5) in the strong-\feld and\nweak-\feld regime, respectively.5\nIII. DYNAMICS IN THE STRONG-FIELD\nREGIME: ATOM PHASE\nThe strong-\feld regime, according to the de\fnition\ngiven in Sec. II C, corresponds to the blue region in the\nlinear stability diagram in the absence of dissipation,\nshown in Fig. 1(b). This region is named atom phase\nin [23, 24], and we hereafter refer to the strong-\feld\nregime as the atom phase. This parameter regime corre-\nsponds to the condition !L\u001d!A;!I. In this regime, the\ncoupling of the magnetic moment \u0016and the anisotropy\naxise3is negligible, and, to \frst approximation, the\nnanomagnet undergoes a free Larmor precession about\nthe local magnetic \feld. In the absence of dissipation,\nthis stabilizes the system in full analogy to magnetic trap-\nping of neutral atoms [72, 73].\nIn Fig. 2(a-c) we show the numerical solution of\nEqs. (1-5) for nanomagnet dimensions a= 2b= 20\nnm and the bias \feld B0= 200 mT. As evidenced by\nFig. 2(a), the magnetization mxof the particle changes\ndirection. During this change, the mechanical angu-\nlar momentum lxchanges accordingly in the manifesta-\ntion of the Einstein{de Haas e\u000bect, such that the to-\ntal angular momentum m+`is conserved4. The dy-\nnamics observed in Fig. 2(a) is indicative of Gilbert-\ndamping-induced magnetization switching, a well-known\nphenomenon in which the projection of the magnetic mo-\nment along the e\u000bective magnetic \feld !e\u000b=\r0changes\nsign [30]. This is expected to happen when the applied\nbias \feldB0is larger than the e\u000bective magnetic \feld\nassociated with the anisotropy, given by \u0018!A=\r0. Mag-\nnetization switching displaces the system from its equi-\nlibrium position on a timescale which is much shorter\nthan the period of center-of-mass oscillations, estimated\nfrom [24] to be \u001ccm\u00181\u0016s. The nanomagnet thus shows\nno signature of con\fnement [see Fig. 2(b)].\nThe timescale of levitation in the atom phase is given\nby the timescale of magnetization switching, which we\nestimate as follows. As evidenced by Fig. 2(a-b), the\ndynamics of the center of mass and the anisotropy axis\nare approximately constant during switching, such that\n!e\u000b\u0019!e\u000b(t= 0). Under this approximation and as-\nsuming\u0011\u001c1, the magnetic moment projection mk\u0011\n!e\u000b\u0001m=j!e\u000bjevolves as\n_mk\u0019\u0011[!L+ 2!Amk](1\u0000m2\nk): (7)\nAccording to Eq. (7) the component mkexhibits switch-\ning ifmk(t= 0)&\u00001 and!L=2!A>1 [30], both of\nwhich are ful\flled in the atom phase. Integrating Eq. (7)\nwe obtain the switching time \u001c[de\fned as mk(\u001c)\u00110],\nwhich can be well approximated by\n\u001c\u0019ln\u0000\n1 +jmk(t= 0)j\u0001\n2\u0011(!L+ 2!A)\u0000ln\u0000\n1\u0000jmk(t= 0)j\u0001\n2\u0011(!L\u00002!A):(8)\n4We always \fnd the transfer of angular momentum to the center\nof mass angular momentum r\u0002pto be negligible.\nFigure 2. Dynamics in the atom phase. (a) Dynamics of\nthe magnetic moment component mx, the mechanical angular\nmomentum component lx, and the anisotropy axis component\ne3;xfor nanomagnet dimensions a= 2b= 20 nm and the bias\n\feldB0= 200 mT. For the initial conditions we consider\ntrapping frequencies !x= 2\u0019\u00022 kHz and!y=!z= 2\u0019\u000250\nkHz. Unless otherwise stated, for the remaining parameters\nthe numerical values are given in Table I. (b) Center-of-mass\ndynamics for the same case considered in (a). (c) Dynamics of\nthe magnetic moment component mk. Line denoted by circle\ncorresponds to the case considered in (a). Each remaining\nline di\u000bers by a single parameter, as denoted by the legend.\nDotted vertical lines show Eq. (8). (d) Switching time given\nby Eq. (8) as a function of the bias \feld B0and the major\nsemi-axisa. In the region left of the thick dashed line the\ndeviation from the exact value is more than 5%. Hatched\narea is the unstable region in the linear stability diagram in\nFig. 1.(b).\nThe estimation Eq. (8) is in excellent agreement with\nthe numerical results for di\u000berent parameter values [see\nFig. 2(c)].\nMagnetization switching characterizes the dynamics of\nthe system in the entire atom phase. In particular, in\nFig. 2(d) we analyze the validity of Eq. (8) for di\u000berent\nvalues of the bias \feld B0and the major semi-axis a, as-\nsumingb=a=2. The thick dashed line shows the region\nwhere Eq. (8) di\u000bers from the exact switching time, as\nestimated from the full dynamics of the system, by 5%;\nleft of this line the deviation becomes increasingly more\nsigni\fcant, with Eq. (8) predicting up to 20% larger val-\nues close to the stability border (namely, for bias \feld\nclose toBatom = 90 mT). We believe that the signi\f-\ncant deviation close to the border of the atom phase is\ndue to the non-negligible coupling to the anisotropy axis,6\nFigure 3. Dynamics in the Einstein{de Haas phase. (a) Motion of the system in the ey-ezplane until time t= 5\u0016s for\nnanomagnet dimensions a= 2b= 2 nm and the bias \feld B0= 0:5 mT. For the initial conditions we consider trapping\nfrequencies !x= 2\u0019\u00022 kHz and !y=!z= 2\u0019\u00021 MHz. For the remaining parameters the numerical values are given\nin Table I. (b) Dynamics of the projection mkand (c) dynamics of the anisotropy axis component e3;x, for the same case\nconsidered in (a). (d) Dynamics of the center-of-mass component ryand (e) dynamics of the magnetic moment component\nmxon a longer timescale, for the same values of parameters as in (a). (f) Escape time t?as a function of gas pressure P, for\ndi\u000berent con\fgurations in the Einstein{de Haas phase. Circles correspond to the case considered in (a). Each remaining case\ndi\u000bers by parameters indicated by the legend. (g) Escape time t?as a function of the major semi-axis a, with the values of\nthe remaining parameters as in (a). Dashed vertical line denotes the upper limit of the Einstein{de Haas phase, given by the\ncritical \feld BEdH,1 [see Fig. 1(b)].\nwhich results in additional mechanisms not captured by\nthe simple model Eq. (7). In fact, it is known that cou-\npling between magnetization and mechanical degrees of\nfreedom might have an impact on the switching dynam-\nics [74]. As demonstrated by Fig. 2(d), the switching\ntime is always shorter than the center-of-mass oscillation\nperiod\u001ccm, and thus no metastability can be observed in\nthe atom phase.\nLet us note that the conclusions we draw in Fig. 2\nremain valid if one varies the anisotropy constant ka,\nGilbert damping parameter \u0011, and the temperature T,\nas we show in App. C. Finally, we note that the dis-\nsipation due to the background gas has negligible ef-\nfects. In particular, for the values assumed in Fig. 2(a-b)\nthe timescale of the gas-induced dissipation is given by\n1=\u0000 = 440\u0016s.\nIV. DYNAMICS IN THE WEAK-FIELD\nREGIME: EINSTEIN{DE HAAS PHASE\nWe now focus on the regime of weak bias \feld, cor-\nresponding to the condition !L\u001c!A. In this regime\nmagnetization switching does not occur, and the dynam-\nics critically depend on the particle size. In the follow-\ning we focus on the regime of small particle dimensions,i.e.!L\u001c!I, which, as we will show, is bene\fcial for\nmetastability. In the absence of dissipation, this regime\ncorresponds to the Einstein{de Haas phase [red region\nin Fig. 1(b)] [23, 24]. The hierarchy of energy scales in\nthe Einstein{de Haas phase (namely, !L\u001c!A;!I) man-\nifests in two ways: (i) the anisotropy is strong enough to\ne\u000bectively \\lock\" the direction of the magnetic moment \u0016\nalong the anisotropy axis e3(!A\u001d!L), and (ii) accord-\ning to the Einstein{de Haas e\u000bect, the frequency at which\nthe nanomagnet would rotate if \u0016switched direction is\nsigni\fcantly increased at small dimensions ( !I\u001d!L),\nsuch that switching can be prevented due to energy con-\nservation [4]. In the absence of dissipation, the combina-\ntion of these two e\u000bects stabilizes the system.\nIn Fig. 3(a-c) we show the numerical solution of\nEqs. (1-5) for nanomagnet dimensions a= 2b= 2 nm\nand the bias \feld B0= 0:5 mT. The nanomagnet is\nmetastable, as evidenced by the con\fned center-of-mass\nmotion shown in Fig. 3(a). In Fig. 3(b-c) we show the\ndynamics of the magnetic moment component mkand\nthe anisotropy axis component e3;x, respectively, which\nindicates that no magnetization switching occurs in this\nregime. We remark that the absence of switching can-\nnot be simply explained on the basis of Eqs. (7-8). In\nfact, the simple model of magnetization switching, given7\nby Eq. (7), assumes that the dynamics of the rotation\nand the center-of-mass motion happen on a much longer\ntimescale than the timescale of magnetization dynam-\nics. However, in this case rotation and magnetization\ndynamics occur on a comparable timescale, as evidenced\nby Fig. 3(b-c). The weak-\feld condition alone ( !L\u001c!A)\nis thus not su\u000ecient to correctly explain the absence of\nswitching, and the role of particle size ( !L\u001c!I) needs\nto be considered.\nLet us analyze the role of Gilbert damping in this case.\nSince in the Einstein{de Haas phase mk\u00181, we de\fne\nm\u0011e3+\u000em, where\u000emrepresents the deviation of m\nfrom the anisotropy axis e3, and we assumej\u000emj\u001cje3j\n[see Fig. 3(b)]. This allows us to simplify Eq. (5) as\n\u000e_m\u0019!e\u000b\u0002\u000em\u0000\u0011[2!A+!3e3\u0001(m+`)]\u000em;(9)\nwhere!3\u0011\u0016=(\r0I3), withI3the principal moment of\ninertia along e3. As evidenced by Eq. (9), the only e\u000bect\nof Gilbert damping is to align mande3on a timescale\ngiven by\u001c0\u00111=[\u0011(2!A+!3)], irrespective of the dy-\nnamics of e3. For the values of parameters considered in\nFig. 3(a-c), \u001c0= 5 ns, and it is much shorter than the\ntimescale of center-of-mass dynamics, given by \u001ccm\u00181\n\u0016s. For all practical purposes, the magnetization in the\nEinstein{de Haas phase can be considered frozen along\nthe anisotropy axis. The nanomagnet in the presence of\nGilbert damping is therefore equivalent to a hard magnet\n(i. e.ka!1 ) [24].\nThe main mechanism behind the instability in the\nEinstein{de Haas phase is thus gas-induced dissipation.\nIn Fig. 3(d-e) we plot the dynamics of the center-of-\nmass component ryand the magnetic moment compo-\nnentmxon a longer timescale, for two di\u000berent values of\nthe pressure P. The e\u000bect of gas-induced dissipation is\nto dampen the center-of-mass motion to the equilibrium\nposition, while the magnetic moment moves away from\nthe equilibrium. Both processes happen on a timescale\ngiven by the dissipation rate \u0000. When ex=mx\u00190, the\nsystem becomes unstable and ultimately leaves the trap\n[see arrow in Fig. 3(d)]. We de\fne the escape time t?as\nthe time at which the particle position is y(t?)\u00115y(0),\nand we show it in Fig. 3(f) as a function of pressure Pfor\ndi\u000berent con\fgurations in the Einstein{de Haas phase,\nand forb=a=2. Fig. 3(f) con\frms that the dissipation\na\u000bects the system on a timescale which scales as \u00181=P.\nThe metastability of the nanomagnet in the Einstein{de\nHaas phase is therefore limited solely by the gas-induced\ndissipation, which can be signi\fcantly reduced in high\nvacuum. Finally, in Fig. 3(g) we analyze the e\u000bect of\nparticle size on metastability. Speci\fcally, we show the\nescape time t?as a function of the major semi-axis aat\nthe bias \feld B0= 0:5 mT, forb=a=2. The escape time\nis signi\fcantly reduced at increased particle sizes. This\ncon\frms the advantage of the Einstein{de Haas phase to\nobserve metastability, even in the presence of dissipation.V. DISCUSSION\nIn deriving the results discussed in the preceding sec-\ntions, we assumed (i) a single-magnetic-domain nanopar-\nticle with uniaxial anisotropy and constant magnetiza-\ntion, with the values of the physical parameters summa-\nrized in Table I, (ii) deterministic dynamics, i. e. the\nabsence of thermal \ructuations, (iii) that gravity can be\nneglected, and (iv) a non-rotating nanomagnet. Let us\njustify the validity of these assumptions.\nWe \frst discuss the values of the parameters given in\nTable I, which are used in our analysis. The material pa-\nrameters, such as \u001aM,\u001a\u0016,kaand\u0011, are consistent with,\nfor example, cobalt [75{78]. We remark that the uniax-\nial anisotropy considered in our model represents a good\ndescription even for materials which do not have an in-\ntrinsic magnetocrystalline uniaxial anisotropy, provided\nthat they have a dominant contribution from the uniaxial\nshape anisotropy. This is the case, for example, for fer-\nromagnetic particles with a prolate shape [75]. We point\nout that the values used here do not correspond to a spe-\nci\fc material, but instead they describe a general order\nof magnitude corresponding to common magnetic materi-\nals. Indeed, our results are general and can be particular-\nized to speci\fc materials by replacing the above generic\nvalues with exact numbers. As we show in App. C, the re-\nsults and conclusions presented here remain unchanged\neven when di\u000berent values of the parameters are con-\nsidered. The values used for the \feld gradient B0and\nthe curvature B00have been obtained in magnetic mi-\ncrotraps [62, 79{82]. The values of the gas pressure P\nand the temperature Tare experimentally feasible, with\nnumerous recent experiments reaching pressure values as\nlow asP= 10\u00006mbar [58, 68, 70, 83{85]. All the values\nassumed in our analysis are therefore consistent with cur-\nrently available technologies in levitated optomechanics.\nThermal \ructuations can be neglected at cryogenic\nconditions (as we argue in Sec. II A), as their e\u000bect is\nweak enough not to destroy the deterministic e\u000bects cap-\ntured by Eqs. (1-5). In particular, thermal activation of\nthe magnetization, as quanti\fed by the N\u0013 eel relaxation\ntime, can be safely neglected due to the large value of\nthe uniaxial anisotropy even for the smallest particles\nconsidered. As for the mechanical thermal \ructuations,\nwe con\frm that they do not modify the deterministic\ndynamics in App. B, where we simulate the associated\nstochastic dynamics.\nGravity, assumed to be along ex, can be safely ne-\nglected, since the gravity-induced displacement of the\ntrap center from the origin is much smaller than the\nlength scale over which the Io\u000be-Pritchard \feld signi\f-\ncantly changes [24]. Speci\fcally, the gravitational poten-\ntialMgx shifts the trap center from the origin r= 0\nalong exby an amount rg\u0011Mg= (\u0016B00), wheregis\nthe gravitational acceleration. On the other hand, the\ncharacteristic length scales of the Io\u000be-Pritchard \feld\nare given by \u0001 r0\u0011p\nB0=B00for the variation along\nex, and \u0001r0\u0011B0=B00for the variation o\u000b-axis. When-8\neverrg\u001c\u0001r0;\u0001r0, gravity has a negligible role in the\nmetastable dynamics of the system. In the parameter\nregime considered in this article, this is always the case.\nWe note that the condition to neglect gravity is the same\nas for a magnetically trapped atom, since both Mand\u0016\nscale with the volume.\nFinally, we remark that the analysis presented here\nis carried out for the case of a non-rotating nanomag-\nnet5. The same qualitative behavior is obtained even in\nthe presence of mechanical rotation (namely, considering\na more general equilibrium con\fguration with `e6= 0).\nThe analysis of dynamics in the presence of rotation is\nprovided in App. C. In particular, the dynamics in the\nEinstein{de Haas phase remains largely una\u000bected, pro-\nvided that the total angular momentum of the system is\nnot zero. In the atom phase, mechanical rotation leads to\ndi\u000berences in the switching time \u001c, as generally expected\nin the presence of magneto-mechanical coupling [74, 88].\nVI. CONCLUSION\nIn conclusion, we analyzed how the stability of a nano-\nmagnet levitated in a static magnetic \feld is a\u000bected by\nthe most relevant sources of dissipation. We \fnd that in\nthe strong-\feld regime (atom phase) the system is un-\nstable due to the Gilbert-damping-induced magnetiza-\ntion switching, which occurs on a much faster timescale\nthan the center-of-mass oscillations, thereby preventing\nthe observation of levitation. On the other hand, the sys-\ntem is metastable in the weak-\feld regime and for small\nparticle dimensions (Einstein{de Haas phase). In this\nregime, the con\fnement of the nanomagnet in a mag-\nnetic trap is limited only by the gas-induced dissipation.\nOur results suggest that the timescale of stable levitation\ncan reach and even exceed several hundreds of periods of\ncenter-of-mass oscillations in high vacuum. These \fnd-\nings indicate the possibility of observing the phenomenon\nof quantum spin stabilized magnetic levitation, which we\nhope will encourage further experimental research.\nThe analysis presented in this article is relevant for\nthe community of levitated magnetic systems. Speci\f-\ncally, we give precise conditions for the observation of\nthe phenomenon of quantum spin stabilized levitation\nunder experimentally feasible conditions. Levitating a\nmagnet in a time-independent gradient trap represents a\nnew direction in the currently growing \feld of magnetic\nlevitation of micro- and nanoparticles, which is interest-\ning for two reasons. First, the experimental observation\nof stable magnetic levitation of a non-rotating nanomag-\nnet would represent a direct observation of the quantum\nnature of magnetization. Second, the observation of such\n5Rotational cooling might be needed to unambiguously identify\nthe internal spin as the source of stabilization. Subkelvin cooling\nof a nanorotor has been recently achieved [86, 87], and cooling\nto\u0016K temperatures should be possible [56].phenomenon would be a step towards controlling and us-\ning the rich physics of magnetically levitated nanomag-\nnets, with applications in magnetometry and in tests of\nfundamental forces [9, 11, 34, 35].\nACKNOWLEDGMENTS\nWe thank G. E. W. Bauer, J. J. Garc\u0013 \u0010a-Ripoll, O.\nRomero-Isart, and B. A. Stickler for helpful discussions.\nWe are grateful to O. Romero-Isart, B. A. Stickler and\nS. Viola Kusminskiy for comments on an early ver-\nsion of the manuscript. C.C.R. acknowledges funding\nfrom ERC Advanced Grant QENOCOBA under the EU\nHorizon 2020 program (Grant Agreement No. 742102).\nV.W. acknowledges funding from the Max Planck So-\nciety and from the Deutsche Forschungsgemeinschaft\n(DFG, German Research Foundation) through Project-\nID 429529648-TRR 306 QuCoLiMa (\"Quantum Cooper-\nativity of Light and Matter\"). A.E.R.L. thanks the AMS\nfor the \fnancial support.\nAppendix A: Rotation to the body frame\nIn this appendix we de\fne the transformation ma-\ntrix between the body-\fxed and the laboratory reference\nframes according to the ZYZ Euler angle convention,\nwith the Euler angles denoted as \n= (\u000b;\f;\r )T. We\nde\fne the transformation between the laboratory frame\nOexeyezand the body frame Oe1e2e3as follows,\n0\n@e1\ne2\ne31\nA=R(\n)0\n@ex\ney\nez1\nA; (A1)\nwhere\nR(\n)\u0011Rz(\u000b)Ry(\f)Rz(\r) =0\n@cos\rsin\r0\n\u0000sin\rcos\r0\n0 0 11\nA\n0\n@cos\f0\u0000sin\f\n0 1 0\n\u0000sin\f0 cos\f1\nA0\n@cos\u000bsin\u000b0\n\u0000sin\u000bcos\u000b0\n0 0 11\nA:(A2)\nAccordingly, the components vj(j= 1;2;3) of a vector\nvin the body frame Oe1e2e3and the components v\u0017\n(\u0017=x;y;z ) of the same vector in the laboratory frame\nOexeyezare related as\n0\n@v1\nv2\nv31\nA=RT(\n)0\n@vx\nvy\nvz1\nA: (A3)\nThe angular velocity of a rotating particle !can be writ-\nten in terms of the Euler angles as != _\u000bez+_\fe0\ny+ _\re3,\nwhere ( e0\nx;e0\ny;e0\nz)T=Rz(\u000b)(ex;ey;ez)Tdenotes the\nframeOe0\nxe0\nye0\nzobtained after the \frst rotation of the9\nlaboratory frame Oexeyezin the ZYZ convention. By\nusing (A1) and (A2), we can rewrite angular velocity in\nterms of the body frame coordinates,\n!= _\u000b2\n4R(\n)\u000010\n@e1\ne2\ne31\nA3\n5\n3+_\f2\n4R(\r)\u000010\n@e1\ne2\ne31\nA3\n5\n2+ _\re3;\n(A4)\nwhich is compactly written as ( !1;!2;!3)T=A(\n)_\n,\nwith\nA(\n) =0\n@\u0000cos\rsin\fsin\r0\nsin\fsin\rcos\r0\ncos\f 0 11\nA: (A5)\nAppendix B: Dynamics in the presence of thermal\n\ructuations\nIn this appendix we consider the dynamics of a lev-\nitated nanomagnet in the presence of stochastic forces\nand torques induced by the surrounding gas. The dy-\nnamics of the system are described by the following set\nof stochastic di\u000berential equations (SDE),\nd~r=!I~pdt; (B1)\nde3=!\u0002e3dt; (B2)\nd~p= [!Lr~r[m\u0001b(~r)]\u0000\u0000cm~p] dt+p\nDcmdWp;(B3)\nd`= [!Lm\u0002b(~r)\u0000_m\u0000\u0000rot`] dt+p\nDrotdWl;\n(B4)\ndm=m\n1 +\u00112\u0002[!e\u000b\u0000\u0011m\u0002(!+!e\u000b+\u0011!\u0002m)]dt;\n(B5)\nwhere we model the thermal \ructuations as uncorrelated\nGaussian noise represented by a six-dimensional vector\nof independent Wiener increments (d Wp;dWl)T. The\ncorresponding di\u000busion rate is described by the tensors\nDcmandDrotwhich, in agreement with the \ructuation-\ndissipation theorem, are related to the corresponding dis-\nsipation tensors \u0000 cmand \u0000 rotasDcm\u00112\u0000cm\u001f;D rot\u0011\n2\u0000rot\u001f, where\u001f\u0011MkBT\r2\n0a2=\u00162.\nIn the following we numerically integrate Eqs. (B1-B5)\nusing the stochastic Euler method implemented in the\nstochastic di\u000berential equations package in MATLAB. As\nthe e\u000bect of thermal noise is more prominent for small\nparticles at weak \felds, we focus on the Einstein-de Haas\nregime considered in Sec. IV. We show that even in this\ncase the e\u000bect of thermal \ructuations leads to dynamics\nwhich are qualitatively very close to the results obtained\nin Sec. IV. In Fig. 4 we present the results of the stochas-\ntic integrator by averaging the solution of 100 di\u000berent\ntrajectories calculated using the same parameters consid-\nered in Fig. 3(a-c). The resulting average dynamics agree\nqualitatively with the results obtained by integrating the\ncorresponding set of deterministic equations Eqs. (1-5)\nFigure 4. Stochastic dynamics of a nanomagnet for the same\nparameter regime as considered in Fig. 3. (a) Average motion\nof the system in the y-zplane until time t= 5\u0016s. (b) Dy-\nnamics of center of mass along the ey(top) and ez(bottom)\ndirections. (c) Dynamics of the anisotropy axis component\ne3;x. (d) Numerical error as function of time. The simulations\nshow the results of the average of 100 di\u000berent realizations of\nthe system dynamics. In panels (b-d) the solid dark lines are\nthe average trajectories, while the shaded area represents the\nstandard deviation.\n[cfr. Fig. 3(a-c)]. The main e\u000bect of thermal excitations\nis to shift the center of oscillations of the particle's de-\ngrees of freedom around the value given by the thermal\n\ructuations. This is more evident for the dynamics of\ne3[cfr. Fig. 4(c) and Fig. 3(c)]. We thus conclude that\nthe deterministic equations Eqs. (1-5) considered in the\nmain text correctly capture the metastable behavior of\nthe system. We emphasize that the results presented in\nthis section include only the noise due to the surround-\ning gas. Should one be interested in simulating the ef-\nfect of the \ructuations of the magnetic moment, the Eu-\nler method used here is not appropriate, and the Heun\nmethod should be used instead [89].\nLet us conclude with a technical note on the numerical\nsimulations. In the presence of dissipation and thermal\n\ructuations the only conserved quantity of the system is\nthe magnitude of the magnetic moment ( jmj= 1). We\nthus use the deviation 1 \u0000jmj2as a measure of the numer-\nical error in both the stochastic and deterministic sim-\nulations presented in this article. For the deterministic\nsimulations the error stays much smaller than any other\nphysical degree of freedom of the system during the whole\nsimulation time. The simulation of the stochastic dynam-\nics shows a larger numerical error [see Fig. 4(d)], which\ncan be partially reduced by taking a smaller time-step\nsize. We note that, for the value of magnetic anisotropy\ngiven in Table I, the system of SDE is sti\u000b. This, together\nwith the requirement imposed on the time-step size by10\nthe numerical error, ultimately limits the maximum time\nwe can simulate to a few microseconds. However, this is\nsu\u000ecient to validate the agreement between the SDE and\nthe deterministic simulations presented in the article.\nAppendix C: Additional \fgures\nIn this appendix we provide additional \fgures.\n1. Dynamics in the atom phase\nIn Fig. 5 we analyze magnetization dynamics in the\natom phase as a function of di\u000berent system parame-\nters. In Fig. 5(a) we show how magnetization switching\nchanges as the anisotropy constant kais varied. We con-\nsider the bias \feld B0= 1100 mT, which is larger than\nthe value considered in the main text. This is done to en-\nsure thatB0>B atom for all anisotropy values. Fig. 5(a)\ndemonstrates that the switching time \u001c, given by Eq. (8),\nis an excellent approximation for the dynamics across a\nwide range of values for the anisotropy constant ka. The\nlarger discrepancy between Eq. (8) and the line showing\nthe case with ka= 106J/m3is explained by the prox-\nimity of this point to the unstable region (in this case\ngiven by the critical \feld Batom = 900 mT), and better\nagreement is recovered at larger bias \feld values.\nIn Fig. 5(b) we analyze the validity of Eq. (8) for dif-\nferent values of the Gilbert damping parameter \u0011and the\ntemperature T. The thick dashed line shows the region\nwhere Eq. (8) di\u000bers from the exact switching time by\n5%; below this line the deviation becomes increasingly\nmore signi\fcant. As evidenced by Fig. 5(b), \u001cshows lit-\ntle dependence on T; its order of magnitude remains con-\nstant over a wide range of cryogenic temperatures. On\nthe other hand, the dependence on \u0011is more pronounced.\nIn fact, reducing the Gilbert parameter signi\fcantly de-\nlays the switching time, leading to levitation times as\nlong as\u00181\u0016s.\nAdditionally, we point out that \u001cdepends on the \feldgradientB0and curvature B00through the initial con-\nditionmk(t= 0). In particular, magnetization switch-\ning can be delayed by decreasing B0, as this reduces\nthe initial misalignment of the magnetization and the\nanisotropy axis (i. e. jmk(t= 0)j!1).\n2. Dynamics in the presence of rotation\nIn Fig. 6 we consider a more general equilibrium con-\n\fguration, namely a nanomagnet initially rotating such\nthat in the equilibrium point Le=\u0000I3!Sex, with!S>0\ndenoting the rotation in the clockwise direction. This\nequilibrium point is linearly stable in the absence of dis-\nsipation [23, 24], with additional stability of the system\nprovided by the mechanical rotation, analogously to the\nclassical magnetic top [25{27].\nIn Fig. 6(a) we analyze how magnetization switching\nin the atom phase changes in the presence of rotation for\ndi\u000berent values of parameters. The rotation has a slight\ne\u000bect on the switching time \u001c, shifting it forwards (back-\nwards) in case of a clockwise (counterclockwise) rotation.\nThis is generally expected in the presence of magneto-\nmechanical coupling [74, 88].\nIn Fig. 6(b) we show the motion in the y-zplane in\nthe Einstein{de Haas phase for both directions of rota-\ntion. This can be compared with Fig. 3(a). 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Mech. , P09008\n(2014)."    },    {        "title": "2008.08043v2.A_class_of_Finite_difference_Methods_for_solving_inhomogeneous_damped_wave_equations.pdf",        "content": "A CLASS OF FINITE DIFFERENCE METHODS FOR SOLVING\nINHOMOGENEOUS DAMPED WAVE EQUATIONS\nFAZEL HADADIFARD, SATBIR MALHI, AND ZHENGYI XIAO\nAbstract. In this paper, a class of \fnite di\u000berence numerical techniques is presented\nto solve the second-order linear inhomogeneous damped wave equation. The consistency,\nstability, and convergences of these numerical schemes are discussed. The results obtained\nare compared to the exact solution, ordinary explicit, implicit \fnite di\u000berence methods,\nand the fourth-order compact method (FOCM). The general idea of these methods is\ndeveloped by using C0-semigroups operator theory. We also showed that the stability\nregion for the explicit \fnite di\u000berence scheme depends on the damping coe\u000ecient.\n1.introduction\nThe damped wave equation is an important evolution model. Physicists and engineers\nwidely use it in describing the propagation of water waves, sound waves, electromagnetic\nwaves, etc. For instance, a model that describes the transverse vibrations of a string of a\n\fnite length in the presence of an external force proportional to the velocity satis\fes the\nfollowing partial di\u000berential equation (PDE)\nutt= \u0001u\u0000\r(x)ut+g(x;t);fora\u0014x\u0014b; t2R; (1)\nwith initial conditions\nu(x;0) =\u001e(x); ut(0;x) = (x);fora\u0014x\u0014b;\nand boundary conditions\nu(a;t) =ua(t)u(b;t) =ub(t); t2R;\nwhere\r\u00150 is the damping force, u(x;t) is the position of a point xin the string, at\ninstantt. The functions \u001e(x); (x) and their derivatives are continuous functions of xand\nthe forcing function g(x;t)2L1\nx(R). The study of the numerical solution of this model\nwill be our main focus in this article.\nIn general, the damping reduces the amplitude of vibration, and therefore, it is desirable\nto have some amount of damping to achieve stability in the system. One can \fnd a detailed\nstudy in [4, 2, 14] of the e\u000bect of damping in the long-time stability of the equation (1).\nAlso, for practical purposes, it is important to know how much damping is needed in the\nsystem to ensure the fastest decay rate in the amplitude of the wave as time evolves. For\nexample, in the case of the 3D tsunami wave, we would like to know the size and structure\nof the damping force to bring the amplitude of the tsunami to a safe level before it hits the\nDate : December 23, 2021.\n2000 Mathematics Subject Classi\fcation. 65M06, 37N30, 65N22 .\nKey words and phrases. damped wave equation, numerical method, Pad\u0013 e approximation, compact \fnite\ndi\u000berence scheme, unconditionally stable, convergence.\n1arXiv:2008.08043v2  [math.NA]  22 Dec 20212 FAZEL HADADIFARD, SATBIR MALHI, AND ZHENGYI XIAO\nshore (see [17] and references within). In the case of the damping terms as a function of\ntime and space, obtaining an analytic solution is a challenging problem. There comes the\nnumerical study to \fnd the approximate solution to such problems. In recent years, much\nattention has been given to studying the behaviours of the numerical solution of (1); see\nfor example [18, 3, 13, 5, 8].\nIn this manuscript, we develop a class of methods based on the properties of C0-\nsemigroups of the evolution equations, as well as the \fnite di\u000berence method (FD). Gen-\nerally speaking, the FD methods are easy to apply to partial di\u000berential equitations, but\nthey may not lead to optimal results depending on the type of equation. The techniques\nused in this article take advantage of the C0-semigroup property and Pad\u0013 e approximation,\nwhich lead to a better performance of new numerical schemes presented in this article.\nAt the time of writing this paper, we became aware of [12] that have a similar approach\nin which the authors drive a fourth-order implicit \fnite di\u000berence scheme to solve a second-\norder telegraph equation with constant coe\u000ecients. However, the author of [12] did not\nconsider the explicit \fnite di\u000berence schemes and used the higher-order approximation\nterms of the space derivative and time integration to attain higher-order accuracy of the\nnumerical solution. In this manuscript, in addition to driving a class of explicit and implicit\nmethods, we discussed the issue of the instability of the explicit \fnite di\u000berence methods.\nMoreover, this paper explains the importance of the non-zero damping term in the existence\nof the stability region as well. We have also shown that the explicit \fnite di\u000berence method\nproduces better results and costs a lot fewer calculations in its stability region.\nAn outline of the contents of this paper is as follows. In section 2, we set our numerical\nschemes and derive our method. Section 3 is devoted to the analytical properties of the\nmethod, i.e., consistency, stability, and convergence. Finally, in section 4, the numerical\nresults of our method are compared with some of the existing methods.\n2.The semigroup approach\nTo present a more convenient form of (1), we de\fne a new vector function\nU(x;t) = (u;ut)T; U 0= (\u001e(x); (x))T: (2)\nWith these changes, the equation (1) turns into an evolution equation of \frst-order in time\nUt=AU+G; (3)\nwhere\nA=0\n@0I\n\u0001\u0000\r(x)1\nA; G (x;t) =0\n@0\ng(x;t)1\nA;\nwith initial condition\nU(x;0) = (u(x;0);ut(x;0))T:\nThe system above is de\fned on a Hilbert space H=H1[a;b]\u0002L2(R). The domain of A\nisD(A) =H2[a;b]\u0002H1(R). Since\u0000Ais a dissipative and invertible operator on a Hilbert\nspace, it generates a C0-semigroup of contractions for t\u00150 by the Lumber-Phillips theoremA CLASS OF FINITE DIFFERENCE METHODS FOR SOLVING INHOMOGENEOUS DAMPED WAVE EQUATIONS 3\n[10]. Also, note that the inclusion D(A),! H is compact by the Rellich-Kondrachiv\ntheorem. Thus, the spectrum of Acontains only eigenvalues of \fnite multiplicity.\n2.1.Discretization. We use the central discretization for the Laplacian operator \u0001 as\n\u0001u(x;t) =u(x\u0000h;t)\u00002u(x;t) +u(x+h;t)\nh2:\nWe set the mesh points\nxi=a+ih; i = 0;1;2:::N; whereh=b\u0000a\nN\nof the interval [ a;b]. Then the continuous operator Acan be approximated by the matrix\noperator\nM(2N\u00002)=2\n40I\n1\nh2A\u0000\u00003\n5;\nwhereIis the identity matrix of order N\u00001, and\nA=2\n6666666664\u00002 1 0\u0001\u0001\u0001 0\n1\u00002 1\u0001\u0001\u0001 0\n0 1\u00002...0\n...............\n0\u0001\u0001\u0001 0 1\u000023\n7777777775\n(N\u00001)\u0002(N\u00001);\u0000 =2\n66666666664\r(x1) 0\u0001\u0001\u0001 0\n0\r(x2)::: 0\n0 0\u0001\u0001\u0001 0\n............\n0\u0001\u0001\u0001 0\r(xN\u00001)3\n77777777775\n(N\u00001)\u0002(N\u00001):(4)\nThe discrete operator M(2N\u00002)is de\fned on the \fnite-dimensional Banach space X(2N\u00002)=\nC(2N\u00002)[a;b].\nLetV2N\u00002(t) =\u0002u(x1;t);u(x2;t):::u(xN\u00001;t);ut(x1;t);\u0001\u0001\u0001ut(xN\u00001)\u0003Tbe a vector which\ndiscretizes the function U(x;t) = (u(x;t);@tu(x;t)) over the interval [ x1;xN\u00001], then (3)\nleads us to the following dynamical system\ndV2N\u00002(t)\ndt=2\n40I\n1\nh2A\u0000\u00003\n5V2N\u00002(t) +2\n40\nG(t)3\n5+2\n40\n1\nh2B(t)3\n5; (5)\nwhereG(t) = [g(x1;t);g(x2;t);:::;g (xN\u00001;t)]T,B(t) =\u0002ua(t);0;0;:::; 0;0;ub(t)\u0003\nand\nthe initial condition\nV2N\u00002(0) =\u0002\u001e(x1);\u001e(x2):::\u001e(xN\u00001); (x1);\u0001\u0001\u0001 (xN\u00001)\u0003T:\nWe will now drop the subscript 2 N\u00002 and write V2N\u00002(x;t) byV(t), andM2N\u00002byM\nin the rest of our presentation.\nSinceMis a bounded linear operator on a \fnite-dimensional space X(2N\u00002)\u0002H1\n0(R), it\ngenerates a C0-semigroup for each N. Then, by using the C0-semigroup theory of inhomo-\ngeneous evolution equations, we can construct the sequences of approximating solutions to\n(5) as\nV(t) =eMtV(0) +Zt\n0eM(t\u0000s)F(s)ds;4 FAZEL HADADIFARD, SATBIR MALHI, AND ZHENGYI XIAO\nwhere\nF(t) =2\n40\nG(t)3\n5+2\n40\n1\nh2B(t)3\n5:\nWe replace tbyt+kin the above equation and use the C0-semigroup property, eM(t+k)=\neMteMk, we get\nV(t+k) =eM(t+k)V(0) +Zt+k\n0eM(t+k\u0000s)F(s)ds\n=eMkeMtV(0) +eMkZt\n0eM(t\u0000s)F(s)ds+eMkZt+k\nteM(t\u0000s)F(s)ds\n=eMk\u0012\nV(t)\u0000Zt\n0eM(t\u0000s)F(s)ds\u0013\n+eMkZt\n0eM(t\u0000s)F(s)ds\n+eMkZt+k\nteM(t\u0000s)F(s)ds:\nThus,\nV(t+k) =eM(k)V(t) +eMkZt+k\nteM(t\u0000s)F(s)ds: (6)\nTo approximate the term eMk, we make use of the rational approximation of exponential\nfunctions, i.e., the Pad\u0013 e approximation.\n2.2.Pad\u0013 e Approximant. The Pad\u0013 e approximation is a rational approximation of a\nfunction of a given order [1]. The technique was developed around 1890 by Henri Pad\u0013 e,\nbut it goes back to F. G. Frobenius who introduced the idea and investigated the features\nof rational approximations of power series. The Pad\u0013 e approximation is usually superior\nwhen functions contain poles because the use of rational function allows them to be well\nrepresented. The Pad\u0013 e approximation often gives a better approximation of the function\nthan truncating its Taylor series, and it may still work where the Taylor series does not\nconverge.\nPad\u0013 e approximation gives the exponential functions e\u0012as\ne\u0012=1 +a1\u0012+a2\u00122+\u0001\u0001\u0001+aT\u0012T\n1 +b1\u0012+b2\u00122+\u0001\u0001\u0001+T\u0012S+cS+T+1\u0012S+T+1+O(\u0012S+T+2);\nwhereCS+T+1,ai's andb's are constants. The rational function\nRS;T(\u0012) :=1 +a1\u0012+a2\u00122+\u0001\u0001\u0001+aT\u0012T\n1 +b1\u0012+b2\u00122+\u0001\u0001\u0001+T\u0012S=PT(\u0012)\nQS(\u0012)(7)\nis the so-called Pad\u0013 e approximation of order ( S;T) toe\u0012with the leading error cS+T+1\u0012S+T+1.\nThe table below gives some Pad\u0013 e approximations of the exponential function[18].A CLASS OF FINITE DIFFERENCE METHODS FOR SOLVING INHOMOGENEOUS DAMPED WAVE EQUATIONS 5\n(S,T) RS;T(\u0012) Leading error\n(0,1) 1 +\u00121\n2\u00122\n(0,2) 1 +\u0012+1\n2\u00122 1\n6\u00123\n(1,0) 1\n1\u0000\u0012\u00001\n2\u00122\n(1,1) 1 +1\n2\u0012\n1\u00001\n2\u0012\u00001\n12\u00123\nNow combining (6) and (7), we get\nQS(Mk)V(t+k) =PT(Mk)V(t) (8)\n+PT(Mk)Zt+k\ntPT(M(t\u0000s))(QS(M(t\u0000s)))\u00001F(s)ds:\nFor the integration term on the right-hand side, one can use the numerical integration\nformula. Here, we will use the Trapezoidal approximation of integration to get the following\nnumerical scheme\nQS(Mk)V(t+k) =PT(Mk)V(t) +k\n2PT(Mk)F(t) +k\n2QS(Mk)F(t+k): (9)\nThis is the general form of our scheme, and each choice of QSandPTproduces explicit and\nimplicit \fnite di\u000berence methods to the solution of the damped wave equation (1). Next,\nwe present two schemes by taking ( S;T) = (0;1) and (S;T) = (1;1). Similarly, we can\ndevelop more schemes of di\u000berent order by taking di\u000berent values of SandTmentioned\nin the table above.\nExplicit Method ( FD\u0000(0;1)): If we set ( S;T) = (0;1) i.e.Q0(\u0012) = 1 andP1(\u0012) = 1+\u0012\nin (9), we will obtain the FD-(0,1) as\n(\nV(t+k) = (1 +Mk)V(t) +k\n2(I+Mk)F(t) +k\n2F(t+k);\nV0= [u1(0);\u0001\u0001\u0001;uN\u00001(0);@tu1(0);\u0001\u0001\u0001;@tuN\u00001(0)]:(10)\nImplicit Method ( FD\u0000(1;1)): By a choice of P1(\u0012) = 1 +1\n2\u0012andQ1(\u0012) = 1\u00001\n2\u0012in\n(9), we will obtain the FD-(1,1) as\n8\n><\n>:\u0000\n1\u00001\n2Mk\u0001\nV(t+k) =\u0000\n1 +1\n2Mk\u0001\nV(t) +k\n2\u0000\nI+1\n2Mk\u0001\nF(t)\n+k\n2\u0000\nI\u00001\n2Mk\u0001\nF(t+k);\nV0= [u1(0);\u0001\u0001\u0001;uN\u00001(0);@tu1(0);\u0001\u0001\u0001;@tuN\u00001(0)]:(11)6 FAZEL HADADIFARD, SATBIR MALHI, AND ZHENGYI XIAO\nIn the case of the implicit method, we need to solve a more extensive system of equations\nin each time step due to the implicit nature of the system. However, the analysis and\nnumerical results suggest that the implicit scheme gives us an accurate approximation\nand, more importantly, an unconditionally stable scheme.\n3.Consistency, Stability and Convergence\nIn this section, we will investigate the analytical properties of our numerical schemes\n(10) and (11). We will prove that the numerical methods (10) and (11) are consistent,\nstable, and hence convergent. We will use the direct analysis to prove the consistency,\nthe matrix method to prove the stability, and the Lax-equivalence theorem to prove the\nconvergence of our numerical schemes.\n3.1.Consistency. Given a partial di\u000berential equation Lu=fand a \fnite di\u000berence\nscheme,Fh;kv=f, we say that the \fnite di\u000berence scheme is consistent with the partial\ndi\u000berential equation if for any smooth function \u001e(x;t),\nL\u001e\u0000Fh;k\u001e!0 ash;k!0;\nor in other words, the local truncation goes to zero as the mesh size handktends to zero.\nThe partial di\u000berential equation\nUt\u00000\n@0I\n\u0001\u0000\r(x)1\nAU\u00000\n@0\ng(x;t)1\nA= 0;\nis approximated at the point ( xi;t) by thenthrow of the following di\u000berence equations\n1\nk(QS(Mk)V(t+k)\u0000PT(Mk)V(t))\u00001\n2PT(Mk)F(t)\u00001\n2QS(Mk)F(t+k) = 0;\nforn= 1;2;\u0001\u0001\u0001;(2N\u00002):\nThen the local truncation error Ti;t(U) is de\fned as the nthrow of\n1\nk(QS(Mk)U(t+k)\u0000PT(Mk)U(t))\u00001\n2PT(Mk)F(t)\u00001\n2QS(Mk)F(t+k);\nforn= 1;\u0001\u0001\u0001;(2N\u00002).\nThe truncated error depends on the choice of QSandPT. Therefore, we should consider\nthem case by case. Here we consider FD\u0000(0;1) andFD\u0000(1;1). The remaining cases\nfollow the same path.\n3.1.1.FD\u0000(0;1).The local truncation error T0;1\ni;t(U) of the explicit FD\u0000(0;1) is de\fned\nas thenthrow of\n1\nk(U(t+k)\u0000(I+Mk)U(t))\u00001\n2(I+Mk)F(t)\u00001\n2F(t+k)\nforn= 1;2\u0001\u0001\u0001;(2N\u00002).\nThus fori= 1;2;\u0001\u0001\u0001N\u00001, we get the following system of (2 N\u00002) equations\nT0;1\ni;t(U) =1\nk(u(xi;t+k)\u0000u(xi;t))\u0000ut(xi;t)\u0000k\n2g(xi;t);A CLASS OF FINITE DIFFERENCE METHODS FOR SOLVING INHOMOGENEOUS DAMPED WAVE EQUATIONS 7\nand\nT0;1\ni+N\u00001;t(U) =1\nkut(xi;t+k)\u00001\nh2(u(xi\u0000h;t)\u00002u(xi;t) +u(xi+h;t))\n\u00001\nk(1\u0000k\r(xi))ut(xi;t)\u0000(1\u0000\r(xi)k)\n2g(xi;t)\u00001\n2g(xi;t+k):\nBy Taylor series expansion, we get\nT0;1\ni;t(U) =k\n2!utt(xi;t) +k2\n3!uttt(xi;t) +\u0001\u0001\u0001\u0000k\n2g(xi;t)\nand\nT0;1\ni+N\u00001;t(U) = (utt(xi;t)\u0000uxx(xi;t) +\r(xi)ut(xi;t)\u0000g(xi;t))\n+k\n2!uttt(xi;t) +O(k2)\u00002h2\n4!uxxxx(xi;t) +O(h4) +\r(xi)k\n2g(xi;t)\n\u0000k\n2gt(xi;t) +O(k2):\nfori= 1;2;\u0001\u0001\u0001;(N\u00001).\nBy (1), the last ( N\u00001), equations can be written as\nT0;1\ni;t(U) =k\n2!uttt(xi;t) +O(k2)\u00002h2\n4!uxxxx(xi;t) +O(h4) +\r(xi)k\n2g(xi;t)\n\u0000k\n2gt(xi;t) +O(k2):\nWe observe as handkgo to zero, the truncation error Ti;t(U)!0. Hence, the numerical\nscheme is consistent. .\n3.1.2.FD\u0000(1;1).The local truncation error T1;1\ni;t(U) of the explicit FD\u0000(1;1) is de\fned\nas thenthrow of\n1\nk\u0012\u0012\nI\u00001\n2Mk\u0013\nU(t+k)\u0000\u0012\nI+1\n2Mk\u0013\nU(t)\u0013\n\u00001\n2\u0012\nI+1\n2Mk\u0013\nF(t)\u00001\n2\u0012\nI\u00001\n2Mk\u0013\nF(t+k)\nforn= 1;2;\u0001\u0001\u0001;(2N\u00002).\nThus fori= 1;2\u0001\u0001\u0001;N\u00001, we get the following system of (2 N\u00002) equations\nT1;1\ni;t(U) =1\nk(u(xi;t+k)\u0000u(xi;t))\u0000ut(xi;t)\u0000k\n4(g(xi;t+k)\u0000g(xi;t));\nand\nT1;1\ni+N\u00001;t(U) =\u0014\n\u00001\n2h2(u(xi\u0000h;t+k)\u00002u(xi;t+k) +u(xi+h;t+k)) +1\nk\u0012\n1 +\rk\n2\u0013\nut(xi;t+k)\u0015\n\u0000\u00141\n2h2(u(xi\u0000h;t)\u00002u(xi;t) +u(xi+h;t)) +1\nk\u0012\n1\u0000\rk\n2\u0013\nut(xi;t)\u0015\n\u00001\n2\u0014\n(1 +\rk\n2)g(xi;t) + (1\u0000\rk\n2)g(xi;t+k)\u0015\n:8 FAZEL HADADIFARD, SATBIR MALHI, AND ZHENGYI XIAO\nBy Taylor series expansion, we get\nT1;1\ni;t(U) =k\n2utt(xi;t)\u0000k2\n4gt(xi;t) +O(k3);\nand\nT1;1\ni+N\u00001;t(U) = (utt(xi;t)\u0000uxx(xi;t) +\r(xi)ut(xi;t)\u0000g(xi;t))\n+k\n2uttt(xi;t) +O(k2)\u0000k\n2uxxt(xi;t) +O(k2)\u0000h2\n2uxxxx(xi;t) +O(h4)\n\u0000kh2\n6uxxxxt +h2O(k2)\u0000k\n2gt(xi;t) +O(k2) +k3\r(xi)\n4gt(xi;t) +O(k3);\nfori= 1;2;\u0001\u0001\u0001;(N\u00001).\nBy (1), the last ( N\u00001), equations can be written as\nT1;1\ni;t(U) =k\n2uttt(xi;t) +O(k2)\u0000k\n2uxxt(xi;t) +O(k2)\u0000h2\n2uxxxx(xi;t) +O(h4)\n\u0000kh2\n6uxxxxt +h2O(k2)\u0000k\n2gt(xi;t) +O(k2) +k3\r(xi)\n4gt(xi;t) +O(k3):\nAshandkgo to zero, the truncation error Ti;t(U)!0. Hence, the numerical scheme is\nconsistent.\n3.2.Stability. To prove the stability of our numerical schemes, we show that there exists\na region \u0003 so that for every h;k2\u0003, all the eigenvalues of the ampli\fcation matrix related\nto the numerical schemes lie in or on the unit circle.\nProposition 1. The explicit FD-(0,1) approximation de\fned in (9)is stable for k <2\n\r\u0003\nandp\nk\nh<p\r\u0003\n2, where\r\u0003= maxx2[a;b]\r(x).\nThe following lemma will be used to prove Proposition 1.\nLemma 1. Letp(x) =ax2+bx+cbe a polynomial function with a >0, then necessary\nand su\u000ecient conditions for the polynomial p(x)to have the modulus of its roots less or\nequal to 1 are\n(i)jcj<a\n(ii)p(1)>0andp(\u00001)>0.\nOne can \fnd the proof of the above lemma in [7, 16].\nProof of proposition 1. The eigenvalues of the ampli\fcation matrix I+kMare the\nroots of the following quadratics equation\n\u00152+ (\u00002 +\r(xn)k)\u0015+ 1\u0000k\r(xn) + 4r2sin2\u0010n\u0019\n2N\u0011\n= 0; n= 1;\u0001\u0001\u0001;(N\u00001);\nwherer=k=h.\nNote for each n, there are two roots of the above polynomial, and hence we have 2 N\u00002\neigenvalues for the matrix I+kM.\nNext, in order to satisfy the conditions ( i) and (ii) of lemma (1), we impose restrictions\non\r\u0003andr. Indeed, the assumption ( i) is satis\fed if\n\u00001<1\u0000k\r(xn) + 4r2sin2\u0010n\u0019\n2N\u0011\n<1; n= 1;2;\u0001\u0001\u0001;N\u00001:A CLASS OF FINITE DIFFERENCE METHODS FOR SOLVING INHOMOGENEOUS DAMPED WAVE EQUATIONS 9\nThe right-hand inequality gives us\n4r2sin2\u0010n\u0019\n2N\u0011\n<k\r (xn)\u0014k\r\u0003;\nr2<k\r\u0003\n4 sin2\u0000n\u0019\n2N\u0001:\nThus,p\nk\nh<p\r\u0003\n2.\nNow, the \frst part of the assumption (ii) is satis\fed if\np(1) = 4r2sin2\u0010n\u0019\n2N\u0011\n>0;\nwhich is true as long as r>0.\nNow, the second part of assumption (ii) is satis\fed if\np(\u00001) = 4\u00002k\r(xn) + 4r2sin2\u0010n\u0019\n2N\u0011\n>0;\nwhich is true if\nk\r\u0003<2:\nHence the second part of the assumption (ii) of lemma (1) is satis\fed for k<2\n\r\u0003.\nProposition (1) tells us that the damping term plays an important role in the stability\nof the explicit method (9). The \fnite di\u000berence scheme (9) will be unstable for any values\nofhandkif the damping term \r(x) is identically zero or handkare out of the required\nbounds of the proposition (1).\nProposition 2. The implicit FD-(1,1) approximation de\fned by (11) is unconditionally\nstable.\nProof. The eigenvalues of the matrix Mare given by\n\u0015\u0006\nn=\u0000\r(xn)\n2\u00061\n2r\n\r(xn)2\u000016\nh2sin2(n\u0019\n2N); n= 1;\u0001\u0001\u0001;N\u00001:\nThen, by using functional calculus, the eigenvalues \u0016\u0006\nnof the matrix ( I\u00001\n2kM)\u00001((I+\n1\n2kM)) are given by\n\u0016\u0006\nn=1 +k\n2\u0015\u0006\nn\n1\u0000k\n2\u0015\u0006\nn; n= 1;\u0001\u0001\u0001;N\u00001:\nAlso, we have Re(\u0015n)\u00140 because\r\u00150. Thus, for any values of n;h;k , and\r(xn), we\ngetj\u0016\u0006\nnj\u00141. Hence, the implicit method (11) is unconditionally stable. \u0003\nA direct application of the Lax Equivalence Theorem [9, 15] leads to the convergence of\nour models.\nCorollary 1. The \fnite di\u000berence explicit FD\u0000(0;1)of(9)and implicit FD\u0000(1;1)of\n(11) are convergent.10 FAZEL HADADIFARD, SATBIR MALHI, AND ZHENGYI XIAO\n4.Performance of Numerical schemes\nIn this section, we will see the performance of each \fnite di\u000berence scheme on a sample\nproblem.\nSample Problem: We consider the following damped wave equation\nutt=uxx\u00002ut;\nover the region \n = [0 \u0014x\u0014\u0019]\u0002(t>0) with initial conditions\nu(x;0) = sin(x); u t(x;0) =\u0000sin(x);\nand boundary conditions\nu(0;t) = 0 =u(\u0019;t);fort>0:\nThe exact solution of the above problem is u(x;t) =e\u0000tsin(x).\nFigure 1. The approximate solution given by explicit FD-(0,1) of (10) at\nt= 1 withk= 0:05,h= 0:13464.\nFigure 2. The approximate solution given by implicit FD-(1,1) of (11) at\nt= 1 withk= 0:05,h= 0:13464.\nFIGURE (1) and (2) show the numerical solutions using \fnite di\u000berence methods (9)\nand (11) at t= 1. From the obtained numerical results, we can conclude that the numerical\nsolutions are in good agreement with the exact solution.A CLASS OF FINITE DIFFERENCE METHODS FOR SOLVING INHOMOGENEOUS DAMPED WAVE EQUATIONS 11\n4.1.Comparison with other methods. In this section, we compare our result with the\nordinary explicit and implicit \fnite di\u000berence methods mentioned below. We also compare\nour result with the FOCM method of [6]. We take the same test example mentioned above\nfor this comparison.\nOrdinary Explicit Finite Di\u000berence Scheme (OEFD): The ordinary explicit \fnite\ndi\u000berence scheme in the matrix form is\n(1 +\rk\n2)u(t+k) = (2I\u0000r2A)u(t) +\u0012\rk\n2\u00001\u0013\nun\u00001+r2B(t); (12)\nwherer=k=h,B(t) =\u0002ua(t);0;0;:::; 0;0;ub(t)\u0003\n, and the matrix Ais de\fned in equation\n(4).\nOrdinary Implicit Finite Di\u000berence Scheme(OIFD): The ordinary implicit \fnite\ndi\u000berence scheme in the matrix form is\n\u0012\n1 +\r(xn)k\n2\u0000r2\n2A\u0013\nu(t+k) =\u0012\n2 +r2\n2A\u0013\nu(t) +\u0012\r(xn)k\n2\u00001\u0013\nu(t\u0000k) (13)\n+r2\n2(B(t+k) +B(t));\nwherer=k=h,B(t) =\u0002ua(t);0;0;:::; 0;0;ub(t)\u0003\n, and the matrix Ais de\fned in equation\n(4). The derivation of these schemes can be found in [11].\nFIGURE (3) and (4) show the performances of our methods (10) and (11) in comparison\nwith \fnite di\u000berence schemes (12) and (13) using k= 0:01 andh= 0:063. The implicit\nFD-(1,1) produces a much better result even for a large value of r. When the values of h\nandkfail to satisfy the stability conditions of the explicit FD-(0,1), it can be seen that the\nnumerical solution became unstable after some time iterations. However, it is interesting\nto see that even in this case the global numerical solution fails to exist, the local numerical\nsolution does exist for a small time and it was very close to the exact solution. It is\napparent that the explicit \fnite di\u000berence scheme (12) and (9) are not stable for large\nvalues ofr. The implicit FD-(1,1) is very stable and produces a much better result when\ncompared to the ordinary implicit \fnite di\u000berence scheme (13).\nFigure 3. The absolute er-\nror of the method (10) and\n(11) forr= 1:5915.\nFigure 4. The absolute er-\nror of the method (12) and\n(13) forr= 1:5915.12 FAZEL HADADIFARD, SATBIR MALHI, AND ZHENGYI XIAO\nFigure 5. The absolute error of the method (10), (11), (12) and (13).\nIn the FIGURE (5), we plotted the absolute error at the four di\u000berent values of r=:016,\nr=:159,r=:995, andr= 1:45. One can see for a small values of r= 0:016, all the four\nschemes produce fairly stable results. This shows that when our explicit \fnite di\u000berence\nFD-(0,1) satis\fes the assumptions of proposition (1), it is stable and produces better results\nthan the other three. However, the performances of the explicit \fnite di\u000berence method\n(9) and implicit \fnite di\u000berence FD-(1,1) (11) are very similar for small values of r.\nTABLE 1 shows the comparison between the errors generated by FOCM, OEFD, OIFD,\nEX\u0000(0;1) andIM\u0000(1;1) att= 0:3 withh=\u0019\n10andk=1\n10.\nTABLE 2 shows the magnitude of the maximum error at time t= 6 between the exact\nsolution and the numerical solution obtained by using FOCM, OEFD, OIFD, FD\u0000(0;1),\nandFD\u0000(1;1) discussed above with di\u000berent values of handk.\n5.Conclusion\nIn this paper, a class of \fnite di\u000berence methods using the C0-semigroup operator theory\nfor solving the inhomogeneous damped wave equation is presented. The stability and\nconsistency of the implicit and explicit methods are proved. Test examples are presented,\nand the results obtained are compared with the exact solutions. The comparison certi\fesA CLASS OF FINITE DIFFERENCE METHODS FOR SOLVING INHOMOGENEOUS DAMPED WAVE EQUATIONS 13\nx FOCM OEFD OIFD EX-(0,1) IM-(1,1)\n0 0 0 0 0 0\n0.314159265 0.00012256 6.29067E-05 0.000135485 0.001494844 1.23932E-05\n0.628318531 0.00022777 0.000119656 0.000257708 0.002843363 2.35734E-05\n0.942477796 0.00031458 0.000164692 0.000354705 0.003913553 3.24459E-05\n1.256637061 0.00036955 0.000193607 0.000416981 0.004600658 3.81425E-05\n1.570796327 0.00038865 0.00020357 0.000438439 0.004837418 4.01054E-05\n1.884955592 0.00036955 0.000193607 0.000416981 0.004600658 3.81425E-05\n2.199114858 0.00031458 0.000164692 0.000354705 0.003913553 3.24459E-05\n2.513274123 0.00022777 0.000119656 0.000257708 0.002843363 2.35734E-05\n2.827433388 0.00012256 6.29067E-05 0.000135485 0.001494844 1.23932E-05\n3.141592654 0 0 0 0 0\nTable 1. Absolute Error\nr EFD IFD EX-(0,1) IM-(1,1)\n1.59 1.00967E+34 0.002547509 9.08234E+13 2.231E-06\n0.53 3.05424E-05 0.00079153 2.18322E+11 1.36036E-05\n0.32 2.04246E-05 0.000473008 3410.243641 1.43835E-05\n0.23 1.76452E-05 0.000339697 0.011310925 1.45754E-05\n0.18 1.64986E-05 0.000266457 7.84447E-05 1.46457E-05\nTable 2. Maximum Error at t= 6\nthat implicit FD-(1,1) gives good results. Summarizing these results, we can say the general\nform of the new \fnite di\u000berence methods has a reasonable amount of calculations and the\nform is easy to use. All results are obtained by using MATLAB version 9.7.14 FAZEL HADADIFARD, SATBIR MALHI, AND ZHENGYI XIAO\nReferences\n[1] GA Baker and PR Graves-Morris. Pad\u0013 e approximants, part i, encycl. math., vol. 13. Reading, MA:\nAddison-Wesley , 7:233{236, 1981.\n[2] Nicolas Burq and Romain Joly. Exponential decay for the damped wave equation in unbounded\ndomains. Communications in Contemporary Mathematics , 18(06):1650012, 2016.\n[3] Ian Christie, David F Gri\u000eths, Andrew R Mitchell, and Olgierd C Zienkiewicz. Finite element meth-\nods for second order di\u000berential equations with signi\fcant \frst derivatives. International Journal for\nNumerical Methods in Engineering , 10(6):1389{1396, 1976.\n[4] Lawrence C. Evans. Partial di\u000berential equations . American Mathematical Society, Providence, R.I.,\n2010.\n[5] Feng Gao and Chunmei Chi. Unconditionally stable di\u000berence schemes for a one-space-dimensional\nlinear hyperbolic equation. Applied Mathematics and Computation , 187(2):1272{1276, 2007.\n[6] M.T. Hussain, A. Pervaiz, Zainulabadin Zafar, and M.O. Ahmad. Fourth order compact method for\none dimensional homogeneous damped wave equation. Pakistan Journal of Science , 64(2):122, 2012.\n[7] Eliahu Jury. On the roots of a real polynomial inside the unit circle and a stability criterion for linear\ndiscrete systems. IFAC Proceedings Volumes , 1(2):142{153, 1963.\n[8] Stig Larsson, Vidar Thom\u0013 ee, and Lars B Wahlbin. Finite-element methods for a strongly damped\nwave equation. IMA journal of numerical analysis , 11(1):115{142, 1991.\n[9] Peter D Lax and Robert D Richtmyer. Survey of the stability of linear \fnite di\u000berence equations.\nCommunications on pure and applied mathematics , 9(2):267{293, 1956.\n[10] Gunter Lumer and Ralph S Phillips. Dissipative operators in a banach space. Paci\fc Journal of\nMathematics , 11(2):679{698, 1961.\n[11] Andrew Ronald Mitchell and David Francis Gri\u000eths. The \fnite di\u000berence method in partial di\u000beren-\ntial equations. Wiley. New York , 1980.\n[12] Akbar Mohebbi. A fourth-order \fnite di\u000berence scheme for the numerical solution of 1d linear hyper-\nbolic equation. Commun. Numer. Anal , 2013.\n[13] Ahmet Ozkan Ozer and E _Inan. One-dimensional wave propagation problem in a nonlocal \fnite\nmedium with \fnite di\u000berence method. In Vibration Problems ICOVP 2005 , 383{388. Springer, 2006.\n[14] Je\u000brey Rauch, Michael Taylor, and Ralph Phillips. Exponential decay of solutions to hyperbolic\nequations in bounded domains. Indiana university Mathematics journal , 24(1):79{86, 1974.\n[15] Robert D Richtmyer and Keith W Morton. Di\u000berence methods for initial-value problems. dmiv , 1994.\n[16] Paul A Samuelson. Conditions that the roots of a polynomial be less than unity in absolute value.\nThe Annals of Mathematical Statistics , 12(3):360{364, 1941.\n[17] Harvey Segur. Waves in shallow water, with emphasis on the tsunami of 2004. In Tsunami and\nnonlinear waves , 3{29. Springer, 2007.\n[18] Gordon Smith. Numerical solution of partial di\u000berential equations: \fnite di\u000berence methods . Oxford\nuniversity press, 1985.\nFazel Hadadifard, Department of Mathematics, Drexel University\nEmail address :fh352@drexel.edu\nSatbir Malhi, Department of Mathematics, Saint Mary's College of California\nEmail address :smalhi@stmarys-ca.edu\nZhengyi Xiao, Department of Mathematics, Franklin & Marshall College\nEmail address :zxiao@fandm.edu"    },    {        "title": "1810.07020v4.Superfluid_spin_transport_in_ferro__and_antiferromagnets.pdf",        "content": "Super\ruid spin transport in ferro- and antiferromagnets\nE. B. Sonin\nRacah Institute of Physics, Hebrew University of Jerusalem, Givat Ram, Jerusalem 91904, Israel\n(Dated: March 25, 2019)\nThis paper focuses on spin super\ruid transport, observation of which was recently reported in\nantiferromagnet Cr 2O3[Yuan et al. , Sci. Adv. 4, eaat1098 (2018)]. This paper analyzes the role of\ndissipation in transformation of spin current injected with incoherent magnons to a super\ruid spin\ncurrent near the interface where spin is injected. The Gilbert damping parameter in the Landau{\nLifshitz{Gilbert theory does not describe dissipation properly, and the dissipation parameters are\ncalculated from the Boltzmann equation for magnons scattered by defects. The two-\ruid theory is\ndeveloped similar to the two-\ruid theory for super\ruids. This theory shows that the in\ruence of\ntemperature variation in bulk on the super\ruid spin transport (bulk Seebeck e\u000bect) is weak at low\ntemperatures. The scenario that the results of Yuan et al. are connected with the Seebeck e\u000bect at\nthe interface between the spin detector and the sample is also discussed.\nThe Landau criterion for an antiferromagnet put in a magnetic \feld is derived from the spectrum\nof collective spin modes. The Landau instability starts in the gapped mode earlier than in the\nGoldstone gapless mode, in contrast to easy-plane ferromagnets where the Goldstone mode becomes\nunstable. The structure of the magnetic vortex in the geometry of the experiment is determined.\nThe vortex core has the skyrmion structure with \fnite magnetization component normal to the\nmagnetic \feld. This magnetization creates stray magnetic \felds around the exit point of the vortex\nline from the sample, which can be used for experimental detection of vortices.\nI. INTRODUCTION\nThe concept of spin super\ruidity is based on the anal-\nogy of the equations of magnetodynamics with the equa-\ntions of super\ruid hydrodynamics.1. The analogy led to\nthe suggestion that in magnetically ordered media persis-\ntent spin currents are possible, which are able to trans-\nport spin on macroscopical distances without essential\nlosses.2\nThe phenomenon of spin super\ruidity has been dis-\ncussed for several decades.2{15We de\fne the term super-\n\ruidity in its original meaning known from the times of\nKamerlingh Onnes and Kapitza: transport of some phys-\nical quantity (mass, charge, or spin) over macroscopical\ndistances without essential dissipation. This requires a\nconstant or slowly varying phase gradient at macroscopic\nscale with the total phase variation along the macroscopic\nsample equal to 2 \u0019multiplied by a very large number.\nSpin super\ruidity assumes the existence of spin current\nproportional to the gradient of the phase (spin super-\ncurrent). In magnetically ordered media the phase is an\nangle of rotation in spin space around some axis (further\nin the paper the axis z). In contrast to the dissipative\nspin-di\u000busion current proportional to the gradient of spin\ndensity, the spin supercurrent is not accompanied by dis-\nsipation.\nSpin super\ruidity require special topology of the order\nparameter space. This topology is realized at the pres-\nence of the easy-plane magnetic anisotropy, which con-\n\fnes the magnetization of the ferromagnet or sublattice\nmagnetizations of the antiferromagnet in an easy plane.\nIn this case one may expect that the current state is sta-\nble with respect to phase slips, which lead to relaxation of\nthe supercurrent. In the phase slip event a vortex with\n2\u0019phase variation around it crosses streamlines of thesupercurrent decreasing the total phase variation across\nstreamlines by 2 \u0019. The concept of the phase slip was\nintroduced by Anderson16for super\ruid4He and later\nwas used in studying spin super\ruidity.2,3\nPhase slips are suppressed by energetic barriers for vor-\ntex expansion. But these barriers disappear when phase\ngradients reach critical values determined by the Landau\ncriterion. The physical meaning of the Landau criterion\nis straightforward: the current state becomes unstable\nwhen there are elementary excitations with negative en-\nergy. So, to check the Landau criterion one must know\nthe full spectrum of collective modes.\nSometimes any presence of spin current proportional\nto the phase gradient is considered as a manifestation\nof spin super\ruidity.17,18However, spin current propor-\ntional to the spin phase gradient is ubiquitous and ex-\nists in any spin wave or domain wall, also in the ground\nstate of disordered magnetic media. In all these cases\nthe total variation of the phase is smaller, or on the or-\nder of\u0019. Connecting these cases with spin super\ruid-\nity makes this phenomenon trivial and already observed\nin old experiments on spin waves in the middle of the\n20th Century. One may call the supercurrent produced\nby the total phase variation of the order or less than\n2\u0019microscopical supercurrent, in contrast to persistent\nmacroscopical supercurrents able to transport spin over\nmacroscopical distances.\nThe analogy with usual super\ruids is exact only if\nthe spin space is invariant with respect to spin rotation\naround the hard axis normal to the easy plane. Then\nthere is the conservation law for the spin component\nalong the hard axis. In reality this invariance is bro-\nken by in-plane anisotropy. But this anisotropy is usu-\nally weak, because it originates from the spin-orbit in-\nteraction, which is relativistically small compared to thearXiv:1810.07020v4  [cond-mat.mes-hall]  22 Mar 20192\nexchange interaction, i.e., inversely proportional to the\nspeed of light.19Macroscopical spin supercurrents are\nstill possible if the energy of supercurrents exceeds the\nin-plane anisotropy energy. Thus, one cannot observe\nmacroscopical spin supercurrents not only at large cur-\nrents as in usual super\ruids, but also at small currents.2\nFrom the time when the concept of spin super\ruidity\n(in our de\fnition of this term) was suggested2, it was\ndebated about whether the super\ruid spin current is a\n\\real\" transport current. As a response to these con-\ncerns, in Ref 2 a Gedanken (at that time) experiment\nfor demonstration of reality of super\ruid spin transport\nwas proposed. The spin is injected to one side of a mag-\nnetically ordered layer of thickness dand spin accumula-\ntion is checked at another side. If the layer is not spin-\nsuper\ruid, then the spin is transported by spin di\u000busion.\nThe spin current and the spin density exponentially de-\ncay at the distance of the spin di\u000busion length, and the\ndensity of spin accumulated at the other side decreases\nexponentially with growing distance d. However, if the\nconditions for spin super\ruidity are realized in the layer,\nthen the super\ruid spin current decays much slower, and\nthe accumulated spin density at the side opposite to the\nside where the spin is injected is inversely proportional\ntod+C, whereCis some constant.\nThe interest to long-distance spin transport, especially\nto spin super\ruid transport, revived recently. Takei and\nTserkovnyak7carried out a microscopic analysis of in-\njection of spin to and ejection of spin out of the spin-\nsuper\ruid medium in an easy-plane ferromagnet justify-\ning the aforementioned scheme of super\ruid spin trans-\nport. Takei et al.8extended this analysis to easy-plane\nantiferromagnets. Finally Yuan et al.20were able to real-\nize the suggested experiment in antiferromagnetic Cr 2O3\nobserving spin accumulation inversely proportional to the\ndistance from the interface where spin was injected into\nCr2O3.\nPreviously Borovik-Romanov et al.21reported evi-\ndence of spin super\ruidity in the Bphase of super\ruid\n3He. They detected phase slips in a channel with su-\nper\ruid spin current close to its critical value. It was\nimportant evidence that persistent spin currents are pos-\nsible. But real long-distance transportation of spin by\nthese currents was not demonstrated. Moreover, it is\nimpossible to do in the nonequilibrium magnon Bose{\nEinstein condensate, which was realized in the Bphase\nof3He super\ruid6and in yttrium-iron-garnet magnetic\n\flms.22The nonequilibrium magnon Bose{Einstein con-\ndensate requires pumping of spin in the whole bulk for\nits existence. In the geometry of the aforementioned spin\ntransport experiment this would mean that spin is per-\nmanently pumped not only by a distant injector but also\nall the way up the place where its accumulation is probed.\nThus, the spin detector measures not only spin coming\nfrom a distant injector but also spin pumped close to\nthe detector. Therefore, the experiment does not prove\nthe existence of long-distance spin super\ruid transport.\nThere were also reports on experimental detection ofspin super\ruidity in magnetically ordered solids17,18, but\nthey addressed microscopical spin supercurrent.23As ex-\nplained above, \\super\ruidity\" connected with such cur-\nrents was well proved by numerous old experiments on\nspin waves and does not need new experimental con\fr-\nmations. The work of Yuan et al.20was the \frst report\non long-distance super\ruid spin transport with spin ac-\ncumulation decreasing with distance from the injector as\nexpected from the theory. Long distance super\ruid spin\ntransport was also recently reported in a graphene quan-\ntum antiferromagnet.24\nThe experiment on super\ruid spin transport20has put\nto rest another old dispute about the spin super\ruidity\nconcept. At studying spin super\ruidity in the Bphase\nof super\ruid3He, it was believed4that spin super\ruidity\nis possible only if there are mobile carriers of spin and\na counter\row of carriers with opposite spins transports\nspin. If so, then spin super\ruidity is impossible in insu-\nlators. Moreover, Shi et al.25argued that it is a critical\n\raw of spin-current de\fnition if it predicts spin currents\nin insulators. Since Cr 2O3is an insulator the experiment\nof Yuan et al.20rules out this presumption.\nBoosted by the super\ruid spin transport experiment20\nthis paper addresses some issues deserving further inves-\ntigation. It is especially needed because Lebrun et al.26\nmade an experiment in an antiferromagnetic iron oxide\nsimilar to that of Yuan et al.20and observed similar de-\npendence of spin accumulation on the distance from the\ninjector. However, Lebrun et al.26explain it not by spin\ntransport from the distant injector but by the Seebeck\ne\u000bect at the detector, which is warmed by the heat \row\nfrom the injector. We shall compare these two interpre-\ntations in Sec. VIII.\nWe analyzed the role of dissipation in the super\ruid\nspin transport. A widely used approach to address dis-\nsipation in magnetically ordered solids is the Landau{\nLifshitz{Gilbert (LLG) theory with the Gilbert damp-\ning parameter. But we came to the conclusion that\nthe Gilbert damping does not provide a proper descrip-\ntion of dissipation processes in easy-plane ferromagnets.\nThe Gilbert damping is described by a single parame-\nter, which scales alldissipation processes independently\nfrom whether they do violate the spin conservation law,\nor do not. Meanwhile, the processes violating the spin\nconservation law, the Bloch spin relaxation in particular,\noriginate from spin-orbit interaction and must be rela-\ntivistically small as explained above. This requires the\npresence of a small factor in the intensity of the Bloch\nspin relaxation, which is absent in the Gilbert damping\napproach. So we determined the dissipation parameters\nfrom the Boltzmann equation for magnons scattered by\ndefects. Dissipation is possible only in the presence of\nthermal magnons, and we developed the two-\ruid theory\nfor easy-plane ferromagnets similar to that in super\ruid\nhydrodynamics for the clamped regime, when the gas of\nquasiparticles cannot freely drift without dissipation in\nthe laboratory frame.\nAs mentioned above, to check the Landau criterion for3\nsuper\ruidity, one must calculate the spectrum of collec-\ntive modes and check whether some modes have nega-\ntive energies. The Landau critical gradient is determined\nby easy-plane crystal anisotropy and was known qualita-\ntively both for ferro- and antiferromagnets long ago.2For\neasy-plane ferromagnets the Landau critical gradient was\nrecently determined quantitatively from the spin-wave\nspectrum in the analysis of ferromagnetic spin-1 BEC\nof cold atoms.15But Cr 2O3, which was investigated in\nthe experiment,20has no crystal easy-plane anisotropy,\nand an \\easy plane\" necessary for spin super\ruidity is\nproduced by an external magnetic \feld. The magnetic\n\feld should exceed the spin-\rop \feld, above which mag-\nnetizations of sublattices in antiferromagnet are kept in\na plane normal to the magnetic \feld. We analyze the\nmagnon spectrum in the spin current states in this situ-\nation. The analysis has shown that the Landau critical\ngradient is determined by the gapped mode, but not by\nthe Goldstone gapless mode as in the cases of easy-plane\nferromagnets.\nWithin the two-\ruid theory the role of spatial temper-\nature variation was investigated. This variation produces\nthe bulk Seebeck e\u000bect. But the e\u000bect is weak because it\nis proportional not to the temperature gradient, but to a\nhigher (third) spatial derivative of the temperature.\nThe transient processes near the interface through\nwhich spin is injected were also discussed. Conversion\nfrom spin current of incoherent thermal magnons to co-\nherent (super\ruid) spin transport is among these pro-\ncesses. The width of the transient layer (healing length),\nwhere formation of the super\ruid spin current occurs,\ncan be determined by di\u000berent scales at di\u000berent condi-\ntion. But at low temperatures it is apparently not less\nthan the magnon mean-free-path.\nIn reality the decay of super\ruid currents starts at val-\nues less than the Landau critical value via phase slips\nproduced by magnetic vortices. The di\u000berence in the\nspectrum of collective modes in ferro- and antiferromag-\nnets leads to the di\u000berence in the structure of magnetic\nvortices. In the past magnetic vortices were investi-\ngated mostly in ferromagnets (see Ref. 15 and references\ntherein). The present work analyzes a vortex in an anti-\nferromagnet. The vortex core has a structure of skyrmion\nwith sublattice magnetizations deviated from the direc-\ntion normal to the magnetic \feld. At the same time\ninside the core the total magnetization has a component\nnormal to the magnetic \feld. In the geometry of the\nCr2O3experiment this transverse magnetization creates\nsurface magnetic charges at the point of the exit of the\nvortex line from the sample. Dipole stray magnetic \felds\nproduced by these charges hopefully can be used for de-\ntection of magnetic vortices experimentally.\nSection II reminds the phenomenological model of\nRef. 2 describing the spin di\u000busion and super\ruid spin\ntransport. Section III reproduces the derivation of the\nspectrum of the collective spin mode and the Landau\ncriterion in a spin current state of an easy-plane ferro-\nmagnet known before15. This is necessary for compari-son with the spectrum of the collective spin modes and\nthe Landau criterion in a spin current state of an easy-\nplane antiferromagnet derived in Sec. IV. Thus, Sec. III,\nas well as Sec. II, do not contain new results, but were\nadded to the paper to make it self-su\u000ecient and more\nreadable. In Sec. V we address two-\ruid e\u000bects and dis-\nsipation parameters (spin di\u000busion and second viscosity\ncoe\u000ecients) deriving them from the Boltzmann equation\nfor magnons. The section also estimates the bulk See-\nbeck e\u000bect and shows that it is weak. Section VI ana-\nlyzes the transient layer near the interface through which\nspin is injected and where the bulk super\ruid spin cur-\nrent is formed. Various scales determining the width of\nthis layer (healing length) are discussed. In Sec. VII the\nskyrmion structure of the magnetic vortex in an anti-\nferromagnets is investigated. The concluding Sec. VIII\nsummarizes the results of the work and presents some\nnumerical estimations for the antiferromagnetic Cr 2O3\ninvestigated in the experiment. The Appendix analyzes\ndissipation in the LLG theory with the Gilbert damping.\nIt is argued that this theory predicts dissipation coe\u000e-\ncients incompatible with the spin conservation law.\nII. SUPERFLUID SPIN TRANSPORT VS SPIN\nDIFFUSION\nHere we remind the simple phenomenological model of\nspin transport suggested in Ref. 2 (see also more recent\nRefs. 5, 7, and 8). The equations of magnetodynamics\nare\ndMz\ndt=\u0000r\u0001J\u0000M0\nz\nT1; (1)\nd'\ndt=\u0000\rM0\nz\n\u001f+\u0010r2': (2)\nHere\u001fis the magnetic susceptibility along the axis z,\n'is the angle of rotation (spin phase) in the spin space\naround the axis z, andM0\nz=Mz\u0000\u001fHis a nonequilib-\nrium part of the magnetization density along the mag-\nnetic \feldHparallel to the axis z. The time T1is the\nBloch time of the longitudinal spin relaxation. The term\n/r2'in Eq. (2) is an analog of the second viscosity in\nsuper\ruid hydrodynamic.27,28The magnetization density\nMzand the magnetization current Jdi\u000ber from the spin\ndensity and the spin current by sign and by the gyromag-\nnetic factor \r. Nevertheless, we shall call the current J\nthe spin current to stress its connection with spin trans-\nport. The total spin current J=Js+Jdconsists of the\nsuper\ruid spin current\nJs=Ar'; (3)\nand the spin di\u000busion current\nJd=\u0000DrMz: (4)4\nJzxJLzxSpin injection\nSpin injectionSpin injectionMedium withoutspin superfluidityMedium withspin superfluiditymxzmz\n0\na)\nb)Spin detection\ndyxzPtPtCr2O3c)H\nc)PtPtCr2O3Hzdxy\nFIG. 1. Long distance spin transport. (a) Spin injection to\na spin-nonsuper\ruid medium. (b) Spin injection to a spin-\nsuper\ruid medium. (c) Geometry of the experiment by Yuan\net al.20. Spin is injected from the left Pt wire and \rows along\nthe Cr 2O3\flm to the right Pt wire, which serves as a detector.\nThe arrowed dashed line shows a spin-current streamline. In\ncontrast to (a) and (b), the spin current is directed along\nthe same axis zas a magnetization parallel to the external\nmagnetic \feld H.\nThe pair of the hydrodynamical variables ( Mz;') is a\npair of conjugate Hamiltonian variables analogous to\nthe pair \\particle density{super\ruid phase\" in super\ruid\nhydrodynamics.1\nThere are two kinds of spin transport illustrated in\nFig. 1. In the absence of spin super\ruidity ( A= 0) there\nis no super\ruid current. Equation (2) is not relevant, and\nEq. (1) describes pure spin di\u000busion [Fig. 1(a)]. Its solu-\ntion, with the boundary condition that the spin current\nJ0is injected at the interface x= 0, is\nJ=Jd=J0e\u0000x=L d; M0\nz=J0r\nT1\nDe\u0000x=L d;(5)\nwhere\nLd=p\nDT1 (6)\nis the spin-di\u000busion length. Thus the e\u000bect of spin injec-\ntion exponentially decays at the scale of the spin-di\u000busion\nlength.However, if spin super\ruidity is possible ( A6= 0), the\nspin precession equation (2) becomes relevant. As a re-\nsult of it, in a stationary state the magnetization M0\nz\ncannot vary in space (Fig. 1b) since according to Eq. (2)\nthe gradient rM0\nzis accompanied by the linear in time\ngrowth of the gradient r'. The requirement of constant\nin space magnetization Mzis similar to the requirement\nof constant in space chemical potential in super\ruids, or\nthe electrochemical potential in superconductors. As a\nconsequence of this requirement, spin di\u000busion current is\nimpossible in the bulk since it is simply \\short-circuited\"\nby the super\ruid spin current. Only in AC processes\nthe oscillating spin injection can produce an oscillating\nbulk spin di\u000busion current coexisting with an oscillating\nsuper\ruid spin current.\nIn the super\ruid spin transport the spin current can\nreach the other boundary opposite to the boundary where\nspin is injected. We locate it at the plane x=d. As a\nboundary condition at x=d, one can use a phenomeno-\nlogical relation connecting the spin current with the mag-\nnetization: Js(d) =M0\nzvd, wherevdis a phenomenologi-\ncal constant. This boundary condition was derived from\nthe microscopic theory by Takei and Tserkovnyak7. To-\ngether with the boundary condition Js(0) =J0atx= 0\nthis yields the solution of Eqs. (1) and (2):\nM0\nz=T1\nd+vdT1J0; Js(x) =J0\u0012\n1\u0000x\nd+vdT1\u0013\n:(7)\nThus, the spin accumulated at large distance dfrom the\nspin injector slowly decreases as the inverse distance 1 =d\n[Fig. 1(b)], in contrast to the exponential decay /e\u0000d=Ld\nin the spin di\u000busion transport [Fig. 1(a)].\nIn Figs. 1(a) and 1(b) the spin \rows along the axis\nx, while the magnetization and the magnetic \feld are\ndirected along the axis z. In the geometry of the experi-\nment of Yuan et al.20the spin \rows along the magnetiza-\ntion axiszparallel to the magnetic \feld. This geometry is\nshown in Fig. 1c. The di\u000berence between two geometries\nis not essential if spin-orbit coupling is ignored. In this\nsection we chose the geometry with di\u000berent directions of\nthe spin current and the magnetization in order to stress\nthe possibility of the independent choice of axes in the\nspin and the con\fgurational spaces. But in Sec. VII ad-\ndressing a vortex in an antiferromagnet we shall switch\nto the geometry of the experiment because in this case\nthe di\u000berence between geometries is important.\nWithout dissipation-connected terms, the phenomeno-\nlogical theory of this section directly follows from the\nLLG theory. For ferromagnets the LLG equation is\ndM\ndt=\r[Heff\u0002M]; (8)\nwhere\nHeff=\u0000\u000eH\n\u000eM=\u0000@H\n@M+rj@H\n@rjM(9)\nis the e\u000bective \feld determined by the functional deriva-\ntive of the Hamiltonian H. For a ferromagnet with uni-5\naxial anisotropy the Hamiltonian is\nH=GM2\nz\n2+AriM\u0001riM\u0000MzH: (10)\nHereHis an external constant magnetic \feld parallel to\nthe axisz, and the exchange constant Adetermines sti\u000b-\nness with respect to deformations of the magnetization\n\feld. In the case of easy-plane anisotropy the anisotropy\nparameter Gis positive and coincides with the inverse\nsusceptibility: G= 1=\u001f.\nSince the absolute value Mof the magnetization is\na constant, one can describe the 3D magnetization vec-\ntorMonly by two Hamiltonian conjugate variables: the\nmagnetization zcomponent Mzand the angle 'of rota-\ntion around the zaxis. Then the LLG theory yields two\nequations\n_Mz=\u0000r\u0001Js; (11)\n_'=\u0000\r\u0016; (12)\nwith the Hamiltonian in new variables\nH=M2\nz\n2\u001f+AM2\n?r'2\n2+AM2(rMz)2\n2M2\n?\u0000MzH: (13)\nHereM?=p\nM2\u0000M2z, and the spin \\chemical poten-\ntial\" and the super\ruid spin current are\n\u0016=\u000eH\n\u000eMz=@H\n@Mz\u0000rj@H\n@rjMz;Js=\r@H\n@r':(14)\nAfter substitution of explicit expressions for functional\nderivatives of the Hamiltonian (13) the equations become\n_Mz\n\r=\u0000r\u0001(AM2\n?r'); (15)\n_'\n\r=\u0000Mz\u00141\n\u001f\u0000A(r')2\u0000AM2(rMz)2\nM4\n?\u0015\n+AM2\nM2\n?r2Mz+H: (16)\nThe equations (1) and (2) without dissipation terms fol-\nlow from Eqs. (15) and (16) after linearization with re-\nspect to small gradients r'and nonequilibrium magne-\ntizationM0\nz=Mz\u0000\u001fHand ignoring the dependence\nof the spin chemical potential \u0016onrMz. ThenA=\n\rAM2\n?, andM?is determined by its valuep\nM2\u0000\u001f2H2\nin the equilibrium.\nIII. COLLECTIVE MODES AND THE LANDAU\nCRITERION IN EASY-PLANE FERROMAGNETS\nTo check the Landau criterion one should know the\nspectrum of collective modes. In an easy-plane ferromag-\nnet the collective modes (spin waves) are determined byEqs. (15) and (16) linearized with respect to weak pertur-\nbations of stationary states. Further the angle variable \u0012\nwill be introduced instead of the variable Mz=Msin\u0012.\nLet us consider a current state with constant gradient\nK=r'and constant magnetization\nMz=Msin\u0012=\u001fH\n1\u0000\u001fAK2: (17)\nTo derive the spectrum of collective modes, we consider\nweak perturbations \u0002 and \b of this state: \u0012!\u0012+ \u0002,\n'!'+ \b. Equations (15) and (16) after linearization\nare:\n_\u0002\u00002\rMzAK\u0001r\u0002 =\u0000\rAM cos\u0012r2\b;\n_\b\u00002\rMzAK\u0001r\b =\n\u0000\rMcos\u0012\n\u001f\u0000\n1\u0000\u001fAK2\u0001\n\u0002 +\rAM cos\u0012r2\u0002:(18)\nFor plane waves/eik\u0001r\u0000i!tthese equations describe the\ngapless Goldstone mode with the spectrum:13,15\n(!+w\u0001k)2= ~c2\nsk2: (19)\nHere\n~cs=r\u001f\n~\u001fcs; (20)\n~\u001f=\u001f\n1\u0000\u001fA\u0010\nK2\u0000M2k2\nM2\n?\u0011; (21)\nand\ncs=\rM?s\nA\n\u001f(22)\nis the spin-wave velocity in the ground state without any\nspin current. In this state the spectrum becomes\n!=csks\n1 +\u001fAM2k2\nM2\n?: (23)\nThe velocity\nw= 2\rMzAK; (24)\ncan be called Doppler velocity because its e\u000bect on the\nmode frequency is similar to the e\u000bect of the mass ve-\nlocity on the mode frequency in a Galilean invariant\n\ruid (Doppler e\u000bect). But our system is not Galilean\ninvariant,13and the gradient Kis present also in the\nright-hand side of the dispersion relation (19).\nIn the long-wavelength hydrodynamical limit magnons\nhave the sound-like spectrum linear in k. Quadratic cor-\nrections/k2become important at k\u0018M?=Mp\u001fA[see\nEq. (23)]. These corrections emerge from the terms in6\nthe Hamiltonian, which depend on rMz. So the hydro-\ndynamical approach is valid at scales exceeding\n\u00180=M\nM?p\n\u001fA; (25)\nwhich can be called the coherence length, in analogy with\nthe coherence length in the Gross{Pitaevskii theory for\nBEC. Also in analogy with BEC, the coherence length\ndiverges at M?!0, i.e., at the second-order phase tran-\nsition from the easy-plane to the easy-axis anisotropy.\nThe same scale determines the Landau critical gradient\nand the vortex core radius. Telling about hydrodynamics\nwe bear in mind hydrodynamics of a perfect \ruid without\ndissipation. Later in this paper we shall discuss hydro-\ndynamics with dissipation. In this case the condition\nk\u001c1=\u00180is not su\u000ecient, and an additional restriction\non using hydrodynamics is determined by the mean-free\npath of magnons.\nAccording to the Landau criterion, the current state\nbecomes unstable at small kwhenkis parallel to wand\nthe frequency !becomes negative. This happens at the\ngradientKequal to the Landau critical gradient\nKc=M?p4M2\u00003M?1p\u001fA\u00181\n\u00180: (26)\nSpin super\ruidity becomes impossible at the phase tran-\nsition to the easy-axis anisotropy ( M?= 0). In the oppo-\nsite limit of small Mz\u001cMthe pseudo-Doppler e\u000bect is\nnot important, and the Landau critical gradient Kcis de-\ntermined from the condition that the spin-wave velocity\n~csvanishes at small k:\nKc=1p\u001fA=\rM\n\u001fcs: (27)\nExpanding the Hamiltonian (13) with respect to weak\nperturbations \u0002 and \b up to the second order one obtains\nthe energy of the spin wave mode per unit volume,\nEsw=M?!(k)\n\rp~\u001fAkj\u0002kj2; (28)\nwherej\u0002kj2is the squared perturbation of the angle \u0012\nwith the wave vector kaveraged over the wave period.\nIn the quantum theory the energy density Eswcorre-\nsponds to the magnon density\nn(k)\nV=Esw\n~!(k)=M?j\u0002kj2\n~\rp~\u001fAk; (29)\nwheren(k) is the number of magnons in the plane-wave\nmode with the wave vector kandVis the volume of the\nsample. Summing over the whole kspace, the averaged\nsquared perturbation is\nh\u00022i=X\nkj\u0002kj2=~\rp\nA\nM?Zp\n~\u001fn(k)kd3k\n(2\u0019)3:(30)Further we proceed within the hydrodynamical ap-\nproach neglecting quadratic corrections to the spectrum.\nThere are quadratic in spin-wave amplitudes corrections\nto the spin super\ruid current and to the spin chemical\npotential:\nJsjsw=\u0000\rM?A(M?h\u00022iK+ 2Mzh\u0002r\bi);(31)\n\u0016jsw=\u0000A(Mzh(r\b)2i+ 2M?K\u0001h\u0002r\bi):(32)\nUsing Eq. (30) and the relation\nr\b =\u0002p\u001fAk\nk; (33)\nwhich follows from the equations of motion (18), one ob-\ntains:\nJsjsw=\u0000\u001f2~c3\ns\n\rM2\n?Z\nn(k)\u0012\nK+2\rMz\n\u001fcsk\nk\u0013\nkd3k\n(2\u0019)3;\n(34)\n\u0016jsw=\u0000\u001f~c2\ns\n\rM2\n?Z\nn(k)\u0012\rMz\n\u001fcs+2K\u0001k\nk\u0013\nkd3k\n(2\u0019)3:\n(35)\nIV. COLLECTIVE MODES AND THE LANDAU\nCRITERION IN ANTIFERROMAGNETS\nFor ferromagnetic state of localized spins the deriva-\ntion of the LLG theory from the microscopic Heisenberg\nmodel was straightforward.29The quantum theory of the\nantiferromagnetic state even for the simplest case of a\ntwo-sublattice antiferromagnet, which was widely used\nfor Cr 2O3, is more di\u000ecult. This is because the state\nwith constant magnetizations of two sublattices is not\na well de\fned quantum-mechanical eigenstate.29Never-\ntheless, long time ago it was widely accepted to ignore\nthis complication and to describe the long-wavelength dy-\nnamics by the LLG theory for two sublattices coupled via\nexchange interaction:30\ndMi\ndt=\r[Hi\u0002Mi]; (36)\nwhere the subscript i= 1;2 points out to which sublattice\nthe magnetization Mibelongs, and\nHi=\u0000\u000eH\n\u000eMi=\u0000@H\n@Mi+rj@H\n@rjMi(37)\nis the e\u000bective \feld for the ith sublattice determined by\nthe functional derivative of the Hamiltonian H. For an\nisotropic antiferromagnet the Hamiltonian is\nH=M1\u0001M2\n\u001f+A(riM1\u0001riM1+riM2\u0001riM2)\n2\n+A12rjM1\u0001rjM2\u0000H\u0001(M1+M2):(38)7\nIn the uniform ground state without the magnetic \feld H\nthe two magnetizations are antiparallel, M2=\u0000M1, and\nthe total magnetization M1+M2vanishes. At H6= 0 the\nsublattice magnetizations are canted, and in the uniform\nground state the total magnetization is parallel to H:\nm=M1+M2=\u001fH: (39)\nThe \frst term in the Hamiltonian (38), which determines\nthe susceptibility \u001f, originates from the exchange inter-\naction between spins of two sublattices. This is the sus-\nceptibility normal to the staggered magnetization (anti-\nferromagnetic vector) L=M1\u0000M2. Since in the LLG\ntheory absolute values of magnetizations M1andM2are\n\fxed the susceptibility parallel to Lvanishes.\nIn the uniform state only the uniform exchange en-\nergy/1=\u001fand the Zeeman energy (the \frst and the\nlast terms) are present in the Hamiltonian, which can be\nrewritten as\nH=\u0000L2\u0000m2\n4\u001f\u0000H\u0001m=\u0000M2\n\u001f+m2\n2\u001f\u0000mHm;(40)\nwhereHm= (H\u0001m)=mis the projection of the mag-\nnetic \feld on the direction of the total magnetization m.\nMinimizing the Hamiltonian with respect to the absolute\nvalue of m(at it \fxed direction, i.e., at \fxed Hm) one\nobtains\nH=\u0000M2\n\u001f\u0000\u001fH2\nm\n2=\u0000M2\n\u001f\u0000\u001fH2\n2+\u001fH2\nL\n2;(41)\nwhereHL= (H\u0001L)=Lis the projection of the magnetic\n\feld on the staggered magnetization L. The \frst two\nterms are constant, while the last term plays the role of\nthe easy-plane anisotropy energy con\fning Lin the plane\nnormal to H. 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sha1_base64=\"iO+t6s0e5FdT3FzK2f3PzRc30JA=\">AAAB/XicbVBNSwMxEJ31s9avqkcvwSJ4KrtF0GPRixehgv2AdilJmm1Dk+ySZIWyFH+DVz17E6/+Fo/+E9N2D7b1wcDjvRlm5pFEcGN9/9tbW9/Y3Nou7BR39/YPDktHx00Tp5qyBo1FrNsEGya4Yg3LrWDtRDMsiWAtMrqd+q0npg2P1aMdJyyUeKB4xCm2Tmp1iUT3vWqvVPYr/gxolQQ5KUOOeq/00+3HNJVMWSqwMZ3AT2yYYW05FWxS7KaGJZiO8IB1HFVYMhNms3Mn6NwpfRTF2pWyaKb+nciwNGYsieuU2A7NsjcV//M6qY2uw4yrJLVM0fmiKBXIxmj6O+pzzagVY0cw1dzdiugQa0ytS2hhC5ETl0mwnMAqaVYrgV8JHi7LtZs8nQKcwhlcQABXUIM7qEMDKIzgBV7hzXv23r0P73PeuublMyewAO/rFxoXlX8=</latexit>\nFIG. 2. Angle variables \u0012and\u00120for the case when the both\nmagnetizations are in the plane xz('0='= 0).\nWe introduce the pairs of angle variables \u0012i,'ideter-\nmining directions of the sublattice magnetizations:\nMix=Mcos\u0012icos'i; Miy=Mcos\u0012isin'i;\nMiz=Msin\u0012i:(43)\nThe equations of motion in the angle variables are\ncos\u0012i_\u0012i\n\r=1\nM\u0012@H\n@'i\u0000r@H\n@r'i\u0013\n;\ncos\u0012i_'i\n\r=\u00001\nM\u0012@H\n@\u0012i\u0000r@H\n@r\u0012i\u0013\n: (44)\nIn the further analysis it is convenient to use other angle\nvariables:\n\u00120=\u0019+\u00121\u0000\u00122\n2; \u0012=\u0019\u0000\u00121\u0000\u00122\n2;\n'0='1+'2\n2; '='1\u0000'2\n2: (45)\nIn these variables the Hamiltonian becomes\nH=\u0000M2\n\u001f(cos 2\u00120cos2'\u0000cos 2\u0012sin2')\u00002HM cos\u0012sin\u00120\n+AM2[(1 + cos 2\u00120cos 2\u0012)r'2\n0+r'2\n2\u0000sin 2\u00120sin 2\u0012r'0\u0001r'+r\u00122\n0+r\u00122]\n+A12M2f(cos 2\u0012sin2'+ cos 2\u00120cos2')(r\u00122\n0\u0000r\u00122)\u0000cos 2\u00120+ cos 2\u0012\n2cos 2'(r'2\n0\u0000r'2)\n\u0000sin 2'[sin 2\u0012(r\u00120\u0001r'0+r\u0012\u0001r') + sin 2\u00120(r\u0012\u0001r'0+r\u00120\u0001r')]g: (46)\nThe polar angles \u0012for the staggered magnetization Land\nthe canting angle \u00120are shown in Fig. 2 for the case when\nthe both magnetizations are in the plane xz('0='=\n0).\nIn the uniform ground state \u0012= 0,'= 0,mz=\n2Msin\u00120=\u001fH, while the angle '0is an arbitrary con-stant. Since we consider \felds Hweak compared to the\nexchange \feld, \u00120is always small. In the state with con-\nstant current K=r'0the magnetization along the\nmagnetic \feld is\nmz=\u001fH\n1\u0000\u001fA\u0000K2=2; (47)8\nwhereA\u0006=A\u0006A12.\nIn a weakly perturbed current state small but nonzero\n\u0012and'appear. Also the angles \u00120and'0di\u000ber from\ntheir values in the stationary current state: \u00120!\u00120+ \u0002,\n'0!'0+ \b. Linearization of the nonlinear equations\nof motion with respect to weak perturbations \u0002, \b, \u0012,\nand'yields decoupled linear equations for two pairs of\nvariables (\u0002 ;\b) and (\u0012;'):\n_\u0002\n\r\u0000A\u0000mzK\u0001r\u0002 =\u0000A\u0000M?r2\b;\n_\b\n\r\u0000A\u0000mzK\u0001r\b =\u0000\u0012\n1\u0000\u001fA\u0000K2\n2\u00132M?\n\u001f\u0002\n+(A+A12cos 2\u00120)\ncos\u00120Mr2\u0002;(48)\n_\u0012\n\r\u0000A+mzK\u0001r\u0012\n=\u00002M?\n\u001f\u0000\n1 +\u001fA12K2\u0001\n'+A+M?r2';\n_'\n\r\u0000A+mzcos\u00120K\u0001r'\n=m2\nz\n2\u001fM?(1 +\u001fA12K2)\u0012\u0000A\u0000K2M?\u0012\n\u0000A\u0000A12cos 2\u00120\ncos\u00120Mr2\u0012: (49)\nFor plane waves/eik\u0001r\u0000i!tEq. (48) describes the gapless\nGoldstone mode with the spectrum:\n(!+\rmzA\u0000K\u0001k)2\n=c2\ns\u0014\n1\u0000\u001fA\u0000K2\n2+\u001f(A+A12cos 2\u00120)k2\n2 cos2\u00120\u0015\nk2:(50)\nHere\ncs=\rM?s\n2A\u0000\n\u001f(51)\nis the spin-wave velocity in the ground state without spin\ncurrent. Apart from quadratic corrections k2to the fre-\nquency, the gapless mode in an antiferromagnet does not\ndi\u000ber from that in a ferromagnet, if one replaces in all\nexpressions for the ferromagnet AbyA\u0000=2 and the pa-\nrameterMby 2M.\nEquation (49) describes the gapped mode with the\nspectrum\n(!+\rmzA+K\u0001k)2=\u0012\n1 +\u001fA12K2+\u001fA+k2\n2\u0013\n\u0002\u0014(1 +\u001fA12K2)\r2m2\nz\n\u001f2\u0000c2\nsK2\n+2\r2M2(A\u0000A12cos 2\u00120)k2\n\u001f\u0015\n:(52)Without spin current and neglecting the term /A+k2\nthe spectrum is\n!=s\n\r2m2z\n\u001f2+c2sk2: (53)\nThis spectrum determines a new correlation length\n\u0018=M\nHs\n2A\u0000\n\u001f=cs\n\rH; (54)\nwhich is connected with the easy-plane anisotropy energy\n(42) and determines the wave vector k= 1=\u0018at which\nthe gap and the kdependent frequency become equal.\nApplying the Landau criterion to the gapless mode one\nobtains the critical gradientp\n2=\u001fA\u0000similar to the value\n(27) obtained for a ferromagnet. But in contrast to a fer-\nromagnet where the susceptibility \u001fis connected with\nweak anisotropy energy, in an antiferromagnet the sus-\nceptibility\u001fis determined by a much larger exchange\nenergy and is rather small. As a result, in an antiferro-\nmagnet the gapless Goldstone mode becomes unstable at\nthe very high value of K. But at much lower values of\nKthe gapped mode becomes unstable. According to the\nspectrum (52), the gap in the spectrum vanishes at the\ncritical gradient\nKc=1\n\u0018=\rH\ncs=\rmz\n\u001fcs: (55)\nV. TWO-FLUID EFFECTS AND DISSIPATION\nFROM THE BOLTZMANN EQUATION FOR\nMAGNONS\nKnowledge of the spectrum of collective modes allows\nto derive the dynamical equations at \fnite temperatures\ntaking into account the presence of thermal magnons.\nFurther we follow the procedure of the derivation of the\ntwo-\ruid hydrodynamics in super\ruids.27We address the\nhydrodynamical limit when all parameters ( Mz,K,T)\nof the system slowly vary in space and time.\nWe shall focus on ferromagnets. The equilibrium\nPlanck distribution of magnons in a ferromagnet with\na small spin current /Kis\nnK=1\ne~!(k)=T\u00001\u0019n0(!0)\u00002\u001fc2\nsMz\n\rM2\n?@n0(!0)\n@!0K\u0001k;\n(56)\nwhere!0=cskand\nn0(!0) =1\ne~!0=T\u00001(57)\nis the Planck distribution in the state without spin cur-\nrent.\nIn the theory of super\ruidity the Plank distribution\nof phonons in general depends not only on density and\nsuper\ruid velocity (analogs of our MzandK) but also on9\nthe normal velocity, which characterizes a possible drift of\nthe gas of quasiparticles with respect to the laboratory\nframe of coordinates. This drift is possible because of\nthe Galilean invariance of super\ruids. In our case the\nGalilean invariance is broken by possible interaction of\nmagnons with defects, and in the equilibrium the drift of\nthe quasiparticle gas is impossible. The case of broken\nGalilean invariance, when the normal velocity vanishes,\nwas also investigated for super\ruids in porous media or\nin very thin channels, when the Galilean invariance is\nbroken by interaction with channel walls. It was called\nthe clamped regime.31,32\nSubstituting the Planck distribution (56) into Eqs. (34)\nand (35) one obtains the contribution of equilibrium\nmagnons to the spin current and the spin chemical po-\ntential:\nJsjeq=\r@\n@K=\u0000\u00192\u001f2T4\n30\rM2\n?~3csK\u0012\n1 +16M2\nz\n3M2\n?\u0013\n;(58)\n\u0016jeq=@\n@Mz=\u00192MzT4\n30~3c3sM2\n?; (59)\nwhere\n\n =TZ\nln(1\u0000e\u0000~!(k)=T)d3k\n(2\u0019)3: (60)\nis the thermodynamical potential for the magnon Bose-\ngas. The contribution (58) decreases the super\ruid spin\ncurrent at \fxed phase gradient K, similarly to the de-\ncrease of the mass super\ruid current after replacing the\ntotal mass density by the lesser super\ruid density.\nYuan et al.20used in their experiment very thin \flm at\nlow temperature, when de Broglie wavelength of magnons\nexceeds \flm thickness, and it is useful to give also the\ntwo-\ruid corrections for a two-dimensional case. Repeat-\ning our calculations after replacing integralsR\nd3k=(2\u0019)3\nby integrals WR\nd2k=(2\u0019)2, one obtains:\nJsjeq=\u0000\u0010(3)\u001f2T3\n\u0019W\rM2\n?~2K\u0012\n1 +6M2\nz\nM2\n?\u0013\n; (61)\n\u0016jeq=\u0010(3)MzT3\n\u0019W~2c2sM2\n?; (62)\nwhere the value of the Riemann zeta function \u0010(3) is\n1.202 andWis the \flm thickness.\nThe next step in derivation of the two-\ruid theory at \f-\nnite temperatures is the analysis of dissipation. A widely\nused approach of studying dissipation in magnetically or-\nder systems is the LLG theory with the Gilbert damp-\ning term added. However, this approach is incompatible\nwith the spin conservation law. This law, although being\napproximate, plays a key role in the problem of spin su-\nper\ruidity. Therefore, we derived dissipation parameters\nfrom the Boltzmann equation for magnons postponing\ndiscussion of the LLG theory with the Gilbert damping\nto the Appendix.Dissipation is connected with nonequilibrium correc-\ntions to the magnon distribution. At low temperatures\nthe number of magnons is small, and magnon-magnon\ninteraction is weak. Then the main source of dissipa-\ntion is scattering of magnons by defects. The Boltzmann\nequation with the collision term in the relaxation-time\napproximation is\n_n+@!\n@k\u0001rn\u0000r!\u0001@n\n@k=\u0000n\u0000nK\n\u001c: (63)\nIf parameters, which determine the magnon distribution\nfunctionn, vary slowly in space and time one can substi-\ntute the equilibrium Planck distribution nKinto the left-\nhand side of the Boltzmann equation (63). This yields:\n@n0\n@!_!+@n0\n@T\u0012\n_T+@!\n@k\u0001rT\u0013\n=\u0000n\u0000n0\n\u001c; (64)\nWe consider small gradients Kwhen the di\u000berence be-\ntweennKandn0is not important. But weak depen-\ndence of!onKis important at calculation of _ !. One\ncan see that at the constant temperature Tin any sta-\ntionary state the left-hand side vanishes, and there is\nno nonequilibrium correction to the magnon distribution.\nCorrespondingly, there is no dissipation. This is one more\nillustration that stationary super\ruid currents do not de-\ncay.\nIn nonstationary cases time derivatives are determined\nby the equations of motions. The equations of motion for\nMzandKare not su\u000ecient, and the equation of heat bal-\nance is needed for \fnding _T. In general the heat balance\nequation is rather complicated since it must take into ac-\ncount interaction of magnons with other subsystems, e.\ng., phonons. Instead of it we consider a simpler case,\nwhen magnons are not important in the heat balance,\ni.e., the temperature does not depend on magnon pro-\ncesses. In other words we consider the isothermal regime\nwhen _T= 0. But we allow slow temperature variation in\nspace.\nThe temporal variation of the frequency !emerges\nfrom slow temporal variation of MzandK, and at small\nK\n_!=@!\n@Mz_Mz+@!\n@K_K=\u0000Mz\nM2\n?\u0012\ncsk_Mz+2\u001fc2\ns\n\rk\u0001_K\u0013\n:\n(65)\nThe partial derivatives @!=@Mzand@!=@Kwere deter-\nmined from the spectrum (19), while the time derivatives\nofMzandKwere found from the linearized equations\n(15) and (16) assuming that r'=Kis small and ig-\nnoring gradients of Mzin the right-hand side of Eq. (16),\nwhich are beyond the hydrodynamical limit. Then\n_!=Mz\nM2\n?c2\ns\u0014\u001f\n\rcskr\u0001K+ 2(k\u0001r)Mz\u0015\n: (66)\nEventually the nonequilibrium correction to the magnon10\ndistribution function is\nn0=n\u0000n0=\u0000Mz\nM2\n?cs\u0014\u001fcs\n\rkr\u0001K\n+2(k\u0001r)Mz\u0000M2\n?\nMzT(k\u0001r)T\u0015\n\u001c@n0\n@k(67)\nSubstituting n0into Eqs. (34) and (35) one obtains dis-\nsipation terms in the spin current and the spin chemical\npotential:\nJd=\u0000D\u0012\nrMz\u00001\n2TM2\n?\nMzrT\u0013\n; (68)\n\u0016d=\u0000\u0010\n\rr\u0001K; (69)\nwhere\nD=\u00002\u001f~c3\ns\n3\u00192M2\nz\nM4\n?Z\n\u001c@n0\n@kk4dk;\n\u0010=\u0000\u001f~c3\ns\n2\u00192M2\nz\nM4\n?Z\n\u001c@n0\n@kk4dk: (70)\nIn addition to the spin di\u000busion current, the dissipative\nspin current Jdcontains also the current proportional\nto the temperature gradient. This is the bulk Seebeck\ne\u000bect. Estimation of the integral in these expressions\nrequires knowledge of possible dependence of the relax-\nation time \u001con the energy. Under the assumption that\n\u001cis independent from the energy,\nD=8\u00192\u001c\r2T4M2\nz\n45~3c3sM2\n?; \u0010=2\u00192\u001c\r2T4M2\nz\n15~3c3sM2\n?; (71)\nor for the two-dimensional case,\nD=16\u0010(3)\u001c\r2T3M2\nz\n3\u0019W~2c2sM2\n?; \u0010=4\u0010(3)\u001c\r2T3M2\nz\n\u0019W~2c2sM2\n?:(72)\nAlthough in antiferromagnets the Landau critical gra-\ndient is connected with the gapped mode, at small phase\ngradients the gapless Goldstone mode has lesser energy,\nand at low temperatures most of magnons belong to this\nmode. Since the Goldstone modes in ferromagnets and\nantiferromagnets are similar, our estimation of dissipa-\ntion coe\u000ecients for ferromagnets is valid also for antifer-\nromagnets after replacing AbyA\u0000=2 andMby 2M.\nThe microscopic analysis of this section agrees with\nthe following phenomenological equations similar to the\nhydrodynamical equations for super\ruids in the clamped\nregime:\n_Mz=\u0000r\u0001Js\u0000@R\n@\u0016+r@R\n@r\u0016; (73)\n_'=\u0000\r\u0016+@R\n@(r\u0001Js); (74)where the spin chemical potential and the super\ruid spin\ncurrent,\n\u0016=\u000eF\n\u000eMz;Js=\r@F\n@r'; (75)\nare determined by derivatives of the free energy\nF=H+ \n\u0000TS: (76)\nThe spin conservation law forbids the term @R=@\u0016 in the\ncontinuity equation (73), because it is not a divergence of\nsome current. Thus, the dissipation function is compati-\nble with the spin conservation law if it depends only on\nthe gradient of the spin chemical potential \u0016, but not on\n\u0016itself. This does not take place in the LLG theory with\nthe Gilbert damping discussed in the Appendix. The\nanalysis of this section assumed the spin conservation\nlaw and corresponded to the dissipation function\nR=\u001fD\n2r\u00162\u0000D\n2TM2\n?\nMzr\u0016\u0001rT+\u0010\n2\rAM2\n?(r\u0001Js)2:\n(77)\nIn general the dissipation function contains also the term\n/rT2responsible for the thermal conductivity. But it\nis important only for the heat balance equation, which\nwas not considered here.\nIf the temperature does not vary in space, then the only\ntemperature e\u000bect is a correction to the spin chemical\npotential. This does not a\u000bect the basic feature of super-\n\ruid spin transport: there is no gradient of the chemical\npotential in a stationary current state, and all dissipation\nprocesses are not e\u000bective except for the relativistically\nsmall spin Bloch relaxation. If there is spatial variation of\ntemperature, then the spin chemical potential also varies\nin space. One can \fnd its gradient by exclusion of r\u0001Js\nfrom Eqs. (73) and (74):\nr\u0016=rMz\n\u001f=\u0000D\u0010\n2\r2AMzTr(r2T): (78)\nNote that the spin chemical potential gradient is propor-\ntional not to the \frst but to the third spatial derivative\nof the temperature. The constant temperature gradi-\nent does not produce spatial variation of the chemical\npotential. This is an analog of the absence of thermo-\nelectric e\u000bects proportional to the temperature gradients\nin superconductors.33Naturally the e\u000bect produced by\nhigher derivatives of the temperature is weaker than pro-\nduced by the \frst derivative.\nThe nonuniform correction to the spin chemical po-\ntential strongly depends on temperature. Assuming the\nT4dependence of the dissipation parameters Dand\u0010in\nEq. (71) the coe\u000ecient before the temperature-gradient\nterm in Eq. (78) is proportional to T8. Now the spin dif-\nfusion current\u0000\u001fDr\u0016does not disappear in the equa-\ntion (73) of continuity for the spin, but it is proportional\ntoT12.\nEarlier Zhang and Zhang34used the Boltzmann equa-\ntion for derivation of the spin di\u000busion coe\u000ecient and11\nthe Bloch relaxation time in an isotropic ferromagnet in\na constant magnetic \feld. We derived the spin di\u000busion\nand the second viscosity coe\u000ecients in an easy-plane fer-\nromagnet with di\u000berent spin-wave spectrum. Two-\ruid\ne\u000bects in easy-plane ferromagnets were investigated by\nFlebus et al.35. They solved the Boltzmann equation\nusing the equilibrium magnon distribution function with\nnonzero chemical potential of magnon (do not confuse it\nwith the spin chemical potential introduced in the present\npaper). In contrast, we assumed complete thermaliza-\ntion of the magnon distribution when the magnon chem-\nical potential vanishes. The thermalization assumption\nis questionable in the transient layer near the interface\nthrough which spin is injected, and in this layer the ap-\nproach Flebus et al.35may become justi\fed. The tran-\nsient layer is discussed in the next section.\nVI. TRANSIENT (HEALING) LAYER NEAR\nTHE INTERFACE INJECTING SPIN\nInjection of spin from a medium without spin super\ru-\nidity to a medium with spin super\ruidity may produce\nnot only a super\ruid spin current but also a spin cur-\nrent of incoherent magnons. But at some distance from\nthe interface between two media, which will be called the\nconversion healing length, the spin current of incoherent\nmagnons (spin di\u000busion current) must inevitably trans-\nform to super\ruid spin current, as we shall show now.\nWe return back to Eqs. (1) and (2) but now we neglect\nthe relativistically small Bloch spin relaxation (the term\n/1=T1). In Sec. II we considered the stationary solution\nof the these equations with constant magnetization and\nabsent spin di\u000busion current. But it is not the only sta-\ntionary solution. Another solution is an evanescent mode\nM0\nz/r'/e\u0000x=\u0015, where\n\u0015=s\n\u001fD\u0010\n\rA(79)\nis the conversion healing length. We look for superposi-\ntion of two solutions, which satis\fes the condition that\nthe injected current J0transforms to the spin di\u000busion\ncurrent, while the super\ruid current vanishes at x= 0:\nJ0=\u0000DrxM0\nz(0);rx'(0) = 0: (80)\nThis superposition is\nM0\nz(x) =M0\nz+\u0015J0\nDe\u0000x=\u0015;rx'(x) =J0\nA(1\u0000e\u0000x=\u0015);\n(81)\nwhereM0\nzin the right-hand side is a constant magneti-\nzation far from the interface x= 0. Thus, at the length\n\u0015the spin di\u000busion current Jddrops from J0to zero,\nwhile the super\ruid spin current grows from zero to J0\nand remains at larger distances constant.\nAs pointed out in the end of Sec. II, the phenomenolog-\nical equations (1) and (2) were derived assuming that thespin chemical potential \u0016=M0\nz=\u001f\u0000Hdoes not depend\non gradients rMz. However, the dissipation coe\u000ecients\nDand\u0010decrease very sharply with temperature, and\nthe conversion healing length eventually becomes much\nsmaller than the scale \u00180[see Eq. (25)], when the depen-\ndence of the free energy and the spin chemical potential\non the gradients rMzbecomes important. But in fact\naddingrMz-dependent terms into the expression for \u0016,\n\u0016=Mz\n\u001f\u0000H\u0000AM2r2Mz\nM2\n?; (82)\ndoes not a\u000bect the expression (79) for the healing length.\nThe generalization of the analysis reduces to replacing of\nM0\nzin Eqs. (1), (2), and (81) by \u001f\u0016.\nTransformation of the injected incoherent magnon spin\ncurrent to the super\ruid spin current is not the only tran-\nsient process near the interface between media with and\nwithout spin super\ruidity. Even in the absence of spin\ncurrent the interface may a\u000bect the equilibrium mag-\nnetic structure. For example, the interface can induce\nanisotropy di\u000berent from easy-plane anisotropy in the\nbulk. Then the crossover from surface to bulk anisotropy\noccurs at the healing length of the order of the correla-\ntion length \u00180determined by Eq. (25) in ferromagnets, or\nthe correlation length \u0018determined by Eq. (54) in anti-\nferromagnets. The similar healing length was suggested\nfor ferromagnets by Takei and Tserkovnyak7and for an-\ntiferromagnets by Takei et al.8although using di\u000berent\narguments.\nThe expression (79) for \u0015was derived within hydrody-\nnamics with dissipation. At distances shorter than the\nmean-free path incoherent magnons are in the ballistic\nregime and cannot converge to the super\ruid current,\nsince conversion is impossible without dissipation. Alto-\ngether this means that the real healing length at which\nthe bulk super\ruid spin current state is formed cannot\nbe less than the longest from three scales: \u0015,\u00180, and the\nmagnon mean-free path cs\u001c. Apparently at low tempera-\ntures and weak magnetization Mzthe latter is the longest\none from three scales. However, close to the phase transi-\ntion to the easy-axis anisotropy ( Mz=M) the coherence\nlength\u00180diverges and becomes the longest scale.\nSolving the Boltzmann equation we assumed complete\nthermalization of the magnon distribution. At low tem-\nperatures when magnon-magnon interaction is weak the\nlength at which thermalization occurs essentially exceeds\nthe mean-free path on defects. It could be that the heal-\ning length would grow up to the thermalization length.\nThis requires a further analysis.\nVII. MAGNETIC VORTEX IN AN\nEASY-PLANE ANTIFERROMAGNET\nLet us consider structure of an axisymmetric vortex in\nan antiferromagnet with one quantum of circulation of\nthe angle'0of rotation around the vortex axis. Now\nwe consider the geometry of the experiment20when the12\nPtPtzxyCr2O3H\n\u0001\u0002\u0001\u0002\u0001\u0002\u0002\u0003\u0001\u0002\u0001\u0004\u0001\u0003\u0002\u0004\u0002\u0003\u0003\u0004\u0002\u0004\u0001\u0006\u0001\u0002\u0002\u0006\u0001\u0002\u0004\u0006\u0001\u0002\u0003\u0006\u0003\u0004\u0001\u0002\u0001\u0002\u0001\u0002\u0001\u0002\u0001\u0002\u0001\u0002\u0004\u0006\u0001\u0002\u0002\u0006\u0001\u0002\u0001\u0006\u0002\u0004\u0005\u0006\u0002\u0003\u0002\u0004\u0001\u0003\u0001\u0004\u0001\u0002\u0001\u0003\u0001\u0004\u0003\u0001a)\nb)\nFIG. 3. Precession of magnetization maround the direction\nof the magnetic \feld Halong the path around the vortex\naxis. (a) The geometry of the experiment20with the magnetic\n\feld (the axis z) in the plane of the Cr 2O3\flm. The vortex\naxis is normal to the \flm (the axis y). (b) Precession of the\nmagnetization mis shown in the plane xz(the plane of the\n\flm). The path around the vortex axis (dashed lines) is inside\nthe vortex core where the total magnetization is not parallel\ntoH(\u0012 6= 0).\nmagnetic \feld H(the axisz) is in the \flm plane. The\nvortex axis is the axis ynormal to the \flm plane (Fig, 3a).\nThe azimuthal component of the angle '0gradient is\nr'0=1\nr: (83)\nAt the same time '= 0 and\u00120is small. Then the Hamil-\ntonian (46) transforms to\nH=2M2\n\u001f\u00122\n0\u00002HM cos\u0012\u00120+A\u0000M2\u0012cos2\u0012\nr2+r\u00122\u0013\n:\n(84)\nMinimization with respect to small \u00120yields\n\u00120=\u001fHcos\u0012\n2M; (85)\nand \fnally the Hamiltonian is\nH=\u0000\u001fH2cos2\u0012\n2+A\u0000M2\u0012cos2\u0012\nr2+r\u00122\u0013\n:(86)The Euler{Lagrange equation for this Hamiltonian de-\nscribes the vortex structure in polar coordinates:\nd2\u0012\ndr2+1\nrd\u0012\ndr\u0000sin 2\u0012\n2\u00121\n\u00182\u00001\nr2\u0013\n= 0; (87)\nwhere the correlation length \u0018is given by Eq. (54) and\ndetermines the size of the vortex core.\nThe vortex core has a structure of a skyrmion, in which\nthe total weak magnetization deviates from the direction\nof the magnetic \feld H(\u00126= 0). The component of\nmagnetization transverse to the magnetic \feld is\nm?=\rHsin 2\u0012\n2: (88)\nThe transverse magnetization creates stray magnetic\n\felds at the exit of the vortex line from the sample. Fig-\nure 3 shows variation of the magnetization inside the core\nalong the path around the vortex axis parallel to the axis\ny. Along the path the magnetization mrevolves around\nthe direction of the magnetic \feld forming a cone. The\nprecession in space creates an oscillating ycomponent\nof magnetization my=m?(r) sin\u001e, where\u001eis the az-\nimuthal angle at the circular path around the vortex line.\nThis produces surface magnetic charges 4 \u0019myat the exit\nof the vortex to the boundary separating the sample from\nthe vacuum. These charges generate the curl-free stray\n\feldh=r . At distances from the vortex exit point\nmuch larger that the core radius the stray \feld is a dipole\n\feld with the scalar potential\n (R) =\u0019\u001fH\n2(R\u0001n)\nR3Z1\n0sin 2\u0012(r)r2dr\n= 1:2\u0019\u001fH\u00183(R\u0001n)\nR3=1:2\u0019\u001fc3\ns\n\r3H2(R\u0001n)\nR3: (89)\nHereR(x;y;z ) is the position vector with the origin in\nthe vortex exit point and nis a unit vector in the plane\nxzalong which the surface charge is maximal ( \u001e=\u0019=2).\nIn our model the direction of nis arbitrary, but it will\nbe \fxed by spin-orbit interaction or crystal magnetic\nanisotropy violating invariance with respect to rotations\naround the axis z. These interactions were ignored in our\nmodel. In principle, the stray \feld can be used for detec-\ntion of vortices nucleated at spin currents approaching\nthe critical value.\nVIII. DISCUSSION AND SUMMARY\nThe paper analyzes the long-distance super\ruid spin\ntransport. The super\ruid spin transport does not require\na gradient of the spin chemical potential (as the electron\nsupercurrent in superconductors does not require a gra-\ndient of the electrochemical potential). As result of it,\nmechanisms of dissipation are suppressed except for weak\nBloch spin relaxation. Other dissipation mechanisms af-\nfect the spin transport only at the transient (healing)13\nlayer close to the interface through which spin is injected,\nor in nonstationary processes.\nThe paper calculates the Landau critical spin phase\ngradient in a two-sublattice antiferromagnet when the\neasy-plane topology of the magnetic order parameter is\nprovided not by crystal magnetic anisotropy but by an\nexternal magnetic \feld. This was the case realized in\nthe experiment by Yuan et al.20. For this goal it was\nnecessary to derive the spectrum of collective modes (spin\nwaves) in spin current states. The Landau instability\ndestroying spin super\ruidity sets on not in the Goldstone\ngapless mode as in easy-plane ferromagnets but in the\ngapped mode, despite that at small spin currents the\nlatter has energy larger than the Goldstone mode.\nThe paper analyzes dissipation processes determining\ndissipation parameters (spin di\u000busion and second viscos-\nity coe\u000ecients) by solving the Boltzmann equation for\nmagnons scattered by defects. The two-\ruid theory sim-\nilar to the super\ruid two-\ruid hydrodynamics was sug-\ngested. It is argued that the LLG theory with the Gilbert\ndamping parameter is not able to properly describe dissi-\npation in easy-plane magnetic insulators. Describing the\nwhole dissipation by a single Gilbert parameter one can-\nnot di\u000berentiate between strong processes connected with\nhigh exchange energy (e.g., spin di\u000busion) and weak pro-\ncesses connected with spin-orbit interaction (Bloch spin\nrelaxation), which violate the spin conservation law.\nThe formation of the super\ruid spin current in the\ntransient (healing) layer near the interface through which\nspin is injected was investigated. The width of this layer\n(healing length) is determined by processes of dissipation,\nand at low temperatures can reach the scale of relevant\nmean-free paths of magnons including those at which the\nmagnon distribution is thermalized.\nThe structure of the magnetic vortex in the geometry\nof the experiment on Cr 2O3is investigated. In the vortex\ncore there is a magnetization along the vortex line, which\nis normal to the magnetic \feld. This magnetization pro-\nduces magnetic charges at the exit of the vortex line from\nthe sample. The magnetic charges create a stray dipole\nmagnetic \feld, which probably can be used for detection\nof vortices.\nWithin the developed two-\ruid theory the paper ad-\ndresses the role of the temperature variation in space on\nthe super\ruid spin transport. This is important because\nin the experiment of Yuan et al.20the spin is created\nin the Pt injector by heating (the Seebeck e\u000bect). Thus\nthe spin current to the detector is inevitably accompa-\nnied by heat \row. The temperature variation produces\nthe bulk Seebeck e\u000bect, which is estimated to be rather\nweak at low temperatures. However, it was argued26that\nprobably Yuan et al.20detected a signal not from spin\ncoming from the injector but from spin produced by the\nSeebeck e\u000bect at the interface between the heated anti-\nferromagnet and the Pt detector. Such e\u000bect has already\nbeen observed for antiferromagnet Cr 2O3.36If true, then\nYuan et al.20observed not long-distance spin transport\nbut long-distance heat transport. It is not supported bythe fact that Yuan et al. observed a threshold for super-\n\ruid spin transport at low intensity of injection, when ac-\ncording to the theory5violation of the approximate spin\nconservation law becomes essential. Investigation of su-\nper\ruid spin transport at low-intensity injection is more\ndi\u000ecult both for theory and experiment. But the exis-\ntence of the threshold is supported by extrapolation of\nthe detected signals from high-intensity to low-intensity\ninjection. According to the experiment, the signal at the\ndetector is not simply proportional to the squared elec-\ntric current j2responsible for the Joule heating in the\ninjector, but to j2+a. The o\u000bset ais evidence of the\nthreshold, in the analogy with the o\u000bset of IVcurves in\nthe mixed state of type II superconductors determining\nthe critical current for vortex deepening. With all that\nsaid, the heat-transport interpretation cannot be ruled\nout and deserves further investigation. According to this\ninterpretation, one can see the signal observed by Yuan\net al.20at the detector even if the Pt injector is replaced\nby a heater, which produces the same heat but no spin.\nAn experimental check of this prediction would con\frm\nor reject the heat-transport interpretation.\nLet us make some numerical estimations for Cr 2O3us-\ning the formulas of the present paper. It follows from\nneutron scattering data37that the spin-wave velocity is\ncs= 8\u0002105cm/sec. According to Foner38, the magne-\ntization of sublattices is M= 590 G and the magnetic\nsusceptibility is \u001f= 1:2\u000210\u00004. Then the total magne-\ntizationmz=\u001fHin the magnetic \feld H= 9 T used\nin the experiment is about 10 G, and the canting an-\ngle\u00120=mz=2M\u00190:01 is small as was assumed in our\nanalysis. The correlation length (54), which determines\nvortex core radius, is about \u0018\u00190:5\u000210\u00006cm. The stray\nmagnetic \feld produced by magnetic charges at the exit\nof the vortex line from the sample is 10( \u00183=R3) G, where\nRis the distance from the vortex exit point. The task to\ndetect such \felds does not look easy, but it is hopefully\npossible with modern experimental techniques.\nACKNOWLEDGMENTS\nI thank Eugene Golovenchits, Wei Han, Mathias Kl aui,\nRomain Lebrun, Allures Qaiumzadeh, Victoria Sanina,\nSo Takei, and Yaroslav Tserkovnyak for fruitful discus-\nsions and comments.\nAppendix: Dissipation in the LLG theory\nFor ferromagnets the LLG equation taking into ac-\ncount dissipation is\ndM\ndt=\r[Heff\u0002M] +\u000b\nM\u0014\nM\u0002dM\ndt\u0015\n; (A.1)\nwhere\u000bis the dimensionless Gilbert damping parameter.\nFor small\u000bthis equation is identical to the equation with14\nthe Landau{Lifshitz damping term:\n1\n\rdM\ndt=\u0014\nM\u0002\u000eH\n\u000eM\u0015\n+\u000b\nM\u0014\nM\u0002\u0014\nM\u0002\u000eH\n\u000eM\u0015\u0015\n:\n(A.2)\nTransforming the vector LLG equation to the equations\nfor two Hamiltonian conjugate variables, the zcompo-\nnentMzof magnetization and the angle 'of rotation\naround the zaxis, one obtains Eqs. (73) and (74) without\nthe term r(@R=@r\u0016) and with the dissipation function\nR=\u000b\rM2\n?\n2M\u00162+\u000bM\n2M2\n?(r\u0001Js)2; (A.3)\nwhich depends on the spin chemical potential \u0016itself,\nbut not on its gradient. Meanwhile, according to the\ntwo-\ruid theory of Sec. V, the r\u0016-dependent term in the\ndissipation function was responsible for the spin-di\u000busion\nterm in the continuity equation for Mz. Indeed, at deriva-\ntion of the continuity equation (1) from the LLG theory\nunder the assumption that \u0016\u0019M0\nz=\u001f=Mz=\u001f\u0000Hthe\nspin di\u000busion term /Ddoes not appear. The term\ndoes appear only if \u0016in the dissipation function (A.3)\nis determined by the more general expression (82) taking\ninto account the dependence on rMz. Then one obtains\nEqs. (1) and (2) with the equal spin di\u000busion and spin\nsecond viscosity coe\u000ecients\nD=\u0010=\u000b\rMA; (A.4)\nand the inverse Bloch relaxation time\n1\nT1=\u000b\rM2\n?\n\u001fM: (A.5)\nThe outcome looks bizarre. The spin di\u000busion emerges\nfrom the\u0016-dependent term in the dissipation function,\nwhich is incompatible with the spin conservation law, asif the spin di\u000busion is forbidden by the spin conserva-\ntion law. Evidently this conclusion is physically incor-\nrect. Moreover, in the analogy of magnetodynamics and\nsuper\ruid hydrodynamics the magnetization Mzcorre-\nsponds to the \ruid density. In hydrodynamics the \ruid\ndensity gradients are usually not taken into account in\nthe Hamiltonian and in the chemical potential since they\nbecome important only at small scales beyond the hydro-\ndynamical approach. This does not rule out the di\u000busion\nprocess. Similarly, one should expect that it is possible\nto ignore the magnetization gradients in the spin chemi-\ncal potential either. It is strange that the spin di\u000busion\nbecomes impossible in the hydrodynamical limit.\nAccording to the Noether theorem the total magnetiza-\ntion along the axis zis conserved if the Hamiltonian is in-\nvariant with respect to rotations around the axis zin the\nspin space. The Landau{Lifshitz theory of magnetism19\nis based on the idea that the spin-orbit interaction, which\nbreaks rotational symmetry in the spin space and there-\nfore violates the spin conservation law, is relativistically\nsmall compared to the exchange interaction because the\nformer is inversely proportional to the speed of light. So,\nalthough the spin conservation law is not exact, it is a\ngood approximation (see Sec. I). Then the spin Bloch\nrelaxation term/1=T1, which violates the spin conser-\nvation law, must be proportional to a small parameter\ninversely proportional to the speed of light and cannot\nbe determined by the same Gilbert parameter as other\ndissipation terms, which do not violate the spin conser-\nvation law\nThe insu\u000eciency of the LLG theory for description of\ndissipation was discussed before, but mostly at higher\ntemperatures. It was suggested to replace of the LLG\nequation by the Landau{Lifshitz{Bloch equation, in\nwhich the Bloch longitudinal spin relaxation is present\nexplicitly (see, e.g., Ref. 39 and references to earlier works\ntherein). Our analysis shows that the problem exists also\nat low temperatures.\n1B. I. Halperin and P. C. Hohenberg, Phys. Rev. 188, 898\n(1969).\n2E. B. Sonin, Zh. Eksp. Teor. Fiz. 74, 2097 (1978), [Sov.\nPhys.{JETP, 47, 1091 (1978)].\n3E. B. Sonin, Usp. Fiz. Nauk 137, 267 (1982), [Sov. Phys.{\nUsp., 25, 409 (1982)].\n4Y. 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Ginzburg and G. F. Zharkov, Usp. Fiz. Nauk 125,\n19 (1978), [Sov. Phys. Usp., 21, 381{404 (1978)].\n34S. S.-L. Zhang and S. Zhang, Phys. Rev. Lett. 109, 096603\n(2012).\n35B. Flebus, S. A. Bender, Y. Tserkovnyak, and R. A. Duine,\nPhys. Rev. Lett. 116, 117201 (2016).\n36S. Seki, T. Ideue, M. Kubota, Y. Kozuka, R. Takagi,\nM. Nakamura, Y. Kaneko, M. Kawasaki, and Y. Tokura,\nPhys. Rev. Lett. 115, 266601 (2015).\n37E. Samuelsen, Physics Letters A 26, 160 (1968).\n38S. Foner, Phys. Rev. 130, 183 (1963).\n39P. Nieves, D. Serantes, U. Atxitia, and O. Chubykalo-\nFesenko, Phys. Rev. B 90, 104428 (2014)."    },    {        "title": "1804.09242v1.Generalisation_of_Gilbert_damping_and_magnetic_inertia_parameter_as_a_series_of_higher_order_relativistic_terms.pdf",        "content": "Generalisation of Gilbert damping and magnetic\ninertia parameter as a series of higher-order\nrelativistic terms\nRitwik Mondalz, Marco Berritta and Peter M. Oppeneer\nDepartment of Physics and Astronomy, Uppsala University, P. O. Box 516, SE-751 20\nUppsala, Sweden\nE-mail: ritwik.mondal@physics.uu.se\nAbstract. The phenomenological Landau-Lifshitz-Gilbert (LLG) equation of motion\nremains as the cornerstone of contemporary magnetisation dynamics studies, wherein\nthe Gilbert damping parameter has been attributed to \frst-order relativistic e\u000bects.\nTo include magnetic inertial e\u000bects the LLG equation has previously been extended\nwith a supplemental inertia term and the arising inertial dynamics has been related\nto second-order relativistic e\u000bects. Here we start from the relativistic Dirac equation\nand, performing a Foldy-Wouthuysen transformation, derive a generalised Pauli spin\nHamiltonian that contains relativistic correction terms to any higher order. Using the\nHeisenberg equation of spin motion we derive general relativistic expressions for the\ntensorial Gilbert damping and magnetic inertia parameters, and show that these ten-\nsors can be expressed as series of higher-order relativistic correction terms. We further\nshow that, in the case of a harmonic external driving \feld, these series can be summed\nand we provide closed analytical expressions for the Gilbert and inertial parameters\nthat are functions of the frequency of the driving \feld.\n1. Introduction\nSpin dynamics in magnetic systems has often been described by the phenomenological\nLandau-Lifshitz (LL) equation of motion of the following form [1]\n@M\n@t=\u0000\rM\u0002He\u000b\u0000\u0015M\u0002[M\u0002He\u000b]; (1)\nwhere\ris the gyromagnetic ratio, He\u000bis the e\u000bective magnetic \feld, and \u0015is an\nisotropic damping parameter. The \frst term describes the precession of the local,\nclassical magnetisation vector M(r;t) around the e\u000bective \feld He\u000b. The second term\ndescribes the magnetisation relaxation such that the magnetisation vector relaxes to the\ndirection of the e\u000bective \feld until \fnally it is aligned with the e\u000bective \feld. To include\nzPresent address: Department of Physics, University of Konstanz, D -78457 Konstanz, GermanyarXiv:1804.09242v1  [cond-mat.other]  3 Apr 20182\nlarge damping, the relaxation term in the LL equation was reformulated by Gilbert [2, 3]\nto give the Landau-Lifshitz-Gilbert (LLG) equation,\n@M\n@t=\u0000\rM\u0002He\u000b+\u000bM\u0002@M\n@t; (2)\nwhere\u000bis the Gilbert damping constant. Note that both damping parameters \u000band\u0015\nare here scalars, which corresponds to the assumption of an isotropic medium. Both the\nLL and LLG equations preserve the length of the magnetisation during the dynamics and\nare mathematically equivalent (see, e.g. [4]). Recently, there have also been attempts\nMHeff\nPrecession\nNutationDamping\nFigure 1. Sketch of extended LLG magnetisation dynamics. The green arrow denotes\nthe classical magnetisation vector which precesses around an e\u000bective \feld. The red\nsolid and dotted lines depict the precession and damping. The yellow path signi\fes\nthe nutation, or inertial damping, of the magnetisation vector.\nto investigate the magnetic inertial dynamics which is essentially an extension to the\nLLG equation with an additional term [5{7]. Phenomenologically this additional term of\nmagnetic inertial dynamics, M\u0002I@2M=@t2, can be seen as a torque due to second-order\ntime derivative of the magnetisation [8{11]. The essence of the terms in the extended\nLLG equation is described pictorially in Fig. 1. Note that in the LLG dynamics the\nmagnetisation is described as a classical vector \feld and not as a quantum spin vector.\nIn their original work, Landau and Lifshitz attributed the damping constant \u0015to\nrelativistic origins [1]; later on, it has been more speci\fcally attributed to spin-orbit\ncoupling [12{15]. In the last few decades, several explanations have been proposed\ntowards the origin of damping mechanisms, e.g., the breathing Fermi surface model\n[16, 17], torque-torque correlation model [18], scattering theory formulation [19], e\u000bective\n\feld theories [20] etc. On the other hand, the origin of magnetic inertia is less discussed\nin the literature, although it's application to ultrafast spin dynamics and switching\ncould potentially be rich [9]. To account for the magnetic inertia, the breathing Fermi\nsurface model has been extended [11, 21] and the inertia parameter has been associated\nwith the magnetic susceptibility [22]. However, the microscopic origins of both Gilbert3\ndamping and magnetic inertia are still under debate and pose a fundamental question\nthat requires to be further investigated.\nIn two recent works [23, 24], we have shown that both quantities are of relativistic\norigin. In particular, we derived the Gilbert damping dynamics from the relativistic\nspin-orbit coupling and showed that the damping parameter is not a scalar quantity\nbut rather a tensor that involves two main contributions: electronic and magnetic\nones [23]. The electronic contribution is calculated as an electronic states' expectation\nvalue of the product of di\u000berent components of position and momentum operators;\nhowever, the magnetic contribution is given by the imaginary part of the susceptibility\ntensor. In an another work, we have derived the magnetic inertial dynamics from a\nhigher-order (1 =c4) spin-orbit coupling and showed that the corresponding parameter\nis also a tensor which depends on the real part of the susceptibility [24]. Both these\ninvestigations used a semirelativistic expansion of the Dirac Hamiltonian employing the\nFoldy-Wouthuysen transformation to obtain an extended Pauli Hamiltonian including\nthe relativistic corrections [25, 26]. The thus-obtained semirelativistic Hamiltonian was\nthen used to calculate the magnetisation dynamics, especially for the derivation of the\nLLG equation and magnetic inertial dynamics.\nIn this article we use an extended approach towards a derivation of the\ngeneralisation of those two (Gilbert damping and magnetic inertia) parameters from\nthe relativistic Dirac Hamiltonian, developing a series to fully include the occurring\nhigher-order relativistic terms. To this end we start from the Dirac Hamiltonian in\nthe presence of an external electromagnetic \feld and derive a semirelativistic expansion\nof it. By doing so, we consider the direct \feld-spin coupling terms and show that\nthese terms can be written as a series of higher-order relativistic contributions. Using\nthe latter Hamiltonian, we derive the corresponding spin dynamics. Our results show\nthat the Gilbert damping parameter and inertia parameter can be expressed as a\nconvergent series of higher-order relativistic terms and we derive closed expressions\nfor both quantities. At the lowest order, we \fnd exactly the same tensorial quantities\nthat have been found in earlier works.\n2. Relativistic Hamiltonian Formulation\nTo describe a relativistic particle, we start with a Dirac particle [27] inside a material,\nand, in the presence of an external \feld, for which one can write the Dirac equation\nasi~@ (r;t)\n@t=H (r;t) for a Dirac bi-spinor  . Adopting furthermore the relativistic\ndensity functional theory (DFT) framework we write the corresponding Hamiltonian as\n[23{25]\nH=c\u000b\u0001(p\u0000eA) + (\f\u0000 1)mc2+V 1\n=O+ (\f\u0000 1)mc2+E; (3)\nwhereVis the e\u000bective unpolarised Kohn-Sham potential created by the ion-ion, ion-\nelectron and electron-electron interactions. Generally, to describe magnetic systems, an4\nadditional spin-polarised energy (exchange energy) term is required. However, we have\ntreated e\u000bects of the exchange \feld previously, and since it doesn't contribute to the\ndamping terms we do not consider it explicitly here (for details of the calculations\ninvolving the exchange potential, see Ref. [23, 25]). The e\u000bect of the external\nelectromagnetic \feld has been accounted through the vector potential, A(r;t),cde\fnes\nthe speed of light, mis particle's mass and 1is the 4\u00024 unit matrix. \u000band\fare the\nDirac matrices which have the form\n\u000b= \n0\u001b\n\u001b0!\n; \f = \n10\n0\u00001!\n;\nwhere\u001b= (\u001bx;\u001by;\u001bz) are the Pauli spin matrix vectors and 1is 2\u00022 unit matrix.\nNote that the Dirac matrices form the diagonal and o\u000b-diagonal matrix elements of\nthe Hamiltonian in Eq. (3). For example, the o\u000b-diagonal elements can be denoted as\nO=c\u000b\u0001(p\u0000eA), and the diagonal matrix elements can be written as E=V 1.\nIn the nonrelativistic limit, the Dirac Hamiltonian equals the Pauli Hamiltonian,\nsee e.g. [28]. In this respect, one has to consider that the Dirac bi-spinor can be written\nas\n (r;t) = \n\u001e(r;t)\n\u0011(r;t)!\n;\nwhere the upper \u001eand lower\u0011components have to be considered as \\large\" and \\small\"\ncomponents, respectively. This nonrelativistic limit is only valid for the case when the\nparticle's momentum is much smaller than the rest mass energy, otherwise it gives\nan unsatisfactory result [26]. Therefore, the issue of separating the wave functions of\nparticles from those of antiparticles is not clear for any given momentum. This is mainly\nbecause the o\u000b-diagonal Hamiltonian elements link the particle and antiparticle. The\nFoldy-Wouthuysen (FW) transformation [29] has been a very successful attempt to \fnd\na representation where the o\u000b-diagonal elements have been reduced in every step of the\ntransformation. Thereafter, neglecting the higher-order o\u000b-diagonal elements, one \fnds\nthe correct Hamiltonian that describes the particles e\u000eciently. The FW transformation\nis an unitary transformation obtained by suitably choosing the FW operator [29],\nUFW=\u0000i\n2mc2\fO: (4)\nThe minus sign in front of the operator is because of the property that \fandO\nanticommute with each other. With the FW operator, the FW transformation of the\nwave function adopts the form  0(r;t) =eiUFW (r;t) such that the probability density\nremains the same, j j2=j 0j2. In this way, the time-dependent FW transformed\nHamiltonian can be expressed as [26, 28, 30]\nHFW=eiUFW\u0012\nH\u0000i~@\n@t\u0013\ne\u0000iUFW+i~@\n@t: (5)5\nAccording to the Baker-Campbell-Hausdor\u000b formula, the above transformed Hamilto-\nnian can be written as a series of commutators, and the \fnally transformed Hamiltonian\nreads\nHFW=H+i\u0014\nUFW;H\u0000i~@\n@t\u0015\n+i2\n2!\u0014\nUFW;\u0014\nUFW;H\u0000i~@\n@t\u0015\u0015\n+i3\n3!\u0014\nUFW;\u0014\nUFW;\u0014\nUFW;H\u0000i~@\n@t\u0015\u0015\u0015\n+:::: : (6)\nIn general, for a time-independent FW transformation, one has to work with@UFW\n@t= 0.\nHowever, this is only valid if the odd operator does not contain any time dependency. In\nour case, a time-dependent transformation is needed as the vector potential is notably\ntime-varying. In this regard, we notice that the even operators and the term i~@=@t\ntransform in a similar way. Therefore, we de\fne a term Fsuch thatF=E\u0000i~@=@t.\nThe main theme of the FW transformation is to make the odd terms smaller in every\nstep of the transformation. After a fourth transformation and neglecting the higher\norder terms, the Hamiltonian with only the even terms can be shown to have the form\nas [26, 30{33]\nH000\nFW= (\f\u0000 1)mc2+\f\u0012O2\n2mc2\u0000O4\n8m3c6+O6\n16m5c10\u0013\n+E\u00001\n8m2c4[O;[O;F]]\n\u0000\f\n8m3c6[O;F]2+3\n64m4c8\b\nO2;[O;[O;F]]\t\n+5\n128m4c8\u0002\nO2;\u0002\nO2;F\u0003\u0003\n:(7)\nHere, for any two operators AandBthe commutator is de\fned as [ A;B] and the\nanticommutator as fA;Bg. As already pointed out, the original FW transformation\ncan only produce correct and expected higher-order terms up to \frst order i.e., 1 =c4\n[26, 30, 33]. In fact, in their original work Foldy and Wouthuysen derived only the\nterms up to 1 =c4, i.e., only the terms in the \frst line of Eq. (7), however, notably\nwith the exception of the fourth term [29]. The higher-order terms in the original FW\ntransformation are of doubtful value [32, 34, 35]. Therefore, the Hamiltonian in Eq. (7)\nis not trustable and corrections are needed to achieve the expected higher-order terms.\nThe main problem with the original FW transformation is that the unitary operators in\ntwo preceding transformations do not commute with each other. For example, for the\nexponential operators eiUFWandeiU0\nFW, the commutator [ UFW;U0\nFW]6= 0. Moreover, as\nthe unitary operators are odd, this commutator produces even terms that have not been\nconsidered in the original FW transformation [26, 30, 33]. Taking into account those\nterms, the correction of the FW transformation generates the Hamiltonian as [33]\nHcorr:\nFW= (\f\u0000 1)mc2+\f\u0012O2\n2mc2\u0000O4\n8m3c6+O6\n16m5c10\u0013\n+E\u00001\n8m2c4[O;[O;F]]\n+\f\n16m3c6fO;[[O;F];F]g+3\n64m4c8\b\nO2;[O;[O;F]]\t\n+1\n128m4c8\u0002\nO2;\u0002\nO2;F\u0003\u0003\n\u00001\n32m4c8[O;[[[O;F];F];F]]: (8)6\nNote the di\u000berence between two Hamiltonians in Eq. (7) and Eq. (8) that are observed\nin the second and consequent lines in both the equations, however, the terms in the\n\frst line are the same. Eq. (8) provides the correct higher-order terms of the FW\ntransformation. In this regard, we mention that an another approach towards the correct\nFW transformation has been employed by Eriksen; this is a single step approach that\nproduces the expected FW transformed higher-order terms [34]. Once the transformed\nHamiltonian has been obtained as a function of odd and even terms, the \fnal form\nis achieved by substituting the correct form of odd terms Oand even termsEin the\nexpression of Eq. (8) and calculating term by term.\nSince we perform here the time-dependent FW transformation, we note that the\ncommutator [O;F] can be evaluated as [ O;F] =i~@O=@t. Therefore, following the\nde\fnition of the odd operator, the time-varying \felds are taken into account through\nthis term. We evaluate each of the terms in Eq. (8) separately and obtain that the\nparticles can be described by the following extended Pauli Hamiltonian [24, 26, 36]\nHcorr:\nFW=(p\u0000eA)2\n2m+V\u0000e~\n2m\u001b\u0001B\u0000(p\u0000eA)4\n8m3c2+(p\u0000eA)6\n16m5c4\n\u0000\u0012e~\n2m\u00132B2\n2mc2+e~\n4m2c2(\n(p\u0000eA)2\n2m;\u001b\u0001B)\n\u0000e~2\n8m2c2r\u0001Etot\u0000e~\n8m2c2\u001b\u0001[Etot\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002Etot]\n\u0000e~2\n16m3c4\u001a\n(p\u0000eA);@Etot\n@t\u001b\n\u0000ie~2\n16m3c4\u001b\u0001\u0014@Etot\n@t\u0002(p\u0000eA) + (p\u0000eA)\u0002@Etot\n@t\u0015\n+3e~\n64m4c4n\n(p\u0000eA)2\u0000e~\u001b\u0001B;~r\u0001Etot+\u001b\u0001[Etot\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002Etot]o\n+e~4\n32m4c6r\u0001@2Etot\n@t2+e~3\n32m4c6\u001b\u0001\u0014@2Etot\n@t2\u0002(p\u0000eA)\u0000(p\u0000eA)\u0002@2Etot\n@t2\u0015\n:\n(9)\nThe \felds in the last Hamiltonian (9) are de\fned as B=r\u0002A, the external magnetic\n\feld,Etot=Eint+Eextare the electric \felds where Eint=\u00001\nerVis the internal \feld\nthat exists even without any perturbation and Eext=\u0000@A\n@tis the external \feld (only\nthe temporal part is retained here because of the Coulomb gauge). It is clear that as the\ninternal \feld is time-independent, it does not contribute to the fourth and sixth lines\nof Eq. (9). However, the external \feld does contribute to the above terms wherever it\nappears in the Hamiltonian.\nThe above-derived Hamiltonian can be split in two parts: (1) a spin-independent\nHamiltonian and (2) a spin-dependent Hamiltonian that involves the Pauli spin matrices.\nThe spin-dependent Hamiltonian, furthermore, has two types of coupling terms. The\ndirect \feld-spin coupling terms are those which directly couples the \felds with the\nmagnetic moments e.g., the third term in the \frst line, the second term in the third\nline of Eq. (9) etc. On the other hand, there are relativistic terms that do not directly\ncouple the spins to the electromagnetic \feld - indirect \feld-spin coupling terms. These7\nterms include e.g., the second term of the second line, the \ffth line of Eq. (9) etc. The\ndirect \feld-spin interaction terms are most important because these govern the directly\nmanipulation of the spins in a system with an electromagnetic \feld. For the external\nelectric \feld, these terms can be written together as a function of electric and magnetic\n\feld. These terms are taken into account and discussed in the next section. The indirect\ncoupling terms are often not taken into consideration and not included in the discussion\n(see Ref. [36, 37] for details). In this context, we reiterate that our current approach of\nderiving relativistic terms does not include the exchange and correlation e\u000bect. A similar\nFW transformed Hamiltonian has previously been derived, however, with a general\nKohn-Sham exchange \feld [23, 25, 26]. As mentioned before, in this article we do not\nintend to include the exchange-correlation e\u000bect, while mostly focussing on the magnetic\nrelaxation and magnetic inertial dynamics.\n2.1. The spin Hamiltonian\nThe aim of this work is to formulate the spin dynamics on the basis of the Hamiltonian\nin Eq. (9). The direct \feld-spin interaction terms can be written together as electric or\nmagnetic contributions. These two contributions can be expressed as a series up to an\norder of 1=m5[36]\nHS\nmagnetic =\u0000e\nmS\u0001\"\nB+1\n2X\nn=1;2;3;4\u00121\n2i!c\u0013n@nB\n@tn#\n+O\u00121\nm6\u0013\n; (10)\nHS\nelectric =\u0000e\nmS\u0001\"\n1\n2mc2X\nn=0;2\u0012i\n2!c\u0013n@nE\n@tn\u0002(p\u0000eA)#\n+O\u00121\nm6\u0013\n; (11)\nwhere the Compton wavelength and pulsation have been expressed by the usual\nde\fnitions \u0015c=h=mc and!c= 2\u0019c=\u0015cwith Plank's constant h. We also have used\nthe spin angular momentum operator as S= (~=2)\u001b. Note that we have dropped\nthe notion of total electric \feld because the the involved \felds ( B,E,A) are external\nonly, the internal \felds are considered as time-independent. The involved terms in the\nabove two spin-dependent Hamiltonians can readily be explained. The \frst term in the\nmagnetic contribution in Eq. (10) explains the Zeeman coupling of spins to the external\nmagnetic \feld. The rest of the terms in both the Hamiltonians in Eqs. (11) and (10)\nrepresent the spin-orbit coupling and its higher-order corrections. We note that these\ntwo spin Hamiltonians are individually not Hermitian, however, it can be shown that\ntogether they form a Hermitian Hamiltonian [38]. As these Hamiltonians describe a\nsemirelativistic Dirac particle, it is possible to derive from them the spin dynamics of\na single Dirac particle [24]. The e\u000bect of the indirect \feld-spin terms is not yet well\nunderstood, but they could become important too in magnetism [36, 37], however, those\nterms are not of our interest here.\nThe electric Hamiltonian can be written in terms of magnetic contributions with\nthe choice of a gauge A=B\u0002r=2. The justi\fcation of the gauge lies in the fact8\nthat the magnetic \feld inside the system being studied is uniform [26]. The transverse\nelectric \feld in the Hamiltonian (10) can be written as\nE=1\n2\u0012\nr\u0002@B\n@t\u0013\n: (12)\nReplacing this expression in the electric spin Hamiltonian in Eq. (11), one can obtain a\ngeneralised expression of the total spin-dependent Hamiltonian as\nHS(t) =\u0000e\nmS\u0001h\nB+1\n21X\nn=1;2;:::\u00121\n2i!c\u0013n@nB\n@tn\n+1\n4mc21X\nn=0;2;:::\u0012i\n2!c\u0013n\u0012\nr\u0002@n+1B\n@tn+1\u0013\n\u0002(p\u0000eA)i\n: (13)\nIt is important to stress that the above spin-Hamiltonian is a generalisation of the two\nHamiltonians in Eqs. (10) and (11). We have already evaluated the Hamiltonian forms\nforn= 1;2;3;4 and assume that the higher-order terms will have the same form [36].\nThis Hamiltonian consists of the direct \feld-spin interaction terms that are linear and/or\nquadratic in the \felds. In the following we consider only the linear interaction terms,\nthat is we neglect the eAterm in Eq. (13). Here, we mention that the quadratic terms\ncould provide an explanation towards the previously unknown origin of spin-photon\ncoupling or optical spin-orbit torque and angular magneto-electric coupling [38{40].\nThe linear direct \feld-spin Hamiltonian can then be recast as\nHS(t) =\u0000e\nmS\u0001h\nB+1\n21X\nn=1;2;:::\u00121\n2i!c\u0013n@nB\n@tn\n+1\n4mc21X\nn=0;2;:::\u0012i\n2!c\u0013n\u001a@n+1B\n@tn+1(r\u0001p)\u0000r\u0012@n+1B\n@tn+1\u0001p\u0013\u001bi\n: (14)\nThis is \fnal form of the Hamiltonian and we are interested to describe to evaluate its\ncontribution to the spin dynamics.\n3. Spin dynamics\nOnce we have the explicit form of the spin Hamiltonian in Eq. (14), we can proceed to\nderive the corresponding classical magnetisation dynamics. Following similar procedures\nof previous work [23, 24], and introducing a magnetisation element M(r;t), the\nmagnetisation dynamics can be calculated by the following equation of motion\n@M\n@t=X\njg\u0016B\n\n1\ni~D\u0002\nSj;HS(t)\u0003E\n; (15)\nwhere\u0016Bis the Bohr magneton, gis the Land\u0013 e g-factor that takes a value \u00192 for electron\nspins and \n is a suitably chosen volume element. Having the spin Hamiltonian in Eq.9\n(14), we evaluate the corresponding commutators. As the spin Hamiltonian involves the\nmagnetic \felds, one can classify the magnetisation dynamics into two situations: (a) the\nsystem is driven by a harmonic \feld, (b) the system is driven by a non-harmonic \feld.\nHowever, in the below we continue the derivation of magnetisation dynamics with the\nharmonic driven \felds. The magnetisation dynamics driven by the non-harmonic \felds\nhas been discussed in the context of Gilbert damping and inertial dynamics where it was\nshown that an additional torque contribution (the \feld-derivative torque) is expected\nto play a crucial role [23, 24, 26].\nThe magnetisation dynamics due to the very \frst term of the Hamiltonian in Eq.\n(14) is derived as [24]\n@M(1)\n@t=\u0000\rM\u0002B; (16)\nwith the gyromagnetic ratio \r=gjej=2m. Here the commutators between two spin\noperators have been evaluated using [ Sj;Sk] =i~Sl\u000fjkl, where\u000fjklis the Levi-Civita\ntensor. This dynamics actually produces the precession of magnetisation vector around\nan e\u000bective \feld. To get the usual form of Landau-Lifshitz precessional dynamics, one\nhas to use a linear relationship of magnetisation and magnetic \feld as B=\u00160(M+H).\nWith the latter relation, the precessional dynamics becomes \u0000\r0M\u0002H, where\r0=\r\u00160\nde\fnes the e\u000bective gyromagnetic ratio. We point out that the there are relativistic\ncontributions to the precession dynamics as well, e.g., from the spin-orbit coupling due\nto the time-independent \feld Eint[23]. Moreover, the contributions to the magnetisation\nprecession due to exchange \feld appear here, but are not explicitly considered in this\narticle as they are not in the focus of the current investigations (see Ref. [23] for details).\nThe rest of the terms in the spin Hamiltonian in Eq. (14) is of much importance\nbecause they involve the time-variation of the magnetic induction. As it has been shown\nin an earlier work [23] that for the external \felds and speci\fcally the terms with n= 1\nin the second terms and n= 0 in the third terms of Eq. (14), these terms together\nare Hermitian. These terms contribute to the magnetisation dynamics as the Gilbert\nrelaxation within the LLG equation of motion,\n@M(2)\n@t=M\u0002\u0012\nA\u0001@M\n@t\u0013\n; (17)\nwhere the Gilbert damping parameter Ahas been derived to be a tensor that has mainly\ntwo contributions: electronic and magnetic. The damping parameter Ahas the form\n[23, 24]\nAij=\u0000e\u00160\n8m2c2X\n`;k\u0002\nhripk+pkrii\u0000hr`p`+p`r`i\u000eik\u0003\n\u0002\u0000\n1+\u001f\u00001\u0001\nkj; (18)\nwhere 1is the 3\u00023 unit matrix and \u001fis the magnetic susceptibility tensor that can be\nintroduced only if the system is driven by a \feld which is single harmonic [26]. Note\nthat the electronic contributions to the Gilbert damping parameter are given by the10\nexpectation value hripkiand the magnetic contributions by the susceptibility. We also\nmention that the tensorial Gilbert damping tensor has been shown to contain a scalar,\nisotropic Heisenberg-like contribution, an anisotropic Ising-like tensorial contribution\nand a chiral Dzyaloshinskii-Moriya-like contribution [23].\nIn an another work, we took into account the terms with n= 2 in the second term\nof Eq. (14) and it has been shown that those containing the second-order time variation\nof the magnetic induction result in the magnetic inertial dynamics. Note that these\nterms provide a contribution to the higher-order relativistic e\u000bects. The corresponding\nmagnetisation dynamics can be written as [24]\n@M(3)\n@t=M\u0002\u0012\nC\u0001@M\n@t+D\u0001@2M\n@t2\u0013\n; (19)\nwith a higher-order Gilbert damping tensor Cijand inertia parameter Dijthat have the\nfollowing expressions Cij=\r0~2\n8m2c4@\n@t( 1+\u001f\u00001)ijandDij=\r0~2\n8m2c4( 1+\u001f\u00001)ij. We note\nthat Eq. (19) contains two fundamentally di\u000berent dynamics { the \frst term on the\nright-hand side has the exact form of Gilbert damping dynamics whereas the second\nterm has the form of magnetic inertial dynamics [24].\nThe main aim of this article is to formulate a general magnetisation dynamics\nequation and an extension of the traditional LLG equation to include higher-order\nrelativistic e\u000bects. The calculated magnetisation dynamics due to the second and third\nterms of Eq. (14) can be expressed as\n@M\n@t=e\nmM\u0002h1\n21X\nn=0;1;:::\u00121\n2i!c\u0013n+1@n+1B\n@tn+1\n+1\n4mc21X\nn=0;2;:::\u0012i\n2!c\u0013n\u001a@n+1B\n@tn+1hr\u0001pi\u0000D\nr\u0012@n+1B\n@tn+1\u0001p\u0013E\u001bi\n: (20)\nNote the di\u000berence in the summation of \frst terms from the Hamiltonian in Eq. (14).\nTo obtain explicit expressions for the Gilbert damping dynamics, we employ a general\nlinear relationship between magnetisation and magnetic induction, B=\u00160(H+M).\nThe time-derivative of the magnetic induction can then be replaced by magnetisation\nand magnetic susceptibility. For the n-th order time-derivative of the magnetic induction\nwe \fnd\n@nB\n@tn=\u00160\u0012@nH\n@tn+@nM\n@tn\u0013\n: (21)\nNote that this equation is valid for the case when the magnetisation is time-dependent.\nSubstituting this expression into the Eq. (20), one can derive the general LLG equation\nand its extensions. Moreover, as we work out the derivation in the case of harmonic\ndriving \felds, the di\u000berential susceptibility can be introduced as \u001f=@M=@H. The\n\frst term ( n-th derivative of the magnetic \feld) can consequently be written by the11\nfollowing Leibniz formula as\n@nH\n@tn=n\u00001X\nk=0(n\u00001)!\nk!(n\u0000k\u00001)!@n\u0000k\u00001(\u001f\u00001)\n@tn\u0000k\u00001\u0001@k\n@tk\u0012@M\n@t\u0013\n; (22)\nwhere the magnetic susceptibility \u001f\u00001is a time-dependent tensorial quantity and\nharmonic. Using this relation, the \frst term and second terms in Eq. (20) assume\nthe form\n@M\n@t\f\f\f\n\frst=e\u00160\n2mM\u00021X\nn=0;1;:::\u00121\n2i!c\u0013n+1nX\nk=0n!\nk!(n\u0000k)!@n\u0000k( 1+\u001f\u00001)\n@tn\u0000k\u0001@k\n@tk\u0012@M\n@t\u0013\n;\n(23)\n@M\n@t\f\f\f\nsecond=e\u00160\n4m2c2M\u0002\n1X\nn=0;2;:::\u00121\n2i!c\u0013nnX\nk=0n!\nk!(n\u0000k)!h@n\u0000k( 1+\u001f\u00001)\n@tn\u0000k\u0001@k\n@tk\u0012@M\n@t\u0013\nhr\u0001pi\n\u0000D\nr\u0012\u001a@n\u0000k( 1+\u001f\u00001)\n@tn\u0000k\u0001@k\n@tk\u0012@M\n@t\u0013\u001b\n\u0001p\u0013Ei\n:(24)\nThese two equations already provide a generalisation of the higher-order magnetisation\ndynamics including the Gilbert damping (i.e., the terms with k= 0) and the inertial\ndynamics (the terms with k= 1) and so on.\n4. Discussion\n4.1. Gilbert damping parameter\nIt is obvious that, as Gilbert damping dynamics involves the \frst-order time derivative of\nthe magnetisation and a torque due to it, kmust take the value k= 0 in the equations\n(23) and (24). Therefore, the Gilbert damping dynamics can be achieved from the\nfollowing equations:\n@M\n@t\f\f\f\n\frst=e\u00160\n2mM\u00021X\nn=0;1;:::\u00121\n2i!c\u0013n+1@n( 1+\u001f\u00001)\n@tn\u0001@M\n@t; (25)\n@M\n@t\f\f\f\nsecond=e\u00160\n4m2c2M\u00021X\nn=0;2;:::\u00121\n2i!c\u0013nh\u0012@n( 1+\u001f\u00001)\n@tn\u0001@M\n@t\u0013\nhr\u0001pi\n\u0000D\nr\u0012\u001a@n( 1+\u001f\u00001)\n@tn\u0001@M\n@t\u001b\n\u0001p\u0013Ei\n: (26)\nNote that these equations can be written in the usual form of Gilbert damping as\nM\u0002\u0000\nG\u0001@M\n@t\u0001\n, where the Gilbert damping parameter Gis notably a tensor [2, 23]. The12\ngeneral expression for the tensor can be given by a series of higher-order relativistic\nterms as follows\nGij=e\u00160\n2m1X\nn=0;1;:::\u00121\n2i!c\u0013n+1@n( 1+\u001f\u00001)ij\n@tn\n+e\u00160\n4m2c21X\nn=0;2;:::\u00121\n2i!c\u0013nh@n( 1+\u001f\u00001)ij\n@tn(hrlpli\u0000hrlpii)i\n: (27)\nHere we have used the Einstein summation convention on the index l. Note that there\nare two series: the \frst series runs over even and odd numbers ( n= 0;1;2;3;\u0001\u0001\u0001),\nhowever, the second series runs only over the even numbers ( n= 0;2;4;\u0001\u0001\u0001). Eq. (27)\nrepresents a general relativistic expression for the Gilbert damping tensor, given as a\nseries of higher-order terms. This equation is one of the central results of this article. It\nis important to observe that this expression provides the correct Gilbert tensor at the\nlowest relativistic order, i.e., putting n= 0 the expression for the tensor is found to be\nexactly the same as Eq. (18).\nThe analytic summation of the above series of higher-order relativistic contributions\ncan be carried out when the susceptibility depends on the frequency of the harmonic\ndriving \feld. This is in general true for ferromagnets where a di\u000berential susceptibility\nis introduced because there exists a spontaneous magnetisation in ferromagnets even\nwithout application of a harmonic external \feld. However, if the system is driven by a\nnonharmonic \feld, the introduction of the susceptibility is not valid anymore. In general\nthe magnetic susceptibility is a function of wave vector and frequency in reciprocal space,\ni.e.,\u001f=\u001f(q;!). Therefore, for the single harmonic applied \feld, we use \u001f\u00001/ei!tand\nthen-th order derivative will follow @n=@tn(\u001f\u00001)/(i!)n\u001f\u00001. With these arguments,\none can express the damping parameter of Eq. (27) as (see Appendix A for detailed\ncalculations)\nGij=e\u00160\n4m2c2\u0014~\ni+hrlpli\u0000hrlpii\u0015\n( 1+\u001f\u00001)ij\n+e\u00160\n4m2c2\"\n(2!!c+!2)~\ni+!2(hrlpli\u0000hrlpii)\n4!2\nc\u0000!2#\n\u001f\u00001\nij: (28)\nHere, the \frst term in the last expression is exactly the same as the one that has been\nderived in our earlier investigation [23]. As the expression of the expectation value\nhripjiis imaginary, the real Gilbert damping parameter will be given by the imaginary\npart of the susceptibility tensor. This holds consistently for the higher-order terms\nas well. The second term in Eq. (28) stems essentially from an in\fnite series which\ncontain higher-order relativistic contributions to the Gilbert damping parameter. As\n!cscales with c, these higher-order terms will scale with c\u00004or more and thus their\ncontributions will be smaller than the \frst term. Note that the higher-order terms will\ndiverge when != 2!c\u00191021sec\u00001, which means that the theory breaks down at the\nlimit!!2!c. In this limit, the original FW transformation is not de\fned any more\nbecause the particles and antiparticles cannot be separated at this energy limit.13\n4.2. Magnetic inertia parameter\nMagnetic inertial dynamics, in contrast, involves a torque due to the second-order time-\nderivative of the magnetisation. In this case, kmust adopt the value k= 1 in the\nafore-derived two equations (23) and (24). However, if k= 1, the constraint n\u0000k\u00150\ndictates that n\u00151. Therefore, the magnetic inertial dynamics can be described with\nthe following equations:\n@M\n@t\f\f\f\n\frst=e\u00160\n2mM\u00021X\nn=1;2;:::\u00121\n2i!c\u0013n+1n!\n(n\u00001)!@n\u00001( 1+\u001f\u00001)\n@tn\u00001\u0001@2M\n@t2; (29)\n@M\n@t\f\f\f\nsecond=e\u00160\n4m2c2M\u00021X\nn=2;4;:::\u00121\n2i!c\u0013nn!\n(n\u00001)!h\u0012@n\u00001( 1+\u001f\u00001)\n@tn\u00001\u0001@2M\n@t2\u0013\nhr\u0001pi\n\u0000D\nr\u0012\u001a@n\u0000k( 1+\u001f\u00001)\n@tn\u0000k\u0001@2M\n@t2\u001b\n\u0001p\u0013Ei\n: (30)\nSimilar to the Gilbert damping dynamics, these dynamical terms can be expressed\nasM\u0002\u0010\nI\u0001@2M\n@t2\u0011\nwhich is the magnetic inertial dynamics [8]. The corresponding\nparameter has the following expression\nIij=e\u00160\n2m1X\nn=1;2;:::\u00121\n2i!c\u0013n+1n!\n(n\u00001)!@n\u00001( 1+\u001f\u00001)ij\n@tn\u00001\n+e\u00160\n4m2c21X\nn=2;4;:::\u00121\n2i!c\u0013nn!\n(n\u00001)!h@n\u00001( 1+\u001f\u00001)ij\n@tn\u00001(hrlpli\u0000hripli)i\n: (31)\nNote that as ncannot adopt the value n= 0, the starting values of nare di\u000berent in\nthe two terms. Importantly, if n= 1 we recover the expression for the lowest order\nmagnetic inertia parameter Dij, as given in the equation (19) [24].\nUsing similar arguments as in the case of the generalised Gilbert damping\nparameter, when we consider a single harmonic \feld as driving \feld, the inertia\nparameter can be rewritten as follows (see Appendix A for detailed calculations)\nIij=\u0000e\u00160~2\n8m3c4( 1+\u001f\u00001)ij\u0000e\u00160~2\n8m3c4\u0012\u0000!2+ 4!!c\n(2!c\u0000!)2\u0013\n\u001f\u00001\nij\n+e\u00160\n8m3c4~\ni(hrlpli\u0000hripli)\u001216!!3\nc\n(4!2\nc\u0000!2)2\u0013\n\u001f\u00001\nij: (32)\nThe \frst term here is exactly the same as the one that was obtained in our earlier\ninvestigation [24]. However, there are now two extra terms which depend on the\nfrequency of the driving \feld and that vanish for !!0. Again, in the limit !!2!c,\nthese two terms diverge and hence this expression is not valid anymore. The inertia\nparameter will consistently be given by the real part of the susceptibility.14\n5. Summary\nWe have developed a generalised LLG equation of motion starting from fundamental\nquantum relativistic theory. Our approach leads to higher-order relativistic correction\nterms in the equation of spin dynamics of Landau and Lifshitz. To achieve this, we have\nstarted from the foundational Dirac equation under the presence of an electromagnetic\n\feld (e.g., external driving \felds or THz excitations) and have employed the FW\ntransformation to separate out the particles from the antiparticles in the Dirac equation.\nIn this way, we derive an extended Pauli Hamiltonian which e\u000eciently describes the\ninteractions between the quantum spin-half particles and the applied \feld. The thus-\nderived direct \feld-spin interaction Hamiltonian can be generalised for any higher-order\nrelativistic corrections and has been expressed as a series. To derive the dynamical\nequation, we have used this generalised spin Hamiltonian to calculate the corresponding\nspin dynamics using the Heisenberg equation of motion. The obtained spin dynamical\nequation provides a generalisation of the phenomenological LLG equation of motion\nand moreover, puts the LLG equation on a rigorous foundational footing. The equation\nincludes all the torque terms of higher-order time-derivatives of the magnetisation (apart\nfrom the Gilbert damping and magnetic inertial dynamics). Speci\fcally, however, we\nhave focussed on deriving an analytic expression for the generalised Gilbert damping\nand for the magnetic inertial parameter. Our results show that both these parameters\ncan be expressed as a series of higher-order relativistic contributions and that they\nare tensors. These series can be summed up for the case of a harmonic driving \feld,\nleading to closed analytic expressions. We have further shown that the imaginary part\nof the susceptibility contributes to the Gilbert damping parameter while the real part\ncontributes to the magnetic inertia parameter. Lastly, with respect to the applicability\nlimits of the derived expressions we have pointed out that when the frequency of the\ndriving \feld becomes comparable to the Compton pulsation, our theory will not be valid\nanymore because of the spontaneous particle-antiparticle pair-production.\n6. Acknowledgments\nWe thank P-A. Hervieux for valuable discussions. This work has been supported\nby the Swedish Research Council (VR), the Knut and Alice Wallenberg Foundation\n(Contract No. 2015.0060), the European Union's Horizon2020 Research and\nInnovation Programme under grant agreement No. 737709 (FEMTOTERABYTE,\nhttp://www.physics.gu.se/femtoterabyte).15\nAppendix A. Detailed calculations of the parameters for a harmonic \feld\nIn the following we provide the calculational details of the summation towards the results\ngiven in Eqs. (28) and (32).\nAppendix A.1. Gilbert damping parameter\nEq. (27) can be expanded as follows\nGij=e\u00160\n2m1\n2i!c( 1+\u001f\u00001)ij+e\u00160\n4m2c2(hrlpli\u0000hrlpii) ( 1+\u001f\u00001)ij\n+e\u00160\n2m1X\nn=1;2;:::\u00121\n2i!c\u0013n+1\n(i!)n\u001f\u00001\nij+e\u00160\n4m2c21X\nn=2;4;:::\u00121\n2i!c\u0013n\n(hrlpli\u0000hrlpii) (i!)n\u001f\u00001\nij\n=e\u00160\n2m1\n2i!c( 1+\u001f\u00001)ij+e\u00160\n4m2c2(hrlpli\u0000hrlpii) ( 1+\u001f\u00001)ij\n+e\u00160\n2m1\n2i!c1X\nn=1;2;:::\u0012!\n2!c\u0013n\n\u001f\u00001\nij+e\u00160\n4m2c21X\nn=2;4;:::\u0012!\n2!c\u0013n\n(hrlpli\u0000hrlpii)\u001f\u00001\nij\n=e\u00160\n4m2c2\u0014~\ni+hrlpli\u0000hrlpii\u0015\n( 1+\u001f\u00001)ij\n+e\u00160\n4m2c2\"\n~\ni1X\nn=1;2;:::\u0012!\n2!c\u0013n\n+ (hrlpli\u0000hrlpii)1X\nn=2;4;:::\u0012!\n2!c\u0013n#\n\u001f\u00001\nij\n=e\u00160\n4m2c2\u0014~\ni+hrlpli\u0000hrlpii\u0015\n( 1+\u001f\u00001)ij\n+e\u00160\n4m2c2\u0014~\ni!\n2!c\u0000!+ (hrlpli\u0000hrlpii)!2\n4!2\nc\u0000!2\u0015\n\u001f\u00001\nij\n=e\u00160\n4m2c2\u0014~\ni+hrlpli\u0000hrlpii\u0015\n( 1+\u001f\u00001)ij\n+e\u00160\n4m2c2\"\n(2!!c+!2)~\ni+!2(hrlpli\u0000hrlpii)\n4!2\nc\u0000!2#\n\u001f\u00001\nij: (A.1)\nWe have used the fact that!\n!c<1 and the summation formula\n1 +x+x2+x3+:::=1\n1\u0000x;\u00001<x< 1: (A.2)REFERENCES 16\nAppendix A.2. Magnetic inertia parameter\nEq. (31) can be expanded as follows\nIij=e\u00160\n2m\u00121\n2i!c\u00132\n( 1+\u001f\u00001)ij+e\u00160\n2m1X\nn=2;3;:::\u00121\n2i!c\u0013n+1n!\n(n\u00001)!@n\u00001( 1+\u001f\u00001)ij\n@tn\u00001\n+e\u00160\n4m2c21X\nn=2;4;:::\u00121\n2i!c\u0013nn!\n(n\u00001)!h@n\u00001( 1+\u001f\u00001)ij\n@tn\u00001(hrlpli\u0000hripli)i\n=e\u00160\n2m\u00121\n2i!c\u00132\n( 1+\u001f\u00001)ij+1X\nn=2;3;:::\u00121\n2i!c\u0013n+1n!\n(n\u00001)!(i!)n\u00001\u001f\u00001\nij\n+e\u00160\n4m2c21X\nn=2;4;:::\u00121\n2i!c\u0013nn!\n(n\u00001)!(hrlpli\u0000hripli) (i!)n\u00001\u001f\u00001\nij\n=\u0000e\u00160~2\n8m3c4( 1+\u001f\u00001)ij+e\u00160\n2m\u00121\n2i!c\u001321X\nn=2;3;:::\u0012!\n2!c\u0013n\u00001n!\n(n\u00001)!\u001f\u00001\nij\n+e\u00160\n4m2c2\u00121\n2i!c\u00131X\nn=2;4;:::\u0012!\n2!c\u0013n\u00001n!\n(n\u00001)!(hrlpli\u0000hripli)\u001f\u00001\nij\n=\u0000e\u00160~2\n8m3c4( 1+\u001f\u00001)ij\u0000e\u00160~2\n8m3c41X\nn=2;3;:::\u0012!\n2!c\u0013n\u00001n!\n(n\u00001)!\u001f\u00001\nij\n+e\u00160\n8m3c4~\ni1X\nn=2;4;:::\u0012!\n2!c\u0013n\u00001n!\n(n\u00001)!(hrlpli\u0000hripli)\u001f\u00001\nij\n=\u0000e\u00160~2\n8m3c4( 1+\u001f\u00001)ij\u0000e\u00160~2\n8m3c4\u001f\u00001\nij\"\n2\u0012!\n2!c\u0013\n+ 3\u0012!\n2!c\u00132\n+ 4\u0012!\n2!c\u00133\n+:::#\n+e\u00160\n8m3c4~\ni(hrlpli\u0000hripli)\u001f\u00001\nij\"\n2\u0012!\n2!c\u0013\n+ 4\u0012!\n2!c\u00133\n+ 6\u0012!\n2!c\u00135\n+:::#\n=\u0000e\u00160~2\n8m3c4( 1+\u001f\u00001)ij\u0000e\u00160~2\n8m3c4\u0012\u0000!2+ 4!!c\n(2!c\u0000!)2\u0013\n\u001f\u00001\nij\n+e\u00160\n8m3c4~\ni(hrlpli\u0000hripli)\u001216!!3\nc\n(4!2\nc\u0000!2)2\u0013\n\u001f\u00001\nij: (A.3)\nHere we have used the formula\n1 + 2x+ 3x2+ 4x3+ 5x4+:::=1\n(1\u0000x)2;\u00001<x< 1: (A.4)\nReferences\n[1] Landau L D and Lifshitz E M 1935 Phys. 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Matter 29\n194002 URL http://stacks.iop.org/0953-8984/29/i=19/a=194002\n[40] Paillard C, Mondal R, Berritta M, Dkhil B, Singh S, Oppeneer P M and Bellaiche\nL 2016 Proc. SPIE 9931 99312E{16 URL http://dx.doi.org/10.1117/12.\n2238196"    },    {        "title": "1412.4032v1.Spin_waves_in_micro_structured_yttrium_iron_garnet_nanometer_thick_films.pdf",        "content": "arXiv:1412.4032v1  [cond-mat.mes-hall]  12 Dec 2014Spin waves in micro-structured yttrium iron garnet nanomet er-thick films\nMatthias B. Jungfleisch,1,a)Wei Zhang,1Wanjun Jiang,1Houchen Chang,2Joseph Sklenar,3Stephen M. Wu,1\nJohn E. Pearson,1Anand Bhattacharya,1John B. Ketterson,3Mingzhong Wu,2and Axel Hoffmann1\n1)Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439,\nUSA\n2)Department of Physics, Colorado State University, Fort Col lins, Colorado 80523,\nUSA\n3)Department of Physics and Astronomy, Northwestern Univers ity, Evanston, Illinois 60208,\nUSA\n(Dated: 1 May 2018)\nWe investigated the spin-wave propagation in a micro-structured y ttrium iron garnet waveguide of 40 nm\nthickness. Utilizing spatially-resolved Brillouin light scattering microsc opy, an exponential decay of the spin-\nwave amplitude of (10 .06±0.83)µm was observed. This leads to an estimated Gilbert damping constant\nofα= (8.79±0.73)×10−4, which is larger than damping values obtained through ferromagnet ic resonance\nmeasurements in unstructured films. The theoretically calculated s patial interference of waveguide modes\nwas compared to the spin-wave pattern observed experimentally b y means of Brillouin light scattering spec-\ntroscopy.\nI. INTRODUCTION\nMagnonics is an emerging field of magnetism study-\ning the spin dynamics in micro- and nanostructured\ndevices aiming for the development of new spintron-\nics applications.1–3Up to now, ferromagnetic metals\n(for example, Permalloy and Heusler alloys) have been\nwidely used for the investigation of magnetization dy-\nnamicson the nanoscale.4–10However, the Gilbert damp-\ning of Permalloy is two orders of magnitude higher than\nthat of ferrimagnetic insulator yttrium iron garnet (YIG,\nY3Fe5O12). Recent progress in the growth of YIG films\nallows for the fabrication of low-damping nanometer-\nthick YIG films,11–14which are well-suited for patterning\nof micro-structured YIG devices. This enables investiga-\ntions of spin-wave propagation in plain YIG microstruc-\ntures of sub-100 nm thicknesses which are a step forward\nfor future insulator-based magnonics applications.\nInthiswork,weexperimentallydemonstratespin-wave\npropagation in a micro-structured YIG waveguide of\n40 nm thickness and 4 µm width. By utilizing spatially-\nresolvedBrillouinlightscattering(BLS) microscopy4–7,11\nthe exponential decay length of spin waves is deter-\nmined. The corresponding damping parameter of the\nmicro-structured YIG is estimated and compared to that\ndetermined from ferromagnetic resonance (FMR) mea-\nsurements. Furthermore, we show that different spin-\nwave modes quantized in the direction perpendicular to\nthe waveguide lead to a spatial interference pattern. We\ncompare the experimental results to the theoretically ex-\npected spatial interference of the waveguide modes.\na)Electronic mail: jungfleisch@anl.govII. EXPERIMENT\nFigure1shows a schematic illustration of the sam-\nple layout. The YIG film of 40 nm thickness was de-\nposited by magnetron sputtering on single crystal pol-\nished gadolinium gallium garnet (GGG, Gd 3Ga4O12)\nsubstrates of 500 µm thickness with (111) orientation\nunder high-purity argon atmosphere. The film was sub-\nsequently annealed in-situ at 800◦C for 4 hours under an\noxygenatmosphereof1.12Torr. Themagneticproperties\nofthe unstructured film werecharacterizedby FMR: The\npeak-to-peak linewidth µ0∆Has a function of the exci-\ntation frequency fis depicted in Fig. 2(a). The Gilbert\ndamping parameter αFMRcan be obtained from FMR\nmeasurements using15\n√\n3µ0∆H=2αFMR\n|γ|f+µ0∆H0, (1)\nFIG. 1. (Color online) Schematic illustration of the sample\nlayout. The 4 µm wide yttrium iron garnet waveguide is\nmagnetized transversally by the bias magnetic field H. Spin\nwaves are excited by a shortened coplanar waveguide and the\nspin-wave intensity is detected by means of spatially-reso lved\nBrillouin light scattering microscopy. Colorbar indicate s spin-\nwave intensity.2\nwhereµ0is the vacuum permeability, γis the gyromag-\nnetic ratio, fis the resonance frequency and µ0∆H0\nis the inhomogeneous linewidth broadening. We find a\ndamping parameter of αFMR= (2.77±0.49)×10−4[fit\nshown as a red solid line in Fig. 2(a)]. The resonance\nfield,µ0H, as a function of the excitation frequency fis\nshown in Fig. 2(b). A fit to16\nf=µ0|γ|\n2π/radicalbig\nH(H+Meff) (2)\nyields an effective magnetization Meff= (122 ±\n0.30) kA/m [solid line in Fig. 2(b)]. In a subsequent fab-\nrication process, YIG waveguides of 4 µm width were\npatterned by photo-lithography and ion milling with\nan Ar plasma at 600 V for 5 min. In a last step, a\nshortened coplanar waveguide (CPW) made of Ti/Au\n(3 nm/150 nm) is patterned on top ofthe YIG waveguide\n(see Fig. 1). The shortened end of the CPW has a width\nof 5µm. The Oersted field of an alternating microwave\nsignal applied to the CPW exerts a torque on the mag-\nnetic moments in the YIG and forces them to precess.\nThe bias magnetic field is applied perpendicular to the\nshortaxisofthe waveguide(Fig. 1) providingefficient ex-\ncitation of Damon-Eshbach spin waves. The microwave\npowerPMW= 1 mW is sufficiently small to avoid pos-\nsible perturbations of spin-wave propagation caused by\nnonlinearities.\nIII. DISCUSSIONS\nIn order to detect spin-wave propagation in the YIG\nwaveguide spatially-resolved BLS microscopy with a res-\nolution of 250 nm is employed. To characterize the prop-\nagating spin waves, the BLS intensity was recorded at\ndifferentdistancesfromtheantenna. Aspatially-resolved\nBLS intensity map is shown in Fig. 3(b) at an exemplary\nexcitation frequency of f= 4.19 GHz. Spin waves are\nexcited near the antenna and propagate towards the op-\nposite end of the waveguide. To further analyze the data\nand to minimize the influence of multi-mode propagation\nin the YIG stripe (this will be discussed below), the BLS\nintensity is integrated over the width of the waveguide.\nThe correspondingBLS intensity as a function of the dis-\ntance from the antenna is illustrated in Fig. 3(c). The\ndecayofthespin-waveamplitudecanbedescribedby:7,11\nI(z) =I0e−2z\nλ+b, (3)\nwherezis the distance from the antenna, λis the de-\ncay length of the spin-wave amplitude and bis an offset.\nFrom Fig. 3(b), it is apparent that the data-points fol-\nlow an exponential behavior. A fit according to Eq. ( 3)\nyields the decay length λ. For an excitation frequency\noff= 4.19 GHz we find λ= (10.06±0.83)µm. This\nvalue is larger than decay lengths reported for Permalloy\n(<6µm, see Ref. 8, 21, and 22), but it is smaller thanFIG. 2. (Color online) (a) Ferromagnetic resonance peak-to -\npeaklinewidth µ0∆Has afunctionoftheresonance frequency\nfof the unstructured 40 nm YIG film. The red solid line\nrepresents a fit to Eq. (1). A Gilbert damping parameter\nα= (2.77±0.49)×10−4is determined. (b) Ferromagnetic\nresonance field µ0Has a function of f. Error bars are smaller\nthan the data symbols.\nthe largest decay length found for the Heusler-compound\nCo2Mn0.6Fe0.4Si (8.7 – 16.7µm, see Ref. 7). Pirro et al.\nreported a decay length of 31 µm in thicker YIG waveg-\nuides(100nm) grownbyliquidphaseepitaxy(LPE)with\na 9 nm thick Pt capping layer.11In order to understand\nthis discrepancy between our and their results, one has\nto take into account two facts: (1) State-of-the-art LPE\nfabrication technology can not be employed to grow film\nthicknesses below ∼100 nm. To date, sputtering offers\nan alternative approach to grow sub-100 nm thick YIG\nfilms with a sufficient quality.23(2) Taking into account\nthe spin-wave group velocity vg=∂ω/∂kand the spin-\nwave lifetime τ, the theoretically expected decay length\nλcan be calculated from λ=vg·τ. The groupvelocity vg\ncan be derived directly from the dispersion relation (see\nFig.4). A thinner YIG film has flatter dispersion and a\nsmallervg. Consequently, the expected decay length is\nsmaller for thinner YIG samples and so it is natural that\nthe decay length reported here is shorter than the one\nfound in Ref. 11 for 100 nm thick YIG waveguides.\nWe estimate the groupvelocity from the spin-wavedis-\npersion to be vg= 0.35−0.40µm/ns. Using our exper-\nimentally found decay length, the spin-wave lifetime is\ndetermined to be τ= 29 ns. We can use the BLS-data\nto determine the corresponding Gilbert damping param-\neterαBLS. In case of Damon-Eshbach spin waves, the\ndamping is given by243\nFIG. 3. (Color online) (a) Calculated spatial interference\npattern of the first two odd waveguide modes ( n= 1 and\nn= 3. (b) Spatially-resolved BLS intensity map at an ex-\ncitation frequency f= 4.19 GHz, applied microwave power\nPMW= 1 mW, biasing magnetic field µ0∆H= 83 mT. The\nnumbers 1 – 4 highlight the main features of the interference\npattern. (c) Corresponding BLS intensity integrated over t he\nentire width of the YIG waveguide. An exponential decay of\nthe spin-wave amplitude λ= (10.06±0.83)µm is found.\nαBLS=1\nτ(γµ0Meff\n2+2πf)−1. (4)\nA Gilbert damping parameter of the micro-structured\nYIG waveguide obtained by BLS characterization is\nfound to be αBLS= (8.79±0.73)×10−4, which is a\nfactor of 3 times larger than that determined by FMR in\nthe unstructuredfilm [ αFMR= (2.77±0.49)×10−4]. This\ndifference mightbe attributed to the micro-structuringof\nthe YIG waveguide by Ar ion beam etching. The etch-\ning might enhance the roughness of the edges of the YIG\nwaveguides and the resist processing could have an in-\nfluence on the surface quality17,18which could possibly\nlead to an enhancement of the two-magnon scattering\nprocess.19It would be desirable to perform FMR mea-\nsurements on the YIG waveguide. However, since the\nstructured bar is very small, the FMR signal is van-\nishingly small which makes it difficult to determine the\nGilbert damping in this way.\nWhile the discussion above only considered the BLS-\nintensity integrated over the waveguide width, we will\nfocus now on the spatial interference pattern shown in\nFig.3(b). The spin-wave intensity map can be under-\nstood by taking into account the dispersion relation of\nmagnetostatic spin waves in an in-plane magnetized fer-\nromagnetic thin film (see Fig. 4). Due to the lateral con-\nfinement, thewavevectorisquantizedacrossthewidthof\nthe YIG waveguide, ky=nπ/w, wheren∈N. The wave\nvectorkzalong the long axis of the waveguide ( z-axis)is assumed to be non-quantized. We follow the approach\npresented in Ref. 8. The dynamic magnetization is as-\nsumed to be pinned at the edges of the waveguide which\ncan be considered by introducing an effective width of\nthe waveguide.8,20\nFigure4shows the calculated dispersion relations of\ndifferent spin-wave modes quantized across the width of\nthe strip. The dashed line represents a fixed excitation\nfrequency. At a particular frequency different spin-waves\nmodes with different wave vectors kzare excited simulta-\nneously. Thisleadstothe occurrenceofspatiallyperiodic\ninterference patterns. In the present excitation config-\nuration, only modes with an odd quantization number\nncan be excited ( ndetermines the number of maxima\nacross the width of the waveguide). Since the intensity\nof the dynamic magnetization of these modes decreases\nwith increasing nas 1/n2, we only consider the first two\nodd modes n= 1 and n= 3.\nAccording to Ref. 8 the spatial distribution of the dy-\nnamic magnetization of the n-th mode can be expressed\nas\nmn(y,z)∝sin(nπ\nwy)cos(kn\nzz−2πft+φn),(5)\nwherefis the excitation frequency, kn\nzis the longitudinal\nwave vector of the n-th spin-wave mode and φnis the\nphase.25The spin-waveintensity distribution Inof then-\nth mode can be derived by averaging mn(y,z)2over one\noscillation period 1 /f. The entire interference pattern\ncan be obtained from the same procedure using the sum\nm1(y,z) +1\n3m3(y,z). The factor 1/3 accounts for the\nlower excitation efficiency of the n= 3 mode. Thus, the\nintensity is given by8\nIΣ(y,z)∝sin(π\nwy)2+1\n9sin(3π\nwy)2\n+2\n3sin(π\nwy)sin(3π\nwy)cos(∆kzz+∆φ),(6)\nwhere ∆kz=k3\nz−k1\nzand ∆φ=φ3−φ1. This pattern re-\npeatsperiodically. Thephaseshift φshiftsthe entirepat-\nFIG. 4. (Color online) Dispersion relations the first five\nwaveguide modes of a transversally magnetized YIG stripe.26\nOnly modes with a odd quantization number ncan be excited\n(solid lines). fdenotes the excitation frequency.4\ntern along the z-direction and the wave-vector difference\n∆kz= 0.97 rad/µm can be calculated from the disper-\nsion relation (Fig. 4). The calculated spatial interference\npattern is depicted in Fig. 3(a) using ∆ φ= 0 and taking\ninto account for the exponential decay of the spin-wave\namplitude by multiplying Eq. ( 6) withe−2z/λusing the\nexperimentallydetermined λ= 10.03µm. As is apparent\nfrom Fig. 3(a) and (b) a qualitative agreement between\ncalculation and experiment is found. (The numbers 1 –\n4 highlight the main features of the interference pattern\nin experiment and calculation.) The small difference in\nFig.3(a) and (b) can be explained by considering the\nfact that in the calculation a spin-wave propagation at\nan angle of exactly 90◦(Damon-Eshbach configuration)\nwith respect to the antenna/externalmagnetic field is as-\nsumed. However, in experiment small misalignments of\nthe external magnetic field might lead to a small asym-\nmetry in the interference pattern.\nIV. CONCLUSION\nIn summary, we demonstrated spin-wave excitation\nand propagation in micro-fabricated pure YIG wave-\nguides of 40 nm thickness. BLS-characterization re-\nvealed a decay length of the spin-wave amplitude of\n10µm leading to an estimated Gilbert damping pa-\nrameter of αBLS= (8.79±0.73)×10−4. This value\nis a factor 3 larger than the one determined for the\nunstructured YIG film by means of FMR techniques\n[αFMR= (2.77±0.49)×10−4]. The difference might be\nattributed to micro-structuring using ion beam etching.\nThe observed spatial spin-wave intensity distribution is\nexplained by the simultaneous excitation of the first two\nodd waveguide modes. These findings are important for\nthe development of new nanometer-thick magnon spin-\ntronics applications and devices based on magnetic insu-\nlators.\nV. ACKNOWLEDGMENTS\nWork at Argonne was supported by the U.S. Depart-\nment of Energy, Office of Science, Materials Science and\nEngineering Division. Work at Colorado State Univer-\nsity was supported by the U.S. Army Research Office,\nand the U.S. National Science Foundation. Lithography\nwas carried out at the Center for Nanoscale Materials,\nwhich is supported by DOE, Office of Science, Basic En-\nergy Sciences under ContractNo. DE-AC02-06CH11357.\n1S. Neusser and D. Grundler, Adv. Mater. 21, 2927 (2009).2V.V. Kruglyak, S.O. Demokritov, D. Grundler, J. Phys. D: App l.\nPhys.43, 264001 (2010).\n3S.O.Demokritov and A.N.Slavin, Magnonics - From Fundamen-\ntals to Applications , (Springer, 2013).\n4K. Vogt, H. Schultheiss, S. Jain, J.E. Pearson, A. Hoffmann,\nS.D. Bader and B. Hillebrands, Appl. Phys. Lett. 101, 042410\n(2012).\n5K. Vogt, F.Y. Fradin, J.E. Pearson, T. Sebastian, S.D. Bader ,\nB. Hillebrands, A. Hoffmann and H. Schultheiss, Nat. Commun.\n5, 3727 (2014).\n6V.E. Demidov, S.O. Demokritov, K. Rott, P. Krzysteczko and\nG. Reiss, J. Phys. D: Appl. Phys. 41, 164012 (2008).\n7T. Sebastian, Y. Ohdaira, T. Kubota, P. Pirro, T. Br¨ acher,\nK. Vogt, A.A. Serga, H. Naganuma, M. Oogane, Y. Ando, and\nB. Hillebrands, Appl. Phys. Lett. 100, 112402 (2012).\n8V.E. Demidov, S.O. Demokritov, K. Rott, P. Krzysteczko, and\nG. Reiss, Phys. Rev. B 77, 064406 (2008).\n9V.E. Demidov, S.O. Demokritov, D. Birt, B. O’Gorman, M. Tsoi\nand X. Li, Phys. Rev. B 80, 014429 (2009).\n10V.E. Demidov, J. Jersch, S.O. Demokritov, K. Rott, P. Krzys-\nteczko, and G. Reiss, Phys. Rev. B 79, 054417 (2009).\n11P. Pirro, T. Br¨ acher, A.V. Chumak, B. L¨ agel, C. Dubs,\nO. Surzhenko, P. G¨ ornert, B. Leven and B. Hillebrands, Appl .\nPhys. Lett. 104, 012402 (2014).\n12T. Liu, H. Chang, V. Vlaminck, Y. Sun, M. Kabatek, A. Hoff-\nmann, L. Deng, and M. Wu, J. Appl. Phys. 115, 17A501 (2014).\n13O. d’Allivy Kelly, A. Anane, R. Bernard, J. Ben Youssef,\nC. Hahn, A H. Molpeceres, C. Carr´ et´ ero, E. Jacquet, C. Der-\nanlot, P. Bortolotti, R. Lebourgeois, J.-C. Mage, G. de Loub ens,\nO. Klein, V. Cros, and A. Fert, Appl. Phys. Lett. 103, 082408\n(2013).\n14Y. Sun, Y.-Y. Song and M. Wu, Appl. Phys. Lett. 101, 082405\n(2012).\n15S.S.Kalarickal, P.Krivosik,M.Wu, C.E.Patton, M.L.Schne ider,\nP.Kabos, T.J.Silva, andJ.P.Nibarger,J.Appl.Phys. 99,093909\n(2006).\n16A. Azevedo, A.B. Oliveira, F.M de Aguiar, and S.M. Rezende,\nPhys. Rev. B, 62, 5331 (2000).\n17B.J.McMorran, A.C.Cochran, R.K.Dumas, KaiLiu, P.Morrow,\nD.T. Pierce and J. Unguris, J. Appl. Phys. 107, 09D305 (2010).\n18O.D. Roshchupkina, J. Grenzer, T. Strache, J. McCord,\nM. Fritzsche, A. Muecklich, C. Baehtz, and J. Fassbender, J.\nAppl. Phys. 112, 033901 (2012)\n19R. Arias and D.L. Mills, Phys. Rev. B 60, 7395 (1999).\n20K.Yu. Guslienko, S.O. Demokritov, B. Hillebrands, and\nA.N. Slavin, Phys. Rev. B 66, 132402 (2002).\n21M. Madami, S. Bonetti, G. Consolo, S. Tacchi, G. Carlotti,\nG. Gubbiotti, F.B. Mancoff, M.A. Yar, and J. ˚Akerman, Nat.\nNano.6, 635 (2011).\n22P. Pirro, T. Br¨ acher, K. Vogt, Bj¨ orn Obry, H. Schultheiss,\nB. Leven, and B. Hillebrands, Phys. Status Solidi B 238, 2404\n(2011).\n23H. Chang, P. Li, W. Zhang, T Liu, A. Hoffmann, L. Deng, and\nM. Wu, IEEE Magnetic Letters 5, 6700104 (2014).\n24D.D. Stancil and A. Prabhakar, Spin Waves - Theory and Ap-\nplications , (Springer, 2009).\n25T. Schneider, A.A. Serga, T. Neumann, B. Hillebrands, and\nM.P. Kostylev, Phys. Rev. B 77, 2144 (2008).\n26For the calculation of the dispersion relations the followi ng pa-\nrameters have been used: external magnetic field µ0H= 83 mT,\nexchange constant A= 3.6 pJ/m, saturation magnetization\nMS=140 kA/m, effective width of the waveguide weff= 3.5µm,\nYIG-film thickness t= 40 nm."    },    {        "title": "1907.04577v2.The_superior_role_of_the_Gilbert_damping_on_the_signal_to_noise_ratio_in_heat_assisted_magnetic_recording.pdf",        "content": "The superior role of the Gilbert damping on the signal-to-noise ratio in\nheat-assisted magnetic recording\nO. Muthsam,1,a)F. Slanovc,1C. Vogler,1and D. Suess1\nUniversity of Vienna, Physics of Functional Materials, Boltzmanngasse 5, 1090 Vienna,\nAustria\n(Dated: 25 September 2019)\nIn magnetic recording the signal-to-noise ratio (SNR) is a good indicator for the quality of\nwritten bits. However, a priori it is not clear which parameters have the strongest in\ruence\non the SNR. In this work, we investigate the role of the Gilbert damping on the SNR. Grains\nconsisting of FePt like hard magnetic material with two di\u000berent grain sizes d= 5 nm and\nd= 7 nm are considered and simulations of heat-assisted magnetic recording (HAMR) are\nperformed with the atomistic simulation program VAMPIRE. The simulations display that\nthe SNR saturates for damping constants larger or equal than 0.1. Additionally, we can\nshow that the Gilbert damping together with the bit length have a major e\u000bect on the SNR\nwhereas other write head and material parameters only have a minor relevance on the SNR.\nI. INTRODUCTION\nThe next generation recording technology to increase\nthe areal storage density of hard drives beyond 1.5 Tb/in2\nis heat-assisted magnetic recording (HAMR)1{6. Higher\nareal storage densities (ADs) require smaller recording\ngrains. These grains need to have high anisotropy to be\nthermally stable. HAMR uses a heat pulse to locally\nenhance the temperature of the high anisotropy record-\ning medium beyond the Curie temperature. Due to the\nheating, the coercivity of the grain drops and it can be\nwritten with the available head \felds. After the grain\nis written, the medium is cooled and the information is\nsafely stored. A good indicator for the quality of the\nwritten bits is the so-called signal-to-noise ratio (SNR)\nwhich gives the power of the signal over the power of the\nnoise7. To achieve high areal storage densities, record-\ning materials that show good magnetic properties even\nat small grain sizes and thus yield high SNR values are\nneeded. However, a priori it is not clear which parame-\nters have the strongest in\ruence on the SNR.\nIn this work, we investigate the e\u000bect of a varying damp-\ning constant on the SNR. HAMR simulations with the\natomistic simulation program VAMPIRE8are performed\nfor cylindrical recording grains with two di\u000berent diame-\ntersd= 5 nm and d= 7 nm and a height h= 8 nm. The\nmaterial parameters of FePt like hard magnetic record-\ning media according to the Advanced Storage Technol-\nogy Consortium (ASTC)9are used. Damping constants\nbetween\u000b= 0:01 and\u000b= 0:5 are considered. Addition-\nally, we present an equation to include the in\ruence of\nthe bit length to the SNR. With this we can explain a\nSNR decrease of about 8.25 dB for 5- nm grains, which\nresults when changing the material and writing parame-\nters in the HAMR simulations from those used in former\nsimulations10{12to those according to the Advanced Stor-\nage Technology Consortium9, with the damping constant\nand the bit length only.\nThe structure of this paper is as follows: In Section II,\nthe HAMR model is introduced and it is explained how\na)Electronic mail: olivia.muthsam@univie.ac.atthe SNR is determined. In Section III, the results are\npresented and in Section IV they are discussed.\nII. HAMR MODEL\nCylindrical recording grains with height h= 8 nm and\ndiametersd= 5 nm and d= 7 nm are considered. One\ngrain can be interpreted as one grain of a state-of-the-\nart granular recording medium. A simple cubic crystal\nstructure is used. The exchange interaction Jijand the\ne\u000bective lattice parameter aare adjusted so that the sim-\nulations lead to the experimentally obtained saturation\nmagnetization and Curie temperature13,14. In the simu-\nlations, only nearest neighbor exchange interactions be-\ntween the atoms are included. A continuous laser pulse\nwith Gaussian shape and the full width at half maximum\n(FWHM) of 60 nm is assumed in the simulations. The\ntemperature pro\fle of the heat pulse is given by\nT(x;y;t ) = (Twrite\u0000Tmin)e\u0000x2+y2\n2\u001b2+Tmin (1)\n=Tpeak(y)\u0001e\u0000x2\n2\u001b2+Tmin (2)\nwith\n\u001b=FWHMp\n8 ln(2)(3)\nand\nTpeak(y) = (Twrite\u0000Tmin)e\u0000y2\n2\u001b2: (4)\nv= 15 m/s is the speed of the write head. xandylabel\nthe down-track and the o\u000b-track position of the grain,\nrespectively. In our simulations both the down-track po-\nsitionxand the o\u000b-track position yare variable. The\nambient and thus minimum temperature of all simula-\ntions isTmin= 300 K. The applied \feld is modeled as\na trapezoidal \feld with a \feld duration of 0.57 ns and a\n\feld rise and decay time of 0.1 ns, resulting in a bit length\nof 10.2 nm. The \feld strength is assumed to be +0 :8 T\nand\u00000:8 T inz-direction. Initially, the magnetization of\neach grain points in + z-direction. The trapezoidal \feldarXiv:1907.04577v2  [physics.app-ph]  24 Sep 20192\ntries to switch the magnetization of the grain from + z-\ndirection to\u0000z-direction. At the end of every simulation,\nit is evaluated if the bit has switched or not.\nThe material and write head parameters according to the\nAdvanced Storage Technology Consortium9are shown\nTable I.\nA. Determination of SNR\nTo calculate the signal-to-noise ratio, the read-back\nsignal of a written bit pattern has to be determined. To\nwrite the bit pattern and get the read-back signal from it,\nthe following procedure is used. First, a switching prob-\nability phase diagram is needed for the writing process of\nthe bit pattern. Since it is very time consuming to com-\npute a switching probability phase diagram with atom-\nistic or micromagnetic simulations, an analytical model\ndeveloped by Slanovc et al15is used in this work. The\nmodel uses eight input parameters (the maximum switch-\ning probability Pmax, the down-track jitter \u001bdown;the o\u000b-\ntrack jitter \u001bo\u000b;the transition curvature c, the bit length\nb, the half maximum temperature F50, the position p2of\nthe phase diagram in Tpeakdirection and the position p3\nof the phase diagram in down-track direction) to deter-\nmine a switching probability phase diagram. Slanovc et\nalshowed that the maximum switching probability Pmax\nand the down-track jitter \u001bdown are the input parameters\nwith the strongest in\ruence on the SNR. Note, that the\nbit lengthbalso has a strong in\ruence on the SNR. In\nthe further course of this work, an equation to include\nthe bit length to the SNR calculations is shown. Thus,\nthe bit length can be assumed constant during the SNR\ndetermination. The transition curvature cdid not show\nstrong in\ruence on the SNR for the used reader model\nand the o\u000b-track jitter \u001bo\u000bis neglectable since the reader\nwidth is with 30 :13 nm smaller than the track width with\n44:34 nm and thus does not sense the o\u000b-track jitter. p2\nandp3only shift the bit pattern and can thus be \fxed\nfor comparability. For this reason, it is reasonable to \fx\nthe input parameters, except for the maximum switch-\ning probability Pmaxand the down-track jitter \u001bdown.\nThe \fxed input parameters are determined by a least\nsquare \ft from a switching probability phase diagram\ncomputed with a coarse-grained Landau-Lifshitz-Bloch\n(LLB) model16for pure hard magnetic grains with mate-\nrial parameters given in Table I. The \ftting parameters\nare summarized in Table II for grain diameters d= 5 nm\nandd= 7 nm.\nFurther, it is necessary to compute the down-track jitter\n\u001bdown and the maximal switching probability Pmaxfor\nthe considered set of material and write head parameters,\nsee Table I. In the simulations, the switching probability\nof a recording grain at various down-track positions xat\na peak temperature Tpeak=Tc+ 60 K is calculated with\nthe atomistic simulations program VAMPIRE8, yielding\na down-track probability function P(x). To get the down-\ntrack jitter and the maximum switching probability, the\nswitching probability curve is \ftted with a Gaussian cu-mulative distribution function\n\b\u0016;\u001b2=1\n2(1 + erf(x\u0000\u0016p\n2\u001b2))\u0001Pmax (5)\nwith\nerf(x) =2p\u0019Zx\n0e\u0000\u001c2d\u001c; (6)\nwhere the mean value \u0016, the standard deviation \u001band\nthe mean maximum switching probability Pmax2[0;1]\nare the \ftting parameters. The standard deviation \u001b,\nwhich determines the steepness of the transition function,\nis a measure for the transition jitter and thus for the\nachievable maximum areal grain density of a recording\nmedium. The \ftting parameter Pmaxis a measure for\nthe average switching probability at the bit center. Note,\nthat the calculated jitter values \u001bdown only consider the\ndown-track contribution of the write jitter. The so-called\na\u0000parameter is given by\na=q\n\u001b2\ndown+\u001b2g (7)\nwhere\u001bgis a grain-size-dependent jitter contribution17.\nThe write jitter can then be calculated by\n\u001bwrite\u0019ar\nS\nW(8)\nwhereWis the reader width and S=D+Bis the grain\nsize, i.e. the sum of the grain diameter Dand the non-\nmagnetic boundary B15,18.\nFor each\u001bdown andPmaxcombination a switching prob-\nability phase diagram is computed with the analytical\nmodel. With the resulting phase diagram, the writing\nprocess of a certain bit pattern is simulated on granu-\nlar recording medium15. Here, the switching probabil-\nity of the grain is set according to its position in the\nphase diagram. The writing process is repeated for 50\ndi\u000berent randomly initialized granular media. Finally,\nthe read-back signal is determined with a reader model\nwhere the reader width is 30.13 nm and the reader res-\nolution in down-track direction is 13 :26 nm. The SNR\ncan then be computed from the read-back signal with\nthe help of a SNR calculator provided by SEAGATE19.\nThe resulting SNR value is given in dB (SNR dB). In the\nfollowing, the SNR dBis simply called SNR unless it is\nexplicitly noted di\u000berent.\nIII. RESULTS\nA. SNR Dependency on Damping\nFirst, the in\ruence of the damping constant on the\nSNR is investigated in more detail. The damping con-\nstant is varied from \u000b= 0:01 to\u000b= 0:5 for two di\u000berent\ngrain sizesd= 5 nm and d= 7 nm. All other parameters\nare taken from Table I. The bit length in the simula-\ntions is 10:2 nm and the track width is 44 :34 nm. The\ndown-track jitter curves are computed at Tpeak= 760 K\nand \ftted with eq. (5). In Figure 1, the SNR over the3\nCurie temp.\nTC[K]Damping\u000bUniaxial anisotropy\nku[J/link]Jij[J/link] \u0016s[\u0016B]v[m/s]\feld duration\n(fd) [ns]FWHM [nm]\n693.5 0.02 9:124\u000210\u0000236:72\u000210\u0000211.6 15 0.57 60\nTABLE I. Material and write head parameters of a FePt like hard magnetic granular recording medium accoring to the Advanced\nStorage Technology Consortium.\ngrain size\n5 nm 7 nm\n\u001bo\u000b[K] 22.5 14.4\nPmax 0.995 0.997\nF50[K] 602 628\nb[nm] 10.2 10.2\nc[10\u00004nm/K2]3.88 4.89\np2[K] 839 830\np3[nm] 27.5 25.8\nTABLE II. Reference parameters that are evaluated via least\nsquare \ft of the simulated phase diagrams for grain sizes 5 nm\nand 7 nm. Details of the parameters can be found in15.\ndamping constants for both grain sizes is visible. Note\nthat the SNR is proportional to the number of grains,\nmeaning that the number of grains per bit has to be kept\nconstant to determine a nearly constant SNR20. How-\never, since the dimensions of the granular media used for\nthe writing and reading process are \fxed, less grains form\none bit ford= 7 nm. Thus, the SNR values for the larger\ngrain size are smaller than for the small grains.\nThe results show that changing the damping constant\nfrom\u000b= 0:01 to\u000b= 0:02 already increases the SNR by\n3.66 dB for 5 nm-grains. For d= 7 nm, the SNR gain is\n1.65 dB. For 5 nm-grains, damping constants \u000b\u00150:1 lead\nto the best results with a total improvement of 6 dB com-\npared to\u000b= 0:01. Surprisingly, enhancing the damping\nconstant beyond 0 :1 does not show any further improve-\nment, the SNR saturates. This behavior is the same for\nthe 7 nm-grains. However, the total betterment of the\nSNR is only 2.24 dB for the larger grains. The SNR sat-\nuration results from the fact that Pmax= 1 for\u000b\u00150:1.\nSimultaneously, the down-track jitter \u001bdown varies only\nmarginally for \u000b\u00150:1 (see Table III) such that it does\nnot alter the SNR. The correlation between the SNR and\nthe maximum switching probability Pmaxis shown in Fig-\nure 2. It shows that the \ftted SNR curve reproduces the\ndata very well.\nBy further studying the switching dynamics of a 5 nm-\ngrain, one can show that the assumed pulse duration of\nthe heat pulse and the applied \feld strength are crucial\nfor the saturation of the SNR. In Figure 3, it is displayed\nhow the duration of the heat pulse in\ruences the maxi-\nmum switching probability and with it the SNR. During\nthe duration of the heat pulse the \feld is considered to\nconstantly point in \u0000z\u0000direction. The results demon-\nstrate that Pmaxdoes not saturate for small pulse du-\nrations. If longer pulse durations \u00150:5 ns are assumed,\naPmaxsaturation can be seen. A similar e\u000bect can be\nseen for a change of the \feld strength when the pulse\nduration is assumed to be 0 :5 ns (see Figure 4). For a\nsmall head \feld with a strength of 0 :5 T,Pmaxshows no\nsaturation whereas it does for larger head \felds. From\nFIG. 1. Resulting SNR for various damping constants \u000bfor\ngrains with two di\u000berent diameters d= 5 nm and d= 7 nm.\nFIG. 2. SNR and Pmaxdepending on the damping constant\n\u000bfor grain size with a diameter of 5 nm.\nthe simulations with varying duration of the heat pulse\nand \feld strength, it can also be seen that the SNR can\nbe improved for smaller damping constants if the dura-\ntion of the heat pulse is increased due to a smaller head\nvelocity or the \feld strength are enhanced.\nB. SNR Dependency on bit length\nThe in\ruence of the bit length on the SNR was al-\nready studied by Slanovc et al15. In this work, the fol-4\n5 nm 7 nm\n\u000b\u001bdown[nm]Pmax\u001bdown[nm]Pmax\n0.01 2.0 0.917 1.13 0.955\n0.02 0.9475 0.974 0.83 0.99\n0.05 0.7 0.989 0.549 1.0\n0.1 0.688 1.0 0.442 1.0\n0.3 0.495 1.0 0.48 1.0\n0.5 0.64 1.0 0.636 1.0\nTABLE III. Resulting down-track jitter parameters and mean maximum switching probability values for pure hard magnetic\nmaterial with di\u000berent damping constants \u000b.\nFIG. 3. Maximum switching probability Pmaxover damping\n\u000bfor di\u000berent pulse lengths of the heat pulse. A \feld strength\nof\u00000:8 T for grains with diameter d= 5 nm is assumed.\nlowing calculation is important. For the SNR calcula-\ntions a bit length b1= 10:2 nm is assumed since this is\nthe bit length resulting from the ASTC parameters. The\ntrack width in the simulations is again 44 :34 nm. How-\never, the bit length can change due to a variation of the\nwrite head parameters (\feld duration and head velocity).\nTherefore, the bit length for the former parameters10{12\nis 22 nm. To write a bit pattern with larger bit lengths\n(b >12 nm) the simulations of new granular media are\nrequired. This is computationally very expensive. Thus,\na di\u000berent approach is needed to qualitatively investi-\ngate the in\ruence of the bit length. For the SNR with\nSNR dB= 10 log10(SNR), there holds18\nSNR/\u0012b\na\u00132\u0012T50\nb\u0013\u0012W\nS\u0013\n(9)\nwith the bit length band the read-back pulse width T50\nwhich is proportional to the reader resolution in down-\ntrack direction. The ratio T50=bis called user bit density\nand is usually kept constant18. Further, the reader width\nWand the grain size Sare constant. Since the aim is to\nqualitatively describe the SNR for a bit length b2from\nSNR calculations with a bit length b1;the a-parameter a\nis also assumed to be constant. The SNR dBfor a di\u000berent\nbit lengthb2can then be calculated by\nFIG. 4. Maximum switching probability Pmaxover damping\n\u000bfor di\u000berent \feld strengths. The durations of the heat pulse\nof 0:5 ns for grains with diameter d= 5 nm is assumed.\nSNR dB(b2)\u0000SNR dB(b1)\n= 10 log10(SNR(b2))\u000010 log10(SNR(b1))\n= 10 log10(b2\n2)\u000010 log10(b2\n1) = 20 log10(b2\nb1) (10)\nsince all other parameters are the same for both bit\nlengths. Thus, one can compute the SNR dBvalue for\na varied bit length b2via the SNR dBof the bit length b1\nby\nSNR dB(b2) = SNR dB(b1) + 20 log10(b2\nb1): (11)\nThe curve achieved by eq. (11) with b1= 10:2 nm agrees\nqualitatively very well with the SNR(bit length) data\nfrom Slanovc et al15. It is thus reasonable to use this\nequation to include the bit length to the SNR.\nC. Combination of damping and bit length5\nCurie temp.\nTC[K]Damping\u000bUniaxial anisotropy\nku[J/link]Jij[J/link] \u0016s[\u0016B]v[m/s]\feld duration\n(fd) [ns]FWHM [nm]\n536.6 0.1 9:12\u000210\u0000235:17\u000210\u0000211.7 20 1.0 20\nTABLE IV. Material and write head parameters of a FePt like hard magnetic granular recording medium that were used in\nformer works10{12.\nParameter set diameter [nm] Tpeak[K]bit length [nm] Pmax\u001bdown [nm] SNR [dB]\nASTC 5 760 10.2 0.974 0.95 17.51\nParameters of former\nworks10{12 5 600 22 0.984 0.384 25.76\nASTC 7 760 10.2 0.99 0.83 15.35\nParameters of former\nworks10{12 7 600 22 1.0 0.44 22.75\nTABLE V. Resulting Pmax; \u001bdown and SNR values for the simulations with ASTC parameters and those used in former\nsimulations.\nThe simulations with write head and material parameters\naccording to the ASTC are compared to simulations with\nparameters used in former works10{12. Main di\u000berences\nto the currently used parameters are the bit length, the\ndamping constant, the height of the grain, the exchange\ninteraction, the atomistic spin moment, the full width at\nhalf maximum, the head velocity and the \feld duration.\nThese former parameters are summarized in Table IV.\nComparing the SNR values of both parameter sets shows\nthat ford= 5 nm the SNR is about 8.25 dB larger for the\nformer used parameters than for the ASTC parameters\nand ford= 7 nm it is\u00187:4 dB larger. The question is\nif the damping and bit length variation can fully explain\nthis deviation.\nIncreasing the damping constant from \u000b= 0:02 to\u000b=\n0:1, yields about +2 :25 dB ford= 5 nm and +0 :72 dB for\nd= 7 nm. Additionally, with the calculations from Sec-\ntion III B, one can show that by changing the bit length\nfromb1= 10:2 nm tob2= 22 nm gives\nSNR dB(b2) = SNR dB(b1) + 6:85 dB: (12)\nCombined, this shows that the di\u000berence in the SNR\ncan be attributed entirely to the damping and the bit\nlength enhancement. Moreover, simulations where the\nother material and write head parameters are changed\none by one con\frm this \fndings. The other write head\nand material parameters that are changed in the simu-\nlations have only minor relevance on the SNR compared\nto the damping constant and the bit length.\nIV. CONCLUSION\nTo conclude, we investigated how the damping con-\nstant a\u000bects the SNR. The damping constant was varied\nbetween\u000b= 0:01 and\u000b= 0:5 for two di\u000berent grain sizes\nd= 5 nm and d= 7 nm and the SNR was determined.\nIn practice, the damping constant of FePt might be in-\ncreased by enhancing the Pt concentration21,22. Another\noption would be to use a high/low Tcbilayer structure23\nand increase the damping of the soft magnetic layer by\ndoping with transition metals24{28. An interesting \fnd-\ning of the study is the enormous SNR improvement of6 dB that can be achieved for 5 nm-grains when enhanc-\ning the damping constant from \u000b= 0:01 to\u000b= 0:1\nand beyond. It is reasonable that the SNR improves\nwith larger damping. This results from the oscillatory\nbehavior of the magnetization for small damping dur-\ning switching. In fact, smaller damping facilitates the\n\frst switching but with larger damping it is more likely\nthat the grain will switch stably during the cooling of the\nthermal pulse29. This leads to a smaller switching time\ndistribution for larger damping constants and in the fur-\nther course to higher SNR values. However, an increase\nof the duration of the heat pulse due to a smaller head\nvelocity or an increase of the \feld strength can improve\nthe SNR even for smaller damping constant.\nFurthermore, the results display a SNR saturation for\ndamping constants \u000b\u00150:1. This SNR saturation can be\nexplained with the saturation of the maximum switching\nprobability and the only marginal change of the down-\ntrack jitter for \u000b\u00150:1. Indeed, one can check that for\nshorter pulse widths and smaller \feld strength, the be-\nhavior is di\u000berent and the SNR does not saturate. In\nthis case, the SNR rises for increasing damping constants.\nSummarizing, the SNR saturation for a varying damping\nconstant depends strongly on the used \feld strength and\nthe duration of the heat pulse.\nThe qualitative behavior for 7 nm-grains is the same. In-\nterestingly, the SNR change for a varying damping con-\nstant is not as signi\fcant as for grains with d= 5 nm.\nThis results from the higher maximum switching proba-\nbility and the smaller down-track jitter \u001bdown for 7 nm-\ngrains even for small damping constants. This is as\nexpected since larger grain sizes lead to an elevated\nmaximum switching probability11and smaller transition\njitter7compared to smaller grain sizes. This limits the\npossible increase of the recording performance in terms\nofPmaxand\u001bdown and thus the possible SNR gain. Ad-\nditionally, the SNR saturation value is smaller for 7 nm-\ngrains since one bit consists of fewer grains.\nThe overall goal was to explain the decrease of the SNR\nby about 8:25 dB and 7 :4 dB ford= 5 nm and d= 7 nm,\nrespectively, when changing from recording parameters\nused in former simulations10{12to the new ASTC pa-\nrameter. Indeed, together with the bit length variation,\nthe SNR variation could be fully attributed to the damp-6\ning enhancement. The other changed parameters like the\natomistic spin moment, the system height, the exchange\ninteraction and the full width at half maximum have only\na minor relevance compared to the in\ruence of the damp-\ning\u000band the bit length.\nIn fact, the variation of the bit length gave the largest\nSNR change. However, since an increase of the bit length\nis not realistic in recording devices, the variation of the\nmaterial parameters, especially the increase of the damp-\ning constant, is a more promising way to improve the\nSNR.\nV. ACKNOWLEDGEMENTS\nThe authors would like to thank the Vienna Sci-\nence and Technology Fund (WWTF) under grant No.\nMA14-044, the Advanced Storage Technology Consor-\ntium (ASTC), and the Austrian Science Fund (FWF)\nunder grant No. I2214-N20 for \fnancial support. The\ncomputational results presented have been achieved us-\ning the Vienna Scienti\fc Cluster (VSC).\n1L. Burns Jr Leslie and others. Magnetic recording system . Google\nPatents, December 1959.\n2G. W. Lewicki and others. Thermomagnetic recording and\nmagneto-optic playback system . Google Patents, December 1971.\n3Mark H. Kryder, Edward C. Gage, Terry W. McDaniel,\nWilliam A. Challener, Robert E. Rottmayer, Ganping Ju, Yiao-\nTee Hsia, and M. Fatih Erden. Heat assisted magnetic recording.\nProceedings of the IEEE , 96(11):1810{1835, 2008.\n4Robert E. Rottmayer, Sharat Batra, Dorothea Buechel,\nWilliam A. Challener, Julius Hohlfeld, Yukiko Kubota, Lei Li,\nBin Lu, Christophe Mihalcea, Keith Mount\feld, and others.\nHeat-assisted magnetic recording. IEEE Transactions on Mag-\nnetics , 42(10):2417{2421, 2006.\n5Hiroshi Kobayashi, Motoharu Tanaka, Hajime Machida, Takashi\nYano, and Uee Myong Hwang. Thermomagnetic recording .\nGoogle Patents, August 1984.\n6C. Mee and G. Fan. 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Journal of Applied Physics ,\n117(16):163913, 2015.\n12O. Muthsam, C. Vogler, and D. Suess. Noise reduction in heat-\nassisted magnetic recording of bit-patterned media by optimizing\na high/low Tc bilayer structure. Journal of Applied Physics ,\n122(21):213903, 2017.\n13Oleg N Mryasov, Ulrich Nowak, K Yu Guslienko, and Roy W\nChantrell. Temperature-dependent magnetic properties of fept:\nE\u000bective spin hamiltonian model. EPL (Europhysics Letters) ,\n69(5):805, 2005.\n14O Hovorka, S Devos, Q Coopman, WJ Fan, CJ Aas, RFL Evans,\nXi Chen, G Ju, and RW Chantrell. The curie temperature dis-\ntribution of fept granular magnetic recording media. Applied\nPhysics Letters , 101(5):052406, 2012.15Florian Slanovc, Christoph Vogler, Olivia Muthsam, and Dieter\nSuess. Systematic parameterization of heat-assisted magnetic\nrecording switching probabilities and the consequences for the\nresulting snr. arXiv preprint arXiv:1907.03884 , 2019.\n16Christoph Vogler, Claas Abert, Florian Bruckner, and Dieter\nSuess. Landau-Lifshitz-Bloch equation for exchange-coupled\ngrains. Physical Review B , 90(21):214431, December 2014.\n17Xiaobin Wang, Bogdan Valcu, and Nan-Hsiung Yeh. Transi-\ntion width limit in magnetic recording. Applied Physics Letters ,\n94(20):202508, 2009.\n18Gaspare Varvaro and Francesca Casoli. Ultra-High-Density Mag-\nnetic Recording: Storage Materials and Media Designs . CRC\nPress, March 2016.\n19S. Hernndez, P. Lu, S. Granz, P. Krivosik, P. Huang, W. Eppler,\nT. Rausch, and E. Gage. Using Ensemble Waveform Analysis to\nCompare Heat Assisted Magnetic Recording Characteristics of\nModeled and Measured Signals. IEEE Transactions on Magnet-\nics, 53(2):1{6, February 2017.\n20Roger Wood. The feasibility of magnetic recording at 1 terabit\nper square inch. IEEE Transactions on magnetics , 36(1):36{42,\n2000.\n21Satoshi Iihama, Shigemi Mizukami, Nobuhito Inami, Takashi Hi-\nratsuka, Gukcheon Kim, Hiroshi Naganuma, Mikihiko Oogane,\nTerunobu Miyazaki, and Yasuo Ando. Observation of Preces-\nsional Magnetization Dynamics in L10-FePt Thin Films with Dif-\nferent L10 Order Parameter Values. Japanese Journal of Applied\nPhysics , 52(7R):073002, June 2013.\n22Ji-Wan Kim, Hyon-Seok Song, Jae-Woo Jeong, Kyeong-Dong\nLee, Jeong-Woo Sohn, Toshiyuki Shima, and Sung-Chul Shin.\nUltrafast magnetization relaxation of L10-ordered Fe50pt50 al-\nloy thin \flm. Applied Physics Letters , 98(9):092509, February\n2011.\n23D. Suess and T. Schre\r. Breaking the thermally induced write er-\nror in heat assisted recording by using low and high Tc materials.\nApplied Physics Letters , 102(16):162405, April 2013.\n24W. Zhang, S. Jiang, P. K. J. Wong, L. Sun, Y. K. Wang, K. Wang,\nM. P. de Jong, W. G. van der Wiel, G. van der Laan, and Y. Zhai.\nEngineering Gilbert damping by dilute Gd doping in soft mag-\nnetic Fe thin \flms. Journal of Applied Physics , 115(17):17A308,\nMay 2014.\n25S. Ingvarsson, Gang Xiao, S. S. P. Parkin, and R. H. Koch. Tun-\nable magnetization damping in transition metal ternary alloys.\nApplied Physics Letters , 85(21):4995{4997, November 2004.\n26J. Fassbender, J. von Borany, A. Mcklich, K. Potzger, W. Mller,\nJ. McCord, L. Schultz, and R. Mattheis. Structural and magnetic\nmodi\fcations of Cr-implanted Permalloy. Physical Review B ,\n73(18), May 2006.\n27W. Bailey, P. Kabos, F. Manco\u000b, and S. Russek. Control of mag-\nnetization dynamics in Ni/sub 81/Fe/sub 19/ thin \flms through\nthe use of rare-earth dopants. IEEE Transactions on Magnetics ,\n37(4):1749{1754, July 2001.\n28J. O. Rantschler, R. D. McMichael, A. Castillo, A. J. Shapiro,\nW. F. Egelho\u000b, B. B. Maranville, D. Pulugurtha, A. P. Chen,\nand L. M. Connors. E\u000bect of 3d, 4d, and 5d transition metal\ndoping on damping in permalloy thin \flms. Journal of Applied\nPhysics , 101(3):033911, February 2007.\n29Roger W Wood, Jim Miles, and Terry Olson. Recording tech-\nnologies for terabit per square inch systems. IEEE Transactions\non Magnetics , 38(4):1711{1718, 2002.\n30R. F. L. Evans, Roy W. Chantrell, Ulrich Nowak, Andreas Ly-\nberatos, and H.-J. Richter. Thermally induced error: Den-\nsity limit for magnetic data storage. Applied Physics Letters ,\n100(10):102402, 2012."    },    {        "title": "2307.05865v1.Asymptotic_behavior_of_solutions_to_the_Cauchy_problem_for_1_D_p_system_with_space_dependent_damping.pdf",        "content": "arXiv:2307.05865v1  [math.AP]  12 Jul 2023Asymptotic behavior of solutions to the Cauchy problem for 1 -D\np-system with space dependent damping\nAkitaka Matsumura and Kenji Nishihara∗\nOsaka University and Waseda University, Japan\nAbstract\nWe consider the Cauchy problem for one-dimensional p-system with damping of space-\ndependent coefficient. This system models the compressible flow thr ough porous media in the\nLagrangeancoordinate. Ourconcernisanasymptoticbehaviorof solutions,whichisexpected to\nbe the diffusion wave based on the Darcy law. In fact, in the constan t coefficient case Hsiao and\nLiu [3] showed the asymptotic behavior under suitable smallness cond itions for the first time.\nAfter this work, there are many literatures, but there are few wo rks in the space-dependent\ndamping case, as far as we know. In this paper we treat this space- dependent case, as a first\nstep when the coefficient is around some positive constant.\n1 Introduction\nWe consider the Cauchy problem for the p-system with space-dependent damping\n(1.1)\n\nvt−ux= 0,(t,x)∈R+×R,\nut+p(v)x=−αu(α=α(x)),\n(v,u)(0,x) = (v0,u0)(x)→(v±,u±), x→ ±∞,(v±>0),\nwhich models the 1-D compressible flow through porous media i n the Lagrangean coordinate,\nwhereu=u(t,x) is the velocity of the flow at time tand position x,v(>0) is the specific volume,\np(>0) is the pressure with p′(v)<0, and the coefficient α=α(x)>0. Our interest is in the large\ntime behavior of the solution ( v,u) to (1.1), which is expected to be the diffusion wave (¯ v,¯u) to\n(1.2)/braceleftBigg\n¯vt−¯ux= 0,\np(¯v)x=−α¯u,\nby the Darcy law. In fact, when αis a constant, Hsiao and Liu [3] showed its asymptotic behavi or\nto (¯v,¯u) under some smallness conditions for the first time. The conv ergence rates of ( v−¯v,u−¯u)\nwere improved by the second author [9]. After these, there ar e many works including the case\nα=α(t). We cite only [4, 5, 6, 8, 10, 13, 14, 15]. See the references t herein, too. However, as\nfar as we know, there are few results on the asymptotics towar d diffusion wave in space-dependent\ndamping case. Concerning the blow-up results, there are som e works(see Chen et al. [1], Sugiyama\n[12] and reference therein).\n∗E-mail addresses: akitaka@muh.biglobe.ne.jp (A. Matsumu ra), kenji@waseda.jp (K. Nishihara)\n1In this paper we consider the case of x-dependent coefficient α=α(x). The Darcy law says\nthat the velocity of flow in porous media is proportional to th e pressure gradient, and so it may be\nreasonable that the coefficient αdepends on the position x. By the situation of media, αwill be\nassumed variously. However, as a first step we treat the simpl est case\n(1.3) α=α(x)→α, x→ ±∞\nfor some constant α>0. When\n(1.4) v+=v−(andu+/ne}ationslash=u−in general) ,\nthe asymptotic profile ¯ vof (1.2) is expected to be a variant of the Gauss function. Whi le, when\n(1.5) v+/ne}ationslash=v−,\n¯vbe an approximate similarity solution, and the treatment be comes complicated.\nIn both cases, the problems are reformulated to the Cauchy pr oblems for the quasilinear wave\nequations with damping of the coefficient α(x), whose results are directly applied to the original\nones. Those details are stated in the next section. As for the results obtained in this paper we be-\nlieve that there are still plenty of rooms for improvement. U nder various assumptions on α=α(x)\norα(t,x) many discussions are expected in the future.\nThe content of this paper is as follows. In Section 2 our probl em is reformulated in each case\nof (1.4) and (1.5) to the Cauchy problem for a quasilinear wav e equation of second order with\ndamping. The case around the constant state of vis treated in Subsection 2.1, while the different\nend states case is mentioned in Subsection 2.2. In each subse ction, the theorem on the reformulated\none will be stated clear, so that our goal for (1.1) in each cas e is obtained as a corollary. For those\nproofs, a standard energy method is applied, for which a seri es of a priori estimates are necessary.\nThe case around the constant state (1.4) is shown in Section 3 . In the final section the convergence\nto the similarity solution in case of (1.5) will be treated.\nNotations . For the function spaces, Lp=Lp(R)(1≤p≤ ∞) is a usual Lebesgue space with\nnorm\n|f|p= (/integraldisplay\nR|f(x)|pdx)1/p(1≤p <∞),|f|∞= sup\nR|f(x)|.\nThe integral domain Ris often abbreviated when it is clear. For any integer l≥0,Hl=Hl(R)\ndenotes the usual l-th order Sobolev space with norm\n/bardblf/bardblHl=/bardblf/bardbll= (l/summationdisplay\nj=0/bardbl∂j\nxf/bardbl2)1/2.\nWhenl= 0andp= 2, weoftenusethenotation /bardblf/bardbl=/bardblf/bardbl0=|f|2. Forbrevity, /bardblf,g,···/bardbl2\nHk×Hl×···\n=/bardblf/bardbl2\nHk+/bardblg/bardbl2\nHl+···=/bardblf/bardbl2\nk+/bardblg/bardbl2\nl+···. The set of k-times continuously differentiable functions\ninRwith compact support is denoted by Ck\n0(R). The space Ck([0,T];X) is a set of k-times\ndifferentiable functions on [0 ,T] to the Hilbert space X. Also, by Corcwe denote a generic\npositive constant independent of the data and time t, whose value may change in line to line.\n22 Reformulation of the problem and results\nIn this section we heuristically reformulate the problem (1 .1) to the quasi-linear wave equation with\ndamping, and state the results on the reformulated one, whic h derive our main results on (1.1).\nRewrite (1.1) as\n(2.1)\n\nvt−ux= 0,\nut+p(v)x+αu= 0,\n(v,u)(0,x) = (v0,u0)(x)→(v±,u±), x→ ±∞,\nunder the assumption (1.3). By the Darcy law, the asymptotic profile (¯v,¯u) is expected to be\ngiven by (1.2) as in the previous papers. However, in the case thatαis depending on xwe meet\ndifficulties to directly construct (¯ v,¯u) by\n(2.2) ¯ vt−/parenleftBig−p(¯v)x\nα/parenrightBig\nx= 0 and ¯ u=−p(¯v)x\nα.\nIn this paper, to avoid the difficulties, we shall introduce a s impler profile in each case of (1.4) and\n(1.5).\n2.1 Reformulation of the problem around the constant state\nStart to discuss the simpler case (1.4). Since both (1.3) and (1.4) are assumed, we here adopt an\nasymptotic profile ( V,U) by\n(2.1.1)/braceleftBigg\nVt−Ux= 0,\np′(v)Vx+αU= 0, V(t,±∞) =v(:=v+=v−),\nor\n(2.1.2)\n\nVt−µVxx= 0, V(t,±∞) =v(µ:=|p′(v)|\nα),\nU=|p′(v)|\nαVx=µVx,\nso that, as an exact solution,\n(2.1.3)\n\nV(t,x) =v+δ0√\n4πµ(1+t)e−x2\n4µ(1+t),\nU(t,x) =−2πδ0x\n(4πµ(1+t))3/2e−x2\n4µ(1+t),\nwhereδ0is some constant, determined later. Then ( V,U) satisfies\n(2.1.4)/braceleftBigg\nVt−Ux= 0,\nUt+p(V)x+αU=Ut+(α−α)U+(p(V)−p′(v)V)x.\nHere, we note that ( V,U)→(v,0) asx→ ±∞.\n3To arrange the condition u(0,x) =u0(x)→u±(x→ ±∞), we introduce a correction function\n(ˆv,ˆu). By (2.1) 2(second equation of (2.1)) we can expect u(t,x)∼e−αtu±asx→ ±∞and so we\ndetermine (ˆ v,ˆu) as\n(2.1.5)\n\nˆvt−ˆux= 0,\nˆut+αˆu= 0,\n(ˆv,ˆu)(t,x)→(0,e−αtu±), x→ ±∞.\nThus we define (ˆ v,ˆu) by\n(2.1.6)\n\nˆu(t,x) =e−αt(u−+(u+−u−)/integraltextx\n−∞m0(y)dy) =:e−αtM0(x),\nˆv(t,x) =∂\n∂x(e−αt\n−αM0(x))\n=e−αt\n−α{(u+−u−)m0(x)−α′(x)M0(x)(1\nα(x)+t)},\nwherem0∈C∞\n0(R) with/integraltext∞\n−∞m0(y)dy= 1. In the result, (2.1.5) holds with\n(2.1.7)/integraldisplay∞\n−∞ˆv(0,x)dx=/integraldisplay∞\n−∞∂\n∂x(1\n−αM0(x))dx=1\n−α(u+−u−).\nCombining (2.1), (2.1.4), (2.1.5), we have\n(2.1.8)\n\n(v−V−ˆv)t−(u−U−ˆu)x= 0,\n(u−U−ˆu)t+(p(v)−p(V))x+α(u−U−ˆu)\n=−{Ut+(α−α)U+(p(V)−p′(v)V)x}.\nSinceu−U−ˆu→0 (x→ ±∞) is expected,\n/integraltext∞\n−∞(v−V−ˆv)(t,x)dx=/integraltext∞\n−∞(v0(x)−V(0,x)−ˆv(0,x))dx\n=/integraltext∞\n−∞{(v0(x)−v)−(V(0,x)−v)}dx+1\nα(u+−u−)\n=/integraltext∞\n−∞(v0(x)−v)dx−δ0+1\nα(u+−u−).\nTherefore, we choose δ0by\n(2.1.9) δ0=/integraldisplay∞\n−∞(v0(x)−v)dx+1\nα(u+−u−),\nso that/integraltext∞\n−∞(v−V−ˆv)(t,x)dx= 0 for any t≥0. Thus, if we define φby\n(2.1.10) φ(t,x) =/integraldisplayx\n−∞(v−V−ˆv)(t,y)dy,\nthen we may expect φ(t,·)∈H1, and it holds that φx=v−V−ˆv. Then, by (2.1.8) 1, it also holds\nφt=u−U−ˆu, and (2.1.8) 2can be written as\n(2.1.11)φtt+(p(V+ ˆv+φx)−p(V))x+αφt\n=−{Ut+(α−α)U+(p(V)−p′(v)V)x}.\n4Modify the second term in (2.1.11) to get the following refor mulated problem:\n(2.1.12)\n\nφtt+(p(V+ ˆv+φx)−p(V+ ˆv))x+αφt\n=−{Ut+(α−α)U+(p(V)−p′(v)V)x+(p(V+ ˆv)−p(V))x}=:F=:4/summationdisplay\ni=1Fi,\nφ(0,x) =φ0(x) :=/integraltextx\n−∞(v0(y)−V(0,y)−ˆv(0,y))dy,\nφt(0,x) =φ1(x) :=u0(x)−U(0,x)−ˆu(0,x).\nOur aim is to show ( φx,φt)(t) =o(t−1/4,t−3/4)inL2-sense and, if possible, o(t−1/2,t−1) in\nL∞-sense as t→ ∞, so that ( V,U+ˆu) is an asymptotic profile of ( v,u), since (ˆ v,ˆu) decays rapidly.\nThus, ournextjobis to showtheexistence and asymptotic beh aviorof theuniqueglobal-in-time\nsolution φto (2.1.12). We assume\n(2.1.13)\n\np∈C3(R+), p′(v)<0 (v >0),\nα∈C2(R), α−α∈L1∩L2,|x|1/2(α−α)∈L2,\n(1+|x|)(|αx|+|αxx|)∈L2andα≥α0>0 (α0: constant) ,\nand\n(2.1.14)v0−v∈L1,/integraldisplayx\n−∞(v0−v)(y)dy∈L2, v0−v∈H2andu0−u±∈L2(R±),u0x∈H1,\nso (φ0,φ1)∈H3×H2. Under these assumptions there exists a unique local-in-ti me solution φin\n∩2\ni=0Ci([0,t0];H3−i) for some t0>0(for the proof, see Matsumura [7]). Main part in the proof fo r\nthe global-in-time solution and its asymptotic behavior is the following a priori estimates.\nProposition 2.1.1 (A priori estimate) Assume (2.1.13) and (2.1.14). Then there exist positive\nconstants ε0andC0such that, if δ1:=|v0−v|1+|u+−u−| ≤ε0and ifφ∈ ∩2\ni=0Ci([0,T];H3−i)\nis a solution to (2.1.12) for some T >0, which satisfies\nsup\n0≤t≤T{/bardbl(φ,φt)(t)/bardblH3×H2+(1+t)/bardbl(φtx,φtxx)(t)/bardbl} ≤ε0,\nthen it holds\n(2.1.15)(1+t)2/bardbl(φtt,φt,φxx)(t)/bardbl2\nH1×H2×H1+(1+t)/bardblφx(t)/bardbl2+/bardblφ(t)/bardbl2\n+/integraltextt\n0{(1+τ)2/bardbl(φtt,φtx,φtxx)(τ)/bardbl2+(1+τ)/bardbl(φt,φxx)(τ)/bardbl2+/bardblφx(τ)/bardbl2}dτ\n≤C0(/bardblφ0,φ1/bardbl2\nH3×H2+δ1),0≤t≤T.\nRemark 2.1.1 Note that none of smallness assumptions on αis not made.\nCombining the local existence and a priori estimates of solu tion, we have theorem for (2.1.12).\nTheorem 2.1.1 Under the assumptions of Proposition 2.1.1, unique global- in-time solution φ∈\n∩2\ni=0Ci([0,∞);H3−i)to (2.1.12) exists and satisfies the decay properties(2.1.1 5) for0≤t <∞.\n5Once we have Theorem 2.1.1, if we newly define ( v,u) by (v,u) = (V+ˆv+φx,U+ˆu+φt), then\nwe can have a unique solution ( v,u) to (2.1), satisfying ( v−V−ˆv,u−U−ˆu)∈C1([0,∞);H2).\nThus, from Theorem 2.1.1 with\n|φx(t)|∞≤C/bardblφx(t)/bardbl1/2/bardblφxx(t)/bardbl1/2≤C(1+t)−3/4,\nwe have main theorem in case of (1.4).\nTheorem 2.1.2 (Around the constant state) Suppose (2.1.13) and (2.1.14). If /bardblφ0,φ1/bardbl2\nH3×H2\n+δ1is suitably small, then the solution (v,u)to (1.1) such that (v−V−ˆv,u−U−ˆu)∈C1([0,∞);H2)\nuniquely exists and it holds that, as t→ ∞,\n(2.1.16)/braceleftBigg\n(v−V)(t,x) =O(t−1/2),\n(u−U−ˆu)(t,x) =O(t−1),inL2-sense,\nand\n(2.1.17) v(t,x) =V(t,x)+O(t−3/4)inL∞-sense.\nSince/bardbl(V,U)(t)/bardbl ∼(t−1/4,t−3/4) and/bardbl(ˆv,ˆu)(t)/bardblH1×L∞=O(e−ct), (V,U+ ˆu) is an asymptotic\nprofile of ( v,u) inL2-framework, and Vis so ofvinL∞-framework.\n2.2 Reformulation around the different end states and result s\nSimilar to the last subsection, as an asymptotic profile we ad opt the similarity solution ( V,U),\nwhich is defined by\n(2.2.1)/braceleftBigg\nVt−Ux= 0,\np(V)x+αU= 0, V(t,±∞) =v±(v+/ne}ationslash=v−),\nor\n(2.2.2)/braceleftBiggVt−(µ(V)Vx)x= 0, V(t,±∞) =v±,\nU=−1\nαp(V)x=−1\nαp′(V)Vx=:µ(V)Vx.\nAs a special solution, (2.2.2) has a similarity solution of t he form\n(2.2.3) V(t,x) =˜V(ξ), ξ= (x−x0)/√\n1+t.\nwherex0is any shift constant(cf. [2]). We denote ˜Vstill byV.\nLemma 2.2.1 The similarity solution Vto (2.2.2) 1satisfies the following:\n(2.2.4)\n\n|V(t,x)−v−| ≤Cδ0e−c(x−x0)2\n1+t,(t,x)∈R+×R−,\n|V(t,x)−v+| ≤Cδ0e−c(x−x0)2\n1+t,(t,x)∈R+×R+,\n|Vx(t,x)| ≤Cδ0(1+t)−1/2e−c(x−x0)2\n1+t,(t,x)∈R+×R,\n|Vxx(t,x)| ≤Cδ0(1+t)−1e−c(x−x0)2\n1+t,(t,x)∈R+×R,\n|Vxxx(t,x)| ≤Cδ0(1+t)−3/2e−c(x−x0)2\n1+t,(t,x)∈R+×R,\nwhereδ0=|v+−v−|.\n6SinceV=V(ξ)→v±(ξ→ ±∞),U=µ(V)Vx→0 asx→ ±∞. On the other hand,\nu(0,x) =u0(x)→u±(x→ ±∞), and so we must introduce the correction function (ˆ v,ˆu), samely\nas in the preceding subsection. We expect u(t,x)∼e−αtu±asx→ ±∞by (2.1) 2, and so we\ndetermine (ˆ v,ˆu) as\n(2.2.5)\n\nˆvt−ˆux= 0,\nˆut+αˆu= 0,\n(ˆv,ˆu)(t,x)→(0,e−αtu±), x→ ±∞.\nThus, we define (ˆ v,ˆu) by\n(2.2.6)\n\nˆu(t,x) =e−αt(u−+(u+−u−)/integraltextx\n−∞m0(y)dy) =:e−αtM0(x),\nˆv=∂\n∂x(e−αt\n−αM0(x))\n=e−αt\n−α{(u+−u−)m0(x)−α′(x)M0(x)(1\nα(x)+t)},\nwherem0∈C∞\n0(R) with/integraltext∞\n−∞m0(y)dy= 1. By (2.1), (2.2.1) and (2.2.5),\n(2.2.7)/braceleftBigg\n(v−V−ˆv)t−(u−U−ˆu)x= 0,\n(u−U−ˆu)t+(p(v)−p(V))x+α(u−U−ˆu) =−{Ut+(α−α)U}.\nSinceu−U−ˆu→0 is expected as x→ ±∞, integrating (2.2.7) 1over [0,t]×R, we have\n/integraldisplay∞\n−∞(v−V−ˆv)(t,x)dx=/integraldisplay∞\n−∞(v0(x)−V(x−x0))dx+1\nα(u+−u−).\nFor a given v0(x),/integraltext∞\n−∞(v0(x)−V(x−x0))dx→ ±∞or→ ∓∞(x0→ ±∞), and so we can choose\nx0as\n(2.2.8)/integraldisplay∞\n−∞(v0(x)−V(x−x0))dx=−1\nα(u+−u−),\n(setx0= 0 without loss of generality) so that, for any t≥0,\n/integraldisplay∞\n−∞(v−V−ˆv)(t,x)dx= 0.\nThus, if we define φby\n(2.2.9) φ(t,x) =/integraldisplayx\n−∞(v(t,y)−V(y√1+t)−ˆv(t,y))dy,\nthen (φx,φt) = (v−V−ˆv,u−U−ˆu) and\nφtt+(p(V+ ˆv+φx)−p(V))x+αφt=−{Ut+(α−α)U}\nor, as in the previous subsection,\n(2.2.10)φtt+(p(V+ ˆv+φx)−p(V+ ˆv))x+αφt\n=−{Ut+(α−α)U+(p(V+ ˆv)−p(V))x}=:G=:3/summationdisplay\ni=1Gi,\n7which is our reformulated problem in the present case. Forma l reformulation is almost similar to\nthe case around the constant state. However, since V(±∞) =v±, v+/ne}ationslash=v−, the energy estimates of\nφare rather different from those in case of v+=v−. Our aim is to derive /bardbl(φx,φt)(t)/bardbl=o(1,t−1/4)\nand, if possible, |(φx,φt)(t)|∞=o(1,t−1/2) ast→ ∞, so that ( V,U+ ˆu) becomes an asymptotic\nprofileof( v,u) inL2andL∞sense, respectively, since(ˆ v,ˆu) decays rapidly. Henceitis importantto\nshow the existence of global-in-time solution φto (2.2.10), combining the existence of local-in-time\nsolution and the a priori estimates with decay rates.\nFor the given data\n(2.2.11) ( φ,φt)(0) = (φ0,φ1)∈H3×H2,\nthere exists a unique local-in-time solution φ∈ ∩2\ni=0Ci([0,t0];H3−i) to (2.2.10) (for the proof, see\nMatsumura [7]).\nAs usual, if we can have a priori estimates\n/bardbl(φ,φt)(t)/bardbl2\nH3×H2≤C(/bardblφ0,φ1/bardbl2\nH3×H2+δ1), δ1=|v+−v−|+|u+−u−|\nfor the solution φ∈ ∩2\ni=0Ci([0,T];H3−i) with some decay properties, then continuation arguments\nare well done and global-in-time solution φis obtained, which may satisfies desired decay rates.\nBut, in this case it seems to be hopeless. In fact, to get the bo undedness of /bardblφ(t)/bardbl, multiplying\n(2.2.10) by φand integrating it over [0 ,t]×R, we have\n/integraldisplay\n(αφ2+φtφ)dx+/integraldisplayt\n0/integraldisplay\n(p(ˆV)−p(ˆV+φx))φxdxdτ≤C(/bardblφ0,φ1/bardbl2+/integraldisplayt\n0/bardblφt/bardbl2dτ+/integraldisplayt\n0/integraldisplay\nGφdxdτ ),\nor\n/bardblφ(t)/bardbl2+/integraldisplayt\n0/bardblφx(τ)/bardbl2dτ≤C(/bardblφ0,φ1/bardbl2+/integraldisplayt\n0/bardblφt(τ)/bardbl2dτ+/integraldisplayt\n0|/integraldisplay\nGφdx|dτ).\nwhereˆV=V+ ˆv. For an example, in the final term we need to estimate G2as\n/integraldisplayt\n0|/integraldisplay\n(α−α)Uφdx|dτ≤/integraldisplayt\n0|U|∞|φ|∞|α−α|1dτ≤Cδ0/integraldisplayt\n0(1+τ)−1/2/bardblφ(τ)/bardbl1/2/bardblφx(τ)/bardbl1/2dτ,\nand to cover this by good terms, which seems to be impossible, even if the other terms can be\nwell-evaluated.\nHowever, fortunately there is not φitself in the nonlinearity of (2.2.10), and so, for the conti n-\nuation arguments the uniform estimate in [0 ,T] of/bardbl(φx,φt)(t)/bardbl2is necessary, while growing up of\n/bardblφ(t)/bardblmay be permitted.\nThus, our a priori estimate is the following.\nProposition 2.2.1 (A priori estimate) Assume that\n/braceleftBigg\np∈C3(R+), p′(v)<0(v >0),andα∈C1(R), α−α∈L1∩L2,αx∈L2\nwithα≥α0>0(α0:constant).\nThen there exist positive constants ε0andC0such that, if δ1:=|v+−v−|+|u+−u−| ≤ε0and if\nφ∈ ∩2\ni=0Ci([0,T];H3−i)is a solution to (2.2.10)-(2.2.11) for some T >0, which satisfies\nsup\n0≤t≤T{(1+t)−γ/2/bardblφ(t)/bardbl+/bardbl(φt,φx)(t)/bardbl2+(1+t)1/2/bardbl(φtxx,φxxx)(t)/bardbl} ≤ε0,\n8then it holds that\n(2.2.12)(1+t)−γ/bardblφ(t)/bardbl2+(1+t)1−γ/bardblφx(t)/bardbl2+(1+t)/bardbl(φt,φtt,φxx)(t)/bardbl2\nH2×H1×H1\n+/integraltextt\n0{(1+τ)−1−γ/bardblφ(τ)/bardbl2+(1+τ)−γ/bardblφx(τ)/bardbl2+(1+τ)1−γ/bardblφt(τ)/bardbl2\n+(1+τ)/bardbl(φtt,φtx)(τ)/bardbl2+/bardbl(φttx,φtxx)(τ)/bardbl2}dτ\n≤C0(/bardblφ0,φ1/bardbl2\nH3×H2+δ1),0≤t≤T.\nProposition 2.2.1 yields the following theorem, together w ith the local existence theorem.\nTheorem 2.2.1 Under the assumption in Proposition 2.2.1, if both /bardblφ0,φ1/bardblH3×H2andδ1=\n|v+−v−|+|u+−u−|are sufficiently small, then there exists a unique global-in- time solution\nφ∈ ∩2\ni=0Ci([0,∞);H3−i)to (2.2.10)-(2.2.11), which satisfies decay property (2.2. 12) for0≤t <\n∞.\nAs same as in the preceding subsection, once we have Theorem 2 .2.1, if we define ( v,u) by\n(v,u) = (V+ ˆv+φx,U+ ˆu+φt), then we can have a unique solution ( v,u) of (2.1) satisfying\n(v−V−ˆv,u−U−ˆu)∈C1([0,∞);H2).\nAlso, since /bardblφx(t)/bardbl ≤C(1+t)−(1−γ)/2,|φx(t)|∞≤C/bardblφx(t)/bardbl1/2/bardblφxx(t)/bardbl1/2≤C(1+t)−(2−γ)/2\nand/bardblφt(t)/bardbl ≤C(1+t)−1/2, the similarity solution ( V,U+ ˆu) is an asymptotic profile of ( v,u) in\nL2-sense, and Vis so ofvinL∞-sense. Thus, our main theorem is the following.\nTheorem 2.2.2 (Main Theorem) Under the assumption in Proposition 2.2.1, suppose that\nv0−V∈L1,/integraldisplay·\n−∞(v0(y)−V(0,y))dy∈L2,(v0−V(0,·),u0−U(0,·)−ˆu(0,·))∈H2×H2.\nIf both/bardbl/integraltext·\n−∞(v0(y)−V(0,y))dy/bardbl+/bardblv0−V(0,·),u0−U(0,·)−ˆu(0,·)/bardblH2×H2andδ1=|v+−\nv−|+|u+−u−|are suitably small, then there exists a unique solution (v,u)to (1.1) satisfying\n(v−V−ˆv,u−U−ˆu)∈C1([0,∞);H2), which satisfies, as t→ ∞,\n(v−V,u−U−ˆu)(t,x) =O(t−(1−γ)/2,t−1/2)inL2-sense,\nand\n(v−V)(t,x) =O(t−(2−γ)/2)inL∞-sense\nwith1/2< γ <1.\n3 A priori estimate around the constant state\nWe prove Proposition 2.1.1 in a series of several steps. To do that, let φbe a smooth solution in\n∩2\ni=0Ci([0,T];H3−i) for some T >0 to (2.1.12) and rewrite (2.1.12) as\n(3.1) φtt+(p(ˆV+φx)−p(ˆV))x+αφt=F,\nwhereˆV:=V+ ˆvand\n(3.2) F=−{Ut+(α−α)U+(p(V)−p′(v)V)x+(p(V+ ˆv)−p(V))x}=:4/summationdisplay\ni=1Fi.\n9Also, denote the a priori assumption as\nδ= sup\n0≤t≤T{/bardbl(φ,φt)(t)/bardblH3×H2+(1+t)/bardbl(φtx,φtxx)(t)/bardbl}(≤1)\ntogether with\nδ1=|v0−v|1+|u+−u−| ≥ |δ0|+(1+1\nα)|u+−u−|\nand\nI0=/bardblφ0,φ1/bardbl2\nH3×H2+δ1(≤1),\nwhereδ0is defined in (2.1.9).\nStep 1.First we multiply (3.1) by φtand integrate it over R:\n(3.3)d\ndt/integraltext1\n2φ2\ntdx+/integraltext\nαφ2\ntdx+/integraltext\n(p(ˆV)−p(ˆV+φx))φtxdx\n≤/integraltext\n|φtF|dx≤ε/integraltext\nφ2\ntdx+Cε/integraltext\nF2dx\nfor small constant ε >0. The integral domain Rand variable tare often abbreviated. The 3-rd\nterm is estimated as\n/integraltext\n(p(ˆV)−p(ˆV+φx))φtxdx\n=/integraltext\n(/integraltextφx\n0(p(ˆV)−p(ˆV+s))ds)tdx+/integraltext\n(p(ˆV+φx)−p(ˆV)−p′(V)φx)ˆVtdx\n≥d\ndt/integraltext\n(/integraltextφx\n0(p(ˆV)−p(ˆV+s))ds)dx−C(|Vt|∞+|ˆvt|∞)/integraltext\nφ2\nxdx\n≥d\ndt/integraltext\n(/integraltextφx\n0(p(ˆV)−p(ˆV+s))ds)dx−Cδ1δ(1+t)−3/2.\nThe estimates of Fare as follow, including those necessary.\nLemma 3.1 (Estimates of F)It holds that, for t∈R+\n(3.4)\n\n/bardblF(t)/bardbl2≤Cδ1(1+t)−5/2,\n/bardblFx(t)/bardbl2≤Cδ1(1+t)−3,\n/bardblFt(t)/bardbl2≤Cδ1(1+t)−9/2,\n/bardblFxt(t)/bardbl2≤Cδ1(1+t)−5.\nThe proof is given in Appendix.\nThus, integrating (3.3) over [0 ,t], we have\n(3.5) /bardbl(φt,φx)(t)/bardbl2+/integraldisplayt\n0/bardblφt(τ)/bardbl2dτ≤C/bardblφ0x,φ1/bardbl2+C(δ1+δδ1)≤CI0.\nNext, multiplying (3.1) by φ, we have\nd\ndt/integraltext\n(α\n2φ2+φφt)dx−/integraltext\nφ2\ntdx+/integraltext\n(p(ˆV)−p(ˆV+φx))φxdx\n≤/integraltext\n|φF|dx≤sup\n0≤t≤T/bardblφ/bardbl/bardblF/bardbl ≤Cδ1(1+t)−5/4,\n10and hence, using (3.5),\n(3.6)/bardblφ(t)/bardbl2+/integraltextt\n0/bardblφx(τ)/bardbl2dτ≤C(/bardblφ0,φ0x,φ1/bardbl2+/bardblφt(t)/bardbl2+/integraltextt\n0/bardblφt(τ)/bardbl2dτ)+Cδ1\n≤CI0.\nUsing (3.6), we can multiply (3.3) by 1+ tand get\n(3.7) /bardblφ(t)/bardbl2+(1+t)/bardbl(φt,φx)(t)/bardbl2+/integraldisplayt\n0(/bardblφx(τ)/bardbl2+(1+τ)/bardblφt(τ)/bardbl2)dτ≤CI0.\nWe repeat the similar procedure to Step 1 for higher derivati ves ofφ.\nStep 2.Differentiate (3.1) in t:\n(3.8) φttt+αφtt+(p′(ˆV+φx)φxt+(p′(ˆV+φx)−p′(ˆV))ˆVt)x=Ft.\nMultiplying (3.8) by φttand integrating it over R, we have\n(3.9)d\ndt/integraltext\n(1\n2φ2\ntt+−p′(ˆV+φx)\n2φ2\nxt)dx+/integraltext\nαφ2\nttdx\n≤Cδ1{(1+t)−4/bardblφx/bardbl2+(1+t)−3/bardblφxx/bardbl2+(1+t)−9/2}+C(δ+δ1)/bardblφxt/bardbl2,\nbecause of (3.4) 3, where the third term of (3.8) is estimated as\n/integraltext\n(−p′(ˆV+φx)φxtφxtt+{(p′(ˆV+φx)−p′(ˆV))ˆVxt+p′′(ˆV+φx)ˆVtφxx\n+(p′′(ˆV+φx)−p′′(ˆV))ˆVxˆVt}φtt)dx\n≥d\ndt/integraltext−p′(ˆV+φx)\n2φ2\nxtdx+/integraltextp′′(ˆV+φx)\n2(ˆVt+φxt)φ2\nxtdx\n−ε/integraltext\nφ2\nttdx−Cε/integraltext\n(ˆV2\nxtφ2\nx+ˆV2\ntφ2\nxx+ˆV2\nxˆV2\ntφ2\nx)dx\n(0< ε≪1) and\n|/integraldisplayp′′(ˆV+φx)\n2(ˆVt+φxt)φ2\nxtdx| ≤C(δ1(1+t)−3/2+δ)/bardblφxt/bardbl2≤C(δ1+δ)/bardblφxt/bardbl2\nby the a priori assumption.\nAlso, the product of (3.8) and φtyields\n(3.10)d\ndt/integraltext\n(α\n2φ2\nt+φtφtt)dx+c\n2/integraltext\nφ2\nxtdx\n≤/integraltext\nφ2\nttdx+Cδ1{(1+t)−9/4/bardblφt/bardbl2+(1+t)−3/bardblφx/bardbl2},\nby|/integraltext\nφtFtdx| ≤C/bardblφt/bardbl/bardblFt/bardbl ≤Cδ1(1+t)−9/4/bardblφt/bardbl. By calculating (3.9)+ λ·(3.10) (0 < λ≪1),\n(3.11)d\ndt/integraltext\n{(1\n2φ2\ntt+λφttφt+λα\n2φ2\nt)+−p′(ˆV+φx)\n2φ2\nxt}dx+/integraltext\n(α\n2φ2\ntt+λc\n2φ2\ntx)dx\n≤C(δ+δ1(1+t)−1)/bardblφxt/bardbl2\n+Cδ1{(1+t)−3/bardblφx/bardbl2+(1+t)−9/4/bardblφt/bardbl2+(1+t)−9/2+(1+t)−3/bardblφxx/bardbl2}.\n11Whenδ+δ1≪1, integrating (3.11), (1+ t)·(3.11) and (1+ t)2·(3.11) over [0 ,t], yields\n(3.12) (1+ t)/bardbl(φtt,φtx,φt)(t)/bardbl2+/integraldisplayt\n0(1+τ)/bardbl(φtt,φtx)(τ)/bardbl2dτ≤CI0\nand\n(3.13) (1+ t)2/bardbl(φtt,φtx,φt)(t)/bardbl2+/integraldisplayt\n0(1+τ)2/bardbl(φtt,φtx)(τ)/bardbl2dτ≤CI0+Cδ1/integraldisplayt\n0/bardblφxx(τ)/bardbl2dτ.\nStep 3.In a similar fashion to x-derivative of (3.1), we have\n(3.14)d\ndt/integraltext\n(1\n2φ2\ntx+−p′(ˆV+φx)\n2φ2\nxx)dx+c/integraltext\nφ2\ntxdx\n≤Cδ1{(1+t)−3/bardblφx/bardbl2+(1+t)−3}+C/bardblφt/bardbl2+Cδ/bardblφxx/bardbl2.\nHere, since we have differentiate (3.1) in x, the additional term αxφtcomes out and\n|/integraldisplay\nαxφtφtxdx| ≤ε/bardblφtx/bardbl2+Cε/bardblαx/bardbl/bardblαxx/bardbl·/bardblφt/bardbl2.\nThough we omit the estimates of the other terms, integrating (3.14) and (1 + t)·(3.14) over [0 ,t]\nand using (3.7), we have\n(3.15) (1+ t)/bardbl(φtx,φxx)(t)/bardbl2+/integraldisplayt\n0(1+τ)/bardblφtx(τ)/bardbl2dτ≤CI0+Cδ/integraldisplayt\n0/bardblφxx(τ)/bardbl2dτ.\nIn (3.13) and (3.15), the final terms are not estimated yet.\nStep 4.Multiplying the variant of (3.1)\n(3.1)′φtt+αφt+p′(ˆV+φx)φxx=F+(p′(ˆV)−p′(ˆV+φx))ˆVx\nby−φxxand integrating it over R, we obtain\n/integraldisplay\nφ2\nxxdx≤ −d\ndt/integraldisplay\nφxφxtdx+/integraldisplay\nφ2\nxtdx+C/integraldisplay\nφ2\ntdx+Cδ1{(1+t)−2/bardblφx/bardbl2+(1+t)−5/2}\nand, also integrating over [0 ,t],\n(3.16)/integraldisplayt\n0/bardblφxx(τ)/bardbl2dτ≤CI0+/bardbl(φt,φxt)(t)/bardbl2+/integraldisplayt\n0/bardbl(φt,φxt)(τ)/bardbl2dτ≤CI0\nby (3.12). Applying (3.16) to (3.13) and (3.15) with δ≪1, we get\n(3.17)(1+t)2/bardbl(φtt,φtx,φt)(t)/bardbl2+(1+t)/bardbl(φx,φxx)(t)/bardbl2+/bardblφ(t)/bardbl2\n+/integraltextt\n0{(1+τ)2/bardbl(φtt,φtx)(τ)/bardbl2+(1+τ)/bardblφt(τ)/bardbl2+/bardbl(φxx,φx)(τ)/bardbl2}dτ≤CI0.\nBy (3.17), integration of (1 + t)2(−φxx)·(3.1)′inRand (1 + t)(−φxx)·(3.1)′over [0,t]×Ralso\nyield\n(3.18) (1+ t)2/bardblφxx(t)/bardbl2+/integraldisplayt\n0(1+τ)/bardblφxx(τ)/bardbl2dτ≤CI0.\n12Step 5. Though the details are omitted, the combination of∂2\n∂t∂x(3.1)·φttx,∂\n∂t(3.1)·(−φxxt) and\n∂\n∂x(3.1)·(−φxxx) yields\n(3.19) (1+ t)2/bardbl(φttx,φtxx,φxxx)(t)/bardbl2+/integraldisplayt\n0{(1+τ)2/bardblφtxx(τ)/bardbl2+(1+τ)/bardblφxxx(τ)/bardbl2}dτ≤CI0.\nThus, we have obtained (2.17) by (3.17) - (3.19) and complete d the proof of Proposition 2.1.\n4 Proof of Proposition 2.2.1 in different end states case\nLetφbe a smooth solution in ∩2\ni=0Ci([0,T];H3−i) for some T >0 to the reformulation problem\n(2.2.10)-(2.2.11), which is re-written as\n(4.1)/braceleftBigg\nφtt+(p(ˆV+φx)−p(ˆV))x+αφt=G,\n(φ,φt)(0) = (φ0,φ1)∈H3×H2,\nwhereˆV=V+ ˆvand\n(4.2) G=3/summationdisplay\ni=1Gi=−{Ut+(α−α)U+(p(V+ ˆv)−p(V))x}.\nTo get the a priori estimate (2.2.12), denote the a priori ass umption as\nδ= sup\n0≤t≤T{(1+t)−γ/2/bardblφ(t)/bardbl+/bardbl(φt,φx)(t)/bardbl2+(1+t)1/2/bardbl(φtxx,φxxx)(t)/bardbl}(≤1),\ntogether with\nI0=/bardblφ0,φ1/bardbl2\nH3×H2+δ1(δ1=|v+−v−|+|u+−u−|).\nNote that the notations δandδ1in Sections 3 and 4 are slightly different from each other.\nSame as in the preceding section, the proof is given in a serie s of several steps.\nStep 1. Multiply (4.1) by φtand integrate it over R:\n(4.3)d\ndt/integraldisplay1\n2φ2\ntdx+/integraldisplay\nαφ2\ntdx−/integraldisplay\n(p(ˆV+φx)−p(ˆV))φxtdx=/integraldisplay\nφtGdx\nand\n(3-rd term) =d\ndt/integraltext /integraltextφx\n0(p(ˆV)−p(ˆV+s))dsdx+/integraltext\n(p(ˆV+φx)−p(ˆV)−p′(ˆV)φx)ˆVtdx\n≥d\ndt/integraltext /integraltextφx\n0(p(ˆV)−p(ˆV+s))dsdx−Cδ1(1+t)−1/integraltext\nφ2\nxdx.\nHence, integrating (4.3) over [0 ,t], we have\n1\n2/bardblφt(t)/bardbl2+/integraltext /integraltextφx\n0(p(ˆV)−p(ˆV+s))dsdx+/integraltextt\n0/integraltext\nαφ2\ntdxdτ\n≤C/bardblφ0,φ1/bardbl2\nH1×L2+Cδ1/integraltextt\n0(1+τ)−1/bardblφx(τ)/bardbl2dx+/integraltextt\n0/integraltext\nφtGdxdt,\n13or\n(4.4)/bardbl(φt,φx)(t)/bardbl2+/integraldisplayt\n0/bardblφt(τ)/bardbl2dτ≤CI0+Cδ1/integraldisplayt\n0(1+τ)−1/bardblφx(τ)/bardbl2dx+C/integraldisplayt\n0/integraldisplay\nφtGdxdt.\nNext, multiply (4.1) by (1+ t)−γφ(1/2< γ <1) and integrate it over [0 ,t]×Rto get\n(4.5)(1+t)−γ/integraltext\n(α\n2φ2+φφt)dx+γ/integraltextt\n0(1+τ)−γ−1/integraltext\n(φ2+φφt)dxdτ+c/integraltextt\n0(1+τ)−γ/integraltext\nφ2\nxdxdτ\n≤CI0+/integraltextt\n0(1+τ)−γ/integraltext\nφ2\ntdxdτ+C/integraltextt\n0/integraltext\n(1+τ)−γφGdxdτ.\nBecause /integraldisplay\nα(1+t)−γφφtdx=d\ndt(1+t)−γ/integraldisplayα\n2φ2dx+αγ\n2(1+t)−γ−1/integraldisplay\nφ2dx,\n/integraldisplay\n(1+t)−γφφttdx=d\ndt(1+t)−γ/integraldisplay\nφφtdx+γ(1+t)−γ−1/integraldisplay\nφφtdx−(1+t)−γ/integraldisplay\nφ2\ntdx,\nand\n−/integraldisplay\n(1+t)−γ(p(ˆV+φx)−p(ˆV))φxdx≥c(1+t)−γ/integraldisplay\nφ2\nxdx.\nAdding (4.4) to λ·(4.5) (0< λ≪1, in particular, Cδ1λ≤c), we have\n(4.6)(1+t)−γ/bardblφ(t)/bardbl2+/bardbl(φt,φx)(t)/bardbl2\n+/integraltextt\n0((1+τ)−1−γ/bardblφ(τ)/bardbl2+(1+τ)−γ/bardblφx(τ)/bardbl2+/bardblφt(τ)/bardbl2)dτ\n≤CI0+C/integraltextt\n0(/integraltext\nφtGdx+(1+τ)−γ/integraltext\nφGdx)dτ.\nItisnecessarytoestimatethelasttermcarefully. Estimat eeachterm, usingtheaprioriassumption\n(2.2.13) with δ0=|v+−v−|:\n(i)/integraltext\nφtG1dx≤Cδ0/integraltext\nφ2\ntdx+Cδ0/integraltext\n(1+τ)−3e−2cx2\n1+τdx≤Cδ0(/bardblφt(τ)/bardbl2+(1+τ)−5/2)\nby|Ut| ≤C(|Vx||Vt|+|Vxt|,\n(ii)/integraltext\nφtG2dx=d\ndt/integraltext\n(α−α)φU dx+/integraltext\n(α−α)φUtdx\nwith\n|/integraltext\n(α−α)φU dx| ≤C|U|∞/bardblφ/bardbl/bardblα−α/bardbl ≤Cδ0δ(1+t)−(1−γ)/2≤Cδ0,\n|/integraltext\n(α−α)φUtdx| ≤ |φ|∞|Ut|∞|α−α|1≤Cδ0(1+τ)−3/2/bardblφ/bardbl1/2/bardblφx/bardbl1/2\n≤Cδ0(1+τ)(2γ−5)/4·(1+τ)−(γ+1)/4/bardblφ/bardbl1/2·(1+τ)−γ/4/bardblφx/bardbl1/2\n≤Cδ0{(1+τ)−(5−2γ)/2+(1+τ)−(γ+1)/bardblφ/bardbl2+(1+τ)−γ/bardblφx/bardbl2},\n(iii)|(1+τ)−γ/integraltext\nφG1dx| ≤Cδ0/integraltext\n|φ|(1+τ)−γ−3/2e−cx2\n1+τdx\n≤Cδ0((1+τ)−γ/bardblφ/bardbl2)1/2(/integraltext\n(1+τ)−γ−3e−2cx2\n1+τdx)1/2≤Cδ0δ(1+τ)−(2γ+5)/4,\n(iv)|(1+τ)−γ/integraltext\nφ(α−α)U dx| ≤(1+τ)−γ|φ|∞|α−α|1|U|∞\n≤Cδ0(1+τ)−γ−1/2/bardblφ/bardbl1/2/bardblφx/bardbl1/2\n=Cδ0(1+τ)−(2γ+1)/4·(1+τ)−(γ+1)/4/bardblφ/bardbl1/2·(1+τ)−γ/4/bardblφx/bardbl1/2\n≤Cδ0{(1+τ)−γ−1/2+(1+τ)−γ−1/bardblφ/bardbl2+(1+τ)−γ/bardblφx/bardbl2},\n14and\n(v) since |G3| ≤C|u+−u−|e−ct,/integraltext\n(φtG3+(1+τ)−γφG3)dxis well estimated.\nThus, the final term in (4.6) is absorbed in the left-hand side ifδ1=δ0+|u+−u−|is small,\nand we have\n(4.7)(1+t)−γ/bardblφ(t)/bardbl2+/bardbl(φt,φx)(t)/bardbl2\n+/integraltextt\n0((1+τ)−1−γ/bardblφ(τ)/bardbl2+(1+τ)−γ/bardblφx(τ)/bardbl2+/bardblφt(τ)/bardbl2)dτ≤CI0.\nStep 2.We want to have decay properties of /bardbl(φt,φx)(t)/bardbland modify Step 1. In fact, multiplying\n(4.1) by (1+ t)1−γφtand integrating it over [0 ,t]×R, we have\n(4.8)(1+t)−γ/bardblφ(t)/bardbl2+(1+t)1−γ/bardbl(φt,φx)(t)/bardbl2\n+/integraltextt\n0((1+τ)−1−γ/bardblφ(τ)/bardbl2+(1+τ)−γ/bardblφx(τ)/bardbl2+(1+τ)1−γ/bardblφt(τ)/bardbl2)dτ\n≤CI0,\ntogether with (4.6). Note that 1 /2< γ <1, and so ( φt,φx)(t) decays in L2-sense.\nLet us derive (4.8). Multiplying (4.3) by (1+ t)1−γyields\nd\ndt/integraltext1\n2(1+t)1−γφ2\ntdx−/integraltext1−γ\n2(1+t)−γφ2\ntdx+/integraltext\nα(1+t)1−γφ2\ntdx\n+d\ndt/integraltext\n(1+t)1−γ/integraltextφx\n0(p(ˆV)−p(ˆV+s))dsdx−(1−γ)/integraltext\n(1+t)−γ/integraltextφx\n0(p(ˆV)−p(ˆV+s))dsdx\n+(1+t)1−γ/integraltext\n(p(ˆV+φx)−p(ˆV)−p′(ˆV)φx)ˆVtdx\n= (1+t)1−γ/integraltext\nφtGdx,\nand hence\nd\ndt(1+t)1−γ{/integraltext1\n2φ2\ntdx+/integraltext /integraltextφx\n0(p(ˆV)−p(ˆV+s))dsdx}+α0(1+t)1−γ/integraltext\nφ2\ntdx\n≤C(1+t)−γ/bardbl(φt,φx)(t)/bardbl2+(1+t)1−γCδ1(1+t)−1/bardblφx(t)/bardbl2+C(1+t)1−γ/integraltext\nφtGdx.\nHere, the final term is estimated as follows and absorbed in th e left-hand side:\n(vi) (1+ t)1−γ/integraltext\nφtG1dx≤(1+t)1−γ/bardblφt|/bardblUt/bardbl ≤Cδ0{/bardblφt(t)/bardbl2+(1+t)−γ−3/2}\n(vii)(1+t)1−γ/integraltext\nφtG2dx=d\ndt(1+t)1−γ/integraltext\n(α−α)φU dx\n−(1−γ)(1+t)−γ/integraltext\n(α−α)φU dx−(1+t)1−γ/integraltext\n(α−α)φUtdx,\nwith\n(1+t)1−γ/integraltext\n(α−α)φU dx≤(1+t)1−γ|φ|∞|U|∞|α−α|1\n≤Cδ0(1+t)−γ+1/2/bardblφ/bardbl1/2/bardblφx/bardbl1/2\n=Cδ0(1+t)−(2γ−1)/4·((1+t)(1−γ)/2/bardblφx/bardbl)1/2·((1+t)−γ/2/bardblφ/bardbl)1/2\n≤Cδ0{(1+t)−γ/bardblφ(t)/bardbl2+(1+t)1−γ/bardblφx(t)/bardbl2+(1+t)−(2γ−1)/2}\nand the same estimates as (iii), (iv).\n(viii) For G3, same as (v).\n15Therefore, using (4.7), we have (4.8).\nWe further need to evaluate higher derivatives of φ.\nStep 3.Differentiate (4.1) in t:\n(4.9) φttt+αφtt+(p′(ˆV+φx)φxt+(p′(ˆV+φx)−p′(ˆV))ˆVt)x=Gt.\nWe multiply (4.9) by φt,φttand (1+ t)φt, (1+t)φtt. First, estimate/integraltext\n(4.9)·φtdx:\n(4.10)d\ndt/integraltext\n(φttφt+α\n2φ2\nt)dx+/integraltext\n|p′(ˆV+φx)|φ2\nxtdx\n−/integraltext\n(p′(ˆV+φx)−p′(ˆV))ˆVtφtxdx=/integraltext\nφtGtdx.\nSince|ˆVt|∞≤Cδ1(1+t)−1≤Cδ1(1+t)−γ,\n|3-rd term | ≤Cδ1/bardblφtx(t)/bardbl2+Cδ1(1+t)−γ/bardblφx(t)/bardbl2.\nIt is easy to show\n(4.11) /bardblGt/bardbl ≤Cδ1(1+t)−3/2\nand|/integraltext\nφtGtdx| ≤Cδ1(1+t)1−γ/bardblφt(t)/bardbl2+Cδ1(1+t)−(4−γ), so that\n(4.12)/integraldisplayt\n0/bardblφtx(τ)/bardbl2dτ≤CI0+C(/bardblφtt(t)/bardbl2+/integraldisplayt\n0/bardblφtt(τ)/bardbl2dτ)\nby (4.8). Secondly, multiplying (4.9) by φtt,\n(4.13)d\ndt/integraltext1\n2φ2\nttdx+/integraltext\nαφ2\nttdx−/integraltext\np′(ˆV+φx)φxtφxttdx\n+/integraltext\n((p′(ˆV+φx)−p′(ˆV))ˆVt)xφttdx=/integraltext\nφttGtdx,\nand\n(3-rd term) =d\ndt/integraltext|p′(ˆV+φx)|\n2φ2\nxtdx+/integraltextp′′(ˆV+φx)\n2(ˆVt+φxt)φ2\nxtdx\n≥d\ndt/integraltext|p′(ˆV+φx)|\n2φ2\nxtdx−C(δ1+δ)/bardblφxt(t)/bardbl2,\n|4-th term | ≤Cδ1/bardblφtt(t)/bardbl2+Cδ1/integraltext\n{(p′(ˆV+φx)−p′(ˆV))ˆVxt\n+p′′(ˆV+φx)φxxˆVt+(p′′(ˆV+φx)−p′′(ˆV))ˆVtˆVx}2dx\n≤Cδ1/bardblφtt(t)/bardbl2+Cδ1{(1+t)−3/bardblφx(t)/bardbl2+(1+t)−2/bardblφxx(t)/bardbl2}.\nHere, by (4.1)\nφxx= (−p′(ˆV+φx))−1(φtt+αφt+(p′(ˆV+φx)−p′(ˆV))ˆVx−G)\nand hence\n(4.14) (1+ t)−2/bardblφxx(t)/bardbl2≤C(1+t)−2(/bardbl(φt,φtt)(t)/bardbl2+/bardblG/bardbl2)+Cδ1(1+t)−3/bardblφx(t)/bardbl2.\nSince\n(4.15) /bardblG/bardbl2≤C(/bardblUt/bardbl2+|U|2\n∞/bardblα−α/bardbl2+|u+−u−|e−ct)≤Cδ1(1+t)−1\n16and also\n|/integraldisplay\nφttGtdx| ≤ε/bardblφtt(t)/bardbl2+Cε/integraldisplay\nG2\ntdx≤ε/bardblφtt(t)/bardbl2+Cδ1(1+t)−3,\nall bad terms are absorbed, and integration of (4.13) over [0 ,t] yields\n(4.16) /bardbl(φtt,φtx)(t)/bardbl2+/integraldisplayt\n0/bardblφtt(τ)/bardbl2dτ≤CI0+C(δ1+δ)/integraldisplayt\n0/bardblφxt(τ)/bardbl2dτ.\nInserting (4.12) to (4.16), we have\n(4.17) /bardbl(φtt,φtx)(t)/bardbl2+/integraldisplayt\n0/bardbl(φtt,φtx(τ)/bardbl2dτ≤CI0,\nprovided that δ1+δis suitably small.\nThirdly, based on (4.17), calculate (1+ t)·(4.10):\n(4.18)d\ndt(1+t)/integraltext\n(φttφt+α\n2φ2\nt)dx+(1+t)/integraltext\n|p′(ˆV+φx)|φ2\nxtdx\n≤/integraltext\n(φttφt+α\n2φ2\nt)dx+(1+t)/integraltext\n(φ2\ntt+(p′(ˆV+φx)−p′(ˆV))ˆVtφtx)dx\n+(1+t)/integraltext\nφtGtdx\n≤C/integraltext\nφ2\ntdx+(1+t)/integraltext\n(φ2\ntt+εφ2\ntx)dx\n+Cε(1+t)|ˆV|2\n∞/integraltext\nφ2\nxdx+C(1+t)1−γ/bardblφt(t)/bardbl2+C(1+t)1+γ/bardblGt/bardbl2\nand\n(1+t)/integraldisplay\nφttφtdx≤ε(1+t)/integraldisplay\nφ2\ntdx+Cε(1+t)/integraldisplay\nφ2\nttdx.\nHence, integrating (4.18) over [0 ,t] and using (4.8), (4.11), we have\n(4.19)(1+t)/bardblφt(t)/bardbl2+/integraltextt\n0(1+τ)/bardblφxt(τ)/bardbl2dτ\n≤CI0+C(1+t)/bardblφtt(t)/bardbl2+C/integraltextt\n0(1+τ)/bardblφtt(τ)/bardbl2dτ.\nFinally, multiplication of (4.13) by 1+ tand integration over [0 ,t] yield\n(1+t)/bardbl(φtt,φtx)(t)/bardbl2+/integraltextt\n0(1+τ)/bardblφtt(τ)/bardbl2dτ\n≤C/bardbl(φtt,φtx)(t)/bardbl2+C(δ1+δ)/integraltextt\n0(1+τ)/bardblφtx(τ)/bardbl2dτ\n+Cδ1/integraltextt\n0((1+τ)/bardblφtt(τ)/bardbl2+(1+τ)−2/bardblφx(τ)/bardbl2+(1+τ)−1/bardblφxx(τ)/bardbl2)dτ.\nBy (4.14),\n/bardblφxx(t)/bardbl2≤C/bardbl(φt,φtt)(t)/bardbl2+Cδ1(1+t)−1/bardblφx(t)/bardbl2+Cδ1(1+t)−1.\nTherefore, by (4.17) we have\n(4.20)(1+t)/bardbl(φtt,φtx)(t)/bardbl2+/integraltextt\n0(1+τ)/bardblφtt(τ)/bardbl2dτ\n≤CI0+C(δ1+δ)/integraltextt\n0(1+τ)/bardblφtx(τ)/bardbl2dτ.\nSubstituting (4.20) into (4.19), we obtain\n(4.21) (1+ t)/bardbl(φt,φtt,φtx)(t)/bardbl2+/integraldisplayt\n0(1+τ)/bardbl(φtt,φtx)(τ)/bardbl2dτ≤CI0,\n17provided that δ1+δis small.\nAdditionally, we have\n(4.22) (1+ t)/bardblφxx(t)/bardbl2≤CI0,\nbecause, by (4.14) and (4.15) with (4.21),\n(1+t)/bardblφxx(t)/bardbl2≤C(1+t)/bardbl(φt,φtt)(t)/bardbl2+Cδ1/bardblφx(t)/bardbl2+Cδ1.\nStep 4.In the final step we estimate the third order derivatives of φ. To do so, differentiate (4.1)\ninxandx,t:\n(4.23)φttx+αxφt+αφtx+(p′(ˆV+φx)φxx)x\n=−((p′(ˆV+φx)−p′(ˆV))x+Gx,\nand\n(4.24)φtttx+αxφtt+αφttx+(p′(ˆV+φx)φtxx)x\n=−(p′′(ˆV+φx)(ˆVt+φtx)φxx)x−((p′(ˆV+φx))ˆVx)tx+Gtx\n=:H+Gtx=:h1+h2+Gtx.\nFirst, multiplying (4.24) by φttxand integrating it over R, we have\nd\ndt/integraltext\n(1\n2φ2\nttx+|p′(ˆV+φx)|\n2φ2\ntxx)dx+/integraltext\nαφ2\nttxdx\n≤ε/integraltext\nφ2\nttxdx+Cε/integraltext\nφ2\nttdx+C|ˆVt+φtx|∞/integraltext\nφ2\ntxxdx+Cε/integraltext\n(H2+G2\ntx)dx,\nwhich derives\n(4.25)d\ndt/integraltext\n(1\n2φ2\nttx+|p′(ˆV+φx)|\n2φ2\ntxx)dx+α0/integraltext\nφ2\nttxdx\n≤C{(δ1+δ)/integraltext\nφ2\ntxxdx+/integraltext\nφ2\nttdx+/bardblH/bardbl2+/bardblGtx/bardbl2},\nsince\n(4.26) |ˆVt+φtx|∞≤C(δ1+/radicalbig\nI0δ)(1+t)−1≤C(δ1+δ)(1+t)−1\nby (2.2.13), (4.21) and ε≪1. Next, multiplying (4.24) by φtx, similarly as above, we have\n(4.27)d\ndt/integraltext\n(φttxφtx+α\n2φ2\ntx)dx−/integraltext\nφ2\nttxdx+/integraltext\n|p′(ˆV+φx)|φ2\ntxxdx\n≤C|αx|∞(/bardblφtt/bardbl2+/bardblφtx/bardbl2)+C(/bardblφtx/bardbl2+/bardblH/bardbl2+/bardblGtx/bardbl2)\n≤C{/bardbl(φtt,φtx)(t)/bardbl2+/bardblH/bardbl2+/bardblGtx/bardbl2}.\nFor small λ >0, add (4.25) to λ·(4.27), and then\n(4.28)d\ndt/integraltext\n{(1\n2φ2\nttx+λφttxφtx+αλ\n2φ2\ntx)+|p′(ˆV+φx)|\n2φ2\ntxx}dx\n+/integraltext\n(α0\n2φ2\nttx+λ\n2|p′(ˆV+φx)|φ2\ntxx)dx\n≤C(/bardbl(φtt,φtx)(t)/bardbl2+/bardblH/bardbl2+/bardblGtx/bardbl2)\nprovided that δ1+δ≤λ\n2|p′(ˆV+φx)|. Here,/bardblH/bardbl2is estimated as the following.\n18Lemma 4.1\n(4.29)/bardblH/bardbl2≤C{(δ1+δ)(1+t)−1/bardbl(φttx,φtxx,φtx,φt)(t)/bardbl2\n+(δ1+δ)(1+t)−2/bardblφxx(t)/bardbl2+δ1(1+t)−3/bardblφx(t)/bardbl2+δ1(1+t)−2}.\nProof of Lemma 4.1 . First, in (4.24)\nh1=−p′′(ˆV+φx)(ˆVt+φtx)φxxx−p′′(ˆV+φx)(ˆVtx+φtxx)φxx\n−p′′′(ˆV+φx)(ˆVx+φxx)(ˆVt+φtx)φxx\n=:h11+h12+h13,\nwith\n/bardblh11/bardbl2≤C(δ1+δ)(1+t)−1/bardblφxxx/bardbl2by (4.26)\n/bardblh12/bardbl2≤C(|ˆVtx|2\n∞/bardblφxx/bardbl2+/bardblφxx/bardbl/bardblφxxx/bardbl/bardblφtxx/bardbl2)\n≤Cδ1(1+t)−3/bardblφxx/bardbl2+(I0+δ)(1+t)−1/bardblφtxx/bardbl2,\n/bardblh13/bardbl2≤C(|ˆVx|2\n∞+/bardblφxx/bardbl/bardblφxxx/bardbl)(|ˆVt|2\n∞+/bardblφtx/bardbl/bardblφtxx/bardbl)/bardblφxx/bardbl2\n≤C(δ1+δ)(1+t)−2/bardblφxx/bardbl2.\nFor the estimate of /bardblφxxx/bardbl2in/bardblh11/bardbl2we back to (4.23):\n−p′(ˆV+φx)φxxx=p′′(ˆV+φx)(ˆVx+φxx)φxx+((p′(ˆV+φx)−p′(ˆV))ˆVx)x\n+φttx+αxφt+αφtx−Gx,\nand hence\n(4.30)/bardblφxxx/bardbl2≤C{(|ˆVx|2\n∞+/bardblφxx/bardbl/bardblφxxx/bardbl)/bardblφxx/bardbl2+(|ˆVxx|2\n∞+|ˆVx|4\n∞)/bardblφx/bardbl2\n+|ˆVx|2\n∞/bardblφxx/bardbl2+/bardblφttx,φtx,φt/bardbl2+/bardblGx/bardbl2}\n≤C{(I0+δ)(1+t)−1/bardblφxx/bardbl2+δ1(1+t)−2/bardblφx/bardbl2\n+/bardblφttx,φtx,φt/bardbl2+δ1(1+t)−1}.\nHere,/bardblGx/bardbl2≤Cδ1(1+t)−1is easily seen. In a similar fashion to the above, we have\n/bardblh2/bardbl2≤Cδ1{(1+t)−1/bardblφtxx/bardbl2+(1+t)−2/bardblφtx,φxx/bardbl2+(1+t)−4/bardblφx/bardbl2}.\nCombining /bardblh1/bardbl2,/bardblh2/bardbl2with (4.30), we get (4.29) and complete the proof of Lemma 4.1 .q.e.d.\nWe also note that\n(4.31) /bardblGtx/bardbl2≤Cδ1(1+t)−3.\nWe now return back to (4.28). Take δ1+δas sufficiently small, then the term φtxxandφttxin\n(4.29) are absorbed into the left hand side, and integration of (4.28) over [0 ,t] yields\n(4.32) /bardbl(φttx,φtxx,φtx)(t)/bardbl2+/integraldisplayt\n0/bardbl(φttx,φtxx)(τ)/bardbl2dτ≤CI0,\nbecause of (4.8), (4.21)-(4.22).\n19To get further decay rate, we want to multiply (4.28) by 1+ t, but, if so, the final term in (4.29)\nis not integrable in t. So, we here use the technique found in Nishikawa [11], that i s, multiply (4.28)\nby (1+t)1+ν(ν >0,notν <0), so that, by (4.32),\n(4.33)(1+t)1+ν/bardbl(φttx,φtxx,φtx)(t)/bardbl2+/integraltextt\n0(1+τ)1+ν/bardbl(φttx,φtxx)(τ)/bardbl2dτ\n≤C{(1+ν)/integraltextt\n0(1+τ)ν/bardbl(φttx,φtxx,φtx)(τ)/bardbl2dτ+/integraltextt\n0(1+τ)1+ν/bardbl(φtt,φtx)(τ)/bardbl2dτ\n+/integraltextt\n0(1+τ)1+ν/bardblH/bardbl2dτ}+Cδ1\nν(1+t)ν.\nDivide (4.33) by (1+ t)ν, and use1+τ\n1+t≤1 and (4.32) just obtained, then we get desired estimate\n(4.34)(1+t)/bardbl(φttx,φtxx,φtx)(t)/bardbl2+/integraltextt\n0/bardbl(φttx,φtxx)(τ)/bardbl2dτ\n≤C{/integraltextt\n0/bardbl(φttx,φtxx,φtx)(τ)/bardbl2dτ+/integraltextt\n0(1+τ)/bardbl(φtt,φtx)(τ)/bardbl2dτ+δ1}\n≤CI0.\nHere, note that, although(1+τ)1+ν\n(1+t)νcomes out in the second term in (4.34), it only holds that\n(1+τ)1+ν\n(1+t)ν≥0(0≤τ≤t).\nAdditionally, multiplying (4.30) by 1+ t, we get\n(4.35) (1+ t)/bardblφxxx(t)/bardbl2≤CI0.\nThus, we have obtained the estimate (2.2.12) and completed t he proof of Proposition 2.2.1.\nAppendix. We prove Lemma 3.1. By\n(A1) V−v=δ0/radicalbig\n4πµ(1+t)e−x2\n4µ(1+t),\n(A2) U=Vx=−2πδ0x\n(4πµ(1+t))3/2e−x2\n4µ(1+t),\nand (2.1.6), we easily know\n(A3)|∂k\nx(V−v)|p≤Cδ0(1+t)−1\n2(1−1\np)−k\n2,\n|∂k\nxU|p=|∂k\nxVx|p≤Cδ0(1+t)−1\n2(1−1\np)−k+1\n2\nand\n(A4) |∂k\nxˆv|p+|∂l\ntˆv|p≤C|u+−u−|e−ct≤Cδ1e−ct(0< c < α 0).\nRewrite (3.2):\n(3.2) F=−{Ut+(α−α)U+(p(V)−p′(v)V)x+(p(V+ ˆv)−p(V))x}=:4/summationdisplay\ni=1Fi.\n20By (A3)-(A4),\n/bardblF1(t)/bardbl2≤ /bardblUt/bardbl2=/bardblVxxx/bardbl2≤Cδ0(1+t)−7/2,\n/bardblF3(t)/bardbl2≤/integraltext\n|p′(V)−p′(v)|2|Vx|2dx≤C|Vx|2\n∞/bardblV−v/bardbl2≤Cδ0(1+t)−5/2,\n/bardblF4(t)/bardbl2≤Cδ1e−ct.\nForF2, by (A2)\n/bardblF2(t)/bardbl2≤/integraltext\n|α−α|2|Vx|2dx≤Cδ0/integraltext|α−α|2|x|\n(1+t)3/2dx·|Vx|∞\n≤Cδ0(1+t)−5/2if|x|1/2(α−α)∈L2.\nHence we have /bardblF(t)/bardbl2≤Cδ1(1+t)−5/2. Next,\n/bardblF1x(t)/bardbl2=/bardblUtx/bardbl2≤Cδ0(1+t)−9/2,\n/bardblF2x(t)/bardbl2≤C/integraltext\n(|αxU|2+|α−α|2U2\nx)dx\n≤Cδ0(1+t)−3(/bardblx·αx/bardbl2+/bardblα−α/bardbl2)≤Cδ0(1+t)−3,\n/bardblF3x(t)/bardbl2≤/integraltext\n(p′′(V)V2\nx+(p′(V)−p′(v))Vxx)2dx≤Cδ0(1+t)−7/2,\n/bardblF4x(t)/bardbl2≤Cδ1e−ct,\nand hence /bardblFx(t)/bardbl2≤Cδ1(1+t)−3. ForFt,\n/bardblF1t(t)/bardbl2=/bardblUtt/bardbl2=/bardblVxtt/bardbl2≤Cδ0(1+t)−11/2,\n/bardblF3t(t)/bardbl2=/bardbl(p′(V)−p′(v))Vtx+p′′(V)VxVt/bardbl2\n≤Cδ0((1+t)−1/2−4+(1+t)−3/2−3) =Cδ0(1+t)−9/2,\n/bardblF4t(t)/bardbl2≤Cδ1e−ct.\nForF2t,\n/bardblF2t(t)/bardbl2=/bardbl(α−α)Ut/bardbl2≤ /bardbl(α−α)Vxxx/bardbl2≤Cδ0(1+t)−5/bardblx·(α−α)/bardbl2\n≤Cδ0(1+t)−5ifx·(α−α)∈L2,\nbecause of Vxxx=Cδ0x·((1+t)−5/2+Cx2(1+t)−7/2)e−x2\n4µ(1+t). Hence/bardblFt(t)/bardbl2≤Cδ1(1+t)−9/2.\nThe estimate of /bardblFtx(t)/bardbl2is done samely as above and omitted.\nAcknowledgment .The authors would like to thank Professor Ming Mei who shortl y visited\nTokyo Institute of Technology in 2022-23. Through discussi ons with him they had a motive to\nconsider the present problem.\nReferences\n[1] S. Chen, H. Li, J. Li, M. Mei and K. Zhang, Global and blow-u p solutions for compressible\nEulerequationswithtime-dependentdamping, J.Differentia l Equations268(2020), 5035-5077.\n21[2] C.J. van Duyn and L.A. Peletier, A class of similarity sol utions of the nonlinear diffusion\nequation, Nonlinear Anal. TMA 1 (1977), 223-233.\n[3] L. Hsiao and T.-P. Liu, Convergence to nonlinear diffusion waves for solutions of a system of\nhyperbolic conservation laws with damping, Commun. Math Ph ys. 143 (1992), 599-605.\n[4] F.M. Huang, P. Marcati and R.H. Pan, Convergence to the Ba renblatt solution for the com-\npressible Euler equation with damping and vacuum, Arch. Rat ion. Mech. Anal. 176 (2005),\n599-605.\n[5] S.JiandM. Mei, Optimaldecay rates ofthecompressibleE ulerequations withtime-dependent\ndamping in Rn: (I) Under-damping case, J. Nonlinear Sci. 33 (2023), 1-47.\n[6] P. Marcati, M. Mei and B. Rubino, Optimal convergence rat es to diffusion waves for solutions\nof the hyperbolic conservation laws with damping. J. Math. F luid Mech.7 (2005)m 224-240.\n[7] A. Matsumura, Global existence and asymptotics of the so lutions of the second-order quasilin-\near hyperbolic equations with the first-order dissipation, Publ. RIMS, Kyoto Univ. 13 (1977),\n349-379.\n[8] M. Mei, Best asymptotic profile for hyperbolic p-system w ith damping, SIAM J Math. Anal.\n42 (2010), 1-23.\n[9] K. Nishihara, Convergence rates to nonlinear diffusion wa ves for solutions of system of hyper-\nbolic conservation laws with damping, J. Differential Equati ons 131 (1996), 171-188.\n[10] K. Nishihara, W.-K. Wang and T. Yang, Lp-convergence rates to nonlinear diffusion waves for\np-system with damping, J. Differential Equations 131 (2000), 1 91-218.\n[11] M. Nishikawa, Convergence rate to the traveling wave fo r viscous conservation laws, Funkcial.\nEkvac. 41 (1998), 107-132.\n[12] Y. Sugiyama, Blow-up for the 1D compressible Euler equa tions with separated time and space\ndependent damping coefficient, preprint. (arXiv:2212.1107 2v1)\n[13] N. Zhang, Optimal convergence rates to diffusion waves fo r solutions of p-system with damping\non quadrant, J. Math. Anal. Appl. 512 (2022), #126118.\n[14] N. Zhang and C. Zhu, Long-time behavior of solutions to t heM1model with boundary effect,\nDisc. Cont. Dyn. Sys. 43 (2023), 1824-1859.\n[15] H. Zhao, Convergence to strong nonlinear diffusion waves for solutions of p-system with damp-\ning, J. Differential Equations 174 (2001), 200-236.\n22"    },    {        "title": "2004.08082v1.Collective_coordinate_study_of_spin_wave_emission_from_dynamic_domain_wall.pdf",        "content": "arXiv:2004.08082v1  [cond-mat.mes-hall]  17 Apr 2020Collective coordinate study of spin wave emission from dyna mic\ndomain wall\nGen Tatara\nRIKEN Center for Emergent Matter Science (CEMS)\n2-1 Hirosawa, Wako, Saitama, 351-0198 Japan\nRubn M. Otxoa de Zuazola\nHitachi Cambridge Laboratory, J. J. Thomson Avenue,\nCB3 OHE, Cambridge, United Kingdom and\nDonostia International Physics Center, 20018 San Sebasti´ an, Spain\n(Dated: April 20, 2020)\nAbstract\nWe study theoretically the spin wave emission from a moving d omain wall in a ferromagnet.\nIntroducing a deformation mode describing a modulation of t he wall thickness in the collective\ncoordinate description, we show that thickness variation c ouples to the spin wave linearly and\ninducesspinwave emission. Thedominant emitted spinwave t urnsout tobepolarized in theout-of\nwall plane ( φ)-direction. The emission contributes to the Gilbert dampi ng parameter proportional\nto/planckover2pi1ωφ/K, the ratio of the angular frequency ωφofφand the easy-axis anisotropy energy K.\n1I. INTRODUCTION\nSpin wave (magnon) is an excitation playing essential roles in the tran sport phenomena\nin magnets, and its control, magnonics, is a hot recent issue. Beside s application interest\nfor devices, behaviours of spin waves have been drawing interests from fundamental science\nview points. Many theoretical studies have been carried out on gen eration of spin waves\nby dynamic magnetic objects such as a domain wall [1–6]. The subject is highly nontrivial\nbecause the wall is a soliton, which is stable in the absence of perturb ation, meaning that\nit couples to fluctuations, spin waves, only weakly in the ideal case, w hile in reality, various\nperturbations and dynamics leads to strong emission of spin waves. There are several pro-\ncesses that lead to the emission, and it is not obvious which is the domin ant process and\nhow large is the dissipation caused by the emission.\nThe low energy behavior of a domain wall in a ferromagnet is described in terms of\ncollective coordinates, its center of mass position Xand angle of the wall plane, φ0[7]. In\nthe absence of a pinning potential, a displacement of the wall costs n o energy owing to the\ntranslational invariance, and it is thus natural to regard Xas a dynamic variable X(t).\nThis is in fact justified mathematically; X(t) is a collection of spin waves that corresponds\nto the translational motion of the wall [8, 9]. It turns out that the c anonical momentum of\nthe ferromagnetic domain wall is the angle φ0. This is because the translational motion of\ncollective spins requires a perpendicular spin polarization, i.e., a tilting o f the wall plane.\nMathematically this is a direct consequence of the spin algebra, and is straightforwardly\nderived based on the equation of motion for spin (Landau-Lifshitz( -Gilbert) equation) [7] or\non the Lagrangian formalism [10]. In the absence of hard-axis anisot ropy energy, φ0is also\na zero mode. As zero modes, X(t) andφ0do not have linear coupling to the fluctuation,\nspin wave, and thus emission of spin wave does not occur to the lowes t order. In this case,\nthe second-order interactions to the spin wave give rise to the dom inant effect. In Ref. [1],\nthe coupled equations of motion for the wall and spin wave modes wer e solved classically\nand demonstrated that a damping indeed arises from the quadratic interaction. In the case\nof a strong hard-axis anisotropy, the plane of the wall is constrain ed near the easy-plane,\nφ0is frozen, resulting in a single variable system described solely by X(t) [9, 11]. The spin\nwave coupling and dissipation in this limit was discussed in Ref. [11].\nIn real materials, hard-axis anisotropy and pinning potential exist , andX(t) andφ0\n2are not rigorously zero modes. In other words, wall dynamics induc es a deformation and\nemission of spin wave is possible due to linear couplings. It was argued in Ref. [2] that there\nemerges a linear coupling when the wall driven by a spin-transfer tor que has a velocity ˙X\ndifferent from the steady velocity determined by the spin-transfe r torque, and the damping\ndue to spin wave emission was discussed. Numerical analysis of Ref. [3 ] revealed that spin\nwave emission occurs by the modulation of the wall thickness during t he dynamics. The\ncoupling to the wall velocity and second order in the spin wave was stu died analytically in\ndetail and dissipation was estimated in Ref. [6]. The energy dissipation proportional to the\nsecond-order in the wall velocity was found.\nIn this paper, we study the spin wave emission extending convention al collective coor-\ndinate representation of the wall [10]. As the domain wall is a soliton, t here is no linear\ncoupling of its center of mass motion to the spin wave field if deformat ion is ignored. We\nthus introduce a deformation mode of the wall, a change of the thick nessλ. This is a natural\nvariable in the presence of the hard-axis anisotropy energy, as th e thickness depends on the\nangleφ0as pointed out in Refs. [12, 13]. Following the prescription of spin wave expansion\n[9], we derive the Lagrangian for the three collective coordinates, t he center of mass position\nX(t), the angle of the wall plane φ0(t) and thickness λ(t), including the spin waves to the\nsecond order. It turns out that Xandφ0and their time-derivatives do not have linear\ncoupling to the spin wave, while ˙λdoes. This result is natural as Xandφ0are (quasi) zero\nmodes, and consistent with numerical observation [3]. It is shown th at the emitted spin\nwave is highly polarized; The dominant emission is the fluctuation of ang leφ, while that\nofθis smaller by the order of the Gilbert damping parameter α. The forward emission of\nwavelength λ∗∝v−1\nw, wherevwis the domain wall velocity, is dominant. The modulation of\nλis induced by the dynamics of φ0, and the contribution to the Gilbert damping parameter\ndue to the spin wave emission from this process is estimated from the energy dissipation\nrate. It was found to be of the order of αφ\nsw≃λ\na/planckover2pi1ωφ\nK, whereωφis the angular frequency of\nthe modulation of φ0,Kis the easy-axis anisotropy energy and ais the lattice constant.\nThis damping parameter contribution becomes very strong of the o rder of unity if /planckover2pi1ωφis\ncomparable to the spin wave gap, K, as deformation of the wall becomes significant in this\nregime.\n3II. COLLECTIVE COORDINATES FOR A DOMAIN WALL\nWe consider a one-dimensional ferromagnet along the x-axis with easy and hard axis\nanisotropy energy along the zandyaxis, respectively. The Lagrangian in terms of polar\ncoordinates ( θ,φ) of spin is\nL=LB−HS (1)\nwhere\nLB=/planckover2pi1S\na/integraldisplay\ndx˙φ(cosθ−1)\nHS=S2\n2a/integraldisplay\ndx/bracketleftbig\nJ[(∇θ)2+sin2θ(∇φ)2]+Ksin2θ(1+κsin2φ)/bracketrightbig\n(2)\nare the kinetic term of the spin (spin Berry phase term) and the Ham iltonian, respectively,\nJ >0,K >0andκK≥0being theexchange, easy-axis anisotropyandhardaxisanisotro py\nenergies, respectively, abeing the lattice constant. A static domain wall solution of this\nsystem is\ncosθ= tanhx−X\nλ0,φ= 0 (3)\nwhereλ0≡/radicalbig\nJ/Kis the wall thickness at rest. The dynamics of the wall is described\nby allowing the wall position Xandφas dynamic variables. This corresponds to treat\nthe energy zero mode of spin waves (zero mode) describing a trans lational motion and its\nconjugate variable φas collective coordinates [9]. This treatment is rigorous in the absenc e\nof pinning and hard-axis anisotropy but is an approximation otherwis e. Most previous\nstudies considered a rigid wall, where the wall thickness is a constant λ0. Here we treat\nthe wall thickness as a dynamic variable λ(t) to include a deformation and study the spin-\nwave emission. This treatment was applied in Ref. [13], but only static s olution of λwas\ndiscussed.\nAs demonstrated in Ref. [9], the spin wave around a domain wall in ferr omagnet is\nconveniently represented using\nξ=e−u(x,t)+iφ0(t)+η(x−X(t),t)(4)\nwhereφ0(t) is the angle of the wall,\nu(x,t) =x−X(t)\nλ(t)(5)\n4θ\nφ\nFIG. 1. Fluctuation corresponding to the real and imaginary part of the spin wave variable\n˜η= ˜ηR+i˜ηI. (a): The profile of ˜ ηantisymmetric with respect to the wall center, which turns\nout to be dominant excitation. (b): The real part ˜ ηRdescribes the deformation within the wall\nplane, i.e., modulation of θ, while the imaginary part ˜ ηIdescribes the out-of plane ( φ) fluctuation\nas shown in (c). Transparent arrows denotes the equilibrium spin configuration.\nandη(x−X,t) describes thespin-wave viewed inthemoving frame. Asitis obvious f romthe\ndefinition, the real and imaginary part of ηdescribe the fluctuation of θandφ, respectively.\nThe fluctuations antisymmetric with respect to the wall center, sh own in Fig. 1, turns out\nto be dominant. The ξ-representation of the polar angles are\ncosθ=1−|ξ|2\n1+|ξ|2, sinθsinφ=−iξ−ξ\n1+|ξ|2. (6)\nA. Domain wall dynamic variables\nWe first study what spin-wave mode the new variable λ(t) couples to, by investigating\nthe ’kinetic’ term of the spin Lagrangian, LB, which is written as\nLB=2i/planckover2pi1Sλ\na/integraldisplay\nduIm[ξ˙ξ]\n1+|ξ|2. (7)\nUsing Eq. (6) and\n∂tu=−1\nλ/parenleftBig\n˙X+u˙λ/parenrightBig\n, ∂ tξ=/parenleftbigg1\nλ/parenleftBig\n˙X+u˙λ/parenrightBig\n+i˙φ0+(∂t−˙X∇x)η/parenrightbigg\nξ,(8)\nwe have\n2iIm[ξ˙ξ] = 2i( ˙ηI+˙φ0−˙X∇xηI)|ξ|2(9)\n5sδ\nFIG. 2. Schematic figure showing the effect of asymmetric perpe ndicular spin polarization δs\ndue to the spin wave mode ϕ. The asymmetric torque (curved arrows) induced by asymmetr icδs\nrotates the spins within the wall plane, resulting in a compr ession of the wall, i.e., to ˙λ.\nwhereηi≡Im[η]. The kinetic term is expanded to the second order in the spin wave as\n(using integral by parts)\nLB=2/planckover2pi1S\na[φ0˙X+ϕ˙λ]+L(2)\nB (10)\nwhere\nϕ≡/integraldisplay\nduu\ncoshu˜ηI, (11)\nrepresents an asymmetric configuration of ˜ ηIand\nL(2)\nB≡2/planckover2pi1Sλ\na/integraldisplay\ndu/bracketleftbigg\n˜ηR↔\n∂t˜ηI−˙X˜ηR↔\n∇x˜ηI−2\nλtanhu/parenleftBig\n2˙X+u˙λ/parenrightBig\n˜ηR˜ηI/bracketrightbigg\n,(12)\nwhere ˜η≡η/(2coshu).\nWhen deriving Eq. (10), the orthogonality of fluctuation and the ze ro-mode,\n/integraldisplay\ndu˜η\ncoshu= 0, (13)\nwas used. Equation (10) indicates that ϕis the canonical momentum of λ. In fact, it\nrepresents the asymmetric deformation of angle φ, as the imaginary part of the spin wave,\n˜ηI, corresponds to fluctuation of φas seen in the definition, Eq. (4). Such an asymmetric\nconfiguration of φexerts a torque that induces a compression or expansion of the do main\nwall (Fig. 2), andthisiswhy ϕandλareconjugatetoeach other. Thecoupling ϕ˙λdescribes\nthe spin wave emission when thickness changes, as we shall argue lat er. The second term\nproportional to ˙Xin the bracket in Eq. (12) represents a magnon current induced in t he\nmoving frame (Doppler shift).\n6The Hamiltonian of the system is similarly written in terms of spin wave va riables to the\nsecond order as\nHS=KS2λ\na/bracketleftBigg/parenleftbiggλ0\nλ/parenrightbigg2\n+1+κsin2φ0/bracketrightBigg\n+2KS2λ\na/integraldisplay\ndutanhu\ncoshu˜ηR/bracketleftBigg\n−/parenleftbiggλ0\nλ/parenrightbigg2\n+1+κsin2φ0/bracketrightBigg\n+H(2)\nS,\n(14)\nwhere\nH(2)\nS≡2KS2λ\na/integraldisplay\ndu/bracketleftbigg\nλ2\n0[(∇˜ηR)2+(∇˜ηI)2]\n+ ˜ηR2/bracketleftbigg\n−λ2\n0\nλ2/parenleftbigg\n1−1\ncosh2u/parenrightbigg\n+/parenleftbigg\n2−3\ncosh2u/parenrightbigg\n(1+κsin2φ0)/bracketrightbigg\n+ ˜ηI2/bracketleftbiggλ2\n0\nλ2/parenleftbigg\n1−2\ncosh2u/parenrightbigg\n+κcos2φ0/bracketrightbigg\n+2κ˜ηR˜ηItanhusin2φ0/bracketrightbigg\n(15)\nIn the case of small κandλ≃λ0, the spin waves are described by a simple Hamiltonian as\nHsw≡2KS2λ\na/integraldisplay\ndu/bracketleftbigg\nλ2[(∇˜ηR)2+(∇˜ηI)2]+( ˜ηR2+ ˜ηI2)/parenleftbigg\n1−2\ncosh2u/parenrightbigg/bracketrightbigg\n+HD,(16)\nwhere\nHD≡2/planckover2pi1Sλ\na˙X/integraldisplay\ndu˜ηR↔\n∇x˜ηI, (17)\nis the Doppler shift term. For a constant wall velocity ˙X, it simply shifts the wave vector\nof the spin wave. Without the Doppler shift, the eigenfunction of th is Hamiltonian (16) is\nlabeled by a wave vector kas\nφk(u) =1√2π˜ωk(−ikλ+tanhu)eikλu, (18)\nwhere\n˜ωk≡1+(kλ)2(19)\nis the dimensionless energy of spin wave.\nDissipation function is\nW=α/planckover2pi1S\n2a/integraldisplay\ndx(˙θ2+sin2θ˙φ2)\n=/planckover2pi1Sλ\n2a\nα/parenleftBigg˙X\nλ/parenrightBigg2\n+α˙φ02+αλ/parenleftBigg˙λ\nλ/parenrightBigg2\n, (20)\n7whereαis the Gilbert damping parameter and αλ≡α/integraltext\nduu2\ncosh2u=π2\n12α.\nAsdrivingmechanisms ofadomainwall, weconsider amagneticfieldandc urrent-induced\ntorque (spin-transfer torque) [9, 14, 15]. A magnetic field applied a long the negative easy\naxis is represented by the Hamiltonian ( γ=e/mis the gyromagnetic ratio)\nHB=/planckover2pi1Sγ\naBz/integraldisplay\ndxcosθ. (21)\nUsing Eqs. (6)(13), we obtain\nHB=−2/planckover2pi1Sγ\naBz/parenleftbigg\nX+λ/integraldisplay\ndutanhu˜η2\nR/parenrightbigg\n. (22)\n(The first term is derived evaluating a diverging integral/integraltext\ndx1\n1+e2u(x)carefully introducing\nthe system size Las/integraltextL/2\n−L/2dx1\n1+e2u(x)and dropping a constant.) The magnetic field therefore\nexerts a force2/planckover2pi1Sγ\naBzon the domain wall.\nThe spin-transfer effect induced by injecting spin-polarized electr ic current is represented\nby a Hamiltonian having the same structure as the spin Berry’s phase termLB[9, 15]\nHSTT=−/planckover2pi1S\navst/integraldisplay\ndxcosθ(∇xφ), (23)\nwherevst≡aP\n2eSjis a steady velocity of magnetization structure under spin polarized current\nPj(Pis the spin polarization and jis the applied current density (one-dimensional)). The\nspin wave expression is\nHSTT=2/planckover2pi1S\navst/bracketleftbigg\nφ0+2/integraldisplay\ndx/parenleftbigg\n˜ηR∇x˜ηI+1\nλtanhu˜ηR˜ηI/parenrightbigg/bracketrightbigg\n. (24)\nAs has been known, a spin-transfer torque contributing to the wa ll velocity and does not\nwork as a force, as the applied current or vstcouples to φ0and not to X.\nThe equation of motion for the tree domain wall variables is therefor e obtained from Eqs.\n(10) (14) (20) and driving terms (22)(24) as\n˙X−αλ˙φ0=vcsin2φ0+2vcsin2φ0ζ+vst\n˙φ0+α˙X\nλ=˜Bz\nαλ˙λ\nλ=KS\n/planckover2pi1/bracketleftBigg/parenleftbiggλ0\nλ/parenrightbigg2\n−(1+κsin2φ0)/bracketrightBigg\n−˙ϕ−2KS\n/planckover2pi1/bracketleftBigg/parenleftbiggλ0\nλ/parenrightbigg2\n+(1+κsin2φ0)/bracketrightBigg\nζ,\n(25)\n8wherevc≡KSκ\n2/planckover2pi1λ,˜Bz≡γBzandϕ(Eq. (11)) and\nζ≡/integraldisplay\ndutanhu\ncoshu˜ηR, (26)\nare contributions linear in spin wave.\nIII. SPIN WAVE EMISSION\nIn this section we study the spin wave emission due to domain wall dyna mics. The\nemission is described by the linear coupling between the spin wave field a nd the domain wall\nin Eqs.(10) (14). Moreover, dynamic second-order couplings in Eqs . (12)(15) leads to spin\nwave excitation. In the first linear process, the momentum and ene rgy of the spin wave is\nsupplied by the dynamic domain wall, while the second process present s a scattering of spin\nwaves where the domain wall transfer momentum and energy to the incident spin wave.\nA. Linear emission\nWe here discuss the emission due to the linear interactions in Eqs.(10) (14) in the labo-\nratory (rest) frame. The laboratory frame is described by replac ingη(x−X(t),t) byη(x,t)\nin the derivation in Sec. IIA. It turns out that the Lagrangian Eq.(1 2) in the laboratory\nframe has no Doppler shift term and the term ˙X˜ηR˜ηIis half. The emitted wave has an\nangular frequency shifted by the Doppler shift from the moving wall. Using the equation of\nmotion, Eq. (25), the spin wave emission arises from the thickness c hange. The interaction\nHamiltonian reads in the complex notation ˜ η= ˜ηR+i˜ηI\nH(1)\nη(t) =˙λ(t)/integraldisplay\ndx(g˜η+g˜η), (27)\nwhere\ng(x)≡2/planckover2pi1S\na1\ncoshx−X(t)\nλ/parenleftbigg\n−αλtanhx−X(t)\nλ+ix−X(t)\nλ/parenrightbigg\n(28)\nLet us study here the emission treating λas a constant as its dynamics is taken account in\nthe first factor in the interaction Hamiltonian (27). The Fourier tra nsform of the interaction\n9is calculated using\n/integraldisplay∞\n−∞duei˜kuu\ncoshu=iπ2\n2sinhπ\n2˜k\ncosh2π\n2˜k\n/integraldisplay∞\n−∞duei˜kutanhu\ncoshu=π˜k\ncoshπ\n2˜k(29)\nas\nH(1)\nη(t) =−π2\n2λ˙λ(t)/summationdisplay\nk1\ncoshπ\n2kλeikX(t)/parenleftbigg\n˜ηIk(t)tanhπ\n2kλ+2\nπαλkλ˜ηRk(t)/parenrightbigg\n,(30)\nWe consider the case where the wall is approximated by a constant v elocityvw, i.e.,X(t) =\nvwt. The frequency representation of time-integral of Eq. (30) is\n/integraldisplay\ndtH(1)\nη(t) =−π2\n2/integraldisplaydΩ\n2π/integraldisplaydω\n2πλ˙λ(Ω)/summationdisplay\nk1\ncoshπ\n2kλ/parenleftbigg\n˜ηIk(t)tanhπ\n2kλ+2αλ\nπkλ˜ηRk(t)/parenrightbigg\nδ(ω−(kvw+Ω)),\n(31)\nIt is seen that the angular frequency of the emitted spin wave ( ω) iskvw+ Ω, i.e., that of\nthe thickness variation ˙λwith a Doppler shift due to the wall motion. The Doppler shift of\nangular frequency, δν≡kvw, is expected tobesignificant; For k= 1/λwithλ= 10−100nm\nandvw= 100 m/s, we have δν= 10−1 GHz. The function g(x) represents the distribution\nof the wave vector k, which has a broad peak at k= 0 with a width of the order of λ−1.\nTo have a finite expectation value ∝angb∇acketleft˜η∝angb∇acket∇ight, the angular frequency ωand wave vector kneeds\nto match the dispersion relation of spin wave, ω=ωk, i.e.,\nkvw+/planckover2pi1Ω =KS(1+(kλ)2). (32)\nThe angular frequency Ω is determined by the equation for λin Eq. (25), and is of the\norder of the angular frequency of φ0,ωφ. (See Sec. VA for more details.) Equation (32) has\nsolution for a velocity larger than the threshold velocity vth≡2KS\n/planckover2pi1λ/radicalBig\n1−/planckover2pi1ωφ\nKS. The emitted\nwave lengths k∗are (plotted in Fig. 4)\nk∗λ=/planckover2pi1vw\n2KSλ\n1±/radicalBigg\n1−/parenleftbiggvth\nvw/parenrightbigg2\n. (33)\nThe sign of k∗(direction of emission) is along the wall velocity, meaning that the emis sion is\ndominantly in the forward direction. The group velocity of the emitte d wave is of the same\n10vwλ∗\nFIG. 3. Schematic figure showing the spin wave emission from a domain wall with thickness\noscillation ( ˙λ) moving with velocity vw. The linear coupling leads to a forward emission of spin\nwave with wave length λ∗≡2π/k∗, wherek∗is defined by Eq. (33).\nvwk*\n•\n•\n•ω~=0.2\nω~=0.5\nω~=0.8\nFIG. 4. Plot of the wave length k∗of the emitted spin wave as function of wall velocity vwfor\n˜ω≡/planckover2pi1ωφ/KS= 0.2,0.5 and 0.8. Dotted line is k∗=/planckover2pi1\nKSλ2vw. Threshold velocity for the emission\nvthis denoted by circles.\norder as the wall velocity;\ndωk\ndk/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nk=k∗=2KS\n/planckover2pi1λ2k∗=vw\n1±/radicalBigg\n1−/parenleftbiggvth\nvw/parenrightbigg2\n. (34)\nThe dominant spin wave emission considered here is the antisymmetric excitation of the\nimaginary part ˜ ηIrepresenting the fluctuation of angle φ. The antisymmetric excitation of\nφis a natural excitation arising from the intrinsic property, the aniso tropy energy. The\neasy-axis anisotropy energy acts as a local potential VKfor each spin in the wall as in Fig.\n5. When the wall moves to the right, the spins ahead of the wall are d riven towards the\nhigh energy state, while the spins behind (left in Fig. 5) are towards lo w energy states. This\nasymmetry leads to an asymmetric local “velocity” of angle θ, and its canonical momentum\n11FIG. 5. The local potential VKfor spins in a domain wall arising from the easy axis anisotro py\nenergy,K. When the wall moves to the right, the spins right (left) of th e wall rotates towards\nhigh (low) energy states, resulting in an asymmetric local v elocity of rotation, exciting the angle\nφasymmetrically with respect to the wall center.\nφ. This role of Kto induce asymmetric φis seen in the equations of motion for polar angles\n[9]: Focusing on the contribution of the easy axis anisotropy, the ve locity of the in-plane\nspin rotation, sin θ˙φ=−KSsinθcosθis asymmetric with the wall center θ=π/2. Faster\nrotation in the left part of the wall (π\n2< θ < π) than the right part (0 < θ <π\n2) indicates\nthat the wall becomes thinner. In the equation of motion for λ(Eq. (25)), this effect is\nrepresented by the term −˙ϕon the right-hand side, meaning that asymmetric deformation\nmodeϕtends to compress the wall.\nB. Green’s function calculation\nWe present microscopic analysis of the spin wave emission using the Gr een’s function.\nWe consider here the slow domain wall dynamics limit compared to the sp in-wave energy\nscale and neglect the time-dependence of the variable uarising from variation of ˙X. The\ncalculation here thus corresponds to the spin wave effects in the mo ving frame with the\ndomain wall. The amplitude of the spin wave, ∝angb∇acketleft˜η∝angb∇acket∇ight, is calculated using the path-ordered\nGreen’s function method as a linear response to the source field ˙λ. The amplitude is\n∝angb∇acketleft˜η(u,t)∝angb∇acket∇ight=−i/integraldisplay\nCdt′˙λ(t′)/integraldisplay\ndu′g(u′)/angbracketleftbig\nTC˜η(u,t)˜η(u′,t′)/angbracketrightbig\n(35)\nwhereCdenotes the contour for the path-ordered (non-equilibrium) Gre en’s function in the\ncomplex time and TCdetnoes the path-ordering. Evaluating the path-order, we obta in the\n12real-time expression of\n∝angb∇acketleft˜η(u,t)∝angb∇acket∇ight=/integraldisplay∞\n−∞dt′˙λ(t′)/integraldisplay\ndu′g(u′)Gr\nη(u,t,u′,t′) (36)\nwhere\nGr\nη(u,t,u′,t′)≡ −iθ(t−t′)/angbracketleftbig\n[˜η(u,t),˜η(u′,t′)]/angbracketrightbig\n(37)\nthe retarded Green’s function of ˜ η. The Green’s function is calculated expressing ˜ ηin terms\nof the orthogonal base for spin wave wave function [9] as\n˜η(u,t) =/summationdisplay\nkηk(t)φk(u), (38)\nwhereφkis the eigenfunction of Eq. (18) and ηkis the annihilation operator satisfying\n[ηk,ηk′] =δk,k′. The time-development of the operator is ηk(t) =e−iωktηk(0), where ωk≡\nKS˜ωkis the energy of spin wave. The retarded Green’s function thus is\nGr\nη(u,t,u′,t′) =−iθ(t−t′)/summationdisplay\nke−iωk(t−t′)φk(u)φk(u′)≡/integraldisplaydω\n2πe−iω(t−t′)Gr\nη(u,u′,ω) (39)\nwhere\nGr\nη(u,u′,ω) =/summationdisplay\nk1\nω−ωk+i0φk(u)φk(u′) (40)\nis the Fourier transform, + i0 denoting the small positive imaginary part. The Green’s\nfunction has a nonlocal nature in space, as seen from the overlap o f the spin wave function\n/summationdisplay\nkφk(u)φk(u′) =a\n2πλ/bracketleftbigg\nδ(u−u′)−1\n2/parenleftBig\ne−|u−u′|(1−tanhutanhu′)+sinh(u−u′)(tanhu−tanhu′)/parenrightBig/bracketrightbigg\n.\n(41)\nHere we use low-frequency approximation, namely, consider the eff ect of high-frequency\nmagnon compared to the wall dynamics and use Gr\nη(u,u′,ω)≃ −/summationtext\nk1\nωkφk(u)φk(u′). The\nretarded Green’s function then becomes local in time as Gr\nη(u,t,u′,t′) =δ(t−t′)Gr\nη(u,u′,ω).\nWe thus obtain\n∝angb∇acketleft˜η(u,t)∝angb∇acket∇ight=−˙λ(t)/summationdisplay\nk1\nωkφk(u)/integraldisplay\ndu′g(u′)φk(u′) (42)\n13withuandu′havingX(t) of the equal time t. The integral/integraltext\ndu′g(u′)φk(u′) describing the\noverlap of spin-wave wave function and the domain wall is calculated u sing\n/integraldisplay\ndutanhu\ncoshuφk(u) =1√2π˜ωkπ\ncoshπ\n2kλ˜ωk\n2/integraldisplay\nduu\ncoshuφk(u) =1√2π˜ωkπ\ncoshπ\n2kλ(43)\nas\n/integraldisplay\ndug(u)φk(u) =2/planckover2pi1S\na1√2π˜ωkπ\ncoshπ\n2kλ/parenleftbigg\ni−αλ˜ωk\n2/parenrightbigg\n(44)\nThe spin wave amplitude emitted by the wall dynamics is therefore\n∝angb∇acketleft˜η(u,t)∝angb∇acket∇ight=−˙λ(t)2/planckover2pi1\nKa/summationdisplay\nk1√2π˜ωkπ\ncoshπ\n2kλ1\n˜ωkφk(u)/parenleftbigg\ni−αλ˜ωk\n2/parenrightbigg\n(45)\nThe integral/summationtext\nk(˜ωk)−β1\ncoshπ\n2kλφk(u) (β=1\n2,3\n2) is real, and thus Re[ ∝angb∇acketleft˜η∝angb∇acket∇ight]/Im[∝angb∇acketleft˜η∝angb∇acket∇ight]≃α. As\n∝angb∇acketleft˜η(u,t)∝angb∇acket∇ightis odd in u, the emitted spin wave is an antisymmetric fluctuation of the angle\nφwith respect to the wall center (Fig. 1). (Because of low frequenc y approximation in\nderiving Eq. (42), the nonlocal nature (Eq. (41)) is smeared out in the result Eq. (45). )\nThe quantities representing the effects of spin wave emission on the wall dynamics in Eq.\n(25) are\nζ=/integraldisplay\ndutanhu\ncoshuRe[˜η] =˙λπ/planckover2pi1\n4Kaαλ/summationdisplay\nk1\ncosh2π\n2kλ≡αµζ˙λ\nλ\nϕ=/integraldisplay\nduu\ncoshuIm[˜η] =−˙λπ/planckover2pi1\nKa/summationdisplay\nk1\n˜ω2\nk1\ncosh2π\n2kλ≡µϕ˙λ\nλ(46)\nwhereµζ≡π3/planckover2pi1λ\n48Ka/summationtext\nk1\ncosh2π\n2kλandµϕ≡π/planckover2pi1λ\nKa/summationtext\nk1\n˜ω2\nk1\ncosh2π\n2kλ. The first integral is evaluated as\n/summationtext\nk1\ncosh2π\n2kλ=a/integraltextdk\n2π1\ncosh2π\n2kλ=2a\nπ2λand the second one is/summationtext\nk1\n˜ω2\nk1\ncosh2π\n2kλ≡2a\nπ2λγϕ, whereγϕ\nis a constant of the order of unity. The constants are therefore\nµζ=π/planckover2pi1\n24K\nµϕ=−2/planckover2pi1γϕ\nπK. (47)\nFrom Eq. (46), the averaged amplitude of the imaginary part of the emitted spin wave is\nof the order of/planckover2pi1˙λ\nKλ(the real part is a factor of αsmaller). As seen from Eq. (25), the time\nscale ofλdynamics is K//planckover2pi1, and thus the emitted spin wave amplitude can be of the order\nof unity if the modulation of λis strong, resulting in a significant damping. (See Eq. (61)\nbelow.)\n14C. Spin wave excitation due to second order interaction\nBesides emission due to the linear order interaction discussed above , spin waves are\nexcited also due to the second order interaction in Eqs.(10) (14) wh en the wall is dynamic.\nHere we focus on the effect of a dynamic potential in the Hamiltonian ( Eq. (16))\nV(x,t)≡4KS2\na1\ncosh2x−X(t)\nλ(48)\nand calculate the excited spin wave density in the laboratory frame b y use of linear response\ntheory. For a constant wall velocity, X(t) =vwt, the Fourier representation of the potential\nis\nVq(Ω) = 8π2KS2λ\naqλ\nsinhπ\n2qλδ(Ω−qvw), (49)\nThe potential thereforeinduces Dopplershift of qvwintheangular frequency ofthescattered\nspin wave. This dynamic potential induces an excited spin wave densit y asδn(x,t) =\niG<\nη(x,t,x,t), where G<\nηis the lesser Green’s function of spin wave. The linear response\ncontribution in the Fourier representation is\nδn(q,Ω) =i/summationdisplay\nk/integraldisplaydω\n2πVq(Ω)(n(ω+Ω)−n(ω))gr\nkωga\nk+q,ω+Ω (50)\nInthisprocess, theexcitedspinwave density hasthesamewavelen gth andangularfrequency\nof the driving potential Vq(Ω). This means that the excitation moves together with the\ndomain wall, and thus this is not an emission process. For slow limit, q≪kand Ω≪ω,\nusingn(ω+Ω)−n(ω) =n(ω+qvw)−n(ω)≃qvwn′(ω), we obtain a compact expression of\nδn(q,Ω) =i4πKS2\navw(qλ)2\nsinhπ\n2qλδ(Ω−qvw)/integraldisplaydω\n2π/summationdisplay\nkn′(ω)|gr\nkω|2(51)\nand the real space profile is\nδn(x,t) =δn0vw\nvatanhx−vwt\nλ\ncosh2x−vwt\nλ(52)\nwhereδn0=−4\nπ(KS)2/integraltextdω\n2π/summationtext\nkn′(ω)|gr\nkω|2andva≡Kλ//planckover2pi1is a velocity scale determined\nby magnetic anisotropy energy. The induced spin wave density has t hus an antisymmetric\nspatial profile with respect to the wall center and propagate with a domain wall velocity in\nthe present slowly varying limit. It is not therefore a spin wave emissio n, but represents the\ndeformation of the wall asymmetric with respect to the center.\n15IV. EQUATION OF MOTION OF THREE COLLECTIVE COORDINATES\nTheequationofmotion(25)including thespinwave emission effectsex plicitly istherefore\n˙X−αλ˙φ0=vcsin2φ0+2vcsin2φ0αµζ˙λ\nλ+vst (53)\n˙φ0+α˙X\nλ=˜Bz\nαλ˙λ\nλ=KS\n/planckover2pi1/bracketleftBigg/parenleftbiggλ0\nλ/parenrightbigg2\n−(1+κsin2φ0)/bracketrightBigg\n−µϕ¨λ\nλ−2KS\n/planckover2pi1/bracketleftBigg/parenleftbiggλ0\nλ/parenrightbigg2\n+(1+κsin2φ0)/bracketrightBigg\nαµζ˙λ\nλ.\n(54)\nThe spin-wave contribution of the first equation, the second term of the right-hand side, is of\nthe order αsmaller than the first term and is neglected. From the equations, we see that the\ndynamics of Xandφare not strongly coupled to the variation of the width. In particular ,\nwhenκis small, the dynamics of the wall center ( Xandφ) governed by the energy scale of\nK⊥=κKis much slower than that of a deformation mode λ, which is of the energy scale\nofK, and thus it is natural that the two dynamics are decoupled. Then κis not small, λ\naffects much the wall center dynamics.\nFor static case of λ, we have\nλ=λ0/radicalbig\n1+κsin2φ0, (55)\nas was argued in Refs. [12, 13]. Using this relation assuming slow dynam ics to estimate the\nspin-wave contribution in the equation for λ, we obtain\nµϕ¨λ+ ˜αλ˙λ=KS\n/planckover2pi1λ/bracketleftBigg/parenleftbiggλ0\nλ/parenrightbigg2\n−(1+κsin2φ0)/bracketrightBigg\n, (56)\nwhere ˜αλ≡αλ/parenleftbig\n1+2S\nπ/parenrightbig\n=π2\n12α/parenleftbig\n1+2S\nπ/parenrightbig\nis the effective damping for the width. The mass\nforλ,µϕ, was induced by the imaginary part of the spin-wave.\nV. DISSIPATION DUE TO SPIN WAVE EMISSION\nConsidering the action, which is a time-integral of the Lagrangian, a nd by use of integral\nby parts with respect to time, the linear interaction Hamiltonian, Eq. (27), is equivalent to\nH(1)\nη=−λFλ, where\nFλ≡2/integraldisplay\nduRe[g˙˜η], (57)\n16is a generalized force for variable λ. Using Eqs. (45)(43), it reads\nFλ=−¨λfλ, (58)\nwhere (neglecting the order of α2)\nfλ≡π/planckover2pi12S\nKa2/summationdisplay\nk1\n˜ω2\nk1\ncosh2π\n2kλ=2/planckover2pi12S\nπKλaγϕ. (59)\nThe energy dissipation rate due to the spin wave emission is therefor e\ndEsw\ndt≡ −˙λFλ=fλ\n2d\ndt˙λ2, (60)\nand thus Esw=fλ\n2˙λ2. As is seen from Eq. (56), the intrinsic energy scale governing the\ndynamics of λisK, and thus the intrinsic scale of ˙λ/λis of the order of K//planckover2pi1. The energy\ndissipation by an intrinsic spin-wave emission is estimated roughly as Ei\nsw≃Kλ\na, which is\nthe typical spin wave energy multiplied by the number of spin waves ex cited in the wall.\nThe quantitydEi\nsw\ndtcorresponds to a dissipation function Wi\nswinduced by the intrinsic spin\nwave emission. Considering the intrinsic frequency of λof the order of K//planckover2pi1, the Gilbert\ndamping parameter induced by the intrinsic emission is\nαi\nsw≃2aλ\n/planckover2pi1SfλK\n/planckover2pi1=4γϕ\nπ. (61)\nThis value is of the order of unity ( γϕis a constant), meaning that spin wave emission from\nthe thickness change is very efficient in dissipating energy from the w all. This result may\nnot be surprising if one notices that the intrinsic energy scale of thic kness change is that of\neasy-axis anisotropy energy K, which is the energy scale where significant deformation of\nthe wall is induced.\nA. Modulation of λdue toφ0dynamics\nIn most cases, the dynamics of λis driven by the time-dependence of φ0as seen in Eq.\n(56). Let us consider this case of a forced oscillation. We consider b y simplyfying φ0grows\nlinear with time, φ0=ωφt,ωφbeing a constant. Linearizing Eq. (56) using λ=λ+δλ,\nwhereλ≡λ0//radicalbig\n1+κ/2 is the average thickness, we have an equation of motion of a force d\noscillation,\nµϕ¨δλ+ ˜αλ˙δλ+µϕ(Ωλ)2δλ=KS\n2/planckover2pi1λκcos2ωφt, (62)\n17where Ω λ=K\n/planckover2pi1/radicalBig\nπS\nγϕ/parenleftbig\n1+κ\n2/parenrightbig\nis an intrinsic angular frequency of δλ. The solution having an\nexternal angular frequency of 2 ωφis\nδλ=δλcos(2ωφt−εφ), (63)\nwhere\nδλ≡κλπS\n4γϕ(K//planckover2pi1)2\n/radicalBig\n(Ω2\nλ−4ω2\nφ)2+4(˜αλωφ\nµϕ)2(64)\nis the amplitude of the forced oscillation and εφ≡tan−12˜αλωφ\nµϕ\nΩ2\nλ−4ω2\nφis a phase shift. A resonance\noccurs for ωφ= Ωλ/2. The energy dissipation rate for the emission due to forced oscillat ion\ninduced by dynamics of φis\ndEφ\nsw\ndt≃λ\na/parenleftbiggδλ\nλ/parenrightbigg2ω3\nφ\nK. (65)\nThe contribution to the Gilbert damping parameter is obtained from t he relationdEφ\nsw\ndt=\nαφ\nsw(˙λ/λ)2as\nαφ\nsw≃λ\na/planckover2pi1ωφ\nK. (66)\nLet us focus on the periodic oscillation of φ0, realized for large driving forces, namely, for\nBz> αKSκ\n2/planckover2pi1γ≡BW(γBz> αvc) for the field-driven case or j >eS2\n/planckover2pi1Pλ\naKκ≡ji(vst> vc) for\nthe current-driven case ( BWis the Walker’s breakdown field and jiis the intrinsic threshold\ncurrent [10]). The solution of the equation of motion (54) then read s\nφ0≃ωφt, (67)\nwhere (jis defined in one-dimension to have the unit of A=C/s)\nωφ≃˜Bz+αvst\nλ=γBz+aP\n2eSλαj. (68)\nTheGilbertdampingconstant duetospinwaveemission, Eq. (66), th usgrowslinearlyinthe\ndriving fields in this oscillation regime. Using current-induced torque f or a pinned domain\nwall would be straightforward for experimental observation of th is behaviour, although the\ncontribution to the Gilbert damping is proportional to αand not large, αφ\nsw≃α/planckover2pi1P\nej\nK(for\nS∼1,P∼1).\n18VI. SUMMARY\nWestudiedspinwaveemissionfromamovingdomainwallinaferromagne tbyintroducing\na deformation mode of thickness modulation as a collective coordinat e. It was shown that\nthe time-derivative of the thickness ˙λhas a coupling linear in the spin wave field, resulting\nin an emission, consistent with previous numerical result [3]. The domin ant emitted spin\nwave is in the forward direction to the moving domain wall and is strong ly polarized in the\nout-of plane direction, i.e., it is a fluctuation of φ. The dynamics of λis induced by the\nvariation of the angle of the wall plane, φ0, as has been noted [12, 13]. For a φ0with an\nangular frequency of ωφ, the Gilbert damping parameter as a result of spin wave emission\nisαφ\nsw≃λ\na/planckover2pi1ωφ\nK, whereKis the easy-axis anisotropy energy ( ais the lattice constant).\nThe present study is in the low energy and weak spin wave regime, and treating the\nhigher energy dynamics with strong spin wave emission is an important future subject.\nACKNOWLEDGMENTS\nGT thanks Y. Nakatani for discussions. This work was supported b y a Grant-in-Aid\nfor Scientific Research (B) (No. 17H02929) from the Japan Societ y for the Promotion of\nScience and a Grant-in-Aid for Scientific Research on Innovative Ar eas (No.26103006) from\nThe Ministry of Education, Culture, Sports, Science and Technolog y (MEXT), Japan.\n[1] D. Bouzidi and H. Suhl, Phys. Rev. Lett. 65, 2587 (1990).\n[2] Y. L. Maho, J.-V. Kim, and G. Tatara, Phys. Rev. B 79, 174404 (2009).\n[3] X. S. Wang, P. Yan, Y. H. Shen, G. E. W. Bauer, and X. R. Wang,\nPhys. Rev. Lett. 109, 167209 (2012).\n[4] X. S. Wang and X. R. Wang, Phys. Rev. B 90, 014414 (2014).\n[5] N. J. Whitehead, S. A. R. Horsley, T. G. Philbin, A. N. Kuch ko, and V. V. Kruglyak,\nPhys. Rev. B 96, 064415 (2017).\n[6] S. K. Kim, O. Tchernyshyov, V. Galitski, and Y. Tserkovny ak,\nPhys. Rev. B 97, 174433 (2018).\n[7] J. C. Slonczewski, Int. J. Magn. 2, 85 (1972).\n19[8] R. Rajaraman, Solitons and Instantons (North-Holland, 1982) p. Chap. 8.\n[9] G. Tatara, H. Kohno, and J. Shibata, Physics Reports 468, 213 (2008).\n[10] G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601 (2004).\n[11] H.-B. Braun and D. Loss, Phys. Rev. B 53, 3237 (1996).\n[12] N. L. Schryer and L. R. Walker, Journal of Applied Physic s45, 5406 (1974).\n[13] A. Thiaville, Y. Nakatani, J. Miltat, and N. Vernier,\nJournal of Applied Physics 95, 7049 (2004), https://doi.org/10.1063/1.1667804.\n[14] L. Berger, Phys. Rev. B 33, 1572 (1986).\n[15] G. Tatara, Physica E: Low-dimensional Systems and Nano structures 106, 208 (2019).\n20"    },    {        "title": "1705.03416v1.Low_spin_wave_damping_in_the_insulating_chiral_magnet_Cu___2__OSeO___3__.pdf",        "content": "Low spin wave damping in the insulating chiral magnet Cu 2OSeO 3\nI. Stasinopoulos,1S. Weichselbaumer,1A. Bauer,2J. Waizner,3\nH. Berger,4S. Maendl,1M. Garst,3, 5C. P\reiderer,2and D. Grundler6,\u0003\n1Physik Department E10, Technische Universit at M unchen, D-85748 Garching, Germany\n2Physik Department E51, Technische Universit at M unchen, D-85748 Garching, Germany\n3Institute for Theoretical Physics, Universit at zu K oln, D-50937 K oln, Germany\n4Institut de Physique de la Mati\u0012 ere Complexe, \u0013Ecole Polytechnique F\u0013 ed\u0013 erale de Lausanne, 1015 Lausanne, Switzerland\n5Institut f ur Theoretische Physik, Technische Universit at Dresden, D-01062 Dresden, Germany\n6Institute of Materials and Laboratory of Nanoscale Magnetic Materials and Magnonics (LMGN),\n\u0013Ecole Polytechnique F\u0013 ed\u0013 erale de Lausanne (EPFL), Station 12, 1015 Lausanne, Switzerland\n(Dated: October 2, 2018)\nChiral magnets with topologically nontrivial spin order such as Skyrmions have generated enor-\nmous interest in both fundamental and applied sciences. We report broadband microwave spec-\ntroscopy performed on the insulating chiral ferrimagnet Cu 2OSeO 3. For the damping of magnetiza-\ntion dynamics we \fnd a remarkably small Gilbert damping parameter of about 1 \u000210\u00004at 5 K. This\nvalue is only a factor of 4 larger than the one reported for the best insulating ferrimagnet yttrium\niron garnet. We detect a series of sharp resonances and attribute them to con\fned spin waves in\nthe mm-sized samples. Considering the small damping, insulating chiral magnets turn out to be\npromising candidates when exploring non-collinear spin structures for high frequency applications.\nPACS numbers: 76.50.+g, 74.25.Ha, 4.40.Az, 41.20.Jb\nThe development of future devices for microwave ap-\nplications, spintronics and magnonics [1{3] requires ma-\nterials with a low spin wave (magnon) damping. In-\nsulating compounds are advantageous over metals for\nhigh-frequency applications as they avoid damping via\nspin wave scattering at free charge carriers and eddy\ncurrents [4, 5]. Indeed, the ferrimagnetic insulator yt-\ntrium iron garnet (YIG) holds the benchmark with a\nGilbert damping parameter \u000bintr= 3\u000210\u00005at room\ntemperature [6, 7]. During the last years chiral mag-\nnets have attracted a lot of attention in fundamental\nresearch and stimulated new concepts for information\ntechnology [8, 9]. This material class hosts non-collinear\nspin structures such as spin helices and Skyrmions be-\nlow the critical temperature Tcand critical \feld Hc2\n[10{12]. Additionally, Dzyaloshinskii-Moriya interaction\n(DMI) is present that induces both the Skyrmion lattice\nphase and nonreciprocal microwave characteristics [13].\nLow damping magnets o\u000bering DMI would generate new\nprospects by particularly combining complex spin order\nwith long-distance magnon transport in high-frequency\napplications and magnonics [14, 15]. At low tempera-\ntures, they would further enrich the physics in magnon-\nphoton cavities that call for materials with small \u000bintrto\nachieve high-cooperative magnon-to-photon coupling in\nthe quantum limit [16{19].\nIn this work, we investigate the Gilbert damping in\nCu2OSeO 3, a prototypical insulator hosting Skyrmions\n[20{23]. This material is a local-moment ferrimagnet\nwithTc= 58 K and magnetoelectric coupling [24] that\ngives rise to dichroism for microwaves [25{27]. The\nmagnetization dynamics in Cu 2OSeO 3has already been\nexplored [13, 28, 29]. A detailed investigation on thedamping which is a key quality for magnonics and spin-\ntronics has not yet been presented however. To eval-\nuate\u000bintrwe explore the \feld polarized state (FP)\nwhere the two spin sublattices attain the ferrimagnetic\narrangement[21]. Using spectra obtained by two di\u000ber-\nent coplanar waveguides (CPWs), we extract a minimum\n\u000bintr=(9.9\u00064.1)\u000210\u00005at 5 K, i.e. only about four times\nhigher than in YIG. We resolve numerous sharp reso-\nnances in our spectra and attribute them to modes that\nare con\fned modes across the macroscopic sample and\nallowed for by the low damping. Our \fndings substanti-\nate the relevance of insulating chiral magnets for future\napplications in magnonics and spintronics.\nFrom single crystals of Cu 2OSeO 3we prepared two\nbar-shaped samples exhibiting di\u000berent crystallographic\norientations. The samples had lateral dimensions of\n2:3\u00020:4\u00020:3 mm3. They were positioned on CPWs that\nprovided us with a dynamic magnetic \feld hinduced by\na sinusoidal current applied to the signal surrounded by\ntwo ground lines. We used two di\u000berent CPWs with ei-\nther a broad [30] or narrow signal line width of ws= 1 mm\nor 20\u0016m, respectively [31]. The central long axis of the\nrectangular Cu 2OSeO 3rods was positioned on the central\naxis of the CPWs. The static magnetic \feld Hwas ap-\nplied perpendicular to the substrate with Hkh100iand\nHkh111ifor sample S1 and S2, respectively. The direc-\ntion ofHde\fned the z-direction. The dynamic \feld com-\nponent h?Hprovided the relevant torque for excita-\ntion. Components hkHdid not induce precessional mo-\ntion in the FP state of Cu 2OSeO 3. We recorded spectra\nby a vector network analyzer using the magnitude of the\nscattering parameter S12. We subtracted a background\nspectrum recorded at 1 T to enhance the signal-to-noisearXiv:1705.03416v1  [cond-mat.str-el]  9 May 20172\nratio (SNR) yielding the displayed \u0001 jS12j. In Ref. [7],\nKlingler et al. have investigated the damping of the in-\nsulating ferrimagnet YIG and found that Gilbert param-\neters\u000bintrevaluated from both the uniform precessional\nmode and standing spin waves con\fned in the macro-\nscopic sample provided the same values. For Cu 2OSeO 3\nwe evaluated \u000bin two ways[32]. When extracting the\nlinewidth \u0001 Hfor di\u000berent resonance frequencies fr, the\nGilbert damping parameter \u000bintrwas assumed to vary\naccording to [33, 34]\n\u00160\r\u0001\u0001H= 4\u0019\u000bintr\u0001fr+\u00160\r\u0001\u0001H0; (1)\nwhere\ris the gyromagnetic factor and \u0001 H0the contri-\nbution due to inhomogeneous broadening. Equation (1)\nis valid when viscous Gilbert damping dominates over\nscattering within the magnetic subsystem [35]. When\nperforming frequency-swept measurements at di\u000berent\n\feldsH, the obtained linewidth \u0001 fwas considered to\nscale linearly with the resonance frequency as [36]\n\u0001f= 2\u000bintr\u0001fr+ \u0001f0; (2)\nwith the inhomogeneous broadening \u0001 f0. The conver-\nsion from Eq. (1) to Eq. (2) is valid when frscales linearly\nwithHandHis applied along a magnetic easy or hard\naxis of the material [37, 38]. In Fig. 1 (a) to (d) we show\nspectra recorded in the FP state of the material using the\ntwo di\u000berent CPWs. For the same applied \feld Hwe ob-\nserve peaks residing at higher frequency fforHkh100i\ncompared to Hkh111i. From the resonance frequencies,\nwe extract the cubic magnetocrystalline anisotropy con-\nstantK= (\u00000:6\u00060:1)\u0001103J/m3for Cu 2OSeO 3[31].\nThe magnetic anisotropy energy is found to be extremal\nforh100iandh111ire\recting easy and hard axes, respec-\ntively [31]. The saturation magnetization of Cu 2OSeO 3\namounted to \u00160Ms= 0:13 T at 5 K[22].\nFigure 1 summarizes spectra taken with two di\u000ber-\nent CPWs on two di\u000berent Cu 2OSeO 3crystals exhibit-\ning di\u000berent crystallographic orientation in the \feld H.\nFor the narrow CPW [Fig. 1 (a) and (c)], we observed a\nbroad peak superimposed by a series of resonances that\nall shifted to higher frequencies with increasing H. The\n\feld dependence excluded them from being noise or arti-\nfacts of the setup. Their number and relative intensities\nvaried from sample to sample and also upon remounting\nthe same sample in the cryostat (not shown). They disap-\npeared with increasing temperature Tbut the broad peak\nremained. For the broad CPW [Fig. 1 (b) and (d)], we\nmeasured pronounced peaks whose linewidths were sig-\nni\fcantly smaller compared to the broad peak detected\nwith the narrow CPW. We resolved resonances below\nthe large peaks [arrows in Fig. 1 (b)] that shifted with\nHand exhibited an almost \feld-independent frequency\no\u000bset from the main peaks that we will discuss later. It\nis instructive to \frst follow the orthodox approach and\nanalyze damping parameters from modes re\recting the\n69121518-0.4-0.20.0(d)Δ  |S12|f\n (GHz)H\n || 〈111〉 \n69121518-6-30(c)Δ |S12| (10-2)f\n (GHz)\n-0.6-0.4-0.20.0H\n || 〈100〉 (b)broad CPWΔ |S12|\n-0.3-0.2-0.10.00\n.35 T(a)narrow CPWΔ |S12|0\n.25 T0\n.45 T0.55 TFIG. 1. (Color online) Spectra \u0001 jS12jobtained at T = 5 K\nfor di\u000berent Husing (a) a narrow and (b) broad CPW when\nHjjh100ion sample S1. Corresponding spectra taken on sam-\nple S2 for Hjjh111iare shown in (c) and (d), respectively.\nNote the strong and sharp resonances in (b) and (d) when us-\ning the broad CPW that provides a much more homogeneous\nexcitation \feld h. Arrows mark resonances that have a \feld-\nindependent o\u000bset with the corresponding main peaks and are\nattributed to standing spin waves. An exemplary Lorentz \ft\ncurve is shown in blue color in (b).\nexcitation characteristics of the CPW [29]. Second, we\nfollow Ref. [7] and analyze con\fned modes.\nLorentz curves (blue) were \ftted to the spectra\nrecorded with the broad CPW to determine resonance\nfrequencies and linewidths. Note that the corresponding\nlinewidths were larger by a factor ofp\n3 compared to the\nlinewidth \u0001 fthat is conventionally extracted from the\nimaginary part of the scattering parameters [39]. The\nextracted linewidths \u0001 fwere found to follow linear \fts\nbased on Eq. (2) at di\u000berent temperatures (details are\nshown in Ref. [31]). In Fig. 2 (a) we show a resonance\ncurve that was obtained as a function of Htaken with\nthe narrow CPW at 15 GHz. The curve does not show\nsharp features as Hwas varied in \fnite steps (symbols).\nThe linewidth \u0001 H(symbols) is plotted in Fig. 2 (b) for\ndi\u000berent resonance frequencies and temperatures. The\ndata are well described by linear \fts (lines) based on\nEq. (1). Note that the resonance peaks measured with\nthe broad CPW were extremely sharp. The sharpness\ndid not allow us to analyze the resonances as a function\nofH. We refrained from \ftting the broad peaks of Fig. 1\n(a) and (c) (narrow CPW) as they showed a clear asym-\nmetry attributed to the overlap of subresonances at \fnite\nwavevector k, as will be discussed below.\nIn Fig. 3 (a) and (b) we compare the parameter \u000bintr\nobtained from both di\u000berent CPWs (circles vs. stars) and\nthe two evaluation routes [40]. For Hkh100i[Fig. 3 (a)],\nbetween 5 and 20 K the lowest value for \u000bintramounts to\n(3.7\u00060.4)\u000210\u00003. This value is three times lower com-\npared to preliminary data presented in Ref. [29]. Beyond3\nFIG. 2. (Color online) (a) Lorentz curve (magenta line) \ftted\nto a resonance (symbols) measured at f= 15 GHz as a func-\ntion ofHat 5 K. (b) Frequency dependencies of linewidths\n\u0001H(symbols) for four di\u000berent T. We performed thep\n3-\ncorrection. The slopes of linear \fts (straight lines) following\nEq. 1 are considered to re\rect the intrinsic damping parame-\nters\u000bintr.\n04812H || 〈100〉 αintr (10-3)Δ H  narrow CPWΔ\n f    broad   CPW\nH || 〈111〉 \n1020304050T\n (K)\n10203040500.00.20.40.60.8Δf0 (GHz)T\n (K)(b)( a)(\nd)( c)\nFIG. 3. (Color online) (a) and (b) Intrinsic damping param-\neters\u000bintrand inhomogeneous broadening \u0001 f0for two di\u000ber-\nent \feld directions (see labels) obtained from the slopes and\nintercepts at fr= 0 of linear \fts to the linewidth data (see\nFig. 2 (b) and Ref. [31]). Dashed lines are guides to the eyes.\n20 K the damping is found to increase. For Hkh111i\n[Fig. 3 (b)] we extract (0.6 \u00060.6)\u000210\u00003as the smallest\nvalue. Note that these values for \u000bintrstill contain an ex-\ntrinsic contribution and thus represent upper bounds for\nCu2OSeO 3, as we will show later. For the inhomogeneous\nbroadening \u0001 f0in Fig. 3 (c) and (d) the datasets are\nconsistent (we have used the relation \u0001 f0=\r\u0001H0=2\u0019\nto convert \u0001 H0into \u0001f0). We see that \u0001 f0increases\nwithTand is small for the broad CPW, independent\nof the crystallographic direction of H. For the narrow\nCPW the inhomogeneous broadening is largest at small\nTand then decreases by about 40 % up to about 50\nK. Note that a CPW broader than the sample is as-\nsumed to excite homogeneously at fFMR [41] transfer-\nring a wave vector k= 0 to the sample. Accordinglywe ascribe the intense resonances of Fig. 1 (b) and (d) to\nfFMR. UsingfFMR= 6 GHz and \u000bintr= 3:7\u000210\u00003at 5\nK [Fig. 3 (a)], we estimate a minimum relaxation time of\n\u001c= [2\u0019\u000bintrfr]\u00001= 6:6 ns.\nIn the following, we examine in detail the additional\nsharp resonances that we observed in spectra of Fig. 1.\nIn Fig. 1 (b) taken with the broad CPW for Hkh100i,\nwe identify sharp resonances that exhibit a characteris-\ntic frequency o\u000bset \u000efwith the main resonance at all\n\felds (black arrows). We illustrate this in Fig. 4(a) in\nthat we shift spectra of Fig. 1 (b) so that the positions of\ntheir main resonances overlap. The additional small res-\nonances (arrows) in Fig. 1 (b) are well below the uniform\nmode. This is characteristic for backward volume magne-\ntostatic spin waves (BVMSWs). Standing waves of such\nkind can develop if they are re\rected at least once at the\nbottom and top surfaces of the sample. The resulting\nstanding waves exhibit a wave vector k=n\u0019=d , with\norder number nand sample thickness d= 0:3 mm. The\nBVMSW dispersion relation f(k) of Ref. [13] provides a\ngroup velocity vg=\u0000300 km/s at k=\u0019=d[triangles in\nFig. 4 (b)]. Hence, the decay length ld=vg\u001camounts\nto 2 mm considering \u001c= 6:6 ns. This is larger than\ntwice the relevant lateral sizes, thereby allowing stand-\ning spin wave modes to form in the sample. Based on\nthe dispersion relation of Ref. [13], we calculated the fre-\nquency splitting \u000ef=fFMR\u0000f(n\u0019=d ) [open diamonds\nin Fig. 4 (b)] assuming n= 1 andt= 0:4 mm for the\nsample width tde\fned in Ref. [13]. Experimental val-\nues (\flled symbols) agree with the calculated ones (open\nsymbols) within about 60 MHz. In case of the narrow\nCPW, we observe even more sharp resonances [Fig. 1 (a)\nand (c)]. A set of resonances was reported previously\nin the \feld-polarized phase of Cu 2OSeO 3[26, 28, 42, 43].\nMaisuradze et al. assigned secondary peaks in thin plates\nof Cu 2OSeO 3to di\u000berent standing spin-wave modes [43]\nin agreement with our analysis outlined above.\n0.30.40.51.101.151.201.25-\n500-300-100100δf (GHz)(b)/s61549\n0H (T)v\ng (km/s)\n-10 -0.8-0.6-0.4-0.20.0f\n - f (0)  (GHz)H || 〈100〉 (a)b\nroad CPWΔ |S12|δ\nf\nFIG. 4. (Color online) (a) Spectra of Fig. 1 (b) replotted as\nf\u0000fFMR(H) for di\u000berent Hsuch that all main peaks are at\nzero frequency and the \feld-independent frequency splitting\n\u000efbecomes visible. The numerous oscillations seen particu-\nlarly on the bottom most curve are artefacts from the cali-\nbration routine. (b) Experimentally evaluated (\flled circles)\nand theoretically predicted (diamonds) splitting \u000efusing dis-\npersion relations for a platelet. Calculated group velocity vg\natk=\u0019=(0:3 mm). Dashed lines are guides to the eyes.4\nThe inhomogeneous dynamic \feld hof the narrow\nCPW provides a much broader distribution of kcom-\npared to the broad CPW. This is consistent with the\nfact that the inhomogeneous broadening \u0001 f0is found to\nbe larger for the narrow CPW compared to the broad\none [Fig. 3 (c) and (c)]. Under these circumstances, the\nexcitation of more standing waves is expected. We at-\ntribute the series of sharp resonances in Fig. 1 (a) and\n(c) to such spin waves. In Fig. 5 (a) and (b) we highlight\nprominent and particularly narrow resonances with #1,\n#2 and #3 recorded with the narrow CPW. We trace\ntheir frequencies fras a function of HforHkh100iand\nHkh111i, respectively. They depend linearly on Hsug-\ngesting a Land\u0013 e factor g= 2:14 at 5 K.\nWe now concentrate on mode #1 for Hk h100iat\n5 K that is best resolved. We \ft a Lorentzian line-\nshape as shown in Fig. 5(c) for 0.85 T, and summarize\nthe corresponding linewidths \u0001 fin Fig. 5(d). The inset\nof Fig. 5(d) shows the e\u000bective damping \u000be\u000b= \u0001f=(2fr)\nevaluated directly from the linewidth as suggested in Ref.\n[29]. We \fnd that \u000be\u000bapproaches a value of about 3.5\n\u000210\u00004with increasing frequency. This value includes\nboth the intrinsic damping and inhomogeneous broad-\nening but is already a factor of 10 smaller compared to\n\u000bintrextracted from Fig. 3 (a). Note that Cu 2OSeO 3\nexhibiting 3.5\u000210\u00004outperforms the best metallic thin-\n\flm magnet [44]. To correct for inhomogeneous broad-\nening and determine the intrinsic Gilbert-type damping,\nwe apply a linear \ft to the linewidths \u0001 fin Fig. 5(d) at\nfr>10:6 GHz and obtain (9.9 \u00064.1)\u000210\u00005. Forfr\u0014\n10.6 GHz the resonance amplitudes of mode #1 were\nsmall reducing the con\fdence of the \ftting procedure.\nFurthermore, at low frequencies, we expect anisotropy to\nmodify the extracted damping, similar to the results in\nRef. [45]. For these reasons, the two points at low frwere\nleft out for the linear \ft providing (9.9 \u00064.1)\u000210\u00005.\nWe \fnd \u0001fand the damping parameters of Fig. 3 to\nincrease with T. It does not scale linearly for Hkh100i\n[31]. A deviation from linear scaling was reported for\nYIG single crystals as well and accounted for by the con-\n\ruence of a low- kmagnon with a phonon or thermally\nexcited magnon [5]. In the case of Hkh111i(cf. Fig. 3\n(b)) we obtain a clear discrepancy between results from\nthe two evaluation routes and CPWs used. We relate\nthis observation to a misalignment of Hwith the hard\naxish111i. The misalignment motivates a \feld-dragging\ncontribution [38] that can explain the discrepancy. For\nthis reason, we concentrated our standing wave analysis\non the case Hkh100i. We now comment on our spectra\ntaken with the broad CPW that do not show the very\nsmall linewidth attributed to the con\fned spin waves.\nThe sharp mode #1 yields \u0001 f= 15:3 MHz near 16 GHz\n[Fig. 5 (d)]. At 5 K the dominant peak measured at 0.55 T\nwith the broad CPW provides however \u0001 f= 129 MHz.\n\u0001fobtained by the broad CPW is thus increased by a\nfactor of eight and explains the relatively large Gilbert\nFIG. 5. (Color online) (a)-(b) Resonance frequency as a func-\ntion of \feld Hof selected sharp modes labelled #1 to #3 (see\ninsets) for Hkh100iandHkh111iat T = 5 K. (c) Exemplary\nLorentz \ft of sharp mode #1 for Hkh100iat 0.85 T. (d) Ex-\ntracted linewidth \u0001f as a function of resonance frequency fr\nalong with the linear \ft performed to determine the intrinsic\ndamping\u000bintrin Cu 2OSeO 3. Inset: Comparison among the\nextrinsic and intrinsic damping contribution. The red dotted\nlines mark the error margins of \u000bintr= (9:9\u00064:1)\u000210\u00005.\ndamping parameter in Fig. 3 (a) and (b). We con\frmed\nthis larger value on a third sample with Hkh100iand ob-\ntained (3.1\u00060.3)\u000210\u00003[31] using the broad CPW. The\ndiscrepancy with the damping parameter extracted from\nthe sharp modes of Fig. 5 might be due to the remaining\ninhomogeneity of hover the thickness of the sample lead-\ning to an uncertainty in the wave vector in z-direction.\nFor a standing spin wave such an inhomogeneity does\nnot play a role as the boundary conditions discretize k.\nAccordingly, Klingler et al. extract the smallest damp-\ning parameter of 2 :7(5)\u000210\u00005reported so far for the\nferrimagnet YIG when analyzing con\fned magnetostatic\nmodes [7].\nTo summarize, we investigated the spin dynamics in\nthe \feld-polarized phase of the insulating chiral mag-\nnet Cu 2OSeO 3. We detected numerous sharp reso-\nnances that we attribute to standing spin waves. Their\ne\u000bective damping parameter is small and amounts to\n3:5\u000210\u00004. A quantitative estimate of the intrinsic\nGilbert damping parameter extracted from the con\fned\nmodes provides even \u000bintr=(9.9\u00064.1)\u000210\u00005at 5 K. The\nsmall damping makes an insulating ferrimagnet exhibit-\ning Dzyaloshinskii-Moriya interaction a promising can-\ndidate for exploitation of complex spin structures and\nrelated nonreciprocity in magnonics and spintronics.\nWe thank S. Mayr for assistance with sample prepa-\nration. Financial support through DFG TRR80, DFG\n1143, DFG FOR960, and ERC Advanced Grant 291079\n(TOPFIT) is gratefully acknowledged.5\n\u0003Electronic mail: dirk.grundler@ep\r.ch\n[1] I. Zutic and H. Dery, Nat. Mater. 10, 647 (2011).\n[2] M. Krawczyk and D. Grundler, J. Phys.: Condens. Mat-\nter26, 123202 (2014).\n[3] A. V. Chumak, V. I. Vasyuchka, A. A. Serga, and\nB. 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Phys. 85(1999)."    },    {        "title": "1507.06748v1.Boosting_Domain_Wall_Propagation_by_Notches.pdf",        "content": "arXiv:1507.06748v1  [cond-mat.mes-hall]  24 Jul 2015Boosting Domain Wall Propagation by Notches\nH. Y. Yuan and X. R. Wang1,2,∗\n1Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong\n2HKUST Shenzhen Research Institute, Shenzhen 518057, China\nWereportacounter-intuitivefindingthatnotchesinanothe rwise homogeneousmagnetic nanowire\ncan boost current-induced domain wall (DW) propagation. DW motion in notch-modulated wires\ncan be classified into three phases: 1) A DW is pinned around a n otch when the current density\nis below the depinning current density. 2) DW propagation ve locity is boosted by notches above\nthe depinning current density and when non-adiabatic spin- transfer torque strength βis smaller\nthan the Gilbert damping constant α. The boost can be manyfold. 3) DW propagation velocity is\nhindered when β > α. The results are explained by using the Thiele equation.\nPACS numbers: 75.60.Ch, 75.78.-n, 85.70.Ay, 85.70.Kh\nI. INTRODUCTION\nMagnetic domain wall (DW) motion along a nanowire\nunderpins many proposals of spintronic devices1,2. High\nDW propagation velocity is obviously important because\nit determines the device speed. In current-driven DW\npropagation,many efforts havebeen devoted to high DW\nvelocity and low current density in order to optimize de-\nvice performance. The issue of whether notches can en-\nhance current-induced DW propagation is investigated\nhere.\nTraditionally, notches are used to locate DW\npositions1–4. Common wisdom expects notches to\nstrengthen DW pinning and to hinder DW motion. In-\ndeed, in the field-driven DW propagation, intentionally\ncreated roughness slows down DW propagation although\nthey can increase the Walker breakdown field5. Unlike\nthe energy-dissipation mechanism of field-induced DW\nmotion6, spin-transfer torque (STT)7–10is the driven\nforce behind the current-driven DW motion. The torque\nconsists of an adiabatic STT and a much smaller non-\nadiabatic STT9,10. In the absence of the non-adiabatic\nSTT, there exists an intrinsic pinning even in a homoge-\nneous wire, below which a sustainable DW motion is not\npossible11,12. Interestingly, there are indications13that\nthe depinningcurrentdensityofaDWtrappedinanotch\nis smaller than the intrinsic threshold current density in\nthe absence of the non-adiabatic STT. Although there is\nno intrinsic pinning1,10in the presence ofa non-adiabatic\nSTT, It is interesting to ask whether notches can boost\nDW propagation in the presence of both adiabatic STT\nand non-adiabatic STT.\nIn this paper, we numerically study how DW propa-\ngates along notch-modulated nanowires. Three phases\nare identified: pinning phase when current density is be-\nlow depinning current density ud; boosting phase and\nhindering phase when the current density is above ud\nandthe non-adiabaticSTT strength βissmallerorlarger\nthan the Gilbert damping constant α, respectively. The\naverage DW velocity in boosting and hindering phases\nis respectively higher and lower than that in the wire\nwithout notches. It is found that DW depinning is facili-tated by antivortex nucleation. In the case of β < α, the\nantivortexgenerationis responsiblefor velocityboost be-\ncause vortices move faster than transverse walls. In the\nother case of β > α, the longitudinal velocity of a vor-\ntex/antivortex is slower than that of a transverse wall in\nahomogeneouswallandnotcheshinderDWpropagation.\nII. MODEL AND METHOD\nWe consider sufficient long wires (with at least 8\nnotches) of various thickness and width. It is well known\nthat14narrowwiresfavoronlytransversewallswhilewide\nwires prefer vortex walls. Transverse walls are the main\nsubjects of this study. A series of identical triangular\nnotches of depth dand width ware placed evenly and\nalternately on the two sides of the nanowires as shown in\nFig. 1a with a typical clockwise transverse wall pinned\nat the center of the first notch. The x−,y−andz−axis\nare along length, width, and thickness directions, respec-\ntively. The magnetization dynamics of the wire is gov-\nerned by the Landau-Lifshitz-Gilbert (LLG) equation\n∂m\n∂t=−γm×Heff+αm×∂m\n∂t−(u·∇)m+βm×(u·∇)m,\nwherem,γ,Heff, andαare respectively the unit vec-\ntor of local magnetization, the gyromagnetic ratio, the\neffective field including exchange and anisotropy fields,\nand the Gilbert damping constant. The third and fourth\nterms on the right hand side are the adiabatic STT and\nnon-adiabatic STT10. The vector uis along the electron\nflow direction and its magnitude is u=jPµB/(eMs),\nwherej,P,µB,e, andMsare current density, current\npolarization,the Bohrmagneton, the electronchargeand\nthe saturation magnetization, respectively. For permal-\nloy ofMs= 8×105A/m,u= 100 m/s corresponds\ntoj= 1.4×1012A/m2. In this study, uis lim-\nited to be smaller than both 850 m/s (corresponding to\nj≃1.2×1013A/m2!) and the Walkerbreakdowncurrent\ndensity because current density above the values gener-\natesintensivespinwavesaroundDWsandnotches,which\nmakes DW motion too complicated to be even described.2\nxy\nz\n(b) (a) \nL\nw\ndj\nFIG. 1. (color online) (a) A notch-modulated nanowire. L\nis the separation between two adjacent notches. The color\ncodes the y−component of mwith red for my= 1, blue for\nmy=−1 and green for my= 0. The white arrows denote\nmagnetization direction. (b) The phase diagram in β−u\nplane. A is the pinning phase; B is the boosting phase; and C\nis the hindering phase. Vortices are (are not) generated nea r\nnotches by a propagating DW in C1 (C2). Inset: The notch\ndepth dependence of depinning current udwhen notch width\nis fixed at w= 48 nm.\nDimensionless quantity βmeasures the strength of non-\nadiabatic STT and whether βis larger or smaller than α\nis still in debate10,15,16. The LLGequation isnumerically\nsolved by both OOMMF17andMUMAX18packages19. The\nelectric current density is modulated according to wire\ncross section area while the possible change of current\ndirection around notch is neglected. The material pa-\nrameters are chosen to mimic permalloy with exchange\nstiffness A= 1.3×10−11J/m,α= 0.02 andβvarying\nfrom 0.002 to 0.04. The mesh size is 4 ×4×4 nm3.\nIII. RESULTS\nA. Transverse walls in wide wires: boosting and\nhindering\nThis is the focus of this work. Our simulations on\nwires of 4 nm thick and width ranging from 32 nm to\n128 nm and notches of d= 16 nm and wvarying from\n16 nm to 128 nm show similar behaviors. Domain walls\nin these wires are transverse. Results presented below\nare on a wire of 64 nm wide and notches of w= 48 nm.\nThree phases can be identified. A DW is pinned at a\nnotch when uis below a depinning current density ud.\nThis pinning phase is denoted as A (green region) in Fig.\n1b. Surprisingly, udincreases slightly with β, indicatingthat the β-term actually hinders DW depinning out of\na notch although it is responsible for the absence of the\nintrinsic pinning in a uniform wire (see discussion below\nfor possible cause). When uis above ud, a DW starts to\npropagate and it can either be faster or slower than the\nDW velocity in the corresponding uniform wire, depend-\ning on relative values of βandα.\nWhenβ < α, DW velocity is boosted through antivor-\ntexgenerationat notches. Thisphaseisdenoted asphase\nB. When β > α, the boosting of DW propagation is sup-\npressed no matter vortices are generated (phase C1) or\nnot (phase C2). The upper bound of the phase plane\nis determined by the Walker breakdown current density\nandu= 850 m/s. If the current density is larger than\nthe upper bound, spin waves emission from DW20and\nnotches are so strong that new DWs may be created.\nAlso, the Walker breakdown is smaller than the depin-\nning value udforβ >0.04. Thus the phase plane in Fig.\n1b is bounded by β= 0.04. Although the general phase\ndiagram does not change, the phase boundaries depend\non the wire and notch specificities. The inset is notch\ndepth dependence of the depinning current when w= 48\nnm andβ= 0.0121.\nBoosting phase: The boost of DW propagation for β < α\ncan be clearly seen in Fig. 2. Figure 2a is the average\nDW velocity ¯ vas a function of notch separation Lfor\nu= 600 m/s > ud. ¯vis maximal around an optimal\nnotch separation Lp, which is close to the longitudinal\ndistance that an antivortex travels in its lifetime. Lp\nincreases with βand it is respectively about 1.5 µm, 2\n��m, and 4 µm forβ= 0.005 (squares), 0.01 (circles) and\n0.015 (up-triangles). This result suggests that the an-\ntivortex generation and vortex dynamics are responsible\nfor the DW propagation boost. Filled symbols in Fig. 2b\nare ¯vfor various current density when Lpis used. For a\ncomparison, DW velocities in the corresponding homoge-\nneous wires are also plotted as open symbols which agree\nperfectlywith ¯ v=βu/αdiscussedbelow. Take β= 0.005\nas an example, ¯ vis zero below ud= 550 m/s and jumps\nto an average velocity ¯ v≃550 m/s at ud, which is about\nfour times of the DW velocity in the homogeneous wire.\nAs the current density further increases, the average ve-\nlocityalsoincreasesandisapproximatelyequalto u. The\ninset of Fig. 2b shows the instantaneous DW velocity for\nβ= 0.005 and u= 600 m/s. Blue dots denote the mo-\nments at which the DW is at notches. Right after the\ncurrent is turned on at t= 0 ns, the instantaneous DW\nvelocity is very low until an antivortex of winding num-\nberq=−123,24is generatednear the notch edge at 0.5ns\n(see discussion and Fig. 9 below). The motion of the an-\ntivortex core drags the whole DW to propagate forward\nat a velocity around 600 m/s. The antivortex core anni-\nhilatesitselfatthebottomedgeofthewireaftertraveling\nabout 1.5 µm and the initial transverse wall reverses its\nchirality at the same time24. Surprisingly, the reversal of\nDW chiralityleads to a significantincreasesofDW veloc-\nity as shown by the peaks of the instantaneous velocity\nat about 2.0ns in the inset. Another antivortex of wind-3\n(a) \n(b) \nFIG. 2. (color online) (a) L−dependence of average DW ve-\nlocity ¯vforu= 600 m/s, α= 0.02, andβ= 0.005 (squares),\n0.01 (circles), 0.015 (up-triangles). The dash lines are βu/α.\n(b)u−dependence of ¯ vforβ= 0.005 (squares), 0.01 (circles),\n0.015 (up-triangles). Open symbols are DW velocity in the\ncorresponding homogeneous wires. Straight lines are βu/α. ¯v\nis above βu/αwhenu > u d. Inset: instantaneous DW speed\nforu= 600 m/s, β= 0.005, and L= 1.5µm. The blue dots\nindicate the moments when the DW is at notches.\ning number q=−1 is generated at the second notch and\nDW propagation speeds up again. Once the antivortex\ncore forms, it pulls the DW out of notch. This process\nthen repeats itself and the DW propagates at an average\nlongitudinal velocity of about 600 m/s. A supplemental\nmovie corresponding to the inset is attached25.\nHindering phase: Things are quite different for β > α.\nFigure 3a shows that ¯ vincreases monotonically with L\nforβ= 0.025, 0.03 and 0.035, which are all larger than\nα. In order to make a fair comparison with the results of\nβ < α, Fig. 3b is the current density dependence of ¯ vfor\nL= 2µm andβ= 0.025 (filled squares), 0.03 (filled cir-\ncles)and0.035(filledup-triangles). Again,DWvelocities\nin the corresponding homogeneouswires are presented as\nopen symbols. Take β= 0.025 as an example, although\nthe average velocity jumps at the depinning current den-\nsity 565 m/s, it’s still well below the DW velocity in the\ncorresponding uniform wire. The inset of Fig. 3b shows(b) (a) \nFIG. 3. (color online) (a) L−dependence of ¯ vforu= 600\nm/s and β= 0.025 (squares), 0.03 (circles), and 0.035 (up-\ntriangles), all larger than α= 0.02. The dash lines are βu/α.\n(b)u−dependence of ¯ vforL= 2µm. Fill symbols (squares\nforβ= 0.025, circles for β= 0.03, and up-triangles for\nβ= 0.035) are numerical data in notched wire of w= 48\nnm andd= 16 nm. Open circles are DW velocity of the cor-\nrespondinghomogeneous wire. Straight lines are βu/α. Inset:\ninstantaneous DW velocity for u= 600 m/s and β= 0.025.\nThebluedotsdenotethemomentswhentheDWisatnotches.\nthe instantaneous DW velocity for u= 600 m/s. An\nantivortex is generated at the first notch. In contrast\nto the case of β < α, the antivortex slows down DW\npropagation velocity below the value in the correspond-\ning uniform wire. Moreover, the transverse wall keeps its\noriginal chirality unchanged when the antivortex is anni-\nhilated at wire edge, and no vortex/antivortex is gener-\nated at the second notch. However, another antivortex\nis generated at the third notch. This is the typical cycle\nof phase C1. As uincreases above 640 m/s, phase C1\ndisappears and the DW passes all the notches without\ngenerating any vortices. This motion is termed as phase\nC2. For β >0.025, only phase C2 is observed. In C2,\nDW profile is not altered, and the average DW velocity\nis slightly below that in a uniform wire.4\n(b) (a) \nFIG. 4. (color online) (a) u−dependence of ¯ vforβ=\n0.01 (filled circles) and 0.015 (filled up-triangles). (b)\nu−dependence of ¯ vforβ= 0.03 (filled squares) and 0.035\n(filled up-triangles). Open symbols are DW velocity in the\ncorresponding homogeneous wires. Straight lines are βu/α. ¯v\nis below βu/αwhenu > u d. The nanowire is 8 nm wide and\n1 nm thick while the notch size is 10 nm wide and 2 nm deep\nfor (a) and 50 nm wide and 2 nm deep for (b). The separation\nof adjacent notches is 100 nm.\nB. Transverse walls in very narrow wires\nOne interesting question is whether notches can boost\nDW propagation in very narrow wires such that the nu-\ncleation of a vortex/antivortex is highly unfavorable. To\naddress this issue, Fig. 4a are u−dependence of the av-\nerage DW velocity on a 8 nm wide wire for β < α(circles\nforβ= 0.01 and up-triangles for β= 0.015) with (filled\nsymbols) and without (open symbols) notches. When\nnotches are placed, notch depth is 2 nm, L= 100 nm,\nw= 10 nm. DW velocity in the corresponding ho-\nmogeneous wire (open symbols) follows perfectly with\n¯v=βu/α(straight lines). It is clear that averaged DW\nvelocity in the notched wire (filled symbols) is below the\nvalues of the DW velocity in the corresponding homo-\ngeneous wire. Take β= 0.015 as an example, ¯ vis zero\nbelowud= 310 m/s and jumps to an average velocity\n¯v≃168 m/s at ud, which is below the DW velocity in\nthe corresponding uniform wire.\nThings are similar for β > α. Figure 4b is the cur-\nrent density dependence of ¯ vforβ= 0.03 (filled squares)\nand 0.035 (filled up-triangles). Again, DW velocities in\nhomogeneous wire are presented as open symbols for a\ncomparison. The averaged DW velocity in the notched\nwire (filled symbols) is below the values of the DW ve-\nlocity in the corresponding homogeneous wire.\nC. Vortex walls in very wide wires\nAlthough our main focus is on transverse walls, it\nshould be interesting to ask whether DW propagation\nboost can occur for vortex walls. It is well-known that\na vortex/antivortex wall is more stable for a much wider\nwire in the absence of a field and a current14. One\nmay expect that DW propagation boost would not oc-\ncur in such a wire because the boost comes from vor-\ntex/antivortex generation near notches and a such vor-\ntex/antivortex exists already in a wider wire even in the\nmy\n+1 -1 0200 nm 1.5 ns 13.5 ns \n18.0 ns 26.0 ns 0 ns \n47.5 ns 14.5 ns 0 ns (c) \n(d) (a) (b) \nFIG. 5. (color online) (a) u−dependence of ¯ vforβ=\n0.01 (filled circles) and 0.015 (filled up-triangles). (b)\nu−dependence of ¯ vforβ= 0.025 (filled squares) and 0.03\n(filled circles). Open symbols are DW velocity in the corre-\nsponding homogeneous wires. Straight lines are βu/α. ¯vis\nabove (below) βu/αwhenu > u dandβ < α(β > α). The\nnanowire is 520 nm wide and 10 nm thick while the rectan-\ngular notch is 160 nm wide and 60 nm deep. The separation\nof adjacent notches is 8 µm. (c) and (d) The spin configu-\nrations in a uniform wire (a) and in a notched wire (b) at\nvarious moments for β= 0.01 andu= 650 m/s. The time\nis indicated on the bottom-right corner of each configuratio n.\nThe color codes the value of myand color bar is shown in the\nbottom-right corner.\nabsence of a current. However, DW propagation boost\nwas still observed as shown in Fig. 5 for a wire of 520\nnm wide and 10 nm thick. Rectangular notches of 60\nnm deep and 160 nm wide are separated by L= 8µm.\nWhenβ < α(Fig. 5a: circles for β= 0.01 and up-\ntriangles for β= 0.015), the average DW propagation\nvelocities in the notched wire (filled symbols) is higher\nthan the DW velocity in the corresponding homogeneous\nwire (open symbols) when u > ud. Figure 5b shows that\nthe average DW propagation velocities in a notched wire\n(filled symbols) is lower than that in the corresponding\nhomogeneous wire (open symbols) for β > α(squares for\nβ= 0.025 and circles for β= 0.03). Figure 5c shows the\nspin configurations of the DW in the homogeneous wire\nofβ= 0.01before a current is applied (the left configura-\ntion) and during the current-driven propagation (middle\nand right configurations). When a current u= 650 m/s\nis applied at 0 ns, a vortex wall moves downward. The\nvortex was annihilated at wire edge, and the vortex wall\ntransformintoatransversewall. TheDWkeepsitstrans-\nverse wall profile and propagates with velocity of βu/α\n(solid lines in Fig. 5a and 5b). The middle and right\nconfigurations are two snapshots at 14.5 ns and 47.5 ns.\nTime is indicated in the bottom-right corner. Figure 5d\nare snapshots of DW spin configurations in the notched\nwire ofβ= 0.01 when a current u= 650 m/s is applied5\nu\n    m\nxxy\nzu\n    m\nx(a) (b)\nFIG. 6. (color online) Directions of vortex core magnetizat ion\n(red symbols) and non-adiabatic torque (blue symbols) for a\nclockwise transverse wall (a) and a counterclockwise trans -\nverse wall (b). The dots (crosses) represent ±z-direction.\natt= 0 ns. At t= 0 ns, a vortex wall is pinned near\nthe first notch. Right after the current is turned on, the\nvortex wall starts to depin and complicated structures\nmay appear during the depinning process as shown by\nthe snapshot at t= 1.5 ns. At t= 13.5 ns, the DW\ntransforms to a transverse wall and propagates forward.\nWhen the transverse wall reaches the second notch at\naboutt= 18.0 ns, new vortex core nucleates near the\nnotch and drags the whole DW to propagate forward. In\ncontrast to the case of homogeneous wire where a prop-\nagating DW prefers a transverse wall profile, DW with\nmore than one vortices can appear as shown by the snap-\nshot att= 26.0 ns. The vortex core in this structure\nboosts DW velocity above the average DW velocity of a\nuniform wire. This finding may also explain a surprising\nobservation in an early experiment4that depinning cur-\nrent does not depend on DW types. A vortex wall under\na current transforms into a transverse wall before depin-\nning from a notch. Thus both vortex wall and transverse\nwall have the same depinning current.\nIV. DISCUSSION\nA. Depinning process analysis\nEmpirically, we found that vortex/antivortex polarity\nis uniquely determined by the types of transverse wall\nand current direction. This result is based on more than\ntwenty simulations that we have done by varying var-\nious parameters like notch geometry, wire width, mag-\nnetic anisotropy, damping etc. Within the picture that\nDW depinning starts from vortex/antivortex nucleation,\ntheβ−dependence of depinning current density udcan\nbe understood as follows. For a clockwise (counter-\nclockwise) transverse wall and current in −xdirection,\np= +1 (p=−1), as shown in Fig. 6. If one as-\nsumes that vortex/antivortex formation starts from the\nvortex/antivortex core, it means that the core spin ro-\ntates into + z-direction for a vortex of p= 1. For a clock-\nwise wall, β-torque ( βm×∂m\n∂x) tends to rotate core spin\nin−z-direction, as shown in Fig. 6a, so the presence of\na smallβ-torque tries to prevent the nucleation of vor-\ntices. Thus, the larger βis, the higher udwill be. This\nmay be the reason why the depinning current density ud\nincreases as βincreases.(a) (b)\nFIG. 7. (a) Depinning current density as a function of an\nexternal field. A 0.4 ns field pulse in the x-direction is turned\non simultaneously with the current. The shape of a pulse of\nH= 100 Oe is shown in the inset. Since the depinning field\nof the wire (64 nm wide and 4 nm thick) is 150 Oe, the field\namplitude is limited to slightly below 150 Oe in the curve. (b )\nDepinningcurrentdensityas afunctionof nanowire thickne ss.\nOur simulations suggest that DW depinning starts\nfrom vortex/antivortexnucleation. Adiabatic spin trans-\nfertorquetendstorotatethespinsattheedgedefectnear\na notch out of plane and to form a vortex/antivortex\ncore. Thus, any mechanisms that help (hinder) the\ncreation of a vortex/antivortex core shall decrease (in-\ncrease) the depinning current density ud. To test this\nhypothesis, we use a magnetic field pulse of 0.4 ns along\n±x−direction (shown in the inset of Fig. 7a) such that\nthe field torque rotates spins out of plane. Figure 7a\nis the numerical results of the magnetic field depen-\ndence of the depinning current density for a 64 nm wide\nwire with triangular notches of 48 nm wide and 16 nm\ndeep. The non-adiabatic coefficient is β= 0.01. As\nexpected, uddecreases (increases) with field when it is\nalong -x−direction (+ x−direction) so that spins rotate\ninto +z−direction (- z−direction). All other parameters\nare the same as those for Fig. 2.\nIf the picture is correct, one should also expect the\ndepinning current density depends on the wire thick-\nness. The shape anisotropy impedes vortex core for-\nmation because it does not favor a spin aligning in the\nz−direction. The shape anisotropy decreases as the\nthickness increases. Thus, one should expect the depin-\nning current density decreases with the increase of wire\nthickness. Indeed, numerical results shown in Fig. 7b\nverifiestheconjecture. All otherparametersarethe same\nas that in Fig. 7a ( H= 0).\nB. Width effects on the depinning current density\nThe DW propagating boost shown above is from the\nwire in which the notch depth (16 nm) is relatively big in\ncomparisonwith wire width (64 nm). Naturally, one may\nask whether the DW propagation boost exists also in a\nwire when the notch depth is much smaller than the wire\nwidth. To address the issue, we fix the notch geometry\nand vary the wire width. Figure 8 is the nanowire width\ndependence of depinning current density when the notch6\n(a) (b) \n(c) (d) 50 nm 50 nm \nFIG. 8. (color online) (a) and (b) are nanowire width de-\npendence of depinning current density for β= 0.005 (a) and\nβ= 0.01 (b), respectively. The wire thickness is 4 nm and\nnotch size is fixed at 48 ×16 nm2. (c) and (d) are the real\nconfigurations of initial domain walls pinned at the notch fo r\n64 nm and 160 nm wide wires, respectively. The color coding\nis the same as that of Fig. 5. The blue jagged lines indicate\nthe profiles of triangular notches.\nsize is fixed at 48 ×16 nm2. Figures 8a and 8b show the\nphase boundary between vortex-assisted boosting phase\nand the pinning phase. DW propagation boost exists\nwhen nanowire width is one order of magnitude larger\nthan the notch depth. The top view of the wire and spin\nconfigurations for 64 nm wide and 160 nm wide wires are\nshown in Fig. 8c and Fig. 8d, respectively.\nC. DW Propagation and vortex dynamics\nDW propagation boost and slow-down by vortices can\nbe understood from the Thiele equation10,26,27,\nF+G×(v−u)+D·(αv−βu) = 0,(1)\nwhereFis the external force related to magnetic field\nthat is zero in our case, Gis gyrovector that is zero for a\ntransverse wall and G=−2πqplM s/γˆ zfor a 2D vortex\nwall, where qis the winding number (+1 for a vortex and\n-1foranantivortex), pisvortexpolarity( ±1forcorespin\nin±zdirection) and lis the thickness of the nanowire. D\nis dissipation dyadic, whose none zero elements for a vor-\ntex/antivortex wall are Dxx=Dyy=−2MsWl/(γ∆)27,\nwhereWis nanowire width and ∆ is the Thiele DW\nwidth26.vis the DW velocity.\nFor a transverse wall, v=βu/α(solid lines) agrees\nperfectly with numerical results (open symbols) in ho-\nmogeneous wires as shown in Figs. 2b and 3b without\nany fitting parameters. For a vortex wall, the DW veloc-\nity is\nvy=1\n1+α2W2/(π2∆2)W\nπqp∆(α−β)u,(2)\nvx=u\n1+α2W2/(π2∆2)/parenleftbigg\n1−β\nα/parenrightbigg\n+βu\nα.(3)vydepends on DW width, αas well as β/α. For a given\nvortexwall, vyhas opposite sign for β < αandβ > α. In\nterms of topological classification of defects23, the edge\ndefect of the transverse DW at the first notch (Fig. 1a)\nhas winding number q=−1/2, and this edge defect can\nonlygivebirthtoanantivortexof q=−1andp= 1while\nitself changes to an edge defect of q= 1/2 as shown in\nFig. 9a. Empirically, we found that antivortexpolarityis\nuniquely determined by the types of transverse wall and\ncurrent direction. A movie visualizing the DW propa-\ngation in boosting phase is shown in the Supplemental\nMovie25. All the parameters are the same as the inset of\nFig. 2b. The three segments of identical length 1200 nm\nare connected in series to form a long wire. When β < α,\nthe antivortex moves downward ( vy<0) to the lower\nedge defect of winding number of q= 1/2. The lower\nedge defect changes its winding number to q=−1/2\nand the transverse DW reverses its chirality24when the\nvortex merges with the edge defect. Then another an-\ntivortex of winding number q=−1 andp=−1 is gen-\nerated at the second notch on the lower wire edge and\nit moves upward ( vy>0). The DW reverses its chiral-\nity again at upper wire edge when the antivortex dies.\nThen this cycle repeats itself. The spin configurations\ncorresponding to various stages are shown in the lower\npanels of Fig. 9a. When β > α, as shown in Fig. 9b,\nthe antivortex of q=−1 andp= +1 moves upward\nsincevy>0. The chirality of the original transverse\nwall shall not change when the antivortex is annihilated\nat the upper edge defect because of winding number con-\nservation. No antivortex is generated at the even number\nnotches and same type of the antivortex is generated at\nodd number notches, hence the transverse wall preserves\nits chirality throughout propagation. The corresponding\nspin configurations are shown in the lower panels of Fig.\n9b.\nThe second term in Eq. (3) (for vx) isβu/α, the same\nas the transverse DW velocity in a homogeneous wire\n(straight lines in Figs. 2b and 3b). The first term de-\npends on DW properties as well as βandα. It changes\nsign atβ=α.vxis larger than βu/αin the presence of\nvortices if β < α. Therefore, in this case vortex genera-\ntions and vortex dynamics boost DW propagation. For\nsmallαand to the leading order correction in αandβ,\nEq. (3) becomes vx=u−(α2−αβ)uW2/(π2∆2). Thus,\nthe longitudinal velocity equals approximately uand de-\npends very weakly on β. This is what was observed in\nFig. 2b. vx=ucorresponds to the complete conversion\nof itinerant electron spins into local magnetic moments.\nAlthough the Thiele equation cannot explain why a DW\ngenerates vortices around notches in phase B, it explains\nwell DW propagation boost for β < α. This result is in\ncontrast to the field-driven DW propagation where vor-\ntex/antivortexgenerationreduces the Walker breakdown\nfield and inevitably slows down DW motion5,24.\nBefore conclusion, we would also like to point out that\nit is possible to realize both β < α(boosting phase) and\nβ > α(hindering phase) experimentally in magnetic ma-7\n(a) \n(b) +1/2 +1/2 +1/2 +1/2 \n+1/2 +1/2 \n+1/2 +1/2 \n+1/2 +1/2 \n-1/2 -1/2 -1/2 \n+1/2 +1/2 +1/2 +1/2 +1/2 +1/2 +1/2 +1/2 -1/2 -1/2 +1/2 -1/2 -1/2 -1/2 \n-1 \n-1 -1 \n-1 -1 \n-1/2 \n(s1) (s2) (s3) (s4) (s5) \ns1 \ns1 (s1) (s2) (s3) (s4) (s5) s2 \ns2 s3 \ns3 s4 \ns4 s5 \ns5 (s6) \n(s6) 50 nm \ns6 \ns6 \nFIG. 9. (color online) (a) Illustrations of changes of topol og-\nical defects (transverse DW edge defects and vortices) duri ng\nthebirthanddeathofvortices inPhase Bas aDWpropagates\nfrom the left to the right and the corresponding spin config-\nurations at various moments. Lines represent DWs. Big blue\ndots for vortices and open circles for edge defects of wind-\ning number −1/2 and filled black circles for edge defects of\nwinding number 1 /2. The color coding is the same as that of\nFig. 5. The blue jagged lines indicate the profiles of trian-\ngular notches. The nanowire is 64 nm wide and 4 nm thick.\nThe notch dimensions are 48 ×16 nm3. The interval between\nadjacent notches is L= 1500 nm. u= 600 m/s, β= 0.005.\n(b) Illustrations of changes of topological defects in Phas e C1\nand the the corresponding spin configurations at various mo-\nments. The nanowire is 64 nm wide and 4 nm thick. The\nnotch dimensions are 48 ×16 nm2. The interval between ad-\njacent notches is L= 2000 nm. u= 600 m/s, β= 0.025.terials like permalloy with damping coefficient engineer-\ning. A recent study28demonstrated that αof permalloy\ncan increaseby four times througha dilute impurity dop-\ning of lanthanides (Sm, Dy, and Ho).\nV. CONCLUSIONS\nIn conclusion, notches can boost DW propagation\nwhenβ < α. The boost is facilitated by antivortex\ngeneration and motion, and boosting effect is optimal\nwhen two neighboring notches is separated by the dis-\ntance that an antivortex travels in its lifetime. In the\nboosting phase, DW can propagate at velocity uthat\ncorresponds to a complete conversion of itinerant elec-\ntron spins into local magnetic moments. When β > α,\nthe notches always hinder DW propagation. According\nto Thiele’s theory, the generation of vortices increases\nDW velocity for β < αand decreases DW velocity when\nβ > α. This explains the origin of boosting phase and\nhindering phase. Furthermore, it is found that a vortex\nwall favored in a very wide wire tends to transform to a\ntransverse wall under a current. This may explain exper-\nimental observation that the depinning current density is\nnot sensitive to DW types.\nVI. ACKNOWLEDGMENTS\nWe thank Gerrit Bauer for useful comments. HYY ac-\nknowledges the support of Hong Kong PhD Fellowship.\nThis work was supported by NSFC of China (11374249)\nas well as Hong Kong RGC Grants (163011151 and\n605413).\n∗Corresponding author: phxwan@ust.hk\n1S. S. P. Parkin, M. Hayashi, and L. Thomas, Science 320,\n190 (2008).\n2D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D.\nPetit, and R. P. Cowburn, Science 309, 1688 (2005).\n3M. Kl¨ aui, C. A. F. Vaz, J. A. C. Bland, W. Wernsdorfer,\nG. Faini, E. Cambril, L. J. Heyderman, F. Nolting, and U.\nR¨ udiger, Phys. Rev. Lett. 94, 106601 (2005).\n4M. Hayashi, L. Thomas, C. Rettner, R. Moriya, X. Jiang,\nand S. S. P. Parkin, Phys. Rev. Lett. 97, 207205 (2006).\n5Y. Nakatani, A. Thiaville, and J. Miltat, Nat. Mater. 2,\n521 (2003).\n6X. R. Wang, P. Yan, J. Lu and C. He, Ann. Phys. (N.Y.)\n324, 1815 (2009); X. R. Wang, P. Yan, and J. Lu, Euro-\nphys. Lett. 86, 67001 (2009).\n7L. Berger, J. Appl. Phys. 55, 1954 (1984).\n8J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996).\n9S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).10A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Euro-\nphys. Lett. 69, 990 (2005).\n11Z. Li and S. Zhang, Phys. Rev. B 70, 024417 (2004).\n12G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601\n(2004).\n13H. Y. Yuan and X. R. Wang, European Physical Journal\nB (in press); arXiv:1407.4559 [cond-mat.mes-hall]\n14R. D. McMichael and M. J. Donahue, IEEE Trans. Magn.\n33, 4167 (1997).\n15G. S. D. Beach, C. Knutson, C. Nistor, M. Tsoi, and J. L.\nErskine, Phys. Rev. Lett. 97, 057203 (2006).\n16L. Thomas, R. Moriya, C. Rettner, and S. S. P. Parkin,\nScience330, 1810 (2010).\n17http://math.nist.gov/oommf.\n18A. Vansteenkiste, J. Leliaert, M. Dvornik, M. Helsen, F.\nGarcia-Sanchez, F. B. V. Waeyenberge, AIP Adbances 4,\n107133 (2014).8\n19OOMMF package was used in the early stage of this re-\nsearch. In order to simulate a long and wide wire, we\nswitched to MUMAX package. Two packages give almost\nidentical results on shorter wires, and the results present ed\nhere were generated from MUMAX.\n20B. Hu and X. R. Wang, Phys. Rev. Lett. 111, 027205\n(2013); X. S. Wang, P. Yan, Y. H. Shen, G. E.W. Bauer,\nand X. R. Wang, Phys. Rev. Lett. 109, 167209 (2012).\n21Notch geometry affects depinning current because of\nthe change of current density and perpendicular shape\nanisotropy (see Ref. 22) in notch area. Both effects help to\ngenerate vortices and thus reduce the depinning current.This may explain the result.\n22A. Aharoni, J. Appl. Phys. 83, 3432 (1998).\n23O. Tchernyshyov and G. -W. Chern, Phys. Rev. Lett. 95,\n197204 (2005).\n24H. Y. Yuan and X. R. Wang, J. Magn. Magn. Mater. 368,\n70 (2014).\n25See Supplemental Material at [URL] for DW propagation\nin boosting phase.\n26A. A. Thiele, Phys. Rev. Lett. 30, 230 (1973).\n27D. L. Huber, Phys. Rev. B 26, 3758 (1982).\n28S. G. Reidy, L. Cheng, and W. E. Bailey, Appl. Phys. Lett.\n82, 1254 (2003)."    },    {        "title": "1603.07977v1.Large_spin_pumping_effect_in_antisymmetric_precession_of_Ni___79__Fe___21___Ru_Ni___79__Fe___21__.pdf",        "content": "Large spin pumping e\u000bect in antisymmetric precession of\nNi79Fe21/Ru/Ni 79Fe21\nH. Yang,1Y. Li,1and W.E. Bailey1,a)\nMaterials Science and Engineering, Dept. of Applied Physics and Applied Mathematics, Columbia University,\nNew York NY 10027\n(Dated: 16 September 2021)\nIn magnetic trilayer structures, a contribution to the Gilbert damping of ferromagnetic resonance arises from\nspin currents pumped from one layer to another. This contribution has been demonstrated for layers with\nweakly coupled, separated resonances, where magnetization dynamics are excited predominantly in one layer\nand the other layer acts as a spin sink. Here we show that trilayer structures in which magnetizations are\nexcited simultaneously, antisymmetrically, show a spin-pumping e\u000bect roughly twice as large. The antisym-\nmetric (optical) mode of antiferromagnetically coupled Ni 79Fe21(8nm)/Ru/Ni 79Fe21(8nm) trilayers shows a\nGilbert damping constant greater than that of the symmetric (acoustic) mode by an amount as large as\nthe intrinsic damping of Py (\u0001 \u000b'0.006). The e\u000bect is shown equally in \feld-normal and \feld-parallel to\n\flm plane geometries over 3-25 GHz. The results con\frm a prediction of the spin pumping model and have\nimplications for the use of synthetic antiferromagnets (SAF)-structures in GHz devices.\nPumped spin currents1,2are widely understood to in-\n\ruence the magnetization dynamics of ultrathin \flms\nand heterostructures. These spin currents increase the\nGilbert damping or decrease the relaxation time for thin\nferromagnets at GHz frequencies. The size of the e\u000bect\nhas been parametrized through the e\u000bective spin mixing\nconductance g\"#\nr, which relates the spin current pumped\nout of the ferromagnet, transverse to its static (time-\naveraged) magnetization, to its precessional amplitude\nand frequency. The spin mixing conductance is inter-\nesting also because it determines the transport of pure\nspin current across interfaces in quasistatic spin trans-\nport, manifested in e.g. the spin Hall e\u000bect.\nIn the spin pumping e\u000bect, spin current is pumped\naway from a ferromagnet / normal metal (F 1/N) in-\nterface, through precession of F1, and is absorbed else-\nwhere in the structure, causing angular momentum loss\nand damping of F1. The spin current can be absorbed\nthrough di\u000berent processes in di\u000berent materials. When\ninjected into paramagnetic metals (Pt, Pd, Ru, and oth-\ners), the spin current relaxes exponentially with para-\nmagnetic layer thickness3{5. The relaxation process has\nbeen likened to spin-\rip scattering as measured in CPP-\nGMR, where spin-\rip events are localized to heavy-metal\nimpurities6and the measurement reveals the spin di\u000bu-\nsion length \u0015SD. When injected into other ferromagnets\n(F2in F 1/N/F 2), the spin current is absorbed through\nits torque on magnetization5,7. A similar process appears\nto be relevant for antiferromagnets as well8.\nIn F 1/N/F 2structures, only half of the total possible\nspin pumping e\u000bect has been detected up until now. For\nwell-separated resonances of F1andF2, only one layer\nwill precess with large amplitude at a given frequency\n!, and spin current is pumped from a precessing F1into\na staticF2. If both layers precess symmetrically, with\na)Electronic mail: Contact author. web54@columbia.eduthe same amplitude and phase, equal and opposite spin\ncurrents are pumped into and out of each layer, causing\nno net e\u000bect on damping. The di\u000berence between the\nsymmetric mode and the uncoupled mode, increased by\na spin pumping damping \u000bspwas detected \frst in epi-\ntaxial Fe/Au/Fe structures9. However, if the magnetiza-\ntions can be excited with antisymmetric precession, the\ncoupled mode should be damped by twice that amount,\n2\u000bsp. Takahashi10has published an explicit prediction of\nthis \"giant spin pumping e\u000bect\" very recently, including\nan estimate of a fourfold enhanced spin accumulation in\nthe central layer.\nIn this paper, we show that a very large spin pump-\ning e\u000bect can be realized in antisymmetric precession of\nPy(8 nm)/Ru(0.70-1.25 nm)/Py(8 nm) synthetic antifer-\nromagnets (SAF, Py=Ni 79Fe21). The e\u000bect is roughly\ntwice that measured in Py trilayers with uncoupled\nprecession. Variable-frequency ferromagnetic resonance\n(FMR) measurements show, for structures with magne-\ntization saturated in the \flm plane or normal to the \flm\nplane, that symmetric (acoustic mode) precession of the\ntrilayer has almost no additional damping, but the an-\ntisymmetric (optical mode) precession has an additional\nGilbert damping of \u00180.006, compared with an uncou-\npled Py(8nm) layer in a F 1/N/F 2structure of\u00180.003.\nThe interaction stabilizes the antiparallel magnetization\nstate of SAF structures, used widely in di\u000berent elements\nof high-speed magnetic information storage, at GHz fre-\nquencies.\nMethod: Ta(5 nm)/Cu(3 nm)/Ni 79Fe21(8\nnm)/Ru(tRu)/Ni 79Fe21(8 nm)/Cu(3 nm)/SiO 2(5 nm),\ntRu= 0.7 - 1.2 nm heterostructures were deposited by\nultrahigh vacuum (UHV) sputtering at a base pressure\nof 5\u000210\u00009Torr on thermally oxidized Si substrates.\nThe Ru thckness range was centered about the second\nantiferromagnetic maximum of interlayer exchange\ncoupling (IEC) for Py/Ru/Py superlattices, 8-12 \u0017A,\nestablished \frst by Brillouin light scattering (BLS)\nmeasurement11. Oscillatory IEC in this system, as aarXiv:1603.07977v1  [cond-mat.mtrl-sci]  25 Mar 20162\nfunction of tRu, is identical to that in the more widely\nstudied Co/Ru( tRu)/Co superlattices12, 11.5 \u0017A, but is\nroughly antiphase to it. An in-plane magnetic \feld bias\nof 200 G, rotating in phase with the sample, was applied\nduring deposition as described in13.\nThe \flms were characterized using variable fre-\nquency, swept-\feld, magnetic-\feld modulated ferromag-\nnetic resonance (FMR). Transmission measurements\nwere recorded through a coplanar waveguide (CPW) with\ncenter conductor width of 300 \u0016m, with the \flms placed\ndirectly over the center conductor, using a microwave\ndiode signal locked in to magnetic \feld bias modulation.\nFMR measurements were recorded for magnetic \feld bias\nHBapplied both in the \flm plane (parallel condition, pc)\nand perpendicular to the plane (normal condition, nc.)\nAn azimuthal alignment step was important for the nc\nmeasurements. For these, the sample was rotated on twoaxes to maximize the \feld for resonance at 3 GHz.\nFor all FMR measurements, the sample magnetization\nwas saturated along the applied \feld direction, simplify-\ning extraction of Gilbert damping \u000b. The measurements\ndi\u000ber in this sense from low-frequency measurements of\nsimilar Py/Ru/Py trilayer structures by Belmenguenai et\nal14, or broadband measurements of (sti\u000ber) [Co/Cu] \u000210\nmultilayers by Tanaka et al15. In these studies, e\u000bects\non\u000bcould not be distinguished from those on inhomo-\ngeneous broadening.\nModel: In the measurements, we compare the mag-\nnitude of the damping, estimated by variable-frequency\nFMR linewidth through \u0001 H1=2= \u0001H0+ 2\u000b!=j\rj, and\nthe interlayer exchange coupling (IEC) measured through\nthe splitting of the resonances. Coupling terms between\nlayersiandjare introduced into the Landau-Lifshitz-\nGilbert equations for magnetization dynamics through\n_mi=\u0000mi\u0002(\riHe\u000b+!ex;imj) +\u000b0mi\u0002_mi+\u000bsp;i(mi\u0002_ mi\u0000mj\u0002_ mj) (1)\nincgsunits, where we de\fned magnetization unit vec-\ntors as m1=M1=Ms;1,m2=M2=Ms;2withMs;ithe\nsaturation moments of layer i. The coupling constants\nare, for the IEC term, !ex;i\u0011\riAex=(Ms;iti), where\nthe energy per unit area of the system can be written\nuA=\u0000Aexmi\u0001mj, andtiis the thickness of layer i. Pos-\nitive values of Aexcorrespond to ferromagnetic coupling,\nnegative values to antiferromagnetic coupling. The spin\npumping damping term is \u000bsp;i\u0011\r\u0016h~gFNF\n\"#=(4\u0019Ms;iti),\nwhere ~gFNF\n\"# is the spin mixing conductance of the tri-\nlayer in units of nm\u00002.\u000b0is the bulk damping for the\nlayer.\nThe collective modes of 1 ;2 are found from small-\namplitude solutions of Equations 1 for i= 1;2. General\nsolutions for resonance frequencies with arbitrary mag-\nnetization alignment, not cognizant of any spin pump-\ning damping or dynamic coupling, were developed by\nZhang et al12. In our experiment, to the extent possi-\nble, layers 1 ;2 are identical in deposited thickness, mag-\nnetization, and interface anisotropy (each with Cu the\nopposite side from Ru). Therefore if !irepresents the\nFMR frequency (dependent on bias \feld HB) of each\nlayeri, the two layers have !1=!2=!0. In this\nlimit, there are two collective modes: a perfectly sym-\nmetric mode Sand a perfectly antisymmetric mode A\nwith complex frequencies f!S= (1\u0000i\u000b0)!0andf!A=\n(1\u0000i\u000b0\u00002i\u000bsp) (!0+ 2!ex). The Gilbert damping for\nthe modes, \u000bk=\u0000Im(f!k)=Re(!k\nf), wherek= (S;A),\nand the resonance \felds Hk\nBsatisfy\nHA\nB=HS\nB+ 2HexHex=\u0000Aex=(MstF) (2)\n\u000bA=\u000bS+ 2\u000bsp\u000bsp=\r\u0016h~gFNF\n\"#=(4\u0019Ms;iti) (3)\nand!ex=\rHex. Note that there is no relationshipin this limit between the strength of the exchange cou-\nplingAexand the spin-pumping damping 2 \u000bspexpressed\nin the antisymmetric mode. The spin pumping damping\nand the interlayer exchange coupling can be read sim-\nply from the di\u000berences in the the Gilbert damping \u000b\nand resonance \felds between the antisymmetric and sym-\nmetric modes. The asymmetric mode will have a higher\ndamping by 2 \u000bspfor anyAexand a higher resonance\n\feld forAex<0, i.e. for antiferromagnetic IEC: because\nthe ground state of the magnetization is antiparallel at\nzero applied \feld, antisymmetric excitation rotates mag-\nnetizations towards the ground state and is lower in fre-\nquency than symmetric excitation.\nResults: Samplepc\u0000andnc\u0000FMR data are shown in\nFigure 1. Raw data traces (lock-in voltage) as a func-\ntion of applied bias \feld HBat 10 GHz are shown in the\ninset. We observe an intense resonance at low \feld and\nresonance weaker by a factor of 20-100 at higher \feld. On\nthe basis of the intensities, as well as supporting MOKE\nmeasurements, we assign the lower-\feld resonance to the\nsymmetric, or \"acoustic\" mode and the higher-\feld res-\nonance to the antisymmetric, or \"optical\" mode. Similar\nbehavior is seen in the nc- andpc\u0000FMR measurements.\nIn Figure 1a) and c), which summarizes the \felds-\nfor-resonance !(HB), there is a rigid shift of the\nantisymmetric-mode resonances to higher bias \felds HB,\nas predicted by the theory. The lines show \fts to\nthe Kittel resonance, !pc=\rr\nHeff\u0010\nHeff+ 4\u0019Meff\ns\u0011\n,\n!nc=\r\u0000\nHeff\u00004\u0019Meff\ns\u0001\nwith an additional e\u000bective\n\feld along the magnetization direction for the antisym-\nmetric mode: Heff;S =HB, andHeff;A =HB\u0000\n8\u0019Aex=(4\u0019MstF).\nIn Figure 2, we show coupling parameters, as a func-3\nHBHeffm(t)\nHBHeff m(t)(a)\n(c)(b)\n(d)10 GHz10 GHz\n1/21/2\nFIG. 1. FMR measurement of Ni 79Fe21(8\nnm)/Ru(tRu)/Ni 79Fe21(8 nm) trilayers; example shown\nfortRu= 1.2 nm. Inset : lock-in signal, transmitted power\nat 10 GHz, as a function of bias \feld HB, for a) pc-FMR\nand c) nc-FMR. A strong resonance is observed at lower HB\nand a weaker one at higher HB, attributed to the symmetric\n(S) and antisymmetric (A) modes, respectively. a), c): Field\nfor resonance !(HB) for the two resonances. Lines are \fts\nto the Kittel resonance expression, assuming an additional,\nconstant, positive \feld shift for !A,Hex=\u00002Aex=(MstF)\ndue to antiferromagnetic interlayer coupling Aex<0. b)\npc-FMR and d) nc-FMR linewidth as a function of frequency\n\u0001Hpp(!) for \fts to Gilbert damping \u000b.\ntion of Ru thickness, extracted from the FMR measure-\nments illustrated in Figure 1. Coupling \felds are mea-\nsured directly from the di\u000berence between the symmet-\nric and antisymmetric mode positions and plotted in\nFigure 2a. We convert the \feld shift to antiferromag-\nnetic IEC constant Aex<0 through Equation 2, us-\ning the thickness tF= 8 nm and bulk magnetization\n4\u0019Ms= 10.7 kG4. The extracted exchange coupling\nstrength in pc-FMR has a maximum antiferromagnetic\nvalue ofAex=\u00000.2 erg/cm2, which agrees to 5% with\nthat measured by Fassbender et al11for [Py/Ru] Nsu-\nperlattices.\nThe central result of the paper is shown in Figure 2 b).\nWe compare the damping \u000bof the symmetric ( S) and an-\ntisymmetric ( A) modes, measured both through pc-FMR\nandnc-FMR. The values measured in the two FMR ge-\nometries agree closely for the symmetric modes, for which\nsignals are larger and resolution is higher. They agree\nroughly within experimental error for the antisymmetric\nmodes, with no systematic di\u000berence. The antisymmetric\nmodes clearly have a higher damping than the symmetric\nmodes. Averaged over all thickness points, the enhanced\ndamping is roughly \u000bA\u0000\u000bS= 0.006.\nDiscussion: The damping enhancement of the anti-\nsymmetric ( A\u0000) mode over the symmetric ( S\u0000) mode,\nshown in Figure 2b), is a large e\u000bect. The value is close\nto the intrinsic bulk damping \u000b0\u00180.007 for Ni 79Fe21.\n0.7 0.8 0.9 1.0 1.1 1.2 1.301002003004005006002Hex (Oe)a)\nnc,Hexpc,HexMOKE,Hex\n0.7 0.8 0.9 1.0 1.1 1.2 1.3\ntRu (nm)0.0060.0080.0100.0120.0140.0160.018α\nα0α0+αspα0+2αspb)pc, S\nnc, Spc, A\nnc, A0.000.050.100.150.20\n−Aex (erg/cm2)FIG. 2. Coupling parameters for Py/Ru/Py trilayers. a):\nInterlayer (static) coupling from resonance \feld shift of an-\ntisymmetric mode; see Fig 1 a),c). The antiferromagnetic\nexchange parameter Aexis extracted through Eq 2, in agree-\nment with values found in Ref11. The line is a guide to the\neye. b) Spin pumping (dynamic) coupling from damping of\nthe symmetric (S) and antisymmetric (A) modes; see Fig 1\nb), d). The spin pumping damping for uncoupled layer pre-\ncession in Py/Ru/Py, \u000bspis shown for comparsion. Dotted\nlines show the possible e\u000bect of \u0018100 Oe detuning for the two\nPy layers. See text for details.\nWe compare the value with the value 2 \u000bspexpected from\ntheory for the antisymmetric mode and written in Eq 3.\nThe interfacial spin mixing conductance for Ni 79Fe21/Ru,\nwas found in Ref.16to be ~gFN\n\"#= 24 nm\u00002. For a F/N/F\nstructure, in the limit of ballistic transport with no spin\nrelaxation through N, the e\u000bective spin mixing conduc-\ntance is ~gFN\n\"#=2: spin current must cross two interfaces\nto relax in the opposite Flayer, and the conductance re-\n\rects two series resistances17. This yields \u000bsp= 0:0027.\nThe observed enhancement matches well with, and per-\nhaps exceeds slightly, the \"giant\" spin pumping e\u000bect of\n2\u000bsp, as shown.\nLittle dependence of the Gilbert damping enhancement\n\u000bA\u0000\u000bSon the resonance \feld shift HA\u0000HScan be ob-\nserved in Figure 2 a,b. We believe that this independence\nre\rects close tuning of the resonance frequencies for Py\nlayers 1 and 2, as designed in the depositions. For \f-\nnite detuning \u0001 !de\fned through !2=!0+ \u0001!and\n!1=!0\u0000\u0001!, the modes change. Symmetric and anti-\nsymmetric modes become hybridized as S0andA0, and4\nthe di\u000berence in damping is reduced. De\fning g\u0001!2=\n(1\u0000i\u000b0\n) (1\u0000i\u000b0\n\u00002i\u000bsp\n) \u0001!2, it is straightfor-\nward to show that for the nc-case, the mode frequen-\ncies are!S0;A0= (f!S+f!A)=2\u0006q\n(f!S\u0000f!A)2=4 +g\u0001!2.\nThe relevant parameter is the frequency detuning nor-\nmalized to the exchange (coupling) frequency, z\u0011\n\u0001!=(2!ex); ifz\u001d1, the layers have well-separated\nmodes, and each recovers the uncoupled damping en-\nhancement of \u000bsp,\u000bS0;A0=\u000b0+\u000bspidenti\fed in Refs5,9.\nThe possibility of \fnite detuning, assuming ( !2\u0000\n!1)=\r= 100 Oe, is shown in Figure 2b), with the dot-\nted lines. The small \u0000zlimit for detuning \fnds sym-\nmetric e\u000bects on damping of the S0andA0modes, with\n\u000bS0=\u000b0+ 2\u000bspz2and\u000bA0=\u000b0+ 2\u000bsp(1\u0000z2), respec-\ntively, recovering perfect symmetric and antisymmetric\nmode values for z= 0. We assume that the \feld split-\nting shown in Figure 2 a) gives an accurate measure of\n2!ex=\r, as supported by the MOKE results. This value\ngoes into the denominator of z. We \fnd a reasonable \ft\nto the dependence of SandAdamping on Ru thickness,\nimplicit in the coupling. For the highest coupling pionts,\nthe damping values closely reach the low- zlimit, and we\nbelieve that the \"giant\" spin pumping result of 2 \u000bspis\nevident here.\nWe would like to point out next that it was not a-\npriori obvious that the Py/Ru/Py SAF would exhibit\nthe observed damping. Ru could behave in two limits in\nthe context of spin pumping: either as a passive spin-\nsink layer, or as a ballistic transmission layer supporting\ntransverse spin-current transmission from one Py layer to\nthe other. Our results show that Ru behaves as the latter\nin this thickness range. The symmetric-mode damping of\nthe SAF structure, extrapolated back to zero Ru thick-\nness, is identical within experimental resolution ( \u001810\u00004)\nto that of a single Py \flm 16 nm thick measured in nc-\nFMR (\"\u000b0\" line in Fig 2b.) If Ru, or the Py/Ru interface,\ndepolarized pumped spin current very strongly over this\nthickness range as has been proposed for Pt18, we would\nexpect an immediate increase in damping of the acous-\ntic mode by the amount of \u0018\u000bsp. Instead, the volume-\ndependent Ru depolarization in spin pumping has an (ex-\nponential) characteristic length of \u0015SD\u001810 nm5, and\nattenuation over the range explored of \u00181 nm is negli-\ngible.\nPerspectives: Finally, we would like to highlight some\nimplications of the study. First, as the study con\frms\nthe prediction of a \"giant\" spin pumping e\u000bect as pro-\nposed by Takahashi10, it is plausible that the greatly en-\nhanced values of spin accumulation predicted there may\nbe supported by Ru in Py/Ru/Py synthetic antiferro-\nmagnets (SAFs). These spin accumulations would di\u000ber\nstrongly in the excitation of symmetric and antisymmet-\nric modes, and may then provide a clear signature intime-resolved x-ray magnetooptical techniques19, similar\nto the observation of static spin accumulation in Cu re-\nported recently20.\nSecond, in most device applications of synthetic an-\ntiferromagnets, it is not desirable to excite the antisym-\nmetric (optical) mode. SAFs are used in the pinned layer\nof MTJ/spin valve structures to increase exchange bias\nand in the free layer to decrease (magnetostatic) stray\n\felds. Both of these functions are degraded if the opti-\ncal, or asymmetric mode of the SAF is excited. Accord-\ning to our results, at GHz frequencies near FMR, the\nsusceptibility of the antisymmetric mode is reduced sub-\nstantially, here by a factor of two (from 1/ \u000b) for nc-FMR ,\ndue to spin pumping. This reduction of \u001fon resonance\nwill scale inversely with layer thickness. The damping,\nand susceptibility, of the desired symmetric (acoustic)\nmode is unchanged, on the other hand, implying that\nspin pumping favors the excitation the symmetric mode\nfor thin Ru, the typical operating point.\nWe acknowledge NSF-DMR-1411160 for support.\n1Y. Tserkovnyak, A. Brataas, and G. E. W. Bauer, Phys. Rev.\nLett. 88, 117601 (2002).\n2Y. Tserkovnyak, A. Brataas, G. Bauer, and B. Halperin, Reviews\nin Modern Physics 77, 1375 (2005).\n3S. Mizukami, Y. Ando, and T. Miyazaki, Journal of Magnetism\nand Magnetic Materials 239, 42 (2002).\n4A. Ghosh, J. F. Sierra, S. Au\u000bret, U. Ebels, and W. E. Bailey,\nApplied Physics Letters 98, (2011).\n5A. Ghosh, S. Au\u000bret, U. Ebels, and W. E. Bailey, Phys. Rev.\nLett. 109, 127202 (2012).\n6J. Bass and W. Pratt, Journal of Physics: Condensed Matter 19,\n41 pp. (2007).\n7G. Woltersdorf, O. Mosendz, B. Heinrich, and C. H. Back, Phys-\nical Review Letters 99, 246603 (2007).\n8P. Merodio, A. Ghosh, C. Lemonias, E. Gautier, U. Ebels,\nV. Baltz, and W. Bailey, Applied Physics Letters 104(2014).\n9B. Heinrich, Y. Tserkovnyak, G. Woltersdorf, A. Brataas, R. Ur-\nban, and G. E. W. Bauer, Phys. Rev. Lett. 90, 187601 (2003).\n10S. Takahashi, Applied Physics Letters 104(2014).\n11J. Fassbender, F. Nortemann, R. Stamps, R. Camley, B. Hille-\nbrands, G. Guntherodt, and S. Parkin, Journal of Magnetism\nand Magnetic Materials 121, 270 (1993).\n12Z. Zhang, L. Zhou, P. E. Wigen, and K. Ounadjela, Phys. Rev.\nB50, 6094 (1994).\n13C. Cheng, N. Sturcken, K. Shepard, and W. Bailey, Review of\nScienti\fc Instruments 83, 063903 (2012).\n14M. Belmeguenai, T. Martin, G. Woltersdorf, M. Maier, and\nG. Bayreuther, Physical Review B 76(2007).\n15K. Tanaka, T. Moriyama, M. Nagata, T. Seki, K. Takanashi,\nS. Takahashi, and T. Ono, Applied Physics Express 7(2014).\n16N. Behera, M. S. Singh, S. Chaudhary, D. K. Pandya, and P. K.\nMuduli, Journal of Applied Physics 117(2015).\n17See eqs. 31, 74, 81 in Ref2.\n18J.-C. Rojas-S\u0013 anchez, N. Reyren, P. Laczkowski, W. Savero, J.-\nP. Attan\u0013 e, C. Deranlot, M. Jamet, J.-M. George, L. Vila, and\nH. Ja\u000br\u0012 es, Phys. Rev. Lett. 112, 106602 (2014).\n19W. Bailey, C. Cheng, R. Knut, O. Karis, S. Au\u000bret, S. Zohar,\nD. Keavney, P. Warnicke, J.-S. Lee, and D. Arena, Nature Com-\nmunications 4, 2025 (2013).\n20R. Kukreja, S. Bonetti, Z. Chen, D. Backes, Y. Acremann, J. A.\nKatine, A. D. Kent, H. A. D urr, H. Ohldag, and J. St ohr, Phys.\nRev. Lett. 115, 096601 (2015)."    },    {        "title": "2206.09969v1.First_principles_calculation_of_the_parameters_used_by_atomistic_magnetic_simulations.pdf",        "content": "arXiv:2206.09969v1  [cond-mat.mtrl-sci]  20 Jun 2022APS/123-QED\nFirst-principles calculation of the parameters used by ato mistic magnetic simulations\nSergiy Mankovsky and Hubert Ebert\nDepartment of Chemistry/Phys. Chemistry, LMU Munich,\nButenandtstrasse 11, D-81377 Munich, Germany\n(Dated: June 22, 2022)\nWhile the ground state of magnetic materials is in general we ll described on the basis of spin den-\nsity functional theory (SDFT), the theoretical descriptio n of finite-temperature and non-equilibrium\nproperties require an extension beyond the standard SDFT. T ime-dependent SDFT (TD-SDFT),\nwhich give for example access to dynamical properties are co mputationally very demanding and can\ncurrently be hardly applied to complex solids. Here we focus on the alternative approach based on\nthe combination of a parameterized phenomenological spin H amiltonian and SDFT-based electronic\nstructure calculations, giving access to the dynamical and finite-temperature properties for example\nvia spin-dynamics simulations using the Landau-Lifshitz- Gilbert (LLG) equation or Monte Carlo\nsimulations. We present an overview on the various methods t o calculate the parameters of the\nvarious phenomenological Hamiltonians with an emphasis on the KKR Green function method as\none of the most flexible band structure methods giving access to practically all relevant parameters.\nConcerning these, it is crucial to account for the spin-orbi t coupling (SOC) by performing rela-\ntivistic SDFT-based calculations as it plays a key role for m agnetic anisotropy and chiral exchange\ninteractions represented by the DMI parameters in the spin H amiltonian. This concerns also the\nGilbert damping parameters characterizing magnetization dissipation in the LLG equation, chiral\nmultispin interaction parameters of the extended Heisenbe rg Hamiltonian, as well as spin-lattice\ninteraction parameters describing the interplay of spin an d lattice dynamics processes, for which an\nefficient computational scheme has been developed recently b y the present authors.\nPACS numbers: 71.15.-m,71.55.Ak, 75.30.Ds\nI. INTRODUCTION\nDensity functional theory (DFT) is a ’formally exact\napproachto the static electronic many-body problem’ for\ntheelectrongasintheequilibrium, whichwasadoptedfor\na huge number of investigations during the last decades\nto describe the ground state of solids, both magnetic and\nnon-magnetic,aswellasvariousgroundstateproperties1.\nHowever, dealing with real systems, the properties\nin an out-of-equilibrium situation are of great interest.\nAn example for this is the presence of external pertur-\nbation varying in time, which could be accounted for\nby performing time-dependent first-principles electronic\nstructure calculations. The time-dependent extension\nof density functional theory (TD-DFT)2is used suc-\ncessfully to study various dynamical processes in atoms\nand molecules, in particular, giving access to the time\nevolution of the electronic structure in a system af-\nfected by a femtosecond laser pulse. However, TD-DFT\ncan be hardly applied to complex solids because of the\nlack of universal parameter-free approximations for the\nexchange-correlation kernel. Because of this, an ap-\nproach based on the combination of simulation methods\nfor spin- and lattice dynamics, using model spin and lat-\ntice Hamiltonians is more popular for the moment. A\ngreat progress with this approach has been achieved dur-\ning last decade due to the availability of parameters for\nthe model Hamiltonians calculated on a first principles\nlevel, that is a central issue of the present contribution.\nAs it was pointed out in Ref. 1, this approach has the ad-\nvantage, that the spin-related many-body effects in thiscase are much simpler to be taken into account when\ncompared to the ab-initio approach. Thus, the isotropic\nexchangecouplingparameters JijfortheclassicalHeisen-\nberg Hamiltonian worked out Liechtenstein et al.3,4have\nbeen successfully used by many authors to predict the\nground state magnetic structure of material and to in-\nvestigateitsfinite-temperatureproperties. Dependingon\nthe materials, the isotropic Jijcan exhibit only spatial\nanisotropy. Extension of the Heisenberg Hamiltonian ac-\ncounting for anisotropy in spin subspace is often done by\nadding the so-called Dzyaloshinskii-Moriya interactions\n(DMI) and the magnetic anisotropy term,\nHH,rel=−/summationdisplay\ni,jJij(ˆei·ˆej)−/summationdisplay\ni,j/vectorDij(ˆei׈ej)+/summationdisplay\niˆeiKiiˆei.\n(1)\nwith ˆei(j)the orientation of the spin magnetic moment at\nsitei(j). Alternatively, one may describe exchange inter-\nactions in the more general tensorial form, Jij, leading\nto:\nHH,rel=−/summationdisplay\ni,jˆeiJijˆej+/summationdisplay\niˆeiKiiˆei,(2)\nIn the second case the DMI is represented as the an-\ntisymmetric part of the exchange tensor, i.e. Dα\nij=\n1\n2(Jβγ\nij−Jγβ\nij)ǫαβγ. It should be stressed, that calcula-\ntions of the spin-anisotropic exchange interaction param-\neters as well as of the magnetic anisotropy parameters\nrequire a relativistic treatment of the electronic struc-\nture in contrast to the case of the isotropic exchange pa-\nrameters which can be calculated on a non-relativistic2\nlevel. Various schemes to map the dependence of the\nelectronicenergyonthemagneticconfigurationweresug-\ngested in the literature to calculate the parameters of the\nspin Hamiltonians5–8, depending of its form given in Eqs.\n(1) or (2).\nDespite of its simplicity, the spin Hamiltonian gives\naccess to a reasonable description of the temperature\ndependence of magnetic properties of materials when\ncombined with Monte Carlo (MC) simulations9, or non-\nequilibrium spin dynamics simulations based on the phe-\nnomenological Landau-Lifshitz-Gilbert equations10,11\n1\nγd/vectorM\ndτ=−/vectorM×/vectorHeff+/vectorM×/bracketleftBigg˜G(/vectorM)\nγ2M2sd/vectorM\ndτ/bracketrightBigg\n.(3)\nHere/vectorHeffis the effective magnetic field defined as /vectorHeff=\n−1\nM∂F\n∂ˆm, whereFis the free energy of the system and\nˆm=/vectorM\n/vectorMswithMsthesaturationmagnetizationtreatedat\nfirst-principles level, and γis the gyromagnetic ratio and\n˜Gis the Gilbert damping parameter. Alternatively, the\neffective magnetic field can be representedin terms ofthe\nspin Hamiltonian in Eq. (2), i.e. /vectorHeff=−1\nM∂/angbracketleftHH,rel/angbracketrightT\n∂ˆm,\nwith/an}b∇acketle{t.../an}b∇acket∇i}htTdenoting the thermal averagefor the extended\nHeisenberg Hamiltonian HH,rel.\nThe first-principles calculation of the parameters for\nthe Heisenberg Hamiltonian as well as for the LLG equa-\ntion for spin dynamics have been reported in the litera-\nture by various groups who applied different approaches\nbased on ab-initio methods. Here we will focus on calcu-\nlations based on the Green function multiple-scattering\nformalism being a rather powerful tool to supply all pa-\nrameters for the extended Heisenberg Hamiltonian as\nwell as for the LLG equation.\nA. Magnetic anisotropy\nLet’s first consider the magnetic anisotropy term in\nspin Hamiltonian, characterized by parameters (written\nintensorialforminEqs.(1)and(2))deducedfromtheto-\ntalenergydependentontheorientationofthemagnetiza-\ntion ˆm. The latter is traditionallysplit into the magneto-\ncrystalline anisotropy(MCA) energy, EMCA(ˆm), induced\nby spin-orbit coupling (SOC) and the shape anisotropy\nenergy,Eshape(ˆm), caused by magnetic dipole interac-\ntions,\nEA(ˆm) =EMCA(ˆm)+Eshape(ˆm). (4)\nAlthough a quantum-mechanical description of the mag-\nneticshapeanisotropydeservesseparatediscussion12this\ncontribution can be reasonably well estimated based on\nclassical magnetic dipole-dipole interactions. Therefore,\nwe will focus on the MCA contribution which is fully\ndetermined by the electronic structure of the considered\nsystem. In the literature the focus is in general on the\nMCA energy of the ground state, which can be estimated\nstraightforwardlyfromthe totalenergycalculatedfordif-\nferent orientations of the magnetization followed by amapping onto a model spin Hamiltonian, given e.g. by\nan expansion in terms of spherical harmonics Ylm(ˆm)13\nEMCA(ˆm) =/summationdisplay\nlevenm=l/summationdisplay\nm=−lκm\nlYlm(ˆm).(5)\nAlternative approach to calculate the MCA parameters\nis based on magnetic torque calculations, using the defi-\nnition\nTˆm(θˆu) =−∂E(ˆm)\n∂θˆu, (6)\navoiding the time-consuming total energy calculations.\nThis scheme is based on the so-called magnetic force the-\noremthatallowstorepresenttheMCAenergyintermsof\na correspondingelectronic single-particleenergies change\nunder rotation of magnetization, as follows14:\n∆ESOC(ˆm,ˆm′) =−/integraldisplayEˆm\nF\ndE/bracketleftBig\nNˆm(E)−Nˆm′(E)/bracketrightBig\n−1\n2nˆm′(Eˆm′\nF)(Eˆm\nF−Eˆm′\nF)2\n+O(Eˆm\nF−Eˆm′\nF)3(7)\nwithNˆm(E) =/integraltextEdE′nˆm(E′) the integrated DOS for\nthe magnetization along the direction ˆ m, andnˆm(E) the\ndensityofstates(DOS) representedin termsofthe Green\nfunction as follows\nnˆm(E) =−1\nπIm TrGˆm(E). (8)\nThis expressioncan be used in a very efficient way within\nthe framework of the multiple-scattering formalism. In\nthis case the Green function is given in terms of the scat-\ntering path operator τ(E)nn′connecting the sites nand\nn′as follows\nG0(/vector r,/vector r′,E) =/summationdisplay\nΛΛ′Zn\nΛ(/vector r,E)τnn′\nΛΛ′(E)Zn′×\nΛ′(/vector r′,E)\n−/summationdisplay\nΛ/bracketleftBig\nZn\nΛ(/vector r,E)Jn×\nΛ(/vector r′,E)Θ(r′−r)\n+Jn\nΛ(/vector r,E)Zn×\nΛ(/vector r′,E)Θ(r−r′)/bracketrightBig\nδnn′,(9)\nwhere the combined index Λ = ( κ,µ) represents the rela-\ntivistic spin-orbit and magnetic quantum numbers κand\nµ, respectively15;Zn\nΛ(/vector r,E) andJn\nΛ(/vector r,E) are the regular\nand irregular solutions of the single-site Dirac equation\n(27)16–18. The scattering path operator is given by the\nexpression\nτ(E) = [m(E)−G0(E)]−1(10)\nwithm(E) =t−1(E) andG0(E) the inverse single-site\nscattering and structure constant matrices, respectively.\nThe double underline used here indicates matrices with\nrespect to site and angular momentum indices17.3\nUsing the Lloyd’s formula that gives the integrated\nDOS in terms of the scattering path operator, Eq. (7)\ncan be transformed to the form\n∆ESOC(ˆm,ˆm′) =−1\nπIm Tr/integraldisplayEF\ndE\n×/parenleftbig\nlnτ(ˆm,E)−lnτ(ˆm′,E)/parenrightbig\n(11)\nwith the scattering path operator evaluated for the mag-\nnetization along ˆ mand ˆm′, respectively.\nWith this, the magnetic torque T(θ) can be expressed\nby means of multiple scattering theory leading for the\ntorque component with respect to a rotation of the mag-\nnetization around an axis ˆ u, to the expression19\nTˆm(θˆu) =−1\nπℑ/integraldisplayEF\ndE∂\n∂θˆu/bracketleftbig\nlndet/parenleftbig\nt(ˆm)−1−G0/parenrightbig/bracketrightbig\n.\n(12)\nMapping the resulting torque onto a corresponding pa-\nrameterized expression as for example Eq. (5), one ob-\ntains the corresponding parameters of the spin Hamilto-\nnian.\nHowever,oneshouldnotethatthemagneticanisotropy\nof materials changes when the temperature increases.\nThis occurs first of all due to the increasing amplitude\nof thermally induced spin fluctuations responsible for a\nmodification of the electronic structure. A correspond-\ning expression for magnetic torque st finite temperature\nwas worked out by Staunton et al.19, on the basis of the\nrelativistic generalization of the disordered local moment\n(RDLM) theory20. To perform the necessary thermal av-\neraging over different orientational configurations of the\nlocal magnetic moments it uses a technique similar to\nthe one used to calculate the configurational average in\nthe case of random metallic alloys, so-called Coherent\nPotential Approximation (CPA) alloy theory21,22. Ac-\ncordingly, the free energy difference for two different ori-\nentations of the magnetization is given by\n∆F(ˆm,ˆm′) =−/integraldisplay\ndEfFD(E,ˆm) (13)\n/bracketleftbigg\n/an}b∇acketle{tNˆm/an}b∇acket∇i}ht(E)−/an}b∇acketle{tNˆm′/an}b∇acket∇i}ht(E)/bracketrightbigg\n.(14)\nBy using in this expression the configurational aver-\naged integrated density of states20,23given by Lloyd’s\nformula, the corresponding expression for the magnetic\ntorque at temperature T\nTˆm,T(θˆu) =−∂\n∂θˆu/parenleftbigg/summationdisplay\ni/integraldisplay\nPˆm\ni(ˆei)/an}b∇acketle{tΩˆm/an}b∇acket∇i}htˆeidˆei/parenrightbigg\n.(15)\ncan be written explicitly as:\nTˆm,T(θˆu) =−1\nπIm/integraldisplayEF\ndEfFD(E,ˆm)\n/parenleftbigg/summationdisplay\ni/integraldisplay∂Pˆm\ni(ˆei)\n∂θˆuln detMˆm\ni(ˆei,E)dˆei/parenrightbigg\n.(16)where\nMˆm\ni(ˆei,E) = 1+([ti(ˆei)]−1−tˆm\ni,c(ˆei)]−1)τˆm\nii,c,(17)\nand\nτˆm\nii,c= ([tˆm\ni,c(ˆei)]−1−G0)−1. (18)\nwhere the index cindicates quantities related to the CPA\nmedium.\nFig. 1 (top) shows as an example the results for\nthe temperature-dependent magnetization ( M(T)) cal-\nculated within the RDLM calculations for L10-ordered\nFePt24. Fig. 1 (bottom) gives the corresponding param-\neterK(T) for a uni-axial magneto-crystalline anisotropy,\nwhich is obviously in good agreement with experiment.\n200 400 600800\nTemperature T (K)00.20.40.60.8M(T)\n0.2 0.4 0.60.8\n(M(T))2-2-1.5-1-0.5∆ESOC (meV)\nFIG. 1. RDLM calculations on FePt. Top: the magneti-\nzationM(T) versus Tfor the magnetization along the easy\n[001] axis (filled squares). The full line shows the mean field\napproximation to a classical Heisenberg model for compar-\nison. Bottom: the magnetic anisotropy energy ∆ ESOCas\na function of the square of the magnetization M(T). The\nfilled circles show the RDLM-based results, the full line giv e\nK(T)∼[M(T)/M(0)]2, and the dashed line is based on the\nsingle-ion model function. All data taken from24.\nB. Inter-atomic bilinear exchange interaction\nparameters\nMost first-principles calculations on the bilinear ex-\nchange coupling parameters reported in the literature,4\nFIG. 2. Adiabatic spin-wave dispersion relations along hig h-\nsymmetry lines of the Brillouin zone for Ni. Broken line:\nfrozen-magnon-torque method, full line: transverse susce pti-\nbility method31. All data are taken from Ref. 31.\nare based on the magnetic force theorem (MFT) by\nevaluating the energy change due to a perturbation on\nthe spin subsystem with respect to a suitable reference\nconfiguration25. Many results are based on calculations\nof the spin-spiral energy ǫ(/vector q), giving access to the ex-\nchange parameters in the momentum space, J/vector q7,26–28,\nfollowed by a Fourier transformation to the real space\nrepresentation Jij. Alternatively, therealspaceexchange\nparameters are calculated directly by evaluating the en-\nergy change due to the tilting of spin moments of inter-\nacting atoms. The corresponding non-relativistic expres-\nsion (so-called Liechtenstein or LKAG formula) has been\nimplemented based on the KKR as well as LMTO Green\nfunction (GF)3,4,25,29band structure methods. It should\nbe noted that the magnetic force theorem provides a rea-\nsonable accuracy for the exchange coupling parameters\nin the case of infinitesimal rotations of the spins close to\nsome equilibrium state, that can be justified only in the\nlong wavelength and strong-coupling limits30. Accord-\ningly, calculations of the exchange coupling parameters\nbeyond the magnetic force theorem, represented in terms\nof the inverse transverse susceptibility, were discussed\nin the literature by various authors25,30–33. Grotheer et\nal., for example, have demonstrated31a deviation of the\nspin-wave dispersion curves away from Γ point in the\nBZ, calculated for fcc Ni using the exchange parameters\nJ/vector q∼χ−1\n/vector q, from the MFT-based results for J/vector q. On the\notherhand, the resultsareclosetoeachotherin the long-\nwavelength limit (see Fig. 2). The calculations beyond\nthestandardDFTaredonebymakinguseoftheso-called\nconstrained-field DFT. The latter theory was also used\nby Bruno33who suggested the ’renormalization’ of the\nexchange coupling parameters expressed in terms of non-\nrelativistic transverse magnetic susceptibility, according\ntoJ=1\n2Mχ−1M=1\n2M(˜χ−1−Ixc)M, with the various\nquantities defined as follows\n˜χ−1\nij=2\nπ/integraldisplayEF\ndE/integraldisplay\nΩid3r/integraldisplay\nΩjd3r′(19)\n×Im[G↑(/vector r,/vector r′,E)G↓(/vector r′,/vector r,E)],(20)Mi=/integraldisplay\nΩid3rm(/vector r), (21)\nand\n˜Ixc\nij=δij∆i\n2Mi, (22)\nwith ∆ i=4\nMi/summationtext\nj˜Jij, where\n˜Jij=1\nπIm/integraldisplayEF\ndE/integraldisplay\nΩid3r/integraldisplay\nΩjd3r′(23)\n×[Bxc(/vector r)G↑(/vector r,/vector r′,E)Bxc(/vector r′)G↓(/vector r′,/vector r,E)].(24)\nThis approach results in a Curie temperature of 634 K\nfor fcc Ni (vs. 350 K based on the MFT) which is in good\nagreement with the experimental value of (621 −631 K).\nAs was pointed out by Solovyev30, such a corrections can\nbe significant only for a certain class of materials, while,\nfor instance, the calculations of spin-wave energies31and\nTC33for bcc Fe demonstrate that these corrections are\nquite small. As most results in the literature were ob-\ntained using the exchange parameters based on the mag-\nnetic force theorem, we restrict below to this approxima-\ntion.\nSimilar to the case of the MCA discussed above, ap-\nplication of the magnetic force theorem gives the energy\nchange due to tilting of two spin moments represented in\nterms of the integrated DOS4. Within the multiple scat-\ntering formalism, this energy can be transformed using\nthe Lloyd’s formula leading to the expression\n∆E=−1\nπIm Tr/integraldisplayEF\ndE/parenleftbig\nlnτ(E)−lnτ0(E)/parenrightbig\n(25)\nwithτ(0)(E) andτ(E) the scattering path operators for\nnon-distorted and distorted systems, respectively.\nAs reported in Ref. 4, the expression for Jijrepresent-\ning the exchange interaction between the spin moments\non sitesiandj, is given by the expression\nJij=−1\n4πImTrL/integraldisplayEF\ndE∆iτ↑\nij∆jτ↓\nji,(26)\nwith ∆i(j)= ([t↑]−1\ni(j)−[t↓]−1\ni(j)), wheret↑\ni(j)andt↓\ni(j)are\nthespin-upandspin-downsingle-sitescatteringmatrices,\nrespectively, while τ↑\nijandτ↓\njiare the spin-up and spin-\ndown, respectively, scattering path operators. As rela-\ntivistic effects are not taken into account, the exchange\ninteractions are isotropic with respect to the orientation\nof the magnetization as well as with respect to the di-\nrection of the spin tilting. On the other hand, spin-orbit\ncoupling gives rise to an anisotropy for exchange inter-\nactions requiring a representation in the form of the ex-\nchange tensor Jijwith its antisymmetric part giving ac-\ncess to the Dzyaloshinskii-Moriya (DM) interaction /vectorDij.\nUdvardi et al.5and later Ebert and Mankovsky6sug-\ngested an extension of the classical Heisenberg Hamilto-\nnianbyaccountingforrelativisticeffects forthe exchange\ncoupling (see also Ref. 25). These calculations are based5\nonafullyrelativistictreatmentoftheelectronicstructure\nobtained by use of of the Dirac Hamiltonian\nHD=−ic/vector α·/vector∇+1\n2c2(β−1)\n+¯V(/vector r)+β/vector σ·/vectorB(/vector r)+e/vector α·/vectorA(/vector r).(27)\nHere,αiandβare the standard Dirac matrices15while\n¯V(/vector r) and/vectorB(/vector r) are the spin independent and spin depen-\ndent parts of the electronic potential.\nConsidering a ferromagnetic (FM) state as a reference\nstate with the magnetization along the zdirection, a tilt-\ning of the magnetic moments on sites iandjleads to a\nmodification ofthe scattering path operatorimplying the\nrelation\nlnτ−lnτ0=−ln/parenleftbig\n1+τ[∆mi+∆mj+...]/parenrightbig\n,(28)\nwithmi=t−1\ni. This allows to write down the expression\nfor the energy change due to a spin tilting on sites iand\njas follows\nEij=−1\nπImTr/integraldisplayEF\ndE∆miτij∆mjτji(29)\nWithin the approach of Udvardi et al.5, the depen-\ndence of the single-site inverse scattering matrix mion\nthe orientation of magnetic moment ˆ eiis accounted for\nby performing a corresponding rotation operation us-\ning the rotation matrix R(θ,φ), i.e., one has mi(θ,φ) =\nR(θ,φ)m0\niR+(θ,φ). The change of the scattering matrix\nmiunder spin rotation, ∆ mi, linearized with respect to\nthe rotation angles, is given by the expression\n∆mi=R(θi,φi)m0\niR+(θi,φi)−m0\ni\n=mθ\niδθi+mφ\niδφi (30)\nwith\nmθ\ni=∂\n∂θmi=∂R\n∂θmiR++Rmi∂R+\n∂θ,\nmφ\ni=∂\n∂φmi=∂R\n∂φmiR++Rmi∂R+\n∂φ.(31)\nTo calculate the derivatives of the rotation matrix, the\ndefinition\nˆR(αˆn,ˆn) =eiαˆn(ˆn·ˆ/vectorJ)(32)\nfor the corresponding operator is used, withˆ/vectorJthe total\nangular momentum operator. ˆR(αˆn,ˆn) describes a rota-\ntion of the magnetic moment ˆ mby the angle αˆnabout\nthe direction ˆ n⊥ˆm, that gives in particular R(θ,ˆn) for\nˆn= ˆyandR(φ,ˆn) for ˆn= ˆz.\nThis leads to the second derivatives of the total energy\nwith respect to the titling angles αi={θi,φi}andβj=\n{θj,φj}\n∂2E\n∂αi∂βj=−1\nπImTr/integraldisplayEF\ndEmα\niτijmβ\njτji(33)As is discussed by Udvardi et al.5, these derivatives give\naccess to all elements Jµν\nijof the exchange tensor, where\nµ(ν) ={x,y,z}. Note, however, that only the tensor el-\nements with µ(ν) ={x,y}can be calculated using the\nmagnetization direction along the ˆ zaxis, giving access to\nthezcomponent Dz\nijof the DMI. In order to obtain all\nother tensor elements, an auxiliary rotation of the mag-\nnetization towards the ˆ xand ˆydirections of the global\nframe of reference is required. For example, the com-\nponentDx\nijif the DMI vector can be evaluated via the\ntensor elements\nJzy\nij=∂2E\n∂θi∂φjandJyz\nij=∂2F\n∂φi∂θj(34)\nforθ=π\n2andφ= 0.\nAn alternative expression within the KKR multiple\nscattering formalism has been worked out by Ebert and\nMankovsky6, by using the alternative convention for the\nelectronicGreenfunction(GF) assuggestedbyDederichs\nand coworkers34. According to this convention, the off-\nsite part of the GF is given by the expression:\nG(/vector ri,/vector rj,E) =/summationdisplay\nΛΛ′Ri\nΛ(/vector ri,E)Gij\nΛΛ′(E)Rj×\nΛ′(/vector rj,E),(35)\nwhereGij\nΛΛ′(E) is the so-called structural Green’s func-\ntion,Ri\nΛisaregularsolutiontothesingle-siteDiracequa-\ntionlabeledbythecombinedquantumnumbersΛ15. The\nenergy change ∆ Eijdue to a spin tilting on sites iandj\n, given by Eq. (29), transformed to the above mentioned\nconvention is expressed as follows\n∆Eij=−1\nπImTr/integraldisplay\ndE∆tiGij∆tjGji,(36)\nwhere the change of the single-site t-matrix ∆ tican be\nrepresented in terms of the perturbation ∆ Vi(/vector r) at site\niusing the expression\n∆ti\nΛ′Λ=/integraldisplay\nd3rRi×\nΛ′(r)∆V(r)Ri\nΛ(r) = ∆V(R)i\nΛ′Λ,(37)\nwherethe perturbation causedby the rotationof the spin\nmagnetic moment ˆ eiis represented by a change of the\nspin-dependent potential in Eq. (27) (in contrast to the\napproach used in Ref. 5)\n∆V(r) =Vˆn(r)−Vˆn0(r) =β/vector σ(ˆn−ˆn0)B(r).(38)\nUsing again the frozen potential approximation implies\nthat the spatial part of the potential Vˆn(r) does not\nchange upon rotation of spin orientation.\nComing back to the convention for the GF used by\nGy¨ orffy and coworkers35according to Eq. (9) the expres-\nsion for the elements of the exchange tensor represented\nin terms of the scattering path operator τij\nΛ′Λ(E) has the\nform\nJαiαj\nij=−1\nπImTr/integraldisplay\ndETαiτijTαjτji,(39)6\nwhere\nTαi\nΛΛ′=/integraldisplay\nd3rZ×\nΛ(/vector r)βσαB(r)ZΛ′(/vector r).(40)\nWhen compared to the approach of Udvardi et al.5,\nthe expression in Eq. (39) is given explicitly in Cartesian\ncoordinates. However, auxiliary rotations of the magne-\ntization are still required to calculate all tensor elements,\nand as a consequence, all components of the DMI vec-\ntor. This can be avoided using the approach reported\nrecently36for DMI calculations.\nIn this case, using the grand-canonical potential in the\noperator form\nK=H−µN, (41)\nwithµthe chemical potential, the variation of single-\nparticle energy density ∆ E(/vector r) caused by a perturbation\nis written in terms of the electronic Green function for\nT= 0 K as follows\n∆E(/vector r) =−1\nπImTr/integraldisplayµ\ndE(E−µ)∆G(/vector r,/vector r,E).(42)\nAssuming the perturbation ∆ Vresponsible for the\nchange of the Green function ∆ G=G−G0(the in-\ndex 0 indicates here the collinear ferromagnetic reference\nstate) to be small, ∆ Gcan be expanded up to any order\nw.r.t. the perturbation\n∆G(E) =G0∆VG0\n+G0∆VG0∆VG0\n+G0∆VG0∆VG0∆VG0\n+G0∆VG0∆VG0∆VG0∆VG0+...,(43)\nleading to a corresponding expansion for the energy\nchange with respect to the perturbation as follows\n∆E= ∆E(1)+∆E(2)+∆E(3)+∆E(4)+...,(44)\nHere and below we drop the energy argument for the\nGreen function G(E) for the sake of convenience. This\nexpression is completely general as it gives the energy\nchange as a response to any type of perturbation. When\n∆Vis associated with tiltings of the spin magnetic mo-\nments, it can be expressed within the frozen potential\napproximation and in line with Eq. (38) as follows\n∆V(/vector r) =/summationdisplay\niβ/parenleftbig\n/vector σ·ˆsi−σz/parenrightbig\nBxc(/vector r).(45)\nWith this, the energy expansion in Eq (44) gives access\nto the bilinear DMI as well as to higher order multispin\ninteractions37. To demonstrate the use of this approach,\nwe start with the xandycomponents of the DMI vector,\nwhich can be obtained by setting the perturbation ∆ Vin\nthe form of a spin-spiral described by the configuration\nof the magnetic moments\nˆmi=/parenleftBig\nsin(/vector q·/vectorRi),0,cos(/vector q·/vectorRi)/parenrightBig\n,(46)with the wave vector /vector q= (0,q,0). As it follows from\nthe spin Hamiltonian, the slope of the spin wave energy\ndispersion at the Γ point is determined by the DMI as\nfollows\nlim\nq→0∂E(1)\nDM\n∂qy= lim\nq→0∂\n∂qy/summationdisplay\nijDy\nijsin(/vector q·(/vectorRj−/vectorRi))\n=/summationdisplay\nijDy\nij(/vectorRj−/vectorRi)y. (47)\nIdentifying this with the corresponding derivative of the\nenergy ∆ E(1)in Eq. 44\n∂∆E(1)\n∂qα/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq→0=∂E(1)\nDM\n∂qα/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq→0, (48)\nand equating the corresponding terms for each atomic\npair (i,j), one obtains the following expression for the y\ncomponent of the DMI vector:\nDy\nij=/parenleftbigg\n−1\n2π/parenrightbigg\nImTr/integraldisplayµ\ndE(E−µ)\n×/bracketleftbigg\nOj(E)τji(E)Ti,x(E)τij(E)\n−Oi(E)τij(E)Tj,x(E)τji(E)/bracketrightbigg\n,(49)\nIn a completely analogous way one can derive the x-\ncomponent of the DMI vector, Dx\nij. The overlap inte-\ngralsOj\nΛΛ′and matrix elements Ti,α\nΛΛ′of the operator\nTi,α=βσαBi\nxc(/vector r) (which are connected with the compo-\nnents of the torque operator β[/vector σ׈m]Bi\nxc(/vector r)) are defined\nas follows:6\nOj\nΛΛ′=/integraldisplay\nΩjd3rZj×\nΛ(/vector r,E)Zj\nΛ′(/vector r,E) (50)\nTi,α\nΛΛ′=/integraldisplay\nΩid3rZi×\nΛ(/vector r,E)/bracketleftBig\nβσαBi\nxc(/vector r)/bracketrightBig\nZi\nΛ′(/vector r,E).(51)\nAs is shown in Ref. 37, the Dz\nijcomponent of the DMI,\nas well isotropic exchange parameter Jijcan also be ob-\ntained on the basis of Eqs. (43) and (44) using the second\norder term w.r.t. the perturbation, for a spin spiral with\nthe form\nˆsi= (sinθcos(/vector q·/vectorR),sinθsin(/vector q·/vectorR),cosθ).(52)\nIn this case case, the DMI component Dz\nijand the\nisotropic exchange interaction are obtained by taking the\nfirst- and second-orderderivatives of the energy ∆ E(2)(/vector q)\n(see Eq. (44)), respectively, with respect to /vector q:\n∂\n∂/vector q∆EH(/vector q)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq→0=−sin2θN/summationdisplay\ni/negationslash=jDz\nijˆq·(/vectorRi−/vectorRj) (53)\nand\n∂2\n∂/vector q2∆EH(/vector q)/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq→0= sin2θ/summationdisplay\ni,jJij(ˆq·(/vectorRi−/vectorRj))2(54)7\nwith ˆq=/vector q/|/vector q|the unit vector giving the direction of the\nwave vector /vector q. Identifying these expressions again with\nthe corresponding derivatives of ∆ E(2)(/vector q), one obtains\nthe following relations for Dz\nij\nDz\nij=1\n2(Jxy\nij−Jyx\nij) (55)\nand forJij\nJij=1\n2(Jxx\nij+Jyy\nij), (56)\nwhere the tensor elements Jαβare given by Eqs. (39)\nand (40).\nSimilar to the magnetic anisotropy, the exchange cou-\npling parameters depend on temperature, that should be\ntaken into account within the finite temperature spin dy-\nnamic simulations. An approach that gives access to\ncalculations of exchange coupling parameters for finite\ntemperature has been reported in Ref. 37. It accounts\nfor the electronic structure modification due to temper-\nature induced lattice vibrations by using the alloy anal-\nogy model in the adiabatic approximation. This implies\ncalculations of the thermal average /an}b∇acketle{t.../an}b∇acket∇i}htTas the configu-\nrational average over a set of appropriately chosen set of\natomic displacements, using the CPA alloy theory38–40.\nTo make use of this scheme to account for lattice vi-\nbrations, a discrete set of Nvvectors ∆/vectorRq\nv(T) is intro-\nduced for each atom, with the temperature dependent\namplitude, which characterize a rigid displacement of\nthe atomic potential in the spirit of the rigid muffin-\ntin approximation41,42. The corresponding single-site t-\nmatrix in the common global frame of the solid is given\nby the transformation:\ntq\nv=U(∆/vectorRv)tq,locU(∆/vectorRv)−1, (57)\nwith the so-called U-transformation matrix U(/vector s) given in\nits non-relativistic form by:41,42\nULL′(/vector s) = 4π/summationdisplay\nL′′il+l′′−l′CLL′L′′jl′′(|/vector s|k)YL′′(ˆs).(58)\nHereL= (l,m) represents the non-relativistic angu-\nlar momentum quantum numbers, jl(x) is a spheri-\ncal Bessel function, YL(ˆr) a real spherical harmonics,\nCLL′L′′a corresponding Gaunt number and k=√\nEis\nthe electronic wave vector. The relativistic version of\nthe U-matrix is obtained by a standard Clebsch-Gordan\ntransformation.15\nEvery displacement characterized by a displacement\nvectors ∆/vectorRv(T) can be treated as a pseudo-component\nof a pseudo alloy. Thus, the thermal averaging can be\nperformed as the site diagonal configurational average\nforasubstitutional alloy,bysolvingthe multi-component\nCPA equations within the global frame of reference40.\nThe same idea can be used also to take into account\nthermalspinfluctuations. Asetofrepresentativeorienta-\ntion vectors ˆ ef(withf= 1,...,Nf) for the local magneticmoment is introduced. Using the rigid spin approxima-\ntion, the single-site t-matrix in the global frame, corre-\nsponding to a given orientation vector, is determined by:\ntq\nf=R(ˆef)tq,locR(ˆef)−1, (59)\nwheretq,locis the single-site t-matrix in the local frame.\nHere the transformation from the local to the global\nframe of reference is expressed by the rotation matrices\nR(ˆef) that are determined by the vectors ˆ efor corre-\nsponding Euler angles.15Again, every orientation can be\ntreated as a pseudo-component of a pseudo alloy, that\nallows to use the alloy analogy model to calculate the\nthermal average over all types of spin fluctuations40.\nThe alloy analogy for thermal vibrations applied to\nthe temperature dependent exchange coupling parame-\nters leads to\n¯Jαiαj\nij=−1\n2πℑ/integraldisplay\ndETrace/an}b∇acketle{t∆Vαiτij∆Vαjτji/an}b∇acket∇i}htc,(60)\nwhere/an}b∇acketle{t.../an}b∇acket∇i}htcrepresents the configurational average with\nrespect to the set of displacements. In case of the ex-\nchange coupling parameters one has to distinguish be-\ntween the averaging over thermal lattice vibrations and\nspin fluctuations. In the first case the configurational av-\nerage is approximated as follows /an}b∇acketle{t∆Viτij∆Vjτji/an}b∇acket∇i}htvib≈\n/an}b∇acketle{t∆Viτij/an}b∇acket∇i}htvib/an}b∇acketle{t∆Vjτji/an}b∇acket∇i}htvib, assuming a negligible impact of\nthe so-called vertex corrections43. This averaging ac-\ncounts for the impact of thermally induced phonons on\nthe exchange coupling parameters for every temperature\nbefore their use in MC or spin dynamics simulations that\ndealsubsequentlywith thethermalaveraginginspinsub-\nspace. The impact of spin fluctuations can be incorpo-\nrated as well within the electronic structure calculations.\nFor a non-polarized paramagnetic reference state, this\ncan be done, e.g., by using the so-called disorder local\nmoment (DLM) scheme formulated in general within the\nnon-relativistic (or scalar-relativistic) framework. Mag-\nnetic disorder in this case can be modeled by creating a\npseudo alloy with an occupation of the atomic sites by\ntwo types of atoms with opposite spin moments oriented\nupwards,M↑and downwards M↓, respectively, i.e. con-\nsidering the alloy M↑\n0.5M↓\n0.5. In the relativistic case the\ncorresponding RDLM scheme has to describe the mag-\nnetic disorder by a discrete set of Nforientation vectors,\nand as a consequence, the average /an}b∇acketle{tτij/an}b∇acket∇i}htspinhas to be\ncalculated taking into account all these orientations. A\ncomparison of the results obtained for the isotropic ex-\nchange coupling constants Jijfor bcc Fe using the DLM\nand RDLM schemes is shown in Fig. 3, demonstrating\nclose agreement, with the small differences to be ascribed\nto the different account of relativistic effects, i.e. in par-\nticular the spin-orbit coupling.8\n11.5 22.5 3\nRij/a051015202530Jij (meV)SR-DLM\nRDLMFe (bcc), T = 1500 K\nFIG. 3. Isotropic exchange coupling parameters calculated\nfor the disordered magnetic state of bcc Fe within the scalar -\nrelativistic approach, using the DLM scheme (circles, SR-\nDLM) and within the fully-relativistic approach, using the\nRDLM scheme19,24(squares, RDLM).\nC. Multi-spin expansion of spin Hamiltonian:\nGeneral remarks\nDespite the obvious success of the classical Heisenberg\nmodel for many applications, higher-order multi-spin ex-\npansionHmsof the spin Hamiltonian H, given by the\nexpression\nHms=−1\n3!/summationdisplay\ni,j,kJijkˆsi·(ˆsj׈sk),\n−2\np!/summationdisplay\ni,j,k,lJs\nijkl(ˆsi·ˆsj)(ˆsk·ˆsl)\n−2\np!/summationdisplay\ni,j,k,l/vectorDijkl·(ˆsi׈sj)(ˆsk·ˆsl)+...,\n=H3+H4,s+H4,a+... (61)\ncan be of great importance to describe more subtle prop-\nerties of magnetic materials44–56.\nThis concerns first of all systems with a non-collinear\nground state characterized by finite spin tilting angles,\nthat makes multispin contributions to the energy non-\nnegligible. Inparticular,manyreportspublishedrecently\ndiscuss the impact of the multispin interactions on the\nstabilization of exotic topologically non-trivial magnetic\ntextures, e.g. skyrmions, hopfions, etc.57–59\nCorresponding calculations of the multi-spin exchange\nparameters have been reported by different groups. The\napproach based on the Connolly-Williams scheme has\nbeen used to calculate the four-spin non-chiral (two-site\nand three-site) and chiral interactions for Cr trimers52\nand for a deposited Fe atomic chain60, respectively, for\nthe biquadratic, three-site four spin and four-site four\nspin interaction parameters58,61. The authors discuss\nthe role of these type of interactions for the stabilization\nof different types of non-collinear magnetic structures as\nskyrmions and antiskyrmions.\nA more flexible mapping scheme using perturbation\ntheory within the KKR Green function formalism wasonly reported recently by Brinker et al.62,63, and by the\npresent authors37. Here we discuss the latter approach,\ni.e. the energy expansion w.r.t. ∆ Vin Eq. (44). One\nhas to point out that a spin tilting in a real system has a\nfinite amplitude and therefore the higher order terms in\nthis expansion might become non-negligible and in gen-\neral should be taken into account. Their role obviously\ndepends on the specific materialandshould increasewith\ntemperature that leads to an increasing amplitude of the\nspin fluctuations. As these higher-order terms are di-\nrectly connected to the multispin terms in the extended\nHeisenberg Hamiltonian, one has to expect also a non-\nnegligibleroleofthe multispin interactionsforsomemag-\nnetic properties.\nExtending the spin Hamiltonian to go beyond the clas-\nsical Heisenberg model, we discuss first the four-spin ex-\nchange interaction terms Jijkland/vectorDijkl. They can be\ncalculated using the fourth-order term of the Green func-\ntion expansion ∆ E(4)given by:\n∆E(4)=−1\nπImTr/integraldisplayEF\ndE\n×(E−EF)∆VG∆VG∆VG∆VG\n=−1\nπImTr/integraldisplayEF\ndE∆VG∆VG∆VG∆VG.\n(62)\nwhere the sum rule for the Green functiondG\ndE=−GG\nfollowed by integration by parts was used to get a more\ncompact expression. Using the multiple-scattering repre-\nsentation for the Green function, this leads to:\n∆E(4)=/summationdisplay\ni,j,k,l−1\nπImTr/integraldisplayEF\ndE\n×∆Viiτij∆Vjjτjk∆Vkkτkl∆Vllτli.(63)\nwith the matrix elements ∆ Vii=/an}b∇acketle{tZi|∆V|Zi/an}b∇acket∇i}ht. Using the\nferromagnetic state with /vectorM||ˆzas a reference state, and\ncreating the perturbation ∆ Vin the form of a spin-spiral\naccording to Eq. (52), one obtains the corresponding /vector q-\ndependent energy change ∆ E(4)(/vector q), written here explic-9\nitly as an example\n∆E(4)=−1\nπ/summationdisplay\ni,j,k,lImTr/integraldisplayEF\ndEsin4θ\n×/bracketleftbigg\nIxxxx\nijklcos(/vector q·/vectorRi)cos(/vector q·/vectorRj)cos(/vector q·/vectorRk)cos(/vector q·/vectorRl)\n+Ixxyy\nijklcos(/vector q·/vectorRi)cos(/vector q·/vectorRj)sin(/vector q·/vectorRk)sin/vector q·/vectorRl)\n+Iyyxx\nijklsin(/vector q·/vectorRi)sin(/vector q·/vectorRj)cos(/vector q·/vectorRk)cos(/vector q·/vectorRl)\n+Iyyyy\nijklsin(/vector q·/vectorRi)sin(/vector q·/vectorRj)sin(/vector q·/vectorRk)sin(/vector q·/vectorRl)\n+Ixyxx\nijklcos(/vector q·/vectorRi)sin(/vector q·/vectorRj)cos(/vector q·/vectorRk)cos(/vector q·/vectorRl)\n+Iyxyy\nijklsin(/vector q·/vectorRi)cos(/vector q·/vectorRj)sin(/vector q·/vectorRk)sin/vector q·/vectorRl)\n+Iyxxx\nijklsin(/vector q·/vectorRi)cos(/vector q·/vectorRj)cos(/vector q·/vectorRk)cos(/vector q·/vectorRl)\n+Ixyyy\nijklcos(/vector q·/vectorRi)sin(/vector q·/vectorRj)sin(/vector q·/vectorRk)sin(/vector q·/vectorRl)+.../bracketrightbigg\n(64)\nwhere\nIαβγδ\nijkl=Ti,α(E)τij(E)Tj,β(E)τjk(E)\n×Tk,γ(E)τkl(E)Tl,δ(E)τli(E).(65)\nAs is shownin Ref. 37, the four-spinisotropicexchange\ninteraction Jijklandz-component of the DMI-like in-\nteraction Dz\nijklcan be obtained calculating the energy\nderivatives∂4\n∂q4∆E(4)and∂3\n∂q3∆E(4)in the limit of q= 0,\nand then identified with the corresponding derivatives of\nthe termsH4,sandH4,ain Eq. (61). These interaction\nterms are given by the expressions\nJs\nijkl=1\n4/bracketleftbigg\nJxxxx\nijkl+Jxxyy\nijkl+Jyyxx\nijkl+Jyyyy\nijkl/bracketrightbigg\n(66)\nand\nDz\nijkl=1\n4/bracketleftbigg\nJxyxx\nijkl+Jxyyy\nijkl−Jyxxx\nijkl−Jyxyy\nijkl)/bracketrightbigg\n,(67)\nwhere the following definition is used:\nJαβγδ\nijkl=1\n2πImTr/integraldisplayEF\ndETα\niτijTβ\njτjkTγ\nkτklTδ\nlτli(68)\nThese expressionobviously give also access to a special\ncases, i.e. the four-spin three-site interactions with l=j,\nand the four spin two-site, socalled biquadratic exchange\ninteractions with k=iandl=j.\nThe scalar biquadratic exchange interaction parame-\ntersJs\nijijcalculated on the basis of Eq. (66) for the three\n3dbulk ferromagnetic systems bcc Fe, hcp Co and fcc Ni\nhave been reported in Ref. 37. The results are plotted in\nFig. 4 as a function of the distance Rij+Rjk+Rkl+Rli.\nFor comparison, the insets give the corresponding bilin-\near isotropic exchange interactions for these materials.\nOne can see rather strong first-neighbor interactions for\nbcc Fe, demonstrating the non-negligible characterof the\n✵ \u0000 ✁\n✶✶ \u0000 ✁\n✷✷\n\u0000 ✁\n✸✸\n\u0000 ✁❘✐ ✂\n✴ ✄\n✵\n✵ \u0000 ✁\n✶\n✶ \u0000 ✁\n✷\n✷\n\u0000 ✁\n✸\n✸\n\u0000 ✁\n✹\n✹\n\u0000 ✁\n✁❏\n☎✆☎✆s\n✥✝✞✟✠✡\n① ①✡\n② ②\n☛☛☞ ✌\n✍✍☞ ✌\n✎✎☞ ✌✏✑ ✒\n✓✔\n✕\n✌\n☛✕\n☛✌✖\n✗✘\n✙✚✛✜✢❜ ✣✣ ✤✦\n(a)✶✶\u0000 ✁\n✷✷\u0000 ✁\n✸✸\u0000 ✁❘✐ ✂\n✴ ✄\n✲ ☎ \u0000 ☎✷\n☎\n☎\u0000 ☎ ✷☎\u0000 ☎✵\n☎\u0000 ☎\n✆\n☎\u0000 ☎\n✝\n☎ \u0000\n✶❏\n✞✟✞✟s\n✥✠✡☛☞\n✌\n① ①✌\n② ②\n✍✍✎ ✏\n✑✑✎ ✏\n✒✒✎ ✏✓✔ ✕\n✖✗\n✘\n✏\n✍✘\n✍✏✙\n✚✛\n✜✢✣✤✦❤ ✧★ ✩ ✪\n(b)✵ \u0000 ✁\n✶✶ \u0000 ✁\n✷✷\n\u0000 ✁\n✸❘✐ ✂\n✴ ✄\n✲\n✵ \u0000 ✵✷\n✲\n✵ \u0000 ✵✶ ✁\n✲\n✵ \u0000 ✵✶\n✲\n✵ \u0000 ✵✵ ✁\n✵❏\n☎✆☎✆s\n✥✝✞✟✠✡\n① ①✡\n② ②\n☛☛☞ ✌\n✍✍☞ ✌\n✎✏✑ ✒\n✓✔\n✕\n☛\n✍\n✎✖\n✗✘\n✙✚✛✜✢❢ ✣✣ ✤✦\n(c)\nFIG. 4. Scalar biquadratic exchange interactions Js\nijijin bcc\nFe (a), hcp Co (b) and fcc Ni (Ni). For comparison, the insets\nshow the bilinear exchange interaction parameters calcula ted\nfor the FM state with the magnetization along the ˆ z-axis. All\ndata are taken from Ref. 37.\nbiquadratic interactions. This is of course a material-\nspecific property, and one notes as decrease for the bi-\nquadratic exchange parameters when going to Co and Ni\nas shown in Fig. 4 (b) and (c), respectively.\nIn order to calculate the xandycomponents of\nthe four-spin and as a special case the three-site-DMI\n(TDMI) and biquadratic-DMI (BDMI) type interactions,\nthe scheme suggested in Ref. 37 for the calculation of the\nDMI parameters36,64can be used, which exploited the\nDMI-governed behavior of the spin-wave dispersion hav-\ning a finite slope at the Γ point of the Brillouin zone.10\nNote, however, that a more general form of perturbation\nisrequiredin thiscasedescribedbya2Dspin modulation\nfield according to the expression\nˆsi=/parenleftbig\nsin(/vector q1·/vectorRi) cos(/vector q2·/vectorRi),sin(/vector q2·/vectorRi),\ncos(/vector q1·/vectorRi)cos(/vector q2·/vectorRi)/parenrightbig\n, (69)\nwhere the wave vectors /vector q1and/vector q2are orthogonal to each\nother, as for example /vector q1=q1ˆyand/vector q2=q2ˆx.\nTaking the second-order derivative with respect to the\nwave-vector /vector q2and the first-order derivative with respect\nto the wave-vectors /vector q1and/vector q2, and considering the limit\nq1(2)→0, one obtains\n∂3\n∂q3\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq2=0H4,a=/summationdisplay\ni,j,k,lDx\nijkl(ˆq2·/vectorRij)(ˆq2·/vectorRlk)2,\nand\n∂\n∂q1/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq1=0∂2\n∂q2\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nq2=0H4,a=/summationdisplay\ni,j,k,lDy\nijkl(ˆq1·/vectorRij)(ˆq2·/vectorRlk)2,\nwhere/vectorRij=/vectorRj−/vectorRiand/vectorRlk=/vectorRk−/vectorRl.\nThe microscopic expressions for the xandycompo-\nnents of/vectorDijkldescribing the four-spin interactions is de-\nrived on the basis of the third-order term in Eq. (43)\n∆E(3)=−1\nπImTr/integraldisplayEF\ndE(E−EF)\n×G0∆VG0∆VG0∆VG0. (70)\nThe final expression for Dα\nijklis achieved by taking the\nsecond-order derivative with respect to the wave-vector\n/vector q2and the first-orderderivative with respect to the wave-\nvectors/vector q1(2), considering the limit q1(2)→0, i.e. equat-\ning within the ab-initio and model expressions the cor-\nresponding terms proportional to ( /vectorRi−/vectorRj)y(/vectorRk−/vectorRl)2\nx\nand (/vectorRi−/vectorRj)x(/vectorRk−/vectorRl)2\nx(we keep a similar form in both\ncasesforthe sakeofconvenience)givestheelements Dy,x\nijkl\nandDy,y\nijkl, as well as Dx,x\nijklandDx,y\nijkl, respectively, of the\nfour-spin chiral interaction as follows\nDα,β\nijkj=ǫαγ1\n8πImTr/integraldisplayEF\ndE(E−EF)\n/bracketleftBig\nOiτijTj,γτjkTk,βτklTl,βτli\n−Ti,γτijOjτjkTk,βτklTl,βτli/bracketrightBig\n+/bracketleftBig\nOiτijTj,βτjkTk,βτklTl,γτli\n−Ti,γτijTj,βτjkTk,βτklOlτli/bracketrightBig\n(71)\nwithα,β=x,y, andǫαγthe elements of the transverse\nLevi-Civita tensor ǫ=/bracketleftbigg\n0 1\n−1 0/bracketrightbigg\n. The TDMI and BDMI\nparameterscan be obtained as the special cases l=jand\nl=j,k=i, respectively, from Eq. (71).The expression in Eq. (71) gives access to the xandy\ncomponents of the DMI-like three-spin interactions\nDα\nijkj=Dα,x\nijkj+Dα,y\nijkj. (72)\nFinally, three-spin chiral exchange interaction (TCI)\nrepresented by first term in the extended spin Hamilto-\nnianhas been discussedin Ref. 37. As it followsfrom this\nexpression, the contribution due to this type of interac-\ntion is non-zero only in case of a non-co-planar and non-\ncollinear magnetic structure characterized by the scalar\nspatial type product ˆ si·(ˆsj׈sk) involving the spin mo-\nments on three different lattice sites.\nIn order to work out the expression for the Jijkinter-\naction, one has to use a multi-Q spin modulation65–67\nwhich ensure a non-zero scalar spin chirality for every\nthree atoms. The energy contribution due to the TCI,\nis non-zero only if Jijk/ne}ationslash=Jikj, etc. Otherwise, the\ntermsijkandikjcancel each other due to the relation\nˆsi·(ˆsj׈sk) =−ˆsi·(ˆsk׈sj).\nAccordingly, the expression for the TCI is derived us-\ning the 2Q non-collinear spin texture described by Eq.\n(69), which is characterized by two wave vectors oriented\nalong two mutually perpendicular directions, as for ex-\nample/vector q1= (0,qy,0) and/vector q2= (qx,0,0). Applying such a\nspin modulation in Eq. (69) for the term H3associated\nwith the three-spin interaction in the spin Hamiltonian\nin Eq. (61), the second-order derivative of the energy\nE(3)(/vector q1,/vector q2) with respect to the wave vectors q1andq2is\ngiven in the limit q1→0,q2→0 by the expression\n∂2\n∂/vector q1∂/vector q2H(3)\n=−/summationdisplay\ni/negationslash=j/negationslash=kJijk/parenleftbig\nˆz·[(/vectorRi−/vectorRj)×(/vectorRk−/vectorRj)]/parenrightbig\n.(73)\nThe microscopic energy term of the electron system,\ngiving access to the chiral three-spin interaction in the\nspin Hamiltonian is described by the second-order term\n∆E(2)=−1\nπImTr/integraldisplayEF\ndE(E−EF)\nG0∆VG0∆VG0 (74)\nof the free energy expansion. Taking the first-order\nderivative with respect to q1andq2in the limit q1→\n0,q2→0, and equating the terms proportional to/parenleftbig\nˆz·[(/vectorRi−/vectorRj)×(/vectorRk−/vectorRj)]/parenrightbig\nwith the correspondingterms\ninthespinHamiltonian,oneobtainsthefollowingexpres-\nsion for the three-spin interaction parameter\nJijk=1\n8πImTr/integraldisplayEF\ndE(E−EF)\n/bracketleftBig\nTi,xτijTj,yτjkOkτki−Ti,yτijTj,xτjkOkτki\n−Ti,xτijOjτjkTk,yτki+Ti,yτijOjτjkTk,xτki\n+OiτijTi,xτjkTk,yτki−OjτijTi,yτjkTk,xτki/bracketrightBig\n,(75)11\ngiving access to the three-spin chiral interaction deter-\nmined asJ∆=Jijk−Jikj. Its interpretation was dis-\ncussed in Ref. 68, where its dependence on the SOC as\nwell as on the topological orbital susceptibility χTO\n∆=\nχTO\nijk−χTO\nikjwas demonstrated. In fact that the expres-\nsion forχTO\nijkworked out in Ref. 68 has a rather similar\nform asJijk, as that can be seen from the expression\nχTO\nijk=−1\n4πImTr/integraldisplayEF\ndE\n×/bracketleftBig\nTi,xτijTj,yτjklk\nzτki−Ti,yτijTj,xτjklk\nzτki\n−Ti,xτijlj\nzτjkTk,yτki+Ti,yτijlj\nzτjkTk,xτki\n+li\nzτijTj,xτjkTk,yτki−li\nzτijTj,yτjkTk,xτki/bracketrightBig\n.\n(76)\nFor everytrimerofatoms, both quantities, χTO\nijkandJijk,\nare non-zero only in the case of non-zero scalar spin chi-\nrality ˆsi·(ˆsj׈sk) and depend on the orientation of the\ntrimermagneticmomentwith respecttothetrimerplain.\nThis is shown in Fig. 668representing ∆ Jand ∆χTOas\na function of the angle between the magnetization and\nnormal ˆnto the surface, which are calculated for the two\nsmallest trimers, ∆ 1and ∆ 2, centered at the Ir atom and\nthe hole site in the Ir surface layer for 1ML Fe/Ir(111),\nrespectively (Fig. 5).\nFIG. 5. Geometry of the smallest three-atom clusters in the\nmonolayer of 3 d-atoms on M(111) surface ( M= Au, Ir): M-\ncentered triangle ∆ 1and hole-centered triangle ∆ 2.\nThe role of the SOC for the three-site 4-spin DMI-like\ninteraction, Dz\nijik, and the three-spin chiral interaction,\nJ∆is shown in Fig. 7. These quantities are calculated\nfor 1ML Fe on Au (111), for the two smallest triangles\n∆1and ∆ 2centered at an Au atom or a hole site, re-\nspectively (see Fig. 5). Here, setting the SOC scaling\nfactorξSOC= 0 implies a suppression of the SOC, while\nξSOC= 1 corresponds to the fully relativistic case. Fig.\n7 (a) shows the three-site 4-spin DMI-like interaction pa-\nrameter, Dz\nijik(ξSOC) when the SOC scaling parameter\nξSOCapplied to all components in the system, shown by\nfull symbols, and with the SOC scaling applied only to\nthe Au substrate. One can see a dominating role of the\nSOC of substrate atoms for Dz\nijik. Also in Fig. 7 (b), a\nnearly linear variation can be seen for J∆(ξSOC) when\nthe SOC scaling parameter ξSOCis applied to all com-\nponents in the system (full symbols). Similar to Dz\nijik,\n✵ ✷✵ ✹✵✻✵\n✽✵❣ ✥ \u0000 ✁✂ ✄\n✵\n✵ ☎ ✆\n✵ ☎ ✷\n✵ ☎ ✝\n✵ ☎ ✹\n✵ ☎ ✞\n✵ ☎ ✻✲✟\n❉ ❚\n✠\n✡☛☞✌✍\n❏✎✶\n✥ ❣ ✄ ✥ ✮ ✏ ✑✒ ✄❏✎\n✓\n✥ ❣ ✄ ✥ ✔ ✑✕ ✖✄❏✎✶\n✥ ✭ ✄ ✗ ✑✘✥ ❣ ✄❏✎\n✓\n✥ ✭ ✄ ✗ ✑✘✥ ❣ ✄\n(a)✵ ✷✵ ✹✵✻✵\n✽✵❣ ✥ \u0000✁✂ ✄\n✵\n✵\n☎✵✵\n✆\n✵\n☎✵\n✝\n✵\n☎✵\n✝✆\n✵\n☎✵✷\n✵\n☎✵✷\n✆\n✵\n☎✵\n✞❝\n❉ ❚\n✟\n✠♠\n❇\n✴✡✡☛☞\n✌✍ ✶\n✥ ❣ ✄ ✥\n✮ ✎ ✏✑✄✌✍ ✒\n✥ ❣ ✄ ✥ ✓\n✏✔ ✕ ✄✌✍ ✶\n✥ ✵ ✄ ✭\n✏✖ ❣✌✍ ✒\n✥ ✵ ✄ ✭\n✏✖ ❣\n(b)\nFIG. 6. (a) Three-spinchiral exchange interaction paramet ers\n−J∆(γ) (’minus’ is used to stress the relation between J∆and\nχTO\n∆), and (bc) topological orbital susceptibility (TOS, for\nSOC = 0), calculated for Fe on Ir (111), as a function of the\nangle between the magnetization and normal ˆ nto the surface,\nfor the smallest triangles ∆ 1and ∆ 2. The dashed lines rep-\nresentJ∆(0) cos(γ) (a) and χTO\n∆(0) cos(γ) (b), respectively.\nAll data are taken from Ref. 68.\nthis shows that the SOC is an ultimate prerequisite for a\nnon-vanishing TCI J∆. When scaling the SOC only for\nAu (open symbols), Fig. 7 (b) show only weak changes\nfortheTCIparameters J∆(ξSOC), demonstratingaminor\nimpact of the SOC of the substrate on these interactions,\nin contrast to the DMI-like interaction shown in Fig. 7\n(a). One can see also that Dz\nijikis about two orders of\nmagnitude smaller than J∆for this particular system.\nThe origin of the TCI parameters have been discussed\nin the literature suggesting a different interpretation of\nthe correspondingterms derivedalsowithin the multiple-\nscattering theory Green function formalism62,69,70. How-\never, the expression worked out in Ref. 69 has obviously\nnot been applied for calculations so far. As pointed out\nin Ref. 68, the different interpretation of this type of in-\nteractions can be explained by their different origin. In\nparticular, one has to stress that the parameters in Refs.\n68 and 69 were derived in a different order of pertur-\nbation theory. On the other hand, the approach used\nfor calculations of the multispin exchange parameters re-\nported in Ref. 62, 69, and 71 is very similar to the one\nused in Refs. 37 and 68. The corresponding expressions\nhave been worked out within the framework of multiple-\nscattering Green function formalism using the magnetic\nforce theorem. In particular, the Lloyd formula has been\nused to express the energy change due to the perturba-\ntion ∆Vleading to the expression\n∆E=−1\nπIm Tr/integraldisplayEF\ndE/summationdisplay\np1\npTr/bracketleftbig\nG(E)∆V/bracketrightbigp.(77)\nUsing the off-site part of the GF in Eq. (35), as defined12\n0 0.2 0.4 0.6 0.8 1\nξSOC-0.004-0.00200.0020.0040.0060.008Dijik  (meV)\nDx(∆1), SOC\nDx(∆1), SOC(Au)Dx(∆2), SOC(Au) Dx(∆2), SOC\n(a)\n0 0.2 0.4 0.6 0.8 1\nξSOC-0.4-0.3-0.2-0.10J∆ (meV)∆1, SOC\n∆2, SOC\n∆1, SOC(Au)\n∆2, SOC(Au)\n(b)\nFIG. 7. (a) Three-site 4-spin DMI-like interaction, Dz\nijkjand\n(c) three-spin chiral exchange interaction (TCI) paramete rs\nJ∆calculated for Fe on Au (111) on the basis of Eq. (75) as\na function of SOC scaling parameter ξSOCfor the smallest\ntriangles ∆ 1and ∆ 2. In figure (b), full symbols represent\nthe results obtained when scaling the SOC for all elements in\nthe system, while open symbols show the results when scaling\nonly the SOC for Au. All data are taken from Ref. 68.\nby Dederichs et al.34, Eq. (77) is transformed to the form\n∆E=−1\nπIm Tr/integraldisplayEF\ndE/summationdisplay\np1\npTr/bracketleftBig\nGstr(E)∆t(E)/bracketrightBigp\n.(78)\nBy splitting the structural Green function Gstr\nijinto a\nspin-dependent ( /vectorBstr\nij) and a spin-independent ( Astr\nij)\nparts according to\nGstr\nij=Astr\nijσ0+/vectorBstr\nij·/vector σ (79)\nand expressing the change of the single-site scattering\nmatrix\n∆ti(E) = (t↑\ni(E)−t↓)δˆsi×/vector σ, (80)\nby means of the rigid spin approximation, the different\nterms in Eq. (78) corresponding to different numbers p\ngive access to corresponding multispin terms, chiral and\nnon-chiral, in the extended spin Hamiltonian. In particu-\nlar, the isotropic six-spin interactions, that are responsi-\nbleforthenon-collinearmagneticstructureofB20-MnGe\naccording to Grytsiuk et al69, is given by the expression\nκ6−spin\nijklmn=1\n3πIm Tr/integraldisplayEF\ndE\n×Aijtσ\njAjktσ\nkAkltσ\nlAlmtσ\nmAmntσ\nnAnitσ\ni.(81)A rather different point of view concerning the multi-\nspin extension of the spin Hamiltonian was adopted by\nStreib et al.72,73, who suggested to distinguish so-called\nlocal and global Hamiltonians. According to that classi-\nfication, a global Hamiltonian implies to include in prin-\nciple all possible spin configurations for the energy map-\nping in orderto calculate exchangeparametersthat char-\nacterize in turn the energy of any spin configuration. On\nthe other hand, a local Hamiltonian is ’designed to de-\nscribe the energetics of spin configurations in the vicinity\nof the ground state or, more generally, in the vicinity of a\npredefined spin configuration’72. This implies that taking\nthe ground state as a reference state, it has to be deter-\nmined first before the calculating the exchange parame-\nters which are in principle applicable only for small spin\ntiltings around the reference state and can be used e.g.\nto investigate spin fluctuations around the ground state\nspin configuration. In Ref. 72, the authors used a con-\nstrainingfieldtostabilizethenon-collinearmagneticcon-\nfiguration. This leads to the effective two-spin exchange\ninteractions corresponding to a non-collinear magnetic\nspin configuration72,73. According to the authors, ’lo-\ncal spin Hamiltonians do not require any spin interac-\ntions beyond the bilinear order (for Heisenberg exchange\nas well as Dzyaloshinskii-Moriya interactions)’ . On the\nother hand, they point out the limitations for these ex-\nchange interactions in the case of non-collinear system in\nthe regime when the standard Heisenberg model is not\nvalid73, and multi-spin interactions get more important.\nII. GILBERT DAMPING\nAnother parameter entering the Landau-Lifshitz-\nGilbert (LLG) equation in Eq. (3) is the Gilbert damping\nparameter ˜Gcharacterizingenergydissipation associated\nwith the magnetization dynamics.\nTheoretical investigations on the Gilbert damping pa-\nrameter have been performed by various groups and ac-\ncordinglythepropertiesofGDisdiscussedindetailinthe\nliterature. Many of these investigations are performed\nassuming a certain dissipation mechanism, like Kamber-\nsky’sbreathingFermisurface(BFS)74,75, ormoregeneral\ntorque-correlationmodels (TCM)76,77to be evaluated on\nthe basis of electronic structure calculations. The earlier\nworks in the field relied on the relaxation time param-\neter that represents scattering processes responsible for\nthe energy dissipation. Only few computational schemes\nfor Gilbert damping parameter account explicitly for dis-\norderin the systems, which is responsible forthe spin-flip\nscatteringprocess. This issuewasaddressedin particular\nby Brataas etal.78who described the Gilbert damping\nmechanism by means of scattering theory. This develop-\nmentsuppliedtheformalbasisforthefirstparameter-free\ninvestigations on disordered alloys38,39,79.\nA formalism for the calculation of the Gilbert damping\nparameter based on linear response theory has been re-\nported in Ref. 39 and implemented using fully relativistic13\nmultiple scattering or Korringa-Kohn-Rostoker (KKR)\nformalism. Considering the FM state as a reference state\nof the system, the energy dissipation can be expressed in\nterms of the GD parameter by:\n˙Emag=/vectorHeff·d/vectorM\ndτ=1\nγ2˙/vector m[˜G(/vector m)˙/vector m].(82)\nOn the other hand, the energy dissipation in the elec-\ntronic system is determined by the underlying Hamilto-\nnianˆH(τ) as follows ˙Edis=/angbracketleftBig\ndˆH\ndτ/angbracketrightBig\n. Assuming a small\ndeviation of the magnetic moment from the equilibrium\n/vector u(τ), the normalized magnetization /vector m(τ) can be written\nin a linearized form /vector m(τ) =/vector m0+/vector u(τ), that in turn leads\nto the linearized time dependent electronic Hamiltonian\nˆH(τ)\nˆH=ˆH0(/vector m0)+/summationdisplay\nµ/vector uµ∂\n∂/vector uµˆH(/vector m0).(83)\nAs shown in Ref. 38, the energy dissipation within the\nlinear response formalism is given by:\n˙Edis=−π/planckover2pi1/summationdisplay\nij/summationdisplay\nµν˙uµ˙uν/angbracketleftBigg\nψi/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ˆH\n∂uµ/vextendsingle/vextendsingle/vextendsingle/vextendsingleψj/angbracketrightBigg/angbracketleftBigg\nψj/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂ˆH\n∂uν/vextendsingle/vextendsingle/vextendsingle/vextendsingleψi/angbracketrightBigg\n×δ(EF−Ei)δ(EF−Ej).(84)\nIdentifying it with the corresponding phenomenological\nquantity in Eq. (82), ˙Emag=˙Edisone obtains for the GD\nparameterαa Kubo-Greenwood-like expression:\nαµν=−/planckover2pi1γ\nπMsTrace/angbracketleftBigg\n∂ˆH\n∂uµImG+(EF)∂ˆH\n∂uνImG+(EF)/angbracketrightBigg\nc,\n(85)\nwhereα=˜G/(γMs), and/an}b∇acketle{t.../an}b∇acket∇i}htcindicates a configura-\ntional average required in the presence of chemical or\nthermally induced disorder responsible for the dissipa-\ntion processes. Within the multiple scattering formalism\nwith the representation of the Green function given by\nEq. (9), Eq. (85) leads to\nαµµ=g\nπµtot/summationdisplay\nnTrace/angbracketleftbig\nT0µ˜τ0nTnµ˜τn0/angbracketrightbig\nc(86)\nwith the g-factor 2(1 + µorb/µspin) in terms of the spin\nand orbital moments, µspinandµorb, respectively, the\ntotal magnetic moment µtot=µspin+µorb, and ˜τ0n\nΛΛ′=\n1\n2i(τ0n\nΛΛ′−τ0n\nΛ′Λ) and with the energy argument EFomit-\nted. The matrix elements Tnµare identical to those oc-\ncurring in the context of exchange coupling6and can be\nexpressed in terms of the spin-dependent part Bof the\nelectronic potential with matrix elements:\nTnµ\nΛ′Λ=/integraldisplay\nd3rZn×\nΛ′(/vector r) [βσµBxc(/vector r)]Zn\nΛ(/vector r).(87)\nAs is discussed in Ref. 39, fora system havingchemical\ndisorder, the configurational average is performed using00.10.2 0.3 0.4 0.5\nconcentration xV02040α × 103without vertex corrections\nwith vertex correctionsFe1-xVx\n(a)\nFIG. 8. The Gilbert damping parameter for (a) bcc Fe 1−xVx\n(T= 0 K) as a function of V concentration. Full (open)\nsymbols give results with (without) the vertex corrections .\nAll data are taken from Ref. 39.\nthe scattering path operators evaluated on the basis of\nthe coherent potential approximation (CPA) alloy the-\nory. In the case of thermally induced disorder, the al-\nloy analogy model is used, which was discussed already\nabove. When evaluating Eq. (86), the so-called vertex\ncorrections have to be included43that accounts for the\ndifference between the averages /an}b∇acketle{tTµImG+TνImG+/an}b∇acket∇i}htcand\n/an}b∇acketle{tTµImG+/an}b∇acket∇i}htc/an}b∇acketle{tTνImG+/an}b∇acket∇i}htc. Within the Boltzmann formal-\nism these corrections account for scattering-in processes.\nThe crucial role of these corrections is demonstrated39\nin Fig. 8 representing the Gilbert damping parameter\nfor an Fe 1−xVxdisordered alloy as a function of the con-\ncentrationx, calculated with and without vertex correc-\ntions. As one can see, neglect of the vertex corrections\nmay lead to the nonphysical result α <0. This wrong\nbehavior does not occur when the vertex corrections are\nincluded, that obviously account for energy transfer pro-\ncesses connected with scattering-in processes.\nThe impact of thermal vibrations onto the Gilbert\ndamping can be taken into account within the alloy-\nanalogy model (see above) by averaging over a discrete\nset of thermal atom displacements for a given temper-\natureT. Fig. 9 represents the temperature dependent\nbehavior of the Gilbert damping parameter αfor bcc Fe\nwith 1% and 5% of impurities of Os and Pt38,39. One can\nseeastrongimpactofimpuritiesonGD.Inthecaseof1%\nof Pt in Fig. 9 (a), αdecreases in the low-temperature\nregime much steeper upon increasing the temperature,\nindicating that the breathing Fermi surface mechanism\ndominates. When the concentration of the impurities in-\ncreases up to 5% (Fig. 9 (a)), the spin-flip scattering\nmechanism takes the leading role for the magnetization\ndissipation practically for the whole region of tempera-\ntures under consideration. The different behavior of GD\nforFe with OsandPt isaresult ofthe different densityof\nstates (DOS) of the impurities at the Fermi energy (see\nRef. 39 for a discussion).\nThe role of the electron-phonon scattering for the ul-\ntrafast laser-induced demagnetization was investigated14\n0100200 300 400 500\ntemperature (K)12345α × 103Fe0.99Me0.01Pt\nOs\n(a)\n0100200 300 400 500\ntemperature (K)22.533.54α × 103Fe0.95Me0.05\nPtOs\n(b)\nFIG. 9. Gilbert damping parameter for bcc Fe 1−xMxwith\nM= Pt (circles) and M= Os (squares) impurities as a func-\ntion of temperature for 1% (a) and 5% (b) of the impurities.\nAll data are taken from Ref. 39.\nby Carva et al.80based on the Elliott-Yafet theory of\nspin relaxation in metals, that puts the focus on spin-\nflip(SF) transitionsupon theelectron-phononscattering.\nAs the evaluation of the spin-dependent electron-phonon\nmatrix elements entering the expression for the rate of\nthe spin-flip transition is a demanding problem, various\napproximations are used for this. In particular, Carva et\nal.80,81use the so-called Elliott approximation to evalu-\nate a SF probability Pb\nS=τ\nτsfwith the spin lifetime τsf\nand a spin-diagonal lifetime τ:\nPb\nS=τ\nτsf= 4/an}b∇acketle{tb2/an}b∇acket∇i}ht (88)\nwith the Fermi-surface averaged spin mixing of Bloch\nwave eigenstates\n/an}b∇acketle{tb2/an}b∇acket∇i}ht=/summationdisplay\nσ,n/integraldisplay\nd3k|bσ\n/vectorkn|δ(Eσ\n/vectorkn−EF).(89)\nIn the case of a non-collinear magnetic structure, the\ndescription of the Gilbert damping can be extended byadding higher-order non-local contributions. The role of\nnon-local damping contributions has been investigated\nby calculating the precession damping α(/vector q) for magnons\nin FM metals, characterized by a wave vector /vector q. Follow-\ning the same idea, Thonig et al.82used a torque-torque\ncorrelationmodel based on atight binding approach,and\ncalculated the Gilbert damping for the itinerant-electron\nferromagnets Fe, Co and Ni, both in the reciprocal, α(/vector q),\nand realαijspace representations. The important role\nof non-local contributions to the GD for spin dynam-\nics has been demonstrated using atomistic magnetization\ndynamics simulations.\nAformalismforcalculatingthe non-localcontributions\nto the GD has been recently worked out within the KKR\nGreen function formalism83. Using linear response the-\nory for weakly-noncollinear magnetic systems it gives ac-\ncess to the GD parameters represented as a function of\na wave vector /vector q. Using the definition for the spin sus-\nceptibility tensor χαβ(/vector q,ω), the Fourier transformation\nof the real-space Gilbert damping can be represented by\nthe expression84,85\nααβ(/vector q) =γ\nM0Vlim\nω→0∂ℑ[χ−1]αβ(/vector q,ω)\n∂ω.(90)\nHereγ=gµBis the gyromagneticratio, M0=µtotµB/V\nis the equilibrium magnetization and Vis the volume of\nthe system. As is shown in Ref. 83, this expression can\nbe transformed to the form which allows an expansion of\nGD in powers of wave vector /vector q:\nα(/vector q) =α+/summationdisplay\nµαµqµ+1\n2/summationdisplay\nµναµνqµqν+....(91)\nwith the following expansion coefficients:\nα0±±\nαα=g\nπµtot1\nΩBZTr/integraldisplay\nd3k/angbracketleftbigg\nTβτ(/vectork,E±\nF)Tβτ(/vectork,E±\nF)/angbracketrightbigg\nc\nαµ±±\nαα=g\nπµtot1\nΩBZTr/integraldisplay\nd3k/angbracketleftbigg\nTβ∂τ(/vectork,E±\nF)\n∂kµTβτ(/vectork,E±\nF)/angbracketrightbigg\nc\nαµν±±\nαα=−g\n2πµtot1\nΩBZ\n×Tr/integraldisplay\nd3k/angbracketleftbigg\nTβ∂τ(/vectork,E±\nF)\n∂kµTβ∂τ(/vectork,E±\nF)\n∂kν/angbracketrightbigg\nc.(92)\nFor the prototype multilayer system\n(Cu/Fe 1−xCox/Pt)nthe calculated zero-order (uni-\nform) GD parameter αxxand the corresponding\nfirst-order (chiral) αx\nxxcorrection term for /vector q/ba∇dblˆxare\nplotted in Fig. 10 top and bottom, respectively, as a\nfunction of the Co concentration x. Both terms, αxx\nandαx\nxx, increase approaching the pure limits w.r.t. the\nFe1−xCoxalloy subsystem. As is pointed out in Ref. 83,\nthis increase is associated with the dominating so-called\nbreathing Fermi-surface damping mechanism due to the\nmodification of the Fermi surface (FS) induced by the\nSOC, which follows the magnetization direction that\nslowly varies with time. As αis caused for a ferromagnet15\n0 0.2 0.4 0.6 0.8 100.20.4αxx\n0 0.2 0.4 0.6 0.8 1xCo0123αxxx (a.u.)\nFIG. 10. The Gilbert damping parameters αxx(top) and\nαx\nxx(bottom) calculated for the model multilayer system\n(Cu/Fe 1−xCox/Pt)nusing first and second expressions in Eq.\n(92), respectively. All data are taken from Ref. 83.\nexclusively by the SOC one can expect that it vanishes\nfor vanishing SOC. This was indeed demonstrated\nbefore39. The same holds also for αxthat is caused by\nSOC as well.\nAlternatively, a real-space extension for classical\nGilbert dampingtensorwasproposedrecentlybyBrinker\net al.86, by introducing two-site Gilbert damping tensor\nGijentering the site-resolved LLG equation\n1\nγd/vectorMi\ndτ=−γ/vectorMi×/parenleftbigg\n/vectorHi,eff+/summationdisplay\nj/bracketleftBigg\nGij(/vectorM)·d/vectorMi\ndτ/bracketrightBigg/parenrightbigg\n,(93)\nwhich is related to the inverse dynamical susceptibility\nχijvia the expression\nd\ndωIm[χ]αβ\nij=δij/parenleftbigg1\nγMiǫαβγ/parenrightbigg\n+/parenleftbigg\nRiGijRT\nj/parenrightbigg\nαβ,(94)\nwhereRiandRjarethe rotationmatricesto gofromthe\nglobal to the local frames of reference for atoms iandj,\nrespectively, assuming a non-collinear magnetic ground\nstate in the system. Thus, an expression for the GD\ncan be directly obtained using the adiabatic approxima-\ntion for the slow spin-dynamics processes. This justifies\nthe approximation ([ χ]−1(ω))′\nω≈([χ0]−1(ω))′\nω, with the\nun-enhanced dynamical susceptibility given in terms ofelectronic Green function Gij\nχαβ\nij(ω+iη) =−1\nπTr/integraldisplayEF\ndE\n/bracketleftbigg\nσαGij(E+ω+iη)σβImGij(E)\n+σαGij(E)σβImGij(E−ω−iη)/bracketrightbigg\n,(95)\nwith the Green function G(E±iη) = (E− H ±iη)−1\ncorresponding to the Hamiltonian H.\nMoreover, this approach allows a multisite expansion\noftheGDaccountingforhigher-ordernon-localcontribu-\ntions for non-collinearstructures86. For this purpose, the\nHamiltonian His split into the on-site contribution H′\nand the intersite hopping term tij, which is spin depen-\ndent in the general case. The GF can then be expanded\nin a perturbative way using the Dyson equation\nGij=G0\niδij+G0\nitijG0\nj+G0\nitikG0\nktkjG0\nj+....(96)\nAs a result, the authors generalize the LLG equation\nby splitting the Gilbert damping tensor in terms pro-\nportional to scalar, anisotropic, vector-chiral and scalar-\nchiral products of the magnetic moments, i.e. terms like\nˆei·ˆej, (ˆnij·ˆei)(ˆnij·ˆej), ˆnij·(ˆei׈ej), etc.\nIt should be stressed that the Gilbert damping param-\neter accounts for the energy transfer connected with the\nmagnetization dynamics but gives no information on the\nangular momentum transfer that plays an important role\ne.g. for ultrafast demagnetization processes. The formal\nbasis to account simultaneously for the spin and lattice\ndegrees of freedom was considered recently by Aßmann\nand Nowak87and Hellsvik et al.88. Hellsvik et al.88,89re-\nportonanapproachsolvingsimultaneouslytheequations\nfor spin and lattice dynamics, accounting for spin-lattice\ninteractions in the Hamiltonian, calculated on a first-\nprinciples level. These interactions appear as a correc-\ntion to the exchange coupling parameters due to atomic\ndisplacements. As a result, this leads to the three-body\nspin-lattice coupling parameters Γαβµ\nijk=∂Jαβ\nij\n∂uµ\nkand four-\nbody parameters Λαβµν\nijkl=∂Jαβ\nij\n∂uµ\nk∂uν\nlrepresented by rank 3\nand rank 4 tensors, respectively, entering the spin-lattice\nHamiltonian\nHsl=−1\n2/summationdisplay\ni,j,k,αβ,µΓαβµ\nijkeα\nieβ\njuµ\nk\n−1\n4/summationdisplay\ni,j,k,l,αβ,µ,νΛαβµν\nijkleα\nieβ\njuµ\nkuν\nl.(97)\nThe parameters Γαβµ\nijkin Ref. 88 are calculated using a\nfinite difference method, using the exchange coupling pa-\nrametersJijfor the system without displacements ( J0\nij)\nand with a displaced atom k(J∆\nij(/vector uk)), used to estimate\nthe coefficient Γαβµ\nijk≈(J∆\nij(/vector uk)−J0\nij)\nuµ.16\nAlternatively, to describe the coupling of spin and spa-\ntial degrees of freedom the present authors (see Ref. 90)\nadopt an atomistic approach and start with the expan-\nsion of a phenomenological spin-lattice Hamiltonian\nHsl=−/summationdisplay\ni,j,α,β/summationdisplay\nk,µJαβ,µ\nij,keα\nieβ\njuµ\nk\n−/summationdisplay\ni,j/summationdisplay\nk,lJαβ,µν\nij,kleα\nieβ\njuµ\nkuν\nl,(98)\nthat can be seen as a lattice extension of a Heisenberg\nmodel. Accordingly, the spin and lattice degrees of free-\ndom are represented by the orientation vectors ˆ eiof the\nmagnetic moments /vector miand displacement vectors /vector uifor\neach atomic site i. The spin-lattice Hamiltonian in Eq.\n(98) is restricted to three and four-site terms. As rel-\nativistic effects are taken into account, the SLC is de-\nscribed in tensorial form with Jαβ,µ\nij,kandJαβ,µν\nij,klrepre-\nsented by rank 3 and rank 4 tensors, similar to those\ndiscussed by Hellsvik et al.88.\nThesamestrategyasforthe exchangecouplingparam-\netersJij4orJαβ\nij5,6, is used to map the free energy land-\nscapeF({ˆei},{/vector ui}) accounting for its dependence on the\nspin configuration {ˆei}as well as atomic displacements\n{/vector ui}, making use of the magnetic force theorem and the\nLloyd formulato evaluate integrated DOS ∆ N(E). With\nthis, the free energy change due to any perturbation in\nthe system is given by Eq. (25).\nUsing as a reference the ferromagnetically ordered\nstate of the system with a non-distorted lattice, and the\nperturbed state characterized by finite spin tiltings δˆei\nand finite atomic displacements /vector uiat sitei, one can\nwrite the corresponding changes of the inverse t-matrix\nas ∆s\nµmi=mi(δˆeµ\ni)−m0\niand ∆u\nνmi=mi(uν\ni)−m0\ni.\nThis allows to replace the integrand in Eq. (11) by\nlnτ−lnτ0=−ln/parenleftBig\n1+τ[∆s\nµmi+∆u\nνmj+...]/parenrightBig\n,(99)\nwhere all site-dependent changes in the spin configura-\ntion{ˆei}and atomic positions {/vector ui}are accounted for in\na one-to-one manner by the various terms on the right\nhand side. Due to the use of the magnetic force theorem\nthese blocks may be written in terms of the spin tiltings\nδˆeµ\niand atomic displacements of the atoms uν\nitogether\nwith the corresponding auxiliary matrices Tµ\niandUν\ni,\nrespectively, as\n∆s\nµmi=δˆeµ\niTµ\ni, (100)\n∆u\nνmi=uν\niUν\ni. (101)\nInserting these expressionsinto Eq. (99) and the result in\nturnintoEq.(25)allowsustocalculatetheparametersof\nthe spin-lattice Hamiltonian as the derivatives of the free\nenergy with respect to tilting angles and displacements.\nThis way one gets for example for the three-site term:\nJαβ,µ\nij,k=∂3F\n∂eα\ni∂eβ\nj∂uµ\nk=1\n2πIm Tr/integraldisplayEF\ndE\n×/bracketleftBig\nTα\niτijTβ\njτjkUµ\nkτki+Tα\niτikUµ\nkτkjTβ\njτji/bracketrightBig\n(102)\nFIG. 11. The absolute values of site-off-diagonal and site-\ndiagonal SLC parameters: DMI |/vectorDx\nij,j|and isotropic SLC\nJiso,x\nij,j(top), anti-symmetric diagonal components Jdia−a,x\nij,j\nandJdia−a,x\nii,k(middle), and symmetric off-diagonal compo-\nnentsJoff−s,x\nij,jandJoff−s,x\nii,k(bottom) for bcc Fe, as a function\nof the interatomic distance rij\nand for the four-site term:\nJαβ,µν\nij,kl=∂4F\n∂eα\ni∂eβ\nj∂uµ\nk∂uν\nl=1\n4πIm Tr/integraldisplayEF\ndE\n×/bracketleftBigg\nUµ\nkτkiTα\niτijTβ\njτjlUν\nlτlk\n+Tα\niτikUµ\nkτkjTβ\njτjlUν\nlτli\n+Uµ\nkτkiTα\niτilUν\nlτljTβ\njτjk\n+Tα\niτikUµ\nkτklUν\nlτljTβ\njτji/bracketrightBigg\n.(103)\nFig. 11 shows corresponding results for the SLC pa-\nrameters of bcc Fe, plotted as a function of the distance\nrijfori=kwhich implies that a displacement along the\nxdirection is applied for one of the interacting atoms.\nThe absolute values of the DMI-like SLC parameters\n(DSLC) |/vectorD|µ=x\nij,k(note that Dz,µ\nij,k=1\n2(Jxy,µ\nij,k− Jyx,µ\nij,k) )\nshow a rather slow decay with the distance rij. The\nisotropic SLC parameters Jiso,µ=x\nij,j, which have only a\nweak dependence on the SOC, are about one order\nof magnitude larger than the DSLC. All other SOC-\ndriven parameters shown in Fig. 11, characterizing the\ndisplacement-induced contributions to MCA, are much\nsmaller than the DSLC.17\nIII. SUMMARY\nTo summarize, we have considered a multi-level atom-\nistic approach commonly used to simulate finite temper-\natureand dynamical magneticpropertiesof solids, avoid-\ning in particular time-consuming TD-SDFT calculations.\nTheapproachisbasedonaphenomenologicalparameter-\nized spin Hamiltonian which allows to separate the spin\nand orbital degrees of freedom and that way to avoid the\ndemanding treatment of complex spin-dependent many-\nbody effects. As these parameters are fully determined\nby the electronic structure of a system, they can be de-\nduced from the information provided by relativistic band\nstructure calculations based on SDFT. 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. . . . . . . . . . . 5\n3.2 Specific Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n4 Data 6\n4.1 Language files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6\n4.2 Sheep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7\n4.2.1 Sheep counting words . . . . . . . . . . . . . . . . . . . . . . 7\n4.2.2 Geographical data . . . . . . . . . . . . . . . . . . . . . . . . 8\n4.3 Colours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9\n4.4 IPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10\n5 Methodology 10\n6 Methods 12\n6.1 Edit Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12\n6.2 Phonetic Substitution Table . . . . . . . . . . . . . . . . . . . . . . . 14\n6.3 Hierarchical Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . 14\n6.3.1 Using the OC program . . . . . . . . . . . . . . . . . . . . . . 14\n6.3.2 Using R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16\n6.4 Further analysis with R . . . . . . . . . . . . . . . . . . . . . . . . . 16\n6.5 Process automation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17\n7 Results 17\n7.1 Sheep . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17\n7.1.1 Analysis of average and subset linguistic distance . . . . . . . 18\n7.2 Hierarchical clustering . . . . . . . . . . . . . . . . . . . . . . . . . . 18\n7.2.1 All to all comparison analysis . . . . . . . . . . . . . . . . . . 19\n7.2.2 Linguistic and Geographical distance relationship . . . . . . . 22\n7.3 Colours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237.3.1 Mean and Standard Deviation . . . . . . . . . . . . . . . . . . 23\n7.3.2 Density Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . 24\n7.3.3 Bhattacharya Coefficients . . . . . . . . . . . . . . . . . . . . 25\n7.3.4 Hierarchical Clustering . . . . . . . . . . . . . . . . . . . . . . 26\n7.4 IPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28\n7.5 Small Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29\n7.5.1 All to all comparison . . . . . . . . . . . . . . . . . . . . . . . 29\n8 Conclusions 30\n9 Further Work 32\n10 Summary of contributions 35\nAcknowledgements 36\nBibliography 37\nAppendix A Phonetic Substitution tables 39\nA.1 Editable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39\nA.2 editableGaby . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41\nAppendix B Dendrograms and Cluster plots 44\nB.1 Sheep counting systems . . . . . . . . . . . . . . . . . . . . . . . . . 44\nB.2 Dendrograms of Bhattacharya scores of colour words . . . . . . . . . 44List of Figures\n1 Indo-European language tree . . . . . . . . . . . . . . . . . . . . . . 4\n2 Indo-European migrations2. . . . . . . . . . . . . . . . . . . . . . . 5\n3 Sheep dialects in Britain . . . . . . . . . . . . . . . . . . . . . . . . . 9\n4 Work-flow of comparison of languages . . . . . . . . . . . . . . . . . 11\n5 Work-flow of relationship analysis . . . . . . . . . . . . . . . . . . . . 12\n6 Table illustrating Grimm’s Law chain shift . . . . . . . . . . . . . . . 15\n7 Statistics of sheep counting systems . . . . . . . . . . . . . . . . . . . 18\n8 Density plot of sheep counting systems . . . . . . . . . . . . . . . . . 19\n9 Dendrogram of sheep counting systems . . . . . . . . . . . . . . . . . 20\n10 Dendrogram of sheep counting systems with the best Silhouette cut . 21\n11 Purity of clusters of sheep counting systems . . . . . . . . . . . . . . 22\n12 Linguistic and geographical distances of sheep counting systems . . . 23\n13 Statistics of “ColoursAll” . . . . . . . . . . . . . . . . . . . . . . . . . 24\n14 Statistics of Indo-European languages (colour words) . . . . . . . . . 25\n15 Statistics of Germanic languages (colour words) . . . . . . . . . . . . 26\n16 Statistics of Romance languages (colour words) . . . . . . . . . . . . 27\n17 Density plot of Germanic languages (colour words) . . . . . . . . . . 28\n18 Density plot of Germanic languages (colour words) . . . . . . . . . . 29\n19 Dendrogram of all languages (colour words) . . . . . . . . . . . . . . 30\n20 Chronological dispersal of the Austronesian people . . . . . . . . . . 31\n21 Dendrogram of Indo-European languages (colour words) . . . . . . . 32\n22 Dendrogram of Germanic languages (colour words) . . . . . . . . . . 33\n23 Dendrogram of Romance languages (colour words) . . . . . . . . . . 34\n24 Cluster plot of sheep counting all to all comparison (k=10) . . . . . 44\n25 Cluster plot of sheep counting all to all comparison (k=10) . . . . . 45\n26 Dendrogram of Bhattacharya scores of “ColoursAll” . . . . . . . . . . 46\n27 DendrogramofBhattacharyascoresofIndo-Europeanlanguages(colours) 47\n28 Dendrogram of Bhattacharya scores of Germanic languages (colours) 48\n29 Dendrogram of Bhattacharya scores of Romance languages (colours) 49\n30 Dendrogram of Bhattacharya scores of Germanic and Romance lan-\nguages (colours) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50List of Tables\n1 Table of Brittonic languages . . . . . . . . . . . . . . . . . . . . . . . 4\n2 Table of phonetic encoding . . . . . . . . . . . . . . . . . . . . . . . 7\n3 Table of Language files . . . . . . . . . . . . . . . . . . . . . . . . . . 8Abstract\nThis research project aimed to overcome the challenge of analysing human\nlanguage relationships, facilitate the grouping of languages and formation of\ngenealogical relationship between them by developing automated comparison\ntechniques. Techniqueswerebasedonthephoneticrepresentationofcertainkey\nwords and concept. Example word sets included numbers 1-10 (curated), large\ndatabase of numbers 1-10 and sheep counting numbers 1-10 (other sources),\ncolours (curated), basic words (curated).\nTo enable comparison within the sets the measure of Edit distance was\ncalculated based on Levenshtein distance metric. This metric between two\nstrings is the minimum number of single-character edits, operations including:\ninsertions, deletions or substitutions. To explore which words exhibit more\nor less variation, which words are more preserved and examine how languages\ncould be grouped based on linguistic distances within sets, several data analyt-\nics techniques were involved. Those included density evaluation, hierarchical\nclustering, silhouette, mean, standard deviation and Bhattacharya coefficient\ncalculations. These techniques lead to the development of a workflow which\nwas later implemented by combining Unix shell scripts, a developed R package\nand SWI Prolog. This proved to be computationally efficient and permitted\nthe fast exploration of large language sets and their analysis.\n1 Introduction\nTheneedtouncoverpresumedunderlyinglinguisticevolutionaryprinciplesandanal-\nyse correlation between world’s languages has entailed this research. For centuries\npeople have been speculating about the origins of language, however this subject is\nstill obscure. Non-automated linguistic analysis of language relationships has been\ncomplicated and very time-consuming. Consequently, this research aims to apply a\ncomputational approach to compare human languages. It is based on the phonetic\nrepresentation of certain key words and concept. This comparison of word similarity\naims to facilitate the grouping of languages and the analysis of the formation of\ngenealogical relationship between languages.\nThis report contains a thorough description of the proposed methods, developed\ntechniques and discussion of the results. During this projects several collections of\nwords were gathered and examined, including colour words and numbers. The meth-\nods included edit distance, phonetic substitution table, hierarchical clustering with\na cut and other analysis methods. They all aimed to provide an insight regarding\nboth technical data summary and its visual representation.\n12 Background\n2.1 Human languages\nFor centuries, people have speculated over the origins of language and its early devel-\nopment. It is believed that language first appeared among Homo Sapiens somewhere\nbetween 50,000 and 150,000 years ago19. However, the origins of human language\nare very obscure.\nTo begin with, it is still unknown if the human language originated from one\noriginal and universal Proto-Language. Alfredo Trombetti made the first scientific\nattempt to establish the reality of monogenesis in languages. His investigation con-\ncluded that it was spoken between 100,000 and 200,000 years ago, or close to the\nfirst emergence of Homo Sapiens22. However it was never accepted comprehensively.\nThe concept of Proto-Language is purely hypothetical and not amenable to analysis\nin historical linguistics.\nFurthermore, there are multiple theories of how language evolved. These could\nbe separated into two distinctly different groups.\nFirstly, some researchers claim that language evolved as a result of other evo-\nlutionary processes, essentially making it a by-product of evolution, selection for\nother abilities or as a consequence of yet unknown laws of growth and form. This\ntheory is clearly established in Noam Chomsky10and Stephen Jay Gould’s work13.\nBoth scientists hypothesize that language evolved together with the human brain, or\nwith the evolution of cognitive structures. They were used for tool making, informa-\ntion processing, learning and were also beneficial for complex communication. This\nconforms with the theory that as our brains became larger, our cognitive functions\nincreased.\nSecondly, another widely held theory is that language came about as an evolu-\ntionary adaptation, which is when a population undergoes a change in process over\ntime to survive better. Scientists Steven Pinker and Paul Bloom in “Natural Lan-\nguage and Natural Selection”20theorize that a series of calls or gestures evolved over\ntime into combinations, resulting in complex communication.\nToday there are 7,111 distinct languages spoken worldwide according to the 2019\nEthnologue language database. Many circumstances such as the spread of old civ-\nilizations, geographical features, and history determine the number of languages\nspoken in a particular region. Nearly two thirds of languages are from Asia and\nAfrica.\nThe Asian continent has the largest number of spoken languages - 2,303. Africa\nfollows closely with 2,140 languages spoken across continent. However, given the\npopulation of certain areas and colonial expansion in recent centuries, 86 percent of\npeople use languages from Europe and Asia. It is estimated that there is around\n4.2 billion speakers of Asian languages and around 1.75 billion speakers of European\nlanguages.\nMoreover, Pacific languages have approximately 1,000 speakers each on average,\n2but altogether, they represent more than a third of our world’s languages. Papua\nNew Guinea is the most linguistically diverse country in the world. This is possibly\ndue to the effect of its geography imposing isolation on communities. It has over\n840 languages spoken, with twelve of them lacking many speakers. It is followed by\nIndonesia, which has 709 languages spoken across the country.\n2.1.1 Indo-European languages and Kurgan Hypothesis\nIndo-European languages is a language family that represents most of the modern\nlanguages of Europe, as well as specific languages of Asia. Indo-European language\nfamily consist of several hundreds of related languages and dialects. Consequently, it\nwas an interest of the linguists to explore the origins of the Indo-European language\nfamily.\nIn the mid-1950s, Marija Gimbutas, a Lithuanian-American archaeologist and\nanthropologist, combined her substantial background in linguistic paleontology with\narchaeologicalevidencetoformulatetheKurganhypothesis12. Thishypothesisisthe\nmostwidelyacceptedproposaltoidentifythehomelandofProto-Indo-European(PIE)\n(ancient common ancestor of the Indo-European languages) speakers and to explain\nthe rapid and extensive spread of Indo-European languages throughout Europe and\nAsia17 18. The Kurgan hypothesis proposes that the most likely speakers of the\nProto-Indo-European language were people of a Kurgan culture in the Pontic steppe,\nby the north side of the Black Sea. It also divides the Kurgan culture into four suc-\ncessive stages (I, II, III, IV) and identifies three waves of expansions (I, II, III). In\naddition, the model suggest that the Indo-European migration was happening from\n4000 to 1000 BC. See figure 2 for visual illustration of Indo-European migration.\nToday there are approximately 445 living Indo-European languages, which are\nspoken by 3.2 billion people, according to Ethnologue. They are divided into the\nfollowing groups: Albanian, Armenian, Baltic, Slavic, Celtic, Germanic, Hellenic,\nIndo-Iranian and Italic (Romance) 116.\n2.1.2 Brittonic languages\nBrittonic or British Celtic languages derive from the Common Brittonic language,\nspoken throughout Great Britain south of the Firth of Forth during the Iron Age and\nRoman period. They are classified as Indo-European Celtic languages9. The family\ntree of Brittonic languages is showed in Table 1. Common Brittonic is ancestral to\nWestern and Southwestern Brittonic. Consequently, Cumbric and Welsh, which is\nspoken in Wales, derived from Western Brittonic. Cornish and Breton, spoken in\nCornwall and Brittany, respectively, originated from Southwestern side.\nToday Welsh, Cornish and Breton are still in use. However, it is worth to point\nout that Cornish is a language revived by second-language learners due to the last\nnative speakers dying in the late 18th century. Some people claimed that the Cornish\nlanguage is an important part of their identity, culture and heritage, and a revival\n3Proto-Indo-Eu ropean\nIrish\nScottishBreton\nManx\nWelshCeltic\nLatin\nIberian\nPortugueseSpanish\nGalicianGalician-\nPortu gueseItalian\nRomania n\nOld French\nMiddle Fren ch\nFrenchLangue d’OïlIberianGallo-\nGallic\nCatala nOccitanOccitanOscanItalic\nEastern\nRomanceGreekClassical\nGreekHellenic Germanic\nWest\nGermanicIcelan dicNorth\nGermanic\nOld East\nNorse\nSwedish\nDanish Norw egianNorseOld West\nModern Engli shMiddle E nglishFrisianAnglo-\nOldOld E nglish\nFrisianOld Dutch\nMiddle Du tch\nDutch Afrikaan sOld High\nGerman\nGermanMiddle High\nYiddish Germa nBalto- Slavic\nBaltic\nSlavicEastSerbia nSlavic\nSlavicWestLatvian\nPolish\nRussia nUkrainianLithuan ian\nCzechCroatian\nPrepared by Jack Lynch. Edited 2 2 Feb . 2014. Jack.Ly nch@rutg ers.ed uHindust aniMarathi\nGujarat i\nHindi UrduPunjabiIndic Irania n\nOld\nPersianAvestan\nFarsiKurdishMiddle\nPersianSanskritIndo-Iranian Armenia n\nArmenianOld Armen ianAlbanian\nAlbanian\nEast\nGermanic\nGothic\nFrisianSouth\nSlavic\nBulgarianFigure 1: Indo-European language tree16\nbegan in the early 20th century. Cornish is currently a recognised minority language\nunder the European Charter for Regional or Minority Languages.\nTable 1: Table of Brittonic languages\nCommon Brittonic\nWestern Brittonic Sothwestern estern Brittonic\nCumbric Welsh Cornish Breton\n2.1.3 Sheep Counting System\nBrittonic Celtic language is an ancestor to the number names used for sheep count-\ning11 3. Until the Industrial Revolution, the use of traditional number systems was\ncommon among shepherds, especially in the fells of the Lake District. The sheep-\ncounting system was referred to as Yan Tan Tethera. It was spread across Northern\nEngland and in other parts of Britain in earlier times. The number names varied\n4Figure 2: Indo-European migrations2\naccording to dialect, geography, and other factors. They also preserved interesting\nindications of how languages evolved over time.\nThe word “yan” or “yen” meaning “one”, in some northern English dialects repre-\nsents a regular development in Northern English15. During the development the Old\nEnglish long vowel / A:/ <¯ a> was broken into /ie/, /ia/ and so on. This explains\nthe shift to “yan” and “ane” from the Old English ¯ an, which is itself derived from the\nProto-Germanic “*ainaz”14.\nIn addition, the counting system demonstrates a clear connection with counting\non the fingers. Particularly after numbers reach 10, as the best known examples are\nformed according to this structure: 1 and 10, 2 and 10, up to 15, and then 1 and\n15, 2 and 15, up to 20. The count variability would end at 20. It might be due to\nthe fact, that the shepherds, on reaching 20, would transfer a pebble or marble from\none pocket to another, so as to keep a tally of the number of scores.\n3 Aims and Objectives\n3.1 Overall Aim\nThe aim of this research was to develop computational methods to compare human\nlanguages based on the phonetic form of single words (i.e. not exploiting grammar).\nThis comparison of word similarity aims to facilitate the grouping of languages, the\nidentification of the the presumed underlying linguistic evolutionary principles and\nthe analysis of the formation of genealogical relationship between languages.\n53.2 Specific Objectives\n1. Devise a way to encode the phonetic representation of words, using:\n(a) an in-house encoding,\n(b) an IPA (International Phonetic Alphabet).\n2. Develop methods to analyze the comparative relationships between languages\nusing: descriptiveandinferentialstatistics, clustering, visualisationofthedata,\nand analysis of the results.\n3. Implement a repeatable process for running the analysis methods with new\ndata.\n4. Analyse the correlation between geographical distance and language similarity\n(linguistic distance), and investigate if it explains the evolutionary distance.\n5. Examine which words exhibit more or less variation and the likely causes of it.\n6. Explore which words are preserved better across the same language group and\npossible reasons behind it.\n7. Explore which language group preserves particular words more in comparison\nto others and potential reasons behind it.\n8. Determine if certain language groups are correct and exploit the possibility of\nforming new ones.\n4 Data\n4.1 Language files\nLanguage file or database is a set of languages, each of which is associated with\nan ordered list of words. All lists of words for a particular data set have the same\nlength. For example:\nnumbers(romani,[iek,dui,trin,shtar,panj,shov,efta,oksto,ena,desh]).\nnumbers(english,[wun,too,three,foor,five,siks,seven,eit,nine,ten]).\nnumbers(french,[un,de,troi,katre,sink,sis,set,wuit,neuf,dis]).\nWords and languages are encoded in this format for later use of Prolog. In Prolog\neach “numbers” line is a fact, which has 2 arguments; the first is the language name\nand the second is a list (indicated in between square brackets) of words. Words\ncan be written down in their original form or encoded phonetically (as shown in the\nexample). Where synonyms for a word are known, then the word itself is represented\nby a list of the synonym words. In the example below, Lithuanian, Russian and\nItalian have two words for the English ‘blue’:\n6words(english,[black,white,red,yellow,blue,green]).\nwords(lithuanian,[juoda,balta,raudona,geltona,[melyna,zhydra],zhalia]).\nwords(russian,[chornyj,belyj,krasnyj,zholtyj,[sinij,goluboj],zeljonyj]).\nwords(italian,[nero,bianco,rosso,giallo,[blu,azzurro],verde]).\nThe main focus of this research was exploring words phonetically. Consequently,\nspecial encoding was used. It consisted of encoding phonemes by using only one\nletter; incorporating capital letters for encoding different sounds (See table 2).\nTable 2: Table of phonetic encoding\nSymbol Meaning\nc ts\nx ks\nC ch as in charity\nk as in cat\nT th\nS sh\nG dzh\nK kh\nZ zh\nD dz\nHSpanish/Portuguese sound of “j”\nA, I, O, U long vowels\nTable 3 summarises the language files that are obtained at the moment.\n4.2 Sheep\n4.2.1 Sheep counting words\nSheep counting numbers were extracted from “Yan Tan Tethera”3page on Wikipedia\nand placed in a Prolog database. Furthermore, data was encoded phonetically using\nthe set of rules provided by Prof. David Gilbert.\nIn the given source, number sets ranged from 1-3 to 1-20 for different dialects.\nThe initial step was to reduce the size of the data to sets of numbers 1-10. This way\naiming:\n(a) to have Prolog syntax without errors (avoided “-”, “ ” as they were common\nsymbols after numbers reached 10);\n(b) to avoid the effects of different methods of forming and writing down num-\nbers higher than 10. (Usually they were formed from numbers 1-10 and a\n7Table 3: Table of Language files\nNameNumber of\nlanguagesNumber of\nwords per\nlanguageDescription\nNumbers Small Collec-\ntion92 10 Numbers 1 to 10\nNumbers Big Collection 3880 10 Numbers 1 to 10\nSheep Counting Num-\nbers54 10 Numbers 1 to 10\nBasic Words 42 13Concept words: sun,\nmoon, rain, water, fire,\nman, woman, mother,\nfather, child, yes, no,\nblood\nColours 23 6Main colours: black,\nwhite, red, yellow, blue,\ngreen\nBasic Words IPA 3 12Concept words as in\n“Basic Words”\nNumbers IPA 3 10 Numbers 1 to 10\nColours IPA 3 6 Colours as in “Colours”\nbase. However, they were written in a different order, making the comparison\ninefficient.)\nIn addition, the Wharfedale dialect was removed since only numbers 1-3 were\nprovided; the Weardale dialect was eliminated as it had a counting system with base\n5. Consequently, the final version of sheep counting numbers database consisted of\n23 observations (dialects) with numbers 1-10.\n4.2.2 Geographical data\nIn order to enable the analysis of linguistic and geographical distance relationship,\na geographical distance database was created. It was done by firstly creating a\npersonalized Google Map with 23 pins, noting the places of different dialects (they\nwere located approximately in the middle of the area) (Figure: 3). Subsequently,\npairwise distances were calculated between all of them (taking walking distance) and\nadded to the database for further use.\n8Figure3: SheepdialectsinBritain. Amapwith23pins, notingtheplacesofdifferent\ndialects\n4.3 Colours\nColour words were extracted from “Colour words in many languages”5page on Om-\nniglot, collected from people and dictionaries. In addition, data was encoded pho-\nnetically using the set of rules provided by Prof. David Gilbert.\nThe latest version of the database consisted of 42 different languages, each con-\ntaining 6 colours: black, white, red, yellow, blue, green. For the purposes of analysis\nthe following groups were created:\n(a) All languages - “ColoursAll” (42 languages)\n(b) Indo-European languages - “ColoursIE” (39 languages)\n(c) Germanic languages - “ColoursPGermanic” (10 languages)\n(d) Romance languages - “ColoursPRomance” (11 languages)\n(e) Germanic and Romance languages - “ColoursPG_R” (21 languages)\n94.4 IPA\n“Automatic Phonemic Transcriber”1was used to create 3 IPA encoded databases:\n(a) “BasicWords” - words in their original form were taken from Prof. David\nGilbert’s database for basic words (including: sun, moon, rain, water, fire,\nman, woman, mother, father, child, yes, no, blood).\n(b) “Numbers” -numbersfrom1-10intheiroriginalformweretakenfromProf.David\nGilbert’s small database of numbers.\n(c) “Colours” - words were taken from the above mentioned database (including\nwords: black, white, red, yellow, blue, green).\nEachoftheabovementioneddatabasesconsistedof3languages: English, Danishand\nGerman (these were the languages the Automatic Phonemic Transcriber provided)\nall encoded in IPA.\nAs the research progressed, the difficulty of obtaining IPA encoding for different\nlanguages was faced. This study could not find a cross-linguistic IPA dictionary\nthat included more than 3 languages. Consequently, the question of its existence\nwas raised.\n5 Methodology\nThere are two main processes to be carried out.\nThe first process (Figure: 4) aims to analyse a databases of words; explore which\nwords exhibit more or less variation, which words are more preserved; examine how\nlanguages could be grouped based on linguistic distances of words.\nItbeginswiththecalculationofpairwiselinguisticdistancesforthegivendatabase\nof words. A Phonetic Substitution Table is used to assign weights during the calcu-\nlation and could possibly be modified. The result is a new distance table which is\nanalysed in the following ways:\n\u000fPerforming “densityP” function. The outcome is density plots for every word\nof a database.\n\u000fPerforming Hierarchical clustering. After, the “Best cut” is determined, which\nis either the best Silhouette value after calculation of all possible cases, or a\nforced number K which is a number of words per language in the language file\n\u000fCalculating Bhattacharya coefficients.\n\u000fPerforming “mean_SD” function.\n10The second process (Figure: 5) targets to investigate the relationship between\ntwo sets of distance data. In this research, it was applied to analyse the relationship\nbetween linguistic and geographical distances.\nIt starts with producing two pairwise distance tables: one of them is calculated\ngeographical distances, another one is calculated linguistic distances. Then the data\nfrom both tables is combined into a data frame for regression analysis in R. The\noutcome is an object of the class “lm” (result of R function “lm” being used), that is\nused for data analysis, and a scatter plot with a regression line for visual analysis.\nBoth processes have been automated, see Section 6.5.\n  Database of words Calculation of Linguistic distances Distance table “densityP” Hierarchical Clustering Density Plots Hierarchical clusters \nK = N Calculation of Silhouette values \n“Best cut” Dendrogram \nSilhouette Plot Cluster Plot Dendrogram + cut “tscore” “mean_SD” \nNormalized table Data frame of mean, SD and mean*SD of every subset Plots “bhatt” Data frame of Bhattacharya Coefficients of all pairs \nsubset \nsubset \nsubset subset;  all words; all to all \nN is the number of words Character weight substitution table \nFigure 4: Work-flow of comparison of languages\n11 \nDatabase of words \nCalculation of Linguistic distances \nDistance table List of places Calculation of Geographical distances Distance table Combining data Data frame Regression analysis Object of class “lm” Scatter plot  + regression line Figure 5: Work-flow of relationship analysis\n6 Methods\n6.1 Edit Distance\nFor the purposes of this research Edit distance (a measure in computer science and\ncomputational linguistics for determining the similarity between 2 strings) was cal-\nculated based on Levenshtein distance metric. This metric between two strings\nis the minimum number of single-character edits, operations including: insertions,\ndeletions or substitutions.\nThe Levenshtein distance between two strings a,b (of length jajandjbj\n12respectively) is given by leva;b(jaj;jbj)where\nleva;b(i; j) =8\n>>>><\n>>>>:max (i; j) If min (i; j)=0\nmin8\n><\n>:leva;b= (i\u00001; j) + 1\nleva;b= (i; j\u00001) + 1 ; otherwise\nleva;b= (i\u00001; j\u00001) + 1 (ai 6=bj)\nwhere 1(ai 6=bj)is the indicator function equal to 0 when ai=bjand equal to 1\notherwise. A normalised edit distance between two strings can be computed by\nlev_norm a;b=leva;b\nmax (jaj;jbj)\nEdit distance was implemented by Prof. David Gilbert using dynamic program-\nming in SWI Prolog23. The program was used to compare two words with the same\nmeaning from different languages. When pairwise comparing two words where either\none or both comprise synonyms, all the alternatives for each word one one language\nare compared with the corresponding (set) of words in the other language, and the\nclosest match is selected. In addition, all to all comparisons were made, i.e. edit\ndistance was calculated for words having different meaning as well. Finally, the edit\ndistance for two languages represented by two lists of equal length of corresponding\nwords was computed by taking the average of the edit distance for each (correspond-\ning) pair of words.\nAn example of pairwise alignments is for the pair of words overa-hofa, where 3\nalignments are produced with the use of gap penalty = 1and substitution penalties\nf$v= 0:2,e$o= 0:2and all other mismatches 1:\n[[-,h],[o,o],[v,f],[e,-],[r,-],[a,a]]\n[[o,-],[v,h],[e,o],[r,f],[a,a]]\n[[o,h],[v,-],[e,o],[r,f],[a,a]]\neach with the raw edit distance of 3.2, and the normalised edit distance of\n3:2\nmax (joveraj;jhofaj)=3:2\n5= 0:64\nFor the sake of clarity we can write the first alignment for example as\n- o v e r a\nh o f - - a\nwhere only 3 letters are directly aligned.\n136.2 Phonetic Substitution Table\nIn order to give a specified weight for different operations (insertion, deletion and\nsubstitution) Phonetic Substitution Table was created by incorporating Grimm’s\nlaw21and extending it in-house.\nGrimm’s Law, principle of relationships in Indo-European languages, describes\na process of the regular shifting of consonants in groups. It consist of 3 phases in\nterms of chain shift7.\n1. Proto-Indo-European voiceless stops change into voiceless fricatives.\n2. Proto-Indo-European voiced stops become voiceless stops.\n3. Proto-Indo-European voiced aspirated stops become voiced stops or fricatives.\nThis is an abstract representation of the chain shift:\nbh > b > p > F\ndh > d > t > T\ngh > g > k > x\ngwh > gw > kw > xw\nFigure 6 illustrates how further consonant shifting following Grimm’s law affected\nwords from different languages4.\nPhonetic substitution table was extended in-house by adding more shifts. In\naddition, it was also written in the way to work with the special encoding described\nin 4.1 section. Find the full table “editable” in Appendix A.\nAnother phonetic substitution table, called “editableGaby”, was made (See Ap-\npendix A). It was extended by adding pairs like “dzh” and “zh”; “dzh” and “ch”; “kh”\nand “g”; as well as “H”(sound of e.g. spannish/portuguese “j”) with “kh”, “g”, “k”,\n“h”. In addition, some of the weights were changed for certain pairs for experimental\npurposes.\n6.3 Hierarchical Clustering\n6.3.1 Using the OC program\nThe OC program6is general purpose hierarchical cluster analysis program. It out-\nputs a list of the clusters and optionally draws a dendrogram in PostScript. It\nrequires complete upper diagonal distance or similarity matrix as an input.\n14 Proto-Indo-European Meaning Non-Germanic (unshifted) cognates Change Proto-Germanic Germanic (shifted) examples *pṓds \"foot\" Ancient Greek: πούς, ποδός (poús, podós), Latin: pēs, pedis, Sanskrit: pāda, Russian: под (pod) \"under; floor\", Lithuanian: pėda, Latvian: pēda, Persian: ﺎﭘ (pa) *p > f [ɸ] *fōt- English: foot, West Frisian: foet, German: Fuß, Gothic: fōtus, Icelandic, Faroese: fótur, Danish: fod, Norwegian, Swedish: fot *trit(y)ós \"third\" Ancient Greek: τρίτος (tritos), Latin: tertius, Welsh: trydydd, Sanskrit: treta, Russian: третий (tretij), Serbo-croatian: трећи (tretji), Lithuanian: trečias, Albanian: tretë *t > þ [θ] *þridjô English: third, Old Frisian: thredda, Old Saxon: thriddio, Gothic: þridja, Icelandic: þriðji, Danish, Swedish: tredje *ḱwón- ~ *ḱun- \"dog\" Ancient Greek: κύων (kýōn), Latin: canis, Welsh: ci (pl. cwn), Persian: ﮓﺳ (sag) *k > h [x] *hundaz English: hound, Dutch: hond, German: Hund, Gothic: hunds, Icelandic, Faroese: hundur, Danish, Norwegian, Swedish: hund *kʷód \"what\" Latin: quod, Irish: cad, Sanskrit: kád, Russian: ко- (ko-), Lithuanian: kas, Serbo-croatian(kajkavian dialect): кај (``kaj``) *kʷ > hw [xʷ] *hwat English: what, Gothic: ƕa(\"hwa\"), Icelandic: hvað, Faroese: hvat, Danish: hvad, Norwegian:  hva  Figure 6: Table illustrating Grimm’s Law chain shift\n156.3.2 Using R\nHierarchical clustering in R was performed by incorporating clustering together with\nSilhouette value calculation and cut performance.\nInordertofulfillagglomerativehierarchicalclusteringmoreefficiently, wecreated\na set of functions in R:\n1. “sMatrix” - Makes a symmetric matrix from a specified column. The function\ntakes a specifically formatted data frame as an input and returns a new data\nframe. Having a symmetric matrix is necessary for “silhouetteV” and “hcutVi-\nsual” functions.\n2. “silhouetteV” -CalculatesSilhouettevalueswith“k” valuevaryingfrom2ton-1\n(n being the number of different languages/number of rows/number of columns\nin a data frame). The function takes a symmetric distance matrix as an input\nand returns a new data frame containing all Silhouette values.\n3. “hcutVisual” - Performs hierarchical clustering and makes a cut with the given\nK value. Makes Silhouette plot, Cluster plot and dendrogram. Returns a\n“hcut” object from which cluster assignment, silhouette information, etc. can\nbe extracted.\nIt is important to note that K-Means clustering was not performed as the algo-\nrithm is meant to operate over a data matrix, not a distance matrix.\n6.4 Further analysis with R\nAnother set of functions was created to analyse collected data further. They target\nto ease the comparison of the mean, standard deviation, Bhattacharya coefficient\nwithin the words or language groups. Including:\n1. “mean_SD” - Calculates mean, standard deviation, product of the mean and\nthe SD multiplication for every column of the input. Visualises all three values\nfor each column and places it in one plot, which is returned.\n2. “densityP” - Makes a density plot for every column of the input and puts it in\none plot, which is returned.\n3. “tscore” - Calculates t-score for every value in the given data frame. (T-score\nis a standard score Z shifted and scaled to have a mean of 50 and a standard\ndeviation of 10)\n4. “bhatt” - Calculates Bhattacharya coefficient (the probability of the two dis-\ntributions being the same) for every pair of columns in the data frame. The\nfunction returns a new data frame.\n166.5 Process automation\nIn order to optimise and perform analysis in the most time-efficient manner processes\nof comparing languages were automated. It was done by creating two shell scripts\nand an R script for each of them.\nThe first shell script named “oc2r_hist.sh” was made to perform hierarchical\nclustering with the best silhouette value cut. This script takes a language database\nas an input and performs pairwise distance calculation. It then calls “hClustering.R”\nR script, which reads in the produced OC file, performs hierarchical clustering and\ncalculates all possible silhouette values. Finally, it makes a cut with the number\nof clusters, which provides the highest silhouette value. To enable this process the\nR script was written by incorporating the functions described in section 6.3.2. The\noutcome of this program is a table of clusters, a dendrogram, clusters’ and silhouette\nplots.\nThe second shell script called “wordset_make_analyse.sh” was made to perform\ncalculations of mean, standard deviation, Bhattacharya scores and produce density\nplots. This script takes a language database as an input and performs pairwise\ndistance calculations for each word of the database. It then calls “rAnalysis.R” R\nscript, whichreadsintheproducedOCfileandperformsfurthercalculations. Firstly,\nit calculates mean, standard deviation and the product of both of each word and\noutputs a histogram and a table of scores. Secondly, it produces density plots of each\nword. Finally,itconvertsscoresintoT-ScoresandcalculatesBhattacharyacoefficient\nfor every possible pair of words. It then outputs a table of scores. To enable this\nprocess the R script was written by incorporating the functions described in section\n6.4.\nFinally, both of the scripts were combined to minimise user participation.\n7 Results\n7.1 Sheep\nThe sheep counting database was evaluated in the following ways:\n\u000fObtaining average pairwise linguistic distance, pairwise linguistic distance of\nsubsets (different words),\n\u000fPerforming all to all comparison (where linguistic distance is calculated be-\ntween words with different meaning, as well as with the same),\n\u000fCollectinggeographicaldataandcomparingrelationshipbetweenlinguisticand\ngeographical distances.\nUpon generation of the above mentioned data, the methods defined in 6 section were\nused.\n177.1.1 Analysis of average and subset linguistic distance\nAfter applying functions “mean_SD” (Figure: 7) and “densityP” (Figure: 8) to the\nlinguistic distances of every word (numbers 1 to 10) in R, the following observations\nwere made. First of all, the most preserved number across all dialects was “10”\nwith distance mean 0.109 and standard deviation 0.129. Numbers “1”, “2”, “3”, “4”\nhad comparatively small distances, which might be the result of being used more\nfrequently. On the other hand, number “6” showed more dissimilarities between\ndialects than other numbers. The mean score was 0.567 and standard deviation -\n0.234. The product scores of mean and standard deviation helped to evaluate both\nat the same time. Moreover, density plots showed significant fluctuation and tented\nto have a few peaks. But in general, conformed with the statistics provided by\n“mean_SD”.\nFigure 7: Mean, SD and mean*SD of every number of sheep counting systems\n7.2 Hierarchical clustering\nHierarchical clustering was performed with the best Silhouette value cut (Figure\n10). The Silhouette value suggested making 9 clusters. In this grouping, the most\ninteresting observation was that Welsh, Breton and Cornish languages were placed\ntogether. It conforms with the fact that all 3 languages descended directly from the\nCommon Brittonic language spoken throughout Britain before the English language\nbecame dominant.\n18Figure 8: Density plots of each number of sheep counting systems\n7.2.1 All to all comparison analysis\nTo enable analysis of clusters of all to all comparison, hierarchical clustering was\nperformed. This was done by two different approaches: calculating a silhouette\nvalue and choosing the number of clusters accordingly; forcing a function to make\n10 clusters due to having numbers from 1 to 10 in the sheep counting database.\nBy using function “silhouetteV” silhouette values were calculated for all possible\nkvalues. The returned data frame indicated the best number of clusters being 70\n(see Appendix B.1 for dendrogram and cluster plot). The suggested clusters were not\ndistinguished with very high clarity in terms of numbers 1-10 perfectly, but they were\ncomparatively good. A pattern that numbers, which had lower mean and standard\ndeviation scores, would result in purer clusters was noticed. Clusters of numbers “1”,\n“2”, ��3”, “4”, “5” and “10” were not as mixed as “6”, “7”, “8”, “9”.\nAnotherwayoflookingatalltoallcomparisondatawasbyproducing10clusters.\nIt was done by using “hcutVisual” and “cPurity” function (see Appendix B.1 cluster\n19welshbretoncornishwestCountryDorsettongswaledaleteesdalelakesrathmellbowlandwensleydalederbyshirenidderdaleeskdalelincolnshirederbyshireDaleswestmorlandborrowdalescotsconistonwiltskirkbyLonsdalesouthWestEngland\n0 0.502745Figure 9: Dendrogram of linguistic distances of 23 dialects\nplot). The results showed high impurities of clusters (Figure: 11). Two out of ten\nclusters were pure, both containing number “5”. Another relatively pure cluster was\ncomposed of number “10” and two entries of number “2”. The rest consisted of up to 7\ndifferent numbers. This shows that sheep counting numbers in different dialects are\ntoo different to form 10 clusters containing each number. However, considering the\npossibility that dialects were grouped and clustering was performed to the smaller\ngroups, they would have reasonably pure clusters. Exploring this grouping options\ncould be a subject for further work.\n20welsh\nbreton\ncornish\nrathmell\nbowland\nwensleydale\nderbyshire\nnidderdale\nconiston\nwilts\nkirkbyLonsdale\nsouthWestEngland\nlakes\neskdale\nlincolnshire\nderbyshireDales\nwestmorland\nborrowdale\nscots\nwestCountryDorset\ntong\nswaledale\nteesdale0100200300HeightCluster DendrogramFigure 10: Dendrogram of sheep counting systems with the best Silhouette cut\n21onextwoxthreex\nfourxfivexsixx\nsevenxeightxninex\ntenx\nnum1\n23\n45\n67\n89\n10Cluster PurityFigure 11: Purity of hierarchical clusters of sheep counting systems all to all com-\nparison. Clusters numbered according to K= 10. Colour key indicates number\nwords.\n7.2.2 Linguistic and Geographical distance relationship\nIn order to investigate the correlation between linguistic and geographical distance,\n“lm” functionwasperformedandascatterplotwascreated. Theregressionlineinthe\nscatter plot suggested that the relationship existed. However, the R-squared value,\nextracted from the “lm” object, was equal to 0.131. This indicated that relationship\nexisted, but was not significant.\nOne assumption made was that Cornish, Breton and Welsh dialects might have\nhad a weakening effect on the relationship, since they had large linguistic distances\ncompared to other dialects. However this assumption could not be validated as the\ncorrelation was less significant after eliminating them. This highlights that although\nthese dialects had large linguistic distance scores, they also had big geographical\ndistances that do not contradict the relationship.\nIn addition, comparison was done between linguistic distance and\nLog 10(GeographicalDistance ). This resulted in an even weaker relationship with\nR-squared being 0.097.\n220300600900\n0.0 0.2 0.4 0.6\nlingDistgeoDistFigure12: Relationshipbetweenlinguisticandgeographicaldistancesofsheepcount-\ning systems\n7.3 Colours\nThe Colours database was evaluated three different ways: getting average pairwise\nlinguistic distance, subset pairwise linguistic distance for every word and performing\nall to all comparison to all groups (All languages, Indo-European, Germanic, Ro-\nmance, Germanic and Romance languages). After the above mentioned data was\ngenerated, the previously defined methods were applied.\n7.3.1 Mean and Standard Deviation\nWhen examining the data calculated for “ColoursAll” none of the colours showed a\nclear tendency to be more preserved than others (Figure: 13). All colours had large\ndistances and comparatively small standard deviation when compared with other\ngroups. Small standard deviation was most likely the result of most of the distances\nbeing large.\nIndo-European language group scores were similar to “ColoursAll”, exhibiting\nslightly larger standard deviation (Figure: 14). Conclusion could be drawn that\nwords for color “Red” are more similar in this group. The mean score of linguistic\ndistances was 0.61, and SD was equal to 0.178, when average mean was 0.642 and\nSD 0.212. However, no colour stood out distinctly.\nGermanic and Romance language groups revealed more significant results. Ger-\n23Figure 13: Mean, SD and mean*SD of every colour of all languages\nmanic languages preserved the colour “Green” considerably (Figure: 15). The mean\nand SD was 0.168 and 0.129, when on average mean was reaching 0.333 and SD\n0.171. In addition, the colour “Blue” had favorable scores as well - mean was 0.209\nand SD was 0.106. Furthermore, Romance languages demonstrated slightly higher\nmeans and standard deviations, on average reaching 0.45 and 0.256 (Figure: 16).\nSimilarly to Germanic, the most preserved colour word in Romance languages was\n“Green” with a mean of 0.296 and SD of 0.214. It was followed by words for “Black”\nand then for “Blue”, both being quite similar.\n7.3.2 Density Plots\nDensity plots of all languages and Indo-European languages were similar: both hav-\ning multiple peaks with the most density around scores of 0.75 (big linguistic dis-\ntance). Moreover, Germanic languages density distribution consisted of two peaks\nfor words “White”, “Blue” and “Green” (Figure: 17). This could possibly be the\nresult of certain weighting in the Phonetic Substitution Table or indicate possible\nfurther grouping of languages. The color “Black” had more normal distribution and\nsmoother bell shape compared to others. Furthermore, Romance languages also ob-\ntained density plots with two peaks for words “White”, “Yellow”, “Blue” (Figure: 18).\nIn contrast, “Black”, “Red” and “Green” distributions were quite smooth.\nIn order to experiment how the Phonetic Substitution Table affects the linguistic\ndistances, “densityP” function was applied to the linguistic distances calculated with\n“GabyTable” substitution table. The aim was to eliminate the two peaks in the\n24Figure 14: Mean, SD and mean*SD of every colour of Indo-European languages\nGermanic language group for word “Green”. In Germanic languages word for green\ntended to begin with either “gr” or “khr” (encoded as “Kr”) - both sounding similar\nphonetically. However, in the original substitution table, a weight for changing\n“K”(kh) to “g” (and the other way around) did not exist. Consequently, a new table\nwas implemented with this substitution. This change resulted in notably smaller\nlinguistic distances - the mean for the word “Green” was 0.099. However, it did not\nsolve the occurrence of two peaks. The density of “Green” again had two main peaks,\nbut differently distributed compared to the previous case.\n7.3.3 Bhattacharya Coefficients\nBhattacharya coefficients were calculated within each group for different pairs of\ncolours. This helped to evaluate which colours were closer in distribution. In addi-\ntion, hierarchical clustering was done with Bhattacharya coefficients (find the den-\ndrograms in the Appendix B.2). However, the potential meaning behind the results\nwas not fully examined.\nAnotherpotentialuseofBhattacharyacoefficientsistheirapplicationtothesame\nword from a different language group. As a result, the preservation of particular\nwords can be analysed across language groups, enabling to compare and evaluate\npotential reasons behind it.\n25Figure 15: Mean, SD and mean*SD of every colour of Indo-European languages\n7.3.4 Hierarchical Clustering\nHierarchical clustering with the best Silhouette value cut was performed in R for\nevery group of formed language groups: all languages, Indo-European, Romance,\nGermanic, and both Germanic and Romance together. It is important to note that\nthe results of the language group “Romance and Germanic” will not be discussed as\nit was used more for testing purposes and as expected resulted in a K=2 cut. After\nmaking the cut, one cluster consisted of Romance languages and another consisted\nof Germanic languages.\nTo begin with, clustering of all languages showed some interesting results that\ncomplied with the grouping of the languages (find the dendrogram in Figure: 19).\nThe suggested cut by Silhouette value was 23. Some of the clusters were more a\ncoincidence than the actual similarity of languages and did not correspond with\nthe existing language grouping. Despite that, most of the clusters resulted in the\nactual language groups, or languages closely related. To begin with, Baltic Romani,\nPunjabi’ and Urdu were placed in the same cluster. Even though Baltic Romani is\nfar away from South Asia geographically, it is believed to have originated from this\narea. Xhosa and Zulu formed another cluster both being the languages of the Nguni\nbranch and spoken in South Africa. Hawaiian, Malagasy and Maori languages were\ngrouped together and they all belong to Austronesian ethnolinguistic group8(see\nfigure 20). Sinhala (language of Sinhalese people, who make up the largest ethnic\ngroup in Sri Lanka), Dhivehi (spoken in Maldives) and Maldivian languages fell in\nthe same group after the cut. They all are spread across islands in the Indian Ocean.\n26Figure 16: Mean, SD and mean*SD of every colour of Indo-European languages\nEstonian and Finnish both being representatives of the Uralic language family were\nthe same cluster.\nMoreover, clusters of Indo-European languages were quite pure as well (groups\nare visible in the dendrogram of all languages, however for clarity see figure 21).\nThere were four larger groups that stood out. First of all, the group of Germanic\nlanguages was produced accurately. It consisted of Faroese, Icelandic, German,\nLuxemburgish, Yiddish, English, Norwegian, Swedish, Afrikaans and Dutch. All of\nthese languages are considered to be in the branch of Germanic languages. Another\nclusterwasSlaviclanguages, whichconsistedofCroatian, Polish, Russian, Slovenian,\nCzech, Slovak and Lithuanian. Lithuanian and Latvian, according some sources, are\nconsidered to be in a separate branch, known as Baltic languages. On the other\nhand, in other sources they are regarded as Slavic languages. In this case, in terms\nof colour words Lithuanian was appointed to the Slavic languages, whereas Latvian\nformed a cluster on its own. In relation to Romance languages, these were divided\ninto two groups. The first one was made of Ladino (language that derived from\nmedieval Spanish), Spanish(Castilian), Galician and Portuguese, forming a group\nof the Western Romance languages. The second one consisted of Sicilian, Italian,\nNeapolitan, Catalan and Romanian and could be called a group of Mediterranean\nRomance languages.\nFurthermore, clustering results of the Germanic languages file (Figure: 22) show\nhigh relation with geographical prevalence of the languages and language develop-\nment history. German, Luxembourgish (has similarities with other varieties of High\nGerman languages) and Yiddish (a High German-based language) were all in the\n27Figure 17: Density plots of each colour of Germanic languages\nsame cluster. Also, Afrikaans and Dutch were placed in the same group, and it is\nknown that Afrikaans derived from Dutch vernacular of South Holland in the course\nof 18th century. Other clusters included Faroese and Icelandic, Swedish and Nor-\nwegian, as well as English forming a cluser on its own. Finally, when looking at\nthe clusters of Romance languages file (Figure 23) it is evident that one cluster,\nconsisting of Ladino, Spanish, Galician and Portuguese, remained the same as in\n“ColoursAll”, “ColoursIE”. Another cluster that was formed from Romance languages\nin these databases was broken down into 3 clusters during separate clustering of Ro-\nmance languages. Romanian and Catalan formed clusters on their own and Italian,\nNeopolitan and Sicilian were members of another cluster. These three languages\nwere close geographically.\n7.4 IPA\nHierarchical Clustering was performed to all three IPA databases and compared\nwith the results of hierarchical clustering of in-house phonetically encoded databases\n(they were created by taking subsets of German, English and Danish languages\nfrom “Basic Words”, “Numbers Small Collection” and “Colours” databases). The\nfirst characteristic noticed was that both IPA and non-IPA databases had the same\ngroupingoflanguages. Thisshowsevidencetowardssubstantiatedphoneticencoding\ndone in-house. Another noted tendency was that pairwise linguistic distance scores\ntented to be higher for IPA databases. This might be due to some graphemes being\nwritten with a few letters in IPA databases, while phonetic encoding done in-house\nexpressed graphemes as one symbol.\n28Figure 18: Density plots of each colour of Romance languages\nPotential further work would be generating an IPA-designated Phonetic Sub-\nstitution table (so far clustering has been done with “editable”) and running the\nroutines with the new weight table. Also, complementing the IPA databases with\nmore languages would be an important step towards receiving more accurate results.\n7.5 Small Numbers\n7.5.1 All to all comparison\nAnalysis was carried out in two ways. First of all, hierarchical clustering was per-\nformed with the best silhouette value cut. For this data set best silhouette value\nwas 0.48 and it suggested making 329 clusters. Clusters did not exhibit high pu-\nrity. However, the ones that did quite clearly corresponded to unique subgroups of\nlanguage families.\nAnotherwayoflookingatalltoallcomparisondatawasbyproducing10clusters.\nThe anticipated outcome was members being distinguished by numbers, forming 10\nclean clusters. However, all the clusters were very impure and consisting multiple\ndifferent numbers. This might be due to different languages having phonetically\nsimilar words for different words, in this case.\nAll to all pairwise comparison could be an advantageous tool when used for lan-\nguage family branches or smaller, but related subsets. It could validate if languages\nbelong to a certain group.\n29faroese\nicelandic\ngerman\nluxembourgish\nyiddish\nenglish\nnorwegian\nswedish\nafrikaans\ndutch\nladino\nspanish\ngalician\nportuguese\nfrench\nsicilian\nitalian\nneapolitan\ncatalan\nromanian\ncroatian\npolish\nrussian\nlithuanian\nslovenian\nczech\nslovak\nromani_baltic\npunjabi\nurdu\nsinhala\ndhivehi\nmaldivian\narabic\nmaltese\nkurdish\nwelsh\nhungarian\nlatvian\nhebrew\narmenian\npersian\nxhosa\nzulu\njapanese\nquechua\nguarani\nmandarin\nalbanian\nbasque\nlatin\nfilipino\nmalay\nestonian\nfinnish\nhawaiian\nmalagasy\nmaori\ngreek\nswahili\nlingala\nsomali0.00.51.01.52.0HeightCluster DendrogramFigure 19: Dendrogram of hierarchical clustering with Silhouette value cut for all\nlanguages (colour words)\n8 Conclusions\nThisprojecthasaimedtodevelopcomputationalmethodstoanalyseandunderstand\nconnections between human languages.\nThe project included collecting words from different languages in order to form\nnew databases, forming rules for phonetic encoding of words and adjusting phonetic\nsubstitution table. Several computational methods of calculating pairwise distance\nbetween two words were taken, includingaverage, subsetand all toall words distance\ncalculation. It was done by incorporating edit distance and phonetic substitution\ntable, and implementing it in SWI Prolog. This was followed by detailed analysis\nof distance scores, which was conducted by the specific automated routines and\ndeveloped R functions. They enabled performing hierarchical clustering with a cut\neither according to silhouette value, or to specified K value. They provided summary\nof mean, standard deviation and other statistics, like Bhattacharya scores. All these\n30Figure 20: Chronological dispersal of the Austronesian people\ntechniques delivered a thorough analysis of data and the automation of processes\nensured they were used efficiently.\nThe resulting outcome of analysis of old sheep counting systems in different\nEnglish dialects was the observation that numbers “1”,“2”,“3”,“4” and “10” were more\nuniform within different dialects than others, posing that they might have been\nthe most frequently used ones. Analysis of all to all comparison did not provide\npure clusters and shows that sheep counting numbers in different dialects are too\ndifferent to form 10 clusters containing each number. This suggests that dialects\nshould be grouped into subsets. Furthermore, hierarchical clustering with the best\nsilhouette cut suggested the potential 9 groups, which consist members with the\nmost similar counting words. Surprisingly, it was not entirely based on location.\nThis corresponded with the difficulty of finding relationship between geographic and\nlinguistic distance, the conducted tests showed it was insignificant.\nAnalysis of colour words revealed that within Indo-European languages words for\ncolour red were moderately more preserved. Both Germanic and Romance language\ngroups tended to have considerably more uniform words for green and blue colours.\nIn addition, Romance language group preserved colour black reasonably well. Analy-\nsis of linguistic distances distribution showed multiple peaks within words for various\nlanguage groups, suggesting that further language grouping could be done. Further-\nmore, the resulting outcome of hierarchical clustering with silhouette cut was known\nand officially accepted language families. Most of the clusters were subgroups of\nexisting language families. Some of them suggested different sub-grouping according\nto colour words (e.g. Lithuanian was appointed to Slavic languages, while Latvian\nformed cluster on its own).\nIPA databases resulted in the same relationships between languages as non-IPA\nphonetically encoded databases. However, to fully explore the potential of IPA-\nencoded databases they ought to be expanded and a customized weights table should\nbe created.\nIn conclusion, this project resulted in creation of several felicitous computational\n31faroese\nicelandic\ngerman\nluxembourgish\nyiddish\nenglish\nnorwegian\nswedish\nafrikaans\ndutch\nladino\nspanish\ngalician\nportuguese\nlatin\nsicilian\nitalian\nneapolitan\ncatalan\nromanian\nlatvian\ncroatian\npolish\nrussian\nlithuanian\nslovenian\nczech\nslovak\npunjabi\nurdu\ndhivehi\nsinhala\nkurdish\narmenian\npersian\nwelsh\nfrench\nalbanian\ngreek0.00.51.01.52.0HeightCluster DendrogramFigure 21: Dendrogram of hierarchical clustering with Silhouette value cut for Indo-\nEuropean languages (colour words)\ntechniques to explore many languages and their correlation all at once.\n9 Further Work\nOne of the areas where further work could be performed is thorough analysis of\nnumbers both Small and Big Collection databases, Basic words database.\nIn addition, analysis routines could be enhanced by adding Bhattacharya scores,\ncalculated in a different manner. In other words, potentially beneficial use of Bhat-\ntacharyacoefficientswouldapplyingthemtothesamewordfromadifferentlanguage\ngroup. As a result, the preservation of particular words could be analysed across\nlanguage groups, enabling to compare and evaluate potential reasons behind it.\n32faroese\nicelandic\ngerman\nluxembourgish\nyiddish\nenglish\nnorwegian\nswedish\nafrikaans\ndutch0.00.20.40.6HeightCluster DendrogramFigure 22: Dendrogram of hierarchical clustering with Silhouette value cut for Ger-\nmanic languages (colour words)\nMoreover, regarding IPA-encoded data potential further work would be gener-\nating a customized IPA Phonetic Substitution table. Also, an important step to-\nwards receiving more accurate and interesting results would be augmenting the IPA\ndatabases with more languages.\nFinally, classifying languages in language databases and automatically analysing\npurity of clusters would be a step forward towards fully automated and consistent\nprocess. Consequently, a list of 118 languages containing their language families\nand branches has been created. It could be incorporated with existing language\ndatabases.\n33ladino\nspanish\ngalician\nportuguese\nfrench\nlatin\nsicilian\nitalian\nneapolitan\ncatalan\nromanian0.000.250.500.75HeightCluster DendrogramFigure 23: Dendrogram of hierarchical clustering with Silhouette value cut for Ro-\nmance languages (colour words)\n3410 Summary of contributions\nMy personal contributions during this undergraduate research assistantship include:\nData Collection.\n\u000fCreated a Sheep counting numbers database.\n\u000fMade geographical data database and a map of dialects.\n\u000fCollectedcolourwordsfrom42differentlanguagesandmadeadatabase. Made\nthefollowingsubsets: Indo-European, Germanic, Romance, RomanceandGer-\nmanic.\n\u000fCreated numbers, colours and basic words databases in IPA encoding.\n\u000fMade a list of 118 languages, their language families and branches.\nTransforming data using phonetics. Transformed sheep counting numbers,\ncolours (including Indo-European, Germanic, Romance, Romance and Germanic\nsubsets) databases using a specified phonetic encoding.\nMean, SD and density analysis. Analysed mean, SD and density of sheep\nnumbers, colours (including all subsets). Produced tables and plots.\nT-Scores and Bhattacharya calculations. Calculated T-Scores and Bhat-\ntacharya coefficients for sheep numbers, colours (including all subsets); Made den-\ndrograms from Bhattacharya scores.\nHierarchical clustering.\n\u000fPerformed hierarchical clustering for sheep numbers, colours (all subsets), IPA\n(all three). Created dendrograms.\n\u000fPerformed hierarchical clustering with the best silhouette cut value for sheep\nnumbers all to all, colours (all subsets), small numbers all to all. Made den-\ndrograms, Silhouette plots, Cluster plots.\n\u000fPerformed hierarchical clustering with k=10 cut for numbers all to all, colours\n(all subsets), small numbers all to all. Made dendrograms, Silhouette plots,\nCluster plots.\n35Code development.\n\u000fCreatedapackageinR“CompLinguistics”,whichconsistedoffunctions: “mean_SD”,\n“densityP”, “sMatrix”, “tscore”, “bhatt”, “silhouetteV”, “hcutVisual”.\n\u000fProduced R script that automates the processes of file reading, generating a\ncertain format data frame, performing hierarchical clustering with the best\nsilhouette value cut. In addition, created another R script, which performed\ncalculations of mean, standard deviation, Bhattacharya scores and analysis of\ndistribution.\n\u000fSeveral shellscrips.\n\u000f“editableGaby” phonetic substitution table.\nAcknowledgements\nGabija Mikulyte was supported by an undergraduate research grant from the De-\npartment of Computer Science at Brunel University London.\n36Bibliography\n[1]Automatic phonemic transcriber . http://tom.brondsted.dk/text2phoneme/.\n(Accessed on 07/20/2019).\n[2]Indo-european migrations . Indo-European migrations. (Accessed on\n07/25/2019).\n[3]Yan tan tethera . https://en.wikipedia.org/wiki/Yan_Tan_Tethera. (Accessed\non 07/18/2019).\n[4]Grimm’s lawchain shiftexamples . https://en.wikipedia.org/wiki/Grimm’s_law,\nJune 2019. (Accessed on 07/18/2019).\n[5]S. Ager , Colour words in many languages .\nhttps://www.omniglot.com/language/colours/multilingual.htm. (Accessed\non 07/18/2019).\n[6]G. J. Barton , Index of /manuals/ oc .\nhttp://www.compbio.dundee.ac.uk/manuals/oc/, August 2002. (Accessed\non 07/18/2019).\n[7]L. Campbell ,Historical linguistics: An Introduction , The MIT Press, 2nd ed.,\n2004.\n[8]G. Chambers and H. Edinur ,The austronesian diaspora: A synthetic total\nevidence model , Global Journal of Anthropology Research, 2 (2016), pp. 53–65.\n[9]A. Champneys ,History of English: A Sketch of the Origin and Development\nof the English Language with Examples, Down to the Present Day , New York,\nPercival and Company, 1893.\n[10]N. Chomsky ,Language and mind , Cambridge University Press, 2006.\n[11]K. Distin ,Cultural Evolution , Cambridge University Press, 2011.\n[12]M. Gimbutas ,The Prehistory of Eastern Europe , no. pt. 1 in American School\nof Prehistoric Research. Bulletin, Peabody Museum, 1956.\n[13]S. Gould, S. Gold, and T. Gould ,The Structure of Evolutionary Theory ,\nHarvard University Press, 2002.\n[14]B. Griffiths ,A Dictionary of North East Dialect , Northumbria University\nPress, 2005.\n[15]D. Leith ,A Social History of English , A Social History of English, Routledge,\n1997.\n37[16]J. Lynch ,Indo-european language tree . http://andromeda.rutgers.edu/jlynch/Lan-\nguageTree.pdf, February 2014. (Accessed on 07/18/2019).\n[17]J. Mallory ,In Search of the Indo-Europeans: Language, Archaeology, and\nMyth, Thames and Hudson, 1991.\n[18]J. Marler ,The beginnings of patriarchy in europe: Reflections on the kurgan\ntheory of marija gimbutas , The Rules of Mars: Readings on the Origins, History\nand Impact of Patriarchy. Manchester: Knowledge, Ideas and Trends, (2005),\npp. 53–75.\n[19]M. S. Perreault, Charles ,Dating the origin of language using phonemic\ndiversity, 7,4 (2012).\n[20]S. Pinker and P. Bloom ,Natural language and natural selection , Behavioral\nand Brain Sciences, 13 (1990), p. 707–727.\n[21]C. U. Press ,The Columbia Encyclopedia , Columbia University Press, 6th ed.,\nJune 2000.\n[22]A. Trombetti ,L’unità d’origine del linguaggio , Libreria Treves di Luigi Bel-\ntrami, 1905.\n[23]J. Wielemaker, T. Schrijvers, M. Triska, and T. Lager ,SWI-Prolog ,\nTheory and Practice of Logic Programming, 12 (2012), pp. 67–96.\n38A Phonetic Substitution tables\nA.1 Editable\nThis table was mostly used for calculations of pairwise linguistic distances. Symbol\n“%” indicates comments.\n%Substitution costs table\nt(S1,S2,D):-\nS1=S2 -> D=0 ; % no cost if same character\n( t1(S1,S2,D) -> true ; ( t1(S2,S1,D) -> true ; % try S1-S2 otherwise S2-S1, in t1/3 table\nD=1)). % else cost=1 if not in t1/3 table\n% old simplistic table\n%t(S1,S2,0):- S1 = S2.\n%t(S1,S2,1):- S1 \\== S2.\n% a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z\n%Grimm’s law\n%close consonants\nt1(b,p,D):- tweight(consonant1,D). % b ->p\nt1(d,t,D):- tweight(consonant1,D). % d -> t\nt1(g,k,D):- tweight(consonant1,D). % g -> k\nt1(p,f,D):- tweight(consonant1,D). % p -> f\nt1(t,’T’,D):- tweight(consonant1,D). % t -> th\nt1(k,’C’,D):- tweight(consonant1,D). % k -> ch\nt1(’C’,h,D):- tweight(consonant1,D). % ch -> h\nt1(b,f,D):- tweight(consonant1x2,D). % b -> p -> f\nt1(d,’T’,D):- tweight(consonant1x2,D). % d -> t -> th\nt1(g,’C’,D):- tweight(consonant1x2,D). % g -> k -> ch\nt1(g,h,D) :- tweight(consonant1x3,D). % g -> k -> ch -> h\nt1(f,v,D):- tweight(consonant1,D).\nt1(g,j,D):- tweight(consonant1,D).\nt1(s,z,D):- tweight(consonant1,D).\nt1(v,w,D):- tweight(consonant1,D).\nt1(f,w,D):- tweight(consonant1x2,D). % f -> v -> w\nt1(’F’,w,D):- tweight(consonant1x2,D). % F -> v -> w\n% from numberslist10\nt1(f,’F’,0). % F from ph in original\nt1(’S’,’š’,0). % ’S’ from sh in original\nt1(’C’,’č’,0). % ’S’ from sh in original\nt1(’T’,’ j’,0). % ’S’ from th in original\n% other close consonants\nt1(’š’,s,D):- tweight(consonant1,D). % sh <-> s\nt1(’S’,s,D):- tweight(consonant1,D). % sh <-> s\nt1(’C’,’S’,D):- tweight(consonant1,D). % ch <-> sh\n39t1(’C’,’š’,D):- tweight(consonant1,D). % ch <-> sh\nt1(’č’,’S’,D):- tweight(consonant1,D). % ch <-> sh\nt1(’č’,’š’,D):- tweight(consonant1,D). % ch <-> sh\nt1(’K’,k,D):- tweight(consonant1,D). % kh <-> k\nt1(’G’,k,D):- tweight(consonant1,D). % gh <-> k\nt1(’G’,g,D):- tweight(consonant1,D). % gh <-> g\nt1(’K’,’G’,D):- tweight(consonant1,D). % kh <->gh\nt1(’Z’,z,D):- tweight(consonant1,D). % zh <-> z\nt1(c,s,D):- tweight(consonant1,D). % ts <-> s\nt1(x,k,D):- tweight(consonant1,D). % ks <-> k\nt1(’D’,d,D):-tweight(consonant1,D). % dh <-> d\n% vowels\n%t1(S1,S2,0.2):- Vowels=[a,e,i,o,u,y], member(S1,Vowels), member(S2,Vowels).\nt1(a,Y,V):- (Y=e;Y=’E’;Y=i;Y=’I’;Y=o;Y=’O’;Y=u;Y=’U’;Y=y;Y=’Y’), tweight(vowel,V).\nt1(e,Y,V):- (Y=a;Y=’A’;Y=i;Y=’I’;Y=o;Y=’O’;Y=u;Y=’U’;Y=y;Y=’Y’), tweight(vowel,V).\nt1(i,Y,V):- (Y=a;Y=’A’;Y=e;Y=’E’;Y=o;Y=’O’;Y=u;Y=’U’;Y=y;Y=’Y’), tweight(vowel,V).\nt1(o,Y,V):- (Y=a;Y=’A’;Y=e;Y=’E’;Y=i;Y=’I’;Y=u;Y=’U’;Y=y;Y=’Y’), tweight(vowel,V).\nt1(u,Y,V):- (Y=a;Y=’A’;Y=e;Y=’E’;Y=i;Y=’I’;Y=o;Y=’O’;Y=y;Y=’Y’), tweight(vowel,V).\nt1(y,Y,V):- (Y=a;Y=’A’;Y=e;Y=’E’;Y=i;Y=’I’;Y=o;Y=’O’;Y=u;Y=’U’), tweight(vowel,V).\n% same vowel\nt1(A1,A2,0):- t_a(A1), t_a(A2).\nt1(E1,E2,0):- t_e(E1), t_e(E2).\nt1(I1,I2,0):- t_i(I1), t_i(I2).\nt1(O1,O2,0):- t_o(O1), t_o(O2).\nt1(U1,U2,0):- t_u(U1), t_u(U2).\nt1(Y1,Y2,0):- t_y(Y1), t_y(Y2).\n% close vowels\nt1(X,Y,V):- tvowel(X), tvowel(Y), tweight(vowel,V).\n% long vowels\nt1(’A’,Y,V):- (Y=’E’;Y=e;Y=’I’;Y=i;Y=’O’;Y=o;Y=’U’;Y=u;Y=’Y’;Y=y), tweight(vowel,V).\nt1(’E’,Y,V):- (Y=’A’;Y=a;Y=’I’;Y=i;Y=’O’;Y=o;Y=’U’;Y=u;Y=’Y’;Y=y), tweight(vowel,V).\nt1(’I’,Y,V):- (Y=’A’;Y=a;Y=’E’;Y=e;Y=’O’;Y=o;Y=’U’;Y=u;Y=’Y’;Y=y), tweight(vowel,V).\nt1(’O’,Y,V):- (Y=’A’;Y=a;Y=’E’;Y=e;Y=’I’;Y=i;Y=’U’;Y=u;Y=’Y’;Y=y), tweight(vowel,V).\nt1(’U’,Y,V):- (Y=’A’;Y=a;Y=’E’;Y=e;Y=’I’;Y=i;Y=’O’;Y=o;Y=’Y’;Y=y), tweight(vowel,V).\nt1(’Y’,Y,V):- (Y=’A’;Y=a;Y=’E’;Y=e;Y=’I’;Y=i;Y=’O’;Y=o;Y=’U’;Y=u), tweight(vowel,V).\n%long-short vowels\nt1(’A’,a,Z):- tweight(longvowel,Z).\nt1(’E’,e,Z):- tweight(longvowel,Z).\nt1(’I’,i,Z):- tweight(longvowel,Z).\nt1(’O’,o,Z):- tweight(longvowel,Z).\nt1(’U’,u,Z):- tweight(longvowel,Z).\nt1(’Y’,y,Z):- tweight(longvowel,Z).\n40%long consonants\nt1(’M’,m,Z):- tweight(longconsonant,Z).\nt1(’N’,n,Z):- tweight(longconsonant,Z).\n% weight table\ntweight(vowel,0.2).\ntweight(longvowel,0.1).\ntweight(consonant1,0.2).\ntweight(consonant1x2,0.4).\ntweight(consonant1x3,0.8).\ntweight(longconsonant,0.05).\ntvowel(V):- t_a(V); t_e(V); t_i(V); t_o(V); t_u(V); t_y(V).\nA.2 editableGaby\nThis table was created based on Editable illustrated before. Comments and “!!”\nsymbol indicates where changes were made.\n% Substitution costs table\nt(S1,S2,D):-\nS1=S2 -> D=0 ; % no cost if same character\n( t1(S1,S2,D) -> true ; ( t1(S2,S1,D) -> true ; % try S1-S2 otherwise S2-S1, in t1/3 table\nD=1)). % else cost=1 if not in t1/3 table\n% old simplistic table\n%t(S1,S2,0):- S1 = S2.\n%t(S1,S2,1):- S1 \\== S2.\n% a,b,c,d,e,f,g,h,i,j,k,l,m,n,o,p,q,r,s,t,u,v,w,x,y,z\n/*\nPhonetic encoding\nc - ts\nx - ks\nC - ch as in charity\nk - as in cat\nT - th\nS - sh\nG - dzh %!!\nK - kh\nZ - zh\nD - dz\nH - spanish/portuguese sound of ’j’ %!!\nF - ph\nA,I,O,U,Y - long vowels\n*/\n41% Grimm’s law\n%close consonants\nt1(b,p,D):- tweight(consonant1,D). % b ->p\nt1(d,t,D):- tweight(consonant1,D). % d -> t\nt1(g,k,D):- tweight(consonant1,D). % g -> k\nt1(p,f,D):- tweight(consonant1,D). % p -> f\nt1(t,’T’,D):- tweight(consonant1,D). % t -> th\nt1(k,’C’,D):- tweight(consonant1x2,D). % k -> ch !!\nt1(’C’,h,D):- tweight(consonant1x2,D). % ch -> h !!\nt1(b,f,D):- tweight(consonant1x2,D). % b -> p -> f\nt1(d,’T’,D):- tweight(consonant1x2,D). % d -> t -> th\nt1(g,’C’,D):- tweight(consonant1x2,D). % g -> ch\nt1(g,h,D) :- tweight(consonant1x1,D). %g->k & g->h & k->h same, ch further !!\nt1(f,v,D):- tweight(consonant1,D).\nt1(g,j,D):- tweight(consonant1,D). %!!\nt1(s,z,D):- tweight(consonant1,D).\nt1(v,w,D):- tweight(consonant1,D).\nt1(f,w,D):- tweight(consonant1x2,D). % f -> v -> w\nt1(’F’,w,D):- tweight(consonant1x2,D). % F -> v -> w\n% from numberslist10\nt1(f,’F’,0). % F from ph in original\nt1(’S’,’š’,0). % ’S’ from sh in original\nt1(’C’,’č’,0). % ’S’ from sh in original\nt1(’T’,’ j’,0). % ’S’ from th in original\n% other close consonents\nt1(’š’,s,D):- tweight(consonant1,D). % sh <-> s\nt1(’S’,s,D):- tweight(consonant1,D). % sh <-> s\nt1(’C’,’S’,D):- tweight(consonant1,D). % ch <-> sh\nt1(’C’,’š’,D):- tweight(consonant1,D). % ch <-> sh\nt1(’č’,’S’,D):- tweight(consonant1,D). % ch <-> sh\nt1(’č’,’š’,D):- tweight(consonant1,D). % ch <-> sh\nt1(’K’,k,D):- tweight(consonant1,D). % kh <-> k\nt1(’K’,g,D):- tweight(consonant1,D). % kh <-> g\nt1(’G’,’Z’,D):- tweight(consonant1,D). % dzh <-> zh !!\nt1(’G’,’C’,D):- tweight(consonant1,D). % dzh <-> ch !!\nt1(’K’,’G’,D):- tweight(consonant1,D). % kh <->gh\nt1(’Z’,z,D):- tweight(consonant1,D). % zh <-> z\nt1(’Z’,s,D):- tweight(consonant1x2,D). % zh -> z -> s !!\nt1(c,s,D):- tweight(consonant1,D). % ts <-> s\nt1(x,k,D):- tweight(consonant1,D). % ks <-> k\nt1(’D’,d,D):-tweight(consonant1,D). % dh <-> d\nt1(’K’,g,D):-tweight(consonant1,D). % kh -> g %!!\nt1(’H’,’K’,D):-tweight(consonant1,D). %!!\nt1(’H’,g,D):-tweight(consonant1,D). %!!\nt1(’H’,k,D):-tweight(consonant1,D). %!!\nt1(’H’,h,D):-tweight(consonant1,D). %!!\n42% vowels\n%t1(S1,S2,0.2):- Vowels=[a,e,i,o,u,y], member(S1,Vowels), member(S2,Vowels).\nt1(a,Y,V):- (Y=e;Y=’E’;Y=i;Y=’I’;Y=o;Y=’O’;Y=u;Y=’U’;Y=y;Y=’Y’), tweight(vowel,V).\nt1(e,Y,V):- (Y=a;Y=’A’;Y=i;Y=’I’;Y=o;Y=’O’;Y=u;Y=’U’;Y=y;Y=’Y’), tweight(vowel,V).\nt1(i,Y,V):- (Y=a;Y=’A’;Y=e;Y=’E’;Y=o;Y=’O’;Y=u;Y=’U’;Y=y;Y=’Y’), tweight(vowel,V).\nt1(o,Y,V):- (Y=a;Y=’A’;Y=e;Y=’E’;Y=i;Y=’I’;Y=u;Y=’U’;Y=y;Y=’Y’), tweight(vowel,V).\nt1(u,Y,V):- (Y=a;Y=’A’;Y=e;Y=’E’;Y=i;Y=’I’;Y=o;Y=’O’;Y=y;Y=’Y’), tweight(vowel,V).\nt1(y,Y,V):- (Y=a;Y=’A’;Y=e;Y=’E’;Y=i;Y=’I’;Y=o;Y=’O’;Y=u;Y=’U’), tweight(vowel,V).\n% same vowel\nt1(A1,A2,0):- t_a(A1), t_a(A2).\nt1(E1,E2,0):- t_e(E1), t_e(E2).\nt1(I1,I2,0):- t_i(I1), t_i(I2).\nt1(O1,O2,0):- t_o(O1), t_o(O2).\nt1(U1,U2,0):- t_u(U1), t_u(U2).\nt1(Y1,Y2,0):- t_y(Y1), t_y(Y2).\n% close vowels\nt1(X,Y,V):- tvowel(X), tvowel(Y), tweight(vowel,V).\n% long vowels\nt1(’A’,Y,V):- (Y=’E’;Y=e;Y=’I’;Y=i;Y=’O’;Y=o;Y=’U’;Y=u;Y=’Y’;Y=y), tweight(vowel,V).\nt1(’E’,Y,V):- (Y=’A’;Y=a;Y=’I’;Y=i;Y=’O’;Y=o;Y=’U’;Y=u;Y=’Y’;Y=y), tweight(vowel,V).\nt1(’I’,Y,V):- (Y=’A’;Y=a;Y=’E’;Y=e;Y=’O’;Y=o;Y=’U’;Y=u;Y=’Y’;Y=y), tweight(vowel,V).\nt1(’O’,Y,V):- (Y=’A’;Y=a;Y=’E’;Y=e;Y=’I’;Y=i;Y=’U’;Y=u;Y=’Y’;Y=y), tweight(vowel,V).\nt1(’U’,Y,V):- (Y=’A’;Y=a;Y=’E’;Y=e;Y=’I’;Y=i;Y=’O’;Y=o;Y=’Y’;Y=y), tweight(vowel,V).\nt1(’Y’,Y,V):- (Y=’A’;Y=a;Y=’E’;Y=e;Y=’I’;Y=i;Y=’O’;Y=o;Y=’U’;Y=u), tweight(vowel,V).\n%long-short vowels\nt1(’A’,a,Z):- tweight(longvowel,Z).\nt1(’E’,e,Z):- tweight(longvowel,Z).\nt1(’I’,i,Z):- tweight(longvowel,Z).\nt1(’O’,o,Z):- tweight(longvowel,Z).\nt1(’U’,u,Z):- tweight(longvowel,Z).\nt1(’Y’,y,Z):- tweight(longvowel,Z).\n%long consonants\nt1(’M’,m,Z):- tweight(longconsonant,Z).\nt1(’N’,n,Z):- tweight(longconsonant,Z).\n% weight table\ntweight(vowel,0.2).\ntweight(longvowel,0.1).\ntweight(consonant1,0.2).\ntweight(consonant1x2,0.4).\ntweight(consonant1x3,0.8).\n43 \n!\n\"#\n$#\n\"!%\n&'\n!\n\"(!\n()*\n+,\n\"−!\n$\n/01\n2\n/%\n&'\n//\n(*\n)3\n2/\n4156\n07\n8\n6\n59\n:6\n56\n;5\n<)=\n>6\n45\n7\n?@A\nBA\n@\n?9\n:7@@\n;\n?\n<C\nDE\n@\nF?B \n!\n\"#\n$%\n&'\n()*\n+,\n−\n/ 01\n23\n456\n789\n:;\n<=\n>?@A\nB\nC\nDE\nF10_borrowdale10_bowland 10_breton\n10_coniston10_cornish\n10_derbyshire\n10_derbyshireDales10_eskdale\n10_kirkbyLonsdale10_lakes\n10_lincolnshire10_nidderdale10_rathmell\n10_scots10_southWestEngland10_swaledale10_teesdale10_tong10_welsh10_wensleydale\n10_westCountryDorset10_westmorland10_wilts\n1_borrowdale1_bowland1_breton\n1_coniston1_cornish\n1_derbyshire\n1_derbyshireDales1_eskdale\n1_kirkbyLonsdale1_lakes\n1_lincolnshire1_nidderdale\n1_rathmell1_scots1_southWestEngland\n1_swaledale1_teesdale1_tong1_welsh\n1_wensleydale1_westCountryDorset\n1_westmorland1_wilts\n2_borrowdale\n2_bowland2_breton\n2_coniston2_cornish\n2_derbyshire2_derbyshireDales\n2_eskdale\n2_kirkbyLonsdale2_lakes\n2_lincolnshire\n2_nidderdale2_rathmell2_scots\n2_southWestEngland2_swaledale\n2_teesdale2_tong2_welsh\n2_wensleydale2_westCountryDorset\n2_westmorland\n2_wilts3_borrowdale3_bowland\n3_breton3_coniston\n3_cornish3_derbyshire\n3_derbyshireDales3_eskdale3_kirkbyLonsdale3_lakes\n3_lincolnshire3_nidderdale\n3_rathmell3_scots\n3_southWestEngland3_swaledale3_teesdale3_tong\n3_welsh3_wensleydale\n3_westCountryDorset3_westmorland3_wilts4_borrowdale4_bowland\n4_breton4_coniston\n4_cornish4_derbyshire\n4_derbyshireDales4_eskdale\n4_kirkbyLonsdale4_lakes4_lincolnshire4_nidderdale4_rathmell\n4_scots\n4_southWestEngland4_swaledale4_teesdale4_tong4_welsh4_wensleydale4_westCountryDorset\n4_westmorland4_wilts5_borrowdale\n5_bowland5_breton 5_coniston5_cornish\n5_derbyshire5_derbyshireDales5_eskdale\n5_kirkbyLonsdale5_lakes5_lincolnshire\n5_nidderdale5_rathmell5_scots\n5_southWestEngland\n5_swaledale5_teesdale\n5_tong5_welsh\n5_wensleydale\n5_westCountryDorset5_westmorland\n5_wilts\n6_borrowdale\n6_bowland6_breton\n6_coniston6_cornish\n6_derbyshire6_derbyshireDales6_eskdale\n6_kirkbyLonsdale6_lakes\n6_lincolnshire\n6_nidderdale6_rathmell6_scots\n6_southWestEngland6_swaledale6_teesdale\n6_tong6_welsh\n6_wensleydale6_westCountryDorset6_westmorland6_wilts7_borrowdale\n7_bowland7_breton7_coniston\n7_cornish\n7_derbyshire7_derbyshireDales7_eskdale\n7_kirkbyLonsdale7_lakes\n7_lincolnshire\n7_nidderdale7_rathmell7_scots\n7_southWestEngland7_swaledale\n7_teesdale7_tong\n7_welsh7_wensleydale\n7_westCountryDorset7_westmorland7_wilts8_borrowdale8_bowland\n8_breton8_coniston\n8_cornish\n8_derbyshire\n8_derbyshireDales8_eskdale\n8_kirkbyLonsdale8_lakes\n8_lincolnshire8_nidderdale\n8_rathmell8_scots\n8_southWestEngland8_swaledale8_teesdale\n8_tong\n8_welsh8_wensleydale\n8_westCountryDorset8_westmorland\n8_wilts\n9_borrowdale9_bowland9_breton\n9_coniston9_cornish\n9_derbyshire\n9_derbyshireDales9_eskdale\n9_kirkbyLonsdale9_lakes9_lincolnshire9_nidderdale\n9_rathmell\n9_scots\n9_southWestEngland9_swaledale9_teesdale\n9_tong9_welsh\n9_wensleydale\n9_westCountryDorset9_westmorland9_wilts\n−10−5051015\n−15 −10 −5 0 5 10 15\nDim1 (40%)Dim2 (17.5%)cluster\na\na\na\na\na\na\na\na\na\na\na\na\na\na\na\na\na\naa\na\na\na\na\na\na\na\na\na\na\na\na\n  a\n!!a\n\"\"a\n##a\n$$a%%a\n&&a\n''a\n((a\n))a\n**a\n++a\n,,a\n−−a\na\n//a\n00a\n11a\n22a\n33a\n44a\n55a\n66a77a\n88a\n99a\n::a\n;;a\n<<a\n==a\n>>a\n??a\n@@a\nAAa\nBBa\nCCa\nDDa\nEEa\nFFa1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\n17\n1819\n20\n21\n22\n23\n24\n25\n26\n27\n28\n29\n30\n31\n32\n33\n34\n35\n3637\n38\n39\n40\n41\n42\n43\n44\n45\n46\n47\n48\n49\n50\n51\n52\n53\n5455\n56\n57\n58\n59\n60\n61\n62\n63\n64\n65\n66\n67\n68\n69\n70Cluster plotFigure 24: Cluster plot of all to all sheep counting systems comparison with best\nsilhouette value cut\ntweight(longconsonant,0.05).\ntvowel(V):- t_a(V); t_e(V); t_i(V); t_o(V); t_u(V); t_y(V).\nB Dendrograms and Cluster plots\nB.1 Sheep counting systems\nFigures 24 and 25.\nB.2 Dendrograms of Bhattacharya scores of colour words\nFigures 26, 27, 28, 29 and 30.\n4410_borrowdale10_bowland 10_breton\n10_coniston10_cornish\n10_derbyshire\n10_derbyshireDales10_eskdale\n10_kirkbyLonsdale10_lakes\n10_lincolnshire10_nidderdale10_rathmell\n10_scots10_southWestEngland10_swaledale10_teesdale10_tong10_welsh10_wensleydale\n10_westCountryDorset10_westmorland10_wilts\n1_borrowdale1_bowland1_breton\n1_coniston1_cornish\n1_derbyshire\n1_derbyshireDales1_eskdale\n1_kirkbyLonsdale1_lakes\n1_lincolnshire1_nidderdale\n1_rathmell1_scots1_southWestEngland\n1_swaledale1_teesdale1_tong1_welsh\n1_wensleydale1_westCountryDorset\n1_westmorland1_wilts\n2_borrowdale\n2_bowland2_breton\n2_coniston2_cornish\n2_derbyshire2_derbyshireDales\n2_eskdale\n2_kirkbyLonsdale2_lakes\n2_lincolnshire\n2_nidderdale2_rathmell2_scots\n2_southWestEngland2_swaledale\n2_teesdale2_tong2_welsh\n2_wensleydale2_westCountryDorset\n2_westmorland\n2_wilts3_borrowdale3_bowland\n3_breton3_coniston\n3_cornish3_derbyshire\n3_derbyshireDales3_eskdale3_kirkbyLonsdale3_lakes\n3_lincolnshire3_nidderdale\n3_rathmell3_scots\n3_southWestEngland3_swaledale3_teesdale3_tong\n3_welsh3_wensleydale\n3_westCountryDorset3_westmorland 3_wilts4_borrowdale4_bowland\n4_breton4_coniston\n4_cornish4_derbyshire\n4_derbyshireDales4_eskdale\n4_kirkbyLonsdale4_lakes4_lincolnshire4_nidderdale4_rathmell\n4_scots\n4_southWestEngland4_swaledale4_teesdale4_tong4_welsh4_wensleydale4_westCountryDorset\n4_westmorland4_wilts5_borrowdale\n5_bowland5_breton 5_coniston5_cornish\n5_derbyshire5_derbyshireDales5_eskdale\n5_kirkbyLonsdale5_lakes5_lincolnshire\n5_nidderdale5_rathmell5_scots\n5_southWestEngland\n5_swaledale5_teesdale\n5_tong5_welsh\n5_wensleydale\n5_westCountryDorset5_westmorland\n5_wilts\n6_borrowdale\n6_bowland6_breton\n6_coniston6_cornish\n6_derbyshire6_derbyshireDales6_eskdale\n6_kirkbyLonsdale6_lakes\n6_lincolnshire\n6_nidderdale6_rathmell6_scots\n6_southWestEngland6_swaledale6_teesdale\n6_tong6_welsh\n6_wensleydale6_westCountryDorset6_westmorland6_wilts7_borrowdale\n7_bowland7_breton7_coniston\n7_cornish\n7_derbyshire7_derbyshireDales7_eskdale\n7_kirkbyLonsdale7_lakes\n7_lincolnshire\n7_nidderdale7_rathmell7_scots\n7_southWestEngland7_swaledale\n7_teesdale7_tong\n7_welsh7_wensleydale\n7_westCountryDorset7_westmorland7_wilts8_borrowdale8_bowland\n8_breton8_coniston\n8_cornish\n8_derbyshire\n8_derbyshireDales8_eskdale\n8_kirkbyLonsdale8_lakes\n8_lincolnshire8_nidderdale\n8_rathmell8_scots\n8_southWestEngland8_swaledale8_teesdale\n8_tong\n8_welsh8_wensleydale\n8_westCountryDorset8_westmorland\n8_wilts\n9_borrowdale9_bowland9_breton\n9_coniston9_cornish\n9_derbyshire\n9_derbyshireDales9_eskdale\n9_kirkbyLonsdale9_lakes9_lincolnshire9_nidderdale\n9_rathmell\n9_scots\n9_southWestEngland9_swaledale9_teesdale\n9_tong9_welsh\n9_wensleydale\n9_westCountryDorset9_westmorland9_wilts\n−10−5051015\n−15 −10 −5 0 5 10 15\nDim1 (40%)Dim2 (17.5%)cluster\na\na\na\na\na\na\na\na\na\na1\n2\n3\n4\n5\n6\n7\n8\n9\n10Cluster plotFigure 25: Cluster plot of all to all sheep counting systems comparison with K=10\ncut\n45redyellowwhitegreenblackblue\n0.820431 0.950568Figure 26: Dendrogram of Bhattacharya scores of “ColoursAll”\n46greenbluewhiteyellowblackred\n0.525667 0.866715Figure27: DendrogramofBhattacharyascoresofIndo-Europeanlanguages(colours)\n47greenbluewhiteyellowblackred\n0.525667 0.866715Figure 28: Dendrogram of Bhattacharya scores of Germanic languages (colours)\n48yellowblueblackgreenwhitered\n0.670457 0.93171Figure 29: Dendrogram of Bhattacharya scores of Romance languages (colours)\n49bluewhitegreenblackredyellow\n0.682005 0.936915Figure 30: Dendrogram of Bhattacharya scores of Germanic and Romance languages\n(colours)\n50"    },    {        "title": "1902.04605v1.Ultra_low_damping_in_lift_off_structured_yttrium_iron_garnet_thin_films.pdf",        "content": "1 \n This article may be downloaded for personal use only. Any other use requires prior permission of the \nauthor and AIP Publishing. This article appeared in Applied Physics Letters 111 (19), 192404 (2017) \nand may be found at https://aip.scitation.org/doi/abs/10.1063/1.5002004  \n \n \nUltra -low damping in lift-off structured y ttrium iron garnet  thin films  \nA. Krysztofik ,1 L. E. Coy,2 P. Kuświk ,1,3 K. Załęski,2 H. Głowiński ,1  \nand J. Dubowik1 \n1Institute of Molecular Physics, Polish Academy of Sciences, PL -60-179 Poznań, Poland  \n2NanoBioMedical Centre, Adam Mickiewicz University, PL -61-614 Poznań, Poland  \n3Centre for Advanced Technology, Adam Mickiewicz University, PL -61-614 Poznań, Poland  \nElectronic mail: adam.krysztofik@ifmpan.poznan.pl , hubert .glowinski @ifmpan.poznan.pl  \n \nWe show that using maskless photolithography and the lift-off technique patterned \nyttrium iron garnet thin films possessing ultra -low Gilbert damping can be  \naccomplished . The films of the 70 nm thickness we re grown on (001)-oriented \ngadolinium gallium garne t by means of pulsed laser deposition and exhibit high \ncrystalline quality, low surface roughness and effective  magnetization of 127 \nemu/cm3. The Gilbert damping parameter is as low as 5×10−4. The obtained \nstructures have well-defined sharp edges  which  along with good structural and \nmagnetic film properties, pave a path in the fabrication of high -quality magnonic \ncircuits as well as  oxide -based  spintronic devices.  \n \n \nYttrium iron garnet (Y 3Fe5O12, YIG) has become an intensively studied material in recent years due \nto exceptionally low damping of magnetization precession and electrical insulation enabling its \napplication in research on spin -wave propagation1–3, spin-wave based logic devices4–6, spin pumping7, \nand thermally -driven spin caloritronics8. These applications inevitably entail film structurization in \norder to construct complex integrated devices . However, the fabrication of high -quality thin YIG films \nrequires  deposition temperatures over 500 C6,9–18 leading to top -down lithographical approach that is \nion-beam etching  of a previously  deposited plain film where as patterned resist layer serves as a mask.  \nConsequently, this metho d introdu ces crystallographic  defects , imperfections to surface structure  and, \nin the case of YIG films, causes significant  increase of the damping parameter .19–21 Moreover, it does \nnot ensure well-defined  structure edges  for insulators , which play a crucial role in devices utilizing 2 \n edge spin waves22, Goos -Hänchen spin wave shifts23,24 or standing spin waves modes25. On the \ncontrary, t he bottom -up structurization  deals with th ese issues  since it allows  for the film grow th in the \nselect ed, patterned  areas followed by  a removal of the resist layer along with redundant film during \nlift-off process.  Additionally, it reduces the patterning  procedure by one step , that is ion etching , and \nimposes room -temperature deposition  which both are particularly important  whenever  low fabrication  \nbudget is required.  \nIn this letter  we report on ultra -low damping in the bottom -up structured YIG film by means of \ndirect writing photolithography technique.  In our case, t he method allows for structure patterning \nwith 0.6 µm resolution across full writing area . In order to not preclude the lift -off process, the pulsed \nlaser deposition (PLD) was conducted at room temperature and since such as -deposited films are \namorphous19,27 the ex-situ annealing was performed for recrystallization.  Note that post -deposition \nannealing of YIG films  is commonly carried out regardless the substrate temperature  during film \ndeposition6,12,13,28,29. As a reference we  investigated  a plain film which was grown in the same \ndeposition process and underwent the same fabrication procedure except for patterning. Henceforth, \nwe will refer to the structured and the plain film as Sample 1 an d Sample 2, respectively . We \nanticipate that such a procedure may be of potential for fabrication of other magnetic oxide structures \nuseful in spintronics.  \nStructural characteriza tion of both samples was performed by means  of X-Ray Diffraction (XRD). \nAtomic force microscopy (AFM) was applied to investigate  surface morphology and the quality of \nstructure edges.  SQUID magnetometry  provided information on the saturation magnetization  and \nmagnetocrystalline anisotropy field . Using a coplanar waveguide connected to a vector network \nanalyzer , broadband ferromagnetic resonance (VNA -FMR) was performed to determine Gilbert \ndamping parameter  and anisotropy fields . All the experiments were co nducted at the room \ntemperature.  \nThe procedure of samples preparation was as follows. The (001) -oriented gadolinium gallium \ngarnet substrates were ultrasonicated in acetone, trichloroethylene and isopropanol to remove surface \nimpurities. After a 1 minute o f hot plate baking for water evaporation, a positive photoresist was spin -\ncoated onto the substrate (Sample 1). Using maskless photolithography an array of 500  μm x 500  μm \nsquares separated over 500 μm was patterned and the exposed areas were developed. Detailed \nparameters of photolithography process can be found in Ref.26. We chose rather large size of the \nsquares to provide a high signal -to-noise ratio in the latter measurements. Thereafter, plasma etching \nwas performed to remove a residual resist. We would like to emphasize the importance of this step in \nthe fabrication p rocedure as the resist residues may locally affect crystalline structure of a YIG film \ncausing an undesirable increase of overall magnetization damping. Both substrates were then placed in \na high vacuum chamber of 9×10-8 mbar base pressure and a film was d eposited from a stoichiometric \nceramic YIG target under 2×10-4 mbar partial pressure of oxygen. We used a Nd:YAG laser (λ  = 355 \nnm) for the ablation with pulse rate of 2 Hz which yielded 1 nm/min growth rate. The target -to-3 \n substrate distance was approximat ely 50 mm. After the deposition the l ift-off process for the Sample 1  \nwas performed using sonication in acetone  to obtain the expected structures. Subsequently, both \nsamples were annealed in a tube furnace under oxygen atmosphere (p ≈ 1 bar) for 30 minutes  at \n850°C. The heating and cooling rates were about 50  C/min and 10  C/min, respectively.  \n \n \n \nFIG. 1.  (a) XRD θ−2θ plot near the (004) reflection of structured ( Sample 1 ) and plain ( Sample 2 ) YIG film. Blue arrows \nshow clear Laue reflections of the plain film. Insets show schematic illustration of the structured and plain film used in this \nstudy. (b) Height  profile (z(x)) taken from the structured sample (left axis), right shows the differential of the p rofile, clearly \nshowing the slope change.  Inset shows 3D map of the structure’s edge.  \n \n \nThe structure of YIG films was determined by the X -ray diffraction.  Although the as-deposited \nfilms were amorphous, with the annealing treatment they inherited the lattice orientation of the GGG \nsubstrate and recrystallized along [ 001] direction. Figure 1 (a) presents diffraction curves  taken in the \nvicinity of ( 004) Bragg reflection.  The ( 004) reflection position of structured YIG well coincide s with \nthe reflection of the plain film. The 2 θ=28.70 9 corresponds to the cubic lattice constant of 12.428  Å. \nA comparison of this value with lattice parameter of a bulk YIG (12.376 Å) suggest distortion of unit \n4 \n cells due to slight nonstoichiometry.16,30 Both samples exhibit distinct Laue oscillation s depicted by \nthe blue arrows,  indicating  film uniformity and high crystalline order , although the structured film \nshowed lower intensity due to the lower mass of the film . From the oscillation period we estimated \nfilm thickness of 73 nm in agreement with the nominal thickness and the value determined using AFM  \nfor Sample 1 ( Fig. 1 (b)). By measuring the diffraction  in the expanded angle range w e also confirmed \nthat no additional phases like Y 2O3 or Fe 2O3 appeared.  \nThe surface morphology of the structured film was investigated by means of AFM. In Fig. 1 (b) \nprofile of a square’s edge is shown. It should be highlighted that no edge irregularities has formed \nduring lift -off process. The horizontal distance between GGG substrate and the surface of YIG film is \nequal to 170 nm as marked in Fig. 1 (b) by the shaded area. A fitting with Gaussian function to the \nderivative of height profile yields the full width at half maximum of 61 nm. This points to the well -\ndefined struct ure edges achieved with bottom -up structurization. Both samples have smooth and \nuniform surface s. The comparable values of root mean square (RMS) roughness (0.306 nm for Sample \n1 and 0.310 nm for Sample 2) indicate that bottom -up structurization process  did not leave any resist \nresidues. Note that a roughness of a bare GGG substrate before deposition was 0.281 nm, therefore, \nthe surface roughness of YIG is increased merely by 10%.  \n \n \nFIG. 2. Hysteresis loops of structured (Sample 1) and plain (Sample 2) YIG films measured by SQUID \nmagnetometry  along [100] direction at the room temperature . \n \nFigure 2 shows magnetization reversal curves measured along [ 100] direction. For each hysteresis \nloop a paramagnetic contribution arising for the GGG substrates was subtracted. The saturation \nmagnetization 𝑀𝑠 was equal to 117 emu/cm3 and 118.5 emu/cm3 for Sample 1 and 2, respectively . \nBoth hysteresis loops demonstrate in -plane anisotropy. For the (001) -oriented YIG the [ 100] direction \nis a “hard” in -plane axis and the magnetization saturates at 𝐻𝑎 = 65 Oe. This value we identify as \n-100 -75 -50 -25 0 25 50 75 100-1.0-0.50.00.51.0  Sample 1\n Sample 2M / MS\nMagnetic Field (Oe)5 \n magnetocrystalline anisotropy field. The VNA -FMR measurements shown in Fig. 3 (a) confirm these \nresults. Using Kitte l dispersion relation, i.e. frequency 𝑓 dependence of resonance magnetic field 𝐻: \n 𝑓=𝛾\n2𝜋√(𝐻+𝐻𝑎cos 4𝜑)(𝐻+1\n4𝐻𝑎(3+cos 4𝜑)+4𝜋𝑀𝑒𝑓𝑓), (1) \n 4𝜋𝑀𝑒𝑓𝑓=4𝜋𝑀𝑠−𝐻𝑢, (2) \nwe derived 𝐻𝑎 and the effective magnetization 𝑀𝑒𝑓𝑓, both comparable to the values determined using \nSQUID and close to the values of a bulk YIG (see Table I .). Here,  the azimuthal angle 𝜑 defines the \nin-plane orientation of the magnetization direction with respect to the [100] axis of YIG  and  𝛾 is the \ngyromagnetic ratio ( 1.77×107𝐺−1𝑠−1). To better compare the values of 𝐻𝑎 between samples and to \ndetermine if the results are influenced by additional anisotropic contribution arising  from  the squares’ \nshape in the  structured film we performed angular resolved resonance measurements  (inset in Fig. \n3(a)) . The fitting according to Eq. (1) gives |𝐻𝑎| equal to 69.5±0.6 for Sample 1 and 69.74±0.28 for \nSample 2 in agreement with the values derived from 𝑓(𝐻) dependence and better accuracy.  Hence, we \nconclude  that the structurization did not affect the in -plane anisotropy. The deviations of  the derived  \n𝑀𝑠 and 𝐻𝑎 from bulk values can be explained in the framework of Fe vacancy model developed for \nYIG films as a result of nonstoichiometry.13,30 For the experimentally determined 𝑀𝑠 and 𝐻𝑎 the \nmodel yields the chemical unit Y 3Fe4.6O11.4 which closely approximates to  the composition of a \nstoichiometric YIG Y 3Fe5O12. \n \n \nTABLE I.  Key parameters reported for PLD and LPE YIG films.  \n  AFM  SQUID  VNA -FMR  \n Film \nthickness  RMS rough -\nness (nm)  Ms  \n(emu/cm3) Ha  \n(Oe) Field  \norientation  Meff \n(emu/cm3) |Ha|  \n(Oe) Hu \n(Oe) α  \n(× 10-4) ΔH 0  \n(Oe) \nSample 1  70 nm  0.306  117±1  65±5  (100):  \n(110):  \n(001):  125±1  \n126±1  \n129±2  64±1  \n63±1  \n− -101±18  \n-113±18  \n-151±28  5.53±0.13  \n5.24±0.12  \n5.19±0.64  1.45±0.09  \n2.86±0.09  \n2.61±0.34  \nSample 2  70 nm  0.310  118.5±2  65±5  (100):  \n(110):  \n(001):  124±1  \n127±1  \n131±2  62±1  \n65±1  \n− -69±28  \n-107±28  \n-157±36  5.05±0.07  \n5.09±0.09  \n5.02±0.18  0.97±0.05  \n1.28±0.06  \n1.48±0.09  \nLPE-YIG31 106 nm  0.3 143 − (112):  − − − 1.2 0.75 \nLPE-YIG30 120 μm  − 139±2  − (111):  133±2  85±6  76±1  0.3 − \n \n \nAlthough the saturation magnetization  of the films  is decreased by 15% with respect to the bulk \nvalue we can expect similar spin wave dynamics since magnon  propagation does not solely depend on \n𝑀𝑠 but on the effective magnetization or equivalently, on the uniaxial anisotropy field  𝐻𝑢.12 \nSubstitution of 𝑀𝑠 into Eq. (2) gives  average values of  𝐻𝑢 equal to -122 Oe and -111 Oe for Sample 1 \nand 2, respectively  (to determine 𝐻𝑢 from the out -of-plane FMR measurements when H || [001] we 6 \n used the 𝑓=𝛾\n2𝜋(𝐻+𝐻𝑎−4𝜋𝑀𝑒𝑓𝑓) dependence13 to fit the data  and assumed  the value of 𝐻𝑎 from \nangular measurements ). As 𝑀𝑒𝑓𝑓𝑆𝑎𝑚𝑝𝑙𝑒  1,2≈𝑀𝑒𝑓𝑓𝑏𝑢𝑙𝑘, it follows  that the low value of 𝑀𝑠 in room -\ntemperature deposited thin films is “compensated ” by uniaxial anisotropy field.  Note that for bulk YIG \nsaturation magnetization is diminished  by 𝐻𝑢/4𝜋 giving  a lower value of 𝑀𝑒𝑓𝑓 while for Sample 1 \nand 2,  𝑀𝑠 is augmented  by 𝐻𝑢/4𝜋 giving a higher value of 𝑀𝑒𝑓𝑓 (Table I .). The negative sign of \nuniaxial anisotropy field  is typical for PLD -grown YIG films  and originates from preferential \ndistribution of Fe vacancies between different si tes of YIG’s octahedral  sublattice.30 This point s to the \ngrowth -induced anisotropy mechanism while the stress -induced contribution is of ≈10 Oe29 and, as it \ncan be estimated  according to Ref.32, the transition layer at the substrate -film interface due to Gd, Ga, \nY ions diffusion is ca. 1.5 nm thick for the 30 min of annealing treatment.  We argue that the growth -\ninduced anisotropy due to ordering of the magnetic ions is related to the growth condition which in our \nstudy is specific. Namely, it is crystallization of an amorphous material.  \nGilbert damping parameter 𝛼 was obtained by fitting dependence of linewidth 𝛥𝐻 (full width at \nhalf maximum ) on frequency 𝑓 as shown in Fig. 3 (b): \n 𝛥𝐻 =4𝜋𝛼\n𝛾𝑓+𝛥𝐻0, (3) \nwhere  𝛥𝐻0 is a zero -frequency linewidth broadening . The 𝛼 parameter of both samples is nearly the \nsame , 5.32×10−4 for Sample 1 and 5.05×10−4 for Sample 2  on average (see Table I.) . It proves \nthat bottom -up patterning does not compromise magnetization damping. The value of 𝛥𝐻0 \ncontribution is around 1.5 Oe although small variations of 𝛥𝐻0 on 𝜑 can be noticed. Additional \ncomments on  angular dependencies of  𝛥𝐻 can be found in the supplementary material. The derived \nvalues of 𝛼 remain one order of magnitude smaller than for soft ferromagnets like Ni 80Fe2033, CoFeB34 \nor Finemet35, and are comparable to values reported for YIG film s deposited at hi gh temperatures \n(from 1×10−4 up to 9×10−4).6,9,11,14,15,17,18 It should be also highlighted  that 𝛼 constant is \nsignificantly increased in comparison to the bulk YIG made by means of Liquid Phase Epitaxy (LPE) . \nHowever, recently reported LPE-YIG films of  nanometer thickness , suffer from  the increased damping \nas well (Table I.) due to impurity elements  present in the high -temperature solutions used in LPE \ntechnique31. As PLD method allow s for a good  contamination control , we attribute the increase as a \nresult of slight nonstoichiometry determined above with Fe vacancy model .30 Optimization of growth \nconditions , which further improve the film composition may resolve this issue and allow to cross the \n𝛼=1×10−4 limit. We also report that additional annealing of the samples (for 2h) did not influence \ndamping nor it improved the value of 𝐻𝑎 or 𝑀𝑒𝑓𝑓 (within 5% accuracy).  7 \n  \nFIG. 3. (a) Kittel dispersion relation s of the structured (Sample 1) and plain (Sample 2) YIG film.  The i nset \nshows angular dependence of resonance field  revealing perfect fourfold anisotropy for both samples . (b) \nLinewidth dependence on frequency fitted with Eq. (3). The inset shows resonance absorptions peaks with very \nsimilar width (5.3 Oe for Sample 1 and 4.7 Oe for Sample 2  at 10 GHz ). Small differences of the resonance field  \noriginate from different values of 4𝜋𝑀𝑒𝑓𝑓. \n \nIn conclusion , the lift-off patterned YIG films possessing low damping have  been presented. \nAlthough the structurization procedure required deposition at room temperature , the 𝛼 parameter does \nnot diverge from those reported for YIG thin films grown at temperatures above 500 C. Using the \nplain, reference film fabricated along with the structured one, we have shown  that structurization does \nnot significantly affect structural nor magnetic properties  of the films,  i.e. out-of-plane lattice constant, \nsurface roughness, saturation magnetization, anisotropy fields and damping. The structures obtain ed \nwith bottom -up structurization indeed possess sharp , well-defined edges . In particular, o ur findings \nwill help in the development of magnonic and spintronic devices utilizing film boundary effects  and \nlow damping of magnetization precession . \n8 \n  \nSupplementary Material  \nSee supplementary material for the angular dependence of resonance linewidth . \n \nThe research received funding from the European Union Horizon 2020 research and innovation \nprogra mme under the Marie Skłodowska -Curie grant agreement No 644348 (MagIC).  We would like \nto thank Andrzej Musiał for the assistance during film annealing.  \n \n1 H. 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Matter 29, 435803 (2017).  \n35 H. Głowiński, I. Gościańska, A. Krysztofik, J. Barnaś, M. Cecot, P. Kuświk, and J. Dubowik, in 21st Int. Conf. Microwave, \nRadar Wirel. Commun. MIKON 2016  (2016).  \n \n \n "    },    {        "title": "1009.3922v1.Diffusive_properties_of_persistent_walks_on_cubic_lattices_with_application_to_periodic_Lorentz_gases.pdf",        "content": "Diffusive properties of persistent walks on cubic lattices with\napplication to periodic Lorentz gases\nThomas Gilberty, Huu Chuong Nguyen y, David P. Sanders z\nyCenter for Nonlinear Phenomena and Complex Systems, Universit ´e Libre de Bruxelles,\nC. P. 231, Campus Plaine, B-1050 Brussels, Belgium\nzDepartamento de F ´ısica, Facultad de Ciencias, Universidad Nacional Aut ´onoma de M ´exico,\n04510 M ´exico D.F., Mexico\nE-mail: thomas.gilbert@ulb.ac.be, hnguyen@ulb.ac.be,\ndps@fciencias.unam.mx\nAbstract. We calculate the diffusion coefficients of persistent random walks on cubic and\nhypercubic lattices, where the direction of a walker at a given step depends on the memory of\none or two previous steps. These results are then applied to study a billiard model, namely a\nthree-dimensional periodic Lorentz gas. The geometry of the model is studied in order to find\nthe regimes in which it exhibits normal diffusion. In this regime, we calculate numerically the\ntransition probabilities between cells to compare the persistent random-walk approximation\nwith simulation results for the diffusion coefficient.\nSubmitted to: J. Phys. A: Math. Theor.\n1. Introduction\nProblems dealing with the persistence of motion of tracer particles – that is, the tendency to\ncontinue or not in the same direction at a scattering event – are encountered in many areas\nof physics; see e.g. [1] and references therein. We are specifically interested in the effect of\npersistence for the motion of random walkers on regular lattices.\nThe diffusive properties of persistent random walks on two-dimensional regular lattices\nwere the subject of a previous paper by two of the present authors [2]. There, we presented\na theory making use of the symmetries of such lattices to derive the transport coefficients of\nwalks with a two-step memory. In the first part of the present paper, we extend this theory to\nhyper-cubic lattices in arbitrary dimensions, which is possible by describing the geometry of\nthe lattices in a suitable way.\nPersistence effects naturally arise in the context of deterministic diffusion [3, 4, 5, 6],\nwhich is concerned with the interplay between dynamical properties at the microscopic scale\nand transport properties at the macroscopic scale. A variety of different techniques are now\navailable, which rely on the chaotic properties of model systems to describe their macroscopic\nproperties [7], [8, chap. 25]. In particular, periodic Lorentz gases and related models, such as\nmulti-baker maps, are simple deterministic dynamical systems with strong chaotic properties\nwhich also exhibit diffusive regimes. Although the transport coefficients of these models can\nbe expressed formally in terms of the microscopic dynamical properties, actually computing\nthem is usually difficult, with the exception of some of the simplest toy models [9, 10, 11].arXiv:1009.3922v1  [cond-mat.stat-mech]  20 Sep 2010Diffusive properties of persistent walks on cubic lattices 2\nOne reason for this is that memory effects can remain important, in spite of the chaotic\ncharacter of the underlying dynamics.\nThe diffusive properties of these models therefore provide ideal applications of the\nformalism presented in this paper. An example of this was illustrated in reference [12], for\na class of two-dimensional periodic billiard tables. Extending these results, in this paper we\napply the formalism to model the diffusive properties of higher-dimensional periodic Lorentz\ngases.\nThe diffusive properties of the three-dimensional periodic Lorentz gas, which consists\nof the free motion of independent tracer particles in a cubic array of spherical obstacles, are\ninteresting in their own right. In two spatial dimensions, the existence of diffusive regimes\nin such systems has been rigorously established [13, 14]. It relies on the finite-horizon\nproperty, which requires that the system admits no ballistic trajectories, i.e. those which never\ncollide with any obstacle. In this case, it is possible to change scales from microscopic to\nmacroscopic, reducing the complicated motion of tracer particles at the microscopic level to\na diffusive equation at the macroscopic level. When the horizon is infinite on the other hand,\nthere is rather a weakly superdiffusive process, with mean-squared displacement growing like\ntlogt[15, 16], as recently shown rigorously in [17].\nThe necessity of finite horizon to have normal diffusion in two dimensions led to the idea\nthat this was also necessary in three dimensions – see, for example, reference [18]. Recently,\nhowever, it was argued by one of the present authors [19] that in higher-dimensional billiards,\nnormal diffusion, by which we mean an asymptotically linear growth in time of the mean-\nsquared displacement, may arise even in the absence of finite horizon. In fact, three different\ntypes of horizon can be identified in the three-dimensional periodic Lorentz gas. The key\nobservation is that it is only “planar” gaps – those with infinite extension in two dimensions\n– which induce anomalous diffusion. If there are only “cylindrical” gaps, whose extension is\nlimited to a single dimension, then the available space in which particles can move ballistically\nis limited. This leads to a decay of correlations which is fast enough to give normal diffusion\nat the level of the mean-squared displacement, although higher moments of the displacement\ndistribution may be non-Gaussian [19].\nThe paper is organized as follows. Section 2 describes the computation of the transport\ncoefficient of walks on hypercubic lattices with one and two-step memories. In the second\npart of this paper, we apply this formalism to the diffusive regimes of the three-dimensional\nperiodic Lorentz gas. In section 3, we give a detailed description of the three-dimensional\nperiodic Lorentz gas introduced in reference [19], in particular delimiting the regimes with\nqualitatively different behaviour in parameter space. We then apply the results on persistent\nrandom walks to the diffusive regimes of this model in section 4. Conclusions are drawn in\nsection 5.\n2. Persistent random walks on cubic lattices\nIn this section, we describe a way to incorporate the specific geometry of cubic and hyper-\ncubic lattices in the framework presented in reference [2] for calculating diffusion coefficients\nfor persistent random walks on lattices.\nWe start by considering the motion of independent walkers on a regular cubic lattice in\nthree dimensions. Given their initial position r0at time t=0, the walkers’ trajectories are\nspecified by the sequence fv0;:::;vngof their successive displacements. Here we consider\ndynamics in discrete time, so that the time sequences are simply assumed to be incremented by\nidentical time steps tas the walkers move from site to site. In the sequel we will loosely refer\nto the displacement vectors as velocity vectors; they are in fact dimensionless unit vectors.Diffusive properties of persistent walks on cubic lattices 3\nThe sequence of successive displacements is determined by the underlying dynamics,\nwhether deterministic or stochastic. At the coarse level of description of the lattice dynamics,\nthis is interpreted as a persistent type of random walk, where some memory effects are\naccounted for: the probability that the nth step is taken in the direction vndepends on the\npast history vn\u00001;vn\u00002;:::.\nThe quantity of interest here is the diffusion coefficient Dof such persistent processes,\nwhich measures the linear growth in time of the mean-squared displacement of walkers. This\ncan be written in terms of velocity autocorrelations using the Taylor–Green–Kubo expression:\nD=`2\n2dt\"\n1+2 lim\nk!¥k\nå\nn=1hv0\u0001vni#\n; (2.1)\nwhere ddenotes the dimensionality of the lattice, here d=3, and `is the lattice spacing. The\n(dimensionless) velocity autocorrelations are computed as averages h\u0001iover the equilibrium\ndistribution, denoted by m, of the underlying process, so that the problem reduces to\ncomputing the quantities\nhv0\u0001vni=å\nv0;:::;vnv0\u0001vnm(fv0;:::;vng): (2.2)\nFollowing the approach of reference [2], we wish to compute the terms in this sum, and\nhence the corresponding diffusion coefficient (2.1), for three different types of random walks,\nnamely those with zero-step, single-step and two-step memories. These cases all involve\nfactorisations of the measure m(fv0;:::;vng)into products of probability measures which\ndepend on a number of velocity vectors, equal to the number of steps of memory of the\nwalkers. These measures will be denoted by pthroughout.\nThe schemes we outline below allow to write equation (2.2) as a sum of powers of\nmatrices, so that (2.1) boils down to a geometric series, which can then be resummed to obtain\nan expression for the diffusion coefficient that is readily computable given the probabilities\nthat characterise the allowed transitions in the process.\n2.1. Description of geometry of cubic lattices\nIt is first necessary to find a succinct description of the geometry of the cubic lattices that\nwe wish to study. The six directions of the three-dimensional cubic lattice and corresponding\ndisplacement vectors are specified in terms of the unit vectors eiof a Cartesian coordinate\nsystem as\u0006ei,i=1;2;3.\nThe crucial property required for the application of our method is that all of these unit\nvectors can be obtained by repeated application of a single transformation G, which generates\nthe cyclic group\nG\u0011fGi\u0011Gi;i=0;:::; 5g: (2.3)\nOne possible choice of Ggives the following group elements:\nG1=\u0000G4=G=0\n@0 0\u00001\n1 0 0\n0 1 01\nA;\nG2=\u0000G5=G2=0\n@0\u00001 0\n0 0\u00001\n1 0 01\nA;\nG3=\u0000G0=G3=0\n@\u00001 0 0\n0\u00001 0\n0 0\u000011\nA:(2.4)Diffusive properties of persistent walks on cubic lattices 4\nFigure 1 displays the six possible directions of a walker on this lattice, numbered\naccording to repeated iterations by G. Thus a walker with incoming direction e1, indicated\nby the arrow, can be deflected to any of the six directions Gie1,i=0;:::; 5, corresponding\nrespectively to e1,e2,e3,\u0000e1,\u0000e2, and\u0000e3.\n012\n3\n4\n5\nFigure 1. The possible directions of motion on a cubic lattice, labelled from 0 to 5 relative\nto the incoming direction shown by the arrow. These directions are obtained by successive\napplications of the transformation Ggiven in equation (2.4).\nA similar transformation Gcan easily be identified for a walk on a d-dimensional hyper-\ncubic lattice:\nG=0\nBBBBB@0 0\u0001\u0001\u0001 0\u00001\n1 0\u0001\u0001\u0001 0 0\n0 1\u0001\u0001\u0001 0 0\n.........\n0 0\u0001\u0001\u0001 1 01\nCCCCCA; (2.5)\nwhich maps the unit vectors onto the 2 d-cycle e17!e27!\u0001\u0001\u00017! ed7!\u0000e17!\u0001\u0001\u00017!\u0000 ed.\n2.2. No-Memory Approximation (NMA)\nWe now proceed to calculate the diffusion coefficient (2.1) for random walks with different\nmemory lengths. The simplest case is that of a Bernoulli process for the velocity trials, so\nthat the walkers have no memory of their history as they proceed to their next position. The\nprobability measure mthus factorises:\nm(fv0;:::;vng) =n\nÕ\ni=0p(vi): (2.6)\nGiven that the lattice is rotation invariant and that pis uniform, the velocity autocorrelation\n(2.2) vanishes:\nhv0\u0001vni=dn;0: (2.7)Diffusive properties of persistent walks on cubic lattices 5\nThe diffusion coefficient of the random walk without memory is then given by\nDNMA=`2\n2dt: (2.8)\n2.3. One-Step Memory Approximation (1-SMA)\nWe now assume that the velocity vectors obey a Markov process, for which vntakes on\ndifferent values according to the velocity at the previous step vn\u00001. We may then write\nm(fv0;:::;vng) =n\nÕ\ni=1P(vijvi\u00001)p(v0): (2.9)\nHere, P(v0jv)denotes the one-step conditional probability that the walker moves with\ndisplacement v0, given that it made a displacement vat the previous step.\nConsidering for definiteness the three-dimensional lattice and using the elements of the\ngroup G, we express each velocity vector vkin terms of the first one, v0, asvk=Gikv0, where\neach ik2f0;:::; 5g. Substituting this into the expression for the velocity autocorrelation\nhv0\u0001vni, equation (2.2), we obtain, using the factorisation (2.9),\nå\nv0;:::;vnv0\u0001vnn\nÕ\ni=1P(vijvi\u00001)p(v0) =6\nå\ni0;:::;in=1v0\u0001Ginv0min;in\u00001\u0001\u0001\u0001mi1;i0pi0: (2.10)\nIn this expression,\nmin;in\u00001\u0011P(Ginv0jGin\u00001v0) (2.11)\nare the elements of the stochastic matrix Mof the Markov chain associated to the persistent\nrandom walk, and pi\u0011p(ei)are the elements of its invariant (equilibrium) distribution,\ndenoted P, evaluated with a velocity in the ith lattice direction. The invariance of Pis\nexpressed as åjmi;jpj=pi. The same notations were used in [2] and will be used throughout\nthis article.\nThe terms involving Min (2.10) constitute the matrix product of ncopies of M.\nFurthermore, since the invariant distribution is uniform over the lattice directions, we can\nchoose an arbitrary direction for v0, and hence write\nhv0\u0001vni=v0\u0001v0m(n)\n1;1+v0\u0001Gv0m(n)\n2;1+\u0001\u0001\u0001+v0\u0001G5v0m(n)\n6;1;\n=m(n)\n1;1\u0000m(n)\n4;1(2.12)\nwhere m(n)\ni;jdenote the elements of Mn.\nThe actual value of the diffusion coefficient depends on the probabilities P(Gjvjv),\nwhich are parameters of the model, subject to the constraints åjP(Gjvjv) =1. To simplify\nthe notation, we assume rotational invariance of the process, i.e. independence with respect to\nthe value of v, and we denote the conditional probabilities of these walks by Pj\u0011P(Gjvjv),\nwhere j=0;:::; 5.\nThe transition matrix Mgiven by (2.11) is thus the cyclic matrix\nM=0\nBBBBBB@P0P1P2P3P4P5\nP5P0P1P2P3P4\nP4P5P0P1P2P3\nP3P4P5P0P1P2\nP2P3P4P5P0P1\nP1P2P3P4P5P01\nCCCCCCA: (2.13)Diffusive properties of persistent walks on cubic lattices 6\nThe matrix Mnshares the same property of cyclicity, so that it also has only six distinct entries.\nIt is thus possible to proceed along the lines described in [2] and obtain the recurrence relation\n0\nB@m(n)\n1;1\u0000m(n)\n4;1\nm(n)\n2;1\u0000m(n)\n5;1\nm(n)\n3;1\u0000m(n)\n6;11\nCA=0\n@P0\u0000P3P1\u0000P4P2\u0000P5\nP5\u0000P2P0\u0000P3P1\u0000P4\nP4\u0000P1P5\u0000P2P0\u0000P31\nA0\nB@m(n\u00001)\n1;1\u0000m(n\u00001)\n4;1\nm(n\u00001)\n2;1\u0000m(n\u00001)\n5;1\nm(n\u00001)\n3;1\u0000m(n\u00001)\n6;11\nCA;\n=0\n@P0\u0000P3P1\u0000P4P2\u0000P5\nP5\u0000P2P0\u0000P3P1\u0000P4\nP4\u0000P1P5\u0000P2P0\u0000P31\nAn\u000010\n@P0\u0000P3\nP1\u0000P4\nP2\u0000P51\nA: (2.14)\n[Note that the left-hand side of this equation was chosen to reduce the size of the matrix\ninvolved and to calculate the element required in (2.12).] As a consequence, we can write for\nthe velocity autocorrelation (2.12)\nhv0\u0001vni=\u0000\n1 0 0\u00010\n@P0\u0000P3P1\u0000P4P2\u0000P5\nP5\u0000P2P0\u0000P3P1\u0000P4\nP4\u0000P1P5\u0000P2P0\u0000P31\nAn\u000010\n@P0\u0000P3\nP1\u0000P4\nP2\u0000P51\nA; (2.15)\nand thus obtain the expression of the diffusion coefficient (2.1) as\nD1SMA\nDNMA=2\n641+2\u0010\n1 0 0\u00110\n@1+P3\u0000P0P4\u0000P1 P5\u0000P2\nP2\u0000P51+P3\u0000P0P4\u0000P1\nP1\u0000P4 P2\u0000P51+P3\u0000P01\nA\u000010\n@P0\u0000P3\nP1\u0000P4\nP2\u0000P51\nA3\n75; (2.16)\nby using the result that å¥\nn=0An= (I\u0000A)\u00001, where Iis the identity matrix, for a square matrix\nAwhose eigenvalues are all strictly less than 1 in modulus,\nThis result easily generalises to a hyper-cubic lattice in any dimension d. Note also that\nfor a symmetric process, in which P1=P4andP2=P5, we recover the diffusion coefficient\nD1SMA =DNMA1+P0\u0000P3\n1\u0000P0+P3; (2.17)\nin agreement with the result stated in [2].\n2.4. Two-Step Memory Approximation (2-SMA)\nLet us now suppose that the velocity vectors obey a random process for which the probability\nofvntakes on different values according to the velocities at the two previous steps, vn\u00001and\nvn\u00002, so that we may write\nm(fv0;:::;vng) =n\nÕ\ni=2P(vijvi\u00001;vi\u00002)p(v0;v1): (2.18)\nThe velocity autocorrelation (2.2) function is then\nhv0\u0001vni=å\nfvn;:::;v0gv0\u0001vnn\nÕ\ni=2P(vijvi\u00001;vi\u00002)p(v0;v1): (2.19)\nSince the probability transitions P(vijvi\u00001;vi\u00002)have symmetries similar to those used\nin reference [2], the computation of equation (2.19) reduces to an expression very similar to\nthat found there for walks on one- and two-dimensional lattices. The details of the derivation\nare a bit more involved than the one-step memory persistent walks, so we will limit ourselves\nto stating the results.Diffusive properties of persistent walks on cubic lattices 7\nLetting z=2ddenote the coordination number of the lattice, and writing‡ Pj;k\u0011\nP(Gz\u0000kGz\u0000jvjGz\u0000jv;v), which is the conditional probability of making a displacement v\ngiven that the two preceding displacements were successively GjGkvandGkv, we define the\nz\u0002zmatrix\nK(f)\u00110\nBBB@P00 P10\u0001\u0001\u0001 Pz\u00001;0\nfP01 fP11\u0001\u0001\u0001 fPz\u00001;1\n............\nfz\u00001P0;z\u00001fz\u00001P1;z\u00001\u0001\u0001\u0001fz\u00001Pz\u00001;z\u000011\nCCCA: (2.20)\nThe argument fin this expression is a complex number such that fz=1. In the case of\ntwo-dimensional lattices, only two of these roots are relevant, corresponding to the complex\nexponential of the smallest angle between two lattice vectors, f=exp(\u00062ip=z). For hyper-\ncubic lattices in arbitrary dimensions, however, we must consider a priori all the zpossible\nroots of unity, fj\u0011exp(2ipj=z),j=0;:::; z\u00001.\nA direct calculation of (2.19) shows that the velocity autocorrelation takes the form\nhv0\u0001vni=\u0010\n1\u0001\u0001\u00011\u0011\"\nz\u00001\nå\nj=0ajK(fj)n\u00001diag(1;fj;:::;fz\u00001\nj)#0\nB@p1\n...\npz1\nCA; (2.21)\nwhere diag (1;fj;:::;fz\u00001\nj)denotes the matrix with elements listed on the main diagonal and\n0 elsewhere. For the three-dimensional cubic lattice, the coefficients ajare found to be\na0=a2=a4=0;\na1=a3=a5=2;(2.22)\nwhich compares to a1=a3=2 and a0=a2=0 in the case of the two-dimensional square\nlattice [2]. In the case of a d-dimensional hyper-cubic lattice, this generalises to\na2j=0;j=0;:::; d\u00001;\na2j+1=2;j=0;:::; d\u00001;(2.23)\nThe diffusion coefficient of a two-step memory persistent random walk on a d-\ndimensional hyper-cubic lattice is thus\nD2SMA\nDNMA=1+4\u0010\n1\u0001\u0001\u00011\u0011(\nd\nå\nj=1[Iz\u0000K(f2j\u00001)]\u00001diag(1;f2j\u00001;:::;fz\u00001\n2j\u00001))0\nB@p1\n...\npz1\nCA;\n(2.24)\nwhere Izdenotes the z\u0002zidentity matrix.\n3. Three-dimensional periodic Lorentz gas\nEquations (2.8), (2.16) and (2.24) can be put to the test to probe the diffusive regimes of\nperiodic Lorentz gases. The diffusive motion of the tracers results from the chaotic nature\nof the microscopic dynamics and the fast decay of correlations, which are in turn due to\nthe convex nature of the obstacles. Taking into consideration the different diffusive regimes\nof these models, which, as we argued earlier, depend on the nature of their horizon, we\n‡ This expression differs from that given in [2] due to a typographical error in that paper – they are really the same.Diffusive properties of persistent walks on cubic lattices 8\ninvestigate how the microscopic dynamical properties of the system determine the diffusion\ncoefficient.\nMachta and Zwanzig [20] addressed this issue in a particular limiting case, showing that,\nin the limit where the obstacles are so close together that a tracer will remain localised on\nekach lattice site for a very long time (compared to the mean time separating two collision\nevents), the process of diffusion on the Lorentz gas is well approximated by the dimensional\nprediction (2.8), where the lattice spacing `is the distance separating two neighbouring\nobstacles and tis the trapping time, which can be computed in terms of the geometry of\nthe billiard as a simple consequence of ergodicity. That is to say, when the geometry of the\nbilliard is such that two neighbouring disks nearly touch, the Lorentz gas is well approximated\nby a Bernoulli process, modeling the random hopping of tracers from cell to cell, with time-\nand length-scales specified according to the geometry of the billiard.\nDifferent approximation schemes have been proposed to go beyond this zeroth-order\napproximation and account for corrections to it [21, 22]; see, in particular, reference [23] for\nan overview. A consistent approach to understanding the effect of these corrections in two-\ndimensional diffusive billiards was described in [12]. The idea is to approximate the hopping\nprocess of tracer particles by persistent random walks with finite memory, and thus estimate\nthe diffusion coefficient of the billiard by the two-dimensional lattice equivalents of the one-\nor two-step formulas (2.16) and (2.24).\nWe discuss below the transposition of these results to the diffusive regimes of the three-\ndimensional periodic Lorentz gas.\n3.1. Geometry of simple three-dimensional periodic Lorentz gas model\nWe begin with a detailed description of the geometry and the different horizon regimes of the\nsystem studied in reference [19]; additional details are given in reference [24].\nThe model consists of a three-dimensional (3D) periodic Lorentz gas constructed out of\ncubic unit cells of side length `, having eight “outer” spheres of radius rout`at its corners\nand a single “inner” sphere of radius rin`at its centre – see figure 2. The infinitely-extended\nperiodic structure formed in this way is symmetric under interchange of rinandrout; without\nloss of generality, we take rout\u0015rin.\nFigure 2. Geometry of the obstacles in a single cell of the 3D periodic Lorentz gas model for\nrout=0:45`andrin=0:30`, in the cylindrical-horizon regime.\nThis model seems to be the simplest one which allows a finite horizon, although this\nis possible only when the spheres are permitted to overlap. It is known that finite-horizonDiffusive properties of persistent walks on cubic lattices 9\nperiodic Lorentz gases with non-overlapping spheres in fact exist in any dimension [25], but\nwe are not aware of any explicit constructions of such models, even in the case of three\ndimensions.\nLattice of outer spheres The spheres of radius rout`form a simple cubic lattice. This lattice\nhas the following properties:\n\u000fWhen rout<1=2, the spheres are disjoint. In this case, there are free planes [25]\nin the structure, that is, infinite planes which do not intersect any of the spheres, in\nparticular there are free planes centered on the faces of the unit cell. In this case, we\nsay that there is a planar horizon (PH). When routis small, there are additional planes at\ndifferent diagonal angles, analogously to the two-dimensional infinite-horizon Lorentz\ngas [15, 16, 17].\n\u000fWhen rout>1=2, the spheres overlap, thereby automatically blocking all planes. The\noverlaps (intersections) of the spheres partially cover the faces of the cubes, leaving a\nspace in between which acts as an exit towards the adjacent cell.\n\u000fWhen rout\u00151=p\n2, the overlaps completely cover the faces of the unit cell, so that it is\nno longer possible to exit the cell.\n\u000fWhen rout\u0015p\n3=2, all of space is covered, and it is no longer possible to define a billiard\ndynamics.\nConditions for normal diffusion: cylindrical horizon As shown in reference [19], the\nnecessary and sufficient condition to have normal diffusion is that all free planes are blocked;\nif there are free planes, then the diffusion is weakly anomalous. The conditions to block all\nplanes are as follows.\n\u000fAll free planes are automatically blocked for rout\u00151=2, when the rout-spheres overlap.\n\u000fIf the rout-spheres do not overlap, then it is necessary to introduce the rin-sphere to block\nplanes which are parallel to the faces of the unit cell. For this blocking to occur, we need\nrin\u00151=2\u0000rout.\n\u000fFurthermore, we must also block diagonal planes at 45 degree angles, which requires\nthatrout\u00151=(2p\n2)orrin\u00151=(2p\n2).\nIf all of these conditions are satisfied, then we no longer have free planes, but may have free\ncylinders (“cylindrical gaps”) in the structure; we then say that there is a cylindrical horizon\n(CH).\nConditions for finite horizon Stronger statistical properties – e.g. faster decay of correlations\n– may be expected when there is a finite horizon [18, 19], i.e. where the length of free paths\nbetween collisions with obstacles is bounded above. To obtain this, not only all planar gaps,\nbut also all cylindrical gaps must be blocked, i.e. all holes viewed from any direction must be\nblocked. To do so, the following conditions must be fulfilled:\n\u000fTherout-spheres must overlap, rout\u00151=2. Furthermore, the projection of the rin-\nsphere on each face of the unit cell must cover the available exit space, as illustrated\nin figure 3(a). Letting dbe the maximum width of overlap of the resulting discs of radius\nrouton a face of the unit cell, we have d2=r2\nout\u00001=4, and we need rin\u00151=2\u0000dto\nblock the space.Diffusive properties of persistent walks on cubic lattices 10\nρin=1\n2−dd\nρout\n(a) Face of unit cell\nd\nrin (b) Mid-plane of unit cell\nFigure 3. Geometry of the 3D periodic Lorentz gas. (a) Cross-section of the unit cell in one\nof its faces. The overlapping outer spheres of radius rout>1=2, give rise to four overlapping\ndiscs (shown in green); the maximum width of their overlap is denoted d. The central disc\n(red) shows the minimum radius rin=1=2\u0000dof the central sphere such that its projection\ncovers the gap between the rout-discs on the face. (b) Geometry of the mid-plane of a unit cell\nfor parameters giving a finite horizon. The outer discs are cross-sections of the overlaps of the\nouter rout-spheres, and have radius dequal to the overlap parameter in (a). The inner disc is\nthe cross-section of the inner rin-sphere.\nFigure 4. Finite horizon can be achieved in a three-dimensional lattice, provided the spheres\nare allowed to overlap. Here the available space for diffusing particles is shown for parameter\nvalues rout=0:65`andrin=0:15`(in the FH1 region) in an unfolded channel.\n\u000fWe must block cylindrical corridors which cross the structure at a 45 degree angle at\nthe level of the mid-plane of a unit cell, which corresponds to the planar cross-section\nwith most available space in the unit cell. The mid-plane has the geometry shown in\nfigure 3(b), with four outer discs of radius d, and a central disc of radius rin; these discs\nare the intersection of the rout-overlaps and of the rin-sphere, respectively, with the mid-\nplane. Free diagonal trajectories in this plane at an angle of 45 degrees give rise to small\ncylindrical corridors. These will be blocked if there is no free line in the mid-plane. This\nblocking occurs provided either d\u00151=(2p\n2), i.e.rout\u0015p\n3=(2p\n2, or if rin\u00151=(2p\n2),Diffusive properties of persistent walks on cubic lattices 11\nthus giving rise to two distinct finite horizon regimes (FH1 and FH2), which are in fact\ndisjoint.\nFigure 4 depicts the space available for tracer particles in a channel of three consecutive cells\nfor a particular finite-horizon case.\nLocalisation of trajectories Having fixed rout, it is also necessary to calculate the value of\nrinabove which the trajectories become localised (L) between neighbouring spheres, and\nare thus no longer able to diffuse. For rout<1=p\n2, when there are still exits available\non the faces of the cubic unit cell, this happens exactly when the discs in the mid-plane\ntouch, i.e. when rin+d=1=p\n2, so that the condition for localised trajectories becomes [19]\nrin\u00151=p\n2\u0000p\nr2out\u00001=4.\nCondition to fill space Finally, we calculate when the spheres fill all space (denoted U, for\nundefined):\n\u000fWhen rout<1=p\n2, this occurs when the rin-spheres are large enough that their\nintersection with each face of the cube, which is a disc of radiusq\nr2\nin\u00001=4, covers\nthe exit on a face left open by the rout-spheres. This gives the condition r2\nin\u0015r2\nout+\n1=4\u0000p\nr2out\u00001=4.\n\u000fWhen rout>1=p\n2, the condition is that rinbe large enough to cover the space left by\ntherout-spheres inside the unit cell. The condition can again be found by looking at\nthe mid-plane, where there is most available space: the disc of radius rinmust cover\nthe space left by the discs of radius d(which are cross-sections of the overlaps of the\nrout-spheres). This occurs when rin\u00151=2\u0000p\nr2out\u00001=2.\nParameter space The complete parameter space of this model is shown in figure 5,\nexhibiting the regions in parameter space corresponding to the regimes of qualitatively\ndifferent behaviour discussed above§. Note that if rin>p\n3=2\u0000rout, then the rout- and rin-\nspheres overlap, and if rin>0:5 then neighbouring rin-spheres also overlap. These conditions\nare marked by the dotted lines in the figure.\n4. Persistence in the diffusive regimes of the three-dimensional Lorentz gas\nIn this section, we study the dependence of the diffusion coefficient on the geometrical\nparameters of the 3D periodic Lorentz gas model in the finite- (FH1) and cylindrical-horizon\n(CH) regimes, comparing the numerical results with the finite-memory approximations (2.8),\n(2.16) and (2.24).\n4.1. Approximation by the NMA process\nThe computation of the dimensional formula (2.8) relies on that of the residence time t. An\nexact formula is available for this quantity [27]:\nt=jQj\nj¶QjjS2j\njB2j; (4.1)\n§ A similar diagram of parameter space for a two-dimensional version of the model was given in reference [26].\nHowever, the symmetry between routandrinwas overlooked there; see also reference [19].Diffusive properties of persistent walks on cubic lattices 12\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8\nrout0.00.10.20.30.40.50.60.70.8rin\nPHCH\nFH1FH2LU\nFigure 5. Parameter space of the three-dimensional periodic Lorentz gas as a function of\nthe geometrical parameters routandrin. Solid lines divide regimes of qualitatively different\nbehaviour, which are also shaded with different colours and labelled as follows: PH: planar\nhorizon; CH: cylindrical horizon; FH1 and FH2: finite horizon; L: localised, non-diffusive\nmotion; U: undefined (all space filled). Note that the FH regime is divided into two disjoint\nregions. The dashed lines mark the different conditions referred to in the text. The diagonal\ndotted line separates regions where the rin-spheres do (above) and do not (below) overlap the\nrout-spheres. The diagram is reflection-symmetric in the line rin=rout, but for clarity only\nthe lower half is shown.\nwherejQjdenotes the volume of the billiard domain outside the obstacles, j¶Qjthe surface\narea of the available gaps separating neighbouring cells, jS2j=4pthe surface area of the unit\nsphere in three dimensions, and jB2j=pthe volume (area) of the unit disk in two dimensions,\nand we assume unit velocity. The explicit formulas giving the values of jQjandj¶Qjare rather\nlengthy and will not be given here; see reference [28].\nThe validity of equation (4.1) can be tested by comparison with numerical computation\nof the residence time, as shown in figure 6. Here, and in the remainder of the paper, we\nrestrict attention to values of routclose to the limiting value 1 =p\n2 and rinclose to 0, so that\nthe geometry is that of a single, cubic unit cell.\n4.2. Approximation by the 1SMA and 2SMA processes\nSingle- and two-step memory processes can be derived as approximations, at the lattice level,\nto the dynamics of the Lorentz gas. This is done by computing numerically the statistics\nof tracer particles as they jump from cell to cell, so as to estimate the single- and two-step\nmemory probability transitions.\nThe results are shown in figure 7 for the single-step memory process, where the six\ntransition probabilities Pi,i=0;:::; 5, are displayed as functions of the outer radius routforDiffusive properties of persistent walks on cubic lattices 13\nááááááááááááááááááááááááááááááááááááááááááóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóárin=0.\nórin=0.15\n0.000.050.100.150.200.00.20.40.60.8\nd=1 2-routt-1\nFigure 6. Residence time t, equation (4.1), compared to direct numerical simulations. The\nresults are shown for two values of the inner radius, rin=0 and rin=0:15, as functions of\nd\u00111=p\n2\u0000rout, which is the characteristic size of the gaps separating neighbouring cells.\nThe curves are very similar since the volume of the inner sphere remains small. In this and the\nfollowing results we take `=1.\ndifferent values of the inner radius rin.\nFor the two-step process, the computation of the transition probabilities Pi;jis shown in\nfigure 8 for rin=0, that is in the absence of a sphere at the center of the cell. The six different\npanels each correspond to a given i=0;:::; 5. The same is shown in figure 9 for rin=0:15.\náááááááááááááááááááááááááááááááááááááááááá óóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóó õõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ\nàààààààààààààààààààààààààààààààààààààààààà òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.\n0.000.050.100.150.200.00.10.20.30.4\nd=1 2-routPk\n(a)\náááááááááááááááááááááááááááááááááááááááááá óóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóó õõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ\nàààààààààààààààààààààààààààààààààààààààààà òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.03\n0.000.050.100.150.200.00.10.20.30.4\nd=1 2-routPk (b)\náááááááááááááááááááááááááááááááááááááááááá óóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóó õõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ\nàààààààààààààààààààààààààààààààààààààààààà òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.06\n0.000.050.100.150.200.00.10.20.30.4\nd=1 2-routPk (c)\náááááááááááááááááááááááááááááááááááááááááá óóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóó õõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ\nàààààààààààààààààààààààààààààààààààààààààà òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò ôôôôôôôôôôôôôôôôô��ôôôôôôôôôôôôôôôôôôôôôôôôrin=0.09\n0.000.050.100.150.200.00.10.20.30.4\nd=1 2-routPk\n(d)\nááááááááááááááááááááááááááááááááááááááááááóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ\nààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.12\n0.000.050.100.150.200.00.10.20.30.4\nd=1 2-routPk (e)\nááááááááááááááááááááááááááááááááááááááááááóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ\nààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.15\n0.000.050.100.150.200.00.10.20.30.4\nd=1 2-routPk (f)\nFigure 7. Numerical computations of the probabilities P0;:::; P5of the single step memory\nprocess, appearing in (2.13). The six panels shown correspond to as many different values of\nrin, where the probabilities are shown as functions of d. The dashed line at Pk=1=6 indicates\nthe value for a memoryless (NMA) walk. Here and in figures 8 and 9, the conventions are\nas follows: Empty squares (blue), P0; empty upward triangles (cyan), P1; empty downward\ntriangles (green), P2; filled squares (red) P3; filled upward triangles (magenta), P4; filled\ndownward triangles (brown), P5. In all cases we verify the symmetry P1=P2=P4=P5,\nwhich also remain close to 1 =6.Diffusive properties of persistent walks on cubic lattices 14\náááááááááááááááááááááááááááááááááááááááááá\nóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ\nààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.\n0.000.050.100.150.200.00.10.20.30.4\nd=1 2-routP0,j\n(a)\nááááááááááááááááááááááááááááááááááááááááááóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ\nààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.\n0.000.050.100.150.200.00.10.20.30.4\nd=1 2-routP1,j (b)\náááááááááááááááááááááááááááááááááááááááááá\nóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ\nààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.\n0.000.050.100.150.200.00.10.20.30.4\nd=1 2-routP2,j (c)\náááááááááááááááááááááááááááááááááááááááááá óóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóó õõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ\nàààààààààààààààààààààààààààààààààààààààààà òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.\n0.000.050.100.150.200.00.10.20.30.4\nd=1 2-routP3,j\n(d)\nááááááááááááááááááááááááááááááááááááááááááóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ\nààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.\n0.000.050.100.150.200.00.10.20.30.4\nd=1 2-routP4,j (e)\nááááááááááááááááááááááááááááááááááááááááááóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ\nààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.\n0.000.050.100.150.200.00.10.20.30.4\nd=1 2-routP5,j (f)\nFigure 8. Numerical computations of the 36 probabilities Pi;jwhich appear in (2.20),\ncorresponding to a cell with no sphere at its center, i.e. the inner radius rin=0. The symmetries\nof the process are reflected by the similarities between figures 8(b), 8(c), 8(e) and 8(f). The\ncolour coding is similar to figure 7.\náááááááááááááááááááááááááááááááááááááááááá óóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóó õõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ\nàààààààààààààààààààààààààààààààààààààààààà òòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.15\n0.000.050.100.150.200.00.10.20.30.4\nd=1 2-routP0,j\n(a)\náááááááááááááááááááááááááááááááááááááááááá óóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ\nààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.15\n0.000.050.100.150.200.00.10.20.30.4\nd=1 2-routP1,j (b)\nááááááááááááááááááááááááááááááááááááááááááóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóó õõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ\nààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.15\n0.000.050.100.150.200.00.10.20.30.4\nd=1 2-routP2,j (c)\nááááááááááááááááááááááááááááááááááááááááááóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ\nààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.15\n0.000.050.100.150.200.00.10.20.30.4\nd=1 2-routP3,j\n(d)\náááááááááááááááááááááááááááááááááááááááááá óóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ\nààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.15\n0.000.050.100.150.200.00.10.20.30.4\nd=1 2-routP4,j (e)\nááááááááááááááááááááááááááááááááááááááááááóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóóó õõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõõ\nààààààààààààààààààààààààààààààààààààààààààòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòòò ôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôôrin=0.15\n0.000.050.100.150.200.00.10.20.30.4\nd=1 2-routP5,j (f)\nFigure 9. Numerical computations of the probabilities 36 Pi;jwhich appear in (2.20),\ncorresponding to inner radius rin=0:15. Here again the symmetries of the process are\nreflected by the similarities between figures 9(b), 9(c), 9(e) and 9(f).Diffusive properties of persistent walks on cubic lattices 15\n4.3. Diffusion coefficient of the billiard\nrin=0.\n0.000.050.100.150.201.01.21.41.61.8\nd=1 2-rout6Dtl-2\n(a)\nrin=0.03\n0.000.050.100.150.201.01.21.41.61.8\nd=1 2-rout6Dtl-2 (b)\n\nrin=0.06\n0.000.050.100.150.201.01.21.41.61.8\nd=1 2-rout6Dtl-2\n(c)\nrin=0.09\n0.000.050.100.150.201.01.21.41.61.8\nd=1 2-rout6Dtl-2 (d)\nrin=0.12\n0.000.050.100.150.201.01.21.41.61.8\nd=1 2-rout6Dtl-2\n(e)\nrin=0.15\n0.000.050.100.150.201.01.21.41.61.8\nd=1 2-rout6Dtl-2 (f)\nFigure 10. Diffusion coefficient normalised with respect to the dimensional prediction DNMA,\nequation (2.8), vs. gap size d=1=p\n2\u0000rout, plotted for different values of the inner radius\nrin=0;:::; 0:15. The symbols (black) correspond to direct numerical computation of this\nquantity, the long dashed (green) lines to the single-step memory diffusion coefficient (2.16),\nand the solid (red) lines to the two-step memory diffusion coefficient (2.24). The vertical\ndotted lines indicate the separation between the finite and infinite horizon regimes.\nHaving computed the probability transitions associated to the single and two-step\nmemory processes, we can compute the invariant distribution Pand substitute the results into\nequations (2.16) and (2.24) to obtain values of the diffusion coefficients. These are compared\nto the diffusion coefficient of the billiard calculated from direct simulations in figure 10.\nWe can draw several conclusions from the results shown in figure 10. Firstly, we remark\nthat in the 3D model studied here there is relatively little back-scattering, i.e. motion inDiffusive properties of persistent walks on cubic lattices 16\nwhich the particle reverses its direction between arriving and leaving a given cell. This gives\nan important contribution to the diffusion coefficient, and, in particular, corresponds to the\nfact that here we find that the diffusion coefficient is larger than the memoryless (NMA)\napproximation, while in reference [12] the diffusion coefficient tended to lie below the results\nof this approximation. Note, however, that this effect depends strongly on the particular model\nused.\nIt is also interesting to note that in the finite-horizon regime, i.e. left of the dotted vertical\nlines in figures 10(b)-10(f), approximating the diffusion coefficient by the one-step memory\nprocess (2.16) is just as good as the two-step process (2.24). In the cylindrical-horizon regime,\nhowever, the two results are different; the single-step approximation gets poorer as rout\ndecreases, whereas the two-step process yields more accurate estimates. This corresponds\nto the fact that correlations decay more slowly in the cylindrical-horizon regime [19], so that\nmemory effects persist for longer.\n5. Conclusions\nThe cyclic structures of certain regular lattices underly symmetries of their statistical\nproperties which can be exploited to greatly simplify their analysis. Examples are two-\ndimensional lattices such as the square, the honeycomb and the triangular lattice, which\nwere studied in reference [2]. Other examples include, in higher dimensions, the hypercubic\nlattices studied in this paper. Having exhibited the cyclic structures of these lattices, we were\nable to extend our previous results to hypercubic lattices with suitable adaptations, in order to\ncalculate the diffusion coefficients of persistent random walks with up to two steps of memory.\nOur method is especially useful to compute the correlations of persistent walks on such\nregular lattices. In particular, the velocity autocorrelations of a two-step persistent walk may\nbe recast in terms of matrix powers, which can then easily be resummed to obtain a readily-\ncomputable expression for the diffusion coefficient.\nAmong the many applications of persistent random walks, deterministic diffusive\nprocesses are ideal candidates to apply our method. The three-dimensional periodic Lorentz\ngas is particularly interesting as it exhibits two distinct types of diffusive regimes, one with\nfinite horizon, where memory effects decay fast, and another with cylindrical horizon, where\nmemory effects can remain important. In this latter case, the approximation of the diffusive\nprocess by a two-step memory walk proves much more accurate than the single-step process.\nWe remark that the application of our formalism to the diffusive properties of Lorentz\ngases relies on the numerical computation of the transition probabilities corresponding to\nthe persistent process with which we approximate the deterministic process. Since there\nare 30 transition probabilities for the two-step memory walk, their analytical calculation is\na daunting task. It relies on knowledge of the statistics of trapped trajectories and involves\ncontributions from different time scales. Nonetheless, this computation is formally possible,\nand is in principle much simpler than that of the actual diffusion coefficient.\nAcknowledgments\nThis research benefited from the joint support of FNRS (Belgium) and CONACYT (Mexico)\nthrough a bilateral collaboration project. The work of TG is financially supported by the\nBelgian Federal Government under the Inter-university Attraction Pole project NOSY P06/02.\nTG is financially supported by the Fonds de la Recherche Scientifique F.R.S.-FNRS. DPSDiffusive properties of persistent walks on cubic lattices 17\nacknowledges financial support from DGAPA-UNAM grant IN105209 and CONACYT grant\nCB101246.\nReferences\n[1] Haus J W and Kehr K W 1987 Diffusion in regular and disordered lattices Phys. Rep. 150263.\n[2] Gilbert T and Sanders D P 2010 Diffusion coefficients for multi-step persistent random walks on lattices J.\nPhys. A Math. Theor. 435001.\n[3] Geisel T and Nierwetberg J 1982 Onset of diffusion and universal scaling in chaotic systems Phys. Rev. Lett.\n487.\n[4] Fujisaka H and Grossmann S 1982 Chaos-induced diffusion in nonlinear discrete dynamics Z. Phys. B 48\n261.\n[5] Schell M, Fraser S and Kapral R 1983 Subharmonic bifurcation in the sine map: An infinite hierarchy of cusp\nbistabilities Phys. Rev. A 28373.\n[6] Grassberger P 1983 New mechanism for deterministic diffusion Phys. Rev. A 283666.\n[7] Gaspard P 1998 Chaos, Scattering and Statistical Mechanics (Cambridge: Cambridge University Press).\n[8] Cvitanovi ´c P and Artuso R 2010 Chapter “Deterministic diffusion” in Cvitanovi ´c P, Artuso R, Mainieri\nR, Tanner G, and Vattay G Chaos: Classical and Quantum ChaosBook.org/version13 (Copenhagen:\nNiels Bohr Institute)\n[9] Dorfman J R 1999 An Introduction to Chaos in Nonequilibrium Statistical Mechanics (Cambridge: Cambridge\nUniversity Press).\n[10] Klages R and Dorfman J R 1995 Simple maps with fractal diffusion coefficients Phys. Rev. Lett. 74387.\n[11] Klages R and Dorfman J R 1999 Simple deterministic dynamical systems with fractal diffusion coefficients\nPhys. Rev. E 595361.\n[12] Gilbert T and Sanders D P 2009 Persistence effects in deterministic diffusion Phys. Rev. E ,8041121.\n[13] Bunimovich L A and Sinai Ya G 1980 Markov partitions for dispersed billiards Commun. Math. Phys. 78\n247.\n[14] Bunimovich L A and Sinai Ya G 1981 Statistical properties of Lorentz gas with periodic configuration of\nscatterers Commun. Math. Phys. 78479.\n[15] Zacherl A, Geisel T, Nierwetberg J, and Radons G 1986 Power spectra for anomalous diffusion in the extended\nSinai billiard Phys. Lett. A 114317.\n[16] Bleher P M 1992 Statistical properties of two-dimensional periodic Lorentz gas with infinite horizon J. Stat.\nPhys. 66315.\n[17] Szasz D and Varj ´u T 2007 Limit laws and recurrence for the planar Lorentz process with infinite horizon J.\nStat. Phys. 12959 2007.\n[18] Chernov N 1994 Statistical properties of the periodic Lorentz gas. Multidimensional case J. Stat. Phys. 7411.\n[19] Sanders D P 2008 Normal diffusion in crystal structures and higher-dimensional billiard models with gaps\nPhys. Rev. E 78060101.\n[20] Machta J and Zwanzig R 1983 Diffusion in a periodic Lorentz gas Phys. Rev. Lett. 501959.\n[21] Klages R and Dellago C 2000 Density-dependent diffusion in the periodic Lorentz gas J. Stat. Phys. 101145.\n[22] Klages R and Korabel N 2002 Understanding deterministic diffusion by correlated random walks J. Phys. A\nMath. Gen. 354823.\n[23] Klages R 2007 Microscopic Chaos, Fractals and Transport in Nonequilibrium Statistical Mechanics\n(Singapore: World Scientific).\n[24] Sanders D P 2008 Deterministic Diffusion in Periodic Billiard Models (PhD thesis, University of Warwick,\n2005) arXiv preprint arXiv:0808.2252 .\n[25] Henk M and Zong C 2000 Segments in ball packings Mathematika ,4731.\n[26] Garrido P L 1997 Kolmogorov–Sinai entropy, Lyapunov exponents, and mean free time in billiard systems J.\nStat. Phys. 88807.\n[27] Chernov N 1997 Entropy, Lyapunov exponents, and mean free path for billiards J. Stat. Phys. ,881.\n[28] Nguyen H C 2010 Etude du comportement diffusif d’un billard chaotique `a trois dimensions et son\napproximation par une marche al ´eatoire persistante Th`ese de Master, Universit ´e Libre de Bruxelles ."    },    {        "title": "2106.09077v1.Sharp_upper_and_lower_bounds_of_the_attractor_dimension_for_3D_damped_Euler_Bardina_equations.pdf",        "content": "arXiv:2106.09077v1  [math.AP]  16 Jun 2021SHARP UPPER AND LOWER BOUNDS OF THE\nATTRACTOR DIMENSION FOR 3D DAMPED\nEULER–BARDINA EQUATIONS\nALEXEI ILYIN1, ANNA KOSTIANKO3,4, AND SERGEY ZELIK1,2,3\nAbstract. The dependence of the fractal dimension of global attrac-\ntors for the damped 3D Euler–Bardina equations on the regularizat ion\nparameterα>0 and Ekman damping coefficient γ >0 is studied. We\npresentexplicitupper boundsforthisdimensionforthecaseofthe whole\nspace, periodic boundary conditions, and the case of bounded dom ain\nwith Dirichlet boundary conditions. The sharpness of these estimat es\nwhenα→0 andγ→0 (which corresponds in the limit to the classi-\ncal Euler equations) is demonstrated on the 3D Kolmogorov flows on a\ntorus.\nContents\n1. Introduction 2\n2. A priori estimates, well-posedness and dissipativity 5\n3. Asymptotic compactness and attractors 8\n4. Upper bounds for the fractal dimension 12\n5. Sharp lower bound on T314\n5.1. Squire’s transformation 16\n5.2. Instability analysis on T217\n5.3. 3D lower bound 20\nAppendix A. Collective Sobolev inequalities for H1-orthonormal families 21\nA.1. The case of the whole space and a domain with Dirichlet BC 21\nA.2. The case of periodic BC: Estimates for the lattice sums 2 4\nAppendix B. A pointwise estimate for the nonlinear term 27\nReferences 28\n2000Mathematics Subject Classification. 35B40, 35B45, 35L70.\nKey words and phrases. Regularized Euler equations, Bardina model, unbounded do-\nmains, attractors, fractal dimension, Kolmogorov flows.\nThis work was supported by Moscow Center for Fundamental and A pplied Mathe-\nmatics, Agreement with the Ministry of Science and Higher Education of the Russian\nFederation, No. 075-15-2019-1623 and by the Russian Science Fo undation grant No.19-\n71-30004 (sections 2-4). The second author was partially suppor ted by the Leverhulme\ngrant No. RPG-2021-072 (United Kingdom).\n12 A. ILYIN, A. KOSTIANKO, AND S. ZELIK\n1.Introduction\nBeing the central mathematical model in hydrodynamics, the Navier-Stokes\nandEulerequationspermanentlyremaininthefocusofbotht heanalysisofPDEs\nand thetheory of infinite dimensional dynamical systems and their attractors, see\n[2,8,13,15,16,25,26,43,44,45]andthereferencestherei nformoredetails. Most\nstudied is the 2D case where reasonable results on the global well-posedness and\nregularity of solutions as well as the results on the existen ce of global attractors\nand their dimension are available. However, the global well -posedness in the 3D\ncase remains a mystery and even listed by the Clay institute o f mathematics as\none of the Millennium problems. This mystery inspires a comp rehensive study of\nvarious modifications/regularizations of the initial Navi er-Stokes/Euler equations\n(suchasLeray- αmodel,hyperviscousNavier-Stokes equations, regulariza tionsvia\np-Laplacian, etc.), many of which have a strong physical back ground and are of\nindependent interest, see e.g. [14, 19, 26, 34, 37] and the re ferences therein.\nIn the present paper we shall be dealing with the following re gularized damped\nEuler system:/braceleftbigg\n∂tu+(¯u,∇x)¯u+γu+∇xp=g,\ndiv¯u= 0, u(0) =u0.(1.1)\nwith forcing gand Ekman damping term γu,γ >0. The dampingterm γumakes\nthe system dissipative and is important in various geophysi cal models [39]. Here\nand below ¯uis a smoothed (filtered) vector field related with the initial velocity\nfielduas the solution of the Stokes problem\nu= ¯u−α∆x¯u+∇xq,div¯u= 0, (1.2)\nwhereα>0 is a given small parameter. In other words,\n¯u= (1−αA)−1u,\nwhereA:= Π∆ xis the Stokes operator and Π is the Helmholtz–Leray projecti on\nto divergent free vector fields in the corresponding domain.\nSystem (1.1), (1.2) (at least in the conservative case γ= 0) is often referred to\nas the simplified Bardina subgrid scale model of turbulence, see [4, 5, 24] for the\nderivation of the model and further discussion, so in this pa per we shall be calling\n(1.1) the damped Euler–Bardina equations. We also mention t hat rewriting (1.1)\nin terms of the variable ¯ ugives\n∂t¯u−α∂t∆x¯u+(¯u,∇x)¯u+γ¯u+∇xp=αγ∆x¯u+g (1.3)\nwhichisadampedversionoftheso-called Navier–Stokes–Vo ight equationsarising\nin the theory of viscoelastic fluids, see [23, 38] for the deta ils.\nOur main interest in the present paper is to study the dimensi on of global\nattractors for system (1.1) in 2D and 3D paying main attentio n to the most\ncomplicated 3D case. Note that, unlike the classical Euler e quations, Bardina-\nEuler equations can be interpreted as an ODE with bounded non lineariry in the\nproper Hilbert space, so no problems with well-posedness ar ise, see [5] and alsoDAMPED 3D EULER–BARDINA EQUATIONS 3\nsection §2 below, so the main aim of our study is to get as sharp as possib le\nbounds for the corresponding global attractors. Each case d= 2 andd= 3 in\nturn is studied in three different settings as far as the bounda ry conditions are\nconcerned. More precisely, the system is studied\n(1) on the torus Ω = Td= [0,2π]d. In this case the standard zero mean\ncondition is imposed on u, ¯uandg;\n(2) in the whole space Ω = Rd;\n(3) in a bounded domain Ω ⊂Rdwith Dirichlet boundary conditions for ¯ u.\nWe denote by Ws,p(Ω) the standard Sobolev space of distributions whose deriv a-\ntives up to order sbelong to the Lebesgue space Lp(Ω). In the Hilbert case p= 2\nwe will write Hs(Ω) instead of Ws,2(Ω). In order to work with velocity vector\nfields, we denote by Hs=Hs(Ω) the subspace of [ Hs(Ω)]dconsisting of diver-\ngence free vector fields. In the case of Ω ⊂Rdwe assume in addition that vector\nfields from Hssatisfy Dirichlet boundary conditions and in the case of per iodic\nboundary conditions Ω = Tdwe assume that these vector fields have zero mean.\nWe also recall that equation (1.1) possesses the standard en ergy identity\n1\n2d\ndt/parenleftig\n/ba∇dbl¯u/ba∇dbl2\nL2(Ω)+α/ba∇dbl∇x¯u/ba∇dbl2\nL2(Ω)/parenrightig\n+γ/parenleftig\n/ba∇dbl¯u/ba∇dbl2\nL2(Ω)+α/ba∇dbl∇x¯u/ba∇dbl2\nL2(Ω)/parenrightig\n= (g,¯u),\nwhere(u,v) isthestandardinnerproductin[ L2(Ω)]d. Forthisreason itisnatural\nto consider problem (1.1) in the phase space H1with norm\n/ba∇dbl¯u/ba∇dbl2\nα:=/ba∇dbl¯u/ba∇dbl2\nL2+α/ba∇dbl∇x¯u/ba∇dbl2\nL2.\nOurfirstmain result is thefollowing theorem which gives an e xplicit upperbound\nfor the fractal dimension of the attractor in the 3D case.\nTheorem 1.1. Letd= 3, letΩbe as described above, and let g∈[L2(Ω)]3(in the\nperiodic case we assume also that ghas zero mean). Then the solution semigroup\nS(t)associated with equation (1.1)possesses a global attractor A⋐H1with\nfinite fractal dimension satisfying the following inequali ty:\ndimFA≤1\n12π/ba∇dblg/ba∇dbl2\nL2\nα5/2γ4. (1.4)\nThe analogue of this estimate for the 2D case reads\ndimFA≤1\n16π/ba∇dblg/ba∇dbl2\nL2\nα2γ4(1.5)\nwith the following improvement for the case when Ω =T2orΩ =R2:\ndimFA≤1\n8π/ba∇dblcurlg/ba∇dbl2\nL2\nαγ4(1.6)\ndue to estimates related with the vorticity equation.4 A. ILYIN, A. KOSTIANKO, AND S. ZELIK\nSince the general case γ >0 is reduced to the particular one with γ= 1\nby scalingt→γ−1t,u→γ−2u,g→γ−2g, the most interesting in estimates\n(1.4), (1.5) and (1.6) is the dependence of the RHS on α. For the viscous case of\nequations (1.1)\n∂tu+(¯u,∇x)¯u+∇xq=ν∆xu+g\nthe following estimate is proved in [5]:\ndimFA≤C/ba∇dblg/ba∇dbl6/5\nL2\nν12/5α18/5\nfor the case Ω = T3. We see that even in the case ν= 1 this estimate gives\nessentially worse dependence on αthan our estimate (1.4). The upper bounds\nfor3DNavier-Stokes-Voight equation obtainedin[23]give evenworsedependence\non the parameter α(likeα−6). Estimates (1.5) and (1.6) have been proved for\nΩ =T2in a recent paper [22]. The sharpness of these estimates in th e limit\nasα→0 was also established there for the case of the 2D Kolmogorov flows.\nHowever, to the best of our knowledge, no lower bounds for the dimension of the\nattractor of the Euler–Bardina equations in 3D are availabl e in the literature.\nOur second main result covers this gap. Namely, we consider t he 3D Kol-\nmogorov flows on the torus Ω = T3for equations (1.1) generated by the family\nof the right-hand sides parameterized by an integer paramet ers∈N:\ng=gs=\n\ng1=γ2λ(s)sin(sx3),\ng2= 0,\ng3= 0,(1.7)\nwheres∼α−1/2andλ(s) is a specially chosen amplitude, see §5. Then, perform-\ning an accurate instability analysis for the linearization of equation (1.1) on the\ncorresponding Kolmogorov flow (in the spirit of [33], see als o [20, 21, 32]), we get\nthe following result.\nTheorem 1.2. LetΩ =T3and letγ >0, andα >0. Then in the limit\nα→0the integer parameter sand the amplitude λ(s)can be chosen so that\nthe corresponding forcing g=gsof the form (1.7)produces the global attractor\nA=As, whose dimension satisfies the following lower bound:\ndimFA≥c/ba∇dblg/ba∇dbl2\nL2\nα5/2γ4, (1.8)\nwherec>0is an absolute effectively computable constant.\nEstimate (1.8) shows that our upper bound(1.4) is optimal. A gain, to the best\nof our knowledge, this is the first optimal two-sided estimat e for the attractor\ndimension in a 3D hydrodynamical problem.\nIn this connection we recall thecelebrated upperboundin [1 1] for the attractor\ndimension of the classical Navier–Stokes system on the 2D to rus, which is stillDAMPED 3D EULER–BARDINA EQUATIONS 5\nlogarithmically larger than the corresponding lower bound in [32]. On the other\nhand,addingtothesystemanarbitraryfixeddampingmakes it possibletoobtain\nthe estimate for the attractor dimension that is optimal in t he vanishing viscosity\nlimit [21].\nWefinallyobservethattheobtainedlowerestimatesforthea ttractordimension\ngrow asα→0 in both 2D and 3D cases (and even are optimal for the case of\ntori), so one may expect that the limit attractor A0(which corresponds to the\ncase of non-modified damped Euler equation) is infinite dimen sional. Indeed,\nthe existence of the attractor A0in the proper phase space is well-known in 2D\nat least ifg∈W1,∞, see [9] and references therein and we expect that some\nweaker version of the limit attractor A0can be also constructed in 3D using the\ntrajectory approach, see [8], and the concept of dissipativ e solutions for 3D Euler\nintroduced by P. Lions, see [31]. However, the situation wit h the dimension is\nmuch more delicate since the obtained lower bounds for the in stability index on\nKolmogorov’s flows are optimal for intermediate values of αonly and do not\nprovide any reasonable bounds for the limit case α= 0. Thus, the question of\nfinite or infinite-dimensionality of the limit attractor rem ains completely open\neven in the 2D case.\nThe paper is organized as follows. The key estimates for the s olutions of\nproblem (1.1) are derived in §2. Global well-posedness and dissipativity are also\ndiscussed there. The existence of a global attractor Ais verified in §4. To make\nthe proof independent of the choice of a (bounded or unbounde d) domain Ω, we\nuse the so called energy method for establishing the asympto tic compactness of\nthe associated semigroup.\nThe upper bounds for its dimension are obtained in §5 via the volume con-\ntraction method [2, 10, 44]. The essential role in getting op timal bounds for the\nglobal Lyapunov exponents is played by the collective Sobol ev inequalities for\nH1-orthonormal families proved in Appendix A based on the idea s of [27]. Their\nrole is somewhat similar to the role of the Lieb–Thirring ine qualities [28, 29] in\nthe dimension estimates of the attractors of the classical N avier–Stokes equations\n[2, 44]. The corresponding inequality in the 2D case has also been used in [22].\nFinally, the sharplower boundsof the dimension for the case Ω =T3are obtained\nin§5 by adapting/extending the ideas of [22, 33] to the 3D case.\n2.A priori estimates, well-posedness and dissipativity\nWe start with the standard energy estimate, which looks the s ame in the 2D\nand 3D cases as well as for the three types of boundary conditi ons.\nProposition 2.1. Letube a sufficiently regular solution of equation (1.1). Then\nthe following dissipative energy estimate holds:\n/ba∇dbl¯u(t)/ba∇dbl2\nα≤ /ba∇dbl¯u(0)/ba∇dbl2\nαe−γt+1\nγ2/ba∇dblg/ba∇dbl2\nL2, (2.1)6 A. ILYIN, A. KOSTIANKO, AND S. ZELIK\nwhere\n/ba∇dbl¯u/ba∇dbl2\nα:=/ba∇dbl¯u/ba∇dbl2\nL2+α/ba∇dbl∇x¯u/ba∇dbl2\nL2. (2.2)\nProof.Indeed, multiplying equation (1.1) by ¯ u, integrating over Ω and using the\nrelation between uand ¯uas well as the standard fact that the inertial term\nvanishes after the integration, we arrive at\nd\ndt/parenleftbig\n/ba∇dbl¯u/ba∇dbl2\nL2+α/ba∇dbl∇x¯u/ba∇dbl2\nL2/parenrightbig\n+2γ/parenleftbig\n/ba∇dbl¯u/ba∇dbl2\nL2+α/ba∇dbl∇x¯u/ba∇dbl2\nL2/parenrightbig\n= 2(g,¯u)≤\n≤2/ba∇dblg/ba∇dblL2/ba∇dbl¯u/ba∇dblL2≤γ/ba∇dbl¯u/ba∇dbl2\nL2+1\nγ/ba∇dblg/ba∇dbl2\nL2.(2.3)\nApplying the Gronwall inequality, we get the desired estima te (2.1) and complete\nthe proof. /square\nThe next corollary is crucial for our upper bounds for the att ractor dimension.\nCorollary 2.2. Letube a sufficiently smooth solution of problem (1.1). Then\nthe following estimate holds:\nlimsup\nt→∞1\nt/integraldisplayt\n0/ba∇dbl∇xu(s)/ba∇dblL2ds≤1\nγ√\n2α/ba∇dblg/ba∇dblL2. (2.4)\nProof.Indeed, integrating estimate (2.3) over t, taking thelimit t→ ∞and using\nthe fact that /ba∇dblu(t)/ba∇dbl2\nαremains bounded (due to estimate (2.1), we arrive at\nlimsup\nt→∞1\nt/integraldisplayt\n0/ba∇dbl∇xu(s)/ba∇dbl2\nL2ds≤1\n2αγ2/ba∇dblg/ba∇dbl2\nL2.\nUsing after that the H¨ older inequality\n1\nt/integraldisplayt\n0/ba∇dbl∇xu(s)/ba∇dblL2ds≤/parenleftbigg1\nt/integraldisplayt\n0/ba∇dbl∇xu(s)/ba∇dbl2\nL2dx/parenrightbigg1/2\n,\nwe get the desired result and finish the proof of the corollary . /square\nWe now turn to the two dimensional case without boundary. In t his case,\nmore accurate estimates are available due to the possibilit y to use the vorticity\nequation. Indeed, applying curl to (1.1) and setting ω= curlu, we obtain the\nvorticity equation for ω:\n∂tω+(¯u,∇x)¯ω+γω= curlg, ω= (1−α∆x)¯ω. (2.5)\nThe estimates for the solution on the torus T2were derived in [22]. Although for\nR2they are formally the same, we reproduce them for the sake of c ompleteness.\nProposition 2.3. Letube a sufficiently smooth solution of (1.1), whereΩ =T2\norR2and letω:= curluand¯ω:= curl¯u. Then, the following dissipative estimate\nholds:\n/ba∇dbl¯ω(t)/ba∇dbl2\nα≤ /ba∇dbl¯ω(0)/ba∇dbl2\nαe−γt+1\nγ2/ba∇dblcurlg/ba∇dbl2\nL2. (2.6)DAMPED 3D EULER–BARDINA EQUATIONS 7\nProof.Taking the scalar product of equation (2.5) with ¯ ω, we see that the non-\nlinear term vanishes and using that\n(ω,¯ω) =/ba∇dbl¯ω/ba∇dbl2\nL2+α/ba∇dbl∇x¯ω/ba∇dbl2\nL2, (2.7)\nwe obtain\n1\n2d\ndt/parenleftbig\n/ba∇dbl¯ω/ba∇dbl2\nL2+α/ba∇dbl∇x¯ω/ba∇dbl2\nL2/parenrightbig\n+γ/parenleftbig\n/ba∇dbl¯ω/ba∇dbl2\nL2+α/ba∇dbl∇x¯ω/ba∇dbl2\nL2/parenrightbig\n= (curlg,¯ω)≤\n≤ /ba∇dblcurlg/ba∇dblL2/ba∇dbl¯ω/ba∇dblL2≤1\n2γ/ba∇dblcurlg/ba∇dbl2\nL2+γ\n2/ba∇dbl¯ω/ba∇dbl2\nL2.(2.8)\nThis gives the desired estimate (2.6) by the Gronwall inequa lity and finishes the\nproof of the proposition. /square\nAnalogously to Corollary 2.2, we get the following estimate .\nCorollary 2.4. LetΩ =T2orR2and letube a sufficiently smooth solution of\nproblem (1.1). Then the following estimate holds:\nlimsup\nt→∞1\nt/integraldisplay1\n0/ba∇dbl∇x¯u(s)/ba∇dblL2ds≤1\nγmin/braceleftbigg\n/ba∇dblcurlg/ba∇dblL2,/ba∇dblg/ba∇dblL2√\n2α/bracerightbigg\n.(2.9)\nIndeed, the second inequality was already proved in Corolla ry 2.2 and the first\none is an immediate corollary of (2.6) and the fact that\n/ba∇dbl∇¯u/ba∇dblL2=/ba∇dbl¯ω/ba∇dblL2.\nLet us conclude this section by discussing the well-posedne ss of problem (1.1)\nand justification of the estimates obtained above. We will co nsider below only\nthe 3D case (the 2D case is analogous and even slightly simple r).\nWe also note from the very beginning that equation (1.1) can b e rewritten\nin the form of an ODE in a Hilbert space with bounded nonlinear iry. Indeed,\napplying the Helmholtz–Leray projection Π to both sides of ( 1.1) together with\nthe operator\nAα:= (1−αA)−1,\nwhereA= Π∆xis the Stokes operator in Ω, we arrive at\n∂t¯u+γ¯u+B(¯u,¯u) =AαΠg,¯u/vextendsingle/vextendsingle\nt=0= ¯u0, (2.10)\nwhereB(¯u,¯v) :=AαΠ((¯u,∇x)¯v).\nIt is natural to consider this system in the phase space ¯ u∈H1(Ω) with norm\n(2.2). Then the nonlinear operator Bis bounded from H1toH3/2:\n/ba∇dblB(¯u,¯v)/ba∇dblH3/2≤Cα/ba∇dbl��u/ba∇dblα/ba∇dbl¯v/ba∇dblα, (2.11)\nwhereCαdependsonlyon α. Indeed,if ¯ u,¯v∈H1, thenbytheSobolevembedding\ntheorem ¯u,¯v∈L6(Ω) and (¯u,∇x)¯v∈L3/2(Ω) by H¨ older’s inequality. Together\nwith the (L3/2→W2,3/2)-boundedness of the operator (1 −αA)−1, we get that\nB(¯u,¯v)∈W2,3/2(Ω). Finally, the Sobolev embedding W2,3/2⊂H3/2proves\nestimate (2.11).8 A. ILYIN, A. KOSTIANKO, AND S. ZELIK\nThus,B(¯u,¯u) isaregularizing operatorin H1andequation (2.10)is anODE in\nH1with bounded nonlineariry. Therefore the local existence a nd uniqueness of a\nsolution as well as (an infinite)differentiability of the corr espondinglocal solution\nsemigroup are straightforward corollaries of the Banach co ntraction principle or\nthe implicit function theorem, see e.g. [18] for the details . Thus, to get the\nglobal well-posedness and dissipativity we only need to ver ify the proper a priori\nestimate. Since this has already been done in Proposition 2. 1, we have proved\nthe following theorem.\nTheorem 2.5. Let¯u0∈H1(Ω),g∈[L2(Ω)]d(in the case of periodic BC we\nalso assume that ghas zero mean). Then there exists a unique global solution\n¯u∈C([0,∞),H1)of problem (2.10)(which is simultaneously the unique solution\nof(1.1)). Moreover, the function\nt→ /ba∇dbl¯u(t)/ba∇dbl2\nL2+α/ba∇dbl∇x¯u(t)/ba∇dbl2\nL2\nis absolutely continuous and the following energy identity holds:\n1\n2d\ndt/parenleftbig\n/ba∇dbl¯u(t)/ba∇dbl2\nL2+α/ba∇dbl∇x¯u(t)/ba∇dbl2\nL2/parenrightbig\n+\n+γ/parenleftbig\n/ba∇dbl¯u(t)/ba∇dbl2\nL2+α/ba∇dbl∇x¯u(t)/ba∇dbl2\nL2/parenrightbig\n= (g,¯u).(2.12)\nIn particular, the dissipative estimate (2.1)holds for any solution uof classu∈\nC([0,∞),H1).\nCorollary 2.6. Let the assumptions of Theorem 2.5 holds. Then equation (2.10)\ngenerates a dissipative solution semigroup\nS(t)¯u0:= ¯u(t), t≥0 (2.13)\nin the phase space H1(Ω). Moreover, S(t)isC∞-differentiable for every fixed t.\nIndeed, the existence of the semigroup is an immediate corol lary of the well-\nposedness proved in the theorem and the differentiability fol lows from the ODE\nstructure of (2.10) and the fact that the map ¯ u→B(¯u,¯u) isC∞-smooth as a\nmap from H1toH1.\n3.Asymptotic compactness and attractors\nIn this section we construct a global attractor for the solut ion semigroup S(t)\ngenerated by problem (1.1). We start with recalling the defin ition of a weak and\nstrong global attractor, see [2, 8] for more details. We will mainly consider below\nthe most complicated case Ω = R3since in the case of a bounded domain the\nasymptotic compactness is an immediate corollary of thefac t thatB(¯u,¯u)∈H3/2\nifu∈H1, see Remark 3.5.\nDefinition 3.1. A set Aw⊂H1is a weak global attractor of the semigroup S(t)\nif\n1)Awis a compact set in H1with weak topology;DAMPED 3D EULER–BARDINA EQUATIONS 9\n2)Awis strictly invariant, i.e., S(t)Aw=Aw;\n3)Awattracts the images of all bounded sets in the weak topology o fH1, i.e.\nfor every bounded set B⊂H1and every neighbourhood O(Aw) of the attractor\nin the weak topology, there exists T=T(O,B) such that\nS(t)B⊂ O(Aw) for allt≥T.\nAnalogously, Asis a strong attractor if it is compact in the strong topology o f\nH1, is strictly invariant and attracts the images of bounded se ts in the strong\ntopology as well. Obviously\nAw=As\nif both attractors exist.\nWe will use the following criterion for verifying the existe nce of an attractor,\nsee [2, 44] for the details.\nProposition 3.2. Let the operators operators S(t)be continuous in the weak\ntopology for every fixed tand let the semigroup S(t)possess a bounded absorbing\nsetB. The latter means that for every bounded B⊂H1there exists T=T(B)\nsuch that\nS(t)B⊂ Bfor allt≥T.\nThen there exists a weak global attractor Awof the semigroup S(t)which is gen-\nerated by all complete (defined for all t∈R) bounded solutions of problem (2.10):\nAw=K/vextendsingle/vextendsingle\nt=0, (3.1)\nwhereK:={¯u∈Cb(R,H1),¯usolves(2.10)}.\nLet, in addition, S(t)be asymptotically compact on B. The latter means that\nfor every sequence ¯un\n0∈ Band every sequence tn→ ∞, the sequence\n{S(tn)¯un\n0}∞\nn=1\nis precompact in the strong topology of H1. Then Awis also a strong global\nattractor for the semigroup S(t).\nWe start with verifying the existence of a weak attractor.\nProposition 3.3. Let the assumptions of Theorem 2.5 hold. Then the solution\nsemigroupS(t)generated by equation (1.1)possesses a weak global attractor Aw\nin the phase space H1.\nProof.The existence of a bounded absorbing set Bis an immediate corollary of\nthe dissipative estimate (2.1). We may take\nB:={¯u∈H1,/ba∇dbl¯u/ba∇dbl2\nL2+α/ba∇dbl∇x¯u/ba∇dbl2\nL2≤2\nγ2/ba∇dblg/ba∇dbl2\nL2}.\nThus, we only need to check the weak continuity. Let ¯ un\n0∈ Bbe a sequence of the\ninitial data weakly converging to ¯ u0: ¯un\n0⇁¯u0inH1. Denote by ¯ un(t) :=S(t)¯un\n010 A. ILYIN, A. KOSTIANKO, AND S. ZELIK\nthe corresponding solutions. We need to check that for every fixedT, ��un(T)⇁\n¯u(T) inH1, where ¯u(t) :=S(T)¯u0.\nTo see this we recall that ¯ unis bounded uniformly with respect to nin\nL∞(0,T;H1) due to estimate (2.1). Moreover, from equation (2.10) we se e also\nthat∂t¯unis uniformly bounded in the same space. Thus, passing to a sub se-\nquence, if necessary, we may assume that ¯ un(t)⇁ v(t) for every t∈[0,T] and\n∂t¯un⇁∂tvinL2(0,T;H1)forsomefunction v(t)suchthatv,∂tv∈L∞(0,T;H1).\nSo, it remains to verify that v(t) =S(t)¯u0by passing to the limit in equations\n(2.10) for functions ¯ un.\nThis passing to the limit is obvious for linear terms, so we on ly need to prove\nthe convergence of the nonlinear term B(¯un,¯un). In turn, this is the same as to\nprove that, in the sense of distributions,\n(¯un,∇x)¯un= div(¯un⊗¯un)⇁div(v⊗v) = (v,∇x)v.\nThe last statement will be proved if we check that\n¯un⊗¯un⇁v⊗vinL2((0,T)×Ω). (3.2)\nTo verify (3.2), we recall that the sequence ¯ un⊗¯unis uniformly bounded in\nL2due to dissipative estimate (2.1) and the embedding H1((0,T)×Ω)⊂L4.\nMoreover, since the embedding H1((0,T)×R3)⊂L2((0,T);L2\nloc(Ω)) is compact,\nwe have the strong convergence ¯ un→vinL2((0,T);L2\nloc(Ω)) and, therefore, the\nconvergence ¯ un→valmost everywhere. Since the sequence ¯ un⊗¯unis uniformly\nbounded in L2((0,T)×Ω), we may assume without loss of generality that it\nis weakly convergent to some ψ∈L2((0,T)×Ω). Along with the established\nconvergence almost everywhere this implies that ψ=v⊗v, see e.g. [30], and\nproves (3.2).\nThus, we have proved that vsolves the equation (2.10) and by the uniqueness\nv(t) = ¯u(t). This finishes the proof of weak continuity of the operators S(t) and\nthe existence of a weak global attractor now follows from Pro position 3.2. The\ntheorem is proved. /square\nWe are now ready to verify the existence of a strong global att ractor.\nProposition 3.4. Let the assumptions of Theorem 2.5 hold. Then the solution\nsemigroupS(t)generated by equation (2.10)possesses a strong global attractor\nA=Asin the phase space H1.\nProof.According to Proposition 3.2, we only need to verify the asym ptotic com-\npactness of S(t) onB. We will use the so-called energy method for this purpose,\nsee [3, 36] for more details.\nLet{¯un\n0} ⊂ B, lettn→ ∞be arbitrary and let ¯ un(t) :=S(tn)¯un\n0. Define\nalso ¯vn(t) := ¯un(t+tn). Then these functions are defined on the time intervals\nt∈[−tn,∞) and, due to the existence of a weak global attractor, withou t loss of\ngenerality, we may assume that ¯ v(t)⇁¯u(t) inH1for allt∈Rto some completeDAMPED 3D EULER–BARDINA EQUATIONS 11\ntrajectory ¯u∈ K. In particular,\n¯vn(0) =S(tn)¯un\n0⇁¯u(0) (3.3)\nand we only need to check that this convergence is strong.\nIt is convenient to use the equivalent norm (2.2) in the space H1. Then, the\nstrong convergence in (3.3) will be proved if we verify that\n/ba∇dbl¯vn(0)/ba∇dbl2\nα→ /ba∇dbl¯u(0)/ba∇dbl2\nα. (3.4)\nTo see this we integrate the energy identity (2.12) for ¯ vn(t) in time and get\n/ba∇dbl¯vn(0)/ba∇dbl2\nα=/ba∇dbl¯un\n0/ba∇dbl2\nαe−2γtn+/integraldisplay0\n−tne2γs(g,¯vn(s))ds. (3.5)\nPassing to the limit n→ ∞in this relation and using the weak convergence of\n¯vnto ¯uand uniform boundedness of ¯ vnand the initial data ¯ un\n0, we conclude that\nlim\nn→∞/ba∇dbl¯vn(0)/ba∇dbl2\nα=/integraldisplay0\n−∞e2γs(g,¯u(s))ds. (3.6)\nOn the other hand, integrating the energy identity for the li mit solution ¯ uin\ntime, we arrive at\n/ba∇dbl¯u(0)/ba∇dbl2\nα=/integraldisplay0\n−∞e2γs(g,¯u(s))ds. (3.7)\nEqualities(3.6)and(3.7)imply(3.4), thereforetheconve rgencein(3.3)isactually\nstrong. Thus, the desired asymptotic compactness is proved and the proposition\nis also proved. /square\nRemark 3.5. Since the operator B(¯u,¯u) is regularizing, one can easily increase\nthe regularity of the global attractor Ausing the decomposition of the semigroup\ninto the decaying linear part and the regularizing nonlinea r part (see [17]):\nS(t) :=L(t)+K(t),\nwherev(t) =L(t)¯u0solves\n∂tv+γv= 0, v/vextendsingle/vextendsingle\nt=0= ¯u0\nandw(t) :=K(t)¯u0satisfies\n∂tw+γw+B(¯u,¯u) =AαΠg, w/vextendsingle/vextendsingle\nt=0= 0.\nCombining this decomposition with bootstrappingargument s, we may check that\ntheregularity oftheattractor Aisrestricted bytheregularity of gonlyanditwill\nbeC∞-smooth ifg∈H∞(R3). Moreover, using the proper weighted estimates,\nsee [35], we may get the estimates on the rate of decay for solu tions belonging to\nthe attractor as |x| → ∞in terms of the decay rate of gwhich clarify the reason\nwhyAis compact. However, all these estimates do not seem very hel pful for\nestimation of the attractor dimension (since they grow rapi dly with respect to\nγ,α→0) and therefore we will not go into more details here.12 A. ILYIN, A. KOSTIANKO, AND S. ZELIK\n4.Upper bounds for the fractal dimension\nIn this section we derive upper bounds for the fractal dimens ion of the at-\ntractor A. As usual for the Navier–Stokes type equations, these bound s will be\nobtained by means of the volume contraction method, see [2, 1 0, 44] and the\nreferences therein. On the analytical side, the Lieb–Thirr ing inequalities for L2-\northonormal families [28, 29] which are an indispensable to ol for the dimension\nestimates of the attractors for the Navier–Stokes equation s are replaced in our\ncase by the collective Sobolev inequalities for H1-orthonormal families and are\nproved in the Appendix A.\nFurthermore, since system (1.1) in the 2D case has already be en studied in [22]\n(for the case Ω = T2), we will concentrate here on the 3D case only.\nTheorem 4.1. Suppose that Ωis either the 3D torus T3, or a bounded domain\nΩ⊂R3(endowed with Dirichlet BC), or the whole space Ω =R3. Letg∈\n[L2(Ω)]d(in the case of T3we also assume that ghas zero mean). Then the\nglobal attractor Acorresponding to the regularized damped Euler system (1.1)\nhas finite fractal dimension satisfying the following estim ate:\ndimFA≤1\n12π/ba∇dblg/ba∇dbl2\nL2\nα5/2γ4. (4.1)\nProof.The solution semigroup S(t) :H1→H1is smooth with respect to the\ninitial data (see Corollary 2.6), so we only need to estimate then-traces for the\nlinearization of equation (2.10) over trajectories on the a ttractor. This lineariza-\ntion of (1.1) reads:/braceleftigg\n∂t¯θ=−γ¯θ−B(¯u(t),¯θ)−B(¯θ,¯u(t)) =:Lu(t)¯θ,\ndiv¯θ= 0,¯θ/vextendsingle/vextendsingle\nt=0=¯θ0∈H1(Ω),(4.2)\nwhereB(¯u,¯v) :=AαΠ((¯u,∇x)¯v). In order to utilize the well-known cancelation\nproperty\n((¯u,∇x)¯θ,¯θ)≡0\nfor the inertial term in the Navier-Stokes equations, it is n atural to endow the\nspaceH1with the scalar product\n(¯θ,¯ξ)α= (¯θ,¯ξ)+α(∇x¯θ,∇x¯ξ) = ((1−αA)¯θ,¯ξ) (4.3)\nassociated with the norm (2.2). Then, using that Π Aα=Aαand Π¯θ=¯θ, we get\nthe cancelation\n(B(¯u,¯θ),¯θ)α=/parenleftbig\nAαΠ(¯u,∇x)¯θ,(1−α∆x)¯θ/parenrightbig\n=\n=/parenleftbig\nAαΠ(¯u,∇x)¯θ,(1−αΠ∆x)¯θ/parenrightbig\n=/parenleftbig\nΠ(¯u,∇x)¯θ,¯θ/parenrightbig\n= ((¯u,∇x)¯θ,¯θ)≡0\nof the most singular term B(¯u,¯θ) and, therefore, only the more regular term\nB(¯θ,¯u) will impact the trace estimates.DAMPED 3D EULER–BARDINA EQUATIONS 13\nFollowing the general strategy, see e.g. [44], the n-dimensional volume contrac-\ntion factors ωn(A) (=the sums of the first nglobal Lyapunov exponents) which\ncontrol the dimension can be estimated from above by the foll owing numbers:\nq(n) := limsup\nt→∞sup\nu(t)∈Asup\n{¯θj}n\nj=11\nt/integraldisplayt\n0n/summationdisplay\nj=1(Lu(τ)¯θj,¯θj)αdτ,\nwhere the first (inner) supremum is taken over all orthonorma l families {¯θj}n\nj=1\nwith respect to the scalar product ( ·,·)αinH1:\n(¯θi,¯θj)α=δij,divθj= 0, (4.4)\nand the second (middle) supremum is over all trajectories u(t) on the attractor\nA. Then, using the cancellation mentioned above together wit h the pointwise\nestimate (B.1) proved in Appendix B, we get\nn/summationdisplay\nj=1(Lu(t)¯θj,¯θj)α=−n/summationdisplay\nj=1γ/ba∇dbl¯θj/ba∇dbl2\nα−n/summationdisplay\nj=1((¯θj,∇x)¯u,¯θj)≤\n≤ −γn+/radicalbigg\n2\n3/integraldisplay\nΩρ(x)|∇x¯u(t,x)|dx≤ −γn+/radicalbigg\n2\n3/ba∇dbl∇x¯u(t)/ba∇dblL2/ba∇dblρ/ba∇dblL2,(4.5)\nwhere\nρ(x) =n/summationdisplay\nj=1|¯θj(x)|2.\nWe now use estimate (A.8) from Appendix A\n/ba∇dblρ/ba∇dblL2≤1\n2√πn1/2\nα3/4(4.6)\nand obtainn/summationdisplay\nj=1(Lu(t)¯θj,¯θj)α≤ −γn+1√\n6πn1/2\nα3/4/ba∇dbl∇x¯u(t)/ba∇dblL2. (4.7)\nFinally, using (2.4), we arrive at\nq(n)≤ −γn+1\n2√\n3πn1/2\nα5/4/ba∇dblg/ba∇dblL2\nγ.\nIt only remains to recall that, according to the general theo ry,ωn(A)≤q(n) and\nany number n∗for whichωn∗(A)≤0 andωn(A)<0 forn > n∗is an upper\nbound both for the Hausdorff [2, 44] and the fractal [6, 7] dime nsion of the global\nattractor A. This gives the desired estimate\ndimFA≤1\n12π/ba∇dblg/ba∇dbl2\nL2\nα5/2γ4\nand finishes the proof of the theorem. /square14 A. ILYIN, A. KOSTIANKO, AND S. ZELIK\nRemark 4.2. Estimates (1.5) and (1.6) for T2(and the fact that it is sharp)\nwere proved in [22]. The upper bound for R2is exactly the same once we now\nknow (A.8) for R2. For a bounded domain we only need to replace in the proof\nin [22] the estimates of the solutions on the attractor by (2. 4). Alternatively, one\ncan go through the proof of Theorem 4.1 and replace the 3D cons tants by their\n2D counterparts accordingly.\n5.Sharp lower bound on T3\nThe aim of this section is to show that estimate (4.1) for syst em (1.1) on\nT3= [0,2π]3is sharp in the limit as α→0.\nWe consider a family of right-hand sides\ng=gs=\n\ng1=γλ(s)sinsx3,\ng2= 0,\ng3= 0,(5.1)\ndepending only on x3and parameterized by s∈N,s≫1. The amplitude\nfunctionλ(s) will be specified in the course of the proof. Corresponding t o the\nfamilygsis the family of stationary solutions of (1.1)\n/vector u0(x3) =\n\nu0(x3) =λ(s)sinsx3,\n0,\n0,(5.2)\nwithp= 0. In fact,\n¯/vector u0= (1−α∆x)−1/vector u0= (¯u0,0,0)T\nalso depends only on x3and therefore ( ¯/vector u0,∇x)¯/vector u0= 0.\nWe now consider system (1.1) linearized on the stationary so lution (5.2)\n\n\n∂tw+ ¯u0∂¯w\n∂x1+ ¯w3∂¯u0\n∂x3e1+γw+∇xq= 0,\ndivw= 0,(5.3)\nwheree1= (1,0,0)Tand ¯w= (1−α∆x)−1w. The standing assumption is/integraldisplay\nT3w(x,t)dx= 0. (5.4)\nWe shall look for the solution of the linear problem (5.3) in t he form\nw(x,t) =\nw1(x3)\nw2(x3)\nw3(x3)\nei(ax1+bx2−act), q(x,t) =q(x3)ei(ax1+bx2−act),(5.5)DAMPED 3D EULER–BARDINA EQUATIONS 15\nwherea,b∈Zso thatwandqare 2π-periodic in each xi. If such a solution\nof (5.3) is found, then substituting (5.5) into (5.3) and set tingt= 0 we see that\nw(x,0) =\nw1(x3)\nw2(x3)\nw3(x3)\nei(ax1+bx2)\nis a vector-valued eigenfunction of the stationary operato r\nL3(/vector u0)w= ¯u0∂¯w\n∂x3¯u+ ¯w3∂¯u0\n∂x3e1+γw+∇xq (5.6)\nandiacis the corresponding eigenvalue. If Re( iac)<0, then the corresponding\nmode is unstable.\nWe substitute (5.5) into (5.3) and obtain the system\n\n−γw1−ia(¯u0¯w1−cw1) =iaq+ ¯w3u′\n0\n−γw2−ia(¯u0¯w2−cw2) =ibq,\n−γw3−ia(¯u0¯w3−cw3) =q′,\niaw1+ibw2+w′\n3= 0,(5.7)\nwhere′=∂/∂x3.\nLemma 5.1. There are no unstable solutions of equation (5.3)\n∂tw=L3(/vector u0)w\nthat can be written in the form (5.5)witha= 0.\nProof.Leta= 0. Then the solutions of (5.3) are sought in the form\nw(x,t) =\nw1(x3)\nw2(x3)\nw3(x3)\nei(bx2−ct), q(x) =q(x3)ei(bx2−ct),\nand (5.7) goes over to\n\n−γw1+icw1= ¯w3u′\n0\n−γw2+icw2=ibq,\n−γw3+icw3=q′,\nibw2+w′\n3= 0.\nLetb/\\e}atio\\slash= 0. Then w2=−w′\n3/(ib). Substituting this into the second equation and\ndifferentiating the third with respect to x3we obtain\nq′′=b2q,\nwhich gives that q= 0, since qis periodic. Since we are looking for unstable\nsolutions, it follows that Re( ic)<0 and therefore −γ+ic/\\e}atio\\slash= 0. This gives that\nw2=w3= 0, and, finally, w1= 0.\nIfa=b= 0, thenw′\n3= 0, andw3= 0 by periodicity and zero mean condition.\nThis givesq= 0 andw1=w2= 0. The proof is complete. /square16 A. ILYIN, A. KOSTIANKO, AND S. ZELIK\n5.1.Squire’s transformation. We now reduce the 3D instability analysis to\nthe instability analysis of the transformed 2D problem. The key role is played by\nthe Squire’s transformation (see [41], [12], [33]).\nSince we a looking for unstable solutions of (5.3), in view of Lemma 5.1 we\nmay assume that a/\\e}atio\\slash= 0 in (5.7). Multiplying the first equation in (5.7) by aand\nthe second by ba adding up the results, we obtain\n\n−/hatwideγ/hatwidew1−i/hatwidea(¯u0¯/hatwidew1−/hatwidec/hatwidew1) =i/hatwidea/hatwideq+¯/hatwidew3u′\n0,\n−/hatwideγ/hatwidew3−i/hatwidea(¯u0¯/hatwidew3−/hatwidec/hatwidew3) =/hatwideq′,\ni/hatwidea/hatwidew1+/hatwidew′\n3= 0,(5.8)\nwhere\n/hatwidea2=a2+b2,/hatwidew1=aw1+bw2\n/hatwidea,/hatwidew3=w3,\n/hatwideγ=γ/hatwidea\na,/hatwideq=q/hatwidea\na,/hatwidec=c.(5.9)\nThe solutions of this problem on the 2d torus\nT2\n|/hatwidea|=x1∈[0,2π/|/hatwidea|], x3∈[0,2π]\nare sought in the form\n/hatwidew(x1,x3,t) =/parenleftigg\n/hatwidew1(x3)\n/hatwidew3(x3)/parenrightigg\nei(/hatwideax1−/hatwidea/hatwidect),/hatwideq(x1,x3,t) =q(x3)ei(/hatwideax1−/hatwidea/hatwidect),(5.10)\nand if such a solution is found, then the vector function\n/hatwidew(x1,x3,0) =/parenleftigg\n/hatwidew1(x3)\n/hatwidew3(x3)/parenrightigg\nei/hatwideax1(5.11)\nis a vector-valued eigenfunction with eigenvalue i/hatwidea/hatwidecof the stationary operator\nL2(/vector u0)/hatwidew=/hatwideγ/hatwidew+ ¯u0∂/hatwide¯w\n∂x1+/hatwide¯w3∂¯u0\n∂x3e1+∇x/hatwideq,div/hatwidew= 0,(5.12)\nonT2\n|/hatwidea|, where the stationary solution and the generating right-ha nd side are\n/vector u0(x3) =/braceleftigg\nu0(x3) =λ(s)sinsx3,\n0,gs(x3) =/braceleftigg\ng1(x3) =λ(s)/hatwideγsinsx3,\n0,(5.13)\nand where as before ¯ u0= ¯u0(x3) = (1−∆x)−1u0.\nTo avoid unnecessary complications we assume in what follow s that/radicalbig\na2+b2=/hatwidea>0, a>0,\nand formulate the main result on the Squire’s reduction of th e 3D instability\nanalysis to the 2D case.DAMPED 3D EULER–BARDINA EQUATIONS 17\nLemma 5.2. Let/hatwidewin(5.11)be an unstable eigenfunction of the operator (5.12)\non the torus T2\n/hatwidea= [0,2π//hatwidea]×[0,2π]. Then for any pair of integers a,b∈Zwith\na2+b2=/hatwidea2\nthere exist an unstable solution of system (5.7)onT3= [0,2π]3.\nProof.Once the/hatwide·-variables are known, q,w3,candγare found from (5.9). It\nremains to find w1andw2. We consider the second equation in (5.7):\nAw2:= (−γ+iac)w2−ia¯u0¯w2=ibq.\nSince/hatwidewis unstable, Re( iac)<0 and therefore Re( −γ+iac)<0. Suppose that\nAw2= 0 for some w2. Taking the scalar product in (complex) L2(0,2π) withw2\nand taking into account that the second term is purely imagin ary we obtain for\nthe real part\nRe(−γ+iac)/ba∇dblw2/ba∇dbl2\nL2= 0,\nwhich gives that w2= 0, andAhas a trivial kernel. In addition, Ais a Fredholm\noperator, since the second term is compact (smoothing). Hen ce it has a bounded\ninverse, and since qis known, we have found w2. Finally,\nw1= (/hatwidea/hatwidew1−bw2)/a.\n/square\n5.2.Instability analysis on T2.We now have to recall the instability analysis\nfor the 2D problem that was carried out in detail in our previo us work [22]. The\nproblem was studied on the standard torus T2= [0,2π]2and we now denote the\nsecond coordinate by x3, so thatx1,x3are the coordinates on T2. The family of\nthe forcing terms and the corresponding stationary solutio ns are as in (5.13) and\nthe linearized stationary operator is precisely (5.12). Ap plying curl to (5.12) we\nobtain the equivalent scalar operator in terms of the vortic ity whose spectrum\nwas studied in [22]\nLsω:=J/parenleftbig\n(∆x−α∆2\nx)−1ωs,(1−α∆x)−1ω/parenrightbig\n+\n+J/parenleftbig\n(∆x−α∆2\nx)−1ω,(1−α∆x)−1ωs/parenrightbig\n+γω=−σω,(5.14)\nwhere\nJ(a,b) =∇a·∇⊥b=∂x1a∂x3b−∂x3a∂x1b,\nand\nωs= curl/vector u0=−λ(s)scossx2, ω= curl/hatwidew.\nThe following result was proved in [22] (see Theorem 4.1 and C orollary 4.2.)\nTheorem 5.3. Given a large integer s>0let a fixed pair of integers t,rbelong\nto a bounded region A(δ)defined by conditions\nt2+r2<s2/3, t2+(−s+r)2>s2, t2+(s+r)2>s2, t≥δs,(5.15)18 A. ILYIN, A. KOSTIANKO, AND S. ZELIK\nwhere0<δ<1/√\n3. There exists an absolute constant c1such that for\nλ≥λ2(s,γ) =c1γ(1+αs2)2\ns(5.16)\nin(5.13)the linear operator Lson the torus T2= [0,2π]2has a real negative\n(unstable) eigenvalue σ <0of multiplicity 2. The corresponding eigenfunctions\nare\nω1(x1,x3) =∞/summationdisplay\nn=−∞at,sn+rcos(tx1+(sn+r)x3),\nω2(x1,x3) =∞/summationdisplay\nn=−∞at,sn+rsin(tx1+(sn+r)x3).(5.17)\nWe now observe that ω1andω2in (5.17) are the real and imaginary parts of\nthe complex-valued eigenfunction\nω1(x1,x3)+iω2(x1,x3) =/bracketleftigg∞/summationdisplay\nn=−∞at,sn+rei(sn+r)x3/bracketrightigg\neitx1.\nRecovering the corresponding divergence free vector funct ion, that is, applying\nthe operator ∇⊥\nx∆−1\nx, we obtain an unstable vector valued eigenfunction of the\noperatorL2(/vector u0) written in the required form (5.11):\nw(x1,x3) =/parenleftigg\nw1(x3)\nw3(x3)/parenrightigg\neitx1.\nFor the 3D instability analysis below we need to repeat the co nstruction of an\nunstable eigenmode on the torus T2\nεwithx1∈[0,2π/ε],x2∈[0,2π], whereε>0\nis arbitrary (not necessarily small).\nProposition 5.4. Letrandt′:=tεbelong to the region A(δ):\nt′2+r2<s2/3, t′2+(−s+r)2>s2, t′2+(s+r)2>s2, t′≥δs.(5.18)\nLetλbe defined in (5.16)and letgsand/vector u0be the same as before but in two\ndimensions:\ngs(x3) = (γλ(s)sinsx3,0)T, /vector u0(x3) = (λ(s)sinsx3,0)T.\nThen there exists an unstable solution\nw(x1,x3) =/parenleftigg\nw1(x3)\nw3(x3)/parenrightigg\neitεx1, x∈T2\nε. (5.19)\nof the form (5.11)of the operator (5.12)on the torus T2\nε.\nProof.Following the proof of Theorem 4.1 in [22] we see that a word fo r word\nrepetition of it shows that if t′=εt,rsatisfy (5.18), then the correspondingDAMPED 3D EULER–BARDINA EQUATIONS 19\noperator (5.14) has an unstable (real negative) eigenvalue of multiplicity two\nwith eigenfunctions\nω1(x1,x3) =∞/summationdisplay\nn=−∞at,sn+rcos(tεx1+(sn+r)x3),\nω2(x1,x3) =∞/summationdisplay\nn=−∞at,sn+rsin(tεx1+(sn+r)x3),\nfrom which we construct the required vector valued complex e igenfunction (5.19)\nas before. /square\nIt is convenient for us to single out a small rectangle Din the (t′,r)-plane\ninside the region A(δ) defined by (5.18), see Fig.1:\n|r| ≤c2s,0<c3s≤t′=tε≤c4s. (5.20)\nHereδ=δ∗∈(0,1/√\n3) is fixed, and all the constants ciare absolute constants,\nwhose explicit values can easily be specified.\nδsr\nt′,tA(δ)\nD\nFigure 1. The region A(δ) and the rectangle D.20 A. ILYIN, A. KOSTIANKO, AND S. ZELIK\n5.3.3D lower bound. We can now formulate the main results of this section.\nTheorem 5.5. We consider the linearized system on the 3D torus T3= [0,2π]3\nwith right-hand side gsand stationary solution /vector u0given in (5.1)and(5.2), where\nλ=λ3(s) =√\n2λ2(s,γ) =√\n2c1γ(1+αs2)2\ns. (5.21)\nThen for each triple of integers a,b,rsatisfying\nc3s≤/hatwidea=/radicalbig\na2+b2≤c4s,|r| ≤c2s,\na≥ |b|,(5.22)\nthere exists an unstable solution of the linearized operato r(5.3).\nProof.We fixa,b,rsatisfying (5.22). Then in view of the first two inequalities\nin (5.22) the pair ( t′,r)∈D⊂A(δ), wheret′=/hatwidea·1 (so that we set t= 1).\nApplying Squire’s transformation we obtain a 2D linearized problem on the torus\nT2\n/hatwideaof the form (5.12) with /hatwideγ=γ/hatwidea/a. Then in view of the third inequality in\n(5.22) we have\nλ=√\n2λ2(s,γ) =λ2(s,√\n2/hatwideγa//hatwidea)≥λ2(s,/hatwideγ).\nWe now see from Proposition 5.4 that the 2D linearized proble m (5.12) has\nan unstable eigenvalue. Lemma 5.2 says, in turn, that so does the linearized\nproblem (5.6) on the standard torus T3= [0,2π]3. /square\nThe number of integers ( a,b,r) satisfying (5.22) is of order c5s3, where\nc5=1\n4πc2(c2\n4−c2\n3).\nTheorem 5.6. Let the right-hand side in (1.1)begsdefined in (5.1)withλ(s)\ndefined in (5.21). Then the dimension of the corresponding attractor A=As\nsatisfies for an absolute constant c6the lower bound\ndimFA≥c6/ba∇dblgs/ba∇dbl2\nL2\nα5/2γ4. (5.23)\nProof.We study our system in the limit α→0. Sincesis at our disposal we set\ns=1√α.\nThenλin (5.21) and /ba∇dblgs/ba∇dbl2\nL2in (5.1) become\nλ=c7γ√α,/ba∇dblgs/ba∇dbl2\nL2=c8γ4α.\nWe can finally write\ndimFA≥c5s3=c51\nα3/2=c5α/ba∇dblgs/ba∇dbl2\nL2\nα5/2/ba∇dblgs/ba∇dbl2\nL2=c5α/ba∇dblgs/ba∇dbl2\nL2\nα5/2c8αγ4,\nwhich gives (5.23) with c6=c5/c8. /squareDAMPED 3D EULER–BARDINA EQUATIONS 21\nRemark 5.7. We recall the sharplower boundin [22] for the attractor dime nsion\nfor system (1.1) on the 2D torus T2:\ndimFA≥cabs/ba∇dblcurlgs/ba∇dbl2\nL2\nαγ4. (5.24)\nSince forλ∼γ√αwe have /ba∇dblcurlgs/ba∇dbl2\nL2∼γ4and/ba∇dblgs/ba∇dbl2\nL2∼γ4α, it follows that\nestimate (5.24) can equivalently be written in terms of the o ther dimensionless\nnumber in (1.8) as follows\ndimFA≥c′\nabs/ba∇dblgs/ba∇dbl2\nL2\nα2γ4.\nAppendix A.Collective Sobolev inequalities for H1-orthonormal\nfamilies\nWe prove here some spectral inequalities for orthonormal fa milies of functions\nwhich are the key technical tool in our derivation of sharp up per bounds for\nthe attractor dimension. These inequalities are somehow co mplementary to the\nclassical Lieb–Thirring inequalities for orthonormal sys tems inL2and in our case\nwe estimate the proper norms of the same quantity ρ(x) =/summationtextn\ni=1|¯θi(x)|2, but for\nfamilies{¯θi}n\ni=1that are orthonormal in H1with norm (2.2) depending on α.\nOur exposition utilizes the ideas from [27] as well as extend s the results of [22]\nto 3D case.\nWe start with the case Ω = Rdor Ω⊂Rdwith Dirichlet boundary condi-\ntions (analogously to the classical Lieb–Thirring inequal ity, the case of Dirichlet\nboundary conditions follows from the case of whole space by z ero extension).\nA.1.The case of the whole space and a domain with Dirichlet BC.\nTheorem A.1. LetΩ⊆Rdbe an arbitrary domain. Let a family of vector\nfunctions {¯θi}n\ni=1∈H1(Ω)withdiv¯θi= 0be orthonormal with respect to the\nscalar product\nm2(¯θi,¯θj)L2+(∇¯θi,∇¯θj)L2=m2(¯θi,¯θj)L2+(curl¯θi,curl¯θj)L2=δij,(A.1)\nThen the function ρ(x) :=/summationtextn\nj=1|¯θj(x)|2satisfies\n/ba∇dblρ/ba∇dblL2≤1\n2√πn1/2\nm, d= 2,\n/ba∇dblρ/ba∇dblL2≤1\n2√πn1/2\nm1/2, d= 3.(A.2)\nProof.We first let Ω = Rdand introduce the operators\nH=V1/2(m2−∆x)−1/2Π,H∗= Π(m2−∆x)−1/2V1/2\nacting in [L2(Rd)]d, whereV∈L2(Rd) is a non-negative scalar function which\nwill be specified below and Π is the Helmholtz–Leray projecti on to divergence22 A. ILYIN, A. KOSTIANKO, AND S. ZELIK\nfree vector fields. Then K=H∗His a compact self-adjoint operator acting from\n[L2(Rd)]dto [L2(Rd)]dand\nTrK2=Tr/parenleftig\nΠ(m2−∆x)−1/2V(m2−∆x)−1/2Π/parenrightig2\n≤\n≤Tr/parenleftbig\nΠ(m2−∆x)−1V2(m2−∆x)−1Π/parenrightbig\n=\n= Tr/parenleftbig\nV2(m2−∆x)2Π/parenrightbig\n,\nwhere we used the Araki–Lieb–Thirring inequality for trace s [1, 29, 40]:\nTr(BA2B)p≤Tr(BpA2pBp), p≥1,\nand the cyclicity property of the trace together with the fac ts that Π commutes\nwith the Laplacian and that Π is a projection: Π2= Π.\nWe want to show that\nTrK2≤\n\n1\n4π1\nm2/ba∇dblV/ba∇dbl2\nL2, d= 2;\n1\n4π1\nm/ba∇dblV/ba∇dbl2\nL2, d= 3.(A.3)\nIndeed, the fundamental solution of ( m2−∆x)2Π inRdis ad×dmatrix\nFd\nij(x) =Gd(x)δij−∂xi∂xj∆−1Gd(x)\nwithRd-trace atx∈Rd\nTrRdFd(x) =dGd(x)−d/summationdisplay\ni=1∂2\nxixi∆−1\nxGd(x) = (d−1)Gd(x),\nwhereGd(x) is the fundamental solution of the scalar operator ( m2−∆x)2in the\nwhole space Rd:\nGd(x) =1\n(2π)d/integraldisplay\nRdeiξxdξ\n(m2+|ξ|2)2=\n\n1\n8π1\nme−|x|m, d = 3;\n1\n4π1\nm2|x|mK1(|x|m), d= 2.(A.4)\nThe first equality here follows from (A.12), while for the sec ond we have (since\nthe function is radial and using formula 13.51(4) in [46])\nG2(x)=1\n2πF−1/parenleftbig\n(m2+|ξ|2)2/parenrightbig\n=1\n2π/integraldisplay∞\n0J0(|x|r)rdr\n(m2+r2)2=1\n4π1\nm2|x|mK1(|x|m),\nwhereK1is the modified Bessel function of the second kind.\nThus, the operator V2(m2−∆x)2Π has the matrix-valued integral kernel\nV(y)2Fd(x−y)DAMPED 3D EULER–BARDINA EQUATIONS 23\nand therefore\nTr(V2(m2−∆x)2Π) =\n=/integraldisplay\nRdTrRd/parenleftbig\nV(y)2Fd(0)/parenrightbig\ndy= (d−1)/ba∇dblV/ba∇dbl2\nL2Gd(0) (A.5)\nwhich along with (A.4) proves the first inequality in (A.3), a nd also the second\none, since ( tK1(t))|t=0= 1.\nWe can now complete the proof as in [27]. Setting\nψi:= (m2−∆x)1/2¯θi,\nwe see from (A.1) that {ψj}n\nj=1is an orthonormal family in L2. We observe that\n/integraldisplay\nRdρ(x)V(x)dx=n/summationdisplay\ni=1/ba∇dblHψi/ba∇dbl2\nL2.\nBy the orthonormality of the ψj’s inL2and the definition of the trace we obtain\nn/summationdisplay\ni=1/ba∇dblHψi/ba∇dbl2\nL2=n/summationdisplay\ni=1(Kψi,ψi)≤n/summationdisplay\ni=1/ba∇dblKψi/ba∇dblL2≤n1/2/parenleftiggn/summationdisplay\ni=1/ba∇dblKψi/ba∇dbl2\nL2/parenrightigg1/2\n=\n=n1/2/parenleftiggn/summationdisplay\ni=1(K2ψi,ψi)/parenrightigg1/2\n≤n1/2/parenleftbig\nTrK2/parenrightbig1/2.(A.6)\nThis gives /integraldisplay\nRdρ(x)V(x)dx≤n1/2/parenleftbig\nTrK2/parenrightbig1/2.\nSettingV(x) :=ρ(x) and using (A.3), we complete the proof of (A.2) for the case\nof Ω =Rd,d= 2,3.\nFinally, if Ω is a proper domain in Rd, we extend by zero the vector functions\n¯θjoutside Ω and denote the results by/tildewide¯θj, so that/tildewide¯θj∈H1(Rd) and div/tildewide¯θj= 0.\nWe further set /tildewideρ(x) :=/summationtextn\nj=1|/tildewide¯θj(x)|2. Then setting /tildewideψi:= (m2−∆x)1/2/tildewide¯θi, we see\nthat the system {/tildewideψj}n\nj=1is orthonormal in L2(Rd) and div/tildewideψj= 0. Since clearly\n/ba∇dbl/tildewideρ/ba∇dblL2(Rd)=/ba∇dblρ/ba∇dblL2(Ω), the proof of estimate (A.2) reduces to the case of Rdand\ntherefore is complete. /square\nThe proved result can be rewritten in terms of orthogonal fun ctions with re-\nspect to inner product (4.3) as follows.\nCorollary A.2. Let the assumptions of Theorem A.1 hold and let {¯θj}n\nj=1,\ndiv¯θj= 0be an orthonormal system with respect to\n(¯θi,¯θj)L2+α(∇¯θi,∇¯θj)L2=δij. (A.7)24 A. ILYIN, A. KOSTIANKO, AND S. ZELIK\nThenρ(x) =/summationtextn\nj=1|¯θj(x)|2satisfies\n/ba∇dblρ/ba∇dblL2≤1\n2√πn1/2\nα1/2, d= 2,\n/ba∇dblρ/ba∇dblL2≤1\n2√πn1/2\nα3/4, d= 3.(A.8)\nIndeed, this statement follows from (A.2) by the proper scal ing.\nA.2.The case of periodic BC: Estimates for the lattice sums. We now\nturn to the case Ω = Td. In this case, we naturally have an extra condition\nthat the considered functions have zero mean. Analogously t o the case Ω = Rd,\nthe Laplacian commutes with the Helholtz–Leray projection , so we may define\nanalogously the operator Kand get exactly the same expression (A.5) for its\ntrace. The only difference is that now Gd(x) =Gd,m(x) a fundamental solution\nof the scalar operator ( m2−∆x)−2on the torus Td(with zero mean condition),\nso the integral in (A.4) should be replaced by the correspond ing sum over the\nlatticeZd\n0=Zd\\{0}:\nGd(x) =1\n(2π)d/summationdisplay\nk∈Zd\n0eik·x\n(m2+|k|2)2. (A.9)\nThus, in order to get estimates (A.2) for the torus T2arguing as in the proof of\nTheorem A.1, we only need to check that\nGd,m(0)<\n\n1\n8π1\nm, d= 3;\n1\n4π1\nm2, d= 2.(A.10)\nfor allm≥0. Unfortunately, we do not know explicit expressions for th e sum\n(A.9), so we need to do rather accurate estimates of the assoc iated lattice sum\nin order get (A.10) based on the Poisson summation formula. I n the cased= 2,\n(A.10) is proved in [22] and the case d= 3 is considered in Proposition A.4 below.\nThus, the following result holds.\nTheorem A.3. Let a family of divergence free vector functions with zero me an\n{¯θi}n\ni=1∈˙H1(T3)be orthonormal with respect to scalar product (A.1). Then\nestimates (A.2)hold. Analogously, if this family is orthonormal with respe ct to\n(A.7), thenρsatisfies inequalities (A.8).\nAs explained before, the case d= 2 is verified in [22] and for proving the result\nford= 3, it is sufficient to prove the following proposition.\nProposition A.4. The following inequality holds for all m≥0:\nF(m) :=m/summationdisplay\nk∈Z3\n01\n(|k|2+m2)2<π2(A.11)DAMPED 3D EULER–BARDINA EQUATIONS 25\nProof.Before we go over to the proof, we first observe that in R3we have the\nequality/integraldisplay\nR3dx\n(|x|2+m2)2=π2\nm\nand, secondly, it is the absence of the term with k= 0 in the sum in (A.11) that\nmakes inequality (A.11) hold at all.\nWe use the Poisson summation formula (see, e.g., [42])\n/summationdisplay\nk∈Znf(k/m) = (2π)n/2mn/summationdisplay\nk∈Zn/hatwidef(2πkm),\nwhere F(f)(ξ) =/hatwidef(ξ) = (2π)−n/2/integraltext\nRnf(x)e−iξxdx. For the function\nf(x) = 1/(1+|x|2)2x∈R3with/hatwidef(ξ) =π2\n(2π)3/2e−|ξ|(A.12)\n(see [42]), and with/integraltext\nR3f(x)dx=π2this gives\nF(m) =1\nm3/summationdisplay\nk∈Z3f(k/m)−1\nm3=\n=π2/summationdisplay\nk∈Z3e−2πm|k|−1\nm3=π2+π2/summationdisplay\nk∈Z3\n0e−2πm|k|−1\nm3.(A.13)\nIn particular, this formula gives a convenient way to comput eF(m) numerically\nfor the case where mis not very small. We start the proof of inequality (A.11)\nwith the case m≤1.\nLemma A.5. Inequality (A.11)holds for all m∈[0,1].\nProof.Since\nF′(m) =/summationdisplay\nk∈Z3\n0/parenleftbigg1\n(|k|2+m2)2−4m2\n(|k|2+m2)3/parenrightbigg\n=/summationdisplay\nk∈Z3\n0|k|2−3m2\n(|k|2+m2)3,\nwe see that all terms in the sum for F(m) with|k|2≥3 are monotone increasing\n(after multiplying by m) with respect to m≤1, so we may write\nF(m)≤6m\n(1+m2)2+12m\n(2+m2)2+/summationdisplay\n|k|2≥31\n(|k|2+1)2≤\n��max\nm∈[0,1]/braceleftbigg6m\n(1+m2)2/bracerightbigg\n+ max\nm∈[0,1]/braceleftbigg12m\n(2+m2)2/bracerightbigg\n+F(1)−6\n4−12\n9=\n=9√\n3\n8+9√\n6\n16−3\n2−4\n3+π2(1.01306)−1 = 9.4915<π2= 9.8696,\nwhere we have used (A.13) in order to compute F(1) =π2(1.01306)−1 (the\ncalculations are reliable since the series has an exponenti al rate of convergence).\nThus, the lemma is proved. /square26 A. ILYIN, A. KOSTIANKO, AND S. ZELIK\nWe now turn to the case m≥1.\nLemma A.6. Inequality (A.11)holds for all m≥1.\nProof.It follows from (A.13) that inequality (A.11) goes over to\nG(m) :=π2m3/summationdisplay\nk∈Z3\n0e2πm|k|−1<0.\nWe use the inequality\n|k| ≥1√\n3(|k1|+|k2|+|k3|)\nfor all terms with |k|>1 and leave the first 6 terms with |k|= 1 unchanged.\nThis gives\nG(m)≤π2m3/summationdisplay\nk∈Z3\n0e2πm(|k1|+|k2|+|k3|)/√\n3−1+6π2m3/parenleftig\ne−2πm−e−2πm/√\n3/parenrightig\nand we only need to prove that the right-hand side of this ineq uality is negative.\nSumming the geometric progression, we get\nG(m)≤G0(m) :=π2m3/parenleftigg/parenleftbigg\n1+2\ne2πm/√\n3−1/parenrightbigg3\n−1/parenrightigg\n−1+\n+6π2m3/parenleftig\ne−2πm−e−2πm/√\n3/parenrightig\n= 6π2m3/parenleftbigg1\ne2πm/√\n3−1−e−2πm/√\n3/parenrightbigg\n+\n+12π2/parenleftigg\nm3/2\ne2πm/√\n3−1/parenrightigg2\n+8π2/parenleftbiggm\ne2πm/√\n3−1/parenrightbigg3\n+6π2m3e−2πm−1 =\n= 6π2ψ1(m)+12π2ψ2(m)2+8π2ψ3(m)3+ψ4(m).\nWe claim that all functions ψi(m) are monotone decreasing for m≥1. Indeed,\nthe function ψ3(m) is obviously decreasing for all m≥0. The function ψ4(m) is\ndecreasing for m≥3\n2π<1. Analogously, as elementary calculations show, the\nsecond function is decreasing for m≥m2<1 where\nm0=√\n3\n4π/parenleftig\n3+2W/parenleftig\n−3e−3/2/2/parenrightig/parenrightig\n≈0.241,\nwhereWis a Lambert W-function. Finally, let us prove the monotonicity of\nψ1(m). Indeed,\nψ′\n1(m) =m22mπ√\n3e−2πm√\n3/3−4πm√\n3−9e−2πm√\n3/3+9\n3(e2πm√\n3/3−1)2\nand we see that\n2mπ√\n3e−2πm√\n3/3−4πm√\n3−9e−2πm√\n3/3+9<\n<2πm√\n3/parenleftig\ne−2πm√\n3/3−1/parenrightig\n+9−2π√\n3<0DAMPED 3D EULER–BARDINA EQUATIONS 27\nifm≥1, since 9 −2π√\n3<0. Thus,ψ′\n1(m)<0 form≥1 andψ1(m) is also\ndecreasing. Thus, G0(m) is decreasing for m≥1 and we only need to note that\nG0(1) =−0.7562<0 and the lemma is proved. /square\nFinally, we have verified (A.11) for all m≥0 and the proof is complete. /square\nRemark A.7. Of course, the estimates obtained above hold for families of scalar\nfunctions {¯θi}n\ni=1∈H1that are orthonormal with respect to (A.7). In this case,\nthe factor ( d−1) in formula (A.5) is replaced by 1, and we get a√\n2-times better\nconstant in the 3D case and the same constant in the 2D case. Na mely, the\nfunctionρ(x) :=/summationtextn\ni=1|¯θi(x)|satisfies\n/ba∇dblρ/ba∇dblL2≤1\n2√πn1/2\nα1/2, d= 2,\n/ba∇dblρ/ba∇dblL2≤1√\n8πn1/2\nα3/4, d= 3.(A.14)\nThese estimates also hold for all three cases Ω = Td, Ω =Rd, and Ω⊂Rdwith\nDirichlet boundary conditions.\nAppendix B.A pointwise estimate for the nonlinear term\nIn this appendix, we prove a pointwise estimate for the inert ial term which\ncorresponds to the Navier–Stokes nonliearity.\nProposition B.1. Let for some x∈Rd,u(x)∈Rdanddivu(x) = 0. Then\n|((θ,∇x)u,θ)(x)| ≤/radicalbigg\nd−1\nd|θ(x)|2|∇xu(x)|, (B.1)\nwhere∇xu(x)is ad×dmatrix with entries ∂iuj, and\n|∇xu|2=d/summationdisplay\ni,j=1(∂iuj)2.\nProof.Basically, this can be extracted from [28]. For the sake of co mpleteness we\nreproduce the details. We suppose first that Ais a symmetric real d×dmatrix\nwith entries aijand with Tr A= 0. Then\n/ba∇dblA/ba∇dbl2\nRd→Rd≤d−1\ndd/summationdisplay\ni,j=1a2\nij. (B.2)\nIn fact, let λ1,...,λ dbe the eigenvalues of Aand letλ1be the largest one in\nabsolute value. Then/summationtextd\nj=1λj= 0 and therefore\n(d−1)d/summationdisplay\nj=2λ2\nj≥\nd/summationdisplay\nj=2λj\n2\n=λ2\n1.28 A. ILYIN, A. KOSTIANKO, AND S. ZELIK\nAdding (d−1)λ2\n1to both sides we obtain\n(d−1)d/summationdisplay\nj=1λ2\nj≥dλ2\n1,\nwhich gives (B.2) since λ2\n1=/ba∇dblA/ba∇dbl2\nRd→Rdand/summationtextd\nj=1λ2\nj= TrA2=/summationtextd\ni,j=1a2\nij. 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Lopes Filho, H. Nussenzveig Lopes, E. Titi, A. Zang. Convergence of the 2D\nEuler-αto Euler equations in the Dirichlet case: indifference to bou ndary layers ,\nPhys. D, vol 292-293, (2015) 51–61.\n[35] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations\nin bounded and unbounded domains , In: Handbook of differential equations: evolu-\ntionary equations. Vol. IV, 103–200, Handb. Differ. Equ., Elsevier/ North-Holland,\nAmsterdam, 2008.\n[36] I. Moise, R. Rosa, and X. Wang. Attractors for non-compact semigroups via energy\nequations , Nonlinearity, vol. 11, no. 5 (1998), 1369–1393.\n[37] E. Olson and E. Titi, Viscosity versus vorticity stretching: global well-posed ness\nfor a family of Navier-Stokes- α-like models , Nonlinear Anal. vol 66, no 11, (2007)\n2427–2458.\n[38] A. Oskolkov, The uniqueness and solvability in the large of boundary valu e problems\nfor the equations of motion of aqueous solutions of polymers , Zap. Nauchn. Sem.\nLOMI, vol. 38, (1973) 98–136.\n[39] J. Pedlosky, Geophysical Fluid Dynamics , Springer, New York, 1979.\n[40] B. Simon, Trace Ideals and Their Applications, 2nd ed. Amer. Math. Soc., Provi-\ndence RI, 2005.\n[41] H.B. Squire, On stability of three-dimensional disturbances of viscous flow between\nparallel walls , Proc. R. Soc. Lond. Ser. A. vol 142(847), (1993) 621–628.\n[42] E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces.\nPrinceton University Press, Princeton NJ, 1972.\n[43] T. Tao, Finite time blowup for an averaged three-dimensional Navie r-Stokes equa-\ntion, J. Amer. Math. Soc., vol 29 (2016), 601–674\n[44] R.Temam, Infinite Dimensional Dynamical Systems in Mechanics and Phy sics,2nd\ned. Springer-Verlag, New York 1997.\n[45] R. Temam, Navier-Stokes equations and nonlinear functional analysi s, vol.66, Siam,\n1995.\n[46] G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. Cambridge\nUniversity Press, Cambridge, 1995.\nEmail address :ilyin@keldysh.ru\nEmail address :aNNa.kostianko@surrey.ac.uk\nEmail address :s.zelik@surrey.ac.uk\n1Keldysh Institute of Applied Mathematics, Moscow, Russia\n2University of Surrey, Department of Mathematics, Guildfor d, GU2\n7XH, United Kingdom.\n3School of Mathematics and Statistics, Lanzhou University, Lanzhou,\n730000, P.R. China\n4Imperial College, London SW7 2AZ, United Kingdom."    },    {        "title": "2107.11699v1.Electron_Phonon_Scattering_governs_both_Ultrafast_and_Precessional_Magnetization_Dynamics_in_Co_Fe_Alloys.pdf",        "content": "1 \n Electron -Phonon Scattering governs both Ultrafast and Precessional Magnetization \nDynamics in Co -Fe Alloys  \nRamya Mohan1, Victor  H. Ortiz2, Luat Vuong2, Sinisa Coh2, Richard B. Wilson1,2* \n1Materials Science & Engineering Program, University of California, Riverside  \n2Department of Mechanical Engineering, University of California, Riverside  \n* Corresponding Author : rwilson@ucr.edu  \n \n \nAbstract  \nRecent investigations have advanced the understanding of how structure -property relationships \nin ferromagnetic metal alloys affect the magnetization dynamics on nanosecond time -scales . A \nsimilar understanding for magnetization dynamics on femto - to pico -second time -scales does not \nyet exist. To address this, we perform time -resolved magneto optic Kerr effect (TRMOKE) \nmeasurements of magnetization dynamics in Co -Fe alloys on femto - to nano-second  regimes . \nWe show that Co -Fe compositions that exhibit low Gilbert damping parameters also feature \nprolonged ultrafast demagnetization upon photoexcitation. We analyze our experimental TR -\nMOKE data with the three-temperature -model (3TM) and the Landau -Lifshitz -Gilbert equation . \nThese analyses  reveal a strong compositional dependence of the dynamics across all time -scales \non the strength of electron -phonon interactions. Our findings are beneficial to the spintronics  and \nmagnonics community , and wil l aid in the quest  for energy -efficient magnetic storage  \napplications.  \nIntroduction  \nLaser excitation of  a magnetic metal causes energy to cascade from photoexcited electrons into \nspin and vibrational degrees of freedom1–3. In ferromagnetic 3d transiti on metals such as Fe, Co, \nand Ni, the rapid increase in thermal energy stored by spin degrees of freedom causes  \nfemtosecond   quenching of the magnetization2,3, followed by a partial recover over the next few \npicoseconds . Subsequently, on nanosecond time-scales , a temperature induced change in \nequilibrium properties causes  oscillatory precessions of the magnetic moment .  \nBoth ultrafast  and precessional magnetization dynamics involve energy exchange between  \nmagnetic and vibrational degrees of freedom . The energy exchange is mediated by quasi -particle \ninteractions . The strength of quasi -particle  interactions in a ferromagnet depends on e lectronic \nband structure4,5. In 3d ferromagnetic alloys, the electronic energy bands  near the Fermi -level \nvary strong ly with composition6. Several  recent investigations  of nanosecond precessional \ndynamics  in ferromagnetic alloy s have explored  the relationship between electronic band \nstructure, quasi -particle interactions , and magnetic damping6–8. Schoen et al.  report a n intrinsic 2 \n damping parameter less than 10-3 for Co 0.25Fe0.756, which is unusually low for  a metal . They \nconclude  that the low damping  in Co0.25Fe0.75 is a result of  a minimization in the density of states \nat the Fermi -level, which decreases the rate of electron -phonon scattering . \nResearchers have not yet reached a  unified understanding of how quasi -particle interactions \ngovern  the magnetization dynamics in the  femtosecond regime2,9–15. Some studies have \nhypothesized that spin-flips caused by electron -phonon interactions are key drivers of \nfemtose cond  magnetization dynamics9,11. Other experimental and theoretical studies have \nexplored the importance of electron -magnon interactions12–15. Encouraged by the recent \nadvances  in the materials science of nanosecond precessional dynamics6–8, we study the \ncompositional dependence of ultrafast magnetization dynamics in Co -Fe alloys. Our  study’s  goal \nis to understand the relationship between electronic band structure , quasi -particle interactions, \nand femto -magnetism properties of ferromagnetic metal alloys.   \nWe perform time -resolved magneto optic Kerr effect  (TR-MOKE)  measurements to characterize  \nthe magnetization dynamics of thin CoxFe1-x alloy  films (capped and seeded with Ta/Cu layers on \na sapphire substrate)  on femto - to nanosecond time-scale s. See Methods for details on sample \ngeometry.  We observe that the ultrafast magnetization dynamics are a strong function of Co -\nconcentration , see Figure . 1a. The ultrafast dynamics of Co xFe1-x differ most significantly from \nthose of Co and Fe at a composition of  x = 0.25.  We also analyze the time -resolved macroscopic \nprecessional dynamics and report the effective damping parameter of our samples , see Figure  2a. \nAfter linewidth analyses, f or CoxFe1-x, we observe  that the  Gilbert  damping parameter varies \nfrom  3.6 ×10−3 to 5.6 ×10−3 for compositions between x = 0 and 1, with a minimum value of \n1.5 ×10−3 at x = 0.25 , in good agreement with previously reported results , see Figure 3b .  \nTo determine the strength and composition dependence of electron -magnon and electron -phonon  \nquasi -particle interactions , we analyze our ultrafast magnetization dynamics  data with a three-\ntemperature -model (3TM)2,16. Our results reveal a strong composition al dependence of  the \nelectron -phonon energy transfer coefficient, 𝑔𝑒𝑝,  suggesting that  the variation in the ultrafast \ndynamics  in Co xFe1-x alloys  occurs primarily due to electron -phonon scattering. We draw this \nconclusion because t he value of 𝑔𝑒𝑝 depends on the rate of phonon emission by hot electrons 17. \nElectron -phonon scattering is also predicted  to govern the dampin g of nanosecond precessional \ndynamics  6,18,19. Therefore, o ur results  demonstrate  that the same  microscopic electron -phonon \ninteractions  responsible for Gilbert damping also play a dominant role in femto -magnetism \nproperties of ferromagnetic alloys.  \nResults  \nUltrafast  Magnetization  Dynamics  \nWe plot the normalized u ltrafast magnetization dynamics  response , ∆M(t), for Co, Fe, and \nCo0.25Fe0.75 as a function of time delay in Figure . 1a. Data for the rest of the Co -Fe compositions \nare plotted  in Supplementary Figure  1. All our measurements were performed with an incident 3 \n laser fluence less than ~15 J/m2. This is a sufficiently  small fluence  for the dynamics in our \nexperiments to follow a  linear regime.  In other words, decreasing the incident f luence by a factor \nof two decreases the optical signal by a factor of two, but does not change the time-dependence  \nof the signal .    \nWe use a polar TR -MOKE configuration t o measure  the ultrafast magnetization dynamics  at \nfemtosecond time delays.  A schemati c of our experimental setup is shown in Supplementary  \nFigure  2a. We apply a n external 2.2 Tesla (T) field perpendicular to the plane of the sample  \nusing  an electromagnet (GMW 3480). This external field is strong enough to effectively \novercome the in-plane shape anisotropy of the Co -Fe alloys and saturate the moment in the out -\nof-plane direction. Since the equilibrium orientation of the moment is in the out -of-plane \ndirection , both, before and after laser irradiation, this geometry allows us to  quan tify the  \nfemtosecond  demagnetization  response  of the Co -Fe alloys , without the presence of macroscopic  \nprecessional dynamics , see schematic in Fig ure 1b.  \nUpon excitation with the pump pulse, the magnetic moment decreases on a sub -picosecond time-\nscale due to the flow of energy from electrons to magnons2,3,16,20,21. Then, on picosecond time-\nscale s, the magnetization partially recovers as energy is transferr ed to the lattice  and temperature \ngradients across the film thickness relax.  After a few picoseconds, the  magnetic film reaches  a \nnew equilibrium at an elevated temperature.  Ultrafast dynamics with sub -picosecond  \ndemagnetization followed by picosecond  re-magnetization  are commonly categorized as “type I” \ndynamics , and are characteristic of 3d ferromagnetic metals such as Fe, Co, and Ni9.  \nTo elucidate  how the de - and re -magnetization dynamics change with composition, we define \ntwo data descriptors : τD and R. We define  the demagnetization time , τD, as the delay time where \nd∆M(t)/dt reaches its maximum value.  We define R as the ratio of the maximum of 𝛥𝑀(𝑡) to \n𝛥𝑀(𝑡≈10ps). We plot τD  and R as a function of composition in Figure  3a. τD varies weakly \nwith composition and has a minimum  value  of 40 fs at x = 0.25. In contrast , we observe that R \nvaries strongly with composition and is a maximum  of 4 at x = 0.25.  \nNanosecond Precessional Dynamics   \nWe show measurements of the  macroscopic  precessional dynamics of Fe, Co, and Co 0.25Fe0.75 in \nFigure  2a. Data for the other Co -Fe compositions are plotted in Supplementary  Figure  3. We use \na polar TR -MOKE experimental setup, with an obliquely angled external magnetic field, to \nmeasure the macroscopic precessional dynamics of our samples. A schematic of our \nexperimental setup is shown in Supplementary  Figure 2b. Tilting the electromagnet to an angle \nof 11° , with respect to the plane of the sample, allows us to apply a canted  external  magnetic \nfield so that the magnetic moment has an out -of-plane component. The equilibrium orientation of \nthe moment depends on the balancing between the applied  external  field and the thin -film shape \nanisotropy field.  The shape anisotropy field in the z -direction is proportional to the  out-of-plane \ncomponent of the  magnetic moment. Upon heating, the  total magnetic moment decreases . This \ndecrease results in an ultrafast change to the  out-of-plane  anisotropy field  and equilibrium 4 \n orientation . As a result, t he magnetic moment will precess to a new equilibrium orientation , see \nschematic in Figure  2b. Our polar TR -MOKE setup detects  changes in the out -of-plane moment , \nso we can sensitively measure the frequency and amplitude of the precessional dynamics.  \nWe collect between  6 and 12 TR-MOKE scans  of precessional dynamics for each sample .  Each \nof these scans is co llected  with a different applied external magnetic field , ranging from 0. 2 T to \n2.2 T. The TR -MOKE signals include precessional dynamics  in addition with a background \nrelated to temperature -induced demagnetization. To analyze the precessional dynamics,  we \nsubtract  the background with  a biexponential  decay  function . We fit the resulting dataset with a \ndamped harmonic function,  V(t)=Asin(ωt+∅)exp (−t/τ). Our fits yield unique  values of A \n(amplitude),  ∅ (the initial phase of the oscillation), T (period), and τ (the exponential decay time \nof the precession). Using these values,  we determine  the effective dimensionless damping \nparameter , αeff = ω.τ-1. \nThe resonance frequency  is a function of applied external magnetic field  and magnetic moment,  \n𝜔=γ √Heff(Heff+μ0Ms). Here, ɣ is the gyromagnetic ratio,  μ0 is the vacuum permeability, \nHeff is the out -of-plane component of the external magnetic field  as measured by a Hall probe , \nand Ms is the saturation magnetization of the sample . We derive the magnetic moment of the \nsample by treating Ms as a fit parameter . We also perform VSM measurements of the moment  of \nsome of the samples and find that the magnetic moment obtained is in good agreement with the \nvalue that we derive by fitting our precessional dynamics data . See Supplementary  Figure 4 for \nmore details . \nThe effective damping parameter α eff that we deduce from our precessional dynamics \nmeasurements includes effects from damping and  inhomogeneous broadening. The effect of \ninhomogeneous  broadening is independent of the applied field at high frequencies22. To obtain \nthe Gilbert  damping parameter  intrinsic to the sample  geometry  (not intrinsic to the material) , we \nplot the effective linewidth, αeff∙f,  as a function of frequency, and linearly fit  to the equation , \nαeff∙f=α∙f+∆H, where ∆H  is the inhomogeneous broadening component and α is the Gilbert \ndamping parameter . Further details can be found in Supplementary Figure 5. \nIn contra st to prior investigations that performed FMR measurements in the frequency range \nfrom 16 -18 GHz8 and 40 GHz6, our TR -MOKE experimental setup allows us to study dynamics \nat frequencies as large as 90 GHz. At such high frequency, we can be confident that our \nmeasured Gilbert damping parameter is dominate d by the intrinsic linewidth over \ninhomogeneous  broadening effects.  \nThe Gilbert damping parameter we observe of α = 1.5 ×10−3 for Co 0.25Fe0.75 is amongst the \nlowest ever reported for a ferromagnetic metal. Schoen et al. report α=2.1 ×10−3 for \nCo0.25Fe0.75.  After accounting for radiative and spin -pumping contributions, they estimate an \nintrinsic damping parameter for Co0.25Fe0.75 to be αint=5 ×10−4 .  Lee et al. 8 performed FMR \nmeasurements of Co0.25Fe0.75 epitaxial films and report α=1.4 ×10−3. Wei et al. report α=5 \n 1.5 ×10−3 for Fe 0.75Al0.25 films 7. We note that our measured damping parameter likely \nincludes  significant contributions from spin -pumping into the adjoining Ta /Cu layers, but we  did \nnot experimentally examine the effect s of spin -pumping in our samples.  \nAnalysis  and Discussion  \nThe c omparison of 𝑅 and 𝛼 in Figure  3a and Figure  3b reveals  that the two quantities depend on \ncomposition in a similar manner.  R is at a maximum and 𝛼 is at a minimum at x = 0.25 .  Fe and \nCoxFe1-x alloys with x ≥ 0.5 have small R and high 𝛼.  Alternatively, C oxFe1-x  alloys with  0.1< x \n< 0.5 have both high 𝑅 and low 𝛼. To confirm this correlation , we performed a hierarchical \ncluster analysis of the raw data  at both femtosecond and nanosecond time-scale s. The clustering \nalgorithm divides  the Co -Fe alloys into groups based on similarit ies in the dynamics data . The \nclustering results as a function of composition are nearly  identical  when based on the femto -\n/pico -second time -scale data vs. the nanosecond time -scale data.  We include further details on \nthe clustering analysis in  Supplementary  Note 1 and Supplementary Figure 6 .  \nWe now explain the correlation between ultrafast and precessional dynamics by considering how \nelectronic scattering processes depend on composition. Similar to prior studies of damping in \nCo-Fe alloys6,7,23, our results for  𝛼 vs. x are in good agreement with the “breathing Fermi \nsurface ” model for damping24. In this model , spin -orbit coupling causes the Fermi -level to shift \nwith the precessi ons of the magnetic moment25. A shift in the equilibrium Fermi -level leads to a \nnonequilibrium electron population . As the Fermi -level repopulates, i ntra-band electron -phonon \nscattering  transfers energy to the lattice .  The “breathing Fermi surface” model predicts  that the  \ndamping parameter  is directly  proportional to 𝐷(𝜀𝑓), because  more electronic states near 𝜀𝑓 leads \nto higher  rates of electron -phonon scattering . We observe  that the  𝛼 value  for Co0.25Fe0.75 is \n~2.5x lower th an 𝛼 for Fe. Density functional theory predicts a ~2x difference in 𝐷(𝜀𝑓) for \nCo0.25Fe0.75 vs. Fe, see Supplementary  Note 2  or Ref.6. Therefore, like prior studies of Co -Fe \nalloys6,7,23, we conclude that intra -band electron -phono n scattering governs precessional \ndamping.  \nTo better understand how composition affects  electron -magnon and electron -phonon energy \ntransfer  mechanisms , we analyze our 𝛥𝑀(𝑡) data with a phenomenological three temperature \nmodel  (3TM) , see Figure  4. The 3TM describes how heat flows  between electrons, phonons, and \nmagnons after laser excitation of the Co-Fe sample . (See Methods for  additional  details. ) The \n3TM predicts  that τD depends on two groupings of model parameters: 𝜏𝑒𝑚≈𝐶𝑚/𝑔𝑒𝑚 and 𝜏𝑒𝑝≈\n𝐶𝑒/𝑔𝑒𝑝.  Here 𝐶𝑚 and 𝐶𝑒 are the magnon and electron heat-capacity per unit volume, and 𝑔𝑒𝑚 \nand 𝑔𝑒𝑝 are the energy transfer coefficients from electrons to magnons an d phonons, \nrespectively.  We estimate v alues for 𝐶𝑒 vs. composition using the Sommerfeld model together \nwith the electronic density of states vs. composition reported in Ref.6. The 3TM also predicts that \nthe parameter R is determined by the following grouping of parameters: 𝑅= 𝐶𝑝𝑔𝑒𝑚/𝐶𝑚𝑔𝑒𝑝 16, \nwhere 𝐶𝑝 is the phonon heat -capacity per unit volume .  We assume  that the  value of  𝐶𝑝 is 3.75 6 \n MJ m-3 K-1 for Co, Fe and Co -Fe alloys.  With these estimates for 𝐶𝑒 and 𝐶𝑝, and other relevant \nmodel parameters, summarized in Supplementary  Table 1, we can deduce unique values for \n𝐶𝑚/𝑔𝑒𝑚 and 𝐶𝑝/𝑔𝑒𝑝  as a function of composition from our TR-MOKE  data, see Figure  4b. \nBased on our 3TM analysis, we conclude that the strong composition dependence of R is due to \nthe composition dependence of 𝑔𝑒𝑝. Boltzmann rate -equation modelling of the nonequilibrium \nelectron dynamics after photoexcitation predicts that the electron -phonon energy -trans fer \ncoefficient  is 𝑔𝑒𝑝=[𝜋ℏ𝑘𝐵𝐷(𝜀𝐹)]𝜆⟨𝜔2⟩ 5. Here,  𝜆⟨𝜔2⟩ is the second frequency moment of the \nEliashberg function  and is a measure of the strength of electron -phonon interactions . Most of the \ncomposition al dependence we observe in 𝑔𝑒𝑝 is explained by the composition al dependence of \n𝐷(𝜀𝑓). To show this, we include a prediction for 𝑔𝑒𝑝 in Figure  4b.  Our prediction uses the \n𝐷(𝜀𝑓) vs. x reported  in6  and treats 𝜆⟨𝜔2⟩ as a composition independent fit parameter . We find \n����⟨𝜔2⟩=260 meV2 provides an excellent fit to our data .  The best-fit value for 𝜆⟨𝜔2⟩ is in good \nagreement with  𝜆⟨𝜔2⟩≈𝜆𝑅Θ𝐷22⁄=280 meV2.  Here, 𝜆𝑅 is derived from electrical resistivity \ndata for Fe  26, and Θ𝐷=470𝐾 is the Debye temperature of Fe.   \nBefore beginning our experimental study, we hypothesized that the energy transfer coefficient \nbetween electrons and magnons, \nemg , would be correlated with the phase -space for electron -\nmagnon scattering . We expected the phase -space for electron -magnon scattering to be a  strong \nfunction of band -structure near the Fermi -level 12–15.  We also expected the phase -space to be \nminimized at a composition of x = 0.25, because of the minimum in the density of states at the \nfermi -level.   To explore how the phase -space for electron -magnon scattering depends on \ncomposition, we performed density functional theory calculations for the electronic band \nstructure with x = 0 and x = 0.25, see Supplementa ry Note 2.  Our DFT calculations suggest  that \nthe phase -space for electron -magnon scattering is an order of magnitude higher for x = 0 vs. \n0.25.  However, we do not see evidence  that this large theoretical difference in electron -magnon \nscattering phase -space  affects ultrafast dynamics . The time -scale for magnons to heat up after \nphotoexcitation, \n/em m emCg , decreases monotonically with increasing x, and does display  \nstructure near x ~ 0.25.  \nSeveral  theoretical models predict a strong correlation between τ D and αint. For example,  \nKoopmans et al.  predicts τ D will be inversely proportional to α by assuming  that the  dissipative \nprocesses responsible for damping also drive ultrafast demagnetization  27.  Alternatively, Fähnle \net al.  predict s that  τD should be proportional to αint 28. In our experiments on Co -Fe thin films, w e \nobserve only a weak correlation between τD and αint. While α int varies  with composition  by a \nfactor of three , τD for 8 of the 9 compositions we study fall within 20% of 75 fs.  The τD value we \nobtained for Fe (=  76 fs) agrees well with experimental results reported in 9,12,29. \n \n 7 \n Conclusions  \nWe have measured the magnetization dynamics of Co xFe1-x thin-films , and we observe that  both \nultrafast and precessional dynamics of Co 0.25Fe0.75 differ significantly from Co and Fe . When the \nmoment of Co0.25Fe0.75 is driven away from its equilibrium  orientation , the time -scale for  the \nmoment  to return to equilibrium is 3 -4x as long as for Fe or Co.  Similarly, when spins of \nCo0.25Fe0.75 are driven into a nonequilibrium state by ultrafast laser heating, the time -scale for \nthermalization with the lattice is 2 -3x as long as for Fe or Co. Through  3TM analyses, we \ndemonstrate that this occurs  primarily  due to the effect of  the electronic band -structure on \nelectron -phonon interactions , consistent with the “breathing Fermi surface” theory . Our findings \nare of fundamental importance to the field of ul trafast magnetism, which seeks to control \nmagnetic order on femto - to picosecond time-scale s. Such control requires  a thorough  \nunderstanding of how and why energy is exchanged between electronic, spin, and vibrational \ndegrees of freedom. Prior studies have  shown  that 𝑔𝑒𝑝 is correlated with a wide range of physical \nproperties, e.g the superconducting transition temperature30, electrical resistivity 26, \nphotoelectron emission31, and the laser fluence required for ablation32. To our knowledge, o ur \nstudy provides the first demonstration that 𝑔𝑒𝑝 in ferromagnetic metals is also correlated to the \nGilbert damping parameter 𝛼.  \nOur findings  also have implications for the ongoing search for magnetic materials with ultrafast \nmagnetic switching functionality. Atomistic spin dynamics simulations predict that the energy \nrequired for ultrafast electrical or optical switching of rare -earth ferromagnetic  alloys, e.g. \nGdFeCo, is governed by the electron -phonon energy transfer coefficient33. To date, most studies \naimed at exploring the materials science of ultrafast switching have used alloy composition as a \nway to control magnetic  properties  34–37. Our work  suggests  an alternative strategy for reducing \nthe energy requirements for ultrafast magnetic switching. The alloy composition should be \nchosen  to minimize the electronic density of states at the Fermi -level. Such metals will have \nlower electron -phonon energy trans fer coefficients, and therefore more  energy efficient ultrafast \nswitching  33.  \nFinally , our findings  offer a new route for discovering ferro magnetic materials with ultra -low \ndamping  as a result of low 𝑔𝑒𝑝. Current methods for identifying low damping materials involve \nlabor -intensive ferromagnetic resonance measurements of one alloy composition at a time.  \nAlternatively, high-throughput  localized measurements of  ultrafast demagnetization  dynamics  of \nsamples produced using combinatorial techniques38 would allow promising alloy compounds \nwith weak electron -phonon interactions to be rapidly identified  39–41.  \n \n \n \n 8 \n Materials and Methods  \nSample Preparation  \nWe sputter deposit the Co -Fe samples onto sapphire substrates with a direct current (DC) \nmagnetron sputtering system (Orion, AJA International). The base pressure prior to deposition is \nless than 3.5  × 10-7 torr. We sputter with an Ar gon pressure of ~3.5 × 10-3 torr. The geometry of \nthe samples is sapphire/Ta(2nm)/Cu(3nm)/Co xFe1-x(15nm)/Cu(3nm)/Ta(1nm). The Co xFe1-x layer \nis deposited  by co -sputtering two 4N purity Co and Fe targets at different powers. We chose this \nfilm geometry to mimic the samples in Ref.6 which demonstrated  low damping at x = 0.25.  \nTo ensure an accurate thickness of each layer in our samples, we calibrate the deposition rates of \neach metal by sputtering individual Co, Fe, Ta, and Cu films onto SiO 2/Si substrates and/or BK -7 \nglass substrates. We use picosecond acoustics42 and time-domain thermo -reflectance  (TDTR) \nmeasurements43,44 to determine the thicknesses of these individual films. We validate the \ncomposition of the Co -Fe alloy layer by perf orming Energy Dispersive X -Ray Spectroscopy \n(EDS) analyses with a scanning electron microscope ( FEI Nova Nano SEM 450) at an operating \nvoltage of 15 kV and working distance of 14 mm. We analyze the EDS data using Aztec Synergy  \nsoftware ( Oxford Instruments ). \nTime -Resolved MOKE Experimental Setup  \nWe use a pump/probe laser system to perform TR -MOKE measurements of the magne tization \ndynamics. The pulsed laser is a Ti:sapphire oscillator with an 80 MHz repetition rate. The laser \nbeam is split into a pump and probe beam, that are modulated to frequencies of 10.7 MHz and \n200 Hz , respectively. A time -delayed  pump beam irradiates the sample surface and heats the \nmetal film. The ultrafast heating causes a change in the magnetic moment. We measure the time -\nevolution of the magnetic moment by monitoring the polarization of the probe beam reflected of f \nthe sample surface. The reflected probe beam’s polarization state is affected by the out -of-plane \nmagnetic moment of the sample due to the polar Kerr effect. Additional details about the MOKE \nexperiment set -up are in Ref.45. \nThe t ime-resolution of our experiment is controlled by the convolution of the intensity vs.  time \nof the pump and probe pulses. The wavelength of our pump and probe beams is tunable. \nEmploying a red  (900 nm ) pump and  blue (450 nm ) probe yields  higher time-resolution  \ncapabilities , allowing us to accurately measure the ultrafast magnetization at fe mtosecond time \ndelays . We measure the full-width -at-half-maximum ( FWHM ) of the convolution of the pump \nand probe pulses by performing an inverse Faraday effect  (IFE)  measurement on Pt . We obtain a \nFWHM  value  of 390 fs for  the convoluted pulses , and a pulse duration of 2 10 fs for the 900 nm \npump/450 nm probe beam setup . For further details on our IFE measurements and pulse duration \ncalculations, please refer to  Supplementary Figure 8.  9 \n To investigate  the precessional dynami cs on longer time -scales, we use a pump and probe \nwavelength of 783 nm. The pulse duration  for this setup  is 610 fs due to pulse broadening from a \ntwo-tint setup we use to prevent pump light from reaching the balanced detector45,46.  \nThree Temperature Modeling  \nTo determine  the electron, phonon, and magnon energy transfer coefficients,  we use t he \nphenomenological  three -temperature model  (3TM),  given by the following set of equations :  \n𝐶𝑒𝑑𝑇𝑒\n𝑑𝑡=𝑔𝑒𝑝(𝑇𝑝−𝑇𝑒)+ 𝑔𝑒𝑚(𝑇𝑚−𝑇𝑒)+Ʌ𝑒𝑑2𝑇𝑒\n𝑑𝑧2+𝑆(𝑧,𝑡)    (1) \n𝐶𝑝𝑑𝑇𝑝\n𝑑𝑡=𝑔𝑒𝑝(𝑇𝑝−𝑇𝑒)+Ʌ𝑝𝑑2𝑇𝑝\n𝑑𝑧2     (2) \n𝐶𝑚𝑑𝑇𝑚\n𝑑𝑡=𝑔𝑒𝑚(𝑇𝑚−𝑇𝑒)+Ʌ𝑚𝑑2𝑇𝑚\n𝑑𝑧2   (3) \n𝑆(𝑧,𝑡)= 𝑆0𝑃(𝑡)𝐴(𝑧)   (4) \nEquations  1 – 3  describe the temperature evolution of electrons (e), phonons (p) and magnons \n(m), as a function of time delay  (t). C, T, and Ʌ are the heat capacity per unit volume, \ntemperature, and thermal conductivity, respectively.  We use the density of states (DOS) at the \nFermi level  as a function of Co -concentration6 to calculate the electronic heat capacity (C e) using \nthe Sommerfeld model . We assume that the phonon -magnon energy transfer is negligible \ncompared to electron -magnon coupling, and  thus,  neglect  𝑔𝑝𝑚.  \nWe calculate the laser energy absorption by electrons  (S), as a function of depth  (z) and time \ndelay  (t), as described in Equation 4. The terms P(t) and A(z) denote the time -dependent laser \npulse intensity and the optical absorption profile as a function of stack thickness. We calculat e \nA(z) us ing the refractive indices of each metal constituent of the stack47–49. The material \nparameters that are used to numerically solve equations 1 – 4 are listed in Supplementary Table \n1. \n \n \n \n \n \n \n \n 10 \n Figures:  \n                   \nFigure 1. Ultrafast magnetization dynamics of Co, Fe, and Co 0.25Fe0.75 thin films (a) Polar TR -\nMOKE data showing ultrafast demagnetization behavior at short delay times. (b) Schematic \nillustration of the three phases of an ultrafast magnetization dynamics experiment. Stage I: A large \nexternal magnetic field  oriented normal to the plane of t he sample leads to an equilibrium moment , \n𝑀⃗⃗  in the out -of-plane direction. Stage II: Upon heating with a pump beam, ultrafast \ndemagnetization ( 𝑀′⃗⃗⃗⃗ ) occurs within ~100s of fs. Energy from hot electrons is transferred to the \nmagnons, increasing the amplitude of precession. Stage III: Over the next few picoseconds, energy \nis transferred from magnons and electrons to the lattice. Additionally, spatial t emperature gradients \nrelax. As a result, magnons cool, i.e. the average precessional amplitude of individual spins \ndecreases. As a result, the magnetization partially recovers to 𝑀′′⃗⃗⃗⃗⃗⃗ . The time -scale for the partial \nrecovery in stage III depends strongly o n the composition.    \n11 \n   \nFigure 2 . Precessional dynamics in Co, Fe, and Co 0.25Fe0.75 thin films (a)  Polar TR -MOKE data \non sub -nanosecond time-scale s. (b) Illustration of the three stages for precessional dynamics after \nlaser excitation . Stage I: Prior to laser excitation, the presence of a canted external magnetic field, \n𝐻𝑒𝑓𝑓⃗⃗⃗⃗⃗⃗⃗⃗⃗ , oriented at an angle θ. This results in the orientation of the out -of-plane moments, 𝑀⃗⃗ 𝑧. Stage \nII: Laser -induced photoexcitation leads to the disorder of the magnetic moment, causing a decay \nin the net magnetization , denoted by 𝑀′⃗⃗⃗⃗ . The net torque imba lance causes macroscopic precessions \nof the magnons, towards equilibrium, 𝐻′𝑒𝑓𝑓⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ , over several ~100s of picoseconds . Stage III : \nEventually, after ~1 ns, the magnetic moment re -equilibrates to 𝐻′𝑒𝑓𝑓⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ . The lifetime of the magnetic \nprecessions depends o n the effective damping parameter, α eff.  The time -scale for the precessional \ndynamics to cease ( in stage III) depends strongly on composition, and is a maximum for x = 0.25.  \n12 \n  \nFigure 3.  Compositional dependence of descriptors for the ultrafast dynamics  data . (a) R \ndescribes the maximum change in the magnetic moment, i.e. how far from equilibrium spin -\ndegrees of freedom are driven after ultrafast excitation. τD describes the lag between zero delay \ntime and demagnetization, as a function of Co -concentration.  (b) α denotes the Gilbert damping \nparameter, as a function of Co concentration. Data obtained from our TR -MOKE experiments \ndescribed in this study (plotted in orange), agree reasonably with data from Ref. [6] (plotted in \ngreen). Co 0.25Fe0.75 features the largest deviation in R and α, when compared to its constituent \nelements Co and Fe.  \n \n \n \n \n \n \n \n \n \n \n \n13 \n  \nFigure 4. Analyses of Ultrafast Demagnetization Results using the Three Temperature Model \n(3TM) in Co -Fe alloys . (a) Polar TR -MOKE dataset of the Co 0.25Fe0.75 composition (black circles) \nwith best -fit results of the 3TM. The 3TM describes the temperature excursions of the electrons \n(blue curve), magnons (red curve) and phonons (green curve) after laser excitation. (b) We treat \n𝑔𝑒𝑝 and 𝑔𝑒𝑚 as fit parameters when solving the 3TM. Using literature values of C p and C m (further \ndetails available in Supplementary Table 1), we calculate and plot the electron -phonon (τ ep) and \nelectron -magnon (τ em) relaxation times, as a function of Co -concentration. The red -line is a best -\nfit value for the electron -phonon relaxation time as a function of composition, with the assumption \nof a composition -independent value for the electron -phonon coupling parameter λ . \n \n \n \n \n \n \n \n \n \n \n \n14 \n References:  \n1. Kirilyuk, A., Kimel, A. V & Rasing, T. 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Optical constants of transition metals. Phys. Rev. B  9, \n5056 –5070 (1974).  \n48. P. B. Johnson and R. W. Chri sty. Optical Constant of the Nobel Metals. Phys. Rev. B  6, 17 \n 4370 –4379 (1972).  \n49. Ordal, M. A., Bell, R. J., Alexander, R. W., Newquist, L. A. & Querry, M. R. Optical \nproperties of Al, Fe, Ti, Ta, W, and Mo at submillimeter wavelengths. Appl. Opt.  27, 1203 \n(1988).  \n \n \n \nAcknowledgements  \nThe work by R. M., V. H. O, and R. B. W. was primarily supported by the U.S. Army Research \nLaboratory and the U.S. Army Research Office under contract/grant number W911NF -18-1-\n0364 and W911NF -20-1-0274. R. M. and R. B. W. also acknowledge support by NSF (C BET – \n1847632). The work by L. V. and S. C. was supported by the U.S. Army Research Laboratory \nand U.S. Army Research Office under contract/grant number W911NF -20-1-0274. Energy \nDispersive X -Ray Spectroscopy (EDS) analyses  were performed at the Central Fac ility for \nAdvanced Microscopy and Microanalysis (CFAMM) at UC Riverside.  \nAuthor Contributions  \nR. M. and R. B. W. designed the experiments. R. M. prepared all the samples and characterized \nthem , and performed TR-MOKE experiments . V. H. O performed VSM measurements. L. V. \nperformed hierarchical clustering analyses. S. C. performed DFT calculations. R. M. and R . B. \nW. analyzed the data and wrote the manuscript, with discussions and contributions from L. V. \nand S. C . \nAdditional Information:  Supplementary information is provided with this manuscript.  \nCompeting Interests:  The authors declare no competing interest.  \nData Availability:  The data that supports the findings of this paper are available from the \ncorresponding author upon reasonable request.  \nCorrespondence:  Correspondence and request for additional information must be addressed to \nrwilson@ucr.edu   "    },    {        "title": "2206.10948v1.Homogenization_of_the_Landau_Lifshitz_Gilbert_equation_with_natural_boundary_condition.pdf",        "content": "arXiv:2206.10948v1  [math.AP]  22 Jun 2022HOMOGENIZATION OF THE\nLANDAU-LIFSHITZ-GILBERT EQUATION WITH\nNATURAL BOUNDARY CONDITION\nJINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN\nAbstract\nThe full Landau-Lifshitz-Gilbert equation with periodic material coe fficients and\nnatural boundary condition is employed to model the magnetization dynamics in\ncomposite ferromagnets. In this work, we establish the converge nce between the\nhomogenized solution and the original solution via a Lax equivalence th eorem kind\nof argument. There are a few technical difficulties, including: 1) it is p roven the\nclassic choice of corrector to homogenization cannot provide the c onvergence re-\nsult in the H1norm; 2) a boundary layer is induced due to the natural boundary\ncondition; 3) the presence of stray field give rise to a multiscale pote ntial problem.\nTo keep the convergence rates near the boundary, we introduce the Neumann cor-\nrector with a high-order modification. Estimates on singular integra l for disturbed\nfunctions and boundary layer are deduced, to conduct consisten cy analysis of stray\nfield. Furthermore, inspired by length conservation of magnetizat ion, we choose\nproper correctors in specific geometric space. These, together with a uniform W1,6\nestimate on original solution, provide the convergence rates in the H1sense.\n1.Introduction\nThe intrinsic magnetic order of a rigid single-crystal ferr omagnet over a\nregion Ω ⊂Rn,n= 1,2,3 is described by the magnetization Msatisfying\nM=Ms(T)m,a.e. in Ω ,\nwhere the saturation magnetization Msdepends on the material and the\ntemperature T. Below Curie temperature, Msis modeled as a constant.\nA stable structure of a ferromagnet is mathematically chara cterized as\nthe local minimizers of the Landau-Lifshitz energy functio nal [7]\nGL[m] :=/integraldisplay\nΩa(x)|∇m|2dx+/integraldisplay\nΩK(x)(m·u)2(m)dx\n−µ0/integraldisplay\nΩhd[Msm]·Msmdx−/integraldisplay\nΩha·Msmdx\n=:E(m)+A(m)+W(m)+Z(m).\nDate: June 23, 2022.\n2010Mathematics Subject Classification. 35B27; 65M15; 82D40.\nKey words and phrases. Homogenization; Landau-Lifshitz-Gilbert equation; Boun dary\nlayer; Magnetization dynamics; Micromagnetics.\n12 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN\nE(m) is the exchange energy, which penalizes the spatial variat ion ofm.\nThe matrix a= (aij)1≤i,j≤3is symmetric, uniformly coercive and bounded,\ni.e.,\n(1)\n\nn/summationdisplay\ni,j=1aij(x)ηiηj≥amin|η|2for anyx∈Rn,η∈Rn,\nn/summationdisplay\ni,j=1aij(x)ηiξj≤amax|η||ξ|for anyx∈Rn,η, ξ∈Rn.\nIntheanisotropyenergy A(m),uistheeasy-axisdirectionwhichdependson\nthecrystallographicstructureofthematerial. Theanisot ropyenergydensity\nis assumed to be a non-negatively even and globally Lipschit z continuous\nfunction that vanishes only on a finite set of unit vectors (th e easy axis).\nThe third term W(m) is the magnetostatic self-energy due to the dipolar\nmagnetic field, also known as the stray field hd[m]. For an open bounded\ndomain Ω with a Lipschitz boundary, the magnetization m∈Lp(Ω,R3)\ngenerates a stray field satisfying\n(2) hd[m] =∇Um,\nwhere the potential Umsolves\n(3) ∆ Um=−div(mXΩ),inD′(R3)\nwithmXΩthe extension of mtoR3that vanishes outside Ω. The exis-\ntence and uniqueness of Umfollows from the Lax-Milgram Theorem and Um\nsatisfies the estimate [ 10]\n(4) /ba∇dblhd[m]/ba∇dblLp(Ω)≤ /ba∇dblm/ba∇dblLp(Ω)1< p <∞.\nThe last term Z(m) is the Zeeman energy that models the interaction be-\ntweenmand the externally applied magnetic field ha.\nFor a composite ferromagnet with periodic micorstructures , the material\nconstants are modeled with periodic material coefficients wi th period ε, i.e.,\naε=a(x/ε),Kε=K(x/ε),Mε=Ms(x/ε), with functions a,K,Ms\nperiodic over Y= [0,1]n. The associated energy reads as\n(5)Gε\nL[m] :=/integraldisplay\nΩaε(x)|∇m|2dx+/integraldisplay\nΩKε(x)(m·u)2dx\n−µ0/integraldisplay\nΩhd[Mεm]·Mεmdx−/integraldisplay\nΩha·Mεmdx.\nIt is proved in [ 2] thatGε\nL[m] is equi-mild coercive in the metric space\n(H1(Ω,S2),dL2(Ω,S2)) and Γ-converges to the functional Ghomdefined as\nGhom[m] =/integraldisplay\nΩa0|∇m|2dx+/integraldisplay\nΩK0(m·u)2dx−µ0(M0)2/integraldisplay\nΩhd[m]·mdx\n−µ0/integraldisplay\nΩ×Y/vextendsingle/vextendsinglem·Hd[Ms(y)](y)/vextendsingle/vextendsingle2dxdy−M0/integraldisplay\nΩha·mdx, (6)HOMOGENIZATION OF THE LLG EQUATION 3\nwherea0is the homogenized tensor\na0\nij=/integraldisplay\nY/parenleftBigg\naij+n/summationdisplay\nk=1aik∂χj\n∂yk/parenrightBigg\ndy,\nthe constants M0andK0are calculated by\nM0=/integraldisplay\nYMs(y)dy, K0=/integraldisplay\nYK(y)dy,\nand the symmetric matrix-valued function Hd[Ms(y)](y) =∇yU(y) with\npotential function given by\n(7)/integraldisplay\nYMs(y)∇yϕ(y)dy=−/integraldisplay\nY∇yU(y)·∇yϕ(y)dy,\nU(y) isY-periodic ,/integraldisplay\nYU(y)dy= 0,\nfor any periodic function ϕ∈H1\nper(Y).\nIn the current work, we are interested in the convergence of t he dynamic\nproblem driven by the Landau-Lifshitz energy ( 5) to the dynamics problem\ndriven by the homogenized energy ( 6) asεgoes to 0. It is well known that\nthe time evolution of the magnetization over Ω T= Ω×[0,T] follows the\nLandau-Lifshitz-Gilbert (LLG) equation [ 7,6]\n(8)\n\n∂tmε−αmε×∂tmε=−(1+α2)mε×Hε\ne(mε) a.e. in Ω T,\nν·aε∇mε= 0,a.e. on∂Ω×[0,T],\nmε(0,x) =mε\ninit(x),|mε\ninit(x)|= 1 a.e. in Ω ,\nwhereα >0isthedampingconstant, andtheeffective field Hε\ne(mε) =−δGε\nL\nδmε\nassociated to the Landau-Lifshitz energy ( 5) is given by\n(9)Hε\ne(mε) = div(aε∇mε)−Kε(mε·u)u+µ0Mεhd[Mεmε]+Mεha.\nMeanwhile, the LLG equation associated to the homogenized e nergy (6)\nreads as\n(10)\n\n∂tm0−αm0×∂tm0=−(1+α2)m0×H0\ne(m0)\nν·a0∇m0= 0,a.e. on∂Ω×[0,T]\nm0(0,x) =m0\ninit(x),|minit(x)|= 1 a.e. in Ω\nwith homogenized effective field H0\ne(m0) =−δGhom\nδm0calculated by\n(11)H0\ne(m0) =div/parenleftbig\na0∇m0/parenrightbig\n−K0(m0·u)u\n+µ0(M0)2hd[m0]+µ0H0\nd·m0+M0ha,\nwhere the matrix H0\nd=/integraltext\nYMs(y)Hd[Ms(y)](y)dy.\nWorks related tothehomogenization oftheLLGequationinth eliterature\ninclude [ 11,5,1,8,9,4]. As for the convergence rate, most relevantly,\nthe LLG equation ( 8) with only the exchange term and with the periodic4 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN\nboundarycondition is studied in [ 8]. Convergence rates between mεandm0\nin time interval [0 ,εσT] are obtained under the assumption\n(12) /ba∇dbl∇mε/ba∇dblL∞(Ω)≤C,for anyt∈[0,εσT],\nwhereCis a constant independent of εandσ∈[0,2). As a special case,\nwhenσ= 0 in assumption ( 12), i.e.,/ba∇dbl∇mε/ba∇dblL∞(Ω)is uniformly boundedover\na time interval independent of ε, it is proven that /ba∇dblmε−m0/ba∇dblL∞(0,T;L2(Ω))=\nO(ε)while/ba∇dblmε−m0/ba∇dblL∞(0,T;H1(Ω))isonlyuniformlyboundedwithoutstrong\nconvergence rate.\nIn this work, we consider the full LLG model ( 8) equipped with the Neu-\nmann boundary condition, which is the original model derive d by Landau\nand Lifshitz [ 7]. We prove the convergence rates between mεandm0in the\nL∞(0,T;H1(Ω)) sense without the strong assumption ( 12). It is worth men-\ntioning that, the trick to improve the convergence result in toH1sense is to\nfind proper correctors m1,m2, such that they satisfy geometric properties\n(13) m0·m1= 0,andm0·m2=−|m1|2,\nwhich are motivated by the length-preserving property of ma gnetization\nand asymptotic expansion. A familiar definition of classic fi rst-order ho-\nmogenization corrector m1in (32) would naturally satisfies first property in\n(13); see [8]. In this article, the suitable corrector m2in (13) is obtained. By\nthe usage of these properties, we are able to derive the estim ate of consis-\ntency error, which is induced by an equivalent form of LLG equ ation, given\nin (22), and a sharper estimate than [ 8] inL∞(0,T;H1(Ω)) sense is finally\nobtained.\nInstead of the assumption ( 12), we prove a weak result that /ba∇dbl∇mε/ba∇dblL6(Ω)\nis uniformly bounded over a time interval independent of ε. Such a uniform\nestimate is nontrivial for the LLG equation, since the stand ard energy es-\ntimate usually transforms the degenerate (damping) term in to the diffusion\nterm and thus the upper bound becomes ε-dependent. To overcome this\ndifficulty, we introduce the interpolation inequality when n≤3\n(14)/ba∇dbldiv(aε∇m)/ba∇dbl3\nL3(Ω)≤C+C/ba∇dbldiv(aε∇m)/ba∇dbl6\nL2(Ω)\n+C/ba∇dblm×∇{div(aε∇m)}/ba∇dbl2\nL2(Ω),\nfor theS2-value function msatisfying homogeneous Neumann boundary\ncondition. This inequality can help us derive a structure-p reserving energy\nestimate, in which the degenerate term is kept in the energy.\nThe full LLG model ( 8) we considered contains the stray field, where an\nindependent homogenization problem of potential function in the distribu-\ntion sense arises, and this complicated the problem when we a rrive at the\nconsistency analysis. By using results in [ 10] and Green’s representation\nformula, the stray field is rewritten as the derivatives of Ne wtonian poten-\ntial. Then we are able to obtain the consistency error by deri ving detailed\nestimate of singular integral for disturbed function and bo undary layer.HOMOGENIZATION OF THE LLG EQUATION 5\nThe effect of boundary layer exists when we apply classic homog eniza-\ntion corrector to the Neumann boundary problem, which would cause the\napproximation deterioration on the boundary. To avoid this , a Neumann\ncorrector is introduced, which is usually used in elliptic h omogenization\nproblems (see [ 12] for example). In this article, we provide a strategy to\napply the Neumann corrector to evolutionary LLG equations, by finding a\nproper higher-order modification. For a big picture, let us w rite ahead the\nlinear parabolic equation of error\n∂teε−Lεeε+fε=0,\nwhose detailed derivation can be found in ( 26). Following the notation of\neε\nb=eε−ωbwith boundary corrector ωb, one can find by above equa-\ntion that an L∞(0,T;H1(Ω)) norm of eε\nbrelies on the boundary data and\ninhomogeneous term induced by ωb, which read as\n(15) /ba∇dblν·aε∇{eε+ωb}/ba∇dblB−1/2,2(∂Ω)and/ba∇dblLεωb/ba∇dblL2(Ω).\nIn this end, we divide the corrector ωbinto two parts as ωb=ωN−ωM,\nsuch that they can control two terms in ( 15) respectively. Here ωNis the\nNeumann corrector used in elliptic problems (see [ 12]), andωMis a modi-\nfication to be determined. We point out the modification ωMis necessary\nsince calculation implies some bad terms in LεωNdo not converge in L2\nsense. Therefore we construct following elliptic problem t o determine ωM:\ndiv(aε∇ωM) =/parenleftBig\nBad Terms in LεωN/parenrightBig\nwith proper Neumann boundary condition. Such a solution ωMcan be\nproved to have better estimates than ωN, by the observation that all “Bad\nTerms in LεωN” can be written in the divergence form. At this point, ωM\ncan be viewed as a high-order modification.\nThis paper is organized as follows. In the next Section, we in troduce the\nmain result of our article and outline the main steps of the pr oof. In Sec-\ntion3, multiscale expansions are used to derive the second-order corrector\nm2. In Section 4, we deduce that the consistency error fεonly relies on the\nconsistency error of the stray field, which can be estimated b y calculation\nof singular integral for disturbed function and boundary la yer. In Section\n5, we introduce the boundary corrector ωb, and derive several relevant es-\ntimates of it. Section 6contains the stability analysis in L2andH1sense\nrespectively. And we finally give a uniform regularity analy sis ofmε, by\nderiving a structure-preserving energy estimate in Sectio n7.\n2.Main result\nTo proceed, we make the following assumption\nAssumption 1.\n1.Smoothness We assume Y-periodic functions a(y) = (aij(y))1≤i,j≤3,\nK(y),Ms(y), and the time-independent external field ha(x), alone with6 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN\nboundary ∂Ω, are sufficiently smooth. These together with the definition in\n(32),(46),(66),(76)leads to the smoothness of m1,m2andωb.\n2.Initial data Assumem0\ninit(x)andmε\ninit(x)are smooth enough and satisfy\nthe Neumann compatibility condition:\n(16) ν·aε∇mε\ninit(x) =ν·a0∇m0\ninit(x) = 0,x∈∂Ω.\nFurthermore, we might as well set them satisfying periodica lly disturbed el-\nliptic problem:\n(17) div(aε∇mε\ninit(x)) = div( a0∇m0\ninit(x)),x∈Ω.\n(16)-(17)implym0\ninit(x)is the homogenization of mε\ninit(x). Note that the as-\nsumption (17)is necessary not only for the convergence analysis in Theorem\n4, but also for the uniform estimate of mεin Theorem 8.\nNow let us state our main result:\nTheorem 1. Letmε∈L∞(0,T;H2(Ω)),m0∈L∞(0,T;H6(Ω))be the\nunique solutions of (8)and(10), respectively. Under Assumption 1, there\nexists some T∗∈(0,T]independent of ε, such that for any t∈(0,T∗)and\nforn= 2,3, it holds\n(18)/ba∇dblmε(t)−m0(t)/ba∇dblL2(Ω)≤β(ε),/ba∇dblmε(t)−m0(t)/ba∇dblH1(Ω)≤Cε1/2,\nwhere\n(19) β(ε) =/braceleftBigg\nCε[ln(ε−1+1)]2,whenn= 2,\nCε5/6,whenn= 3.\nIn the absence of the stray field, i.e., µ0= 0, then it holds for any t∈(0,T∗)\nand forn= 1,2,3\n(20)/ba∇dblmε(t)−m0(t)/ba∇dblL2(Ω)≤Cε[ln(ε−1+1)]2,\n/ba∇dblmε(t)−m0(t)−(Φ−x)∇m0(t)/ba∇dblH1(Ω)≤Cε[ln(ε−1+1)]2,\nwherexis spatial variable, Φ= (Φi)1≤i≤nis the corrector defined in (67).\nConstant Cdepends on the initial data mε\ninitandm0\ninit, but is independent\nofε.\nRemark 2.1. Comparing (18)and(20), one can see that in the L2norm,\nthe stray field makes little influence when n= 2, but causes 1/6-order loss\nof rate when n= 3. In the H1norm, however, the stray field leads to 1/2-\norder loss of rate in both cases. Such a deterioration of conv ergence rate is\ninduced since the zero-extension has been applied for stray field(3), which\nintroduces a boundary layer.\nRemark 2.2. The logarithmic growth [ln(ε−1+1)]2in(20)is caused by the\nNeumann corrector (Φ−x)∇m0. For problems (8)and(10)with periodic\nboundary condition over a cube, by replacing the Neumann cor rector in (20)HOMOGENIZATION OF THE LLG EQUATION 7\nwith the classical two-scale corrector, a similar argument in the current work\nleads to\n(21) /ba∇dblmε−m0−χ∇m0/ba∇dblH1(Ω)≤Cε,\nwhereχ= (χi)1≤i≤nis defined in (33).\nNote that (21)is consistent with the L2result in [8]. However, only the\nuniform boundedness in H1has been shown in [8], while our results (20)and\n(21)imply that it maintains the same convergence rate in L2andH1norm,\nby choosing the correctors satisfying specific geometric pr operty(13).\n2.1.Some notations and Lax equivalence type theorem. Recall that\na classical solution to ( 8) also satisfies an equivalent form of equation, reads\n(22)LLLG(mε) :=∂tmε−αHε\ne(mε)+mε×Hε\ne(mε)−αgε\nl[mε]mε= 0,\nwhere the gε\nl[·] is the energy density calculated by\n(23)gε\nl[mε] =aε|∇mε|2+Kε(mε·u)u−hd[Mεmε]·Mεmε−ha·Mεmε.\nFor convenience, we also define a bilinear operator deduced f rom (23), which\nreads\nBε[m,n] =aε∇m·∇n+Kε(m·u)(n·u)−µ0hd[Mεm]·Mεn.\nNow let us set up the equation of error, in terms of Lax equival ence\ntheorem kind of argument. Define the approximate solution\n(24) /tildewidemε(x) =m0(x)+εm1(x,x\nε)+ε2m2(x,x\nε),\nwherem0is the homogenized solution to ( 10),m1is the first-order corrector\ndefinedin( 32), andm2isthesecond-ordercorrectordeterminedbyTheorem\n2. Then replacing mεby/tildewidemεin (22) provides the notation of consistence\nerrorfε:\n(25) LLLG(/tildewidemε) =fε.\nTogether ( 22) and (25), we can obtain the equation of error eε=mε−/tildewidemε,\ndenoted by\n(26) ∂teε−Lεeε+fε=0,\nwhereLεis second-order linear elliptic operator depending on mεand/tildewidemε,\nthat can be characterized as\n(27) Lε(eε) =α/tildewideHε\ne(eε)−D1(eε)−D2(eε).\nHere/tildewideHε\ne(mε) is the linear part of Hε\ne(mε), i.e.,\n/tildewideHε\ne(mε) :=Hε\ne(mε)−Mεha,\nprocession term D1is calculated by\n(28)D1(eε) =mε×Hε\ne(mε)−/tildewidemε×Hε\ne(/tildewidemε)\n=mε×/tildewideHε\ne(eε)+eε×Hε\ne(/tildewidemε),8 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN\nand the degeneracy term D2reads as\nD2(eε) =−αgε\nl[mε]mε+αgε\nl[/tildewidemε]/tildewidemε\n=−α/parenleftbig\nBε[eε,mε]+Bε[/tildewidemε,eε]+Mε(ha·eε)/parenrightbig\nmε−αgε\nl[/tildewidemε]eε.\nMoreover, we define a correctional error eε\nbas\n(29) eε\nb=eε−ωb,\nwhereωbis the boundary corrector satisfying ωb=ωN−ωM, forωNthe\nNeumann corrector given in ( 66), andωMthe modification determined in\n(76). Then equation ( 26) leads to\n(30) ∂teε\nb−Lεeε\nb+/parenleftbig\n∂tωb−Lεωb+fε/parenrightbig\n= 0.\nBy the Lax equivalence theorem kind of argument, the estimat e of error\neε\nbfollows from consistency analysis of ( 25), energy estimate of boundary\ncorrector, and stability analysis of ( 30).\n2.2.Proof of Theorem 1.\nProof.Following the above notations, for the consistency error fε, Theorem\n3says that it can be divided as fε=f0+/tildewidef, satisfying /ba∇dbl/tildewidef(t)/ba∇dblL2(Ω)≤Cε,\nand\n/ba∇dblf0(t)/ba∇dblL2(Ω)= 0,whenµ0= 0,\n/ba∇dblf0(t)/ba∇dblLr(Ω)≤Crµ0/parenleftbig\nε1/r+εln(ε−1+1)/parenrightbig\n,whenµ0>0,n/ne}ationslash= 1,\nwhere constants CrandCare independent of ε, for any t∈(0,T), and\n1≤r <+∞. Consideringtheboundarycorrectortermsin( 30), byTheorem\n5there exists C=C(/ba∇dbl∇mε/ba∇dblL2(Ω)) such that\n/ba∇dbl∂tωb(t)/ba∇dblL2(Ω)≤Cεln(ε−1+1),\n/ba∇dblLεωb(t)/ba∇dblL2(Ω)≤Cε[ln(ε−1+1)]2+C/ba∇dbleε\nb(t)/ba∇dblH1(Ω),\nfor anyt∈(0,T). As for initial-boundary data of eε\nb, using Theorem 4we\nwrite with C=C(/ba∇dbl∇mε/ba∇dblL2(Ω)),\n/ba∇dbleε\nb(x,0)/ba∇dblH1(Ω)+/ba∇dbl∂\n∂νεeε\nb/ba∇dblW1,∞(0,T;B−1/2,2(∂Ω))≤Cεln(ε−1+1).\nNow let us turn to stability analysis of ( 30). For the L∞(0,T;L2(Ω))\nnorm, let σ= 1 when n= 1,2, andσ= 6/5 whenn= 3, we can apply\nTheorem 6to derive for n= 1,2,3\n(31)/ba∇dbleε\nb/ba∇dbl2\nL∞(0,T;L2(Ω))+/ba∇dbl∇eε\nb/ba∇dbl2\nL2(0,T;L2(Ω))\n≤Cδ/parenleftBig\n/ba∇dbleε\nb(x,0)/ba∇dbl2\nL2(Ω)+/ba∇dbl∂\n∂νεeε\nb/ba∇dbl2\nL2(0,T;B−1/2,2(∂Ω))+/ba∇dbl/tildewidef/ba∇dbl2\nL2(0,T;L2(Ω))\n+/ba∇dbl∂tωb/ba∇dbl2\nL2(0,T;L2(Ω))+γ(ε)/ba∇dblf0/ba∇dbl2\nL2(0,T;Lσ(Ω))/parenrightBig\n+δ/ba∇dblLεωb/ba∇dbl2\nL2(0,T;L2(Ω))+ε2/ba∇dblAεeε\nb/ba∇dbl2\nL2(0,T;L2(Ω)).HOMOGENIZATION OF THE LLG EQUATION 9\nwith /braceleftBigg\nγ(ε) = 1, whenn= 1,3,\nγ(ε) = [ln(ε−1+1)]2,whenn= 2.\nConstant Cδ=Cδ(/ba∇dbl∇mε/ba∇dblL4(Ω)). Now taking δsmall enough in ( 31), and\nusing the fact\n/ba∇dblAεeε\nb/ba∇dblL2(0,T;L2(Ω))≤Cln(ε−1+1)\nwithC=C/parenleftbig\n/ba∇dblAεmε/ba∇dblL2(Ω)/parenrightbig\nfrom Theorem 5, we finally obtain\n/ba∇dbleε\nb/ba∇dblL∞(0,T;L2(Ω))≤/braceleftBiggβ(ε), whenµ0>0,n= 2,3,\nCε[ln(ε−1+1)]2,whenµ0= 0,n= 1,2,3,\nwhereβ(ε) satisfies ( 19). Using the fact mε−m0=eε\nb+εm1+ε2m2+ωb,\nalong with the estimates of εm1,ε2m2,ωbin Lemma 4-5, we obtain the L2\nestimates in Theorem 1.\nAs for the stability of ( 30) inL∞(0,T;H1(Ω)) norm, we can apply The-\norem7to obtain for n= 1,2,3\n/ba∇dbl∇eε\nb/ba∇dbl2\nL∞(0,T;L2(Ω))≤C/parenleftBig\n/ba∇dbleε\nb(x,0)/ba∇dbl2\nH1(Ω)+/ba∇dbl∂\n∂νεeε\nb/ba∇dbl2\nH1(0,T;B−1/2,2(∂Ω))\n+/ba∇dblLεωε\nb/ba∇dbl2\nL2(0,T;L2(Ω))+/ba∇dblfε/ba∇dbl2\nL2(0,T;L2(Ω))+/ba∇dbl∂tωb/ba∇dbl2\nL2(0,T;L2(Ω))/parenrightBig\n,\nwhere constant C=C(/ba∇dblAεmε/ba∇dblL2(Ω),/ba∇dbl∇mε/ba∇dblL4(Ω)). Together with above\nresults, and estimate for /ba∇dbl∇eε\nb/ba∇dbl2\nL2(0,T;L2(Ω))in (31), we arrive at\n/ba∇dbl∇eε\nb(t)/ba∇dbl2\nL∞(0,T;L2(Ω))≤/braceleftBigg\nCε1/2, whenµ0>0,n= 2,3,\nCε[ln(ε−1+1)]2,whenµ0= 0,n= 1,2,3,\nby the representation of eε\nbin (80), together with estimate of m2andωM\nin Lemma 5, it leads to the H1estimates in Theorem 1.\nNotice that all the constants in our estimate depend on the va lue of\n/ba∇dblAεmε(t)/ba∇dblL2(Ω)and/ba∇dbl∇mε(t)/ba∇dblL4(Ω), which from Theorem 8are uniformly\nbounded with respect to εandtfor anyt∈(0,T∗), with some T∗∈(0,T].\nThis completes the proof. /square\n3.The Asymptotic Expansion\nIn this section, we derive the second-order corrector using the formal\nasymptotic expansion. First, let us define the first-order co rrectorm1by\n(32) m1(x,y) =n/summationdisplay\nj=1χj(y)∂\n∂xjm0(x),\nwhereχj,j= 1,...,nare auxiliary functions satisfying cell problem\n(33)\n\ndiv/parenleftbig\na(y)∇χj(y)/parenrightbig\n=−n/summationdisplay\ni=1∂\n∂yiaij(y),\nχjY-periodic ,10 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN\nsuch that the first geometric property in ( 13) holds. As for the second-order\ncorrector m2, we assume it as a two-scale function satisfying\n/braceleftBigg\nm2(x,y) is defined for x∈Ω andy∈Y,\nm2(·,y) isY-periodic .\nFor notational convenience, given a two-scale function in t he form of\nm(x,x\nε), we denote the fast variable y=x\nεand have the following chain\nrule\n(34) ∇m(x,x\nε) = [(∇x+ε−1∇y)m](x,y).\nMoreover, denoting Aε= div(aε∇), one can rewrite\nAεm(x,x\nε) = [(ε−2A0+ε−1A1+A2)m](x,y),\nwhere\n(35)\n\nA0= divy/parenleftbig\na(y)∇y/parenrightbig\n,\nA1= divx/parenleftbig\na(y)∇y/parenrightbig\n+divy/parenleftbig\nA(y)∇x/parenrightbig\n,\nA2= divx/parenleftbig\na(y)∇x/parenrightbig\n.\nThe procedure to determine m2is standard. With the notation in ( 24),\nassumemεcan be written in form of\n(36) mε(x) =/tildewidemε(x)+o(ε2).\nOne can derive m2by substituting ( 36) into (8) and comparing like terms of\nε. However, it is a bit fussy in the presence of stray field. Let u s outline the\nmain steps here. Revisiting the stray field hd[Mεmε(x)] =∇Uεin (2)-(3),\none finds that the potential function Uε=Uε[Mεmε(x)] satisfies\n(37) ∆Uε=−div(Ms(x\nε)mεXΩ).\nSubstituting Uε= Σ2\nj=0εjUj(x,x\nε)+o(ε2) and (36) into (37) and combining\nlike terms of εleads to\n(38)\n\ndivy(∇yU0(x,y)) = 0,\ndivy(∇yU1(x,y)) =−divy(Ms(y)m0(x)XΩ(x)),\ndivx(∇xU0(x,y))+2div y(∇xU1(x,y))+div y(∇yU2(x,y))\n=−Ms(y)divxm0(x)XΩ(x)−divy(Ms(y)m1(x,y)XΩ(x)).\nThe first equation in ( 38) implies that U0(x,y) =U0(x) since the Lax-\nMilgram Theorem ensures the uniqueness and existence of sol ution (up to a\nconstant). Integrating the third equation in ( 38) with respect to yyields\n∆U0(x) =−div(M0m0XΩ).\nAn application of ( 2)-(3) implies that U0is actually the potential function\nofhd[Mhm0], i.e.,\n(39) ∇U0(x) =hd[Mhm0] =Mhhd[m0].HOMOGENIZATION OF THE LLG EQUATION 11\nWith notation given in ( 7), one can deduce from the second equation in ( 38)\nthatm0(x)XΩ(x)U(y) =U1(x,y) up to a constant in the H1(Y) space.\nHence it follows that by ( 7)\n(40) ∇yU1(x,y) =XΩ(x)m0(x)·Hd[Ms(y)](y).\nSubstituting ( 39) and (40) into the expansion of Uε, one can deduce that,\nforx∈Ω,\n(41)hd[Mεmε] =∇Uε=hd[Mhm0]+m0(x)·Hd[Ms(y)](x\nε)+O(ε).\nSubstituting ( 36), (32), (35), (41) into (8) and collecting terms of O(ε0),\nwe obtain the following equations\n(42)/braceleftBigg\n∂tm0−αm0×∂tm0=−(1+α2)m0×{A0m2+Ha\ne},\nm2Y-periodic in y,\nwhere\n(43)Ha\ne=A1m1+A2m0−Kε(m0·u)u\n+µ0Mshd[Mhm0]+µ0Msm0·Hd[Ms(y)]+Msha.\nSubstituting ( 10) into (42) leads to\n(44)/braceleftBigg\nm0×A0m2=m0×/braceleftbig\nH0\ne(m0)−Ha\ne/bracerightbig\n,\nm2Y-periodic in y.\n(44) is the degenerate system that determines m2in terms of m0.\n3.1.Second-order corrector. Thewell-posednessof ( 44)isnontrivial due\ntothedegeneracy. InthefollowingTheorem, bysearchingas uitablesolution\nsatisfying ( 13), we give the existence result, andderive an explicit expre ssion\nofm2in terms of m0and some auxiliary functions.\nTheorem 2. Givenm0∈L∞/parenleftbig\n[0,T];H2(Ω)/parenrightbig\nthe homogenization solution\nandm1calculated in (32), define\nV=/braceleftBig\nm∈H2(Y)∩H1\nper(Y) :m·m0=−1\n2|m1|2a.e. inΩ×Y/bracerightBig\n,\nthen(44)admits a unique solution m2(x,y)∈ V/Tm0(S2), with notation\nTm0(S2)denoting the tangent space of m0.\nProof.Assume m2(x,y)∈ V, i.e.,m2·m0=−1\n2|m1|2. Applying A0to\nboth sides of it yields\n(45) m0·A0m2=−a(y)∇ym1·∇ym1−m1·A0m1.\nTaking the cross-product with m0to (44) and substituting ( 45) lead to\n(46)A0m2=−{Ha\ne−H0\ne(m0)}+/braceleftbig\nm0·/parenleftbig\nHa\ne−H0\ne(m0)/parenrightbig/bracerightbig\nm0\n−(m1·A0m1+a(y)∇ym1·∇ym1)m0.12 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN\nNow using the fact\n(47)/integraldisplay\nYH0\ne(m0)−Ha\nedy= 0,\ntogether with ( 45), one can check equation ( 46) satisfies the compatibility\ncondition for Y-periodic function m2iny. Thus by the application of Lax-\nMilgram Theorem andsmoothnessassumption, ( 46) admitsauniqueregular\nsolution up to a function independent of y, denoted by m2(x,y)+/tildewidem2(x).\nMoreover, one can determine /tildewidem2(x) by\nm0·(m2+/tildewidem2) =−1\n2|m1|2,\nsuch that m2+/tildewidem2∈ V, and therefore is also a solution to ( 44) by taking\nthe above transformation inversely.\n/square\nRemark 3.1. One can check that equation (46)has a solution\n(48)\nm2=n/summationdisplay\ni,j=1θij∂2m0\n∂xi∂xj+n/summationdisplay\ni,j=1(θij+1\n2χiχj)/parenleftbigg∂m0\n∂xi·∂m0\n∂xj/parenrightbigg\nm0+Tlow−(m0·Tlow)m0\nwith low-order terms Tlowcalculated by\nTlow=−κ(m0·u)u+µ0ρhd[Mhm0]+µ0m0·Λ+Msha,\nwhereθijandκ,ρ,Λare given by\n(49)\n\nA0θij=a0\nij−/parenleftbig\naij+n/summationdisplay\nk=1aik∂χj\n∂yk/parenrightbig\n−n/summationdisplay\nk=1∂(aikχj)\n∂yk,\nA0ρ=Ms(y)−M0,A0κ=K(y)−K0,\nA0Λ=Ms(y)Hd[Ms(y)](y)−H0\nd,\nθij, κ, ρ,Λ,areY-periodic .\nMoreover, one can find m2defined above satisfies geometric property (13),\ntherefore is also the solution to equation (44). In the following, we may\nassume second-order correct m2takes the form in (48).\n4.Consistency Estimate\nIn this section, we aim to estimate the consistence error fεdefined in\n(25). Following the notation in ( 34)-(35), by the definition of /tildewidemε, (25) can\nbe written in terms of\nfε=ε−2f−2+ε−1f−1+f0+εf1+ε2f2.\nIt is easy to check that f−2=f−1=0by the definition of m0,m1in\nSection3. Along the same line, by the H¨ older’s inequality, one has\n/ba∇dblf1/ba∇dblL2(Ω)+/ba∇dblf2/ba∇dblL2(Ω)≤C,HOMOGENIZATION OF THE LLG EQUATION 13\nwhereCdepends on the L2(Ω) andL∞(Ω) norms of mi(x,x\nε),∇xmi(x,x\nε),\n∇ymi(x,x\nε),i= 0,1,2, and thus is bounded from above by /ba∇dbl∇m0/ba∇dblH4(Ω)\nwith the help of smoothness assumption and Sobolev inequali ty.\nIt remains to estimate f0, let us prove that f0only depends on the con-\nsistence error of stray field, by the help of geometric proper ty (13). Denote\nthe consistence error of stray field by\n(50)/tildewideh=µMεhd[(Mε−Mh)m0]−µMεHd[Ms(y)](x\nε)·m0\nwithHdgiven in ( 7). After some algebraic calculations and the usage of\n(42) and (43), one has\n(51)f0=∂tm0−α/braceleftBig\nA0m2+Ha\ne+/tildewideh/bracerightBig\n+m0×/braceleftBig\nA0m2+Ha\ne+/tildewideh/bracerightBig\n−αgε\nl[m0]m0−(aε∇ym1·∇ym1)m0−2(aε∇ym1·∇m0)m0.\nNotice that the classical solution m0to (10) also satisfies the equivalent\nform\n∂tm0−αH0\ne(m0)+m0×H0\ne(m0)−αg0\nl[m0]m0= 0, (52)\nwhere\ng0\nl[m] :=a0|∇m|2+K0(m·u)u−µ0(M0)2hd[m]·m\n−µ0m·H0\nd·m−ha·M0m.\nSubstituting ( 52) into (51) and using ( 46) lead to\nf0=−α/tildewideh+m0×/tildewideh−α/braceleftbig\nm0·/parenleftbig\nHa\ne−H0\ne(m0)/parenrightbig/bracerightbig\nm0\n+α(m1·A0m1−2aε∇ym1·∇m0)m0+αg0\nl[m0]m0−αgε\nl[m0]m0. (53)\nNote that A2m0=Aεm0−ε−1A1m0, one can deduce\nHa\ne=Hε\ne(m0)+A1m1+ε−1A0m1−/tildewideh.\nSubstituting it into ( 53), and using the fact\nm0·Hε\ne(m0) =−gε\nl[m0],m0·H0\ne(m0) =−g0\nl[m0],\none has\n(54)f0=−α/tildewideh+m0×/tildewideh−α/braceleftbig\nm0·/parenleftbig\nA1m1+ε−1A0m1−/tildewideh/parenrightbig/bracerightbig\nm0\n+α(m1·A0m1−2aε∇ym1·∇m0)m0.\nApplyA0andA1to both sides of m0·m1= 0 respectively, and substitute\nresulting equations into ( 54). After simplification, we finally obtain\n(55) f0=−α/tildewideh+m0×/tildewideh+α/parenleftBig\nm0·/tildewideh/parenrightBig\nm0.\n(55) implies that the convergence of fεdepends on the convergence of\nstray field error /tildewideh. In fact, we have14 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN\nLemma 1. For any 1≤r <∞, andn= 2,3, it holds that\n(56)/vextenddouble/vextenddoubleMεhd[(Mε−Mh)m0]−MεHd[Ms(y)](x\nε)·m0/vextenddouble/vextenddouble\nLr(Ω)\n≤Cε1/r+Cεln(ε−1+1),\nwhereCdepends on /ba∇dbl∇m0/ba∇dblW1,∞(Ω),/ba∇dbl∇Ms(y)/ba∇dblH1(Y)and is independent of\nε.\nProof of Lemma 1will be given in Section 4.2. Lemma 1directly leads to\nthe consistency error:\nTheorem 3. (Consistency) Given fεdefined in (25), it can be divided as\nfε=f0+/tildewidef, satisfying /ba∇dbl/tildewidef/ba∇dblL2(Ω)≤Cε, and\n(57)/ba∇dblf0/ba∇dblL2(Ω)= 0,whenµ0= 0,\n/ba∇dblf0/ba∇dblLr(Ω)≤Crµ0/parenleftbig\nε1/r+εln(ε−1+1)/parenrightbig\n,whenµ0>0,n/ne}ationslash= 1,\nfor any1≤r <+∞. Here constant CandCrdepend on /ba∇dbl∇m0/ba∇dblH4(Ω),\n/ba∇dbl∇Ms(y)/ba∇dblH1(Y), and are independent of ε.\n4.1.Estimate of some singular integral. The strategy to prove Lemma\n1is to rewrite the stray field into derivatives of Newtonian po tential, thus\nthe consistency estimate turns into the estimate of singula r integrals. The\nfollowing Lemmas introduce the estimate of singular integr al in terms of\ndistribution function and boundary layer. We will use the cu t-off function\nηεwithin the interior of area away from boundary:\n(58)\n\n0≤ηε≤1,|∇ηε| ≤Cε−1,\nηε(x) = 1 if dist( x,∂Ω)≥2\n3ε,\nηε(x) = 0 if dist( x,∂Ω)≤1\n3ε.\nwhere dist( x,∂Ω) denotes the distance between xand∂Ω, and cut-off func-\ntionφεin a small ball:\n(59)\n\n0≤φε≤1,|∇φε| ≤Cε−1,\nφε(x) = 1 if |x| ≤1\n3ε,\nφε(x) = 0 if |x| ≥2\n3ε.\nDenote the boundary layer Ωεas\nΩε={x∈Ω,dist(x,∂Ω)≤ε}.\nLemma 2. Assume that scalar functions f(y)∈C1(Rn)isY-periodic,\ng(x)∈C1(¯Ω), define for x∈Ω\nu(x) =/integraldisplay\nΩ/vextendsingle/vextendsinglef(x\nε)−f(z\nε)/vextendsingle/vextendsingle\n|x−z|ndz, v(x) =/integraldisplay\nΩε|g(x)−g(z)|\n|x−z|ndz,\nthenu(x)∈L∞(Ω)logarithmically grows with respect to ε, satisfying\n/ba∇dblu/ba∇dblL∞(Ω)≤Cln(ε−1+1)/ba∇dblf(y)/ba∇dblL∞(Y)+C/ba∇dbl∇f(y)/ba∇dblL∞(Y),HOMOGENIZATION OF THE LLG EQUATION 15\nandv(x)∈Lr(Ω)decreases at speed of O(ε1/r)for any1≤r <∞, satisfying\n/ba∇dblv/ba∇dblLr(Ω)≤Cε1/rln(ε−1+1)/parenleftbig\n/ba∇dblg(x)/ba∇dblL∞(Ω)+ε/ba∇dbl∇g(x)/ba∇dblL∞(Ω)/parenrightbig\n.\nConstant Cis independent of ε.\nProof.Splitting the integral in uinto/integraltext\nΩ−B(x,ε)+/integraltext\nB(x,ε), one can estimate\nit by\n|u(x)| ≤C/integraldisplay\nΩ−B(x,ε)/ba∇dblf(y)/ba∇dblL∞(Y)\n|x−z|ndz+Cε−1/integraldisplay\nB(x,ε)/ba∇dbl∇f(y)/ba∇dblL∞(Y)\n|x−z|n−1dz,\ntherefore the estimate of uin Lemma follows by simple integral. As for the\nestimate of v, by application of cut-off function φε=φε(x−z), one has\n|v(x)|=/integraldisplay\nΩεφε|g(x)−g(z)|\n|x−z|ndz+/integraldisplay\nΩε(1−φε)|g(x)−g(z)|\n|x−z|ndz\n≤C/ba∇dbl∇g/ba∇dblL∞(Ω)/integraldisplay\nΩεφε\n|x−z|n−1dz+C/ba∇dblg/ba∇dblL∞(Ω)/integraldisplay\nΩε1−φε\n|x−z|ndz\n=R1+R2.\nForR1, one can write by Fubini’s Theorem\n/ba∇dblR1/ba∇dblr\nLr(Ω)≤C/ba∇dbl∇g/ba∇dblr\nL∞(Ω)/integraldisplay\nΩ/parenleftBig/integraldisplay\nΩεφε\n|x−z|n−1dz/parenrightBigr\ndx\n≤C/ba∇dbl∇g/ba∇dblr\nL∞(Ω)sup\nx∈Ω/parenleftBig/integraldisplay\nΩεφε\n|x−z|n−1dz/parenrightBigr−1\n×sup\nz∈Ωε/integraldisplay\nΩφε\n|x−z|n−1dx/integraldisplay\nΩε1dz\n≤C/ba∇dbl∇g/ba∇dblr\nL∞(Ω)·Cεr−1·Cε·Cε.\nAs forR2, applying the same argument leads to\n/ba∇dblR2/ba∇dblr\nLr(Ω)≤C/ba∇dblg/ba∇dblr\nL∞(Ω)sup\nx∈Ω/parenleftBig/integraldisplay\nΩε1−φε\n|x−z|ndz/parenrightBigr−1\n×sup\nz∈Ωε/integraldisplay\nΩ1−φε\n|x−z|ndx/integraldisplay\nΩε1dz\n≤C/ba∇dblg/ba∇dblr\nL∞(Ω)·C[ln(ε−1+1)]r−1·Cln(ε−1+1)·Cε.\n/square\nLemma 3. Assume that a scalar function fε(x)∈L∞(Ω)satisfiesfε(x) =\n0whenx∈Ω−Ωε, which means fεis nonzero only in boundary layer. Let\nw(x)be the Newtonian potential of fεinΩ, i.e.,\nw(x) =/integraldisplay\nΩΦ(x−z)fε(z)dz,x∈Ω,\nwhereΦis the fundamental solution of Laplace’s equation. Then w(x)∈\nW2,p(Ω)satisfies for any 1≤p <+∞\n/ba∇dbl∇2w/ba∇dblLp(Ω)≤C/parenleftbig\nε1/p+εln(ε−1+1)/parenrightbig/parenleftbig\n/ba∇dblfε(x)/ba∇dblL∞(Ω)+ε/ba∇dbl∇fε(x)/ba∇dblL∞(Ω)/parenrightbig\n.16 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN\nConstant Cis independent of ε.\nProof.The case of 1 < p <+∞follows directly by the property of Newto-\nnian potential:\n/ba∇dbl∇2w/ba∇dblLp(Ω)≤C/ba∇dblfε/ba∇dblLp(Ω)≤C/ba∇dblfε/ba∇dblLp(Ωε),\nand the fact\n(60) /ba∇dblfε/ba∇dblLp(Ωε)≤ |Ωε|1/p/ba∇dblfε/ba∇dblL∞(Ω).\nNow let us consider the case of p= 1 and write\n∂2w\n∂xi∂xj=/integraldisplay\nΩ∂2\n∂xi∂xj/braceleftbig\nΦ(x−z)/bracerightbig\n·/braceleftbig\nfε(z)−fε(x)/bracerightbig\ndz\n+fε(x)/integraldisplay\n∂Ωνi·∂\n∂xj/braceleftbig\nΦ(x−z)/bracerightbig\ndz\n=:S1+S2.\nForS1, one can apply Lemma 2to derive\n/ba∇dblS1/ba∇dblL1(Ω)≤Cεln(ε−1+1)/parenleftbig\n/ba∇dblfε(x)/ba∇dblL∞(Ω)+ε/ba∇dbl∇fε(x)/ba∇dblL∞(Ω)/parenrightbig\n.\nAs forS2, we can split the integral into/integraltext\n∂Ω−B(x,ε)+/integraltext\n∂Ω∩B(x,ε), and write\n/ba∇dblS2/ba∇dblL1(Ω)≤sup\nx∈Ω/integraldisplay\n∂Ω−B(x,ε)νi·∂\n∂xj/braceleftbig\nΦ(x−z)/bracerightbig\ndz×/integraldisplay\nΩfε(x)dx\n+ sup\nz∈∂Ω/integraldisplay\nΩ∩B(z,ε)νi·∂\n∂xj/braceleftbig\nΦ(x−z)/bracerightbig\n·fε(x)dx×/integraldisplay\n∂Ω1dz\n≤Cln(ε−1+1)×ε/ba∇dblfε/ba∇dblL∞(Ω)+Cε/ba∇dblfε/ba∇dblL∞(Ω),\nhere in the second line we have used the Fubini’s theorem. Thu s the Lemma\nis proved. /square\n4.2.Consistency error of stray field. Now we are ready to prove the\nconsistency error of stray field /tildewidehin Lemma 1. The idea is to use result in\n[10] and Green’s representation formula, to rewrite /tildewidehinto singular integral\nthat of the types estimated in above Lemmas.\nProof.(Proof of Lemma 1) Recall from ( 2) the stray field in LLG equation\ncan be calculated by\n(61) hd[(Mε−Mh)m0] =∇U,\nwhereU=U[(Mε−Mh)m0] satisfies\n∆U=−div[(Mε−Mh)m0XΩ] inD′(Rn).\nDenotes the ith component of m0bym0,i. Using the fact |m0|= 1, one can\nwrite [10]\nU(x) =−n/summationdisplay\ni=1/integraldisplay\nΩ∂\n∂xiΦ(x−z)(Ms(z\nε)−Mh)m0,i(z)dz.HOMOGENIZATION OF THE LLG EQUATION 17\nSubstituting above representation of U(x) into (61) and making the use of\ncut-off function ηεdefined in ( 58), one can derive\n(62)hd[(Mε−Mh)m0]\n=−∇/parenleftBign/summationdisplay\ni=1/integraldisplay\nΩ∂\n∂xiΦ(x−z)ηε(z)(M(z\nε)−Mh)m0,i(z)dz/parenrightBig\n−∇/parenleftBign/summationdisplay\ni=1/integraldisplay\nΩε∂\n∂xiΦ(x−z)(1−ηε(z))(M(z\nε)−Mh)m0,i(z)dz/parenrightBig\n=:Pε+/tildewidePε,\nwhere/tildewidePεis the derivative of Newtonian potential in boundary layer t hat\ncan be estimated by Lemma 3. Define/tildewideU(y) as the solution of\n(63) ∆/tildewideU(y) =−(Ms(y)−Mh), U(y) isY-periodic in y,\nthen one can write from ( 7) and (63) that\n(64)Hd[Ms(y)](x\nε) =ε2∇2/tildewideU(x\nε)\n=ε2∇2/braceleftbig\nηε(x)/tildewideU(x\nε)/bracerightbig\n+ε2∇2/braceleftbig\n(1−ηε(x))/tildewideU(x\nε)/bracerightbig\n.\nNote that by Green’s representation formula,\nε2ηε(x)/tildewideU(x\nε) =−/integraldisplay\nΩΦ(x−z)∆/parenleftbig\nε2ηε(z)/tildewideU(z\nε)/parenrightbig\ndz.\nSubstituting the above formula into ( 64) and using the fact of /tildewideU\n−∆/parenleftbig\nε2ηε(z)/tildewideU(z\nε)/parenrightbig\n=ηε(M(z\nε)−Mh)−/braceleftbig\nε2∆ηε(z)·/tildewideU(z\nε)+2ε2∇ηε(z)·∇/tildewideU(z\nε)/bracerightbig\n,\nwe finally obtain\nm0·Hd[Ms(y)](x\nε)\n=m0·∇2/integraldisplay\nΩΦ(x−z)ηε(z)(M(z\nε)−Mh)dz+m0·/braceleftBig\nε2∇2/braceleftbig\n(1−ηε(x))/tildewideU(x\nε)/bracerightbig\n+∇2/integraldisplay\nΩΦ(x−z)/braceleftbig\nε2∆ηε(z)·/tildewideU(z\nε)+2ε2∇ηε(z)·∇/tildewideU(z\nε)/bracerightbig\ndz/bracerightBig\n=:Qε+/tildewideQε,\nwhere the boundary layer term /tildewideQεcan be estimated by Lemma 3and (60).\nNow in order to estimate the left-hand side of ( 56) in the Lemma, it only\nremains to consider the term Pε−Qε. Notice that one can write\n(65)Pε−Qε=n/summationdisplay\ni=1/integraldisplay\nΩ∂\n∂xi/braceleftbig\n∇xΦ(x−z)/bracerightbig\nηε(z)\n×(M(z\nε)−Mh)/parenleftbig\nm0,i(x)−m0,i(z)/parenrightbig\ndz.18 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN\nWith the notation ( 63), one has\n(M(z\nε)−Mh) =∇z·/braceleftbig\n∇z(ε2/tildewideU(z\nε))−∇x(ε2/tildewideU(x\nε))/bracerightbig\n.\nAfter substituting it into ( 65) and applying integration by parts, the leading\nintegrals are estimated directly by application of Lemma 2. /square\n5.Boundary Corrector\n5.1.Neumann corrector. Let usgive thedefinition of Neumanncorrector\nωNas\n(66) ωN=n/summationdisplay\ni=1/parenleftbig\nΦi−xi−εχε\ni/parenrightbig∂m0\n∂xi\nwith the notation χε\ni(x) =χi(x\nε),xiis theith component of spatial variable,\nand (Φ i)1≤i≤nis given by\n(67)\n\ndiv(aε∇Φi) = div(a0∇xi) in Ω,\n∂\n∂νεΦi=∂\n∂νhxion∂Ω.\nHere we denote∂\n∂νε=ν·aε∇,∂\n∂νh=ν·a0∇. Thusxiis the homogenized\nsolution of Φ ifrom (67). Since Φ iis unique up to a constant, one may\nassume Φ i(˜x)−˜x= 0 for some ˜x∈Ω. We introduce that Φ i−xi−εχε\nihas\nfollowing property.\nLemma 4. ForΦigiven in (67), under the smoothness assumption on A(y)\nand∂Ω, it holds that (see [12])\n(68) /ba∇dbl∇Φi−∇xi−ε∇χε\ni/ba∇dblL∞(Ω)≤C,/ba∇dbl∇2Φi/ba∇dblL∞(Ω)≤C,\nand\n(69) /ba∇dblΦi−xi/ba∇dblL∞(Ω)≤Cεln(ε−1+1),\nwhereCis independent of ε.\nProof.In fact, one has the estimate\n/vextendsingle/vextendsingle∇Φi−∇xi−ε∇χε\ni/vextendsingle/vextendsingle≤Cmax{1,ε[dist(x,∂Ω)]−1}\nfrom Lemma 7 .4.5 in [12]. This, together with the fact Φ i(˜x)−˜x= 0 , yields\n(69) by following integrals:\n|Φi(x)−xi|=/vextendsingle/vextendsingle/vextendsingle/integraldisplay1\n0d\nds/braceleftBig\nΦi/parenleftbig\n˜x+s(x−˜x)/parenrightbig\n−/parenleftbig\n˜xi+s(xi−˜xi)/parenrightbig/bracerightBig\nds/vextendsingle/vextendsingle/vextendsingle\n≤C/integraldisplay1\n0max{1,ε(1−s)−1}ds≤Cεln(ε−1+1),\nfor anyx∈Ω.HOMOGENIZATION OF THE LLG EQUATION 19\nAs for the second inequality in ( 68), we prove by making use of the Neu-\nmann function for operator Aεfrom [12] Section 7.4, denoted by Nε(x,z),\nand write from ( 67) that\n(70) Φi(x) =−n/summationdisplay\nk=1/integraldisplay\n∂Ωνk·a0\nkiNε(x,z)dz+1\n|∂Ω|/integraldisplay\n∂ΩΦi(z)dz.\nLet us denote the projection of∂\n∂xjalong∂\n∂νεbyPxj, and define P⊥\nxj=\n∂\n∂νε−Pxj, one can write for z∈∂Ω\n∂\n∂zjNε(x,z) =/parenleftbig\nPzj+P⊥\nzj/parenrightbig\nNε(x,z) =P⊥\nzjNε(x,z).\nNow applying∂2\n∂xl∂xjto both sides of ( 70), using above formula and integra-\ntion by parts on ∂Ω lead to\n(71)∂2\n∂xl∂xjΦi(x) =−n/summationdisplay\nk=1/integraldisplay\n∂ΩP⊥\nzlP⊥\nzjνk(z)·a0\nkiNε(x,z)dz,\nhere we have used the fact that P⊥\nzlis a tangential derivative on the bound-\nary, and Nε(x,z) =Nε(z,x) by the symmetry of Aε. (71) implies the\nsecond inequality in ( 68) by the smoothness assumption of boundary. /square\n5.2.A high-order modification. As noted in Section 1, we use ωNto\ncontrol the Neumann boundary data, and use a modification fun ctionωM\nto control the inhomogeneous term that induced by ωN, written in ( 15)\nseparately. In order to explain the construction of the modi fication function,\nwe point out that there are some bad terms appear whenwe calcu lateLεωN,\nwhich have no convergence in L2norm. Denote the bad terms by T1\nbadand\nT2\nbad, then they can be written as\n(72)T1\nbad=2n/summationdisplay\ni,j,k=1∂\n∂xk/braceleftBig\naε\nki/parenleftbig\nΦj−xj−εχε\nj/parenrightbig\n·∂2m0\n∂xi∂xj/bracerightBig\n−n/summationdisplay\ni,j,k=1/braceleftBig∂\n∂xkaε\nik·/parenleftbig\nΦj−xj−εχε\nj/parenrightbig/bracerightBig∂2m0\n∂xi∂xj,\nand\nT2\nbad=αn/summationdisplay\ni,j=1/parenleftbig\naε\nij∂ωN\n∂xi·∂{2/tildewidemε+ωN}\n∂xj/parenrightbig/parenleftbig/tildewidemε+ωN/parenrightbig\n.\nNotice that these terms cannot converge for the existence of∂ωN\n∂xiand∂aε\nik\n∂xk.\nNow let us rewrite T1\nbadandT2\nbadinto divergence form up to a small term.\nForT1\nbad, notice that/summationtextn\nk=1∂aε\nik\n∂xk=Aε(εχε\ni) from (33), substitute it into the20 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN\nsecond term on the right-hand side of ( 72), it leads to\n(73)T1\nbad=n/summationdisplay\nk,l=1∂\n∂xk/parenleftbig\naε\nklG1\nl(x)/parenrightbig\n+n/summationdisplay\ni,j=1εχε\ni·Aε/braceleftBig/parenleftbig\nΦj−xj−εχε\nj/parenrightbig\n·∂2m0\n∂xi∂xj/bracerightBig\n,\nwhereG1\nl(x) in the divergence term reads\nG1\nl(x) =2n/summationdisplay\nj=1/parenleftbig\nΦj−xj−εχε\nj/parenrightbig∂2m0\n∂xl∂xj+n/summationdisplay\ni,j=1/braceleftBig∂\n∂xl(εχε\ni)\n·/parenleftbig\nΦj−xj−εχε\nj/parenrightbig\n−εχε\ni·∂\n∂xl/parenleftbig\nΦj−xj−εχε\nj/parenrightbig/bracerightBig∂2m0\n∂xi∂xj.\nAs forT2\nbad, a direct calculation implies it can be rewritten as\nT2\nbad=n/summationdisplay\nk,l=1∂\n∂xk/parenleftbig\naε\nklG2\nl(x)/parenrightbig\n−αn/summationdisplay\ni,j=1aε\nij/parenleftbig\nωN·∂{2/tildewidemε+ωN}\n∂xj/parenrightbig/parenleftbig\n/tildewidemε+ωN/parenrightbig\n−αn/summationdisplay\ni,j=1/parenleftbig\nωN·Aε{2/tildewidemε+ωN}/parenrightbig/parenleftbig\n/tildewidemε+ωN/parenrightbig\n, (74)\nwhereG2\nl(x) in the divergence term can be calculated by\nG2\nl(x) =α/parenleftbig\nωN·∂{2/tildewidemε+ωN}\n∂xl/parenrightbig/parenleftbig\n/tildewidemε+ωN/parenrightbig\n.\nMoreover, one can apply Lemma 4to deduce from ( 73) and (74) that for\ni= 1,2\n(75)/vextenddouble/vextenddoubleTi\nbad−n/summationdisplay\nk,l=1∂\n∂xk/parenleftbig\naε\nklGi\nl(x)/parenrightbig/vextenddouble/vextenddouble\nL2(Ω)≤Cε[ln(ε−1+1)]2,\nhere we have use the fact Aε(Φi−xi−εχε\ni) = 0. Constant Cdepends on\n/ba∇dbl∇m0/ba∇dblW2,∞(Ω),/ba∇dblAε/tildewidemε/ba∇dblL∞(Ω), but is independent of ε.\nNow we define the modification function ωM=ω1\nM+ω2\nM, whereωi\nM,\ni= 1,2 satisfies\n(76)\n\nAεωi\nM=n/summationdisplay\nk,l=1∂\n∂xk/parenleftbig\naε\nklGi\nl(x)/parenrightbig\nin Ω,\n∂\n∂νεωi\nM=n/summationdisplay\nk,l=1νk·aε\nklGi\nl(x) on∂Ω,\nhereνkis thek-th component of vector ν. By the Lax-Milgram theorem,\nonecan obtain theexistence anduniquenessof ωi\nM,i= 1,2 upto aconstant.\nLet/integraltext\n∂Ωωi\nMdx= 0, then the correctors yield the following estimate.HOMOGENIZATION OF THE LLG EQUATION 21\nLemma 5. Forωi\nM,i= 1,2defined in (76), under smooth assumption of\nm0and∂Ω, it holds that for n≤3\n(77) /ba∇dblωi\nM/ba∇dblL∞(Ω)≤Cε,/ba∇dbl∇ωi\nM/ba∇dblL∞(Ω)≤Cεln(ε−1+1),\nwhereCdepends on /ba∇dbl∇m0/ba∇dblW3,∞(Ω)and is independent of ε.\nProof.Here we use the Neumann function Nε(x,z) for operator Aε, see [12]\nSection 7.4. ( 76) implies for i= 1,2\nωi\nM=n/summationdisplay\nk,l=1/integraldisplay\nΩaε\nkl∂\n∂zk/braceleftbig\nNε(x,z)/bracerightbig\nGi\nl(z)dz.\nUsing the fact ∇zNε(x,z)≤C|x−z|1−n, see [12] p.159, we can derive\n/ba∇dblωi\nM/ba∇dblL∞(Ω)≤C/ba∇dblGi\nl/ba∇dblL∞(Ω)≤Cεln(ε−1+1).\nAs for the second inequality in ( 77), it follows from [ 12], Lemma 7.4.7:\n/ba∇dbl∇ωi\nM/ba∇dblL∞(Ω)≤Cln(ε−1+1)/ba∇dblGi\nl/ba∇dblL∞(Ω)+Cε/ba∇dbl∇Gi\nl/ba∇dblL∞(Ω)\nwith the estimate\n/ba∇dbl∇Gi\nl/ba∇dblL∞(Ω)≤C,\nfrom Lemma 4. Here constant Cdepends on /ba∇dbl∇m0/ba∇dblW3,∞(Ω),/ba∇dbl∇(Φj−xj−\nεχε\nj)/ba∇dblL∞(Ω), but is independent of εby Lemma 4. /square\n5.3.Estimates of initial-boundary data.\nTheorem 4. Foreε\nbgiven in (29), withωb=ωN−ωMgiven in (66), under\nthe smooth assumption, it holds that initial data of eε\nbsatisfies\n(78) /ba∇dbleε\nb(x,0)/ba∇dblH1(Ω)≤Cεln(ε−1+1),\nwhereCdepends on /ba∇dbl∇2m0\ninit/ba∇dblH1(Ω)and is independent of ε. And for the\nboundary data, it holds that\n/ba∇dbl∂\n∂νεeε\nb/ba∇dblW1,∞(0,T;B−1/2,2(∂Ω))≤Cεln(ε−1+1), (79)\nwhereCdepends on /ba∇dbl∇2m0/ba∇dblW1,∞(0,T;B−1/2,2(∂Ω))and is independent of ε.\nProof.We rewrite eε\nbfrom its definition as\n(80) eε\nb=mε−m0−n/summationdisplay\ni=1(Φi−xi)∂m0\n∂xi−ε2m2+ωM.\nFirst, let us prove ( 78). By the initial condition of mεandm0, along with\nthe smoothness condition, one can check\n(81)eε\nb(x,0) =mε\ninit−m0\ninit−n/summationdisplay\ni=1(Φi−xi)∂m0\ninit\n∂xi−ε2m2,init+ωM,init\nwith notation m2,init,ωM,initdefined the same as m2,ωMexcept we replace\nm0bym0\ninit. From the assumption ( 16)-(17),m0\ninitis the homogenized22 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN\nsolution of mε\ninit, by classical homogenization theorem of elliptic problems\nin [12], one has\n/ba∇dblmε\ninit−m0\ninit−(Φ−x)∇m0\ninit/ba∇dblH1(Ω)≤Cεln(ε−1+1).\nAlso note that by definition of m2and Lemma 5, one has\n/ba∇dblε2m2,init/ba∇dblH1(Ω)+/ba∇dblωM,init/ba∇dblH1(Ω)≤Cεln(ε−1+1)/ba∇dbl∇2m0\ninit/ba∇dblH1(Ω).\nTherefore the inequality ( 78) follows from ( 81) and above estimates.\nNotice that by the boundary condition of m0in (10) and Φ kin (67), we\nhave\nn/summationdisplay\nk=1∂\n∂νε(Φk−xk)·∂m0\n∂xk=−∂\n∂νεm0,x∈∂Ω,\ntherefore applying∂\n∂νεto both sides of ( 80) leads to\n(82)\n∂\n∂νεeε\nb=−n/summationdisplay\nk=1(Φk−xk)·∂\n∂νε(∂m0\n∂xk)−ε2∂\n∂νεm2+∂\n∂νεωM,x∈∂Ω.\nUnder the smoothness assumption of m0andaε, we can also derive the\nsmoothness of ( Φ−x),m2andωMover¯Ω. Thus by Lemma 4and Lemma\n5, one can directly obtain from ( 82)\n/ba∇dbl∂\n∂νεeε\nb/ba∇dblB−1/2,2(∂Ω)≤C/ba∇dblΦ−x/ba∇dblL∞(Ω)+Cε2/ba∇dbl∇m2/ba∇dblL∞(Ω)+C/ba∇dbl∇ωM/ba∇dblL∞(Ω)\n≤Cεln(ε−1+1),\nwhereCdepends on /ba∇dbl∇2m0/ba∇dblB−1/2,2(∂Ω)and/ba∇dbl∇2(∂tm0)/ba∇dblB−1/2,2(∂Ω). The\nsame argument for /ba∇dbl∂\n∂νε(∂teε\nb)/ba∇dblB−1/2,2(∂Ω)leads to ( 79).\n/square\n5.4.Estimates of inhomogeneous terms. Fromtheabovedefinitionand\nproperty, we get the main result of this section.\nTheorem 5. Foreε\nbgiven in (29), withωb=ωN−ωMgiven in (66)and\nLεdefined in (27), under the smooth assumption, it holds that\n/ba∇dbl∂tωb/ba∇dblL2(Ω)≤Cεln(ε−1+1), (83)\n/ba∇dblLεωb/ba∇dblL2(Ω)≤Cε[ln(ε−1+1)]2+C/ba∇dblmε−/tildewidemε−ωb/ba∇dblH1(Ω), (84)\nwhereCdepends on /ba∇dblmε/ba∇dblH1(Ω),/ba∇dbl∇2m0/ba∇dblW2,∞(Ω),/ba∇dbl∇(∂tm0)/ba∇dblW1,∞(Ω)and is\nindependent of ε. Moreover, one has the estimate\n(85) /ba∇dblAεeε\nb/ba∇dblL2(Ω)≤Cln(ε−1+1),\nwhereCdepends on /ba∇dblAεmε/ba∇dblL2(Ω),/ba∇dbl∇2m0/ba∇dblW2,∞(Ω),/ba∇dbl∇(∂tm0)/ba∇dblW1,∞(Ω)and\nis independent of ε.HOMOGENIZATION OF THE LLG EQUATION 23\nProof.Inordertoestimateleft-handsideof ( 84), wesplititas /ba∇dblLεωb/ba∇dblL2(Ω)=\n/ba∇dblLεωN−LεωM/ba∇dblL2(Ω)≤R1+R2+R3, with\n\n\nR1=/vextenddouble/vextenddoubleLεωN−/braceleftbig\nAεωN−mε×AεωN−D2(ωN)/bracerightbig/vextenddouble/vextenddouble\nL2(Ω),\nR2=/vextenddouble/vextenddouble/braceleftbig\nAεω1\nM−mε×Aεω1\nM−Aεω2\nM/bracerightbig\n−LεωM/vextenddouble/vextenddouble\nL2(Ω),\nR3=/vextenddouble/vextenddoubleAε(ωN−ω1\nM)−mε×Aε(ωN−ω1\nM)−/parenleftbig\nD2(ωN)−Aεω2\nM/parenrightbig/vextenddouble/vextenddouble\nL2(Ω).\nOne can check by definition of LεthatR1does not have derivative of ωN,\nandR2onlycontains first-orderderivativeof ωM, thustheycanbeestimated\nby Lemma 4and Lemma 5as\nR1+R2≤Cεln(ε−1+1),\nwhereCdepends on /ba∇dblmε/ba∇dblH1(Ω),/ba∇dbl∇2m0/ba∇dblW2,∞(Ω). As forR3, in the view of\n(75), it can be bounded from above by\n/ba∇dblAεωN−T1\nbad/ba∇dblL2(Ω)+/ba∇dblmε×(AεωN−T1\nbad)/ba∇dblL2(Ω)+/ba∇dblD2(ωN)−T2\nbad/ba∇dblL2(Ω).\nIn above terms, the first term can be estimated by applying Aε(Φi−xi−\nεχε\ni) = 0 to derive that /ba∇dblAεωN− T1\nbad/ba∇dblL2(Ω)≤Cεln(ε−1+ 1), with C\nindependent of ε. The same result holds for the second term. Now let us\nestimate the last term. We assert that\n(86)/ba∇dblD2(ωN)−T2\nbad/ba∇dblL2(Ω)≤Cεln(ε−1+1)+C/ba∇dblmε−/tildewidemε−ωb/ba∇dblH1(Ω),\nwhereCdepends on /ba∇dbl∇2m0/ba∇dblW1,∞(Ω)and/ba∇dblmε/ba∇dblH1(Ω). In fact, we denote the\nterms in D2(ωN) that contain derivatives of ωNby/tildewideD2(ωN), then it reads\n/tildewideD2(ωN) =αn/summationdisplay\ni,j=1/parenleftbig\naε\nij∂ωN\n∂xi·∂mε\n∂xj+aε\nij∂/tildewidemε\n∂xi·∂ωN\n∂xj/parenrightbig\nmε,\nand one can check the remaining terms satisfy\n/ba∇dblD2(ωN)−/tildewideD2(ωN)/ba∇dblL2(Ω)≤C(1+/ba∇dbl∇m0/ba∇dbl2\nL∞(Ω))/ba∇dblωN/ba∇dblL2(Ω)≤Cεln(ε−1+1).\nSubstituting mε= (mε−/tildewidemε−ωN)+(/tildewidemε+ωN) into/tildewideD2(ωN), one can write\n/tildewideD2(ωN) =T2\nbad+αn/summationdisplay\ni,j=1/parenleftbig\naε\nij∂ωN\n∂xi·∂{mε−/tildewidemε−ωN}\n∂xj/parenrightbig\nmε\n+αn/summationdisplay\ni,j=1/parenleftbig\naε\nij∂ωN\n∂xi·∂{/tildewidemε+ωN}\n∂xj/parenrightbig/parenleftbig\nmε−/tildewidemε−ωN/parenrightbig\n.\nHence it follows that\n/ba∇dbl/tildewideD2(ωN)−T2\nbad/ba∇dblL2(Ω)≤C(1+/ba∇dbl∇m0/ba∇dbl2\nL∞(Ω))/ba∇dblmε−/tildewidemε−ωN/ba∇dblH1(Ω).\nThe assertion is proved.24 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN\nAs for (83), one can deduce from Lemma 4and the proof of Lemma 5to\nobtain\n/ba∇dbl∂tωN/ba∇dblL∞(Ω)≤Cεln(ε−1+1),\n/ba∇dbl∂tωi\nM/ba∇dblL∞(Ω)≤C/ba∇dbl∂tGi\nl/ba∇dblL∞(Ω)≤Cεln(ε−1+1),\nwhereCdepends on /ba∇dbl∇(∂tm0)/ba∇dblW1,∞(Ω). In order to prove ( 85), we use\nLemma4, Lemma 5and definition of /tildewidemε, to deduce the estimates\n/ba∇dblAε/tildewidemε/ba∇dblL2(Ω)+/ba∇dblAεωb/ba∇dblL2(Ω)≤Cln(ε−1+1),\nthen (85) follows with some constant Cdepending on /ba∇dblAεmε/ba∇dblL2(Ω). There-\nfore Theorem is proved. /square\n6.Stability Analysis\nIn this section, we will discuss the stability of following i nitial-boundary\nproblem, which is motivated by equation ( 26):\n(87)\n\n∂te−Lε(e) =Fin Ω,\nν·aε∇e=gon∂Ω,\ne(0,x) =hin Ω.\nThe following two inequalities will be used. The first inequa lity is motivated\nbyW1,pestimate for oscillatory elliptic problem.\nLemma 6. Assumeu∈H2(Ω),ν·aε∇u=gon∂Ω, withg∈B−1/2,2(∂Ω),\nthen it holds that for n≤3,\n/ba∇dbl∇u/ba∇dblL6(Ω)≤C/ba∇dblAεu/ba∇dblL2(Ω)+C/ba∇dblg/ba∇dblB−1/2,2(∂Ω),\nmoreover, if g= 0, then one has for n≤3\n/ba∇dbl∇u/ba∇dblL6(Ω)≤C/ba∇dblAεu/ba∇dblL2(Ω).\nConstant Cis independent of ε.\nProof.We refer that Lemma 6is a direct corollary of Theorem 6.3.2 in [ 12].\nOne can find the proof in [ 12] [Pages 144-152]. /square\nWe also introduce Sobolev inequality with small coefficient w henn= 2.\nLemma 7. For any function f∈H2(Ω), one has when n= 2\n/ba∇dblf/ba∇dblL∞(Ω)≤Cln(ε−1+1)/ba∇dblf/ba∇dblH1(Ω)+ε/ba∇dblAεf/ba∇dblL2(Ω),\nwhere constant Cis independent of ε.\nProof.Using the Neumann function, see [ 12] Section 7.4, one has\nf=−/integraldisplay\nΩ∇zNε(x,z)·aε∇f(z)dz+1\n|∂Ω|/integraldisplay\n∂Ωfdz=:P1+P2.HOMOGENIZATION OF THE LLG EQUATION 25\nApplying cut-off function φε=φε(x−z), the first term yields by integration\nby parts\nP1=−/integraldisplay\nΩ(1−φε)∇zNε(x,z)·aε∇f(z)dz+/integraldisplay\nΩφεNε(x,z)·Aεf(z)dz\n+/integraldisplay\nΩ∇zφε(x−z)·Nε(x,z)·aε∇f(z)dz\n≤Cε/ba∇dblAεf/ba∇dblL2(Ω)+Cln(ε−1+1)/ba∇dbl∇f/ba∇dblL2(Ω),\nhere in the last line we have used the fact ∇zNε(x,z)≤C|x−y|−1and\nNε(x,z)≤C{1+ln[|x−z|−1]}forn= 2, see [ 12] page 159. As for P2, one\nhas by trace inequality that P2≤C/ba∇dblf/ba∇dblH1(Ω). The Lemma is proved. /square\nNow let us give the stability of system ( 87) in terms of h,g,Fin\nL∞(0,T;L2(Ω)) and L∞(0,T;H1(Ω)) norm, respectively.\n6.1.Stability in L∞(0,T;L2(Ω)).\nTheorem 6. Lete∈L∞(0,T;H2(Ω))be a strong solution to (87). Assume\nh∈L2(Ω),g∈L∞(0,T;B−1/2,2(∂Ω)), andF=F1+F2satisfies\n(88) F1∈L2(0,T;Lσ(Ω)),F2∈L2(0,T;L2(Ω))\nwithσ= 1whenn= 1,2, andσ= 6/5whenn= 3, then it holds that, for\nany0≤t≤T\n(89)/ba∇dble/ba∇dbl2\nL∞(0,T;L2(Ω))+/ba∇dbl∇e/ba∇dbl2\nL2(0,T;L2(Ω))\n≤Cδ/parenleftBig\n/ba∇dblh/ba∇dbl2\nL2(Ω)+/ba∇dblg/ba∇dbl2\nL2(0,T;B−1/2,2(∂Ω))+γ(ε)/ba∇dblF1/ba∇dbl2\nL2(0,T;Lσ(Ω))/parenrightBig\n+δ/ba∇dblF2/ba∇dbl2\nL2(0,T;L2(Ω))+ε2/ba∇dblAεe/ba∇dbl2\nL2(0,T;L2(Ω)),\nfor any small δ >0, where\n/braceleftBigg\nγ(ε) = 1, whenn= 1,3,\nγ(ε) = [ln(ε−1+1)]2,whenn= 2.\nCδis a constant depending on /ba∇dbl∇mε/ba∇dblL4(Ω),/ba∇dbl∇/tildewidemε/ba∇dblL4(Ω), but is independent\noftandε.\nProof.The inner product between ( 87) andeinL2(Ω) leads to\n(90)1\n2d\ndt/integraldisplay\nΩ|e|2dx−α/integraldisplay\nΩ/tildewideHε\ne(e)·edx\n=−/integraldisplay\nΩD1(e)·edx−/integraldisplay\nΩD2(e)·edx−/integraldisplay\nΩF·edx.\nNow let us give the estimates to ( 90) term by term. Integration by parts for\nthe second term on the left-hand side yields\n−/integraldisplay\nΩ/tildewideHε\ne(e)·edx≥n/summationdisplay\ni,j=1/integraldisplay\nΩaε\nij∂e\n∂xi·∂e\n∂xjdx−/integraldisplay\n∂Ωg·edx−C/integraldisplay\nΩ|e|2dx26 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN\nwith the boundary term satisfying\n/integraldisplay\n∂Ωg·edx≤ /ba∇dblg/ba∇dblB−1/2,2(∂Ω)/ba∇dble/ba∇dblB1/2,2(∂Ω)≤C/ba∇dblg/ba∇dblB−1/2,2(∂Ω)/ba∇dble/ba∇dblH1(Ω).\nBy integration by parts and the same argument for the boundar y term, the\nfirst term on the right-hand side can be estimated as\n−/integraldisplay\nΩD1(e)·edx≤C/integraldisplay\nΩ|e|2dx+δC/integraldisplay\nΩ|∇e|2dx−C/ba∇dblg/ba∇dblB−1/2,2(∂Ω)/ba∇dble/ba∇dblH1(Ω).\nFor the second term on the right-hand side of ( 90), using the estimates\n/integraldisplay\nΩ/parenleftbig\nBε[e,mε]/parenrightbig\nmε·edx≤C/ba∇dbl∇mε/ba∇dblL4(Ω)/ba∇dble/ba∇dblL4(Ω)/ba∇dbl∇e/ba∇dblL2(Ω)\n+C/ba∇dblmε/ba∇dblL4(Ω)/ba∇dble/ba∇dblL4(Ω)/ba∇dble/ba∇dblL2(Ω),/integraldisplay\nΩgε\nl[/tildewidemε]e·edx≤C/ba∇dbl∇/tildewidemε/ba∇dbl2\nL4(Ω)/ba∇dble/ba∇dbl2\nL4(Ω)+C/ba∇dbl/tildewidemε/ba∇dbl2\nL4(Ω)/ba∇dble/ba∇dbl2\nL4(Ω),\nand the same argument can be applied to the other terms, we fina lly obtain\nby Sobolev inequality\n−/integraldisplay\nΩD2(e)·edx≤C+C/integraldisplay\nΩ|e|2dx+δC/integraldisplay\nΩ|∇e|2dx,\nwhereC=C0/parenleftbig\n1+/ba∇dbl∇mε/ba∇dbl2\nL4(Ω)+/ba∇dbl∇/tildewidemε/ba∇dbl2\nL4(Ω)/parenrightbig\n. For the last term in ( 90),\nby the assumption ( 88), we apply Sobolev inequality for n= 1,3, and apply\nLemma7forn= 2, it follows that\n−/integraldisplay\nΩF1·edx≤\n\nC/ba∇dblF1/ba∇dbl2\nL1(Ω)+δ/ba∇dble/ba∇dbl2\nH1(Ω), n = 1\nC[ln(ε−1+1)]2/ba∇dblF1/ba∇dbl2\nL1(Ω)\n+δ/ba∇dble/ba∇dbl2\nH1(Ω)+ε2/ba∇dblAεe/ba∇dbl2\nL2,n= 2\nC/ba∇dblF1/ba∇dbl2\nL6/5(Ω)+δ/ba∇dble/ba∇dbl2\nH1(Ω), n = 3\n−/integraldisplay\nΩF2·edx≤δ∗/ba∇dblF2/ba∇dbl2\nL2(Ω)+C/ba∇dble/ba∇dblL2(Ω),\nwith any small δ,δ∗>0. Substituting above estimates, one can derive from\n(90) that\n1\n2d\ndt/integraldisplay\nΩ|e|2dx+(αamin−2δ)/integraldisplay\nΩ|∇e|2dx≤C/integraldisplay\nΩ|e|2dx+C/ba∇dblg/ba∇dbl2\nB−1/2,2(∂Ω)\n+C[ln(ε−1+1)]2/ba∇dblF1/ba∇dbl2\nLσ(Ω)+δ∗/ba∇dblF2/ba∇dbl2\nL2(Ω)+ε2/ba∇dblAεe/ba∇dbl2\nL2.\nThen (89) follows directly by taking δsmall enough, and the application of\nGr¨ onwall’s inequality. /squareHOMOGENIZATION OF THE LLG EQUATION 27\n6.2.Stability in L∞(0,T;H1(Ω)).\nTheorem 7. Lete∈L∞(0,T;H2(Ω))be a strong solution to (87). Assume\nh∈H1(Ω),g∈H1(0,T;B−1/2,2(∂Ω)), andF∈L2(0,T;L2(Ω)), it holds\n(91)\n/ba∇dbl∇e/ba∇dbl2\nL∞(0,T;L2(Ω))≤C/parenleftbig\n/ba∇dblh/ba∇dbl2\nH1(Ω)+/ba∇dblg/ba∇dbl2\nH1(0,T;B−1/2,2(∂Ω))+/ba∇dblF/ba∇dbl2\nL2(0,T;L2(Ω))/parenrightbig\n,\nwhereCdepends on /ba∇dbl∇mε/ba∇dblL4(Ω),/ba∇dbl∇/tildewidemε/ba∇dblL4(Ω)and/ba∇dblHε\ne(/tildewidemε)/ba∇dblL4(Ω), but is\nindependent of tandε.\nProof.The inner product between ( 87) and/tildewideHε\ne(e) inL2(Ω) leads to\n(92)−/integraldisplay\nΩ∂te·/tildewideHε\ne(e)dx+α/integraldisplay\nΩ/tildewideHε\ne(e)·/tildewideHε\ne(e)dx\n=/integraldisplay\nΩD1(e)·/tildewideHε\ne(e)dx+/integraldisplay\nΩD2(e)·/tildewideHε\ne(e)dx+/integraldisplay\nΩF·/tildewideHε\ne(e)dx.\nIn the following we give the estimates to ( 92) term by term. Note that\nintegration by parts yields\n−/integraldisplay\nΩ∂te·/tildewideHε\ne(e)dx=d\ndtGε\nL[e]−/integraldisplay\n∂Ω∂te·gdx,\n=d\ndtGε\nL[e]−∂t/integraldisplay\n∂Ωe·gdx−C/ba∇dbl∂tg/ba∇dblB−1/2,2(∂Ω)/ba∇dble/ba∇dblH1(Ω).\nUsing the fact mε×/tildewideHε\ne(e)·/tildewideHε\ne(e) = 0, the first term on the right-hand side\nof (92) can be estimate by Sobolev inequality as\n/integraldisplay\nΩD1(e)·/tildewideHε\ne(e)dx≤C/ba∇dblHε\ne(/tildewidemε)/ba∇dblL4(Ω)/ba∇dble/ba∇dblL4(Ω)/ba∇dbl/tildewideHε\ne(e)/ba∇dblL2(Ω)\n≤C/ba∇dble/ba∇dbl2\nH1(Ω)+δC/ba∇dbl/tildewideHε\ne(e)/ba∇dbl2\nL2(Ω),\nwhereC=C0/parenleftbig\n1+/ba∇dblHε\ne(/tildewidemε)/ba∇dbl2\nL4(Ω)/parenrightbig\n. For the second term on the right-hand\nside of (92), note that we have the estimate\n/integraldisplay\nΩ/parenleftbig\nBε[e,mε]/parenrightbig\nmε·/tildewideHε\ne(e)dx≤C/ba∇dbl∇mε/ba∇dblL4(Ω)/ba∇dbl∇e/ba∇dblL4(Ω)/ba∇dbl/tildewideHε\ne(e)/ba∇dblL2(Ω)\n+C/ba∇dblmε/ba∇dblL4(Ω)/ba∇dble/ba∇dblL4(Ω)/ba∇dbl/tildewideHε\ne(e)/ba∇dblL2(Ω),\nin which one can deduce\n/ba∇dbl∇e/ba∇dblL4(Ω)≤/ba∇dbl∇e/ba∇dbl1/3\nL2(Ω)/ba∇dbl∇e/ba∇dbl2/3\nL6(Ω)\n≤C/ba∇dbl∇e/ba∇dbl1/3\nL2(Ω)(1+/ba∇dbl/tildewideHε\ne(e)/ba∇dblL2(Ω)+/ba∇dblg/ba∇dblB−1/2,2(∂Ω))2/3,\nusing interpolation inequality and Lemma 6. The other terms can be esti-\nmated in the same fashion. After the application of Young’s i nequality, one\nfinally obtains\n/integraldisplay\nΩD2(e)·/tildewideHε\ne(e)dx≤C/ba∇dble/ba∇dbl2\nH1(Ω)+δC/ba∇dbl/tildewideHε\ne(e)/ba∇dbl2\nL2(Ω)+C/ba∇dblg/ba∇dbl2\nB−1/2,2(∂Ω),28 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN\nwhereC=C0(1+/ba∇dbl∇mε/ba∇dbl2\nL4(Ω)+/ba∇dbl∇/tildewidemε/ba∇dbl2\nL4(Ω)). Substitutingaboveestimates\ninto (92), we arrive at\nd\ndtGε\nL[e]+(α−Cδ)/integraldisplay\nΩ|/tildewideHε\ne(e)|2dx−∂t/integraldisplay\n∂Ωe·gdx\n≤C/parenleftbig\n/ba∇dble/ba∇dbl2\nH1(Ω)+/ba∇dblF/ba∇dbl2\nL2(Ω)+/ba∇dblg/ba∇dblB−1/2,2(∂Ω)+/ba∇dbl∂tg/ba∇dblB−1/2,2(∂Ω)/parenrightbig\n.\nIntegrating the above inequality over [0 ,t] with 0 < t < T and using the\nfacts/integraldisplay\n∂Ωe·gdx≤δ/ba∇dble/ba∇dbl2\nH1(Ω)+C/ba∇dblg/ba∇dbl2\nB−1/2,2(∂Ω),\nGε\nL[e]≥amin\n2/ba∇dbl∇e/ba∇dbl2\nL2(Ω)−C,\none can finally derive\n(amin\n2−δ)/ba∇dbl∇e(t)/ba∇dbl2\nL2(Ω)+(α−Cδ)/integraldisplayt\n0/ba∇dbl/tildewideHε\ne(e)/ba∇dbl2\nL2(Ω)dτ\n≤C/integraldisplayt\n0/parenleftbig\n/ba∇dble/ba∇dbl2\nL2(Ω)+/ba∇dbl∇e/ba∇dbl2\nL2(Ω)+/ba∇dblF/ba∇dbl2\nL2(Ω)+/ba∇dbl∂tg/ba∇dblB−1/2,2(∂Ω)/parenrightbig\ndτ+J(h),\nwhereJ(h) yields\nJ(h) =Gε\nL[h]−/integraldisplay\n∂Ωh·∂\n∂νεhdx≤C/ba∇dblh/ba∇dbl2\nH1(Ω)+/ba∇dbl∂\n∂νεh/ba∇dbl2\nB−1/2,2(∂Ω).\n(91) is then derived after taking δsmall enough and the application of\nGr¨ onwall’s inequality. /square\n7.Regularity\nIn the estimate of boundary corrector and stability analysi s by Theorem\n5, Theorem 6, Theorem 7, the constant we deduced rely on the value of\n/ba∇dblAεmε/ba∇dblL2(Ω)and/ba∇dbl∇mε/ba∇dblL6(Ω). In this section, we introduce the uniform\nregularity on mε, over a time interval independent of ε. For this purpose,\nwe intend to derive a structure-preserving energy inequali ty, in which the\ndegenerate term are kept in the energy.\nFirst, let us introduce an interpolation inequality of the e ffective field\nHε\ne(mε) for some S2-valued function mε, which is the generalization of ( 14).\nThe following estimates will be used:\na−1\nmax/ba∇dblmε·Aεmε/ba∇dbl3\nL3(Ω)≤ /ba∇dbl∇mε/ba∇dbl6\nL6(Ω)≤a−1\nmin/ba∇dblAεmε/ba∇dbl3\nL3(Ω), (93)\n/ba∇dblAεmε/ba∇dblLp(Ω)−Cp≤ /ba∇dblHε\ne(mε)/ba∇dblLp(Ω)≤ /ba∇dblAεmε/ba∇dblLp(Ω)+Cp, (94)\nwith 1< p <+∞, here the first line follows from the fact −aε|∇mε|2=\nmε·Aεmεby|mε|= 1 and assumption of aεin (1), and in second line the\nestimate ( 4) is used. We also introduce a orthogonal decomposition to an y\nvectoraas\n(95) a= (mε·a)mε−mε×(mε×a).HOMOGENIZATION OF THE LLG EQUATION 29\nLemma 8. Givenmε∈H3(Ω)that satisfies |mε|= 1and Neumann bound-\nary condition ν·aε∇mε= 0, then it holds for n≤3and any 0< δ <1\n(96)/ba∇dblHε\ne(mε)/ba∇dbl3\nL3(Ω)≤Cδ+Cδ/ba∇dblHε\ne(mε)/ba∇dbl6\nL2(Ω)+δ/ba∇dblmε×∇Hε\ne(mε)/ba∇dbl2\nL2(Ω),\nwhereCδis a constant depending on δbut independent of ε.\nProof.Applying decomposition ( 95) by taking a=Hε\ne(mε), one can write\n(97)/ba∇dblHε\ne(mε)/ba∇dbl3\nL3(Ω)≤/integraldisplay\nΩ|mε·Hε\ne(mε)|3dx\n+/integraldisplay\nΩ|mε×Hε\ne(mε)|3dx=:I1+I2.\nNow let us estimate the right-hand side of ( 97) separately. For I1, we apply\n(93) and Remark 6to derive\nI1≤C+C/ba∇dbl∇mε/ba∇dbl6\nL6(Ω)≤C+C/ba∇dblHε\ne(mε)/ba∇dbl6\nL2(Ω).\nAs forI2, we have by Sobolev inequality for n≤3\nI2≤C+C/ba∇dblmε×Hε\ne(mε)/ba∇dbl6\nL2(Ω)+δ∗/ba∇dblmε×Hε\ne(mε)/ba∇dbl2\nH1(Ω),\nhere in the last term, we can apply ( 93)-(94) to derive:\nδ∗/ba∇dbl∇mε×Hε\ne(mε)/ba∇dbl2\nL2(Ω)≤δ∗/ba∇dbl∇mε/ba∇dbl6\nL6(Ω)+δ∗/ba∇dblHε\ne(mε)/ba∇dbl3\nL3(Ω)\n≤C+Cδ∗/ba∇dblHε\ne(mε)/ba∇dbl3\nL3(Ω).\nNow let us turn back to ( 97), we finally obtain\n(1−Cδ∗)/ba∇dblHε\ne(mε)/ba∇dbl3\nL3(Ω)≤C+C/ba∇dblHε\ne(mε)/ba∇dbl6\nL2(Ω)+δ∗/ba∇dblmε×∇Hε\ne(mε)/ba∇dbl2\nL2(Ω).\nLetδ∗<1\n2C, one can derive ( 96) withδ=δ∗/(1−Cδ∗)<1. /square\nNow let us recall some energy property of LLG equation, and gi ve the\nuniform regularity result. Using the formula of vector oute r production\n(98) a×(b×c) = (a·c)b−(a·b)c,\none can rewrite LLG equation ( 22) into a degenerate form\n(99) ∂tmε+αmε×/parenleftbig\nmε×Hε\ne(mε)/parenrightbig\n+mε×Hε\ne(mε) = 0.\nMultiplying ( 99) byHε\ne(mε) and integrating over (0 ,t), we derive the energy\ndissipation identity\n(100) Gε\nL[mε(t)]+α/integraldisplayt\n0/ba∇dblmε×Hε\ne(mε)/ba∇dbl2\nL2(Ω)dτ=Gε\nL[mε(0)],\ntogether ( 99) and (100) leads to the integrable of kinetic energy\nα\n1+α2/integraldisplayt\n0/ba∇dbl∂tmε/ba∇dbl2\nL2(Ω)dτ≤ Gε\nL[mε(0)]. (101)\nTheenergyidentity ( 100)impliestheuniformregularityof /ba∇dblmε×Aεmε/ba∇dbl2\nL2(Ω),\nhowever, this is not enough to obtain the regularity of /ba∇dblAεmε/ba∇dbl2\nL2(Ω)due to\nthe degeneracy. In this end we introduce that:30 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN\nTheorem 8. Letmε∈L2([0,T];H3(Ω))be a solution to (8). Assume n≤\n3, then there exists T∗∈(0,T]independent of ε, such that for 0≤t≤T∗,\n/ba∇dblAεmε(t)/ba∇dbl2\nL2(Ω)+/integraldisplayt\n0/vextenddouble/vextenddoublemε×∇Hε\ne(mε)(τ)/vextenddouble/vextenddouble2\nL2(Ω)dτ≤C,\nand therefore, by the Sobolev-type inequality in Remark 6,\n/ba∇dbl∇mε(·,t)/ba∇dbl2\nL6(Ω)≤C,\nwhereCis a constant independent of εandt.\nProof.Applying ∇to (99) and multiplying by aε∇Hε\ne(mε) lead to\n(102)−/integraldisplay\nΩ∇(∂tmε)·aε∇Hε\ne(mε)dx\n=α/integraldisplay\nΩ∇/parenleftbig\nmε×(mε×Hε\ne(mε))/parenrightbig\n·aε∇Hε\ne(mε)dx\n+n/summationdisplay\ni,j=1/integraldisplay\nΩ∂\n∂ximε×Hε\ne(mε)·aε\nij∂\n∂xjHε\ne(mε)dx=:J1+J2.\nDenote Γε(mε) =Hε\ne(mε)−Aεmε. After integration by parts, the left-hand\nside of (102) becomes\n−/integraldisplay\nΩ∇(∂tmε)·aε∇Hε\ne(mε)dx=/integraldisplay\nΩAε(∂tmε)·/parenleftbig\nAεmε+Γε(mε)/parenrightbig\ndx,\nwhere the right-hand side can be rewritten as\n1\n2d\ndt/integraldisplay\nΩ|Aεmε|2dx+d\ndt/integraldisplay\nΩAεmε·Γε(mε)dx−/integraldisplay\nΩAε(mε)·Γε(∂tmε)dx.\nNow let us consider the right-hand side of ( 102). ForJ1, one can derive by\nswapping the order of mixed product\nJ1=−αn/summationdisplay\ni,j=1/integraldisplay\nΩ/parenleftbig\nmε×∂\n∂xiHε\ne(mε)/parenrightbig\n·aε\nij/parenleftbig\nmε×∂\n∂xjHε\ne(mε)/parenrightbig\ndx+F1,\nherethefirstterm onright-handsideissign-preservedduet otheuniformco-\nerciveness of aεin (1). As for J2, we apply ( 95) by taking a=aε\nij∂jHε\ne(mε),\nit leads to\n(103)J2=n/summationdisplay\ni,j=1/integraldisplay\nΩmε×/parenleftbig∂\n∂ximε×Hε\ne(mε)/parenrightbig\n·/parenleftbig\nmε×aε\nij∂\n∂xjHε\ne(mε)/parenrightbig\ndx\n−n/summationdisplay\ni,j=1/integraldisplay\nΩ/parenleftbig\nmε×Hε\ne(mε)·∂\n∂ximε/parenrightbig\nmε·aε\nij∂\n∂xjHε\ne(mε)dx.HOMOGENIZATION OF THE LLG EQUATION 31\nUsingpropertyof vector outer production( 98) forfirstterm, andintegration\nby parts for the second term, ( 103) becomes\nJ2=2n/summationdisplay\ni,j=1/integraldisplay\nΩ/parenleftbig\nmε·Hε\ne(mε)/parenrightbig/parenleftbig\nmε×aε\nij∂\n∂xjHε\ne(mε)·∂\n∂ximε/parenrightbig\ndx+F2\n≤C/ba∇dbl∇mε/ba∇dbl6\nL6(Ω)+C/ba∇dblHε\ne(mε)/ba∇dbl3\nL3(Ω)+δ/ba∇dblmε×∇Hε\ne(mε)/ba∇dbl2\nL2(Ω)+F2.\nHere low-order terms Fi,i= 1,2 satisfies by ( 94) and H¨ older’s inequality\nFi≤C+C/ba∇dbl∇mε/ba∇dbl6\nL6(Ω)+C/ba∇dblHε\ne(mε)/ba∇dbl3\nL3(Ω).\nSubstituting above estimates into ( 102), applying estimate ( 93) and Lemma\n8, we finally arrive at\n(104)1\n2d\ndt/ba∇dblAεmε/ba∇dbl2\nL2(Ω)+(αamin−Cδ)/ba∇dblmε×∇Hε\ne(mε)/ba∇dbl2\nL2(Ω)\n≤C+C/ba∇dblAεmε/ba∇dbl6\nL2(Ω)+C/ba∇dbl∂tmε/ba∇dbl2\nL2(Ω)−d\ndt/integraldisplay\nΩAεmε·Γε(mε)dx,\nIntegrating ( 104) over [0 ,t], using the integrability of kinetic energy ( 101)\nand the following inequality\n/integraldisplay\nΩAεmε·Γε(mε)dx≤C/ba∇dblΓε(mε)/ba∇dbl2\nL2(Ω)+1\n4/ba∇dblAεmε/ba∇dbl2\nL2(Ω),\none has for any t∈(0,T]\n(105)1\n4/ba∇dblAεmε(t)/ba∇dbl2\nL2(Ω)≤C+C/integraldisplayt\n0/ba∇dblAεmε(τ)/ba∇dbl6\nL2(Ω)dτ,\nwhereCdependson /ba∇dblAεmε\ninit/ba∇dblL2(Ω),Gε\nL[mε\ninit] thus is independent of εandt\nby assumption ( 16)-(17) and Lemma 6. Denote the right-hand side of ( 105)\nbyF(t) and write\nd\ndtF(t)≤CF3(t).\nBy the Cauchy-Lipshitz-Picard Theorem [ 3] and comparison principle, there\nexistsT∗∈(0,T] independent of ε, such that F(t) is uniformly bounded on\n[0,T∗], thus/ba∇dblAεmε(t)/ba∇dbl2\nL2(Ω)is uniformly bounded by ( 105). The Lemma is\nproved. /square\nAcknowledgments\nJ. Chen was supported by National Natural Science Foundatio n of China\nvia grant 11971021. J.-G. Liu was supported by Natural Scien ce Foun-\ndation via grant DMS-2106988. Z. Sun was supported by the Pos tgradu-\nate Research & Practice Innovation Program of Jiangsu Provi nce via grant\nKYCX21 2934.32 JINGRUN CHEN, JIAN-GUO LIU, AND ZHIWEI SUN\nReferences\n1. Francois Alouges, AnneDeBouard, Benoˆ ıt Merlet, andL´ e aNicolas, Stochastic homog-\nenization of the Landau-Lifshitz-Gilbert equation , Stochastics and Partial Differential\nEquations: Analysis and Computations 9(2021), 789–818.\n2. Francois Alouges and Giovanni Di Fratta, Homogenization of composite ferromagnetic\nmaterials , Proceedings of the Royal Society A: Mathematical, Physica l and Engineer-\ning Sciences 471(2015), 20150365.\n3. Haim Brezis and Haim Br´ ezis, Functional analysis, sobolev spaces and partial differ-\nential equations , vol. 2, Springer, 2011.\n4. Jingrun Chen, Rui Du, Zetao Ma, Zhiwei Sun, and Zhang Lei, On the multiscale\nLandau-Lifshitz-Gilbert equation: Two-scale convergenc e and stability analysis , Mul-\ntiscale Modeling & Simulation (2022), in press.\n5. Catherine Choquet, Mohammed Moumni, and Mouhcine Tiliou a,Homogenization of\nthe Landau-Lifshitz-Gilbert equation in a contrasted comp osite medium , Discrete &\nContinuous Dynamical Systems-S 11(2018), 35.\n6. T. L.Gilbert, A Lagrangian formulation of gyromagnetic equation of the ma gnetization\nfield, Physical Review D 100(1955), 1243–1255.\n7. Lev Davidovich Landau and E Lifshitz, On the theory of the dispersion of mag-\nnetic permeability in ferromagnetic bodies , Physikalische Zeitschrift der Sowjetunion\n8(1935), 153–169.\n8. Lena Leitenmaier and Olof Runborg, On homogenization of the Landau-Lifshitz equa-\ntion with rapidly oscillating material coefficient , Communications in Mathematical\nSciences 20(2022), 653–694.\n9. ,Upscaling errors in heterogeneous multiscale methods for t he Landau-Lifshitz\nequation, Multiscale Modeling & Simulation 20(2022), 1–35.\n10. Dirk Praetorius, Analysis of the operator delta div arising in magnetic model s,\nZeitschrift Fur Analysis Und Ihre Anwendungen 23(2004), 589–605.\n11. K´ evin Santugini-Repiquet, Homogenization of the demagnetization field operator in\nperiodically perforated domains , Journal of Mathematical Analysis and Applications\n334(2007), 502–516.\n12. Zhongwei Shen, Periodic homogenization of elliptic systems , Springer, 2018.\nSchool of MathematicalSciences, Universityof Science and Technology of\nChina, Hefei, Anhui 230026, China; Suzhou Institute for Adv anced Research,\nUniversityof Science and Technology of China, Suzhou, Jian gsu 215123, China\nEmail address :jingrunchen@ustc.edu.cn\nDepartment of Mathematics and Department of Physics, Duke U niversity,\nBox 90320, Durham NC 27708, USA\nEmail address :jliu@phy.duke.edu\nSchool of Mathematical Sciences, Soochow University, Suzh ou, Jiangsu\n215006, China\nEmail address :20194007008@stu.suda.edu.cn"    },    {        "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": "2202.02834v1.Enhancing_Perpendicular_Magnetic_Anisotropy_in_Garnet_Ferrimagnet_by_Interfacing_with_Few_Layer_WTe2.pdf",        "content": "   \n \n1 \n Enhancing Perpendicular Magnetic Anisotropy in Garnet Ferrimagnet \nby Interfacing with Few -Layer WTe 2 \nGuanzhong Wu1*, Dongying Wang1, Nishchhal Verma1, Rahul Rao2, Yang Cheng1, Side Guo1, \nGuixin Cao1, Kenji Watanabe3, Takashi Taniguchi4, Chun Ning Lau1, Fengyuan Yang1,  \nMohit Randeria1, Marc Bockrath1, and P. Chris Hammel1 \n1.Department of Physics, The Ohio State University, Columbus, OH, 43210, USA  \n2.Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright -Patterson Air \nForce Base, Dayton, OH, 45433, USA  \n3. Research Center for Functional Materials,  National Institute for Materials Science, 1 -1 Namiki, \nTsukuba 305 -0044, Japan  \n4. International Center for Materials Nanoarchitectonics,  Nationa l Institute for Materials Science,  1-1 \nNamiki, Tsukuba 305 -0044, Japan  \n \n \n \n \n \n \n \n \n \n \n \n*wu.2314@osu.edu     \n \n2 \n Abstract:  \nEngineering magnetic anisotropy in a ferro - or ferrimagnetic (FM) thin film is crucial in \nspintronic device. One way to modify the magnetic anisotropy is through the surface of the FM \nthin film. Here, we report the emergence of a perpendicular magnetic anisotropy (PMA) induced \nby interfacial interactions in a heterostructure comprised of a garnet ferrimagnet, Y 3Fe5O12 \n(YIG), and the low -symmetry, high spin orbit coupling (SOC) transition metal dichalcogenide, \nWTe 2.  At the same time, we also observed an enhancement in Gilbert damping in the WTe 2 \ncovered YIG area. Both the magnitude of interface -induced PMA and the Gilb ert damping \nenhancement have no observable WTe 2 thickness dependence down to single quadruple -layer, \nindicating that the interfacial interaction plays a critical role. The ability of WTe 2 to enhance the \nPMA in FM thin film, combined with its previously rep orted capability to generate out -of-plane \ndamping like spin torque, makes it desirable for magnetic memory applications.   \n \nKey words:  perpendicular magnetic anisotropy, magnetic resonance force microscope, transition metal \ndichalcogenides, ferrimagnetic i nsulator      \n \n3 \n Perpendicular magnetic anisotropy  (PMA)  in a ferromagnetic thin film is of great interest in \nspintronics research and application s. Ferromagnetic nano -element s with PMA overcome  their shape \nanisotropy , greatly ease the memory cell size reduction and improves  memory retention . These \nexceptional properties, improving the performance  of magnetic devices , make PMA highly desirable for \nmagnetic memory  application s.  PMA becomes even more important in the recent development of solid \nstate magnetic random -access memory (MRAM) since it allows MRAM to have lower switching current \nand faster switching speed compare d to in-plane magnetized materials 1, 2.  \nMagnetic storage devices generally rely on metallic magnetic material s due to their robust \nelectrical response . Interfacial magnetic anisotropy plays a critical role in generat ing PMA in metallic \nferromagnet s. When interfacing with a nonmagnetic material (NM), electron orbital angular momentum \nof the magnetic ions at the ferromagnet surface will be modified, in some cases  enabling strong covalent \nbonding, resulting in distinct magnetic properties compare d to the single layer 3-6. However, spintronics \ndevices made of metallic magnetic materials are inherently energy consumptive due to resistive losses. \nRecently, complex oxide ferro - or ferrimagnet  insulator s (FMI)  have  attracted substantial  interest  due to \ntheir ability to transport spin excitation s with low dissipation  7. Inducing PMA in FMIs naturally \nbecomes an important topic both for scientific and technologic al reasons . Several successful route s to \nachiev ing PMA in FMIs ha ve been reported using bulk intrinsic anisotropy 8 or lattice strain 9-12. But in \nmost experiments,  the sign of the resulting interfacial anisotropy in FMI/NM heterostructures is such as \nto enhance  the easy-plane anisotropy  13-15. Only one recent experiment has shown the possibility of \ngenerating interfacial PMA, and this was attributed to topological surface states  16. Nevertheless, t hese \nresults demonstrate  the possibility of controlling magnetic anisotropy through interfac ial interaction s in    \n \n4 \n FMI/NM heterostructures . Here, we report a study on YIG/WTe 2/hBN heterostructures, which shows  \nthat when interfacing with a low symmetry nonmagnetic  van der Waals  material , WTe 2, an additional  \ninterfac e-induced  PMA  (iPMA) term  emerges  in the magnetic anisotropy  of the YIG thin film . The \nabsence of topological surface states at room temperature  in WTe 2 17, 18 forces us to seek an explanation \nfor our observation of enhanced PMA that is distinct from that proposed for top ological insulator/YIG \nbilayers 16. We therefore turn to an analysis of the broken symmetries in WTe 2. We point out that low \nsymmetry WTe 2 has recently shown the capability of generating both in -plane and out -of-plane spin \npolarization in charge -spin conversion experiments 19-22. It also enables field-free switching of PMA \nmagnet ic material , which ease s the application of PMA material s in MRAM application s 23-25.   \nFerrimagnetic insulator  YIG is of significant research interest in spintronics  due to its \nexceptionally  low Gilbert damping  26, which describes the  relaxation rate  of magnetization precession . \nAnd 1T’-WTe 2 is a semi -metallic transition metal dichalcogenide (TMD) layered material with strong \nSOC  27, 28. The crystal structure of 1T’-WTe 2 lacks twofold rotational symmetry about the c -axis (Fig. \n1a). The only symmetry in the  WTe 2 crystal lattice  ab plane is the mirror symmetry about  the bc plane  \n29. This u nique symmetry breaking allows  out-of-plane damping -like torque  to be generated 30, 31, \nenabling efficient switching of the  out-of-plane magnetization  of the adjacent magnetic material  24.  \nA 20nm thick YIG thin film used in our experiment is epitaxially grown on (111) -oriented  \nGd3Ga5O12 (GGG) substrate  by off -axis sputtering  32. WTe 2 flakes are then mechanically exfoliated \nfrom a flux-grown crystal, and dry transferred on to the clean top YIG surface without touching any \nother substances. This whole process is carried out in an Ar -filled glove box with <0.1 ppm of H 2O and    \n \n5 \n O2 to protect the flakes from degradation and  ensure  the clean liness  of the YIG/WTe 2 interface. We \nemploy hexagonal boron nitride ( hBN ) encapsulation to protect the WTe 2 flakes from oxidation after \nbeing removed from the glove box. We make two samples  and focus on the  data taken from sample 1 in \nthe main text. The raw data taken from sample 2 can be found  in Supporting Information  Fig. S 2.  \n \nFig. 1 Crystal structure of WTe 2 and sample schematic . a) Crystal lattice structure of WTe 2 viewed  \nfrom the top along  the c-axis and looking from the side along the a-axis. The black dashed box in the \nside view indicates a monolayer of WTe 2. b) Schematic of the ferromagnetic resonance force \nmicroscope. RF excitation is generated by a stripline underneath the sample , where the hBN \nencapsulation is not shown . The  region of  localized mode is shown as a yellow dot adjacent to the  WTe 2 \nflake, and the probe magnetic moment is shown as a yellow arrow on the particle. The cantilever \noscillation is detected by a fiber laser interferometer.  \nFigure 2a shows an optical image  of the sample 1. Due to the small  lateral size of the exfoliate d \nWTe 2 and hBN  flake s having  length  scale s of 10 μm, we use a home -built ferromagnetic resonance \nforce microscope (FMRFM) to measure  the local ferromagnetic resonance (FMR) signal. FMRFM is a \nsensitive technique to detect  the local magnetic properties with high spatial and spectral resolution  33. In \nour FMRFM, the external magnetic field 𝐻⃗⃗ ext is aligned perpendicular to  the sample plane. The \ncantilever tip holds a  high coercivity SmCo 5 magnetic particle , whose  moment  is magnetized in the  \n   \n \n6 \n direction opposite to 𝐻⃗⃗ ext to create a magnetic field well . The field well supports a set of  localized \nstanding spin wave modes (LMs). During the measurement, we excite spin precession uniformly by a \nstripline underneath the sample at a fixed RF frequency  (2 GHz)  and sweep the magnetic field. The \nresonance of each LM  generate s a stray field, whic h can then  be detected by the SmCo 5 magnetic \nparticle attached on the cantilever through their magnetic dipole -dipole interaction (Fig. 1b). During the \nmeasurement, we keep the probe -to-sample separation around 4 μm. The operation  of FMRFM is \ndescribed in detail in Ref s. 34-36. For reference, w e separate  a region of  YIG that does not contain  \nWTe 2/hBN heterostructures and measur e its Gilbert damping using  broadband FMR. To eliminate two -\nmagnon scattering, w e perform broadband FMR in the out -of-plane  field geometry. The FMR linewidth \nas a function of frequency measured on bare YIG (sample 1) shows a linear dependence (Fig. 2b), from \nwhich we can extract the Gilbert damping  of bare YIG  𝛼YIG=1.05×10−3. We also confirm that the \nWTe 2 used in the experiment is indeed  the 1T’ phase  through polarized Raman measurements. The \npolarization angle dependence of the Raman peak at 212 cm-1 (spectrum is shown in Fig. S4) exhibits \nminimum intensity when the excitation laser polarization is along the crystallographic a axis of WTe 2 37 \nas shown by the polar plot in Fig. 2c and Raman intensity plot in Fig. 2d .  \nWe find t he position of the YIG/WTe 2/hBN heterostructure with the assist ance of magnetic \nalignment markers  (Fig. 2a) . Figure 2e shows t wo raw FMRFM scans taken in the region  of YIG/hBN \nand YIG/WTe 2/hBN , indicated by the blue and the red dot in Fig. 2a, respectively , which  reveals  the \nchange in FMRFM spectra  at two different location s. Here we focus on the 𝑛=1 LM because it has the \nmode radius of around 1 μm and gives the highest spatial resolution. Higher order modes  have \nincreasing  mode radius and therefore, detect  less local ized magnetic  properties.  This is the reason why    \n \n7 \n the quasi -uniform mode at ~  3325 Oe does not show obvious change in resonance field or signal \namplitude.  We further take a  line scan across the edge of WTe 2 flake (Fig. 2 f) to resolve the spatial \nevolution of FMRFM spectra . The line scan in Fig. 2f (along the dashed line shown in Fig. 2a) shows \nthree main features : first, the magnitude of the LM resonance signal is reduced in the YIG/WTe 2/hBN \nregion compare d to the YIG/hBN region; second, the LM resonance field for all LMs is decreased by \n~40 Oe in the YIG/WTe 2/hBN region; third , the LMs  show complex  splitting and crossing when the \nprobe is close to the boundary (−5 μm<𝑋<10 μm).  \n \nFig. 2 FMRFM and Raman measurement data . a) An optical micrograph of the YIG/WTe 2/hBN \nheterostructure under study . WTe 2 crystal a and b axis are labeled.  b) Broadband FMR measurement of \nthe frequency -dependent linewidth of the YIG thin film. The measurement is done on the same piece of \nYIG used to make sample shown in Fig. 1b. c) Polar plot of the 21 2 cm-1 peak  Raman  intensity. Angle \ndenote s the relative angle between the measurement laser polarization and the WTe 2 a axis. d) 2D \nintensity plot showing Raman peak intensities versus polarization angle . e) FMRFM spectra, one over \nthe YIG/hBN  region (blue line)  and the second over the YIG/WTe 2/hBN  region (red line); these \nlocations are  indicated by the  blue and red dot s in Fig. 2a respectively . f) Color plot of field -dependence \n   \n \n8 \n FMRFM scans as a function of position along the trace indicated by the black dashed line in Fig. 2a. A \nconstant background is subtracted to show only the signal from the several LM resonance s. \nIn the following, we will explain the origin of the three observed effects using spin pumping and \nmagnetic anisotropy. The first effect , i.e. signal reduction in  the YIG/WTe 2/hBN area relative to the \nYIG/hBN  area, is the result of enhanced relaxation due to spin pumping from YIG to WTe 2 38. The 𝑛=\n1 LM resonance signal amplitude ∆𝐴 is inversely proportional to the square of Gilbert damping , 𝛼2. We \ndetermine  the Gilbert damping  constant  𝛼 for YIG/WTe 2/hBN  using 𝛼YIG/WTe2/hBN=\n𝛼YIG/hBN×√∆𝐴YIG/hBN∆𝐴YIG/WTe2/hBN ⁄  (see Ref. 39), where 𝛼YIG/hBN is assumed to be the same as \n𝛼YIG=1.05×10−3 due to  the low SOC and insulating character of hBN . The second  effect is  the \ndecrease of 𝑛=1 LM resonance field 𝐻r,1 by ~40 Oe . And the third effect is  splitting and crossing  of \ncomplex modes  in the region −5 μm<𝑋<10 μm. The second and the third effects  are due  to an \nabrupt change of uniaxial anisotropy across the boundary  separating the YIG/WTe 2/hBN and YIG/hBN \nregions  15. Here, the uniaxial anisotropy refers to the magnetic free energy  depends on the angle between \nmagnetization and sample normal  ℱu=−𝐾u𝒎z2, where 𝒎z is the component  of magnetization unit \nvector in the direction normal to sample plane  and 𝐾u is the uniaxial anisotropy constant specific to \nsample  and depends on the total interaction in the sample . When 𝐾u is positive, ℱu is called to be of \nPMA type, on the other hand, if 𝐾u is negative, ℱu is called to be of easy -plane  type.  This uniaxial \nanisotropy will lead to an effective uniaxial magnetic field 𝑯u=−𝜕ℱu𝜕𝑴⁄ , where 𝑴 is the \nmagnetization . And therefore, a change in 𝐾u can modify the resonance field in a FMR measurement . In \nFMRFM spatial mapping, a n abrupt change in 𝐾u spatially could disturb the LM and lead to mode \nsplitting and crossing as described in Ref. 15. Moreover, i n striking contrast to the previously studied    \n \n9 \n YIG/Au interface 15, which result s in a 32 Oe increase of 𝐻r,1 due to  the enhanced easy -plane \nanisotropy, the observed decrease of 𝐻r,1 indicates that the WTe 2 overlayer induces  an iPMA  in YIG. \nWe note that the  magnitude of the shift in  𝐻r,1 is comparable to the easy -plane anisotropy induced by a \nheavy metal 15, 40 or the iPMA generated by topological surface state 16 on garnet ferrimagnetic material .  \nIn order to probe  the global effect of  a WTe 2 overlayer on YIG, we spatial ly map 𝐻r,1 using  the \n𝑛=1 LM. Figure 3a present s an optical image of WTe 2 flakes on a Si/SiO 2 (285nm ) substrate , where \ndifferent c olors of WTe 2 flakes indicat e different WTe 2 thickness es. Figure 3b and 3 c show spatial  maps \nof magnetic properties  in the region enclosed by the black dashed rectangle  in Fig. 3a .  We acquire the \nmaps using the procedure described in Ref. 39, i.e., simultaneously  measuring spatial variation of  the \nmagnetic anisotropy and Gilbert damping using the  𝑛=1 LM resonance field 𝐻r,1 and signal amplitude  \n∆𝐴. The  entire  WTe 2-covered area show s uniformly lower ed 𝐻r,1 and increased Gilbert damping relative \nto the area without WTe 2. In Fig. 3c, despite the not great signal to noise ratio in damping imaging, the re \nis a clear  Gilbert damping enhancement in WTe 2-covered area . The averaged Gilbert damping of YIG in \nWTe 2-covered area is 𝛼̅YIG/WTe2/hBN≈1.30×10−3, about 24% higher than  𝛼YIG. We note that due to \nthe slight relative tilting of the scan plane and the sample plane, there is a color shift in Fig. 3b  that \nmight conceal the contrast difference in different WTe 2 thickness region . Therefore, to study the WTe 2 \nthickness dependence, we will show fine line scans across edge s of flakes  having  different WTe 2 \nthickness es.     \n \n10 \n  \nFig. 3 Two -dimensional FMRFM scan resolving the spatial variation of magnetic anisotropy and \nGilbert damping. a ) Optical micrograph showing the color contrast of different thickness WTe 2 flake s \n(ranging from 4.7 nm to 44.8 nm) on Si/SiO 2(300 nm). Black dashed box outlines the FMRFM scanned \narea for 2D mapping. b) 2D map of the 𝑛=1 LM resonance field . The d ashed line s labeled 1 -4 \ncorrespond to the  four line-scans shown in Fig. S1a-S1d. c) 2D mapping of the Gilbert damping \nextracted from  the 𝑛=1 LM resonance peak amplitude.  \n  \n   \n \n11 \n Next, we want to understand what gives rise to the PMA in WTe 2/YIG. We rule out the effect \ninduced by a modification of the gyromagnetic ratio by showing the resonance field shift across the \nWTe 2 edge does not depend on RF excitation frequency (See Fig. S3). We also exclude a strain induced \neffect given the absence of an epitaxial relation and the weakness of the van der Waals interaction \nbetween YIG and WTe 2. We further note that we can ignore the role of topological surface states  16 in \nour analysis; they are not relevant for our room temperature experiment since WTe 2 is a topological \nWeyl semimetal only below 100 K  17, 18. \nWe show how an analysis based on symmetry and the nature of the interfacial SOC , generalizing \nthe theory in Ref. 41, gives insight into the PMA observed in our experiment. This will also help us \nunderstand why the easy-axis anisotropy we observe in WTe 2/YIG is so different from the results of \nRef. 13, 15  on YIG interfaces with a dozen different metallic and semiconducting materials, all of which \nexhibit interface -induced easy-plane  anisotropy, as is predicted by theory  41.  \nYIG is a ferrimagnetic Mott insulator, with two inequivalent Fe -sites coupled via \nantiferromagnetic (AFM) superexchange interactions. We focus on how interfacial SOC impacts AFM \nsuperexchange in YIG and show that it leads to a very specific form of the mag netic anisotropy that is \ngoverned by the direction of the effective B-field (see Supporting Information for a details).  \nBefore turning to WTe 2/YIG, it is useful to first consider the simpler case when the only broken \nsymmetry is the mirror plane defined b y the interface.  The abrupt change in lattice potential then results \nin an effective electric field that points normal to the interface, which in turn leads to an effective \nmagnetic field in the rest frame of the electron that couples to its spin. Since t he E-field points normal to    \n \n12 \n the interfacial plane in which the electron moves, the resulting B-field arising from SOC lies within  the \ninterfacial plane. As we show in the SI, this leads to a SOC -induced correction to AFM superexchange \nthat necessarily lead s to an easy-plane  anisotropy.  \nIn the case of WTe 2/YIG, however, when there are additional broken symmetries. Not only does \nthe interface break inversion  symmetry , but the crystal structure of WTe 2 itself breaks in -plane inversion  \nsymmetry . The electric field is now no longer normal to the interface, and the effective B-field arising \nfrom SOC necessarily has an out -of-plane component, as shown in Fig S5b in SI. Thus, we see why the \nlower symmetry of WTe 2/YIG can naturally result in an easy-axis or perpendicular magnetic anisotropy  \n(PMA);  see Supporting Information for details.  \nWe note that the lack of two -fold rotational symmetry in the ab plane in WTe 2 that plays a \ncritical role in our understanding of PMA in WTe 2/YIG, has also been pointed  out be crucial for the out -\nof-plane damping -like torque in WTe 2/Permalloy30. We note, however, that the out -of-plane damping -\nlike torque necessarily involves current flow in WTe 2, while the PMA is an equilibrium property of the \nsystem independent of current flow.  \nWe further demonstrate the interfacial  origin of the observed effect by studying the influence of \nWTe 2 thickness.  We show four  line-scans , labeled in Fig. 3b, across the edges of WTe 2 with different \nthickness es, ranging from 4. 7 nm to 44.8 nm . From these four line -scans, we extract  the 𝑛=1 LM \nresonance field 𝐻r,1 and the 𝑛=1 LM resonance signal amplitude  ∆𝐴. Figures S1a-d in the Supporting \nInformation  show the evolution of 𝐻r,1 and ∆𝐴 along the traces labeled correspondingly . The thickness \nof WTe 2 at each measurement location is later measured using atomic force microscop y. From these    \n \n13 \n line-scans , we choose the region s where the probe is far away from the edge of WTe 2 so that the \nmagnetic propert ies are uniform,  to obtain  spatial average s of 𝐻r,1 and ∆𝐴, which are denoted  \n𝐻̅r,1,YIG/hBN and ∆𝐴̅̅̅̅YIG/hBN in the YIG/hBN region , and 𝐻̅r,1,YIG/WTe2/hBN and ∆𝐴̅̅̅̅YIG/WTe2/hBN in the \nYIG/WTe 2/hBN region , respectively . We further extract the 𝑛=1 LM resonance field difference \nbetween two regions using ∆𝐻r,1=𝐻̅r,1,YIG/hBN−𝐻̅r,1,YIG/WTe2/hBN, as well as the Gilbert damping \ndifference using ∆𝛼=𝛼YIG×(√∆𝐴̅̅̅̅YIG/hBN ∆𝐴̅̅̅̅YIG/WTe2/hBN ⁄ −1) as a function of the WTe 2 \nthickness . We note  that the hBN overlayer does not change the Gilbert damping in YIG . The \nsummarized results containing the data from both sample 1 and sample 2  are shown in Fig s. 4a and 4 b. \nRaw data from sample 2 can be found in Supporting Information  Fig. S 2. The thinnest WTe 2 acquired in \nthe experiment is 3.2nm from sample 2, which is approximately the thickness of  a quadruple -layer \nWTe 2.  \nFigures 4a and 4 b indicate  that both ∆𝐻r,1 and ∆𝛼 have  almost no WTe 2 thickness  dependence . \nThere is a  small sample -to-sample variation possibly  due to different YIG/WTe 2 interfacial quality . The \nchange of 𝑛=1 LM resonance field,  ∆𝐻r,1, is as large as ~38 Oe even when the WTe 2 thickness \napproaches the quadruple -layer thickness . This indicates that the modification of magnetic anisotropy is \ndue to the YIG/WTe 2 interfac ial interaction , with no bulk contribution. For the increase of Gilbert \ndamping ∆𝛼, no obvious thickness dependence is observed when comparing the data from the same \nsample. In sample 2, the Gilbert damping enhancement  due to the quadruple -layer WTe 2 has almost the \nsame value as the 50 nm thick WTe 2 flake, indicating that no thickness dependence of spin pumping  can \nbe resolved from our measurement. There are two possible interpretations of these results . First, if the    \n \n14 \n spin current injected into WTe 2 is mainly relaxed due to spin relaxation in the bulk, then the \nexperimental result is a demonstration of ultra -short spin diffusion length along the c axis38, smaller or \ncomparable to the thinnest WTe 2 flake (3.2 nm), employed in this experiment . It is much smaller than \nthe 8nm spin diffusio n length  in the  in-plane direction measured using inverse spin Hall effect  22. Note \nthat due to the chang e in mo bility and the metal -insulator transition  in few layer WTe 2 when its \nthickness reduces  42, the spin diffusion length approximated here could be inaccurate . Alternatively , it is \npossible that the spin relaxation is primarily due to the interfacial SOC induced by inversion symmetry \nbreaking at the interface and in the WTe 2 crystal lattice. In this case, the Gilbert damping enhancement \nwill have no WTe 2 thickness dependence.     \n \n15 \n  \nFig. 4  WTe 2 thickness dependence  of resonance field and damping enhancement . a) 𝐻r,1 in the \nYIG/hBN and YIG/WTe 2/hBN regions  are averaged  respectively to get 𝐻̅r,1,YIG and 𝐻̅r,1,YIG/WTe2, and \n∆𝐻r,1=𝐻̅r,1,YIG−𝐻̅r,1,YIG/WTe2. b) ∆𝛼 as a function of WTe 2 thickness , and  ∆𝛼=𝛼YIG/WTe2−𝛼YIG \nwhere 𝛼YIG is the Gilbert damping of bare YIG measured using broadband FMR for each sample , and \n𝛼YIG/WTe2=𝛼YIG×(√∆𝐴̅̅̅̅YIG/hBN ∆𝐴̅̅̅̅YIG/WTe2/hBN ⁄ ).   \nIn conclusion, we have shown that the YIG/WTe 2 interface plays a critical  role in both interfacial \nmagnetic anisotropy and spin relaxation , making WTe 2 a promising material  in magnetic memory \n   \n \n16 \n application s. Combining the iPMA  created by WTe 2 with the out-of-plane spin orbit torque  generated by \nflowing a charge current along  the a axis of WTe 2, one can possibly achieve field -free switching of a \nPMA magnetic cell  for magnetic memory application s. It will improv e the  scalability , reduc e the  power \nconsumption  and increas e operation speed  of magnetic solid -state devices . Our result reveals new \npossibilities in selecting materials and designing spintronic devices. For example, one can consider other \nmaterials with low lattice symmetry and strong SOC to induce larger PMA type interfacial ani sotropy in \nFMIs. To achieve a fully PMA material, one could utilize thinner FMIs to magnify the role of iPMA. \nMoreover, interfacial SOC also plays an important role in  generat ing topologically protected magnetic \ntextures in the FMIs  43. These findings will motivate further research to reveal the fundamental  physics \narising at the  interface between FMIs and nonmagnetic materials.  \n \nData availability:  \nThe data generated by the present study are available from the corresponding author on request.  \nSupporting Information:  \nA description of raw data on WTe 2 thickness dependence, a FMRFM measurement on a second sample, \na FMRFM measurement at different RF frequency, a description of polarized Raman measurement \nresult, and a detailed illustration of impact of broken  mirror reflection symmetries on the magnetic \nanisotropy.  \nAckno wledgements:     \n \n17 \n This work was primarily supported by the Center for Emergent Materials: an NSF MRSEC under award \nnumber DMR -2011876 (GW, NV , YC, SG, FY , MR and PCH) . KW and TT acknowledge support from \nthe Elemental Strategy Initiative conducted by the MEXT, Japan (Grant Number JPMXP0112101001) \nand JSPS KAKENHI (Grant Numbers 19H05790, 20H00354 and 21H05233).  DW, GC, CNL, and MB \nare supported by NSF under award DMR -2004801.  We gr atefully acknowledge N. Trivedi for insightful \ndiscussions. Fabrication and some characterization were performed in the Ohio State University \nNanoSystems Laboratory.  \n     \n \n18 \n Methods:  \nSample fabrication  \nOur YIG/WTe 2/hBN heterostructure was prepared by means of dry transfer and stacking 44. hBN crystals \nwere mechanically exfoliated under ambient conditions onto SiO 2/Si substrates (285 nm thick SiO 2). 20-\n40 nm thick hBN flakes were identified under an optical microscope and used for the capping lay er for \nthe stack. The hBN was picked up using a polymer -based dry transfer technique and then moved into an \nAr-filled glove box with oxygen and water level below 0.1 ppm. Flux -grown WTe 2 crystals 45 were \nexfoliated inside the glove box and flakes with different thicknesses were optically identified and \nquickly picked up with the capping hBN layer then transferred to the YIG substrate. Finally, we removed \nthe fully encapsulated sample from the glove b ox and performed the e -beam lithography and \nmetallization (Ni/Au) step for alignment in our ferromagnetic resonance force microscope (FMRFM).  \n \nPolarized Raman measurement  \nPolarized Raman spectra from the WTe 2 sample were collected using 633 nm excitation w avelength in \nan inVia Renishaw Raman microscope.  The sample was loaded onto the microscope stage and \npositioned in such a way that the long edge of the flake was aligned parallel to the laser polarization ( θ = \n0°). In this configuration, the incident illu mination is polarized vertically coming out of the laser and is \naligned with the long axis of the WTe 2 flake. The polarization of the incident laser was rotated from 0 to \n360° by 10° increments using a polarization rotator, while an analyzer was set to onl y allow vertically \npolarized light to enter the spectrometer.  Raman spectra were collected at each polarization for 3 \nacquisitions with a 20 s time per acquisition.  The laser power was set to 0.5 mW at the sample to avoid \nany damage by heating.  Followin g spectral collection, the (baseline corrected) integrated intensities \nunder each peak were calculated to make the contour plots and polar plots in Fig. 2c and 2d.  \n \nFMRFM measurement and signal fitting  \nOur FMRFM perform s local ly measures  FMR at room temper ature in vacuum. The cantilever has \nnatural frequency of ~18 KHz, spring constant of 0.2 N/m and Q factor of ~20000, resulting in force \ndetection sensitivity of  10-15 N/Hz1/2. The SmCo 5 magnetic particle attached on the cantilever has a \nmagnetic moment of ~4 nemu.  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B 2015,  92, (4), 041104.  \n \n     \n \n23 \n Enhancing  Perpendicular Magnetic Anisotropy in Garnet  Ferrimagnet \nby Interfacing with Few-Layer  WTe 2 \nGuanzhong Wu1*, Dongying Wang1, Nish chhal  Verma1, Rahul Rao2, Yang Cheng1, Side Guo1, \nGuixin Cao1, Kenji Watanabe3, Takashi Taniguchi4, Chun  Ning Lau1, Fengyuan Yang1,  \nMohit Randeria1, Marc Bockrath1, and P. Chris Hammel1 \n1.Department of Physics, The Ohio State University, Columbus, OH, 43210, USA  \n2.Materials and Manufacturing Directorate, Air Force Research Laboratory, Wright -Patterson Air \nForce Base, Dayton, OH, 45433, USA  \n3. Research Center for Functional Materials,  National Institute for Materials Science, 1 -1 Namiki, \nTsukuba 305 -0044, Japan  \n4. International Center for Materials Nanoarchitectonics , National Institute for Materials Science,  1-1 \nNamiki, Tsukuba 305 -0044, Japan  \n*wu.2314@osu.edu  \n     \n \n24 \n FMRFM  line-scan across edge of WTe 2 with different thickness  \n \nFig. S1 a-d, FMRFM line-scans along the traces 1~4 indicated in Fig. 3b respectively.  The gray shaded \narea in four figures are outlining the location of  WTe2 flake . 𝐻r,1 and ∆𝐴 at each position are derived by \nfitting the 𝑛=1 LM to a Lorentzian line shape. The thickness of WTe2 flake at each location are \nmeasured by atomic force microscope.  \n  \n   \n \n25 \n FMRFM measurement on Sample 2  \n \nFig. S 2 FMRFM measurement on sample 2. a, The optical picture of the YIG/WTe 2/hBN \nheterostructure. b, 2D mapping of the 𝑛=1 LM resonance field in the black dash line circled area.  c, \n2D mapping of the Gilbert damping extracted from 𝑛=1 LM resonance peak amplitude in the black \ndash line circled area. d, FMRFM line scan along the trace indicated by the solid black line in Fig. S 2a. \nA constant background is subtracted to show only the signal from the LMs resonance. e, Fine scan \nzoomed in o n the quadruple layer WTe 2 stripe area  \n  \n   \n \n26 \n FMRFMR measurement at 4 GHz  \n \nFig. S 3 FMRFM measurement across WTe 2 edge at 4 GHz. FMRFM line scan is measured at 4 GHz \nacross the WTe 2 flake edge. The shift of the resonance field 𝐻r,1 is 36 Oe, similar to the 𝐻r,1 shift \nmeasured at 2 GHz. This result excludes the possibility that the resonance field shift arises from \nmodification of the gyromagnetic ratio.  \n \n  \n   \n \n27 \n Polarized Raman measurement  \n \nFig. S 4 Polarized Raman measurement. As shown by the red curve, the Raman  spectrum taken on \nGGG/YIG/WTe 2/hBN heterostructure contains more peaks than WTe 2. The Raman spectrum taken in \nthe GGG/YIG/hBN area identifies the peaks arising from the  substrate GGG/YIG or top hBN \nencapsulation layer. By subtracting the Raman spectrum in the GGG/YIG/hBN area, the Raman spectra \nfrom WTe 2 layer are extracted and plotted in Fig. 2d. The black dash line are the markers indicating the \nRaman peaks of WTe 2  \n   \n \n28 \n Impact of  broken mirror reflection  symmetr ies on the  magnetic anisotropy  \nWe describe theo retical constraints on the interface -induced  magnetic anisotropy in the WTe 2/YIG \nbilayer. We first show that symmetry arguments alone do not provide strong constraints on the anisotropy \ntensor, given that we are dealing with an interface between two crystalline materials at an arbitrary \norientation with respect to each other . We then present qualitative arguments, based on the interfacia l spin -\norbit coupling, that give insight into the magnetic anisotropy in WTe 2/YIG. This helps us understand why \nthe easy-axis anisotropy that we observe  in WTe 2/YIG differ s from the results  of Lee et al. [1] on YIG \ninterfaces with a dozen different metallic and semiconducting materials , all of which exhibit interface -\ninduced easy-plane anisotropy  as predicted  by theory  [2]. \nOn general grounds, the  anisotropy (free) energy can be written as   \nℱ𝑎𝑛𝑖𝑠= ∑ 𝐾𝑎𝑏𝑎,𝑏 𝑚𝑎 𝑚𝑏,           (S1) \nwhere a and b take on values x,y,z. We focus here on the leading term, quadratic in the magnetization, and \nignore higher order anisotropy terms like (mx4+my4+mz4) or (mx2my2+my2mz2+mz 2mx2). The form of \n 𝐾𝑎𝑏=  𝐾𝑏𝑎 is constrained by symmetry. Let us consider three cases , going from the most symmetric to \nthe least .  \nCase I: The only broken symmetry is interfacial inversion (z →  - z), which  is relevant for the \nexperiments of Ref. [1]. The magnetization is an axial vector (or pseudovector) that transforms under  \nrotation s like a vector but is unchanged  under inversion . Thus (𝑚𝑥 ,𝑚𝑦 ,𝑚𝑧)→ (𝑚𝑥 ,− 𝑚𝑦 ,−𝑚𝑧) under \nreflection in  a mirror plane with normal 𝑥̂. Using reflection symmetry in mirror planes normal  to 𝑥̂ and to    \n \n29 \n 𝑦̂ , we can see that all off -diagonal components of  𝐾𝑎𝑏  vanish. Further, f our-fold rotational symmetry \nabout the 𝑧̂ axis shows that  𝐾𝑥𝑥=  𝐾𝑦𝑦. Using mx2+my2+mz2=1, we write  𝐾𝑥𝑥(m𝐱2+m𝐲2) in terms \nof m𝑧2, and d efining 𝐾𝑢=  (𝐾𝑥𝑥−  𝐾𝑧𝑧), we obtain  \nℱ𝑎𝑛𝑖𝑠= −  𝐾𝑢 m𝑧2.         (S2) \nThis symmetry analysis only constrains the form  of the anisotropy energy, but not the sign of  𝐾𝑢. We will \ngive below a simple microscopic argument [2] that shows that   𝐾𝑢<0  (easy plane)  for Case I.  \nCase II: In addition to broken interfacial inversion (z → - z), let u s also break reflection symmetry \nin the plane normal to 𝑥̂. This would be the case if the crystalline axes of WTe 2 were aligned with YIG.  \nThis also breaks four-fold rotational symmetry about 𝑧̂, so that   𝐾𝑥𝑥 ≠ 𝐾𝑦𝑦. However, we  can still use \nreflection symmetry in the plane normal to 𝑦̂ to conclude that  𝐾𝑥𝑦=  𝐾𝑦𝑧=0. Thus  we find that  \n      K = (𝐾𝑥𝑥0 𝐾𝑥𝑧\n0 𝐾𝑦𝑦0\n 𝐾𝑥𝑧0 𝐾𝑧𝑧)                                                               (S3) \nCase III: When the crystalline axes of WTe 2 are not aligned with YIG, which is the experimentally \nrelevant case, all mirror reflection and rotation symmetries are broken. Then there are no symmetr y \nconstrain ts on   𝐾𝑎𝑏 and all six components of this symmetric tensor are in general non -zero.      \nLet us now see how, despite the lack of general symmetry -based constrai nts, we can still get some \nqualitative insight about the form of the anisotropy from simple microscopic considerations informed by \nsymmetry. YIG is a ferrimagnetic Mott insulator, with two inequivalent Fe -sites coupled via    \n \n30 \n antiferromagnetic (AFM) superexch ange interactions. We thus focus on how interfacial spin -orbit \ncoupling (SOC) impacts AFM superexchange.   \nThe broken symmetry at the interface leads to an electric field ℇ=−𝛁𝑉(𝒓), whose direction will \nbe discussed in detail below for three cases. This in turn produces a magnetic field in the rest frame of the \nelectron  which underlies SOC. As the electron moves along  𝐫̂ij from site i to j, it experiences an  SOC field  \nin the direction 𝐝̂ij  which is determined by  ℇ ×𝐫̂ij . The SOC Hamiltonian  is thus given by \n−𝑖𝜆∑𝑐𝐢𝛼†(𝐝̂ij∙𝝈𝛼𝛽)𝑐𝐣𝛃 𝛼𝛽 . Including the effect of this term in addition to the usual hopping t and Hubbard \nU in the standard strong coupling expansion calculation leads to the Hamiltonian  \n                 ℋex=J∑𝐒i∙𝐒j <𝐢,𝐣>+D∑𝐝̂ij∙𝐒i×𝐒j <𝐢,𝐣> +K0∑(𝐝̂ij∙𝐒i)(𝐝̂ij∙𝐒j). <𝐢,𝐣>                  (S4) \nHere  the spin  𝐒i   at site i is coupled to its neighbors via  the AFM superexchange  𝐽 ~𝑡2\n𝑈  and the \nDzyaloshinskii -Moriya interaction (DMI)  𝐷 ~𝑡𝜆\n𝑈. The K0 term will be the focus of our attention belo w as \nit leads  to magnetic anisotropy.  We note that the general form of ℋex is in fact substantially independent \n[2] of the microscopic mechanism and very similar results are obtained not only for superexchange but \nalso for Zener double exchange and RKKY interactions.  \nCase I: Let us again return to the simplest case with broken interfacial inversion (z → - z). This \nleads to an electric field ℇ=−𝛁𝑉(𝒓)  along ẑ , the normal to the interface.  The SOC magnetic field \ndirection is then given by  𝐝̂ij= ẑ ×𝐫̂ij ; see Figure S4(a). This is the well -known Rashba SOC at \ninterfaces . We note in passing that 𝐝̂ij is antis ymmetric under the interchange of i and j, and thus  leads to \na DMI term where 𝐒i×𝐒j is also antisymmetric.     \n \n31 \n  \n \nFig. S 5 Symmetry based selection of magneto -crystalline anisotropy. Interfacial SOC originates from \nan effective Electric field ℇ=−𝛁𝑉(𝒓) whose direction is determined by the broken mirror planes in the \nsystem. This electric field leads to  spin-orbit coupling (SOC), with the 𝐝̂ij= ℇ̂ ×𝐫̂ij, the direction of the \nSOC magnetic  field. Note that the direction of the electron hop 𝐫̂ij lies in the xy plane of the interface. As \nshown in the text 𝐝̂ij controls the interface -induced magnetic anisotropy. (a) When only surface inversion \nis broken, 𝐝𝐢𝐣 is constrained to lie in the int erface and interfacial SOC leads to easy -plane anisotropy. (b) \nIf there are other broken mirror planes, the 𝐝𝐢𝐣 must lie outside the interfacial plane. This can lead to a \nperpendicular magnetic anisotropy in systems like YIG/WTe 2 bilayers.  \n \nWe see that in Case I,  𝐝̂ij lies in the plane  of the interface, and the third term in eq. (S 4) then takes \nthe form K0∑(S𝐢𝑥𝑆𝐢+𝑦𝑥+S𝐢𝑦𝑆𝐢+𝑥𝑦) 𝐢  for a square lattice . To make the connection with  magnetic anisotropy, \nwe look at a continuum approximation with a s lowly  varying magnetization 𝐦(𝐫). We make a Taylor \nexpan sion of  𝐒r in terms of its value at 𝒓, denoted by  𝐦(𝐫), and its spatial derivatives . The exchange and \nDMI terms involve gradients of 𝐦(𝐫), but we focus here on local terms that  do not involve derivatives  to \n   \n \n32 \n understand the magnetic anisotropy . The leading term is + K0(m𝐱2+m𝐲2)  which can be rewritten as \n– K0 m𝑧2 using the fact that mx2+my2+mz2=1 at each 𝒓. Thus, we may identify K0 with the anisotropy  \nK𝑢 defined in eq. (S2).  \nThe microscopic analysis leads to the result K0= − 𝜆2\n𝑈   < 0 and this explains the easy-plane \nanisotropy  arising Rashba SOC at the interface . The easy-plane nature of the anisotropy  is in fact a general \nfeature of various microscopic models as emphasized  in Ref. [2]. We note however that these author s use d \nthe opposite sign convention for anisotropies  from the one we use here . The easy plane vs. easy -axis \ncharacter is , of course, independent of sign conventions.  The FMR experiments of Ref. [1] have seen the \ninterface -induced easy-plane anisotropy  predicted by the theory in a YIG interfaces with several  metallic \nand semiconducting materials . \nThe key difference between the YIG/WTe 2 bilayer studied here and systems  studied earlier [1] is \nthat WTe 2 has a broken mirror plane  (the ac plane ) as shown in  Fig. 1(a)  of the paper . We now  look at the \neffect of this lower symmetry on the microscopic analysis.  \nCase II: Let us break reflection symmetry in the plane normal to 𝑥̂ in addition to broken interfacial \ninversion. We choose x̂ parallel to the b axis, ŷ parallel to a, and ẑ parallel to c. Reflection symmetry in \nthe ŷ mirror plane constrains the  electric field ℇ =−𝛁𝑉(𝒓) to lie in the xz plane, at an angle 𝜃 from the \nz-axis as shown in Fig. S 5(b). Thus  \n𝐝ij=(sin𝜃𝑥̂+cos𝜃𝑧̂)×𝐫̂ij     (S5)    \n \n33 \n where 𝐫̂ij is a vector in the interface  (xy plane ) and 0≤𝜃≤𝜋. Using eq. (S5), we may rewrite  the last \nterm in the  Hamiltonian (S4) as \n K0 sin2𝜃∑(S𝐢𝑧𝑆𝐢+𝑦𝑧)\n𝐢+K0 cos2𝜃∑(S𝐢𝑥𝑆𝐢+𝑦𝑥+S𝐢𝑦𝑆𝐢+𝑥𝑦)\n𝐢\n−K0sin𝜃cos𝜃∑(S𝐢𝑧𝑆𝐢+𝑦𝑥+S𝐢𝑥𝑆𝐢+𝑦𝑧)\n𝐢  \nAs before, we  make  a continuum approximation with a smoothly varying 𝐦(𝐫) and focus only on the \nlocal terms, without gradients, to obtain the magnetic anisotropy . We find that the leading order \ncontribution to anisotropy is −K0cos2𝜃m𝑧2+K0sin2𝜃 mzmx. This analysis correctly captures the non -\nzero K𝑥𝑧 expected on general grounds; see eq. (S3). We did not include here , for simplicity, the effects of \nbroken four -fold rotation that would have led to  𝐾𝑥𝑥 ≠ 𝐾𝑦𝑦. \nCase III: When we lose all mirror symmetries, the case relevant to the YIG/WTe 2 experiment, the \nelectric field ℇ =−𝛁𝑉(𝒓) will point in a general direction specified by 0≤𝜃≤𝜋 and 0≤𝜑≤2𝜋, and \nthere will be no symmetry constraints on the anisotropy tensor  𝐾𝑎𝑏. \n Let us conclude by highlighting the  key qualitative difference  between Case I  on the one hand and \nCases II and III  on the other . In Case I, the only broken symmetry is interfaci al inversion (z → - z). Then \nsymmetry constrains the  𝐝̂ij, the direction of the SOC B-field,  to lie in the plane  of the interface  and this \nleads to easy -plane anisotropy as described above. In Cases II and III, there are other additional broken \nmirror planes, and this leads to the 𝐝̂ij  vector being pulled out of the plane of the interface. This \nimmediately leads to the possibility of an easy -axis like character to the anisotropy, although in the general \ncase one has a non -trivial anisotropy tensor  𝐾𝑎𝑏.     \n \n34 \n Reference  \n[1] Lee, A. J.; Ahmed, A. S.; McCullian, B. A.; Guo, S. D.; Zhu, M. L.; Yu, S. S.; Woodward, P. M.; \nHwang, J.; Hammel, P. C.; Yang, F. Y . Interfacial Rashba -Effect-Induced Anisotropy in Nonmagnetic -\nMaterial -Ferrimagnetic -Insulator Bilayers. Phys. Rev. Lett. 2020,  124, (25), 257202.  \n[2] Banerjee, S.; Rowland, J.; Erten, O.; Randeria, M. Enhanced Stability of Skyrmions in Two -\nDimensional Chiral Magnets with Rashba Spin -Orbit Coupling. Physical Review X 2014,  4, (3), 031045.  \n \n "    },    {        "title": "1906.08987v1.Unique_determination_of_the_damping_coefficient_in_the_wave_equation_using_point_source_and_receiver_data.pdf",        "content": "arXiv:1906.08987v1  [math.AP]  21 Jun 2019UNIQUE DETERMINATION OF THE DAMPING COEFFICIENT IN THE WAVE\nEQUATION USING POINT SOURCE AND RECEIVER DATA\nMANMOHAN VASHISTH\nAbstract. Inthisarticle, weconsidertheinverseproblemsofdetermi ningthedampingcoefficientappearing\nin the wave equation. We prove the unique determination of th e coefficient from the data coming from a\nsingle coincident source-receiver pair. Since our problem is under-determined, so some extra assumption on\nthe coefficient is required to prove the uniqueness.\nKeywords : Inverse problems, wave equation, point source-receiver, d amping coefficient\nMathematics subject classification 2010: 35L05, 35L10, 35R30, 74J25\n1.Introduction\nWe consider the following initial value problem (IVP),\n(/square−q(x)∂t)u(x,t) =δ(x,t) (x,t)∈R3×R\nu(x,t)|t<0= 0 x∈R3(1)\nwhere/square:=∂2\nt−∆xdenotes the wave operator and the coefficient q∈C∞(R3) is known as damping\ncoefficient. In this paper, we study the problem of determinat ion of coefficient qappearing in (1) from\nthe knowledge of solution measured at a single point for a cer tain period of time. We are interested in\nthe uniqueness of determination of coefficients qfrom the knowledge of u(0,t) fort∈[0,T] withT >0 in\nEquation (1). The problem studied here is motivated by geoph ysics, where geophysicists wish to determine\nthe properties of earth structure by sending the waves from t he surface of the earth and measuring the\ncorrespondingscattered responses(see [2, 24] andreferen ces therein). Sincethecoefficient tobedetermined\nhere depends on three variables while the given data depends on one variable as far as the parameter count\nis concerned, the problem studied here is under-determined . Thus some extra assumptions on coefficient q\nare required in order to make the inverse problem solvable. W e prove the uniqueness result for the radial\ncoefficient.\nThere are several results related to the inverse problems fo r the wave equation with point source. We list\nthem here. Romanov in [18] considered the problem for determ ining the damping and potential coefficient\nin the wave equation with point source and proved unique dete rmination of these coefficients by measuring\nthe solution on a set containing infinite points. In [12] the p roblem of determining the radial potential\nfrom the knowledge of solution measured on a unit sphere for s ome time interval is studied. Rakesh and\nSacks in [16] established the uniqueness for angular contro lled potential in the wave equation from the\nknowledge of solution and its radial derivative measured on a unit sphere. In the above mentioned works\nthe measurement set is an infinite set. Next we mention the wor k where uniqueness is established from\nthe measurement of solution at a single point. Determinatio n of the potential from the data coming from\na single coincident source-receiver pair is considered in [ 15] and the uniqueness result is established for the\npotentials which are either radial with respect a point differ ent from source location or the potentials which\nare comparable. Recently author in [25] extended the result of [15] to a separated point source and receiver\ndata. To the best of our understanding, very few results exis t in the literature involving the recovery of\nthe damping coefficient from point source and receiver data. O ur result, Theorem 1.1, is work in this\ndirection. In the 1-dimensional inverse problems context, several results exist involving the uniqueness of\n12 VASHISTH\nrecovery of the coefficient which depends on the space variabl e corresponding to the first order derivative;\nsee [9, 10, 11, 13, 19, 22]. We refer to [1, 3, 8, 14, 17] and refe rences therein for more works related to the\npoint source inverse problems for the wave equation.\nWe now state the main results of this article.\nTheorem 1.1. Suppose qi(x)∈C∞(R3),i= 1,2withqi(x) =Ai(|x|)for some C∞function Aion[0,∞).\nLetuibe the solution of the IVP\n(/square−qi(x)∂t)ui(x,t) =δ(x,t) (x,t)∈R3×R\nui(x,t)|t<0= 0 x∈R3.(2)\nIfu1(0,t) =u2(0,t)for allt∈[0,T]for some T >0, thenq1(x) =q2(x)for allxwith|x| ≤T/2, provided\nq1(0) =q2(0).\nThe proof of the above theorem is based on an integral identit y derived using the solution to an adjoint\nproblem as used in [21] and [23]. This idea was used in [4, 17, 2 5] as well.\nThe article is organized as follows. In Section 2, we state th e existence and uniqueness results for the\nsolution of Equation (1), the proof of which is given in [5, 8, 20]. Section 3contains the proof of Theorem\n1.1.\n2.Preliminaries\nProposition 2.1. [5, pp.139,140] Suppose q∈C∞(R3)andu(x,t)satisfies the following initial value\nproblem\nPu(x,t) := (/square−q(x)∂t)u(x,t) =δ(x,t),(x,t)∈R3×R\nu(x,t)|t<0= 0, x ∈R3(3)\nthenu(x,t)is given by\nu(x,t) =R(x,t)δ(t−|x|)\n4π|x|+v(x,t) (4)\nwherev(x,t) = 0fort <|x|and in the region t >|x|,v(x,t)is aC∞solution of the characteristic\nboundary value problem (Goursat Problem)\nPv(x,t) = 0, for t > |x|\nv(x,|x|) =−R(x,|x|)\n8π1/integraldisplay\n0PR(sx,s|x|)\nR(sx,s|x|)ds,∀x∈R3(5)\nandR(x,t)is given by [5, pp. 134]\nR(x,t) = exp\n−1\n21/integraldisplay\n0q(sx)tds\n. (6)\n3.Proof of Theorem 1.1\nIn this section, we prove Theorem 1.1. We will first prove an in tegral identity which will be used to\nprove our main result.\nLemma3.1.Letui(x,t)fori= 1,2be the solution to Equation (2). Then the following integral identity\nholds for all σ≥0/integraldisplay\nR3/integraldisplay\nRq(x)∂tu2(x,t)u1(x,2σ−t)dtdx=u(0,2σ) (7)\nwhereq(x) :=q1(x)−q2(x)andu(x,t) = (u1−u2)(x,t).AN INVERSE PROBLEM WITH UNDER-DETERMINED DATA 3\nProof.Here we have usatisfies the following IVP\n/squareu(x,t)−q1(x)∂tu(x,t) =q(x)∂tu2(x,t) (x,t)∈R3×R\nu(x,t)|t<0= 0 x∈R3.(8)\nMultiplying Equation (8) by u1(x,2σ−t) and integrating over R3×R, we have\n/integraldisplay\nR3/integraldisplay\nRq(x)∂tu2(x,t)u1(x,2σ−t)dtdx=/integraldisplay\nR3/integraldisplay\nR(/squareu(x,t)−q1(x)∂tu(x,t))u1(x,2σ−t)dtdx\n=/integraldisplay\nR3/integraldisplay\nRu(x,t)(/squareu(x,t)−q1(x)∂tu1(x,2σ−t))dxdt\nwhere in the last step above we have used integration by parts and the properties of vin Proposition 2.1.\nThus finally using the fact that u1is solution to (2), we get\n/integraldisplay\nR3/integraldisplay\nRq(x)∂tu2(x,t)u1(x,2σ−t)dtdx=u(0,2σ); for all σ≥0.\nThis completes the proof of the lemma. /square\nUsing Lemma 3.1and the fact that u(0,t) = 0 for all t∈[0,T], we see that\n/integraldisplay\nR3/integraldisplay\nRq(x)∂tu2(x,t)u1(x,2σ−t)dtdx= 0; for all σ∈[0,T/2].\nNow using Equation (4), we get\n/integraldisplay\nR3/integraldisplay\nRq(x)∂t/parenleftBigg\nR2(x,t)δ(t−|x|)\n4π|x|+v2(x,t)/parenrightBigg/parenleftBigg\nR1(x,2σ−t)δ(2σ−t−|x|)\n4π|x|/parenrightBigg\ndtdx\n+/integraldisplay\nR3/integraldisplay\nRq(x)v1(x,2σ−t)∂t/parenleftBigg\nR2(x,t)δ(t−|x|)\n4π|x|+v2(x,t)/parenrightBigg\ndtdx= 0.4 VASHISTH\nThis gives\n/integraldisplay\nR3/integraldisplay\nRq(x)∂tR2(x,t)R1(x,2σ−t)δ(t−|x|)δ(2σ−t−|x|)\n16π2|x|2dtdx\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI1\n+/integraldisplay\nR3/integraldisplay\nRq(x)R2(x,t)R1(x,2σ−t)∂tδ(t−|x|)δ(2σ−t−|x|)\n16π2|x|2dtdx\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI2\n+/integraldisplay\nR3/integraldisplay\nRq(x)∂t/parenleftBigR2(x,t)δ(t−|x|)\n4π|x|/parenrightBig\nv1(x,2σ−t)dtdx\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI3\n+/integraldisplay\nR3/integraldisplay\nRq(x)∂tv2(x,t)R1(x,2σ−t)δ(2σ−t−|x|)\n4π|x|dtdx\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI4\n+/integraldisplay\nR3/integraldisplay\nRq(x)∂tv2(x,t)v1(x,2σ−t)dtdx\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nI5= 0; for all σ∈[0,T/2].(9)\nIn a compact form, this can be written as\nI1+I2+I3+I4+I5= 0. (10)\nNext we simplify each Ijwithj= 1,2,.....,5. We will use the fact that vi(x,t) = 0 for t <|x|.\nWe have\nI1=/integraldisplay\nR3/integraldisplay\nRq(x)∂tR2(x,t)R1(x,2σ−t)δ(t−|x|)δ(2σ−t−|x|)\n16π2|x|2dtdx\n=/integraldisplay\n|x|=σq(x)∂tR2(x,|x|)R1(x,|x|)\n16π2|x|2dSx\n=−/integraldisplay\n|x|=σq(x)R1(x,|x|)R2(x,|x|)\n32π2|x|2/parenleftBig1/integraldisplay\n0q2(sx)ds/parenrightBig\ndSx.\nNext we simplify the integral I2. We use the following formula [7, Page 231, Eq.(10)]\n/integraldisplay\nδ′(r−|x|)ϕdx=−1\n|x|2/integraldisplay\n|x|=r∂\n∂r/parenleftbig\nϕr2/parenrightbig\ndSx. (11)\nNote that from this formula, by a change of variable, we have\n/integraldisplay\nδ′(2r−2|x|)ϕdx=−1\n2|x|2/integraldisplay\n|x|=r∂\n∂r/parenleftbig\nϕr2/parenrightbig\ndSx. (12)AN INVERSE PROBLEM WITH UNDER-DETERMINED DATA 5\nNow\nI2=/integraldisplay\nR3/integraldisplay\nRq(x)R2(x,t)R1(x,2σ−t)∂tδ(t−|x|)δ(2σ−t−|x|)\n16π2|x|2dtdx\n=/integraldisplay\nR3/integraldisplay\nRq(x)R2(x,t)R1(x,2σ−t)δ′(t−|x|)δ(2σ−t−|x|)\n16π2|x|2dtdx\n=/integraldisplay\nR3q(x)R2(x,2σ−|x|)R1(x,|x|)δ′(2σ−2|x|)\n16π2|x|2dx\n=−1\n32π2σ2/integraldisplay\n|x|=σ∂\n∂r{q(x)R1(x,|x|)R2(x,2σ−|x|)}dSx.\nIn the last step above, we used Equation (12).\nNext we have\nI3=/integraldisplay\nR3/integraldisplay\nRq(x)∂t/parenleftBigR2(x,t)δ(t−|x|)\n4π|x|/parenrightBig\nv1(x,2σ−t)dxdt.\nWe can view the derivative above as a limit of the difference quo tients in the distribution topolgy [6, pp.48].\nCombining this with the fact that v1isC2in{(x,t) :|x| ≤t}, we get,\nI3=−/integraldisplay\nR3/integraldisplay\nRq(x)R2(x,t)δ(t−|x|)\n4π|x|∂t/parenleftBig\nv1(x,2σ−t)/parenrightBig\ndxdt\n=/integraldisplay\nR3q(x)R2(x,|x|)∂tv1(x,2σ−|x|)\n4π|x|dx.\nAgain using the fact that v1(x,t) = 0 for t <|x|, we get,\nI3=/integraldisplay\n|x|≤σq(x)R2(x,|x|)∂tv1(x,2σ−|x|)\n4π|x|dx.\nNext we simplify I4. Similiar to I3, we have\nI4=/integraldisplay\nR3/integraldisplay\nRq(x)∂tv2(x,t)R1(x,2σ−t)δ(2σ−t−|x|)\n4π|x|dtdx\n=/integraldisplay\n|x|≤σq(x)R1(x,|x|)∂tv2(x,2σ−|x|)\n4π|x|dx.\nFinally, we have\nI5=/integraldisplay\nR3/integraldisplay\nRq(x)∂tv2(x,t)v1(x,2σ−t)dtdx\n=/integraldisplay\n|x|≤σ2σ−|x|/integraldisplay\n|x|q(x)∂tv2(x,t)v1(x,2σ−t)dtdx.6 VASHISTH\nNow, we use the fact that qiis a radial function, that is, qi(x) =Ai(|x|). Then note that\nRi(x,|x|) = exp\n−|x|\n21/integraldisplay\n0qi(sx)ds\n= exp\n−|x|\n21/integraldisplay\n0Ai(s|x|)ds\n\nis also radial. For simplicity, we denote R(x,|x|) byR(|x|).\nWith this, we have\nI1=−A(σ)R1(σ)R2(σ)\n8π1/integraldisplay\n0A2(sσ)ds.\nNext we consider I2. First let us consider the derivative:\nDr:=∂\n∂r(A(r)R1(x,r)R2(x,2σ−r)).\nAfter a routine calculation, we get,\nDr=A′(r)R1(x,r)R2(x,r)−1\n2A(r)2R1(x,r)R2(x,2σ−r)\n−σA(r)R1(x,r)R2(x,2σ−r)1/integraldisplay\n0A′\n2(rs)sds\n=A′(r)R1(x,r)R2(x,r)−1\n2A(r)2R1(x,r)R2(x,2σ−r)\n−A(r)R1(x,r)R2(x,2σ−r)\nσ\nr\nA2(r)−1/integraldisplay\n0A2(rs)ds\n\n.\nOn|x|=σ, we have\nDr||x|=σ=R1(σ)R2(σ)\nA′(σ)−1\n2A(σ)2−A(σ)A2(σ)+A(σ)1/integraldisplay\n0A2(sσ)ds\n\n=R1(σ)R2(σ)\nA′(σ)−1\n2A(σ)(A1+A2)(σ)+A(σ)1/integraldisplay\n0A2(sσ)ds\n.\nHence\nI2=−1\n8π\nR1(σ)R2(σ)\nA′(σ)−1\n2A(σ)(A1+A2)(σ)+A(σ)1/integraldisplay\n0A2(sσ)ds\n\n.\nLet us denote\n˜A(σ) =A(σ)R1(σ)R2(σ).\nThen\nI2=−1\n8πd\ndσ˜A(σ)−1\n8π˜A(σ)1/integraldisplay\n0A2(sσ)ds.\nTherefore\nI1+I2=−1\n8π\n2˜A(σ)1/integraldisplay\n0A2(sσ)ds+d\ndσ˜A(σ)\n.AN INVERSE PROBLEM WITH UNDER-DETERMINED DATA 7\nConsidering the following integrating factor for I1+I2\nexp\n2σ/integraldisplay\n01/integraldisplay\n0A2(ts)dtds\n,\nwe have\nI1+I2=−1\n8πexp\n−2σ/integraldisplay\n01/integraldisplay\n0A2(ts)dtds\nd\ndσ\nexp\n2σ/integraldisplay\n01/integraldisplay\n0A2(ts)dtds\n˜A(σ)\n.\nNow from Equation (10), we have\n1\n8πd\ndσ\n˜A(σ)exp\n2σ/integraldisplay\n01/integraldisplay\n0A2(st)dsdt\n\n\n= exp\n2σ/integraldisplay\n01/integraldisplay\n0A2(st)dsdt\n/bracketleftBigg/integraldisplay\n|x|≤σq(x)R2(x,|x|)∂t{R1v1}(x,2σ−|x|)\n4π|x|dx\n+/integraldisplay\n|x|≤σq(x)R1(x,|x|)∂tv2(x,2σ−|x|)\n4π|x|dx\n+/integraldisplay\n|x|≤σ2σ−|x|/integraldisplay\n|x|q(x)∂tv2(x,t)v1(x,2σ−t)dtdx/bracketrightBigg\nfor allσ∈[0,T/2].(13)\nIntegrating on both sides with respect to σunder the assumption that ˜A(0) = 0, we get\nexp\n˜σ/integraldisplay\n01/integraldisplay\n02A2(st)dsdt\n˜A(˜σ)\n=˜σ/integraldisplay\n0exp\nσ/integraldisplay\n01/integraldisplay\n02A2(st)dsdt\n/braceleftBigg/integraldisplay\n|x|≤σq(x)R2(x,|x|)∂tv1(x,2σ−|x|)\n4π|x|dx\n+/integraldisplay\n|x|≤σq(x)R1(x,|x|)∂tv2(x,2σ−|x|)\n4π|x|dx\n+/integraldisplay\n|x|≤σ2σ−|x|/integraldisplay\n|x|q(x)∂tv2(x,t)v1(x,2σ−t)dtdx/bracerightBigg\ndσ,for all ˜σ∈[0,T/2].\nNow using the fact that R′\nisare continuous, non-zero functions, and v′\nisare continuous, we have the\nfollowing inequality:\n|˜A(˜σ)| ≤C˜σ/integraldisplay\n0|˜A(r)|drfor all ˜σ∈[0,T/2].\nNow by Gronwall’s inequality, we have ˜A(σ) = 0 for all ˜ σ∈[0,T/2], which gives us q1(x) =q2(x) for all\nx∈R3such that |x| ≤T/2. This completes the proof.8 VASHISTH\nAcknowledgement\nThe author would like to thank Dr. Venky Krishnan for useful d iscussions. He is supported by NSAF\ngrant (No. U1530401).\nReferences\n[1] T. Aktosun, A. Machuca and P. Sacks; Determining the shap e of a human vocal tract from pressure measurements at the\nlips, Inverse Problems, vol. 33, (2017), 115002, 33 pages.\n[2] K.P. Bube and R. Burridge; The one-dimensional inverse p roblem of reflection seismology. SIAM Rev. 25 (1983), no. 4,\n497–559.\n[3] R. Burridge; The Gel’fand-Levitan, the Marchenko, and t he Gopinath-Sondhi integral equations of inverse scatteri ng\ntheory, regarded in the context of inverse impulse-respons e problems. Wave Motion 2 (1980), no. 4, 305–323.\n[4] E. Bl˚ asten; Well-posedness of the Goursat problem and s tability for point source inverse backscattering, 2017 Inv erse\nProblems 33 125003.\n[5] F.G. Friedlander; The wave equation on a curved space-ti me. Cambridge University Press, Cambridge-New York-\nMelbourne, 1975. Cambridge Monographs on Mathematical Phy sics, No. 2.\n[6] F.G. Friedlander and M. Joshi; The Theory of Distributio ns, 2nd Edition, Cambridge University Press, 1998.\n[7] I.M. Gel’fand and G. E. Shilov; Generalized functions. V ol. 1. Properties and operations. Translated from the 1958 R ussian\noriginal by Eugene Saletan.\n[8] M. M. Lavrent’ev, V. G. Romanov and S. P. Shishat ·ski˜i; Ill-posed problems of mathematical physics and analysis .\nTranslated from the Russian by J. R. Schulenberger.\n[9] W. Ning and M. Yamamoto; The GelfandLevitan theory for on e-dimensional hyperbolic systems with impulsive inputs,\nInverse Problems 24 (2008) 025004 (19pp).\n[10] Rakesh; An inverse impedance transmission problem for the wave equation. Comm. Partial Differential Equations 18\n(1993), no. 3-4, 583–600.\n[11] Rakesh and P. Sacks; Impedance inversion from transmis sion data for the wave equation. Wave Motion 24 (1996), no. 3,\n263–274.\n[12] Rakesh; Inversion of spherically symmetric potential s from boundary data for the wave equation. Inverse Problems 14\n(1998), no. 4, 999–1007.\n[13] Rakesh; Characterization of transmission data for Web ster’s horn equation. Inverse Problems 16 (2000), no. 2, L9– L24.\n[14] Rakesh; An inverse problem for a layered medium with a po int source. Problems 19 (2003), no. 3, 497–506.\n[15] Rakesh; Inverse problems for the wave equation with a si ngle coincident source-receiver pair. Inverse Problems 24 (2008),\nno. 1, 015012, 16 pp.\n[16] Rakesh and P. Sacks; Uniqueness for a hyperbolic invers e problem with angular control on the coefficients. J. Inverse\nIll-Posed Probl. 19 (2011), no. 1, 107–126.\n[17] Rakesh and G. Uhlmann; The point source inverse back-sc attering problem. Analysis, complex geometry, and mathema t-\nical physics: in honor of Duong H. Phong, 279–289.\n[18] V.G. Romanov; On the problem of determining the coefficie nts in the lowest order terms of a hyperbolic equation.\n(Russian. Russian summary) Sibirsk. Mat. Zh. 33 (1992), no. 3, 156–160, 220; translation in Siberian Math. J. 33 (1992),\nno. 3, 497–500.\n[19] V. G. Romanov and D. I. Glushkova; The problem of determi ning two coefficients of a hyperbolic equation. (Russian)\nDokl. Akad. Nauk 390 (2003), no. 4, 452–456.\n[20] V.G.Romanov; Integralgeometryandinverseproblemsf or hyperbolicequations, volume26.SpringerScienceandBu siness\nMedia, 2013.\n[21] F. Santosa and W.W. Symes; High-frequency perturbatio nal analysis of the surface point-source response of a layer ed\nfluid. J. Comput. Phys. 74 (1988), no. 2, 318–381.\n[22] M.M. Sondhi; A survey of the vocal tract inverse problem : theory, computations and experiments. Inverse problems o f\nacoustic and elastic waves (Ithaca, N.Y., 1984), 1–19, SIAM , Philadelphia, PA, 1984.\n[23] P.D. Stefanov; A uniqueness result for the inverse back -scattering problem. Inverse Problems 6 (1990), no. 6, 1055 –1064.\n[24] W. W. Symes; The seismic reflection inverse problem. Inv erse Problems 25 (2009), no. 12, 123008, 39 pp.\n[25] M. Vashisth; An inverse problems for the wave equation w ith source and receiver at distinct points, Journal of Inver se\nand Ill-posed Problems, http://doi.org/10.1515/jiip-20 18-0004.\nBeijing Computational Science Research Center, Beijing 10 0193, China.\nE-mail: manmohanvashisth@gmail.com"    },    {        "title": "2305.10111v1.Material_Parameters_for_Faster_Ballistic_Switching_of_an_In_plane_Magnetized_Nanomagnet.pdf",        "content": "arXiv:2305.10111v1  [cond-mat.mes-hall]  17 May 2023Journal of the Physical Society of Japan FULL PAPERS\nMaterial Parameters for Faster Ballistic Switching of an In -plane Magnetized\nNanomagnet\nToshiki Yamaji1*and Hiroshi Imamura1 †\n1National Institute of Advanced Industrial Science and Tech nology (AIST), Tsukuba, Ibaraki 305-8568, Japan\nHigh-speed magnetization switching of a nanomagnet is nece ssary for faster information processing. The ballistic\nswitching by a pulsed magnetic filed is a promising candidate for the high-speed switching. It is known that the switch-\ning speed of the ballistic switching can be increased by incr easing the magnitude of the pulsed magnetic field. However\nit is difficult to generate a strong and short magnetic field pulse in a sm all device. Here we explore another direction\nto achieve the high-speed ballistic switching by designing material parameters such as anisotropy constant, saturati on\nmagnetization, and the Gilbert damping constant. We perfor m the macrospin simulations for the ballistic switching of\nin-plane magnetized nano magnets with varying material par ameters. The results are analyzed based on the switching\ndynamics on the energy density contour. We show that the puls e width required for the ballistic switching can be re-\nduced by increasing the magnetic anisotropy constant or by d ecreasing the saturation magnetization. We also show that\nthere exists an optimal value of the Gilbert damping constan t that minimizes the pulse width required for the ballistic\nswitching.\n1. Introduction\nIn modern information technologies huge amount of data\nare represented as the direction of the magnetization in a sm all\nmagnet such as magnetic grains in magnetic tapes or hard\ndisk drives. To write information on the conventional mag-\nnetic recording media an external magnetic field is applied i n\nthe opposite direction of the magnetization to switch the di -\nrection of the magnetization. During the switching the mag-\nnetization undergoes multiple precessions around the loca l ef-\nfective field consisting of the external field, anisotropy fie ld,\nand demagnetizing field. The typical switching time or write\ntime is of the order of nanoseconds.\nTo meet the growing demand for fast information process-\ning it is important to develop a faster switching scheme. The\nballistic switching is a promising candidate for high-spee d\nswitching, and much e ffort has been devoted to developing\nthe ballistic switching both theoretically1–8)and experimen-\ntally.9–16)In ballistic switching a pulsed magnetic field is ap-\nplied perpendicular to the easy axis to induce the large-ang le\nprecession around the external magnetic field axis. The dura -\ntion of the pulse is set to a half of the precession period. Aft er\nthe pulse the magnetization relaxes to the equilibrium dire c-\ntion opposite to the initial direction. The switching speed of\nthe ballistic switching can be increased by increasing the m ag-\nnitude of the pulsed field. However, it is di fficult to generate a\nstrong and short field pulse in a small device. It is desired to\nfind a way to speed up the ballistic switching without increas -\ning magnetic field.\nThe magnetization dynamics of the ballistic switching is\ndetermined by the torques due to the external magnetic field,\nthe uniaxial anisotropy field, the demagnetizing field, and t he\nGilbert damping. The torques other than the external mag-\nnetic field torque are determined by the material parameters\nsuch as the anisotropy constant, the saturation magnetizat ion,\nand the Gilbert damping constant. There is room to speed up\n*toshiki-yamaji@aist.go.jp\n†h-imamura@aist.go.jpthe ballistic switching by designing the appropriate mater ial\nparameters.\nIn the early 2000s the several groups each independently\nreported the optical microscope measurements of the ballis -\ntic switching by picosecond pulse magnetic field.9–13)Then\nthe mechanism of a ballistic switching was analyzed in terms\nof the nonlinear dynamics concepts such as a fixed point, at-\ntractors, and saddle point.2, 3, 6)Especially the minimal field\nrequired for a ballistic switching was investigated by comp ar-\ning the so-called Stoner-Wohlfarth (SW) type.2, 3)The damp-\ning constant dependence of the minimal switching field was\nalso studied.2)The characteristics of the parameters of a pulse\nmagnetic field, i.e., magnitude, direction, and rise /fall time on\nthe mechanism of a ballistic switching had been also studied\nby the simulations and experiments.6, 7, 14, 15)\nAs described above, in 2000s and 2010s a ballistic switch-\ning technique had received much attention for the fast magne -\ntization reversal with ringing suppression by fine-tuning t he\nmagnetic pulse parameters. Due to the recent advance of an\nultra-fast measurement17)the studies of a ballistic switching\nhave attracted much attention again. Last year the in-plane\nmagnetization switching dynamics as functions of the pulse\nmagnetic field duration and amplitude was calculated and\nanalyzed by using the conventional Landau-Lifshitz-Gilbe rt\n(LLG) equation and its inertial form, the so-called iLLG\nequation.16)The solutions of both equations were compared\nin terms of the switching characteristics, speed and energy\ndensity analysis. Both equations return qualitatively sim ilar\nswitching dynamics. However the extensive material param-\neter dependences of a ballistic switching region have not\nyet been sufficiently explored. Therefore it is worth clearing\nthe extensive material parameter dependences of the ballis tic\nswitching of an in-plane magnetized nanomagnet.\nIn this paper, we study the ballistic switching of an in-\nplane magnetized nanomagnet with systematically varying\nthe material parameters by using the macrospin simulations .\nThe results show that the pulse width required for the bal-\nlistic switching can be reduced by increasing the magnetic\n1J. Phys. Soc. Jpn. FULL PAPERS\nHp\nmz\nyx(a)\n(c) my at t = 10 ns (b) \n(d) 0 200 400-1 01\nt [ps]my\ntp [ps]0 1 2 3 4 5tSW  [ps] \n110 10 210 3\ntl tutSW \n0 1 -1 \n0 1 2 3 4 502.55.010.0\n7.5\ntp [ps]Hp [T] \nFig. 1. (a) Schematic illustration of the in-plane magnetized nano magnet.\nThe pulse field, Hp, is applied along the x-direction. The initial direction of\nthe magnetization is in the positive y-direction. (b) Gray scale map of myat\nt=10 ns as a function of the pulse field width, tp, and Hp. The black and\nwhite regions represent the success and failure of switchin g. The parameters\nareµ0Ms=0.92 T,µ0HK=0.1 T, andα=0.023. (c) Typical example of\nthe time evolution of mywhen the magnetization switches ( Hp=5 T and tp\n=0.4 ps). The switching time, tSW, is defined as the time when mychanges\nthe sign. (d) tpdependence of tSWalong the dashed horizontal line at Hp=5\nT shown in Fig. 1(b). tlandtuare 3.15 ps and 3.93 ps, respectively. tSWat\ntl≤tp≤tuis 1.7 ps.\nanisotropy constant or by decreasing the saturation magnet i-\nzation. There exists an optimal value of the Gilbert damping\nconstant that minimizes the pulse width required for ballis -\ntic switching. The simulation results are intuitively expl ained\nby analyzing the switching trajectory on the energy density\ncontour.\n2. Model and Method\nIn this section we show the theoretical model, the numer-\nical simulation method, and the analysis using the trajecto ry\nin the limit ofα→0. The macrospin model of the in-plane\nmagnetized noanomagnet and the equations we solve to simu-\nlate the magnetization dynamics are given in Sec. 2.1. In Sec .\n2.2 we show that the switching conditions can be analyzed by\nusing the trajectory on the energy density contour in the lim it\nofα→0 if theα≪1.\n2.1 Macrospin Model Simulation\nFigure 1(a) shows the schematic illustration of the in-\nplane magnetized nanomagnet. The pulsed magnetic field,\nHp, is applied along the x-direction. The unit vector m=\n(mx,my,mz) indicates the direction of the magnetization. The\nsize of the nanomagnet is assumed to be so small that the dy-\nnamics of mcan be described by the macrospin LLG equation\ndm\ndt=−γm×/parenleftBigg\nHeff−α\nγdm\ndt/parenrightBigg\n, (1)\nwhere tis time,γis the gyromagnetic ratio, αis the Gilbert\ndamping constant. The e ffective field, Heff=Hp+HK+Hd,\ncomprises the pulse field, Hp, the anisotropy field, HK, andthe demagnetizing field, Hd. The anisotropy field and the de-\nmagnetizing field are defined as\nHK=/bracketleftbig2K/(µ0Ms)/bracketrightbigmyey, (2)\nand\nHd=µ0Msmzez, (3)\nrespectively, where Kis the uniaxial anisotropy constant, µ0\nis the magnetic permeability of vacuum, Msis the saturation\nmagnetization, and ejis the unit vector along the j-axis ( j=\nx,y,z).\nThe switching dynamics are calculated by numerically\nsolving the LLG equation. The initial ( t=0) direction is set\nasmy=1. The rectangular shaped pulse magnetic field with\nduration of tpis applied at t=0. The time evolution of magne-\ntization dynamics are calculated for 10 ns. Success or failu re\nof switching is determined by whether my<−0.5 att=10\nns.\nFigure 1(b) shows the gray scale plot of myatt=10 ns\non the tp-Hpplane. Following Ref. 16 the parameters are as-\nsumed to beµ0Ms=0.92 T, K=2.3 kJ/m3, i.e.µ0HK=\n0.1 T, andα=0.023. The black and white regions represent\nthe success and failure of switching, respectively. The wid e\nblack region at upper right of Fig. 1(b) represents the balli stic\nswitching region (BSR). A typical example of the time evolu-\ntion of mywhen the magnetization switches is shown in Fig.\n1(c). The switching time, tSW, is defined as the time when my\nchanges the sign. Figure 1(d) shows the tpdependence of tSW\nalong the horizontal line shown in Fig. 1(b), i.e. at Hp=5\nT. The BSR indicated by shade appears between tl=3.15\nps and tu=3.93 ps, where tSW=1.7 ps independent of tp.\nThe lower and upper boundary of the BSR are represented by\ntlandtu, respectively. We investigate the material parameter\ndependence of tlandtuwith keeping Hp=5 T.\n2.2 Analysis of the Switching Conditions for α≪1\nIf the Gilbert damping constant is much smaller than unity\nthe approximate value of tlandtucan be obtained without\nperforming macrospin simulations. In the limit of α→0, the\ntrajectory is represented by the energy contour because the en-\nergy is conserved during the motion of m. The energy density,\nE, of the nanomagnet is defined as18)\nE=1\n2µ0M2\nscos2θ+K(1−sin2θsin2φ), (4)\nwhereθandφare the polar and azimuthal angles of the mag-\nnetization, respectively. The color plot of the energy dens ity\ncontour is shown in Fig. 2. The separatrix representing the\nenergy contour with E=Kis indicated by the white curve,\nwhich is expresses as\n1\n2µ0M2\nscos2θ−Ksin2θsin2φ=0. (5)\nThe green dot indicates the initial direction of matt=0. The\nblack curve represents the trajectory of munder the pulse field\nofHpin the limit ofα→0. Under the pulse field the energy\ndensity is given by\nE=1\n2µ0M2\nscos2θ+K(1−sin2θsin2φ)\n−µ0MsHpsinθcosφ. (6)\n2J. Phys. Soc. Jpn. FULL PAPERS\n01 5 4 3 26E/K\ntltu\nθ\nφ\nFig. 2. (Color online) Color plot of the energy density contour give n by\nEq. (4).θandφare the polar and azimuthal angles of the magnetization, re-\nspectively. The material parameters, MsandKare same as in Fig. 1. The\nseparatrix given by Eq. (5) is indicated by the white curve. T he initial direc-\ntion of mis indicated by the green dot at ( θ,φ)=(π/2,π/2). The black curve\nrepresents the trajectory of the magnetization under the fie ld of Hp=5 T in\nthe limit ofα→0, which is given by Eq. (7). The yellow stars indicate the\nintersection points of the separatrix and the trajectory, w hich correspond to tp\n=tlandtu. If the pulse is turned o ffattl≤t≤tu, the magnetization switches\nballistically. The yellow triangle indicates the turning p oint of the trajectory\nof the magnetization near mz=1, at whichφ=0.\nSince the energy density of the initial direction, θ=φ=π/2,\nisE=0, the trajectory under the pulse field is expressed as\n1\n2µ0M2\nscos2θ+K(1−sin2θsin2φ)\n−µ0MsHpsinθcosφ=0. (7)\nThe yellow stars indicate the points where the trajectory\ncrosses the separatrix surrounding the equilibrium point a t\nφ=−π/2. The upper and lower points indicates the direc-\ntion of mat the end of the pulse with tp=tuandtl, re-\nspectively. The corresponding angles ( θl,φl) and (θu,φu) can\nbe obtained by solving Eqs. (5) and (7) simultaneously. If\ntl≤tp≤tu, the magnetization relaxes to the equilibrium di-\nrection at (θ,φ)=(π/2,−π/2) after the pulse to complete the\nswitching. We can obtain the approximate expressions of tl\nandtuas follows. Assuming that the pulse field is much larger\nthan the other fields, the angular velocity of the precession ,ω,\nis approximated as γHp/(1+α2), and tlandtuare analytically\nobtained as\ntl=π−2θturn\nω−1\n2∆θ\nω, (8)\nand\ntu=π−2θturn\nω+1\n2∆θ\nω, (9)\nwhere∆θ=θu−θl, andθturnis the polar angle at the turning\npoint (φ=0) indicated by the yellow triangle.3. Results and Discussion\nIn this section we discuss the dependence of the BSR on\nthe material parameters by analyzing the numerical simula-\ntion results and Eqs. (8) and (9). The results for the variati on\nof the magnetic anisotropy constant, K, saturation magnetiza-\ntion, Ms, and the Gilbert damping constant, α, will be given\nin Secs. 3.1, 3.2, and 3.3, respectively.\n3.1 Anisotropy Constant Dependence of the BSR\nFigure 3(a) shows the anisotropy constant, K, dependence\nof the BSR. The parameters are Hp=5 T,µ0Ms=0.92 T, and\nα=0.023. The simulation results of tlandtuare indicated\nby the orange and blue dots, respectively. The analytical ap -\nproximations of tlandtuobtained by solving Eqs. (5),(7),(8),\nand (9) are represented by the orange and blue curves, respec -\ntively. The simulation and analytical results agree well wi th\neach other because the Gilbert damping constant is as small a s\n0.023. As shown in Fig. 3(a), tlis a monotonically decreasing\nfunction of Kwhile tuis a monotonically increasing function\nofK. As a result the width of the BSR, tu-tl, is a monoton-\nically increasing function of Kas shown in the inset of Fig.\n3(a).\nIn the left panel of Fig. 3(b) the separatrix and the trajecto ry\nwithα=0 for K=2.3 kJ/m3are shown by the blue and\nblack curves, respectively. The same plot for K=9.3 kJ/m3\nis shown in the right panel. As shown in these panels, the\nincrease of Kdoes not change the trajectory much. However,\nthe increase of Kchanges the separatrix significantly through\nthe second term of Eq. (5). Assuming that the angular velocit y\nof the precession is almost constant, the spread of the area\nsurrounded by the separatrix results in the spread of the tim e\ndifference between tlandtu. As a result the BSR is spread by\nthe increase of Kas shown in Fig. 3(a)\n3.2 Saturation Magnetization Dependence of the BSR\nFigure 4(a) shows the saturation magnetization dependence\nof the BSR obtained by the numerical simulation and the ana-\nlytical approximations. The horizontal axis represents th e sat-\nuration magnetization in unit of T, i.e µ0Ms. The parameters\nareHp=5 T,K=2.3 kJ/m3, andα=0.023. The symbols are\nthe same as in Fig. 3(a). The lower boundary of the BSR, tl,\nincreases as theµ0Msincreases while the upper boundary of\nthe BSR, tu, decreases with increase of µ0Ms. Therefore, the\nfaster switching is available for smaller Ms. Theµ0Msdepen-\ndence of the BSR ( tu-tl) is also shown in the inset of Fig. 4(a).\nThe BSR decreases with increase of µ0Ms. In other words, the\nwider BSR is obtained for smaller Ms.\nIn the right panel of Fig. 4(b) the separatrix and the trajec-\ntory withα=0 forµ0Ms=0.35 T are shown by the blue and\nblack curves, respectively. The same plot for µ0Ms=0.92 T is\nshown in the left panel. As shown in these panels, the increas e\nofMsdoes not change the trajectory much but decrease the\nseparatrix significantly through the first term of Eq. (5). As -\nsuming that the angular velocity of the precession is almost\nconstant, the reduction of the area surrounded by the separa -\ntrix results in the reduction of the time di fference between tl\nandtu. As a result the BSR decreases with increase of Msas\nshown in Fig. 4(a)\n3J. Phys. Soc. Jpn. FULL PAPERStl, t u [ps] \n0 10 20 30 40 2.03.04.05.0\nK [kJ/m3]ballistic switching region (a)\ntltu\ntu - t l [ps] \nK [kJ/m 3]0 10 20 30 40 0.0 0.5 1.0 1.5 2.0 2.5 \n0π\nπ/2 \n0 0 π π -π -π π /2 π/2 -π /2 -π /2 θ\nφ φˑˑ\nˑˑ(b) K = 2.3 kJ/m3K = 9.3 kJ/m3\ntltltutu\nFig. 3. (Color online) (a) Anisotropy constant, K, dependence of the BSR\n(orange shade). Simulation results of tlandtuare plotted by the orange and\nblue dots, respectively. The analytical results are indica ted by the solid curves\nwith the same color. The parameters are Hp=5 T,µ0Ms=0.92 T, andα=\n0.023. In the inset the simulation and analytical results of the width of the\nBSR, tu-tl, are plotted by the dots and the solid curve, respectively. ( b)\nTypical examples of the trajectory of the magnetization (bl ack curve) and the\nseparatrix (blue curve). The left and right panels show the r esults for K=2.3\nkJ/m3andK=9.3 kJ/m3, respectively. The orange and blue stars indicate\nthe direction at t=tlandtu, respectively. The green dots indicate the initial\ndirection of m.\n3.3 Gilbert Damping Constant Dependence of the BSR\nFigure 5(a) shows the simulation results of the Gilbert\ndamping constant, α, dependence of the BSR. The width of\nthe BSR is shown in the inset. The symbols are the same as\nin Fig. 3(a). The approximate values obtained by Eqs. (8) and\n(9) are not shown because the αis not limited toα≪1. The\nparameters are Hp=5 T, K=2.3 kJ/m3, andµ0Ms=0.92\nT. There exists an optimal value of αthat minimizes tl. The\noptimum value in Fig. 5 (a) is αopt=0.35.\nTo understand the mechanism for minimization of tlat a\ncertain value ofαone need to consider two di fferent effects of\nαon the magnetization dynamics. One e ffect is the decrease\nof the precession angular velocity with increase of α. The pre-\ncession angular velocity around the e ffective field of Heffis\ngiven by (γHeff)/(1+α2), which decreases with increase of α.\nThis effect causes the increase of tlandtu.\nThe other effect is the increase of the energy dissipation rate\nwith increase ofα. Let us consider the trajectory in the cases\nof small damping ( α=0.023) and large damping ( α=αopt).\nIn Fig. 5 (b) the typical examples of the trajectory for the\nsmall damping are shown by the yellow and green curves\nand dots on the energy density contour. The pulse widths are\ntp=tl(=3.15 ps) and 3.14 ps. The trajectories during the\npulse are represented by the solid curves and the trajectori es\nafter the pulse are represented by the dots. The white curve\nshows the separatrix and the black dot indicates the initial di-\ntl, t u [ps] \n2.03.04.05.0\n0.0 0.3 0.6 0.9 1.2\nμ0Ms [T]ballistic switching region \ntltu(a)\n(b) μ0Ms = 0.92 T μ0Ms = 0.35 T\n0π\nπ/2 \n0 0 π π -π -π π /2 π/2 -π /2 -π /2 θ\nφ φˑˑ\nˑˑ\ntl tltu tu1.5tu - t l [ps] \n0123\n0.0 0.3 0.6 0.9 1.2 \nμ0Ms [T] 1.5 \nFig. 4. (Color online) (a) Saturation magnetization dependence of the\nBSR. The horizontal axis represents the saturation magneti zation in unit of T,\ni.eµ0Ms. The parameters are Hp=5 T, K=2.3 kJ/m3, andα=0.023. The\nsymbols are the same as in Fig. 3 (a). (b) Typical examples of t he trajectory\nof the magnetization (black curve) and the separatrix (blue curve). The right\nand left panels show the results for µ0Ms=0.35 T and 0.92 T, respectively.\nThe symbols are the same as in Fig. 3 (b).\nrection. The yellow and green stars indicate the points wher e\nthe trajectories cross the separatrix surrounding the targ et and\ninitial states, respectively. The arrows indicate the dire ction\nof the movement of the magnetization. For the small damp-\ning, even very close to the separatrix around the target stat e at\nthe end of the pulse, the magnetization flows to the sepatrari x\naround the initial state and relax to the initial state after many\nprecessions with the slow energy dissipation.\nFigure 5 (c) shows the tpdependence of tSWat the large\ndamping (α=αopt). All parameters except αare the same\nas in Fig. 1 (d). t′\nl,tl, and tuare 0.82 ps, 1.98 ps, and 4.54\nps, respectively. t′\nlis the time when for the large damping the\nmagnetization goes across the e ffective separatrix around the\ninitial state during the pulse duration. In Fig. 5 (d) the typ ical\nexamples of the trajectory for the large damping are shown\nby the yellow ( tp=0.9 ps), green ( tp=tl=1.98 ps), and\npurple ( tp=4.55 ps) curves and dots on the energy density\ncontour. The symbols are the same as in Fig. 5 (b). In the\nregion 1 ( tp<t′\nl) of Fig. 5 (c) the magnetization is located on\nthe inside of the effective separatrix at the end of the pulse and\nreturn to the initial state. The trajectory is not shown in Fi g. 5\n(d). In the region 2 ( t′\nl≤tp<tl) of Fig. 5 (c) after the pulse is\nremoved the magnetization move toward the target state unde r\nthe effective field of Heffand goes across the separatrix. Then\nthe magnetization finishes the switching. The typical examp le\nof the trajectory is shown by the yellow curve and dots in Fig.\n5 (d). In the region 3 ( tl≤tp≤tu) after the pulse is removed\nthe magnetization ballistically switches. The typical exa mple\n4J. Phys. Soc. Jpn. FULL PAPERS\ntl, t u [ps] \n2.04.06.08.0\n0.0 0.2 0.4 0.6 0.8\nαballistic switching region \ntltu(a)\n0\ntu - t l [ps] \n0.00123\n0.2 0.4 0.6 0.8 \nα45\nαopt \nαopt\n01 5 4 3 26E/K \nα = 0.023 \ntp = t l = 3.15 ps \ntp = 3.14 ps (b)\n0 -π/2 π/2 π -π0π/2 π\nφθ\n< <<\n< <\nˑ\nˑ\n(c)\ntp [ps] 0 1 2 3 4 5tSW  [ps] \n110 10 210 3\ntl tu1 2 4 3\ntl’\n01 5 4 3 26E/K \nα = 0.35 \ntp = 0.9 ps \ntp = t l = 1.98 ps \ntp = 4.55 ps (d)\n0 -π/2 π/2 π -π0π/2 π\nφθ\n<\n<\n<ˑˑ <\n< <\nˑ\nFig. 5. (Color online) (a) Simulation results of the tlandtuas a function\nofα. The parameters are Hp=5.0 T, K=2.3 kJ/m3, andµ0Ms=0.92\nT. The symbols are the same as in Fig. 3 (a). (b) The trajectori es atα=\n0.023 with tp=3.15 ps (yellow) and 3.14 ps (green) on the energy density\ncontour. The trajectory during the pulse is represented by t he solid curve. The\ntrajectory after the pulse is represented by the dots. The wh ite curve shows\nthe separatrix.The direction of the trajectory is indicate d by the arrow. The\nstar indicates the intersection point of the trajectory and the separatrix. The\ninitial direction is indicated by the black dot. (c) tpdependence of tSWat\nα=αopt. All parameters except αare the same as in Fig. 1 (d). t′\nl,tl, and tu\nare 0.82 ps, 1.98 ps, and 4.54 ps, respectively. tSWattl≤tp≤tuis 1.98 ps.\n(d) The trajectories at α=αoptwith tp=0.9 ps (yellow), 1.98 ps (green), and\n4.55 ps (purple) on the energy density contour. The symbols a re the same as\nin Fig. 5 (b).\nof the trajectory is shown by the green curve and dots in Fig.5 (d). As explained in Sec. 2.1, in this study the tSWis defined\nas the time when mychanges the sign, in other words, the\nmagnetization reaches the turning point ( φ=0) on the energy\ndensity contour. Therefore the tlfor the large damping is equal\nto the ballistic tSW, i.e. the time when under the pulse field the\nmagnetization reaches the turning point. The ballistic tSWat\nthe region 3 is 1.98 ps.\nAs described above, for the large damping the magnetiza-\ntion can relax to the target state even at greater distances f rom\nthe separatrix by moving under Heffwith the fast energy dis-\nsipation. The effect can be regarded as the e ffective spread\nof the separatrix and results in the decrease of tland the in-\ncrease of tuwith increase ofα. Therefore, there exists the\noptimal value ofαthat minimizes tlwhile tumonotonically\nincreases with increase of αas shown in Fig. 5(a). In the re-\ngion 4 ( tp>tu) after the pulse is removed the magnetization\nmoves toward the separatrix around the initial state under Heff\nand relaxes to the initial state. We find that the BSR for the\nlarge damping can be explained by the anisotropic spread of\nthe effective separatrix with increasing α, which is fundamen-\ntally due to the breaking of the spatial inversion symmetry o f\nthe spin dynamics. The broken symmetry of the spatial inver-\nsion of the spin dynamics for the large damping can be easily\nconfirmed by comparing Fig. 5 (c) with Fig. 1 (d).\n4. Summary\nIn summary, we study the material parameter dependence\nof the ballistic switching region of the in-plane magnetize d\nnanomagnets based on the macrospin model. The results show\nthat the pulse width required for the ballistic switching ca n be\nreduced by increasing the magnetic anisotropy constant or b y\ndecreasing the saturation magnetization. The results also re-\nvealed that there exists an optimal value of the Gilbert damp -\ning constant that minimizes the pulse width required for the\nballistic switching. The simulation results are explained by\nanalyzing the trajectories on the energy contour. The resul ts\nare useful for further development of the high-speed inform a-\ntion processing using the ballistic switching of magnetiza tion.\nThis work is partially supported by JSPS KAKENHI Grant\nNumber JP20K05313.\n1) L. He and W. D. Doyle: J. Appl. Phys. 79(1996) 6489.\n2) Z. Z. Sun and X. R. Wang: Phys. Rev. B 71(2005) 174430.\n3) D. Xiao, M. Tsoi, and Q. Niu: Journal of Applied Physics 99(2006)\n013903.\n4) Y . Nozaki and K. Matsuyama: Jpn. J. Appl. Phys. 45(2006) L758.\n5) Y . Nozaki and K. Matsuyama: Journal of Applied Physics 100(2006)\n053911.\n6) Q. F. Xiao, B. C. Choi, J. Rudge, Y . K. Hong, and G. Donohoe: J ournal\nof Applied Physics 101(2007) 024306.\n7) P. P. Horley, V . R. Vieira, P. M. Gorley, V . K. Dugaev, J. Ber akdar, and\nJ. Barna´ s: Journal of Magnetism and Magnetic Materials 322(2010)\n1373.\n8) Y . B. Bazaliy: Journal of Applied Physics 110(2011) 063920.\n9) T. Gerrits, H. A. M. van den Berg, J. Hohlfeld, L. B¨ ar, and T . 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Scalera, M. d’Aquino, C. S erpico,\nO. Hellwig, J.-E. Wegrowe, M. Gensch, and S. Bonetti: Nat. Ph ys.17\n(2021) 245.\n18) W. F. Brown: Phys. Rev. 130(1963) 1677.\n6"    },    {        "title": "2206.02460v2.Probing_spin_dynamics_of_ultra_thin_van_der_Waals_magnets_via_photon_magnon_coupling.pdf",        "content": "Probing spin dynamics of ultra-thin van der Waals magnets via\nphoton-magnon coupling\nChristoph W. Zollitsch,1,a)Safe Khan,1Vu Thanh Trung Nam,2Ivan A. Verzhbitskiy,2Dimitrios Sagkovits,1, 3\nJames O’Sullivan,1Oscar W. Kennedy,1Mara Strungaru,4Elton J. G. Santos,4, 5John J. L. Morton,1, 6Goki\nEda,7, 2, 8and Hidekazu Kurebayashi1, 6, 9\n1)London Centre for Nanotechnology, University College London, 17-19 Gordon Street, London, WCH1 0AH,\nUK\n2)Department of Physics, Faculty of Science, National University of Singapore, 2 Science Drive 3, Singapore 117542,\nSingapore\n3)National Physical Laboratory, Hampton Road, Teddington TW11 0LW, UK\n4)Institute for Condensed Matter Physics and Complex Systems, School of Physics and Astronomy, The University of Edinburgh,\nEdinburgh EH9 3FD, UK\n5)Higgs Centre for Theoretical Physics, The University of Edinburgh, Edinburgh EH9 3FD,\nUK\n6)Department of Electronic & Electrical Engineering, UCL, London WC1E 7JE, United Kingdom\n7)Centre for Advanced 2D Materials, National University of Singapore, 6 Science Drive 2, Singapore 117546,\nSingapore\n8)Department of Chemistry, Faculty of Science, National University of Singapore, 3 Science Drive 3, Singapore 117543,\nSingapore\n9)WPI Advanced Institute for Materials Research, Tohoku University, 2-1-1, Katahira, Sendai, 980- 8577,\nJapan\n(Dated: 1 May 2023)\nLayered van der Waals (vdW) magnets can maintain a\nmagnetic order even down to the single-layer regime and\nhold promise for integrated spintronic devices. While the\nmagnetic ground state of vdW magnets was extensively\nstudied, key parameters of spin dynamics, like the Gilbert\ndamping, crucial for designing ultra-fast spintronic de-\nvices, remains largely unexplored. Despite recent studies\nby optical excitation and detection, achieving spin wave\ncontrol with microwaves is highly desirable, as modern in-\ntegrated information technologies predominantly are op-\nerated with these. The intrinsically small numbers of\nspins, however, poses a major challenge to this. Here, we\npresent a hybrid approach to detect spin dynamics medi-\nated by photon-magnon coupling between high-Q super-\nconducting resonators and ultra-thin flakes of Cr 2Ge2Te6\n(CGT) as thin as 11 nm. We test and benchmark our tech-\nnique with 23 individual CGT flakes and extract an upper\nlimit for the Gilbert damping parameter. These results are\ncrucial in designing on-chip integrated circuits using vdW\nmagnets and offer prospects for probing spin dynamics of\nmonolayer vdW magnets.\nINTRODUCTION\nvan der Waals (vdW) materials1–3consist of individual\natomic layers bonded by vdW forces and can host different\ntypes of collective excitations such as plasmons, phonons and\nmagnons. Strong coupling between these excitation modes\nand electromagnetic waves (i.e. photonic modes) creates con-\nfined light-matter hybrid modes, termed polaritons. Polaritons\na)Electronic mail: c.zollitsch@ucl.ac.ukin vdW materials are an ideal model system to explore a va-\nriety of polaritonic states5,6, e.g. surface plasmon polaritons\nin graphene7,8and exciton polaritons in a monolayer MoS 2\nembedded inside a dielectric microcavity9. These states can\nbe further modified by electrostatic gating16, as well as by\nhetero-structuring with dissimilar vdW layers1.\nNumerous studies on magnon polaritons (MPs)11,12have\nbeen using macroscopic yttrium iron garnet (YIG) cou-\npled to either three-dimensional cavities13or to on-chip\nresonators14,15, with potential applications in ultra-fast infor-\nmation processing, non-reciprocity or microwave to optical\ntransduction. By reducing the number of excitations, MPs\nfind application in the quantum regime e.g., magnon number\ncounting via an electromagnetically coupled superconducting\nqubit16,17or as a building block for Bell state generation18.\nThe rapidly developing research around polaritons and\nspecifically MPs has so far, been little studied in magnetic\nvdW materials due to the relatively recent discoveries of long-\nrange magnetic order in vdW systems at the few monolayer\nregime9,20,21, in addition to its technically challenging real-\nization. Stable MP states are formed by strongly coupling the\nmagnetic field oscillation of a resonant photon to the collec-\ntive magnetization oscillation in a magnetic material. This\nstrong coupling is achieved when the collective coupling rate\ngeffis larger than the average of both system loss rates. In a\nsimplified picture, geffscales linearly with the strength of the\noscillating magnetic field of a resonator and the square root\nnumber of spins14. For studies involving bulk magnetic mate-\nrials and low quality and large microwave resonators, strong\ncoupling is achieved when geff=2pis in the MHz range, which\nis accomplished with relative ease due to the abundance of\nspins in bulk magnetic materials. A reduction of the bulk di-\nmensions down from mm to mm and nm scales, the typical\nlateral dimensions and thickness of vdW material monolay-arXiv:2206.02460v2  [cond-mat.mtrl-sci]  28 Apr 20232\ners, results in a decrease of the coupling strength by at least 6\norders of magnitude. Commonly used microwave resonators\nare not able to produce strong enough oscillating magnetic\nfields to compensate for such a reduction in absolute number\nof spins. Only by advanced resonator design and engineering\nthe regime of strongly coupled MPs in monolayer vdW mag-\nnetic materials can be accomplished, granting access to spin\ndynamic physics at a true 2d monolayer limit and research on\nMPs in nano-scale devices where the whole range of on-chip\ntuning and engineering tools, such as electric fields or device\ndesign, are available.\nMagnons or magnon polaritons have been observed in mag-\nnetic vdW materials, but it had been restricted to either to the\noptical frequency range22,23or a large thickness limit24,25, re-\nspectively. Here, we present our attempt of detecting spin\ndynamics in ultra-thin vdW magnetic materials and the cre-\nation of MPs by magnon-photon coupling in the microwave\nfrequency range, using superconducting resonators optimized\nfor increased magnon-photon coupling. By using microwave\nresonators with a small mode volume, we not only increased\nits oscillating magnetic field strength but also matched it more\nefficiently to the size of nanoscale vdW flakes. Our work\npresents a fundamental cornerstone for a general blueprint\nfor designing and developing magnon-photon hybrids for any\ntype of ultra-thin or monolayer vdW magnetic material, en-\nabling research on on-chip microwave applications for (quan-\ntum) information processing.\nRESULTS\nIn this article, we report on the observation of spin dynam-\nics and the creation of MPs at the onset of the high cooper-\nativity regime with the vdW ferromagnet CGT of nm scale\nthickness, demonstrating a pathway towards stable magnon-\nphoton polariton creation. We combine a precise transfer\nprocess of exfoliated CGT flakes and high sensitivity su-\nperconducting resonators, to access and study the dynami-\ncal response of coupled photon-magnon states in a small-\nvolume (nm-thick and \u0016m-sized) CGT flake (illustrated in\nFig. 1 (a)). High-quality-factor superconducting lumped el-\nement resonators are chosen to be the counterpart due to\ntheir extremely small mode volume ( \u00196000\u0016m3) and con-\nsequently strong oscillating magnetic fields ( B1\u001925nT, see\nSI for resonator quality-factors and B1-field distributions), re-\nsulting in high spin sensitivities4,26. At cryogenic temper-\natures, we perform low-power microwave spectroscopy on\nmultiple resonator-vdW-flake hybrids, covering a frequency\nrange from 12GHz to 18GHz for a variety of thickness. Sam-\nples consist of up to 12 resonators on a single chip, all capac-\nitively coupled to a common microwave transmission line for\nread-out (see SI for details). Multiple peaks of spin-wave res-\nonances are observed for each CGT flake measured. The spin-\nwave modes are closely spaced in frequency and show a large\noverlap. We employ a semi-optimized fitting model to pro-\nduce a good estimate for the collective coupling strength and\nmagnetic linewidth. By taking the resonance value of the most\nprominent peak of each spectrum, we find that all measuredpoints can be fitted very well by a single curve calculated by\nthe Kittel formula with bulk CGT parameters. Furthermore,\nwe extracted the linewidth for the thinnest CGT flake inves-\ntigated, 11nm or 15 monolayers (ML), the only device ex-\nhibiting well separated spin-wave modes. This allowed a fully\nquantitative analysis and we determined an upper limit of the\nGilbert damping parameter of 0 :02. This value is comparable\nto the damping reported for 3d transition metal ferromagnets,\nsuggesting that magnetic vdW flakes have the potential for the\nfabrication of functional spintronic devices.\nWe investigate the dynamics of nm-thick CGT flakes, us-\ning superconducting lumped element resonators made of NbN\n(see methods for fabrication details and SI and Ref. [28] for\nmore performance details). The advantages of a lumped ele-\nment design are the spatial separation of the oscillating mag-\nnetic field B1and electric field E1and the concentration of\nB1within a narrow wire section of the resonators, as indi-\ncated in Fig. 1 (a). Additionally, the B1field distribution is\nhomogeneous along the length of the narrow wire section (see\nfinite element simulations in SI). This magnetic-field concen-\ntration is our primary reason to use this type of resonator in\norder to reduce the photon mode volume as well as achieve\na considerable mode overlap between the resonator photon\nmode and CGT magnon mode, and consequently, a large cou-\npling strength. We therefore transfer CGT flakes onto these\n5 μm\nB0CGTB1a\nbcE1\n0 4 8\nx (μm)y (nm)\n102030\n1240\nMCGT\nCrGeTe\nB1,extent  ≈ 2 μm\nFIG. 1. Magnon-photon coupling between thin CGT and a super-\nconducting resonator. a Schematic of a resonator shows the design\nin detail, indicating the areas of high E1-field (yellow) and B1-field\n(green) intensities, as well as the orientation of the externally applied\nfield B0. Finally, a schematic zoom in of the section loaded with a\nCGT flake is shown. The collective coupling between a microwave\nphoton and the magnetization of the CGT is illustrated, as well as the\napproximate extent of the microwave B1-field. bMicrograph image\nof a CGT flake transferred onto the narrow section of a resonator. c\nAFM image of the CGT flake together with a height profile along the\nblue solid line in the AFM image. The red solid line is a fit to the\nflake thickness. The results of this resonator are presented in Fig. 2.3\n12.8112.8212.83 ω/2π (GHz)\n560 580 600 620\nMagnetic Field (mT)640 66012.841.0 0.9 0.8|S21|20.7\n|S21|21.0\n0.9\n0.8a b\nc\n580 600\nMagnetic Field (mT)620234\nκeff/2π (MHz)0510 ωres/2π (MHz)\nd\n640\n+ 12820 MHz\n0.7550 mT\n598 mT\n614 mT\n670 mT\n12.81 12.82 12.83\nω/2π (GHz)12.84\nFIG. 2. Magnon-photon coupling observed in resonator microwave transmission. a jS21j2as a function static magnetic field B0and\nfrequency, with the microwave transmission encoded in the color. The results are obtained from the resonator shown in Fig 1 (b) and (c),\nfeaturing a loaded quality factor of QL=4600. bjS21j2as a function of frequency at fixed magnetic fields, indicated in aby dashed vertical\nlines. canddResonance frequency wresand effective loss rate keffas a function of magnetic field. Note the multiple resonance peaks,\nindicating multiple CGT FMRs. The dashed orange lines are results from the semi-optimized fit. dexemplary includes the individual peaks of\nwhich the orange dashed lines consists. The green bar in canddhighlights the main mode.\nnarrow sections (Fig. 1 (b)). Details of CGT flake transfers\nare described in the methods section. Optical imaging and\natomic force microscopy (AFM) measurements are used to\ncharacterise the size and thickness of the CGT flakes (see\nFig. 1 (c)). Measured thicknesses range from 153 \u000623nm\ndown to 11\u00061:8nm (15 ML), enabling a thickness dependent\nstudy of CGT flakes and their coupling to the resonators.\nWe measured the microwave transmission jS21j2as a func-\ntion of frequency and externally applied magnetic field B0for\neach resonator at a temperature of 1 :8K, using a microwave\npower of approximately \u000080dBm at the resonator chip. Fig-\nure 2 (a) shows the resulting 2D plot of jS21j2for a resonator\nloaded with a 17nm \u00060:8nm thick CGT flake (see Fig. 1 (b)\nand (c) for the respective micrograph and AFM images). A\nresonator peak can be clearly observed for each magnetic\nfield, with its resonance frequency wresdecreasing with in-\ncreasing magnetic field. The reduction of the frequency is\na result of a slow degradation of the superconductivity by\nB0, which in general exhibits a parabolic dependence29. For\n580mT\u0014B0\u0014630mT the resonator prominence is reduced,\nhighlighted byjS21j2as a function of frequency for four con-\nstant B0values in Fig. 2 (b). Within this field range, the mode\nresonance has been modified due to its hybridization with the\nmagnetic modes of the CGT flake. To further quantify the in-\nteraction, we fit each jS21j2profile by a Fano resonance line-\nshape (solid orange lines in Fig. 2 (b)) to account for an asym-\nmetric resonance peak due to additional microwave interfer-\nence in the circuitry30,31,\njS21j2=S0+A(qkeff=2+w\u0000wres)2\n(keff=2)2+ (w\u0000wres)2: (1)\nHere, S0is the microwave transmission baseline, Athe peak\namplitude, qdescribes the asymmetry of the lineshape and\nkeffrepresents the effective loss rate of the hybrid system (seeSI for resonator parameters before and after CGT transfer for\nall resonators). Figure 2 (c) shows wresof the hybrid system\nas a function of B0.wresexperiences a dispersive shift when\nthe photon mode and the magnon mode hybridize, indicating\nan onset of a strong interaction between the two individual\nsystems14,17,32–34. We observe multiple shifts in wres, suggest-\ning an interaction of several magnon modes with the resonator\nin our experiment.\nSignatures of the resonator–CGT-flake coupling are also\ncharacterised by keffof the hybrid system (Fig. 2 (d)). keffis\nenhanced from the value of the resonator loss rate k0due to\nan additional loss introduced by the magnon system charac-\nterized by the loss rate g14,32,35. Consistent with the B0de-\npendence of wres,keffshows a rich structure, having its main\npeak at 598mT, together with less prominent peaks distributed\naround it. Based on a formalism for coupled-harmonic-\noscillator systems in the high cooperativity regime32–34, we\nuse the following to analyse our experimental results with\nmultiple peaks:\nwres=wres;0+mB2\n0++n\nå\nk=\u0000ng2\neff;kDk\nD2\nk+g2; (2)\nkeff=k0++n\nå\nk=\u0000ng2\neff;kg\nD2\nk+g2: (3)\nwith the detuning factor for each resonance as Dk=\ngCGTmB\n¯h\u0000\nB0\u0000BFMR ;k\u0001\n. Here, wres;0is the resonator resonance\nfrequency at B0=0T and mrepresents the curvature of the\nresonance frequency decrease due to the applied magnetic\nfield. BFMR ;kis the CGT FMR field, gCGT the g-factor of\nCGT and geff;kgives the collective coupling strength between\nphoton and magnon mode. The summation is over all reso-\nnance modes kpresent on the low or high field (frequency)4\nside of the main resonance mode, where ngives the number\nof modes on one side. For simplicity, we assume a symmet-\nric distribution of modes about the main mode. The large\nnumber of multiple modes and their strong overlap prevent\na reliable application of a fully optimized fit to the data, due\nto the large number of free parameters required. In an ef-\nfort to gain a good estimate of the model parameters we ap-\nply the model functions Eq. (2) and (3) in a two-step semi-\noptimized fashion (see SI for details). With this approach, we\narrive at a model in good agreement with wresandkeff(see\norange dashed lines in Fig. 2 (c), (d), exemplary showing the\nindividual peaks of the orange dashed line in Fig. 2 (d) and\nthe SI for additional results and data). We can reproduce the\ndata using g=2p=94:03\u00065:95MHz and a collective cou-\npling strength of the main mode of 13 :25\u00061MHz. Together\nwith k0=2p=1:4\u00060:02MHz the system resides at the onset\nof the high cooperativity regime, classified by the cooperativ-\nityC=g2\neff=k0g=1:3>113,32. In this regime, magnon polari-\ntons are created and coherently exchange excitations between\nmagnons and resonator photons on a rate given by geff. The\ncreated MPs are, however, short lived and the excitations pre-\n100 200 500 700\nResonance Field BFMR (mT)300 400 600 0051015ωFMR/2π (GHz)500\nResonance Field BFMR (mT)600 7001518ωFMR/2π (GHz)\n12a\nb\n11 31 51 71 91 111 131151Flake Thickness (nm)\nFIG. 3. Summary of CGT-FMR conditions. a Extracted CGT res-\nonance fields and frequencies from the set of resonators loaded with\nCGT flakes of different thickness. Resonance values are taken from\nthe most prominent peaks in keff. The solid curve is calculated us-\ning the Kittel formalism presented in10, using same parameters, with\ngCGT =2:18,m0Ms=211:4mT and Ku=3:84\u0002104J=m3.bWider\nmagnetic field range of awhere the CGT flake thickness for the dif-\nferent symbols is indicated by the color gradient given in a.dominately dissipate in the magnonic system, as geff\u001cg.\nOur analysis suggests that the separation of the different\nFMR modes is of the same order of magnitude as the loss rate\n(see SI for additional data). We consider that these are from\nstanding spin wave resonances, commonly observed for thin\nmagnetic films12and with one reported observation in bulk of\nthe vdW material CrI 338. In thin-film magnets under a static\nmagnetic field applied in-plane, the magnetic-dipole interac-\ntion generates two prominent spin wave branches for an in-\nplane momentum, the backward volume spin wave (BVMSW)\nand magnetostatic surface spin wave (MSSW) modes39,40.\nThese spin wave modes have different dispersion relations,\nhaving higher (MSSWs) and lower (BVMSWs) resonance\nfrequencies with respect to that of the uniform FMR mode.\nWe calculate the distance of these standing spin-wave modes\nbased on magnetic parameters of bulk CGT as well as the lat-\neral dimensions of the flakes (see SI for more details). We\ncan find spin waves having a frequency separation within\n100MHz and 200MHz (3 :3mT to 6 :6mT in magnetic field\nunits), which are consistent with our experimental observa-\ntion in terms of its mode separation. However, the irregular\nshape of the CGT flakes renders exact calculations of spin\nwave mode frequencies very challenging. We also consid-\nered a possibility that each layer of CGT might have different\nmagnetic parameters (e.g. chemical inhomogeneity), and thus\nproducing different individual resonance modes. Our numer-\nical simulations based on atomistic spin dynamics14,15rule\nout this possibility, as resonance modes from individual lay-\ners average to a single mode as soon as a fraction of 10% of\ninter-layer exchange coupling is introduced (see SI for more\ndetails). Therefore, we speculate that the multiple mode na-\nture we observe in our experiments is likely originating from\nintrinsic properties of the CGT flakes.\nFigure 3 shows the extracted wFRM as a function of BFMR\nfor each resonator–CGT-flake hybrid. The experimental val-\nues are in excellent agreement with a curve calculated by the\nKittel equation with magnetic parameters for bulk CGT10,\nfrom which the data exhibits a standard deviation of less than\n5%. This agreement, achieved by independent characteri-\nzations of 23 CGT flakes measured by superconducting res-\nonators, is experimental evidence that the magnetic parame-\nters that determine the dispersion of wFRM (BFMR), i.e. the\nCGT g-factor gCGT, saturation magnetization Msand uniaxial\nanisotropy Ku, exhibit little thickness dependence in exfoli-\nated CGT flakes, and are not disturbed by the transfer onto\nthe resonator structure. We note, that this demonstrates that\nvdW magnetic materials are particularly attractive for device\napplications, as they are less prone for contamination from\nexfoliation.\nFinally, we present our analysis of kefffor a resonator with\na 11\u00061:8nm CGT flake in Fig.4. With the thickness of a\nsingle layer of CGT being 0 :7nm9, this flake consists of 15\nmonolayers and is the thinnest in our series. Figure 4 (a) and\n(b) show wresandkeffas a function of B0, respectively. While\nthe response of the CGT flake shows a prominent signature\ninkeff, the CGT FMR is considerably more subtle in wres.\nThis highlights the excellent sensitivity of the high-Q super-\nconducting resonators in our study. kefffeatures five well-5\nseparated peaks with the main peak at B0=547mT, which\nenables us to perform a single-peak fully optimized analy-\nsis for each, in contrast to our multi-step analysis for the re-\nmainder of the devices. We assume the additional peaks are\nBVMSW modes, as discussed in the previous section. How-\never, the splitting is about four times larger than compared to\nall other investigated devices, which would result in a signifi-\ncantly shorter wavelength. Thickness steps can lead to a wave-\nlength down-conversion13, however, due to the irregular shape\nandB1inhomogeneities it is difficult to exactly calculate the\nspin wave frequencies (see SI for further details). From the\nmain peak profile, we extract geff=2p=3:61\u00060:09MHz,\ng=2p=126:26\u00068:5MHz and k0=2p=0:92\u00060:05MHz. We\ncompare the experimental value of geffwith a numerically cal-\nculated geff;simu, using the dimensions of the CGT flake de-\ntermined by AFM measurements (see SI for details). The\ncalculation yields geff;simu=2p=8:94MHz, lying within the\nsame order of magnitude. The overestimation is likely due\nto in-perfect experimental conditions, like non-optimal place-\nment of the flake, uncertainties in the thickness and dimen-\nsion determination as well as excluding the additional modes\nin the calculation (see SI). With g\u001dgeffandC=0:11, the\nhybrid system is in the weak coupling regime13, but due to\nthe highly sensitive resonator with its small k0the response\nfrom the magnon system can still be detected. With the ex-\ntracted g=2pwe can give an upper limit of the Gilbert damp-\ning in CGT, by calculating aupper =g=wFMR. We find aupper as\n0:021\u00060:002, which is comparable to other transition metal\nmagnetic materials44, and is in very good agreement with a\npreviously reported effective Gilbert damping parameter de-\ntermined by laser induced magnetization dynamics45. Here,\nwe emphasise that the actual Gilbert damping value is lower\ndue to a finite, extrinsic inhomogeneous broadening contribu-\ntion.\nWe further use these results to benchmark the sensitivity\nof our measurement techniques. The detection limit is given\nby comparing the main peak height characterised by g2\neff=g\nand the median noise amplitude which is 18kHz in Fig. 4 (b)\nwhere g2\neff=2pg= 103 kHz. By assuming the same lateral di-\nmensions and scale the thickness down to a single monolayer,\nwhile keeping gconstant, we calculate the expected signal re-\nduction numerically by geff;simu;1ML=geff;simu;15ML to 0.26. We\nobtain (0:26geff)2=2pg=7kHz for the monolayer limit. Al-\nthough this suggests the noise amplitude is greater than the\nexpected peak amplitude, we can overturn this condition by\nimproving the coupling strength by optimising the resonator\ndesign, enhancing the exfoliation and flake transfer as well as\nby reducing the noise level by averaging a number of mul-\ntiple scans. Superconducting resonators with mode volumes\nof about 10 \u0016m3have been realised46, a reduction of 2 orders\nof magnitude compared to our current design. This would\ntranslate to an order of magnitude improvement in geff. Fur-\nthermore, this flake covers about 4% of the resonator. By\nassuming maximised coverage a 5 times enhancement of geff\ncan be achieved. Both approaches would make the detection\nof monolayer flakes possible.\nIn summary, we provide the first demonstration of photon-\nmagnon coupling between a superconducting resonator and\n520 540 600 6400.900.951.001.05\nκeff/2π (MHz)\nMagnetic Field (mT)560 580 620\nωres/2π (MHz)\n122801229012300a\nbFIG. 4. Magnon-photon coupling for the thinnest CGT flake. a\nResonance frequency wresandbeffective loss rate keffas a func-\ntion of magnetic field of a resonator loaded with the thinnest CGT,\nconsisting of 15 ML. The resonator’s loaded quality factor is 6938.\nThe solid orange lines are results a fit to Eq.(2) and (3), respectively.\nThe errorbars in brepresent the standard deviation from the Fano\nresonance lineshape fit to the resonator transmission.\nnm-thick vdW flakes of CGT, using a total of 23 devices\nwith different CGT flakes of thickness from 153nm down to\n11nm. By employing a coupled-harmonic-oscillator model,\nwe extract the coupling strength, magnetic resonance field\nand relaxation rates for both photon and magnon modes in\nour devices. From our semi-broadband experiments, we find\nthat the magnetic properties of exfoliated CGT flakes are ro-\nbust against the transfer process, with a standard deviation of\nless than 5% to expected resonance values from bulk param-\neters. Notably, this suggests that vdW magnetic materials can\nbe pre-screened at bulk to identify the most promising mate-\nrial for few layer device fabrication. The upper limit of the\nGilbert damping in the 15 ML thick CGT flake is determined\nto be 0 :021, which is comparable to commonly used ferro-\nmagnetic thin-films such as NiFe and CoFeB and thus mak-\ning CGT attractive for similar device applications. We high-\nlight that the damping parameter is key in precessional mag-\nnetisation switching47,48, auto-oscillations by dc currents49,50,\nand comprehensive spin-orbit transport in vdW magnetic sys-\ntems51. The presented techniques are readily transferable\nto other vdW magnetic systems to study spin dynamics in\natomically-thin crystalline materials. While creating stable\nmagnon polaritons is still an open challenge due to the large\nloss rate gof the CGT magnon system, this work offers an\nimportant approach towards its achievement. There are still\npotential improvements to the measurement sensitivity such\nas resonator mode volume reduction by introducing nm scale\nconstrictions52,53and use of exfoliation/transfer techniques to\nproduce larger flakes to enhance the mode overlap (hence cou-6\npling strength)54,55. With concerted efforts, the formation of\nmagnon polaritons in few layers vdW materials will become\nfeasible.\nMETHODS\nSuperconducting Resonators: The resonators were fab-\nricated by direct laser writing and a metal lift-off process.\nThe individual 5mm \u00025mm chips are scribed from an in-\ntrinsic, high resistivity ( r>5000Wcm) n-type silicon wafer\nof 250 \u0016m thickness. For a well defined lift-off, we use a\ndouble photoresist layer of LOR and SR1805. The resonator\nstructures are transferred into the resist by a Heidelberg Di-\nrect Writer system. After development, \u001850nm NbN are de-\nposited by magnetron sputtering in a SVS6000 chamber, at\na base pressure of 7 \u000210\u00007mbar, using a sputter power of\n200W in an 50:50 Ar/N atmosphere held at 5 \u000210\u00003mbar,\nwith both gas flows set to 50 SCCM28. Finally, the lift-off is\ndone in a 1165 solvent to release the resonator structures.\nCGT Crystal Growth: CGT crystals used in this study\nwere grown via chemical vapour transport. To this end, high-\npurity elemental precursors of Cr (chips, \u001599:995%), Ge\n(powder,\u001599:999%), and Te (shots, 99 :999%) were mixed\nin the molar weight ratio Cr:Ge:Te = 10:13.5:76.5, loaded into\na thick-wall quartz ampule and sealed under the vacuum of\n\u001810\u00005mbar. Then, the ampule was loaded into a two-zone\nfurnace, heated up and kept at 950\u000eC for 1 week to homog-\nenize the precursors. To ensure high-quality growth, the am-\npule was slowly cooled (0 :4\u000eC=h) maintaining a small tem-\nperature gradient between the opposite ends of the ampule.\nOnce the ampule reached 500\u000eC, the furnace was turned off\nallowing the ampule to cool down to room temperature nat-\nurally. The large ( \u00181cm) single-crystalline flakes were ex-\ntracted from the excess tellurium and stored in the inert envi-\nronment.\nCGT Flake Transfer: Devices for this study were made\nvia transfer of single-crystalline thin flakes on top of the super-\nconducting resonators. The flakes were first exfoliated from\nbulk crystals on the clean surface of a home-cured PDMS\n(polydimethylsiloxane, Sylgard 184) substrate. The thickness\nof the CGT flakes on PDMS was estimated through the con-\ntrast variation with transmission optical microscopy. Then,\nthe selected flake was transferred to a resonator. The trans-\nfer was performed in air at room temperature. To minimize\nthe air exposure, the entire process of exfoliation, inspection\nand transfer was reduced to 10-15 min per resonator. For\nthe flakes thicker than 50nm, the strong optical absorption of\nCGT prevented the accurate thickness estimation with optical\ncontrast. 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E.J.G.S. acknowledges computational resources\nthrough CIRRUS Tier-2 HPC Service (ec131 Cirrus Project)\nat EPCC (http://www.cirrus.ac.uk) funded by the University\nof Edinburgh and EPSRC (EP/P020267/1); ARCHER UK Na-\ntional Supercomputing Service (http://www.archer.ac.uk) via\nProject d429. E.J.G.S acknowledges the Spanish Ministry\nof Science’s grant program “Europa-Excelencia” under grant\nnumber EUR2020-112238, the EPSRC Early Career Fellow-\nship (EP/T021578/1), and the University of Edinburgh for\nfunding support. D.S. acknowledges EPSRC funding through\nthe Centre for Doctoral Training in Advanced Characteri-\nsation of Materials (EP/L015277/1) and European Union’s\nHorizon 2020 Research and Innovation program under grantagreement GrapheneCore3, number 881603 and the Depart-\nment for Business, Energy and Industrial Strategy through the\nNPL Quantum Program.\nAUTHOR CONTRIBUTION\nC.W.Z, S.K. and H.K. conceived the experimental project.\nResonator design and optimization was done by J.O’S.,\nO.W.K, C.W.Z and supervised by J.J.L.M. Resonator fabri-\ncation and characterization was done by C.W.Z. CGT crystals\nwere grown by I.A.V . and exfoliated and transferred by I.A.V .\nand N.V .T.T. and supervised by E.G.. D.S. measured AFM on\nthe CGT flakes on the resonators. C.W.Z. performed the ex-\nperiments and the data analysis with input from S.K. and H.K.\nAtomistic spin dynamics simulations were carried out by M.S.\nsupervised by E.J.G.S.. C.W.Z., M.S., I.A.V . and H.K. wrote\nthe manuscript with input from all authors.\nCOMPETING INTERESTS\nThe Authors declare no conflict of interests.\nSupplemental Material - Probing spin dynamics of ultra-thin van der Waals magnets via\nphoton-magnon coupling\nI. MICROWAVE SETUP AND MEASUREMENT\nVNA\nMW out MW in\n-20\ndB+32\ndB\nDUT\nB0Cryostat\nFIG. S1. Microwave delivery and detection setup. Schematic of the microwave delivery and detection circuit. The image shows the coplanar\nwaveguide transmission line. A resonator chip is placed on top of the transmission line for read out. On the right, a schematic layout of the\nresonators on a single chip is shown.2\nFigure S1 shows a schematic of the used microwave measurement setup. We are using a Keysight E5071C vector network\nanalyzer (VNA) to deliver and detect microwaves. The VNA is connected to a low temperature probe, fitted into a closed\ncycle helium cryostat and cooled to a base temperature of about 1 :8K. The microwave signal is transmitted into the cryostat\nand is attenuated by \u000020dB. The attenuator is positioned just before the sample box and provides a thermal anchoring for\nthe center conductor of the coaxial cable to minimize the thermal load onto the sample. The output line is equipped with a\nLow Noise Factory LNC6_20C cryogenic amplifier, operating between 6 \u000020GHz with an average amplification of +32dB.\nThe transmitted and amplified signal is finally detected by the VNA. Figure S1 also shows an image of the coplanar waveguide\ntransmission line PCB, loaded with a resonator ship, of which a schematic shows the resonator layout on a single chip. The\nresonators on the chip are capacitively coupled to the transmission line PCB. Upon resonance the transmission through the\nPCB is reduced, indicating the resonator resonance. The cryostat is equipped with a mechanical rotation stage and prior to the\nmeasurements the superconducting resonators are carefully aligned to the externally applied static magnetic field B0, such that\nthe field is in the plane of the superconductor and along the narrow section of the resonators.\nFigure S2 shows the raw uncalibrated microwave transmission, ranging from 10GHz to 18GHz. The transmission is domi-\nnated by imperfections in our microwave circuitry, masking the small signals from the superconducting resonators. Thus, we\nperformed a simple thru calibration of the microwave transmission to remove contributions from the setup, prior each magnetic\nfield dependent measurement. Here, we exploit the magnetic field tunability of our superconducting resonators. Before calibra-\ntion, we set the frequency range of the measurement. We change the applied magnetic field such that the resonator’s resonance\nfrequency is tuned out of the set frequency range. With a frequency window just showing the transmission of the setup we\nperform the thru calibration. After calibration we set the magnetic field back to its starting value, resulting in a background\ncorrected spectrum with just the resonator feature on it.\n10 11 12 13 14 15 16 17 18-1010\n0\nRaw Transmission |S21| (dB)\nFrequency (GHz)\nFIG. S2. Raw broadband microwave transmission signal. Logarithmic microwave transmission jS21jas a function of frequency between\n10GHz and 18GHz at a temperature of 1 :8K.\nII. RESONATOR CHARACTERIZATION\nIn this study, we fabricate twelve superconducting lumped element resonators on each of three resonator chips were fabricated\nusing the same design (see schematic Fig. 1 (a) in the main text). Prior to transfer of the CGT flakes, we characterized the res-\nonators at a temperature of 1 :8K and zero applied magnetic field, using microwave powers of about \u000080dBm at the resonators,\nwhich is well below the bifurcation limit starting above \u000060dBm. Due to finite fabrication tolerances the resonator parameters\nhave some variation, while some didn’t work at all. However, the targeted resonance frequencies are well reproducible and very\nsimilar for the 3 different chips. We compare the resonator parameters before and after transfer of the CGT flakes and collate the\nparameters in Tab. I. Note, the resonator parameters with the CGT flakes on were obtained with a static magnetic field applied\nin the plane of the superconductor, but far detuned from the CGT FMR. In addition, we add the respective thickness of the flake\non each resonator, acquired from AFM measurements. Here, we give the values of the thickest region of a given flake on a\nresonator, as the thickest region will dominate the FMR signal. Due to the arbitrary shape of exfoliated flakes, some exhibit\nregions of different thickness, as seen e.g. in Fig. S5 (h) and (i).3\nTABLE I. Resonator Parameters\nChip Number wres;before (MHz) QL;before wres;after(MHz) QL;after CGT Thickness (nm)\n1 12165 1978 12063 5733 16.2\u00061.3\n1 13303 7357 13177 4950 -\n1 13968 5575 13860 4679 49.4\u00063.5\n1 14184 6492 14048 5627 153.1\u000623.3\n1 16648 6606 16470 5021 23.5\u00062.5\n1 17431 3215 17237 6826 23.8\u00066.4\n1 17959 7595 17790 3963 26.2\u00064.1\n2 12285 360 12153 7135 49.1\u00069.1\n2 12669 3600 12548 6693 102.8\u00065.6\n2 12782 3448 12648 6557 105.9\u00063.9\n2 13393 4643 13244 4501 34.4\u00064.1\n2 13760 6858 13620 5488 95.9\u00065.9\n2 14395 9048 14201 4139 36.7\u00064.3\n2 16075 7283 - - -\n2 17048 6541 16811 4241 75.5\u00065.4\n3 12043 6114 11899 6044 59.7\u000632.8\n3 12456 2716 12314 6938 11.4\u00061.8\n3 12996 5828 12848 4600 17\u00060.8\n3 13422 6517 13272 5461 89.8\u00067.5\n3 13719 6800 13582 6608 -\n3 14238 9184 14064 5420 73.5\u00068.4\n3 15390 8680 15219 6030 30.5\u00064.2\n3 15821 2386 15604 4769 33.1\u00069.9\n3 16430 7518 16193 5780 30.1\u000638.1\n3 17308 6521 17054 5569 137.9\u00063.4\n3 18111 3542 17870 4643 50.2\u00066.9\nIII. RESONATOR AND COUPLING SIMULATION\nWe use finite element and numerical simulations to optimize our resonator design. Key requirements of our resonators are\na strong resilience to externally applied static magnetic fields and a small mode volume. To achieve a large field resilience we\nreduced the area of the resonator to minimize effects of the magnetic field on the superconducting film. Further, we designed the\nresonators such that they act as lumped element resonators. Here, the resonance frequency is given by the total capacitance and\ninductance of the structure, with wres=1=p\nLC, analogues to a parallel LC circuit. This allows us to locally separate oscillating\nelectric and magnetic fields and also to concentrate the magnetic fields in more confined regions, resulting in very small mode\nvolumes. To verify the lumped element nature of our resonators we performed finite element simulations, using CST Microwave\nStudio. Figure S3 shows the resulting magnitude of the E-field (left side) and H-field (right side) distribution along the resonator\nstructure for the resonator design producing the results shown in Fig. 2 in the main text. The E-field is concentrated along the\nparallel running wire sections, with its strength approaching zero along the narrow wire section. The opposite is the case for the\nH-field, where it is zero along the parallel wire sections and strongly concentrated along the narrow wire section. Note, that the\nH-field magnitude is homogeneous along the whole of the narrow wire section.\nThe CST Microwave Studio at hand allowed us a simulation with perfect electric conductors. This is sufficient to model\nthe general electric and magnetic energy distributions and resonance frequencies, however, not to simulate the corresponding\noscillating magnetic field distribution, created by a superconducting rectangular wire. To this end, we numerically solve the\nBiot-Savart law for a rectangular wire cross-sectionS1, assuming a superconducting current distribution Jx;zS2,\nB1;x;z=m0\n2pZw=2\n\u0000w=2Zd=2\n\u0000d=2J\u0002r\n(x\u0000x0)2+ (z\u0000z0)2dx0dz0; (S1)\nwith the vectors as J= (0;J(x;z);0)Tandr= (x\u0000x0;0;z\u0000z0)Tandm0being the magnetic constant. The integration is performed\nover the cross-section of the wire, of width wand thickness d. We define the wire cross-section in the x-z-plane, with win x-\ndirection and din z-direction. The length of the wire is along the y-direction. For a superconducting wire, the current is\nnot homogeneously distributed over the cross-section of the wire. Current is only flowing on the surface and is exponentially\ndecaying towards the center of the wire. The characteristic length scale is given by the London penetration depth lL. We use the4\nFIG. S3. Finite element simulations of resonator. CST Microwave Studio simulation of the distribution of E-fields and H-fields across the\nresonator structure. The color encoded fields represent the magnitude values.\nfollowing expression for the current distributionS2\nJ(x;z) =J1 \ncoshz0=lL\ncoshd=lL\"\nCcoshx0=l1\ncoshw=l1+1\u0000cosh x0=l2=coshw=l2p\n1\u0000(x0=w)2#\n+J2\nJ1coshx0=lL\ncoshw=lL!\n; (S2)\nwhere\nJ2\nJ1=1:008\ncoshd=lLs\nw=l?\n4\u0003l?=lL\u00000:08301lL=l?;\nC=\u0010\n0:506p\nw=2l?\u00110:75\n;\nl1=lLp\n2lL=l?;\nl2=0:774 l2\nL=l?+0:5152l?;\nl?=lL=2d:\nThe prefactors J1andJ2define the amplitude of the current density and hence the absolute value of the oscillating magnetic field\nB1. We define J1by normalizing the vacuum B1field to the energy density stored in the resonatorS3,S4\n1\n2¯hwres\n2=1\n2m0Z\nB2\n1dV=1\n2m0B2\n1Vm; (S3)\nwith Vmrepresenting the resonator mode volume. The additional factor of1=2on the left hand side of S3 takes into account that\nonly half of the total energy is stored in the magnetic fieldS5. As our resonator design is a quasi 1-dimensional structure we have\nto define boundaries for the mode volume in the x- and z-direction. A common assumption is to use the width of the conductor\nwire wS6. For simplicity, we approximate the x-z-area of the mode distribution with the area of an ellipse. For the last dimension\nwe use the length of the narrow wire section, supported by the CST Microwave Studio simulations (see Fig. S3). In total we find5\nthe mode volume to be Vm= ((p3:0\u0016m\u00022:025\u0016m)\u0000w\u0002d)\u0002300\u0016m=5696\u0016m3. Figure S4 shows the resulting distribution\nof the oscillating magnetic field for the cross-section of the rectangular wire of width w=2\u0016m and thickness d=50nm. The\nmagnitudejB1;x;zjis encoded in the color and the arrows indicate the B1;xandB1;zcomponents of the oscillating field.\n-2000200z (nm)400\n-400\nx (μm)0 -1 1 2 3 -2 -320\n15\n10\n5\n|B| (nT)\n30\n25\nFIG. S4. Cross-section of resonator magnetic field distribution. Calculated magnitude of the magnetic field distribution around the cross-\nsection of a rectangular superconducting wire. The wire cross-section lies in the center, indicated by the grey rectangular. The red arrows show\nthe direction of the magnetic field.\nWith the simulated B1field distribution we can calculate the position dependent single photon - single spin coupling strength\ng0(r)S3,S4for each magnetic moment per unit cell of CGT (ab-plane 0 :68nmS7,S8, along the c-axis 0 :7nmS9). Summation over\nall CGT unit cells Nwithin the mode volume of the resonator results in the collective coupling strength\ngeff=s\nN\nå\ni=1jg0(ri)j2=gCGTmB\n2¯hs\nN\nå\ni=1jB1(ri)j2=gCGTmB\n2¯hNys\nN\nå\ni=1h\n(B2\nx;i+B2\nz;i)i\n: (S4)\nHere, mBis the Bohr magneton, Nyis the number of unit cells along the y-direction and gCGTis the g-factor for CGT for which\na value of 2 :18S10is used. Note, we give the collective coupling strength for spin1=2and for linear polarized microwavesS3. For\nthe calculation of gefffor the resonator loaded with 15 monolayers of CGT we extracted its lateral dimension from the AFM\nmeasurements (see Fig. S4 (g)) to 2 \u0016m along the x-direction and 12 \u0016m along the y-direction. The flake is assumed to lie directly\non top of the superconducting wire without any gap in between. For these values the simulation yields geff=2p=8:94MHz,\nwhich is about a factor 2 :5 larger than the experimentally determined value of 3 :61MHz. The overestimation of the simulation\nmost likely results from non-ideal conditions in the experiment. The corresponding flake lies at the top end of the resonators\nnarrow wire section (see Fig. S5 (g)), where B1is concentrated. The finite element simulations show that in this area the field\nstrength is already declining, resulting in a reduced coupling strength. Further, AFM can overestimate the thickness of a flake\nslightly for when there is a gap between resonator surface and flakeS9. The calculation also not includes the multiple peaks\nobserved in the experiment, which - depending on their real nature - can distribute the magnon density over all resonant peaks.\nNevertheless, we can use the simulation to estimate the signal reduction by scaling down the thickness of the flake to a single\nmonolayer. Reducing the simulation to a single monolayer, while keeping the lateral dimensions, results in geff=2p=2:33MHz,\na reduction by a factor of 0 :26.\nIV. AFM MEASUREMENTS ON CGT FLAKES\nAfter the transfer of the CGT flakes onto the individual resonators and after measuring FMR, we characterized the thickness\nof the flakes by AFM. Figure S5 shows a selection of height profile maps from the three resonator chips, including a height\nprofile along the inductor wire of the resonator (blue line in the AFM profile images in Fig. S5). To extract the thickness we fit\nthe steps in the height profile (red or green lines in the height profiles in Fig. S5). Note, the height values are relative values with\nan arbitrary offset. Figure S5 (g) shows the thinnest flake of this study, where the processed FMR data is shown in Fig. 4 in the\nmain text.6\n10 μm\n5 μm0 5\nx (μm)y (nm)40506070\n5 μmy (nm)\n02040\n0 4\nx (μm)2 6\n5 μmy (nm)\n506070\n0 4\nx (μm)2 680\n8\n5 μm\ny (nm)\n02040\n0 10\nx (μm)5 1560\n2080\ny (nm)050100\n0 10\nx (μm)5\n5 μm\n5 μm\ny (nm)\n02040\n0 10\nx (μm)5 15\n5 μm 10 μmy (nm)101520\n0 6\nx (μm)325\ny (nm)\n03060\n0 20\nx (μm)1090\ny (nm)50100\n0 6\nx (μm)3150 120\n30(c) (b) (a)\n(f) (e) (d)\n(i) (h) (g)9\nFIG. S5. AFM measurements. AFM profile images with respective height profile (above) along the resonator inductor wire (blue and\npurple lines in profile images, with the arrow indicating scan direction). a-cfigures for resonator chip 1 (refer to Tab. I), having resonance\nfrequencies with CGT of 17237MHz, 17790MHz and 16470MHz, respectively. d-ffigures for resonator chip 2 (refer to Tab. I), having\nresonance frequencies with CGT of 13244MHz, 12063MHz and 13620MHz, respectively. g-iimages for resonator chip 3 (refer to Tab. I),\nhaving resonance frequencies with CGT of 12314MHz, 13272MHz and 17054 ;MHz, respectively. The red and green solid lines are fits to the\nheight profiles.\nV. ANALYSIS AND ADDITIONAL FMR DATA\nWe analyze our experimental data, using the model functions (2) and (3) from the main text in a two-step semi-optimized\nfashion. The main intention for this approach is to minimize the number of free parameters in our model functions. In a first7\ncoarse step, we match the collective coupling strength geff;kto fit the experimental data, assume a constant separation between\nthe individual magnon modes at BFMR ;kand the same magnon loss rate gfor all modes and determine the resonator loss rate\nk0from the resonator transmission far detuned from the FMR with the CGT flakes. This results in 3 free parameters for the\nfirst stage of our analysis, the magnon loss rate g,BFMR of the main mode and the constant separation between the BFMR ;k.\nAfter this first step we arrive at a best fit to the envelope of the experimental data, however with not matching amplitudes. In a\nconsecutive second step, we manually optimize the geff;kto arrive at a model in good agreement with wresandkeff(see dashed\nlines in Fig. S6).\nFig. S6 shows additional results from the corresponding FMR measurements performed on the in Fig. S5 showed resonators.\nAs described in the main text, the measurements were performed at a temperature of 1 :8K and recording the microwave trans-\nmissionjS21j2as a function of the static magnetic field. Analyzing the microwave transmission by fitting a Fano resonance\nlineshape to it we extract the effective loss rate of the resonator, interacting with the CGT keff. Figure S6 shows the resulting\nkeffas a function of the magnetic field. In general, the response of the CGT FMR is complex and varies for the different res-\nonators. The resonance lineshape is not well described by just a single Lorentzian and requires multiple peaks to produce a\ngood agreement. For some resonators, keffexhibits obvious peaks, residing on a broader spectrum (see Fig. S6 (c), (f) and (i)).\nTogether with the observation of well and clearly separated peaks for the resonator loaded with the thinnest CGT flake of 11nm,\nwe motivating the multiple peak analysis as presented in the main text. However, as the individual peaks are overlapping for the\nremainder of the resonators we only applied a qualitative analysis.\nκeff/2π (MHz)580 600\nMagnetic Field (mT)62036912\n525 550\nMagnetic Field (mT)5751.61.82.0\n600\n625\n675 700\nMagnetic Field (mT)725246\n7508\n10\n500 525\nMagnetic Field (mT)5501.251.301.35\n5751.40\n520 560\nMagnetic Field (mT)6000.900.951.00\n6401.05\n650 675\nMagnetic Field (mT)7004812\n72516\n675 700\nMagnetic Field (mT)7251.41.61.8\nκeff/2π (MHz)\n750 650 700\nMagnetic Field (mT)750369\n800\n12\n500 520\nMagnetic Field (mT)540123\n560\n(c) (b) (a)\n(f) (e) (d)\n(i) (h) (g)\nκeff/2π (MHz)\nκeff/2π (MHz)\nκeff/2π (MHz)\nκeff/2π (MHz)\nκeff/2π (MHz)\nκeff/2π (MHz)\nκeff/2π (MHz)\nFIG. S6. Additional data on magnon-photon coupling of CGT-resonator devices. Results from FMR measurements with effective loss rate\nkeff=2pas a function of the static magnetic field. a-cresults for resonator chip 1 (refer to Tab. I), having resonance frequencies with CGT\nof 17237MHz, 17790MHz and 16470MHz, respectively. d-fresults for resonator chip 2 (refer to Tab. I), having resonance frequencies with\nCGT of 13244MHz, 12063MHz and 13620MHz, respectively. g-iresults for resonator chip 3 (refer to Tab. I), having resonance frequencies\nwith CGT of 12314MHz, 13272MHz and 11899MHz, respectively. The orange solid lines are semi-optimized fits, as described in the main\ntext. The errorbars in the figures represent the standard deviation from the Fano resonance lineshape fit to the respective resonator transmission.\nFigure S7 shows the extracted collective coupling strength geffas a function of the square root of the FMR active volume. We\ndefine the active volume as the overlap of the oscillating magnetic field B1and the CGT flake lying on the resonator. The B1\nfield distribution, discussed in Sec. III, is used to estimate the extend of the B1and is taken as 2 \u0016m. From AFM measurements\nand microscope images we extract the thickness and lateral dimensions of the flakes to calculate the final active volume. As\nthe collective coupling is proportional to the square root of the number of magnetic momentsS3, which are interacting with the\nresonator field, it follows that geffscales linearly with the square root of the active volume. This linear trend is highlighted by\nthe orange solid line in Fig. S7. The majority of the extracted data follows this linear trend very well, corroborating our analysis.\nOnly 3 data points deviate strongly from the rest of the data, which we attribute to significant inhomogeneities in the CGT-flakes,\nmaking the volume estimation inaccurate. These data points are highlighted in red in Fig. S7.8\n0 1 2 3 4 51030\n20Collective Coupling geff/2π (MHz)\nSquare Root of Active FMR Volume ( μm3/2)\nFIG. S7. Scaling of the collective coupling. Collective coupling strength geffas a function of the FMR active CGT-flake volume. The orange\nline highlights the linear trend of geffwith increasing volume. The red symbols are regarded as outliers, as these flakes show inhomogeneities,\nleading to inaccurate volume estimations. The star symbol represents data from the thinnest flake (see data in Fig. 4 in the main text) and the\npentagon symbol data from the 17nm flake (see data in Fig. 2 in the main text) The errorbars give confidence values for the extracted values.\nVI. MAGNETO-STATIC SPIN-WAVE DISPERSION IN THIN-FILM MAGNETS WITH PERPENDICULAR ANISOTROPY\nHere we describe the spin-wave mode frequency in a thin-film magnet with perpendicular anisotropy along the film normal.\nWe consider this at the magnetic-dipole limit where the wavelength is relatively large and the exchange interaction contribution\nto the spin-wave dispersion is neglected. Furthermore, standing spin-wave modes along the thickness direction are also ruled out\nsince these modes only appear at much higher frequencies than the main mode, where we consistently observe additional peaks\nat both higher and lower frequencies from the main mode. The mode (angular) frequency ( w) for wavevector k=0 when we\napply a magnetic field Balong one of the film plane directions can be given by Eq. 3d in Ref.S11as:\n\u0012w\ng\u00132\n=B\u0012\nB+m0Ms\u00002Ku\nMs\u0013\n: (S5)\nHere, g,MsandKuare the gyromagnetic ratio, saturation magnetization and the perpendicular anisotropy energy density, respec-\ntively. Note, that the total field within m0Ms\u00002Ku\nMsis negative for perpendicularly-magnetized films which we consider in this\nsection. Within the magnetic-dipole limit, the demagnetization term m0Msis modified for spin-waves with finite k, depending\non the relative orientation between the Msandkdirections. Here we follow the expression given in Serga et al.S12. For pure\nbackward volume magnetostatic modes where kkMs(illustrated in Fig. S8), the mode frequency becomes:\n\u0012wBVMSW\ng\u00132\n=B\u0012\nB+m0Ms\u00121\u0000e\u0000kt\nkt\u0013\n\u00002Ku\nMs\u0013\n; (S6)\nwhere tis the thickness of the magnet. Note, that this expression is only valid for the case where Msis colinear to B, meaning that\njBj>jm0Ms\u00002Ku\nMsj. To the limit of k!0, the term (1\u0000e\u0000kt)=ktis reduced to unity, consistent to Eq. (S5). When kis nonzero,\nwe can observe that wBVMSW becomes smaller than that for k= 0, exhibiting a negative group velocity for this spin-wave mode.\nAs the opposite extreme where k?Ms(illustrated in Fig. S8), the resonance frequency becomes larger than that for k= 0 and is\ncalled magneto-static surface spin-wave mode. The mode frequency expression for this mode is given by:\n\u0012wMSSW\ng\u00132\n=B\u0012\nB+m0Ms\u00002Ku\nMs\u0013\n+m2\n0M2\ns\u0010\n1\u0000e\u00002kt\u0011\n: (S7)\nHere, m2\n0M2\ns\u0000\n1\u0000e\u00002kt\u0001\nis the spin-wave correction term which goes to zero for k!0 (hence consistent to Eq. (S5)) and\nbecomes positive for k>0, meaning that wMSSW becomes larger as soon as spin-waves gain momentum along this direction.\nWe use these two expressions in an effort to explain the origin of the multiple peaks in our experiments. Figure S8 plots the\ncalculated wBVMSW =2pandwMSSW =2pas a function of wavevector k. The range of wavevector is chosen such that the resulting\nresonance frequencies are within the same order of magnitude as the observed mode splittings in the experiment ( µ100MHz).9\nWavevector (μm-1)6 4 8 10 12 2 012.712.812.9Resonance Frequency (GHz)13.0\n12.6\n12.5\n12.4Kittel\nBVMSW\nMSSWB0B0100 MHz\nFIG. S8. Spin-wave dispersion. Spin-wave resonance frequency for BVMSW (green solid line) and MSSW (yellow solid line) as a function\nof wavevector. The dashed blue line is the resonance frequency of the k=0 main mode. The parameters used are B0=598mT, gCGT =2:18,\nm0Ms=194:3mT and Ku=3:84\u0002104J=m3and a thickness of 17nm. The grey area highlights a 100MHz margin relative to the main mode,\nindicating the order of magnitude of the mode splitting observed in the experiment. The arrows on the right hand side illustrate the relative\nwavevector orientations of the BVMSW and MSSW spin-wave modes with respect to the static magnetic field.\nThe corresponding wavelength to a 100MHz resonance offset to the main mode are about 2 :2\u0016m and 620nm for wBVMSW\nandwMSSW , respectively. These values are within a reasonable scale for our different lateral CGT flake dimensions under\ninvestigation. This suggests that spin-wave modes are likely the origin of the multiple resonance peaks observed.\nThe thinnest CGT flake shows, however, a deviation from this behaviour. We only observe modes at lower frequencies, which\nwould indicate to BVMSW modes. Calculating the respective shortest wavelength results in 225nm, which is significantly\nshorter than for the other devices. We assume that the placement and irregular shape are likely to cause this difference. First,\nthis flake is placed at the very edge of the inductor wire, where the B1field strength is declining (see Fig. S3), reducing the FMR\nactive area. Thickness steps can lead to a wavelength down-conversionS13, however, with the overall irregular shape of the flake\nit is difficult to define a length scale for a standing spin wave mode.\nVII. ATOMISTIC SPIN DYNAMICS SIMULATIONS OF FMR\nTo study the ferromagnetic resonance in CGT we perform atomistic spin dynamics simulationsS14,S15. The magnetic Hamil-\ntonian employed in the simulations is given by:\nH=\u00001\n2å\ni;jSiJi jSj\u0000å\niDi(Si·e)2\u0000å\nimiSi·(B0+B1) (S8)\nwhere i,jrepresent the atoms index, Ji jrepresents the exchange interaction tensor, Dithe uniaxial anisotropy, which for\nCGT is orientated out of plane ( e= (0;0;1)) and B0the external static magnetic field applied in-plane during the ferromagnetic\nresonance simulations and B1=B1sin(2pnt)the oscillating field applied perpendicular with respect to B0. The CGT system\nhas been parameterized from first principle methodsS9, up to the third nearest neighbor intralayer and interlayer exchange. The\nexchange values have also been re-scaled by Gong et al.S9with a 0.72 factor to obtain the experimental TCand multiplied by\nS2to match the magnetic Hamiltonian. The magnetic moment or Cr is considered 3.26 mBS16and the uniaxial anisotropy has\na value of 0 :05 meV as extracted from first principle methodsS9. The parameters used in the simulations are given in Table II.\nFMR calculations have previously been employed for atomistic models, and can reproduced well the variation of linewidth with\ntemperature, for example, in recording media systemsS17. Hence, in the current simulations we use the same setup of frequency\nswept FMRS17and we obtain the spectra by performing a Fourier transform of the magnetisation component parallel to the\noscillating field. Since these calculations are done close to 0K, no averaging is require to reduce the thermal noise. To excite the\nFMR mode, we apply a DC field in-plane of 0.9 T on x-direction and an AC field perpendicular to the DC field, on y-direction.\nThe Fourier transform has been performed for the y-component of magnetisation for 5ns after an initial 1ns equilibration time.\nA thermal bath coupling has been chosen in agreement with the upper limit of the Gilbert damping observed in experiments.\nThe system size we performed FMR on is a 4-layer CGT system, with lateral size of 6 :91nm\u000211:97nm, periodic boundary\nconditions in xy and total of 1600 atoms. The small system size has been used to reduce the computational cost associated10\nQuantity Symbol quantity units\nTimestep ts 0.1 fs\nThermal bath coupling a 0.02\nGyromagnetic ratio ge 1.760859\u00021011rad s\u00001T\u00001\nMagnetic moment mB 3.26S16mB\nUniaxial anisotropy Di 0.05S9meV/link\nSimulation temperature T 0.001 K\nStatic magnetic field B0 0.9, 0.7 T\nOscillating magnetic field amplitude B0 0.001 T\nFMR frequency n varied GHz\nIntralayer exchange, NN J1 2.71S9meV/link\nIntralayer exchange, 2NN J2 - 0.058S9meV/link\nIntralayer exchange, 3NN J3 0.115S9meV/link\nInterlayer exchange, NN Jz\n1-0.036S9meV/link\nInterlayer exchange, 2NN Jz\n20.086S9meV/link\nInterlayer exchange, 3NN Jz\n30.27S9meV/link\nTABLE II. Simulation parameters for FMR on CGT system\n.\nwith FMR simulations. Experiments have showed modified g-factors due to photon-magnon coupling hence hereby we propose\na simple model where the properties of the individual layers have been modified to include different gyromagnetic ratio, as\nillustrated in Fig. S9 a.\nWe can define the resonance frequencies for each magnetic layer using the Kittel equation in the case of in-plane applied field\nwith perpendicular anisotropy B?u:\nw=gp\nB0(B0\u0000B?u) (S9)\nWe next investigate the FMR signal for a few cases assuming the CGT monolayers at low or strong interlayer exchange cou-\nplings J0\nz=0;0:1%;10%;100% Jz, where Jzcorresponds to the pristine interlayer exchange (Fig. S9 b-c). In the low interlayer\nexchange regime ( J0\nz=0;0:1%Jz), the CGT presents multiple peaks with each frequency corresponding to the layer dependent\ngyromagnetic ratio, g-n(g1) =16:81GHz, n(g2) =25:22GHz, n(g3) =33:62GHz. At J0\nz=0:1%J0\nz(Fig. S9 b) we can still\nobserve resonance peaks corresponding to each individual layer. However by increasing the exchange coupling to 10% J0\nzor\nhigher (Fig. S9 c) there is a single FMR peak indicating that the system behave coherently with all layers having the same FMR\nfrequency. The single FMR frequency corresponds to the average magnetic properties of the CGT layers. Small variations of\nthe resonance frequency as function of the inter-layer exchange coupling can be observed which these being correlated to the\ntransition of the system from the multi-peaks regime to a coherent excitation. By calculating the damping of the highest reso-\nnance peaks from a Lorenzian fit, we reobtain the damping corresponding to the input thermal bath coupling, 0 :02 with a relative\ntinny error\u00185%. Overall, the interlayer exchange coupling locks the dynamics of individual layers coherently together without\nallowing multiple frequencies at the FMR signalS18.\n[S1]A. E. Primenko, M. A. Osipov, and I. A. Rudnev, Technical Physics 62, 1346 (2017).\n[S2]L. H. Lee, T. P. 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Cao, W. Bao, C. Wang, Y . Wang, Z. Q. Qiu, R. J. Cava, S. G. Louie, J. Xia, and X. Zhang, Nature 546,\n265 (2017).\n[S10]S. Khan, C. W. Zollitsch, D. M. Arroo, H. Cheng, I. Verzhbitskiy, A. Sud, Y . P. Feng, G. Eda, and H. Kurebayashi, Physical Review B 100, 134437 (2019).\n[S11]M. Farle, Reports on Progress in Physics 61, 755 (1998).\n[S12]A. A. Serga, A. V . Chumak, and B. Hillebrands, Journal of Physics D: Applied Physics 43, 264002 (2010).\n[S13]J. Stigloher, T. Taniguchi, M. Madami, M. Decker, H. S. Körner, T. Moriyama, G. Gubbiotti, T. Ono, and C. H. Back, Applied Physics Express 11, 053002\n(2018).\n[S14]D. A. Wahab, M. Augustin, S. M. Valero, W. Kuang, S. Jenkins, E. Coronado, I. V . Grigorieva, I. J. Vera-Marun, E. Navarro-Moratalla, R. F. Evans, et al. ,\nAdvanced Materials 33, 2004138 (2021).\n[S15]A. Kartsev, M. Augustin, R. F. Evans, K. S. Novoselov, and E. J. G. Santos, npj Computational Materials 6, 1 (2020).\n[S16]I. A. Verzhbitskiy, H. Kurebayashi, H. Cheng, J. Zhou, S. Khan, Y . P. Feng, and G. Eda, Nature Electronics 3, 460 (2020).11\nFIG. S9. Atomistic simulations. a, Schematic of the crystal structure of CGT with atoms defined by different colours. b,FMR spectra of 4\nlayer CGT where the layers are low interayer exchange coupled (0 ;0:1%J0z, where J0zis the pristine CGT interlayer exchange). c,Similar as b,\nbut with the layers at a strong exchange coupling (10% ;100% J0z). The solid lines in b-crepresent a Lorenzian fit to the numerical data.\n[S17]M. Strungaru, S. Ruta, R. F. Evans, and R. W. Chantrell, Physical Review Applied 14, 014077 (2020).\n[S18]Data inputs/plots utilised for Supplementary Figure S7 (atomistic simulations) can be found at the following GitHub repository."    },    {        "title": "2309.11152v1.Evaluating_Gilbert_Damping_in_Magnetic_Insulators_from_First_Principles.pdf",        "content": "Evaluating Gilbert Damping in Magnetic Insulators from First Principles\nLiangliang Hong,1, 2Changsong Xu,1, 2and Hongjun Xiang1, 2,∗\n1Key Laboratory of Computational Physical Sciences (Ministry of Education), Institute of Computational Physical Sciences,\nState Key Laboratory of Surface Physics, and Department of Physics, Fudan University, Shanghai 200433, China\n2Shanghai Qi Zhi Institute, Shanghai 200030, China\n(Dated: September 21, 2023)\nMagnetic damping has a significant impact on the performance of various magnetic and spin-\ntronic devices, making it a long-standing focus of research. The strength of magnetic damping is\nusually quantified by the Gilbert damping constant in the Landau-Lifshitz-Gilbert equation. Here\nwe propose a first-principles based approach to evaluate the Gilbert damping constant contributed\nby spin-lattice coupling in magnetic insulators. The approach involves effective Hamiltonian mod-\nels and spin-lattice dynamics simulations. As a case study, we applied our method to Y 3Fe5O12,\nMnFe 2O4and Cr 2O3. Their damping constants were calculated to be 0 .8×10−4, 0.2×10−4,\n2.2×10−4, respectively at a low temperature. The results for Y 3Fe5O12and Cr 2O3are in good\nagreement with experimental measurements, while the discrepancy in MnFe 2O4can be attributed\nto the inhomogeneity and small band gap in real samples. The stronger damping observed in Cr 2O3,\ncompared to Y 3Fe5O12, essentially results from its stronger spin-lattice coupling. In addition, we\nconfirmed a proportional relationship between damping constants and the temperature difference\nof subsystems, which had been reported in previous studies. These successful applications suggest\nthat our approach serves as a promising candidate for estimating the Gilbert damping constant in\nmagnetic insulators.\nI. INTRODUCTION\nRecent decades have witnessed rapid developments in\nmagnetics and spintronics [1–3]. A long-time pursuit in\nspintronics is to actively control and manipulate the spin\ndegrees of freedom in solid-state systems. Related fun-\ndamental studies involve spin transport, spin dynamics\nand spin relaxation [4]. Within these domains, magnetic\ndamping often plays a crucial role. Generally, stronger\ndamping enables a faster writing rate for magnetic mem-\nories, while lower damping leads to a longer propagation\ndistance of spin waves. Therefore, it is always essential\nto accurately evaluate the magnetic damping in different\nmaterials. For instance, yttrium iron garnet (YIG) is a\nhighly promising spintronic material due to its ultra-low\nmagnetic damping [5–7]. However, the intrinsic mecha-\nnism behind its unique property has yet to be fully eluci-\ndated, which partly motivates us to carry out this work.\nAt present, magnetic damping is typically represented\nby a phenomenological term in the well-known Landau-\nLifshitz-Gilbert (LLG) equation, which has been widely\nemployed to simulate magnetization dynamics [8, 9]. A\nbasic form of this equation can be written as,\n∂ ⃗ m\n∂t=−γ ⃗ m×⃗B+α\nm⃗ m×∂ ⃗ m\n∂t(1)\nwhere ⃗Brepresents the total magnetic field acting on the\nlocal dipole ⃗ m,mdenotes the norm of ⃗ m,γis the gyro-\nmagnetic ratio, and αis the Gilbert damping constant.\nThe second term on the right side, as we mentioned, leads\n∗hxiang@fudan.edu.cndirectly to the relaxation process, in which the rate of en-\nergy dissipation is determined by the damping constant.\nGiven the importance of αin magnetization dynamics,\nits origin has been extensively studied in the literature\n[10–13]. To the best of our knowledge, both intrinsic and\nextrinsic mechanisms contribute to the damping. Specif-\nically, the intrinsic factors include spin-lattice and spin-\nelectron couplings, while the extrinsic contributions pri-\nmarily involve lattice imperfections and eddy currents\n[14, 15].\nTwo types of first-principles based methods have been\ndeveloped to calculate the damping constants in the past.\nOne approach involves the breathing Fermi surface model\n[16, 17] and the torque correlation model [18, 19], while\nthe other is based on the scattering theory from linear\nresponse [20–22]. These methods have demonstrated re-\nmarkable success in studying the magnetic damping in\ntransition metals such as Fe, Co, and Ni. Despite be-\ning free from complicated experiments, which are mostly\nbased on ferromagnetic resonance, these theoretical ap-\nproaches still exhibit several limitations. Firstly, when\ndealing with complex systems, we often have to spend a\nsignificant amount of computing resources on the first-\nprinciples calculations. In addition, these methods are\nmore suitable for calculating the electronic contribution\nto Gilbert damping in metallic magnets, thus rarely tak-\ning the effect of spin-lattice coupling into consideration\n[14, 23].\nRecently, spin-lattice dynamics (SLD) simulations [24]\nhave been adopted as an alternative method to evaluate\nthe Gilbert damping parameters. In Ref. [23], the au-\nthors constructed an empirically parameterized Hamil-\ntonian model for a cobalt cluster. They coupled a pre-\nheated lattice with a fully ordered spin state, then per-\nformed SLD simulation. During the relaxation process,arXiv:2309.11152v1  [cond-mat.mtrl-sci]  20 Sep 20232\nthe energy of lattice and spin subsystems were recorded\nand fitted to the following logistic functions,\nUlat=Ulat\n0−∆U0\n1 + exp[ −η∆U0t−Θ](2)\nUmag=Umag\n0+∆U0\n1 + exp[ −η∆U0t−Θ](3)\nfrom which they extracted the relaxation rate Γ = η∆U0\nand calculated the damping constant α=ηµS/γ. Here,\nµSdenotes the magnitude of magnetic moments. In Ref.\n[25], the authors also built an empirical potential model\nfor a periodic bcc Fe system. They firstly applied an ex-\nternal magnetic field in the z-direction and thermalized\nthe system to a finite temperature. Then, the magnetiza-\ntion orientation of each atom was rotated artificially by\na same angle. Afterwards, the system would relax back\nto equilibrium, during which the averaged z component\nof atomic magnetization was recorded and fitted to the\nfollowing function,\nmz(t) = tanh\u0014α\n1 +α2γBext(t+t0)\u0015\n(4)\nwhere αwas exactly the Gilbert damping parameter to\nbe estimated. Since these works selected transition met-\nals as the research object, their results were both orders\nof magnitude smaller than the experimental values. In\naddition, the use of empirically parameterized models re-\nduced the accuracy of their simulated results.\nIn this work, we combine SLD simulations with first-\nprinciples based effective Hamiltonian models to evalu-\nate the damping constants in magnetic insulators, where\nthe dominant contribution results from spin-lattice cou-\nplings. Compared to the previous studies, our work has\nmade improvements mainly in two aspects. Firstly, the\nutilization of first-principles based Hamiltonian models\nin simulations enhances the reliability of our conclusions.\nBesides, the better choice of research objects allows for\ndemonstrating the superiority of SLD simulations. In\nparticular, the microscopic origin of low damping in YIG\nwill be investigated. The paper is organized as follows.\nIn Sec. II, we introduce our effective Hamiltonian model,\nparameterization methods, and a scheme for evaluating\nGilbert damping parameters. Then, both the validation\nand application of our method are presented in Sec. III.\nFinally, we summarize this work and give a brief outlook\nin Sec. IV.\nII. MODEL AND METHODS\nThis section is split into three parts. Firstly (in Sec.\nII A), we introduce a generic form of our effective Hamil-\ntonian model. Then, methods involving the calculation\nof model parameters are presented in Sec. II B. At the\nlast part (Sec. II C), we propose a novel scheme to de-\ntermine the Gilbert damping constant through dynamics\nsimulations.A. The Hamiltonian Model\nSince our purpose is to evaluate the contribution of\nspin-lattice coupling to magnetic damping, the effective\nHamiltonian model must incorporate both spin and lat-\ntice degrees of freedom. A concise and generic formula\nthat meets our basic requirements consists of the three\nterms as follows:\nH=HL({ui,α}) +HS({⃗ sj}) +HSLC({ui,α,⃗ sj}) (5)\nwhere αabbreviates three orthogonal axes, ui,αrepre-\nsents the displacement of atom i, and ⃗ sjis a unit vector\nthat represents the direction of spin j.\nThe first term HLin Hamiltonian model describes the\ndynamical behavior of individual phonons. Technically,\nwe take the atomic displacements as independent vari-\nables and expand the Hamiltonian to the second order\nwith Taylor series. Then, we have the form as,\nHL=1\n2X\nijX\nαβKij,αβui,αuj,β+1\n2X\ni,αMi˙ui,α˙ui,α(6)\nwhere Kij,αβ denotes the force constant tensor and Mi\nrepresents the mass of atom i.\nSimilarly, the second term HSdescribes the dynami-\ncal behavior of individual magnons. For simplicity but\nno loss of accuracy, we only considered the Heisenberg\nexchange interactions between neighbor magnetic atoms\nin this work, though more complex interactions could be\ntaken into account in principle. Therefore, this term can\nbe expressed as,\nHS=X\n⟨i,j⟩Jij⃗Si·⃗Sj (7)\nwhere Jijdenotes the isotropic magnetic interaction co-\nefficient.\nThe third term HSLCrepresents the coupling of spin\nand lattice subsystems, and is expected to describe the\nscattering process between phonons and magnons. As\nan approximation of the lowest order, this term can be\nwritten as,\nHSLC=X\n⟨i,j⟩X\nkα\u0012∂Jij\n∂uk,αuk,α\u0013\n⃗Si·⃗Sj (8)\nAccording to the theory of quantum mechanics, this\ncoupling term provides a fundamental description of the\nsingle-phonon scattering process, which is believed to be\ndominant among all scatterings in the low-temperature\nregion. This type of relaxation mechanism in ferromag-\nnetic resonance was systematically studied by Kasuya\nand LeCraw for the first time [26]. It’s worth noting that\na higher order of Taylor expansion could have been con-\nducted to improve the accuracy of Hamiltonian models\ndirectly. For instance, the scattering between individual\nphonons can be adequately described by the anharmonic\nterms. However, as one always has to make a trade-off3\nbetween the precision and complexity of models, in this\nwork we choose to neglect the high order terms since the\nanharmonic effects in current investigated systems are\nnot important.\nIn this study, we adopted the symmetry-adapted clus-\nter expansion method implemented in the Property Anal-\nysis and Simulation Package for Materials (PASP) [27]\nto build the Hamiltonian model presented above. This\npackage can identify the nonequivalent interactions and\nequivalent atom clusters in a crystal system by analyz-\ning its structural properties based on the group theory.\nA significant benefit of working with PASP is we are en-\nabled to describe the target system with the least number\nof parameters. In the next section, we will discuss how\nto calculate the model parameters for different materials.\nB. Calculation of Model Parameters\nFirstly, the Heisenberg exchange coefficients Jijand\nspin-lattice coupling constants ∂Jij/∂uk,αcan be calcu-\nlated with the four-state method [28, 29]. The basic flow\nis to construct four artificially designated spin states of\nthe target system, calculate the corresponding energies\nand forces based on the density functional theory (DFT),\nthen determine the parameters by proper combination of\nthose results. At the last step, the following formulas will\nbe used,\nJij=E↑↑+E↓↓−E↑↓−E↓↑\n4S2(9)\n∂Jij\n∂uk,α=F↑↑\nk,α+F↓↓\nk,α−F↑↓\nk,α−F↓↑\nk,α\n4S2(10)\nwhere Sis the spin quantum number of magnetic atoms,\nEis the total energy of system and Fk,αrefers to one\ncomponent of the force on atom k. The superscripts ( ↑↑,\n↓↓,↑↓,↓↑) specify the constrained spin states of system\nin the calculation. More technical information about the\nfour-state method can be found in the references [28, 29].\nCompared to other approaches, the four-state method of-\nfers an obvious advantage in that no additional DFT cal-\nculations are needed to determine the coupling constants\n∂Jij/∂uk,αonce the exchange coefficients Jijhave been\nobtained. This is because the energy and forces are typ-\nically provided simultaneously by one DFT calculation.\nSince atomic masses Mican be directly obtained from\nthe periodic table, more efforts are needed to deal with\nthe force constant tensor Kij,αβ. Currently, there are two\ncommonly adopted ways to calculate the force constant\ntensor: density functional perturbation theory (DFPT)\nand finite displacement method. Both of these methods\nare applicable to our task.\nHowever, we cannot directly take the force constant\ntensor obtained from first-principles calculations as the\nmodel parameter. This is because in dynamics simula-\ntions we usually expand crystal cells to reduce the un-\ndesired influence of thermal fluctuations, which leads toa conflict between the periodic boundary condition and\nthe locality (also known as nearsightedness [30, 31]) of\nmodels. To be more specific, when calculating the con-\ntribution of one atom or spin to the total energy, we tend\nto set a well designed cutoff radius and ignore the inter-\nactions beyond it. This step is essential when dealing\nwith a large-scale system, otherwise we will suffer from\nthe model complexity and the computational cost. Nev-\nertheless, if we set the elements of Kij,αβ that represent\nout-of-range interactions to be zero and leave the others\nunchanged, we may violate the so-called acoustic sum-\nmation rules:\nX\niKij,αβ = 0 for all j, α, β. (11)\nIt should be pointed out that a straightforward en-\nforcement of the acoustic summation rules, achieved by\nsubtracting errors uniformly from force constants, will\nbreak the inherent crystal symmetry inevitably, which is\nthe technique employed in phonopy [32]. To address the\nabove issues, we adopted a more appropriate method in\nthis work. Before a detailed introduction, it’s necessary\nto recall that not every element of the force constant ten-\nsor serves as an independent variable due to the crystal\nsymmetries. Taking the cubic cell of Y 3Fe5O12(contain-\ning 160 atoms) for example, there are 230400 elements in\nthe tensor. After symmetry analyses, we find that only\n597 independent variables {pn}are needed to adequately\ndetermine all the tensor elements {Kij,αβ({pn})}, where\nthe effect of locality is already considered. Afterwards,\nour method is to set a correction factor xnfor each vari-\nablepnand minimize the deviation of parameters under\nthe constraints of Eq. (11). A mathematical reformula-\ntion of this method can be written as,\nmin\n{xn}X\nn(xn−1)2,with\nX\niKij,αβ({xnpn}) = 0 for all j, α, β.(12)\nIn the case of Y 3Fe5O12, there are only 18 linearly inde-\npendent constraints, which allow the extremum problem\nto be solved rigorously. The modified force constant ten-\nsor restores positive definiteness and translational sym-\nmetry while maintaining the crystal symmetries. There-\nfore, the modified tensor meets the requirements for dy-\nnamics simulations. In Sec. III B, the effectiveness of this\napproximate method will be demonstrated through a spe-\ncific example.\nAll the first-principles calculations mentioned in this\nsection are carried out using the Vienna ab initial simu-\nlation package (VASP) [33–35]. The force constants and\nphonon spectra are obtained by phonopy [32]. The opti-\nmizations formulated in (12) are accomplished with the\nfunction optimize.minimize implemented in SciPy [36].4\nC. Evaluation of Damping Constants\nAfter the construction and parameterization of Hamil-\ntonian models, we are finally able to perform spin-lattice\ndynamics simulations. Before the evaluation of Gilbert\ndamping constants, we briefly introduce the framework\nof SLD to cover some relevant concepts. In practice, the\nmotion of magnetic moments follows the stochastic Lan-\ndau–Lifshitz–Gilbert (SLLG) equation [14],\nd⃗ mi\ndt=−γL⃗ mi×\u0010\n⃗Bi+⃗Bfl\ni\u0011\n−γLα⃗ mi\n|⃗ mi|×h\n⃗ mi×\u0010\n⃗Bi+⃗Bfl\ni\u0011i\n(13)\nwhere γLis the renormalized gyromagnetic ratio, ⃗Bi=\n−∂H/∂ ⃗ m iis the effective local magnetic field and ⃗Bfl\ni\nrefers to a stochastic field introduced by Langevin ther-\nmostat. At the same time, the motion of atoms obeys\nthe Newton’s equation,\nd˙ui,α\ndt=1\nMi\u0010\n⃗Fi,α+⃗Ffl\ni,α\u0011\n−ν˙ui,α (14)\nwhere νis the damping constant and ⃗Ffl\ni,αrefers to a\nstochastic force caused by thermal fluctuations. In this\nwork, ⃗Bfl\niand⃗Ffl\ni,αare modeled as normally distributed\nnoises with temperature-dependent variances,\nBfl\ni,β∼N\u0010\n0,p\n2αkBTS/γ|⃗ mi|δt\u0011\n(15)\nFfl\ni,β∼N\u0010\n0,p\n2νMikBTL/δt\u0011\n(16)\nwhere TSandTLrefer to the equilibrium temperature of\nspin and lattice subsystems respectively. During simula-\ntions, we can also measure the transient temperature of\neach subsystem with the following formulas [37],\nTS=P\ni|⃗ mi×⃗Bi|2\n2kBP\ni⃗ mi·⃗Bi, TL=1\n2kBNX\ni,αMi˙u2\ni,α (17)\nIn this work, the LLG equation is numerically solved\nwith the semi-implicit SIB method proposed by Mentink\net al. [38]. The Newton’s motion equation is integrated\nusing the Grønbech-Jensen-Farago Verlet-type method\n[39]. To ensure the stability of those algorithms, a step\nlength of 0 .5 or 0 .2 fs is adopted [40], where the shorter\none is used in energy-conserving simulations.\nBased on the combination of atomistic spin dynamics\n(ASD) and SLD simulations, a new scheme is proposed\nto evaluate the damping constant in magnetic materials.\nHere is the basic flow of this method and more details of\na specific application are presented in Sec. III B.\n1. Freeze the spin degree of freedom and thermalize\nthe lattice from 0 to TLin the simulation.\n2. Fix atomic positions and raise the temperature of\nspin to TS> TL. Compared to TL> TS, this type\nof nonequilibrium state is more common in actual\nscenarios.3. Perform an energy-conserving SLD simulation to\nrelax the system. Normally, the spin temperature\nwill decrease to the same as lattice and stay there\ntill the end.\n4. Conduct a series of ASD simulations with different\nGilbert damping constants. The initial states are\nthe same as in step 3 and the equilibrium temper-\natures are set to be TL.\n5. Compare the cooling rates ∂TS/∂tof spin system\nbetween SLD and ASD simulations to evaluate the\nequivalent Gilbert damping constant contributed\nby spin-lattice coupling.\nThe key point behind step 5 is that the cooling rates\nobserved in ASD simulations are related to the assigned\ndamping constants, while in SLD simulation the cooling\nrate is determined by the strength of spin-lattice cou-\npling. Note that the former relation can be viewed as a\nnatural deduction of the LLG equation,\n∂TS\n∂t=1\nCV∂Emag\n∂t∝ −1\nCV\u0012∂ ⃗ m\n∂t·⃗B\u0013\n∝ −1\nCV\u0014\u0012α\nm⃗ m×∂ ⃗ m\n∂t\u0013\n·⃗B\u0015\n∝α (18)\nwhere we have used Eq. (1) and simplified the formula of\nmagnetic energy as Emag∝ −⃗ m·⃗B.\nIII. RESULTS\nThis section is divided into four parts. In Sec. III A,\nseveral test results are presented to validate the accu-\nracy of SLD simulations, which are implemented in the\nPASP package. Subsequently, detailed calculations on\nthree magnetic materials, namely Y 3Fe5O12, MnFe 2O4\nand Cr 2O3, are discussed in the rest parts.\nA. Validations\nIn order to guarantee the reliability of our conclusions\nobtained from dynamics simulations, a series of pretests\nwere carried out. We select some representative results\nand present them in Fig. 1, where Cr 2O3is taken as the\nobject to be studied.\nFirstly, we set the ground state of Cr 2O3as the ini-\ntial state and performed a NVT simulation with Tset=\n150K. As shown in Fig. 1(a), the temperature of spin\nand lattice subsystems increased to 150 Kin less than 5\nps and stayed there till the end. Since we can approxi-\nmate Ek= 0.5ELandEp= 0.5EL+ES, Fig. 1(b) also\nindicates that the contribution of phonons and magnons\nto the excited state energy is around 87.5% and 12.5%\nrespectively. This result could be verified from another\nperspective. Note that there are totally 10 atoms in the5\nFIG. 1. NVT and NVE relaxations of a spin-lattice coupled system (Cr 2O3) within the framework of spin-lattice dynamics.\nThe top row plots the time evolution of temperatures and the bottom row shows the variation of potential, kinetic and total\nenergies. (a) & (b): NVT thermalization from TL=TS= 0KtoTL=TS= 150 K. (c) & (d): NVE relaxation with TL= 30K,\nTS= 175 Kinitially. (e) & (f): NVE relaxation with TL= 180 K,TS= 30Kinitially.\nunit cell of Cr 2O3, which contribute 30 kBto the heat ca-\npacity. Meanwhile, the 4 magnetic atoms will contribute\nanother 4 kBin the low temperature region. Therefore,\nwe can estimate that the contribution of magnons to the\ntotal heat capacity is close to 11.8%, which is consistent\nwith the result from dynamics simulations.\nIn Figs. 1(c) & 1(d), the initial state was set to be a\nnonequilibrium state with TL= 30KandTS= 175 K. As\nwe expected, the total energy was well conserved when\nthe system evolved to equilibrium. In addition, the final\ntemperature fell within the range of 48 K∼55K, which\nagrees with our previous analysis of the heat capacities.\nLastly, we simulated the relaxation process using an-\nother nonequilibrium excited state with TL= 180 Kand\nTS= 30Kas the initial state. As shown in Figs. 1(e) &\n1(f), the temperature of spin system increased gradually\nto equilibrium with the total energy conserved through-\nout the simulation. Also, the final temperature is around\n160K, which matches well with our analysis. It should be\npointed out that there exist two notable differences be-\ntween this case and the previous. Firstly, the subsystems\nultimately evolved to a same temperature in a finite time,which alleviated our concerns about the accuracy of SLD\nsimulations. Besides, the relaxation time ( τ2) was much\nlonger than that ( τ1) in Fig. 1(c). For this phenomenon,\na qualitative explanation is presented below.\nBased on the theory of second quantization, the Hamil-\ntonian model presented in Sec. II A can be expressed in\nthe following form [41, 42],\nHL=X\nqpℏωqp(b†\nqpbqp+ 1/2) (19)\nHS=X\nλϵλa†\nλaλ+Const. (20)\nHSLC=X\nλ,qpMλ,qpa†\nλ−qaλ\u0000\nb†\nqp−b−qp\u0001\n(21)\nwhere bqpdenotes the annihilation operator of phonons\nwith wave vector qin branch p, and aλrepresents the an-\nnihilation operator of magnons with wave vector λ. All\nthe parameters, namely ωqp,ϵλandMλ,qp, can be deter-\nmined from the effective Hamiltonian model in principle.\nAccording to the Fermi’s golden rule, we have\nW{nλ−q, nλ, Nqp→nλ−q+ 1, nλ−1, Nqp+ 1}=2π\nℏ|Mλ,qp|2(nλ−q+ 1)( nλ)(Nqp+ 1)δ(ϵλ−q−ϵλ+ℏωqp) (22)\nW{nλ−q, nλ, N−qp→nλ−q+ 1, nλ−1, N−qp−1}=2π\nℏ|Mλ,qp|2(nλ−q+ 1)( nλ)(N−qp)δ(ϵλ−q−ϵλ−ℏω−qp) (23)6\nFIG. 2. (a) The primitive cell of Y 3Fe5O12. The golden balls\nrepresent iron atoms, the cyan ball stand for yttrium atoms,\nand the red balls refer to oxygen atoms. (b) The magnetic\nground state of YIG. The arrows of different colors represent\nthe spin directions of Fe atoms. (c) The density of states ob-\ntained by DFT calculations. (d) The temperature dependence\nof average magnetization measured in MC and ASD simula-\ntions. For YIG, the phase transition point from ferrimagnetic\nto paramagnetic lies in 530 K approximately.\nwhere Wrepresents the probability of one-phonon emis-\nsion or absorption, nλdenotes the occupation number of\nmagnons and Nqpstands for phonons. Both nλandNqp\ncan be evaluated approximately using the Bose–Einstein\ndistribution. According to the above formulas, the scat-\ntering rate Wgrows linearly with Nand quadratically\nwith n. Compared to Fig. 1(c), there are more phonons\nbut fewer magnons in the case of Fig. 1(e), thus leading\nto a lower transition probability and a longer relaxation\ntime. More technical details about the second quantiza-\ntion of interactions between phonons and magnons can\nbe found in Ref. [41, 42].\nB. Damping constants in Y 3Fe5O12\nIn the field of spintronics, Y 3Fe5O12(yttrium iron gar-\nnet, YIG) has gained much attention due to its ultra-low\nmagnetic damping [5–7]. The unique property of this\nmaterial motivated us to investigate the intrinsic mecha-\nnism behind it. The crystal structure of YIG is presented\nin Fig. 3(a). There are totally 80 atoms in the primitive\ncell, of which 12 Fe ions are located in the center of oxy-\ngen tetrahedrons while the other 8 Fe ions are sited in\noxygen octahedrons. The magnetic ground state of YIG\nis illustrated in Fig. 3(b). The Fe ions situated in differ-\nent chemical environments contribute spins in opposite\ndirections, which makes YIG a typical ferrimagnetic ma-\nterial.TABLE I. The Heisenberg exchange coefficients J of YIG,\nwhere an effective spin S= 1 is adopted. For the FeO−FeO\npairs, the Greek letters ( α&β) refer to different chemical\nenvironments. All the results are calculated with the four-\nstate method.\nSpin Pair. Distance (Angst) J (meV)\n1NN FeT−FeO3.445 47.414\n1NN FeT−FeT3.774 2.399\n1NN FeO−FeO(α) 5.337 0.538\n1NN FeO−FeO(β) 5.337 5.055\n2NN FeT−FeO5.555 0.285\n2NN FeT−FeT5.765 3.437\nIn order to evaluate the Gilbert damping constants in\nYIG, our first step is to prepare an effective Hamilto-\nnian model. Considering the balance between precision\nand efficiency, the cutoff radius of interactions was set\nto be 11.0 Bohr for atomic pairs and 6.7 Bohr for 3-\nbody clusters. After symmetry analyses, we identified\n612 nonequivalent interactions in total, which included\n6 Heisenberg exchange terms and 9 spin-lattice coupling\nterms.\nTo determine the interaction parameters, we carried\nout a series of first-principles calculations, where a cu-\nbic cell was adopted to reduce the interference between\nadjacent cells caused by periodic boundary conditions.\nFollowing the settings in Ref. [43], we utilized the pro-\njector augmented-wave (PAW) method [44] and revised\nPerdew-Burke-Ernzerhof exchange-correlation functional\nfor solids (PBEsol) [45] in our calculations. Besides, the\nDFT+U method in its simplified form [46] was employed\nwhere the effective Hubbard U parameter was set to be\n4 eV for the 3 delectrons of Fe ions. In addition, a cutoff\nenergy of 520 eV for plane wave basis and a Γ-centered\n2×2×2 mesh of k-points were used in the DFT calcu-\nlations.\nIn Figure 2(c), we present the density of states (DOS)\nfor YIG. With a band gap of 1.863 eV, there is hardly\nany electric current occurring in the low temperature re-\ngion. Moreover, the Heisenberg exchange coefficients of\nYIG is listed in Table I. To verify the accuracy of these\nparameters, we conducted both Monte Carlo (MC) and\nASD simulations. The temperature dependence of aver-\nage magnetization is shown in Fig. 2(d), which reveals\nthe critical temperature of YIG to be 530 K. This result\nis slightly lower than the measured Curie temperature,\nTC= 560 K[5], but falls within our tolerance. The cal-\nculated results of coupling constants are provided in the\nsupplementary material.\nNext, we come to deal with the force constant tensor.\nIn order to demonstrate the impact of locality and val-\nidate the effectiveness of our optimization method, we\npresent some results pertaining to the tensor of YIG in\nTable II. Here we use “VASP” to tag the original tensor7\nTABLE II. The force constant tensor of YIG. The columns\nlabeled by A represent the sorted absolute values ofP\niKij,αβ\nand the columns labeled by B list the sorted eigenvalues of\nKij,αβ. For the cubic cell of YIG, we obtained the original\ntensor with the VASP package. Then, we eliminated the el-\nements that represent interactions beyond the cutoff radius.\nThis step was done by PASP. Finally, the tensor was modified\nto meet the requirement of translational symmetry through\nthe optimization formulated in (12).\nVASP PASP Modified\nNo. A B A B A B\n1 0.000 0.000 1.587 -0.102 0.000 0.000\n2 0.000 0.000 1.587 -0.102 0.000 0.000\n3 0.000 0.000 1.587 -0.102 0.000 0.000\n4 0.000 1.065 1.587 0.643 0.000 0.444\n5 0.000 1.065 1.587 0.643 0.000 0.444\n6 0.000 1.065 1.587 0.643 0.000 0.444\nobtained from DFT calculations, “PASP” to label the\nmodified tensor in which interactions beyond the cutoff\nradius are eliminated, and “Modified” to label the tensor\nafter optimization of independent variables. As shown in\nTable II, the “PASP” tensor violated the acoustic sum\nrule and was not positive semi-definite, whereas these is-\nsues were resolved for the “Modified” tensor. Although\nan obvious difference existed between the “PASP” and\n“Modified” tensor in terms of their eigenvalues, we still\nassumed the target system could be reasonably described\nby the “Modified” tensor and the validity of this assump-\ntion would be verified by the calculated results of damp-\ning constants. Additional details regarding the selection\nof tensor elements and the deviation of phonon spectra\nare provided in Fig. 3. According to figure 3(b) and 3(c),\nthe major deviation in phonon spectra resulted from the\nelimination of tensor elements, rather than the subse-\nquent modification.\nCompleting the preparation of Hamiltonian model, we\napplied the scheme proposed in Sec. II C to our first ob-\nject, Y 3Fe5O12. An instance is presented in Figure 4. We\nsetTL= 30K,TS= 180 Kfor the initial nonequilibrium\nstate and adopted an expanded supercell which contains\n12800 atoms in the simulation. Fig. 4(a) shows the time\nevolution of spin temperature in different types of simu-\nlations. By comparing the curves, we could roughly esti-\nmate that the equivalent damping constant in SLD simu-\nlation fell within the range of 10−3∼10−4. To make the\nestimation more precise, we calculated the initial cool-\ning rates ∂TS/∂t|t=0through polynomial (or exponen-\ntial) fittings and plotted them in Fig. 4(b). Afterwards,\na linear regression was performed to determine the quan-\ntitative relation between lg( −∂TS/∂t|t=0) and lg( α). As\nwe expected, the cooling rates in ASD simulations were\nproportional to the assigned damping constants. Then,\nwe combined the results of SLD and ASD simulations toevaluate the equivalent damping constant. This step was\naccomplished by identifying the intersection of red and\nblue lines in Figure 4(b). Finally, the damping constant\nwas determined to be αf= (2.87±0.13)×10−4in this\ncase. To verify our method and result, we present a com-\nparison between SLD and ASD (where we set α=αf)\nsimulations in Fig. 4(c). The curves agree well with each\nother in the initial stage but deviate in the second half.\nThis phenomenon is within our expectation, because in\nthe SLD simulation the lattice heats up as the spin cools\ndown, thereby slowing the energy transfer between two\nsubsystems.\nIn addition to the above case, we have measured the\nequivalent damping constants under different conditions\nto investigate the temperature dependence of magnetic\ndamping. The final results are summarized in Figure 5.\nDetails about the estimation of uncertainties are given in\nthe supplementary material. For Y 3Fe5O12, the damping\nconstants at different temperatures stay on the order of\n10−4, which is in good agreement with the experimental\nresults (3 .2×10−4[47], 2 .2×10−4[48], 1 .2–1.7×10−4\n[49]). For example, the damping constant in bulk YIG\nwas reported as 0 .4×10−4in Ref. [50]. Meanwhile, our\ncalculations yielded α= (2.8±0.3)×10−5at ∆T= 15\nK and α= (7.0±0.7)×10−5at ∆T= 30 K, where both\nTL= 0 K. Therefore, the experimental value corresponds\nroughly to the temperature region of ∆ T= 15∼30 K in\nour study. We believe such extent of thermal excitation\nis quite common in all kinds of spintronics experiments.\nMoreover, Fig. 5 indicates that αis approximately pro-\nportional to the temperature difference between subsys-\ntems. This outcome is also consistent with some com-\nputational works in the past [23, 25]. By comparing the\nsubfigures in Figure 5, we found that αhas little depen-\ndence on the lattice temperature, although here TLcould\nbe viewed to some extent as the ambient temperature of\nthe spin system.\nAs a supplement to Sec. III A, we further validate our\nsimulations by analyzing the measured cooling rates in\nFig. 5(a). By subtracting Eq. (23) from Eq. (22), the\ntransfer rate of energy between magnon and phonon sys-\ntems can be expressed as,\n˙Q=X\nqpℏωqp⟨˙Nqp⟩=X\nλ,qpTλ,qp (24)\nwhere Tλ,qpdenotes different transfer channels,\nTλ,qp∝(nλ−nλ−q)Nqp+nλ−qnλ+ 1 (25)\nAccording to the Bose–Einstein distribution, the number\nof magnons and phonons can be expressed as,\nnλ=1\neϵλ/kBTS−1, Nqp=1\neℏωqp/kBTL−1(26)\nWhen TSis high enough and TLis close to zero, we can\napproximate nλ=kBTS/ϵλ∝TSandNqpclose to zero.\nUnder these conditions, we have ˙Q∝T2\nS. This relation8\nFIG. 3. (a) The selection of force constant tensor elements for the cubic cell of YIG. An 160 ×160 zero-one matrix is used\nto show the result of selection, in which ’1’ denotes the interactions within cutoff radius and ’0’ represents the elements that\nare artificially eliminated. (b) The phonon spectrum calculated from the force constant tensor before and after the elimination\nof tensor elements. (c) The phonon spectrum calculated from the force constant tensor before and after the optimization of\nindependent variables.\nFIG. 4. (a) The time evolution of spin temperature in SLD and ASD simulations. The gray line represents the SLD simulation\nwhile the others refer to the ASD simulations with different damping constants. (b) The initial cooling rates ∂TS/∂t|t=0with\nrespect to the damping constants α, where the scaling of axis is set to be logarithm. The gray squares refer to the results of\nASD simulations and the blue line acts as the linear regression. The red circle is plotted by intersection of the blue line and\nthe horizontal red dash line, which represents the initial cooling rate in the SLD simulation. Then we can obtain the equivalent\ndamping constant from the abscissa of the red circle. (c) The comparison between ASD and SLD simulations. In the ASD\nsimulation, the Gilbert damping constant is set to be α= 2.87×10−4, which is exactly the result of our evaluation from the\nSLD simulation.\nFIG. 5. The temperature dependence of Gilbert damping constants for Y 3Fe5O12. The label of abscissa axis ∆ Trefers to\nTS−TLof the initial state in dynamical simulations. Measurements on the magnetic damping are performed under different\ninitial conditions of the lattice temperature: (a) TL= 0, (b) TL= 30K, (c)TL= 60K.9\nFIG. 6. The relation between damping constants αand spin-\nlattice coupling constants ∂Jij/∂uk,αin YIG. Through a lin-\near fitting, the slope is determined to be 2 .01, which agrees\nwell with our theoretical predictions.\nis well verified by linear regressions and the details are\nprovided in the supplementary material.\nFurthermore, the accuracy of our simulations can also\nbe proved from another perspective. According to Eqs.\n(22) and (23), the scattering rate Wgrows quadratically\nwith the coupling parameters Mλ,qp. Based on the theory\nof second quantization, Mλ,qpshall be proportional to\nthe coupling constants ∂Jij/∂uk,α. Therefore, under a\ndefinite condition of temperature, we have:\nα∝˙Q∝∆W∝M2\nλ,qp∝(∂Jij/∂uk,α)2(27)\nIn order to verify this relation, we adjusted the spin-\nlattice coupling constants of YIG coherently while keep-\ning the other model parameters unchanged. Then, SLD\nsimulations were carried out to evaluate the correspond-\ning damping constants. The result is plotted in Fig. 6,\nwhere the x-label “slcc” stands for the spin-lattice cou-\npling constants and the subscript “0” refers to the orig-\ninal situation. From a linear fitting, the slope is deter-\nmined to be 2 .01, which agrees well with our prediction.\nC. Damping constants in MnFe 2O4\nAfter the calculation on YIG, we applied our method\nto MnFe 2O4(MFO), which was reported to possess a\nlarge Gilbert damping constant in the literature [13, 51].\nAs shown in Fig. 7(a), MnFe 2O4has a typical structure\nof spinels, where A sites are surrounded by four oxygen\natoms and B sites are located in octahedrons. Generally,\nspinels can be classified into normal and inverse struc-\ntures according to the distribution of divalent and triva-\nlent cations between A/B sites. In experiments, MFO\nusually crystallizes into a mixed phase where the normal\nstructure occupies the major part (80% in bulk MFO\n[52]). Here, we only considered its normal structure in\nthis work. Also, the magnetic ground state of MFO is\nshown in Fig. 22(b), where the magnetic moments are\nantiparallel between A/B sites.\nFIG. 7. (a) The cubic cell of MnFe 2O4. The purple balls rep-\nresent manganese atoms, the golden balls refer to iron atoms,\nand the red balls stand for oxygen atoms. (b) The magnetic\nground state of MFO. The arrows of different colors repre-\nsent the spin directions of Mn and Fe atoms separately. (c)\nThe density of states obtained by DFT calculations. (d) The\ntemperature dependence of average magnetization measured\nin MC and ASD simulations. For MnFe 2O4, the phase tran-\nsition point from ferrimagnetic to paramagnetic lies in 730K\napproximately.\nFirstly, we started to construct an effective Hamilto-\nnian model for MFO. With the same cutoff settings for\nYIG, we found 105 nonequivalent interactions, including\n4 Heisenberg exchange terms and 10 spin-lattice coupling\nterms. Subsequently, DFT calculations were carried out\nto determine the interaction parameters. In these calcu-\nlations, we adopted a cubic cell containing 56 atoms and\na Γ-centered 4 ×4×4 grid mesh in the reciprocal space.\nBesides, UMn= 3.3 eV and UFe= 3.6 eV were used as the\neffective Hubbard parameters [52]. With the exception of\naforementioned settings, all the relevant first-principles\ncalculations were performed under the same conditions\nas in Sec. III B.\nThe DOS of MnFe 2O4is plotted in Fig. 7(c), yielding\na calculated band gap of 0.612 eV. This value does not\nmatch with the result of transport experiments, which re-\nported a much smaller band gap (0 .04–0.06 eV) [53]. In\naddition, MC and ASD simulations were performed using\nthe Heisenberg exchange coefficients listed in Table III.\nThe temperature dependence of average magnetization,\nshown in Fig. 7(d), suggests the critical temperature to\nbe around 730 K. This result is significantly higher than\nthe measured value of 573 K [54]. Both of the above dis-\ncrepancies may be attributed to the inevitable difference\nbetween the ideal normal spinel structure in calculations\nand the partially disordered samples in reality. Despite\nthis problem, we proceeded to describe the target system\nwith our Hamiltonian model and expected to see how far\nthe calculated results of damping constants would differ10\nTABLE III. The exchange coefficients J of MnFe 2O4, where\nan effective spin S= 1 is adopted.\nSpin Pair. Distance (Angst) J (meV)\n1NN Fe-Fe 3.003 6.835\n1NN Mn-Fe 3.521 33.224\n1NN Mn-Mn 3.667 3.956\n2NN Fe-Fe 5.201 0.929\nfrom experimental values.\nAfter the preparation of Hamiltonian model, we con-\nducted dynamics simulations to evaluate the equivalent\ndamping parameters in MFO at different temperatures.\nA supercell containing 13440 atoms was adopted in the\nsimulation, and the results are summarized in Fig. 10.\nThe average of calculated damping constants is around\n8×10−5, which is much smaller than the measured value,\n1.0×10−2[13, 51]. Two factors may account for this in-\nconsistency. Firstly, the inhomogeneity in real MnFe 2O4\nsamples greatly enhances the scattering of magnons and\nphonons, thereby increasing the damping constants. Ad-\nditionally, due to the narrow band gap observed in ex-\nperiments, eddy currents can arise at finite temperatures,\nwhich leads to a rapid loss of energy in the form of joule\nheat. As the result of these factors, we failed to obtain a\nreasonable estimation of Gilbert damping constants for\nMnFe 2O4with our methodology. On the other side, the\ncontribution of different relaxation mechanisms to FMR\nlinewidth has been studied comprehensively for MnFe 2O4\nin Ref. [53], which further confirms our analyses.\nD. Damping constants in Cr 2O3\nChromia (Cr 2O3) is a well-known collinear magneto-\nelectric antiferromagnet, which holds great prospects in\nthe field of spintronics [55–57]. As shown in Fig. 8(a),\nthe primitive cell of Cr 2O3contains 10 atoms, with each\nchromium atom bonded to the six oxygen atoms around\nit. Additionally, Fig. 8(b) displays the magnetic ground\nstate of Cr 2O3, where the spins of two nearest neighbor-\ning Cr atoms are oriented in opposite directions.\nAs a preliminary step in constructing the Hamiltonian\nmodel, we set the cutoff radius of interactions to be 11.0\nBohr for atomic pairs and 7.0 Bohr for 3-body clusters.\nThrough symmetry analyses, we identified 319 nonequiv-\nalent interactions, including 5 Heisenberg exchange terms\nand 21 spin-lattice coupling terms.\nAfterwards, a series of first-principles calculations were\nperformed to determine the model parameters. Following\nthe settings in Ref. [58], we adopted a hexagonal cell of\nCr2O3which contained a total of 90 atoms in the calcula-\ntions. Additionally, we used the LSDA+U method in its\nfull spherically symmetric form [59]. As to the Hubbard\nparameters, Jwas fixed at its recommended value of 0.6\nFIG. 8. (a) The primitive cell of Cr 2O3. The dark blue balls\nrepresent chromium atoms, and the red balls stand for oxygen\natoms. (b) The magnetic ground state. The arrows of differ-\nent colors represent the spin directions of Cr atoms. (c) The\ndensity of states obtained by DFT calculations. (d) The tem-\nperature dependence of sublattice magnetization measured in\nMC and ASD simulations. For Cr 2O3, the phase transition\npoint from ferrimagnetic to paramagnetic lies in 310K approx-\nimately.\nTABLE IV. The exchange coefficients J of Cr 2O3, in which\nan effective spin S= 1 is adopted.\nSpin Pair. Distance (Angst) J (meV)\n1NN Cr-Cr 2.640 44.778\n2NN Cr-Cr 2.873 29.269\n3NN Cr-Cr 3.411 -0.182\n4NN Cr-Cr 3.635 0.007\n5NN Cr-Cr 4.137 -0.500\neV, and Uwas adjusted to fit the N´ eel temperature ob-\nserved in experiments [60]. We found U= 2.0 eV was the\noptimal value for 3 delectrons of Cr ions. Except for the\nsettings specified above, all the DFT calculations were\nconducted under the same conditions as in Sec. III C.\nThe DOS of Cr 2O3is plotted in Fig. 8(c), which yields\na calculated band gap of 1.935 eV. This value indicates\nthat the energy dissipation of electric currents can be ne-\nglected in this system. Additionally, we list the Heisen-\nberg exchange coefficients of chromia in Table IV. Both\nMC and ASD simulations were performed to investigate\nthe temperature dependence of sublattice magnetization.\nAccording to Fig. 8(d), the critical point was determined\nto be 310 K approximately, which was quite consistent\nwith experimental observations. Also, the force constants\nof Cr 2O3went through the modification formulated in\nSec. II B, and the spin-lattice coupling parameters are\nprovided in the supplementary material.\nAfter the construction of Hamiltonian model, we con-\nducted a series of dynamics simulations to evaluate the11\nFIG. 9. (a) The 1NN FeT-FeOpair in Y 3Fe5O12. (b) The\n1NN Cr-Cr pair in Cr 2O3. The steel blue arrow stands for\nthe orientation of ∂J/∂u and the red number along with it\nrepresents the magnitude in unit of meV/Angst.\nequivalent damping parameters in Cr 2O3. An expanded\nhexagonal cell containing 14400 atoms was adopted for\nthe simulation, and the results are summarized in Fig. 11.\nAs two specific cases, our calculation yielded α= (1.31±\n0.14)×10−4at ∆T= 15 K and α= (2.7±0.3)×10−4\nat ∆T= 30 K, where both TL= 0 K. Therefore, the\ncalculated damping constants within ∆ T= 15∼30 K\nare quite close to 2 ×10−4, which is the estimated value\nreported in Ref. [61].\nFurthermore, the damping constants in Cr 2O3exhibit\na significant non-linear relation with the temperature dif-\nference of subsystems. Through logarithmic fittings, we\ncalculated the power exponents for Figures 11(a) to 11(c),\nand the results were 1.17, 1.62, 1.38. If we disregard the\ndifference between ∆ TandTfor the moment, these val-\nues are in good agreement with the theoretical prediction\nof Kasuya and LeCraw [26]. According to their study, the\nrelaxation rate varies as Tnwhere n= 1∼2 while n= 2\ncorresponds to a larger regime of temperatures.\nCompared to YIG, the greater magnetic damping ob-\nserved in chromia can be attributed to its significantly\nstronger spin-lattice coupling. As shown in Fig. 9, the\nmagnitude of principal spin-lattice coupling constant in\nCr2O3is two or three times larger than that in YIG. This\ncould be explained by the fact that direct exchange in-\nteraction between two magnetic atoms decreases rapidlywith their distance [62]. Therefore, owing to the shorter\ndistance of Cr-Cr pair, the direct exchange interaction\nbetween neighboring Cr atoms is believed to have a great\ncontribution to the spin-lattice coupling in Cr 2O3.\nIV. CONCLUSIONS\nIn summary, we propose a scheme to evaluate the con-\ntribution of spin-lattice coupling to the Gilbert damp-\ning in insulating magnetic materials. Our methodology\ninvolves first-principles based Hamiltonian models and\nspin-lattice dynamics simulations. Following a series of\nvalidations, we applied our method to three magnetic ma-\nterials, namely Y 3Fe5O12, MnFe 2O4and Cr 2O3. Their\ndamping constants were estimated separately, and the\nresults show that, in general, αis approximately propor-\ntional to the temperature difference between spin and\nlattice subsystems. Under the condition of ∆ T= 30\nK, the calculated damping constants are averaged to be\n0.8×10−4for YIG, 0 .2×10−4for MFO and 2 .2×10−4\nfor Cr 2O3. The results for YIG and Cr 2O3are in good\nagreement with experimental measurements, while the\ndiscrepancy for MFO can be attributed to the inhomo-\ngeneity and small band gap in real samples. Overall, the\napproach presented in this work holds great promise for\naccurately predicting the Gilbert damping constants for\nmagnetic insulators.\nACKNOWLEDGMENTS\nThis work is supported by the National Key R&D\nProgram of China (No. 2022YFA1402901 ), the Na-\ntional Natural Science Foundation of China (Grant Nos.\n11825403, 11991061, and 12188101), the Guangdong Ma-\njor Project of the Basic and Applied Basic Research\n(Future functional materials under extreme conditions–\n2021B0301030005).\n[1] S. Wolf, D. Awschalom, R. Buhrman, J. Daughton,\nS. von Molnar, M. Roukes, A. Chtchelkanova, and\nD. Treger, Spintronics: A spin-based electronics vision\nfor the future, SCIENCE 294, 1488 (2001).\n[2] T. Jungwirth, X. Marti, P. Wadley, and J. Wunderlich,\nAntiferromagnetic spintronics, NATURE NANOTECH-\nNOLOGY 11, 231 (2016).\n[3] L. Smejkal, Y. Mokrousov, B. Yan, and A. H. MacDon-\nald, Topological antiferromagnetic spintronics, NATURE\nPHYSICS 14, 242 (2018).\n[4] I. Zutic, J. 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Akansel1, A. Kumar1, V.A.Venugopal2, R.Banerjee3, C. Autieri3, R.Brucas1, N. Behera1, M. \nA. Sortica3, D. Primetzhofer3, S. Basu2, M.A. Gubbins2, B. Sanyal3, and P. Svedlindh1 \n1Department of Engineering Sciences , Uppsala University, Box 534, SE -751 21 Uppsala, Sweden  \n2Seagate Technology, BT48  0BF, Londonderry, United Kingdom  \n3Department of Physics and Astronomy, Uppsala University, Box 516, SE -751 20 Uppsala,  \nSweden  \n \nThe effect s of rhenium doping  in the range 0 – 10 at% on the static and dynamic magnetic \nproperties of Fe65Co35 thin films have been studied experimentally as well as with first principles \nelectronic structure calculations  focussing on the change of the saturation magnetization (𝑀𝑠) and \nthe Gilbert damping parameter ( 𝛼) Both experiment al and theoretical results show that 𝑀𝑠 \ndecreases with increasing Re doping level, while at the same time 𝛼 increases. The experimental \nlow temperature saturation magnetic induction exhibits a 2 9% decrease,  from 2.3 1T to 1. 64T, in \nthe investigated doping concentration range , which is more than predicted by the theoretical \ncalculations. The room temperature value of the damping parameter obtained from ferromagnetic \nresonance measurements , correcting for extrinsic contributions  to the damping,  is for the undoped \nsample 2.7×10−3, which is close to the theoretically calculated Gilbert damping parameter . With \n10 at% Re doping , the damping parameter increases to 9.0×10−3, which is in good agreement \nwith the theoretical value of 7.3×10−3. The increase in damping parameter with Re doping is \nexplained by the increase in density of states at Fermi level, mostly contributed by the s pin-up \nchannel of Re. Moreover, both experimental and theoretical values  for the da mping parameter are \nobserved to be weakly decreas ing with decreasing temperature .  \n  1. INTRODUCTION  \nDuring the last decades , thin films of soft magnetic alloys such as NiFe and FeCo  have been in \nfocus  due to possible use in applications such as spin valves ,1,2 magnetic tunneling junctions ,3,4,5 \nspin injectors ,6 magnetic storage technologies  and in particular in magnetic recording write heads .7 \nBeside s spintronic and magnetic memory devices , such materials are useful for shielding \napplications that are necessary in order to reduce the effect of electromagnetic fields created  by \nelectronic devices. The magnetic damping parameter  of the material play s a critical role for the \nperformance of such spintronic and memory devices as well as for shielding applications.  On the \none hand, a low damping parameter is desired in order to get low critical switching current in \nspintronic devices .8,9,10 On the other hand , a high damping parameter is necessary in order to \nreduce the magetization switching time in magnetic memory devices and to be able to operate \ndevices at high speeds .11 FeCo alloys are promising materials for  high frequency spintronic \napplications and magnetic recording devices due to their high saturation magnetization  (𝑀𝑠), high \npermeability, thermal stability and comparably high resistivity .12,13,14 One possible drawback is \nthat FeCo alloy s exhibit  high coercivity  (𝐻𝑐), which is not favorable for such applications , however \nthis problem can be solved by thin film growth  on suitable buffer layer s.15,16,12 Except coercivity \nproblems, the damping parameter of these materials should be increased to make them com patible \nfor high speed devices . \nDynamic properties of  magnetic materials are highly dependent on the damping parameter. This \nparameter is composed of both intrinsic and extrinsic contributions. The intrinsic contribution is \ncalled  the Gilbert damping and  depends  primarily  on the spin-orbit coupling .17 Intrinsic damping \nis explained as scattering of electrons by phonons and magnons .18,19 Beside s electron scattering , \ndue to the close relation between magnetocrystalline  anisotropy and spin-orbit coupling , it can be \nassumed that the intrinsic damping is also related to the magnetocrystalline anisotropy constant .20 \nRegarding extrinsic damping , there can be a number of different contributions. The most common \ncontribution originates  from two magnon scattering  (TMS) .21 However , this contribution vanishes \nwhen ferromagnetic resonance  (FMR) measurements are performed by applying the static \nmagnetic field  along  the film normal  in inplane anisotropic thin films .22 Beside s TMS , there are \nsome other extrinsic contributions to the damping that are not possible to get rid of  by changing \nthe measurement configuration . One of these contributions  is radiative damping , which arises from \ninductive coupling between  the precessing magnetization and the waveguide  used for FMR \nmeasurem ents.23 Another contribution  for metallic ferromagnetic films is the eddy current \ndamping related to microwave magnetic field induced eddy currents in the thin film s during  \nmeasurement s.23,24 \nIn order to make a soft magnetic thin film  suitible for  a specific  applica tion, taking into account  \nrequirements set by the device application , its damping paramete r should be tailored. As mentioned \nabove , an increased damping parameter is necesssary for devices requiring high switching speed . \nSeveral efforts have been made  for enhanching the damping parameter of soft magnetic materials. \nNiFe alloys constitu te one of the most studied systems in this respect . The most common way to enhance  the intrinsic damping of an all oy is to dope it with differ ent elements . Rare earth elements  \nwith large spin-orbit coupling  have revealed  promising results  as dopant s in terms of increas ed \ndampin g parameter .25,26,27 3d, 4d and 5d transition metals  dopants have also been studied \nexperimentally , revealing an increase of the damping parameter .28,29 Beside s experimental results , \ntheoretical calculations support the idea that transition metals and especially 5d elements can \nenhance the damping parameter of NiF e alloys due to   scattering in presence of chemical disorde r \n, as well as due to the effect of spin -orbit coupling .30 \nAlthough NiFe alloys have been the focus in several extensive studies, FeCo alloys have so far not \nbeen studied to the same extent . Attempts have been made  to  dope  FeCo with Yb,20 Dy,31 Gd,32 \nand Si ,33 where in all cases an increase of the  damping parameter  was observed . Apart from  doping \nof alloys , the addition of  adjacent layers to NiFe and CoFe has also been studied . In particular , \nadding layers consisting of rare earth elem ents with large orbital moment s  gave positive results in \nterms of increased damping parameter .34 \nFe65Co35 alloy s are attractive material s because of high 𝑀𝑠 and reduced 𝐻𝑐 values. However , not \nmuch is known about the magnetic damping mechanism s for this composition . Since it is of \ninterest for high data rate magnetic memory devices, the damping parameter should be increased \nin order to make the magnetic switching faster. To the best of our knowledge , systematic doping \nof Fe 65Co35 with 5d elements has not been studied so far  experimentally . Some of us have found \nfrom ab initio  calculations  that 5d transition metal dopants can increase the damping parameter \nand Re is one of the potential candidates.35 Re is particularly interesting as it has a nice compromise \nof having not so much reduced saturation magnetization and a quite enh anced damping parameter. \nIn this work, we have perfomed a systematic ab initio study of Fe65Co35 doped with increasing Re \nconcentration to find an increasing  damping parameter . The theoretical prediction s are confirmed  \nby results obtained from temperature  dependent FMR measurements performed on Re doped  \nFe65Co35 films.  \n \n2. EXPERIMENTAL  AND THEORETIC AL METHOD S \nRhenium  doped Fe 65Co35 samples were prepared by varying the Re concentration from 0 to 10.23  \nat%. All samples were deposited using DC magnetron sputtering on Si/SiO 2 substrate s. First a 3 \nnm thick  Ru seed layer was deposited on the Si/SiO 2 substrate followed by room temperature \ndeposition of  20 nm  and 40 nm  thick Re -doped  Fe65Co35 films by co -sputtering between Fe 65Co35 \nand  Re target s. Finally, a 3 nm thick Ru layer was deposited as a capping layer over  the Re -doped  \nFe65Co35 film. The nominal Re concentration was derived from the calibrated deposition rate used \nin the deposition system. The nominal Re doping concentration s of the Fe65Co35 samples are as \nfollows ; 0, 2.62, 5.45 and  10.23  at%.  \nThe crystalline structure  of the fims were  investigated by utilizing grazing incident X -Ray \ndiffraction  (GIXRD). The i ncidence angle was fixed at  1o during  GIXRD measurements  and a CuKα source was used.  Accurate values for film thickness and interface roughness were  \ndetermined by X -ray reflectivity  (XRR) measurements.   \nBeside XRD , composition and areal density of the films were deduced by Rutherford \nbackscattering spectrometry36 (RBS)  with ion beams of 2 MeV 4He+ and 10 MeV 12C+. The beams \nwere provided by a 5 MV 15SDH -2 tandem accelerator at the Tandem Laboratory at Uppsala \nUniversity. The experiments were performed with the incident beam at 5° with respect to the \nsurface normal and scattering angles of 170° and 120° . The experimental data was evaluated with \nthe SIMNRA program .37 \nIn-plane magnetic hysteresis measurments were performed using a Magnetic Property \nMeasurement System (MPMS, Quantum Design) .  \nFerromagnetic  resonance  measurements were  performed using  two different techniques. First in-\nplane X -band (9.8 GHz) cavity FMR measurements were performed . The setup  is equipped with a \ngoniometer making it possible to rotate the sample with respect  to the applied magnetic field; in \nthis way the in -plane anisotropy fields of the different samples have been determine d. Beside s \ncavity FMR  studies , a setup for broadband out-of-plane FMR measurements have been utilized . \nFor out -of-plane measurements a vector network analyzer (VNA) was used. Two ports of the VNA \nwere connected to a coplanar waveguide (CPW) mounted on a Ph ysical Property Measurement \nSystem (PPMS, Quantum Design) multi -function probe . The PPMS is equipped with a 9 T \nsuperconducting magnet, which is needed to saturate Fe65Co35 films out -plane and to detect the \nFMR signal. The broadband FMR  measurements were carried out a t a fixed microwave frequency \nusing the field -swept mode, repeating the measurement for different f requencies in the range 15 – \n30GHz.  \nThe theoretical calculations are based on spin -polarized relativistic m ultiple scattering theory using \nthe Korringa -Kohn -Rostoker (KKR) formalism implemented in the  spin polarized relativistic \nKKR code  (SPR-KKR) . The Perdew -Burke -Ernzerhof (PBE) exchange -correlation functional \nwithin generalized gradient approximation was used. The equilibrium lattice parameter s were \nobtained by energy minimization for each composition. Substitutional disorder was treated within \nthe Coherent Potential Approximation (CPA). The damping parameters were calcu lated by the \nmethod proposed by Mankovsky et al.,38 based on the ab initio Green's function technique and  \nlinear res ponse formalism where one takes into consi deration scattering processes as well as spin -\norbit coupling built in Dirac's relativistic formulation. The calculations of Gilbert damping \nparameters at finite temperatures were  done using an alloy -analo gy model of atomic displacements \ncorresponding to the thermal average of the root mean square displacement at a given temperature.  \n3. RESULTS AND  DISCUSSION  \nRe concentrations and layer thickness (areal densities) of the 20 nm  doped films were obtained by \nRBS experiments. RBS employing a beam of 2 MeV He primary ions was used to deduce the areal \nconcentration of each layer. Additional measurements with 10 MeV C probing particles permit to resolve the atomic fractions of Fe,  Co and Re. The spectra for the  samples with different Re \nconcentration are shown in Fig. A1 . The measured Re concentrations are 3.0±0.1  at%, 6.6±0.3  at% \nand 12.6±0.5  at%. Moreover, the results for Fe and Co atomic fractions show that there is no \npreferential replacement by Re , implying that the two elements are replaced according to their \nrespective concentration . \nFigure 1 (a) shows GIXRD spectra in the 2𝜃-range from 20o to 120o for the Fe65Co35 films with \ndifferent Re concentration. Diffraction peaks corresponding to the body centered cubic Fe 65Co35 \nstructure have been indexed in the figure; no other diffraction peaks appear in the different spectra.  \nDepending on the Re -dopant concentration shi fts in the peak positions are observed, the diffraction \npeaks are suppressed to lower 2𝜃-values with increasing dopant concentration . The shift for the \n(110) peak for the different dopant concentrations is given as an inset in Fig. 1 (a). Similar shifts \nare observed for the other diffraction peaks. This trend in peak shift is an experimental evidence \nof an increasing amount of Re dopant within the deposited thin films. Since the peaks are shifted \ntowards lower 2𝜃-values  with increasing amount of Re dopant , the lattice parameter increases with \nincreasing Re concentration.39 Figure 1 (b) shows the experimental as well as theoretically \ncalculated lattice parameter versus Re concentration.  The qualitative agreement between theory \nand experiment is obtained. However, t he rate of lattice parameter increase with increasing Re \nconcentration is larger for the theoretically calculated lattice parameter. This is not unexpected as \nthe generalized gradie nt approximation for the exchange -correlation potential has a tendency to \noverestimate the lattice parameter. Another possible explanation for the difference in lattice \nparameter is that the increase of the lattice parameter for the Re -doped Fe 65Co35 films  is held back \nby the compressive strain due to lattice mismatch with Si/SiO 2/Ru. XRR measurements revealed \nthat the surface roughness of the Fe 65Co35 films is less than 1 nm , which cannot affect static and \nmagnetic properties drastically. Results from XRR measurements are given in table 1.  \nRoom temperature normalized magnetization curves for the Re-doped Fe 65Co35 films  are shown  \nin Fig. 2 (a) . The coercivity for  all films is in the range of 2 mT and all films, except for the 1 2.6 \nat% Re doped film that show a slightly rounded hysteresis loop, exhibit rectangular hysteresis \nloops. The low value for the coercivity is expected for seed layer grown films .15 The \nexperimentally determined  low temperature saturation magnetization  together with the \ntheoretically calculated magnetization versus Re concentra tion are shown in Fig. 2 (b). As \nexpected, both experimental and theoretical r esults show that the saturation magnetization \ndecreases with increasing Re concentration . A linear decrease in magnetization is observed in the \ntheoretical calculations whereas a non -linear behavior is seen in the experimental data.  \nAngle resolved cavity FMR measurements were used to study the in -plane magnetic anisotropy . \nThe angular -dependent resonant field ( 𝐻𝑟) data was analyzed using the following equation ,40 \n 𝑓=µ0𝛾\n2𝜋[{𝐻𝑟cos(𝜙𝐻−𝜙𝑀)+𝐻𝑐\n2cos4(𝜙𝑀−𝜙𝐶)+𝐻𝑢cos2(𝜙𝑀−𝜙𝑢)}{𝐻𝑟cos(𝜙𝐻−\n𝜙𝑀)+𝑀𝑒𝑓𝑓+𝐻𝑐\n8(3+cos4(𝜙𝑀−𝜙𝐶))+𝐻𝑢cos2(𝜙𝑀−𝜙𝑢)}]12⁄\n,  (1) where  𝑓 is the cavity resonance frequency  and 𝛾 is the gyromagnetic ratio . 𝜙𝐻, 𝜙𝑀, 𝜙𝑢 and 𝜙𝐶 \nare the in -plane directions for the magnetic field,  magnetization, uniaxial anisotropy and cubic \nanisotropy, respectively, with respect to  the [100 ] direction of the Si substrate.   𝐻𝑢=2𝐾𝑢\nµ0𝑀𝑠 and \n𝐻𝑐=4𝐾𝑐\nµ0𝑀𝑠  are the uniaxial and cubic anisotropy fields, where 𝐾𝑢 and 𝐾𝑐 are the uniaxial and cubic \nmagnetic anisotropy constants , and  𝑀𝑒𝑓𝑓 is the effective magnetization.  Fitting parameters were \nlimited to 𝑀𝑒𝑓𝑓,  𝛾 and 𝐻𝑢, since the Hr versus ϕH curves did not give any indication of a cubic \nanisotropy.  \nFigure 3 shows 𝐻𝑟 versus 𝜙𝐻 extracted from the angular -dependent FMR measurements together \nwith fits according to Eq. (1), clearly revealing dominant twofold uniaxial in -plane magnetic \nanisotropy. Extracted anisotropy field and effective magnetization values are given in  Table 2 . The \nresults show that 𝐻𝑢 is within the accuracy of the experiment independent of the Re concentration . \nTemperature dependent o ut-of-plane FMR measurements  were  performed in the temperature range \n50 K to 300 K  recording the complex transmission coefficient 𝑆21. Typical field -swept results for \nthe r eal and imaginary components of  𝑆21 for the undoped and 1 2.6 at% Re-doped samples are \nshown  in Fig. 4. The field -dependent 𝑆21 data was fitted to the following set of equations,41  \n𝑆21(𝐻,𝑡)=𝑆210+𝐷𝑡+𝜒(𝐻)\n𝜒̃0    \n𝜒(𝐻)=𝑀𝑒𝑓𝑓(𝐻−𝑀𝑒𝑓𝑓)\n(𝐻−𝑀𝑒𝑓𝑓)2−𝐻𝑒𝑓𝑓2−𝑖𝛥𝐻 (𝐻−𝑀𝑒𝑓𝑓) .  (2) \nIn these equations 𝑆210 corresponds to  the non-magnetic contribution to the complex transmission \nsignal , 𝜒̃0 is an imaginary function of the microwave frequency and film thickness  and 𝜒(𝐻) is the \ncomplex susceptibility  of the magnetic film. The term 𝐷𝑡 accounts for a linear drift of the recorded \n𝑆21 signal.  𝑀𝑒𝑓𝑓=𝑀𝑠−𝐻𝑘⫠, where 𝐻𝑘⫠ is the perpendicular anisotropy field  and 𝐻𝑒𝑓𝑓=2𝑓\n𝛾µ0. The \n𝑆21 spectra were  fitted to Eq. (2 ) in order to extract the linewidth 𝛥𝐻 and 𝐻𝑟 values. Fits t o Eq. (2) \nare shown as solid lines in Fig. 4.  \nThe experimentally measured total d amping parameter ( 𝛼𝑡𝑜𝑡𝑎𝑙 ), including both the intrinsic \ncontribution  (Gilbert damping) and extrinsic contributions , was extracted by fitting 𝛥𝐻 versus \nfrequency to the following expression, 41 \nµ0𝛥𝐻=4𝛼𝑡𝑜𝑡𝑎𝑙 𝑓\n𝛾+µ0𝛥𝐻0 ,   (3) \nwhere 𝛥𝐻0 is the frequency independent linewidth broadening  due to sample inhomogeneity . \nBeside s 𝛼𝑡𝑜𝑡𝑎𝑙 , 𝑀𝑒𝑓𝑓 can also be  extracted by fitting the 𝐻𝑟 versus frequency results to the \nexpression  µ0𝐻𝑟=2𝜋𝑓\n𝛾+µ0𝑀𝑒𝑓𝑓 .   (4) \nTypical temperature dependent results for 𝑓  versus 𝐻𝑟  and 𝛥𝐻 versus 𝑓  are shown in Fig. 5 for \nthe 1 2.6 at% Re -doped Fe65Co35 film. Extracted values of 𝑀𝑒𝑓𝑓  at different temperatures are given \nin Table 3 for all samples . As expected, the results show that 𝑀𝑒𝑓𝑓 decreas es with increasing \ndopant concentration.  Since 𝑀𝑒𝑓𝑓=𝑀𝑠−𝐻𝑘⫠  and the film thickness is large enough to make a \npossible contribution from out -of-plane anisotropy negligible one can make the justified \nassumption that 𝑀𝑒𝑓𝑓≈𝑀𝑠. The analysis using Eqs. (2) – (4) also give values for the Land é 𝑔-\nfactor ( 𝛾=𝑔µ𝐵\nħ), yielding 2.064 and 2.075 for the undoped and 12.6 at% doped samples, \nrespectively (similar values are obtained at all temperatures).  \nAs indicated above, the d amping parameters extracte d from FMR measurements ( 𝛼𝑡𝑜𝑡𝑎𝑙) include  \nboth intrinsic and extrinsic contributions. One of the most common extrinsic contribution s is TMS , \nwhich is avoided in this study by measuring FMR with the magnetic field applied out of the film \nplane. Except TMS , extrinsic contributions  such as eddy curr ent damping  and radiative damping \nare expected to contribute the measured damping . In a metallic  ferromagnet, which is placed on \ntop of a CPW , precession  of spin waves induces AC currents in the ferromagnet ic film, thereby \ndissipating energy . The radiative  damping has similar origin as the eddy current damping, but here \nthe precession of the magnetization induces microwave -frequency currents in the CPW where \nenergy is dissipated. Thus,  there are two extrinsic  contributions to the measured damping ; one that \nis caused by eddy currents in the ferromagnet ic film (𝛼𝑒𝑑𝑑𝑦) and  another one caused by  eddy \ncurrents in the CPW ( 𝛼𝑟𝑎𝑑).23 In order to  obtain the reduced  damping of the films  (𝛼𝑟𝑒𝑑), which \nwe expect to be close to the intrinsic damping of the films, the extrinsic contributions should be \nsubtracted from  𝛼𝑡𝑜𝑡. We have neglected any contribution to the measured damping originating \nfrom spin -pumping into seed and capping layers. However, since spin -pumping in low spin -orbit \ncoupling materials like Ru with thickness quite less than the spin -diffusion length is quit e small, \nthe assumption  of negligible contribution from spin -pumping  is justified.  The t otal damping can \nthus be given as  𝛼𝑡𝑜𝑡=𝛼𝑟𝑒𝑑+𝛼𝑟𝑎𝑑+𝛼𝑒𝑑𝑑𝑦 .  \nWhen the precession of the magnetization is assumed to be uniform in the sample , the expression \nfor  radiative damping can be given as23 \n𝛼𝑟𝑎𝑑=𝜂𝛾µ02𝑀𝑠𝛿𝑙\n2𝑍0𝑤 ,  (5) \nwhere 𝑍0 =50 Ω is the waveguide impedance, 𝑤=240 µm is the width  of the CPW center \nconductor , 𝜂 is a dimensionless parameter that accounts for FMR mode profile, δ is the thickness \nand 𝑙 is the length of the sample. The l ength of all samples were  4mm and the thickness  20nm for  \nthe undoped and 1 2.6 at% Re-doped films and 40nm for  the 3.0 at% and 6.6 at% Re-doped films. \nTemperature dependent radiative damping contributions for all Fe 65Co35 films are given  in Table \n4. Beside s 𝛼𝑟𝑎𝑑, the 𝛼𝑒𝑑𝑑𝑦 contribution should also be calculated and extracted from 𝛼𝑡𝑜𝑡𝑎𝑙 to extract \nthe reduced  damping parameter. 𝛼𝑒𝑑���𝑦 can be estimated by the expression23 \n𝛼𝑒𝑑𝑑𝑦 =𝐶𝛾µ02𝑀𝑠𝛿2\n16𝜌 ,  (6) \nwhere  𝐶 is a parameter describing the  distribution of eddy current s within the films  and its value \nis 0.5 in our studied samples and 𝜌 is the resistivity of the films. Resistivity is measured for all \nfilms with different dopant concentrations at different temperatures. It is in the range of 8.2×10-8 \nto 5.6 ×10-8 𝛺𝑚 for undoped, 5.7 ×10-7 to 5.3 ×10-7 𝛺𝑚 for 3.0  at% doped , 6.9 ×10-7 to 6.1 ×10-\n7 𝛺𝑚 for 6.6  at% doped and 3.9×10-7 to 3.6 ×10-7 𝛺𝑚 for 12.6 at% doped films. Temperature \ndependent eddy current damping contributions , which are negligible,  for all Fe 65Co35 films are \ngiven in Table  5. \n𝛼𝑡𝑜𝑡 (filled symbols) and 𝛼𝑟𝑒𝑑  (open symbols) versus temperature for the differently Re -doped \nFe65Co35 films are shown in Fig. 6 .  Both  damping parameter s slowly decrease  with decreasing \ntempera ture. Moreover, the damping parameter increases with increasing Re concentration; the \ndamping parameter is 4 times as large for  the 12.6 at% Re -doped sample compared to the undoped \nsample . Since the damping parameter depends both on disorder induced scattering and spin-orbit \ncoupling, the observed enhanc ement of the damping parameter can emerge from the electronic \nstructure of the  alloy and  large spin -orbit coupling of Re.  \nA c omparison between temperature dependent experimental 𝛼𝑡𝑜𝑡 and 𝛼𝑟𝑒𝑑 values and \ntheoretically calculated intrinsic damping parameters is shown in Fig. 7 for the undoped and 12.6 \nat% Re -doped Fe 65Co35 films.  In agreement with the experimental results, the theoretically \ncalculated damping parameters decrease in magnitude with decreasing temperature . It has been \nargued  by Schoen et al., 42 that the contribution to the intrinsic Gilbert damping parameter comes \nprimarily from the strong electron -phonon coupling at high temperatures  due to interband \ntransition whereas at a low temperature, density of states at Fermi level (𝑛(𝐸𝐹)) and spin -orbit \ncoupling  give rise to intraband transition.  In Fig. 8, we show the correspondence between the \ncalculated damping parameter at 10 K with the density of states (spin up +spin down) at Fermi \nlevel for varying Re concentration. The increasing trend in both  properties is obviously seen. The \nincrease in DOS mainly comes from increasing DOS at Re sites in the spin -up channel. In the \ninset, the calculated spin -polarization as a function of Re concentration is shown. Spin polarization \nis defined as   𝜁=𝑛(𝐸𝐹)↑−𝑛(𝐸𝐹)↓\n𝑛(𝐸𝐹)↑+𝑛(𝐸𝐹)↓ where the contribution from both spin channels are seen. It is \nclearly observed that Re doping decreases the spin polarization.  \n \nOne should note that a quantitative comparison between theory and experiment requires more \nrigoro us theoretical considerations. The difference  between experimental and theoretical results \nfor the damping parameter may be  explained by the incompleteness of the model used to calculate \nthe Gilbert damping parameter  by neglecting several complex scatterin g processes. Firstly, the effect of spin fluctuations was neglected, which in principle could be considered in the present \nmethodology if the temperature dependent magnetization and hence information about the \nfluctuations of atomic moments were available from Monte -Carlo simulations. Other effects such \nas non-local damping and more sophisticated treatment of atomic displac ements in terms of \nphonon self -energies40  that may contribute to the relaxation of the magnetization in magnetic thin \nfilm materials  have been neglected . Nevertheless, a qualitative agreement has been achieved where  \nboth experimental  and theoretical results show that there is a significant increase of the damping \nparameter with increasing concentration of Re.  \n \n4. CONCLUSION  \nStatic and dynamic magnetic properties of rhenium doped Fe 65Co35 thin films have been \ninvestigated and clarified in a combined experimental and theoretical study. Results from first \nprinciples theoretical calculations show that the saturation magnetization gradually decreases with \nincreasing Re concentration, from 2.3T  for the undoped sample to 1.95T for the 10% Re -doped \nsample. The experimental results for the dependence of the saturation magnetization on the Re -\ndoping are in agreement with the theoretical results, although indicating a more pronounced \ndecrease of the saturation magnetization for the largest doping concentrations. The theoretical \ncalculations show that the intrinsic Gilbert damping increases with increasing Re concentration; at \nroom temperature the damping parameter  is 2.8×10−3, which increases to 7.3×10−3 for the 10  \nat% Re -doped sample. Moreover, temperature dependent calculations of the Gilbert damping \nparameter reveal a weak decrease of the value with decreasing temperature . At a low temperature, \nour theoretical analysis showed the prominence of intra band contribution arising from an increase \nin the density of states at Fermi level. The experimental results for the damping parameter were \ncorrected for radiative and eddy current contributions to the measured damping parameter and \nreveal similar trends as observed in the theoretical results; the damping parameter increases with \nincreasing Re concentration and the damping parameter value decreases with decreasing \ntemperature. The room temperature value for the reduced damping paramet er was  2.7×10−3 for \nthe undoped sample, which increased to 9.0×10−3 for the 1 2.6 at% Re -doped film. The \npossibility to e nhanc e the damping parameter for Fe65Co35 thin films is a  promising result since \nthese materials are used in magnetic memory applications and higher data rates are achievable if \nthe damping parameter of the material is increased.   \n \nACKNOWLEDGEMENT  \nThis work is supported by the Knut and Alice Wallenberg (KAW) Fou ndation, Grant No. KAW \n2012.0031 and by the Marie Curie Action “Industry -Academia Partnership and Pathways” (ref. \n612170, FP7 -PEOPLE -2013 -IAPP). The authors acknowledge financial support from Swedish \nResearch Council (grant no. 2016 -05366) and Carl Tryggers Stiftelse (grant no. CTS 12:419 and \n13:413). The simulations were performed on resources provided by the Swedish National Infrastructure (SNIC) at National Supercomputer Centre at Link öping University (NSC). M. 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Nembach, O. Eriksson, O. \nKaris, and J. M. Shaw, Nature Physics, 12, 839 –842 (2016).  \n \n     \nFigure 1 (a) GIXRD plot for Fe 65Co35 films with dif ferent Re concentrations. S hift of (110) peak \ndiffraction peak with Re concentration is given as insert . (b) Lattice parameter versus Re \nconcentration.  Circles are lattice parameters extracted from XRD measurements and squares are \ncalculated th eoretical values. Line s are guide to the eye. \n  \nFigure 2 (a) Normalized room temperature magnetization versus magnetic field for Fe 65Co35 \nfilms with different Re concentration . (b) Low temperature saturation magnetization versus Re \nconcentration. Circles  are experimental data and squares  corresponding calculated results.  \nExperimental 𝝁𝟎𝑴𝒔 values were extracted from temperature dependent FMR results. Lines are \nguide s to the eye .  \n  \n \n \nFigure 3 𝝁𝟎𝑯𝒓 versus in -plane angle of magnetic field 𝝓𝑯 for different dopant concentrations  of \nRe. Black squares are experimental  data and red line s are fits to Eq. (1).  \n  \n \n \nFigure 4 Room temperature  real (a and c) and imaginary  (b and d)  𝑺𝟐𝟏 components versus out -\nof-plane magnetic field  for Fe65Co35 thin  films with 0% and 12.6 at% Re recorded at 20GHz . \nBlack squares are data points and red lines are fit s to Eq. (2). \n \n  \n \n \nFigure 5 (a) Frequency versus  𝝁𝟎𝑯𝒓 values at different temperatures for  the Fe65Co35 thin film \nwith 12.6 at% Re. Coloured lines correspond to fits to Eq. ( 4). (b) Linewidth 𝝁𝟎∆𝑯 versus \nfrequency at different temperatures for the same Re doping concentration. Coloured  lines \ncorrespond to  fits to Eq. ( 3). Symbols represent  experimental  data. \n  \n \n \nFigur e 6  𝜶𝒕𝒐𝒕 versus  temperature for Fe 65Co35 thin films with different concentration of Re. \nBesides showing  𝜶𝒕𝒐𝒕 , reduced 𝜶𝒓𝒆𝒅 values are also plotted obtained by subtraction of radiative \ndamping and eddy current damping contributions from  𝜶𝒕𝒐𝒕. Error bars are  given for measured \n𝜶𝒕𝒐𝒕 (same size as symbol size ). \n  \n \nFigur e 7 𝜶𝒕𝒐𝒕 versus  temperature for Fe 65Co35 thin films with 0  at% and 1 2.6 at% concentration \nof Re. Beside s showing  𝜶𝒕𝒐𝒕 , reduced 𝜶𝒓𝒆𝒅 values are also plotted obtained by subtraction of \nradiative damping and eddy current damping contribution s from  𝜶𝒕𝒐𝒕. In addition to \nexperimental results theoretically calculated intrinsic damping parameters  are given for the \nsimilar  concentrations  of Re . Error bars are given for  measured 𝜶𝒕𝒐𝒕 (same size as symbol size) . \n  \n \n \nFigure 8  Calculated density of states at Fermi level (left axis) and damping parameter (right \naxis) are shown as a function of Re concentration. In the inset, spin -polarization is shown  as a \nfunction of Re concentration.  \n  \n0 0.03 0.06 0.09 0.12\nRe concentration0.90.951DOS at EF (States/eV)\n0 0.03 0.06 0.09 0.12\nRe concentration0.350.40.450.50.55Spin polarization\n0123456\nDamping parameter (x 10-3)Re \n(at%)  𝑡𝑅𝑢,𝑐𝑎𝑝 \n(nm)  𝜎  \n(nm)  𝑡𝐹𝑒𝐶𝑜 \n(nm)  𝜎  \n(nm)  𝑡𝑅𝑢,𝑠𝑒𝑒𝑑 \n(nm)  \n  \n(nm)  \n0 2.46 1.89 39.71  0.67 2.74 0.66 \n3.0 2.47 1.80 37.47 0.59 2.45 1.03 \n6.6 1.85 0.50 37.47 0.51 2.13 0.90 \n12.6 2.15 1.49 37.38 0.64 1.89 1.03 \nTable 1 Thickness and roughness (𝝈) values for different layers in films  extracted  from XRR \ndata. Error margin is 0.02nm for all thickness and roughness values.   \n \nRe (at%) 𝜇0𝐻𝑢 (mT) 𝜇0𝑀𝑒𝑓𝑓 (T) \n0 2.20 2.31 \n3.0 2.10 2.12 \n6.6 2.30 1.95 \n12.6 2.20 1.64 \nTable 2 Room temperature  𝝁𝟎𝑴𝒆����𝒇 and  𝝁𝟎𝑯𝒖 values  for Fe 65Co35 films with different \nconcentration  of Re extracted by fitting the angle dependent cavity FMR data to Eq. (1). \n \n \nTemperature  (K) 0% Re  3.0 at% Re 6.6 at% Re 12.6 at% Re \n𝜇0𝑀𝑒𝑓𝑓  (T) 𝜇0𝑀𝑒𝑓𝑓 (T) 𝜇0𝑀𝑒𝑓𝑓 (T) 𝜇0𝑀𝑒𝑓𝑓 (T) \n300 2.29 2.16 1.99 1.61 \n200 2.31 2.16 2.04 1.67 \n150 2.33 2.24 2.06 1.70 \n100 2.36 2.25 2.07 1.72 \n50 2.36 2.27 2.08 1.74 \nTable 3 Temperature dependent  𝝁𝟎𝑴𝒆𝒇𝒇 values for Fe65Co35 films with  different concentrati on \nof Re extracted by fitting broadband out -of-plane FMR data to Eq. (4). Error margin is about  10 \nmT. \n \n \nTemperature(K)  𝛼𝑟𝑎𝑑 (×10-3) \n0% Re  3.0 at% Re 6.6 at% Re 12.6 at% Re \n300 0.218  0.482  0.438  0.154  \n200 0.222  0.494  0.450  0.160  \n150 0.216  0.499  0.454  0.162  \n100 0.225  0.502  0.456  0.219  \n50 0.221  0.505  0.458  0.166  \nTable 4 Temperature dependent r adiative damping contribution to total damping parameter for \nFe65Co35 films with  different concentration of Re  calculated  using Eq.  (5). \n  \nTemperature(K)  𝛼𝑒𝑑𝑑𝑦  (×10-3) \n0% Re  3.3 at% Re 6.6 at% Re 12.6 at% Re \n300 0.038  0.077  0.064  0.006  \n200 0.047  0.081  0.067  0.006  \n150 0.050  0.084  0.070  0.006  \n100 0.055  0.084  0.073  0.007  \n50 0.058  0.086  0.075  0.007  \nTable 5 Temperature dependent eddy current damping contribution to total damping parameter \nfor Fe 65Co35 films with  different concentration of Re calculated using Eq. ( 6).     \n \nFigure A1  RBS spectra for the Re -doped Fe 65Co35 films.  \n"    },    {        "title": "1408.3499v1.Linear_hyperbolic_equations_with_time_dependent_propagation_speed_and_strong_damping.pdf",        "content": "arXiv:1408.3499v1  [math.AP]  15 Aug 2014Linear hyperbolic equations with time-dependent\npropagation speed and strong damping\nMarina Ghisi\nUniversit` a degli Studi di Pisa\nDipartimento di Matematica\nPISA (Italy)\ne-mail:ghisi@dm.unipi.itMassimo Gobbino\nUniversit` a degli Studi di Pisa\nDipartimento di Matematica\nPISA (Italy)\ne-mail:m.gobbino@dma.unipi.itAbstract\nWe consider a second order linear equation with a time-dependent co efficientc(t) in\nfront of the “elastic” operator. For these equations it is well-know n that a higher space-\nregularity of initial data compensates a lower time-regularity of c(t).\nIn this paper we investigate the influence of a strong dissipation, na mely a friction\nterm which depends on a power of the elastic operator.\nWhat we discover is a threshold effect. When the exponent of the ela stic operator\nin the friction term is greater than 1/2, the damping prevails and the equation behaves\nas if the coefficient c(t) were constant. When the exponent is less than 1/2, the time-\nregularity of c(t) comes into play. If c(t) is regular enough, once again the damping\nprevails. On the contrary, when c(t) is not regular enough the damping might be\nineffective, and there are examples in which the dissipative equation b ehaves as the\nnon-dissipative one. As expected, the stronger is the damping, th e lower is the time-\nregularity threshold.\nWe also provide counterexamples showing the optimality of our result s.\nMathematics Subject Classification 2010 (MSC2010): 35L20, 35L80, 35L90.\nKey words: linear hyperbolic equation, dissipative hyperbolic equation, strong d amp-\ning, fractional damping, time-dependent coefficients, well-posedn ess, Gevrey spaces.1 Introduction\nLetHbe a separable real Hilbert space. For every xandyinH,|x|denotes the norm\nofx, and/a\\}⌊ra⌋ketle{tx,y/a\\}⌊ra⌋ketri}htdenotes the scalar product of xandy. LetAbe a self-adjoint linear\noperator on Hwith dense domain D(A). We assume that Ais nonnegative, namely\n/a\\}⌊ra⌋ketle{tAx,x/a\\}⌊ra⌋ketri}ht ≥0 for every x∈D(A), so that for every α≥0 the power Aαxis defined\nprovided that xlies in a suitable domain D(Aα).\nWe consider the second order linear evolution equation\nu′′(t)+2δAσu′(t)+c(t)Au(t) = 0, (1.1)\nwith initial data\nu(0) =u0, u′(0) =u1. (1.2)\nAs far as we know, this equation has been considered in the literatur e either in the\ncase where δ= 0, or in the case where δ >0 but the coefficient c(t) is constant. Let us\ngive a brief outline of the previous literature which is closely related to our results.\nThe non-dissipative case Whenδ= 0, equation (1.1) reduces to\nu′′(t)+c(t)Au(t) = 0. (1.3)\nThis is the abstract setting of a wave equation in which c(t) represents the square of\nthe propagation speed.\nIf the coefficient c(t) is Lipschitz continuous and satisfies the strict hyperbolicity\ncondition\n0<µ1≤c(t)≤µ2, (1.4)\nthen it is well-know that problem (1.3)–(1.2) is well-posed in the classic e nergy space\nD(A1/2)×H(see for example the classic reference [14]).\nIf the coefficient is not Lipschitz continuous, things are more comple x, even if (1.4)\nstill holds true. This problem was addressed by F. Colombini, E. De Gior gi and S. Spag-\nnolo in the seminal paper [6]. Their results can be summed up as follows ( we refer to\nsection 2 below for the precise functional setting and rigorous sta tements).\n(1) Problem (1.3)–(1.2) has always a unique solution, up to admitting t hat this solu-\ntion takes its values in a very large Hilbert space (ultradistributions) . This is true\nfor initial data in the energy space D(A1/2)×H, but also for less regular data,\nsuch as distributions or ultradistributions.\n(2) If initial data are regular enough, then the solution is regular as well. How much\nregularity is required depends on the time-regularity of c(t). Classic examples are\nthe following. If c(t) is just measurable, problem (1.3)–(1.2) is well-posed in the\nclass of analytic functions. If c(t) isα-H¨ older continuous for some α∈(0,1),\nproblem (1.3)–(1.2) is well-posed in the Gevrey space of order (1 −α)−1.\n1(3) If initial data are not regular enough, then the solution may exh ibit a severe\nderivative loss for all positive times. For example, for every α∈(0,1) there exist a\ncoefficientc(t) which isα-H¨ older continuous, and initial data ( u0,u1) which are in\nthe Gevrey class of order βfor everyβ >(1−α)−1, such that the corresponding\nsolution to (1.3)–(1.2) (which exists in the weak sense of point (1)) is not even a\ndistribution for every t>0.\nIn the sequel we call (DGCS)-phenomenon the instantaneous loss of regularity de-\nscribed in point (3) above.\nThe dissipative case with constant coefficients Ifδ >0 andc(t) is a constant function\n(equal to 1 without loss of generality), equation (1.1) reduces to\nu′′(t)+2δAσu′(t)+Au(t) = 0. (1.5)\nMathematical models with damping terms of this form were proposed in [1], and\nthen rigorously analyzed by many authors from different points of v iew. The first\npapers [2, 3, 4], and the more recent [10], are devoted to analyticity properties of the\nsemigroup associated to (1.5). The classic assumptions in these pap ers are that the\noperatorAis strictly positive, σ∈[0,1], and the phase space is D(A1/2)×H. On a\ndifferent side, the community working on dispersive equations consid ered equation (1.5)\nintheconcretecasewhere σ∈[0,1]andAu=−∆uinRnorspecialclassesofunbounded\ndomains. They proved energy decay and dispersive estimates, but exploiting in an\nessential way the spectral properties of the Laplacian in those do mains. The interested\nreader is referred to [11, 12, 13, 19] and to the references quot ed therein.\nFinally, equation (1.5) was considered in [9] in full generality, namely fo r every\nσ≥0 and every nonnegative self-adjoint operator A. Two different regimes appeared.\nIn the subcritical regime σ∈[0,1/2], problem (1.5)–(1.2) is well-posed in the classic\nenergy space D(A1/2)×Hor more generally in D(Aα+1/2)×D(Aα) withα≥0. In the\nsupercritical regime σ≥1/2, problem (1.5)–(1.2) is well-posed in D(Aα)×D(Aβ) if and\nonly if\n1−σ≤α−β≤σ. (1.6)\nThis means that in the supercritical regime different choices of the p hase space are\npossible, even with α−β/\\e}atio\\slash= 1/2.\nThe dissipative case with time-dependent coefficients As far as we know, the case of a\ndissipative equation with a time-dependent propagation speed had n ot been considered\nyet. The main question we address in this paper is the extent to which the dissipative\nterm added in (1.1) prevents the (DGCS)-phenomenon of (1.3) fro m happening. We\ndiscover a composite picture, depending on σ.\n•In the subcritical regime σ∈[0,1/2], if the strict hyperbolicity assumption (1.4)\nis satisfied, well-posedness results do depend on the time-regularit y ofc(t) (see\nTheorem 3.2). Classic examples are the following.\n2–Ifc(t) isα-H¨ older continuous for some exponent α >1−2σ, then the dis-\nsipation prevails, and problem (1.1)–(1.2) is well-posed in the classic en ergy\nspaceD(A1/2)×Hor more generally in D(Aβ+1/2)×D(Aβ) withβ≥0.\n–Ifc(t) is no more than α-H¨ older continuous for some exponent α <1−2σ,\nthenthedissipationcanbeneglected, sothat(1.1)behavesexact lyasthenon-\ndissipative equation (1.3). This means well-posedness in the Gevrey s pace of\norder (1−α)−1and the possibility to produce the (DGCS)-phenomenon for\nless regular data (see Theorem 3.10).\n–The case with α= 1−2σis critical and also the size of the H¨ older constant\nofc(t) compared with δcomes into play.\n•In the supercritical regime σ >1/2 the dissipation prevails in an overwhelming\nway. In Theorem 3.1 we prove that, if c(t) is just measurable and satisfies just the\ndegenerate hyperbolicity condition\n0≤c(t)≤µ2, (1.7)\nthen (1.1) behaves as (1.5). This means that problem (1.1)–(1.2) is w ell-posed in\nD(Aα)×D(Aβ) if and only if (1.6) is satisfied, the same result obtained in [9] in\nthe case of a constant coefficient.\nThe second issue we address in this paper is the further space-reg ularity of solutions\nfor positive times, since a strong dissipation is expected to have a re gularizing effect\nsimilar to parabolic equations. This turns out to be true provided tha t the assumptions\nof our well-posedness results are satisfied, and in addition σ∈(0,1). Indeed, we prove\nthat in this regime u(t) lies in the Gevrey space of order (2min {σ,1−σ})−1for every\nt>0. We refer to Theorem 3.8 and Theorem 3.9 for the details. This effec t had already\nbeen observed in [15] in the dispersive case.\nWe point out that the regularizing effect is maximum when σ= 1/2 (the only case in\nwhich solutions become analytic with respect to space variables) and disappears when\nσ≥1, meaning that a stronger overdamping prevents smoothing.\nOverview of the technique The spectral theory reduces the problem to an analysis of\nthe family of ordinary differential equations\nu′′\nλ(t)+2δλ2σu′\nλ(t)+λ2c(t)uλ(t) = 0. (1.8)\nWhenδ= 0, a coefficient c(t) which oscillates with a suitable period can produce\na resonance effect so that (1.8) admits a solution whose oscillations h ave an amplitude\nwhich grows exponentially with time. This is the primordial origin of the ( DGCS)-\nphenomenon for non-dissipative equations. When δ >0, the damping term causes\nan exponential decay of the amplitude of oscillations. The competition between the\n3exponential energy growth due to resonance and the exponent ial energy decay due to\ndissipation originates the threshold effect we observed.\nWhenc(t) is constant, equation (1.8) can be explicitly integrated, and the ex plicit\nformulae for solutions led to the sharp results of [9]. Here we need th e same sharp\nestimates, but without relying on explicit solutions. To this end, we int roduce suitable\nenergy estimates.\nIn the supercritical regime σ≥1/2 we exploit the following σ-adapted “Kovaleskyan\nenergy”\nE(t) :=|u′\nλ(t)+δλ2σuλ(t)|2+δ2λ4σ|uλ(t)|2. (1.9)\nIn the subcritical regime σ≤1/2 we exploit the so-called “approximated hyperbolic\nenergies”\nEε(t) :=|u′\nλ(t)+δλ2σuλ(t)|2+δ2λ4σ|uλ(t)|2+λ2cε(t)|uλ(t)|2,(1.10)\nobtained by adding to (1.9) an “hyperbolic term” depending on a suita ble smooth ap-\nproximation cε(t) ofc(t), which in turn is chosen in a λ-dependent way. Terms of this\ntype are the key tool introduced in [6] for the non-dissipative equa tion.\nFuture extensions We hope that this paper could represent a first step in the theory\nof dissipative hyperbolic equations with variable coefficients, both line ar and nonlinear.\nNext steps could be considering a coefficient c(x,t) depending both on time and space\nvariables, and finally quasilinear equations. This could lead to improve t he classic\nresults by K. Nishihara [16, 17] for Kirchhoff equations, whose linear ization has a time-\ndependent coefficient, and finally to consider more general local no nlinearities, in which\ncase the linearization involves a coefficient c(x,t) depending on both variables.\nInadifferent direction, thesubcritical case σ∈[0,1/2]withdegeneratehyperbolicity\nassumptions remains open and could be the subject of further res earch, in the same way\nas [7] was the follow-up of [6].\nOn the other side, we hope that our counterexamples could finally dis pel the dif-\nfuse misconception according to which dissipative hyperbolic equatio ns are more stable,\nand hence definitely easier to handle. Now we know that a friction ter m below a suit-\nable threshold is substantially ineffective, opening the door to patho logies such as the\n(DGCS)-phenomenon, exactly as in the non-dissipative case.\nStructure of the paper This paper is organized as follows. In section 2 we introduce\nthe functional setting and we recall the classic existence results f rom [6]. In section 3 we\nstate our main results. In section 4 we provide a heuristic descriptio n of the competition\nbetween resonance and decay. In section 5 we prove our existenc e and regularity results.\nIn section 6 we present our examples of (DGCS)-phenomenon.\n42 Notation and previous results\nFunctional spaces LetHbe a separable Hilbert space. Let us assume that Hadmits\na countable complete orthonormal system {ek}k∈Nmade by eigenvectors of A. We\ndenote the corresponding eigenvalues by λ2\nk(with the agreement that λk≥0), so that\nAek=λ2\nkekfor everyk∈N. In this case every u∈Hcan be written in a unique way\nin the form u=/summationtext∞\nk=0ukek, whereuk=/a\\}⌊ra⌋ketle{tu,ek/a\\}⌊ra⌋ketri}htare the Fourier components of u. In\nother words, the Hilbert space Hcan be identified with the set of sequences {uk}of real\nnumbers such that/summationtext∞\nk=0u2\nk<+∞.\nWe stress that this is just a simplifying assumption, with substantially no loss of\ngenerality. Indeed, according to the spectral theorem in its gene ral form (see for ex-\nample Theorem VIII.4 in [18]), one can always identify HwithL2(M,µ) for a suitable\nmeasure space ( M,µ), in such a way that under this identification the operator Aacts\nas a multiplication operator by some measurable function λ2(ξ). All definitions and\nstatements in the sequel, with the exception of the counterexamp les of Theorem 3.10,\ncan be easily extended to the general setting just by replacing the sequence {λ2\nk}with\nthe function λ2(ξ), and the sequence {uk}of Fourier components of uwith the element\n/hatwideu(ξ) ofL2(M,µ) corresponding to uunder the identification of HwithL2(M,µ).\nThe usual functional spaces can be characterized in terms of Fou rier components as\nfollows.\nDefinition 2.1. Letube a sequence {uk}of real numbers.\n•Sobolev spaces . For every α≥0 it turns out that u∈D(Aα) if\n/⌊ard⌊lu/⌊ard⌊l2\nD(Aα):=∞/summationdisplay\nk=0(1+λk)4αu2\nk<+∞. (2.1)\n•Distributions . We say that u∈D(A−α) for someα≥0 if\n/⌊ard⌊lu/⌊ard⌊l2\nD(A−α):=∞/summationdisplay\nk=0(1+λk)−4αu2\nk<+∞. (2.2)\n•Generalized Gevrey spaces . Letϕ: [0,+∞)→[0,+∞) be any function, let r≥0,\nand letα∈R. We say that u∈ Gϕ,r,α(A) if\n/⌊ard⌊lu/⌊ard⌊l2\nϕ,r,α:=∞/summationdisplay\nk=0(1+λk)4αu2\nkexp/parenleftbig\n2rϕ(λk)/parenrightbig\n<+∞. (2.3)\n•Generalized Gevrey ultradistributions . Letψ: [0,+∞)→[0,+∞)beanyfunction,\nletR≥0, and letα∈R. We say that u∈ G−ψ,R,α(A) if\n/⌊ard⌊lu/⌊ard⌊l2\n−ψ,R,α:=∞/summationdisplay\nk=0(1+λk)4αu2\nkexp/parenleftbig\n−2Rψ(λk)/parenrightbig\n<+∞. (2.4)\n5Remark 2.2. Ifϕ1(x) =ϕ2(x) for every x >0, thenGϕ1,r,α(A) =Gϕ2,r,α(A) for every\nadmissible value of randα. For this reason, with a little abuse of notation, we consider\nthe spaces Gϕ,r,α(A) even when ϕ(x) is defined only for x >0. The same comment\napplies also to the spaces G−ψ,R,α(A).\nThe quantities defined in (2.1) through (2.4) are actually norms which induce a\nHilbert space structure on D(Aα),Gϕ,r,α(A),G−ψ,R,α(A), respectively. The standard\ninclusions\nGϕ,r,α(A)⊆D(Aα)⊆H⊆D(A−α)⊆ G−ψ,R,−α(A)\nhold true for every α≥0 and every admissible choice of ϕ,ψ,r,R. All inclusions\nare strict if α,randRare positive, and the sequences {λk},{ϕ(λk)}, and{ψ(λk)}are\nunbounded.\nWe observe that Gϕ,r,α(A) is actually a so-called scale of Hilbert spaces with respect\nto theparameter r, withlarger values of rcorresponding to smaller spaces. Analogously,\nG−ψ,R,α(A) is a scale of Hilbert spaces with respect to the parameter R, but with larger\nvalues ofRcorresponding to larger spaces.\nRemark 2.3. Let us consider the concrete case where I⊆Ris an open interval,\nH=L2(I), andAu=−uxx, with periodic boundary conditions. For every α≥0, the\nspaceD(Aα) is actually the usual Sobolev space H2α(I), andD(A−α) is the usual space\nof distributions of order 2 α.\nWhenϕ(x) :=x1/sfor somes>0, elements of Gϕ,r,0(A) withr>0 are usually called\nGevrey functions of order s, the cases= 1 corresponding to analytic functions. When\nψ(x) :=x1/sforsomes>0, elements of G−ψ,R,0(A)withR>0areusually called Gevrey\nultradistributions of order s, the cases= 1 corresponding to analytic functionals. In\nthis case the parameter αis substantially irrelevant because the exponential term is\ndominant both in (2.3) and in (2.4).\nFor the sake of consistency, with a little abuse of notation we use th e same terms\n(Gevrey functions, Gevrey ultradistributions, analytic functions and analytic function-\nals) in order to denote the same spaces also in the general abstrac t framework. To be\nmore precise, we should always add “with respect to the operator A”, or even better\n“with respect to the operator A1/2”.\nContinuity moduli Throughout this paper we call continuity modulus any continuous\nfunctionω: [0,+∞)→[0,+∞) such that ω(0) = 0,ω(x)>0 for every x >0, and\nmoreover\nx→ω(x) is a nondecreasing function , (2.5)\nx→x\nω(x)is a nondecreasing function. (2.6)\nA function c: [0,+∞)→Ris said to be ω-continuous if\n|c(a)−c(b)| ≤ω(|a−b|)∀a≥0,∀b≥0. (2.7)\n6More generally, a function c:X→R(withX⊆R) is said to be ω-continuous if it\nsatisfies the same inequality for every aandbinX.\nPrevious results We are now ready to recall the classic results concerning existence ,\nuniqueness, and regularity for solutions to problem (1.1)–(1.2). We state them using our\nnotations which allow general continuity moduli and general spaces of Gevrey functions\nor ultradistributions.\nProofs are a straightforward application of the approximated ene rgy estimates in-\ntroduced in [6]. In that paper only the case δ= 0 is considered, but when δ≥0 all new\nterms have the “right sign” in those estimates.\nThe first result concerns existence and uniqueness in huge spaces such as analytic\nfunctionals, with minimal assumptions on c(t).\nTheorem A (see [6, Theorem 1]) .Let us consider problem (1.1)–(1.2) under the fol-\nlowing assumptions:\n•Ais a self-adjoint nonnegative operator on a separable Hilbe rt spaceH,\n•c∈L1((0,T))for everyT >0(without sign conditions),\n•σ≥0andδ≥0are two real numbers,\n•initial conditions satisfy\n(u0,u1)∈ G−ψ,R0,1/2(A)×G−ψ,R0,0(A)\nfor someR0>0and someψ: (0,+∞)→(0,+∞)such that\nlimsup\nx→+∞x\nψ(x)<+∞.\nThen there exists a nondecreasing function R: [0,+∞)→[0,+∞), withR(0) =R0,\nsuch that problem (1.1)–(1.2) admits a unique solution\nu∈C0/parenleftbig\n[0,+∞);G−ψ,R(t),1/2(A)/parenrightbig\n∩C1/parenleftbig\n[0,+∞);G−ψ,R(t),0(A)/parenrightbig\n.(2.8)\nCondition (2.8), with the range space increasing with time, simply mean s that\nu∈C0/parenleftbig\n[0,τ];G−ψ,R(τ),1/2(A)/parenrightbig\n∩C1/parenleftbig\n[0,τ];G−ψ,R(τ),0(A)/parenrightbig\n∀τ≥0.\nThis amounts to say that scales of Hilbert spaces, rather than fixe d Hilbert spaces,\nare the natural setting for this problem.\nInthesecondresultweassumestricthyperbolicityand ω-continuityofthecoefficient,\nand we obtain well-posedness in a suitable class of Gevrey ultradistrib utions.\n7Theorem B (see [6, Theorem 3]) .Let us consider problem (1.1)–(1.2) under the fol-\nlowing assumptions:\n•Ais a self-adjoint nonnegative operator on a separable Hilbe rt spaceH,\n•the coefficient c: [0,+∞)→Rsatisfies the strict hyperbolicity assumption (1.4)\nand theω-continuity assumption (2.7) for some continuity modulus ω(x),\n•σ≥0andδ≥0are two real numbers,\n•initial conditions satisfy\n(u0,u1)∈ G−ψ,R0,1/2(A)×G−ψ,R0,0(A)\nfor someR0>0and some function ψ: (0,+∞)→(0,+∞)such that\nlimsup\nx→+∞x\nψ(x)ω/parenleftbigg1\nx/parenrightbigg\n<+∞. (2.9)\nLetube the unique solution to the problem provided by Theorem A.\nThen there exists R>0such that\nu∈C0/parenleftbig\n[0,+∞),G−ψ,R0+Rt,1/2(A)/parenrightbig\n∩C1([0,+∞),G−ψ,R0+Rt,0(A)).\nThe third result we recall concerns existence of regular solutions. The assumptions\nonc(t) are the same as in Theorem B, but initial data are significantly more r egular\n(Gevrey spaces instead of Gevrey ultradistributions).\nTheorem C (see [6, Theorem 2]) .Let us consider problem (1.1)–(1.2) under the fol-\nlowing assumptions:\n•Ais a self-adjoint nonnegative operator on a separable Hilbe rt spaceH,\n•the coefficient c: [0,+∞)→Rsatisfies the strict hyperbolicity assumption (1.4)\nand theω-continuity assumption (2.7) for some continuity modulus ω(x),\n•σ≥0andδ≥0are two real numbers,\n•initial conditions satisfy\n(u0,u1)∈ Gϕ,r0,1/2(A)×Gϕ,r0,0(A)\nfor somer0>0and some function ϕ: (0,+∞)→(0,+∞)such that\nlimsup\nx→+∞x\nϕ(x)ω/parenleftbigg1\nx/parenrightbigg\n<+∞. (2.10)\n8Letube the unique solution to the problem provided by Theorem A.\nThen there exist T >0andr>0such thatrT <r 0and\nu∈C0/parenleftbig\n[0,T],Gϕ,r0−rt,1/2(A)/parenrightbig\n∩C1([0,T],Gϕ,r0−rt,0(A)). (2.11)\nRemark 2.4. The key assumptions of Theorem B and Theorem C are (2.9) and (2.10 ),\nrespectively, representing the exact compensation between spa ce-regularity of initial\ndataandtime-regularityofthecoefficient c(t)requiredinordertoobtainwell-posedness.\nThese conditions do not appear explicitly in [6], where they are replace d by suitable\nspecific choices of ω,ϕ,ψ, which of course satisfy the same relations. To our knowledge,\nthose conditions were stated for the first time in [8], thus unifying se veral papers that in\nthelast 30 years hadbeendevoted tospecial cases (see forexam ple [5] andthe references\nquoted therein).\nRemark 2.5. The standard example of application of Theorem B and Theorem C is\nthe following. Let us assume that c(t) isα-H¨ older continuous for some α∈(0,1),\nnamelyω(x) =Mxαfor a suitable constant M. Then (2.9) and (2.10) hold true with\nψ(x) =ϕ(x) :=x1−α. This leads to well-posedness both in the large space of Gevrey\nultradistributions of order (1 −α)−1, and in the small space of Gevrey functions of the\nsame order.\nRemark 2.6. The choice of ultradistributions in Theorem B is not motivated by the\nsearchforgenerality, butitisinsomesense theonlypossibleonebec auseofthe(DGCS)-\nphenomenon exhibited in [6], at least in the non-dissipative case. When δ= 0, if initial\ndata are taken in Sobolev spaces or in any space larger than the Gev rey spaces of\nTheorem C, then it may happen that for all positive times the solution lies in the space\nof ultradistributions specified in Theorem B, and nothing more. In ot her words, for\nδ= 0 there is no well-posedness result in between the Gevrey spaces o f Theorem C\nand the Gevrey ultradistributions of Theorem B, and conditions (2.9 ) and (2.10) are\noptimal.\nThe aim of this paper is to provide an optimal picture for the case δ >0.\n3 Main results\nIn this section we state our main regularity results for solutions to ( 1.1)–(1.2). To this\nend, we need some further notation. Given any ν≥0, we write Has an orthogonal\ndirect sum\nH:=Hν,−⊕Hν,+, (3.1)\nwhereHν,−is the closure of the subspace generated by all eigenvectors of Arelative to\neigenvalues λk<ν, andHν,+is the closure of the subspace generated by all eigenvectors\nofArelative to eigenvalues λk≥ν. For every vector u∈H, we writeuν,−anduν,+\nto denote its components with respect to the decomposition (3.1). We point out that\n9Hν,−andHν,+areA-invariant subspaces of H, and thatAis a bounded operator when\nrestricted to Hν,−, and a coercive operator when restricted to Hν,+ifν >0.\nIn the following statements we provide separate estimates for low- frequency compo-\nnentsuν,−(t) and high-frequency components uν,+(t) of solutions to (1.1). This is due to\nthe fact that the energy of uν,−(t) can be unbounded as t→+∞, while in many cases\nwe are able to prove that the energy of uν,+(t) is bounded in time.\n3.1 Existence results in Sobolev spaces\nThe first result concerns the supercritical regime σ≥1/2, in which case the dissipation\nalways dominates the time-dependent coefficient.\nTheorem 3.1 (Supercritical dissipation) .Let us consider problem (1.1)–(1.2) under\nthe following assumptions:\n•Ais a self-adjoint nonnegative operator on a separable Hilbe rt spaceH,\n•the coefficient c: [0,+∞)→Ris measurable and satisfies the degenerate hyper-\nbolicity assumption (1.7),\n•σandδare two positive real numbers such that either σ >1/2, orσ= 1/2and\n4δ2≥µ2,\n•(u0,u1)∈D(Aα)×D(Aβ)for some real numbers αandβsatisfying (1.6).\nLetube the unique solution to the problem provided by Theorem A.\nThenuactually satisfies\n(u,u′)∈C0/parenleftbig\n[0,+∞),D(Aα)×D(Aβ)/parenrightbig\n. (3.2)\nMoreover, for every ν≥1such that 4δ2ν4σ−2≥µ2, it turns out that\n|Aβu′\nν,+(t)|2+|Aαuν,+(t)|2≤/parenleftbigg\n2+2\nδ2+µ2\n2\nδ4/parenrightbigg\n|Aβu1,ν,+|2+3/parenleftbigg\n1+µ2\n2\n2δ2/parenrightbigg\n|Aαu0,ν,+|2(3.3)\nfor everyt≥0.\nOur second result concerns the subcritical regime σ∈[0,1/2], in which case the\ntime-regularity of c(t) competes with the exponent σ.\nTheorem 3.2 (Subcritical dissipation) .Let us consider problem (1.1)–(1.2) under the\nfollowing assumptions:\n•Ais a self-adjoint nonnegative operator on a separable Hilbe rt spaceH,\n•the coefficient c: [0,+∞)→Rsatisfies the strict hyperbolicity assumption (1.4)\nand theω-continuity assumption (2.7) for some continuity modulus ω(x),\n10•σ∈[0,1/2]andδ>0are two real numbers such that\n4δ2µ1>Λ2\n∞+2δΛ∞, (3.4)\nwhere we set\nΛ∞:= limsup\nε→0+ω(ε)\nε1−2σ, (3.5)\n•(u0,u1)∈D(A1/2)×H.\nLetube the unique solution to the problem provided by Theorem A.\nThenuactually satisfies\nu∈C0/parenleftbig\n[0,+∞),D(A1/2)/parenrightbig\n∩C1([0,+∞),H).\nMoreover, for every ν≥1such that\n4δ2µ1≥/bracketleftbigg\nλ1−2σω/parenleftbigg1\nλ/parenrightbigg/bracketrightbigg2\n+2δ/bracketleftbigg\nλ1−2σω/parenleftbigg1\nλ/parenrightbigg/bracketrightbigg\n(3.6)\nfor everyλ≥ν, it turns out that\n|u′\nν,+(t)|2+2µ1|A1/2uν,+(t)|2≤4|u1,ν,+|2+2(3δ2+µ2)|A1/2u0,ν,+|2(3.7)\nfor everyt≥0.\nLet us make a few comments on the first two statements.\nRemark 3.3. Inbothresultsweprovedthatasuitablehigh-frequencycompone nt ofthe\nsolution can be uniformly bounded in terms of initial data. Low-frequ ency components\nmight in general diverge as t→+∞. Nevertheless, they can always be estimated as\nfollows.\nLet us just assume that c∈L1((0,T)) for every T >0. Then for every ν≥0 the\ncomponent uν,−(t) satisfies\n|u′\nν,−(t)|2+|A1/2uν,−(t)|2≤/parenleftbig\n|u1,ν,−|2+|A1/2u0,ν,−|2/parenrightbig\nexp/parenleftbigg\nνt+ν/integraldisplayt\n0|c(s)|ds/parenrightbigg\n(3.8)\nfor everyt≥0. Indeed, let F(t) denote the left-hand side of (3.8). Then\nF′(t) =−4δ|Aσ/2u′\nν,−(t)|2+2(1−c(t))/a\\}⌊ra⌋ketle{tu′\nν,−(t),Auν,−(t)/a\\}⌊ra⌋ketri}ht\n≤2(1+|c(t)|)·|u′\nν,−(t)|·ν|A1/2uν,−(t)|\n≤ν(1+|c(t)|)F(t)\nfor almost every t≥0, so that (3.8) follows by integrating this differential inequality.\n11Remark 3.4. The phase spaces involved in Theorem 3.1 and Theorem 3.2 are exactly\nthe same which are known to be optimal when c(t) is constant (see [9]). In particular,\nthe only possible choice in the subcritical regime is the classic energy s paceD(A1/2)×H,\nor more generally D(Aα+1/2)×D(Aα). This “gap1/2” between the powers of Ainvolved\nin the phase space is typical of hyperbolic problems, and it is the same which appears\nin the classic results of section 2.\nOn the contrary, in the supercritical regime there is an interval of possible gaps,\ndescribed by (1.6). This interval is always centered in 1/2, but also d ifferent values are\nallowed, including negative ones when σ>1.\nRemark 3.5. The classic example of application of Theorem 3.2 is the following. Let\nus assume that c(t) isα-H¨ older continuous for some α∈(0,1), namely ω(x) =Mxαfor\nsome constant M. Then problem (1.1)–(1.2) is well-posed in the energy space provided\nthat either α>1−2σ, orα= 1−2σandMis small enough. Indeed, for α>1−2σwe\nget Λ∞= 0, and hence (3.4) is automatically satisfied. For α= 1−2σwe get Λ ∞=M,\nso that (3.4) is satisfied provided that Mis small enough.\nIn all other cases, namely when either α <1−2σ, orα= 1−2σandMis large\nenough, only Theorem B applies to initial data in Sobolev spaces, prov iding global\nexistence just in the sense of Gevrey ultradistributions of order ( 1−α)−1.\nRemark 3.6. Let us examine the limit case σ= 0, which falls in the subcritical regime.\nWhenσ= 0, assumption (3.4) is satisfied if and only if c(t) is Lipschitz continuous\nand its Lipschitz constant is small enough. On the other hand, in the Lipschitz case it\nis a classic result that problem (1.1)–(1.2) is well-posed in the energy s pace, regardless\nof the Lipschitz constant. Therefore, the result stated in Theor em 3.2 is non-optimal\nwhenσ= 0 andc(t) is Lipschitz continuous.\nA simple refinement of our argument would lead to the full result also in this case,\nbut unfortunately it would be useless in all other limit cases in which c(t) isα-H¨ older\ncontinuous with α= 1−2σandσ∈(0,1/2]. We refer to section 4 for further details.\nRemark 3.7. Let us examine the limit case σ= 1/2, which falls both in the subcritical\nand in the supercritical regime, so that the conclusions of Theorem 3.1 and Theorem 3.2\ncoexist. Both of them provide well-posedness in the energy space, but with different\nassumptions.\nTheorem 3.1 needs less assumptions on c(t), which is only required to be measurable\nand to satisfy the degenerate hyperbolicity assumption (1.7), but it requires δto be\nlarge enough so that 4 δ2≥µ2.\nOn the contrary, Theorem 3.2 needs less assumptions on δ, which is only required to\nbe positive, but it requires c(t) to be continuous and to satisfy the strict hyperbolicity\nassumption (1.4). Indeed, inequality (3.4) is automatically satisfied in the caseσ= 1/2\nbecause Λ ∞= 0.\nThe existence of two different sets of assumptions leading to the sa me conclusion\nsuggests the existence of a unifying statement, which could proba bly deserve further\ninvestigation.\n123.2 Gevrey regularity for positive times\nA strong dissipation in the range σ∈(0,1) has a regularizing effect on initial data,\nprovided that the solution exists in Sobolev spaces. In the following t wo statements we\nquantify this effect in terms of scales of Gevrey spaces.\nBoth results can be summed up by saying that the solution lies, for po sitive times,\nin Gevrey spaces of order (2min {σ,1−σ})−1. It is not difficult to show that this order\nis optimal, even in the case where c(t) is constant.\nTheorem 3.8 (Supercritical dissipation) .Let us consider problem (1.1)–(1.2) under\nthe same assumptions of Theorem 3.1, and let ube the unique solution to the problem\nprovided by Theorem A.\nLet us assume in addition that either σ∈(1/2,1), orσ= 1/2and4δ2>µ2. Let us\nsetϕ(x) :=x2(1−σ), and\nC(t) :=/integraldisplayt\n0c(s)ds. (3.9)\nThen there exists r>0such that\n(u,u′)∈C0/parenleftbig\n(0,+∞),Gϕ,α,rC(t)(A)×Gϕ,β,rC(t)(A)/parenrightbig\n, (3.10)\nand there exist ν≥1andK >0such that\n/⌊ard⌊lu′\nν,+(t)/⌊ard⌊l2\nϕ,β,rC(t)+/⌊ard⌊luν,+(t)/⌊ard⌊l2\nϕ,α,rC(t)≤K/parenleftbig\n|Aβu1,ν,+|2+|Aαu0,ν,+|2/parenrightbig\n(3.11)\nfor everyt>0. The constants r,ν, andKdepend only on δ,µ2, andσ.\nOf course, (3.10) and (3.11) are nontrivial only if C(t)>0, which is equivalent to\nsaying that the coefficient c(t) is not identically 0 in [0 ,t]. On the other hand, this weak\nform of hyperbolicity is necessary, since no regularizing effect on u(t) can be expected\nas long asc(t) vanishes.\nTheorem 3.9 (Subcritical dissipation) .Let us consider problem (1.1)–(1.2) under the\nsame assumptions of Theorem 3.2, and let ube the unique solution to the problem\nprovided by Theorem A.\nLet us assume in addition that σ∈(0,1/2](instead of σ∈[0,1/2]), and let us set\nϕ(x) :=x2σ.\nThen there exists r>0such that\nu∈C0/parenleftbig\n(0,+∞),Gϕ,1/2,rt(A)/parenrightbig\n∩C1((0,+∞),Gϕ,0,rt(A)),\nand there exist ν≥1andK >0such that\n/⌊ard⌊lu′\nν,+(t)/⌊ard⌊l2\nϕ,0,rt+/⌊ard⌊luν,+(t)/⌊ard⌊l2\nϕ,1/2,rt≤K/parenleftbig\n|u1,ν,+|2+|A1/2u0,ν,+|2/parenrightbig\n(3.12)\nfor everyt>0. The constants r,ν, andKdepend only on δ,µ1,µ2,σandω.\nTheestimateswhichprovideGevreyregularityofhigh-frequencyc omponentsprovide\nalso the decay of the same components as t→+∞. We refer to Lemma 5.1 and\nLemma 5.2 for further details.\n133.3 Counterexamples\nThe following result shows that even strongly dissipative hyperbolic e quations can ex-\nhibit the (DGCS)-phenomenon, provided that we are in the subcritic al regime.\nTheorem 3.10 ((DGCS)-phenomenon) .LetAbe a linear operator on a Hilbert space\nH. Let us assume that there exists a countable (not necessaril y complete) orthonormal\nsystem{ek}inH, and an unbounded sequence {λk}of positive real numbers such that\nAek=λ2\nkekfor everyk∈N. Letσ∈[0,1/2)andδ>0be real numbers.\nLetω: [0,+∞)→[0,+∞)be a continuity modulus such that\nlim\nε→0+ω(ε)\nε1−2σ= +∞. (3.13)\nLetϕ: (0,+∞)→(0,+∞)andψ: (0,+∞)→(0,+∞)be two functions such that\nlim\nx→+∞x\nϕ(x)ω/parenleftbigg1\nx/parenrightbigg\n= lim\nx→+∞x\nψ(x)ω/parenleftbigg1\nx/parenrightbigg\n= +∞. (3.14)\nThen there exist a function c:R→Rsuch that\n1\n2≤c(t)≤3\n2∀t∈R, (3.15)\n|c(t)−c(s)| ≤ω(|t−s|)∀(t,s)∈R2, (3.16)\nand a solution u(t)to equation (1.1) such that\n(u(0),u′(0))∈ Gϕ,r,1/2(A)×Gϕ,r,0(A)∀r>0, (3.17)\n(u(t),u′(t))/\\e}atio\\slash∈ G−ψ,R,1/2(A)×G−ψ,R,0(A)∀R>0,∀t>0.(3.18)\nRemark 3.11. Due to (3.15), (3.16), and (3.17), the function u(t) provided by Theo-\nrem 3.10 is a solution to (1.1) in the sense of Theorem A with ψ(x) :=x, or even better\nin the sense of Theorem B with ψ(x) :=xω(1/x).\nRemark 3.12. Assumption (3.13) is equivalent to saying that Λ ∞defined by (3.5) is\nequal to + ∞, so that (3.4) can not be satisfied. In other words, Theorem 3.2 giv es\nwell-posedness in the energy space if Λ ∞is 0 or small, while Theorem 3.10 provides\nthe (DGCS)-phenomenon if Λ ∞= +∞. The case where Λ ∞is finite but large remains\nopen. We suspect that the (DGCS)-phenomenon is still possible, bu t our construction\ndoes not work. We comment on this issue in the first part of section 6 .\nFinally, Theorem 3.10 shows that assumptions (2.9) and (2.10) of The orems B and C\nare optimal also in the subcritical dissipative case with Λ ∞= +∞. If initial data are in\nthe Gevrey space with ϕ(x) =xω(1/x), solutions remain in the same space. If initial are\nin a Gevrey space corresponding to some ϕ(x)≪xω(1/x), then it may happen that for\npositive times the solution lies in the space of ultradistributions with ψ(x) :=xω(1/x),\nbut not in the space of ultradistributions corresponding to any give nψ(x)≪xω(1/x).\n144 Heuristics\nThefollowingpicturessummarizeroughlytheresultsofthispaper. I nthehorizontalaxis\nwe represent the time-regularity of c(t). With some abuse of notation, values α∈(0,1)\nmean that c(t) isα-H¨ older continuous, α= 1 means that it is Lipschitz continuous,\nα >1 means even more regular. In the vertical axis we represent the s pace-regularity\nof initial data, where the value sstands for the Gevrey space of order s(so that higher\nvalues ofsmean lower regularity). The curve is s= (1−α)−1.\nα 1s\n1\nδ= 0Potential (DGCS)-phenomenon Well-posedness\nα 1−2σ/Bullets\n1\nδ >0,0<σ<1/2α 1s\n1\nδ >0, σ>1/2\nForδ= 0 we have the situation described in Remark 2.5 and Remark 2.6, name ly\nwell-posedness provided that either c(t) is Lipschitz continuous or c(t) isα-H¨ older con-\ntinuous and initial data are in Gevrey spaces of order less than or eq ual to (1 −α)−1,\nand (DGCS)-phenomenon otherwise. The same picture applies if δ >0 andσ= 0.\nWhenδ >0 and 0< σ <1/2, the full strip with α >1−2σfalls in the well-\nposedness region, as stated in Theorem 3.2. The region with α <1−2σis divided as\nin the non-dissipative case. Indeed, Theorem C still provides well-po sedness below the\ncurve and on the curve, while Theorem 3.10 provides the (DGCS)-ph enomenon above\nthe curve. What happens on the vertical half-line which separates the two regions is\nless clear (it is the region where Λ ∞is positive and finite, see Remark 3.12).\nFinally, when δ >0 andσ>1/2 well-posedness dominates because of Theorem 3.1,\neven in the degenerate hyperbolic case.\nNow we present a rough justification of this threshold effect. As alr eady observed,\nexistence results for problem (1.1)–(1.2) are related to estimates for solutions to the\nfamily of ordinary differential equations (1.8).\nLet us consider the simplest energy function E(t) :=|u′\nλ(t)|2+λ2|uλ(t)|2, whose\ntime-derivative is\nE′(t) =−4δλ2σ|u′\nλ(t)|2+2λ2(1−c(t))uλ(t)u′\nλ(t)\n≤ −4δλ2σ|u′\nλ(t)|2+λ(1+|c(t)|)E(t). (4.1)\n15Sinceδ≥0, a simple integration gives that\nE(t)≤ E(0)exp/parenleftbigg\nλt+λ/integraldisplayt\n0|c(s)|ds/parenrightbigg\n, (4.2)\nwhich is almost enough to establish Theorem A.\nIfinaddition c(t)isω-continuousandsatisfiesthestricthyperbolicitycondition(1.4),\nthen (4.2) can be improved to\nE(t)≤M1E(0)exp(M2λω(1/λ)t) (4.3)\nfor suitable constants M1andM2. Estimates of this kind are the key point in the proof\nof both Theorem B and Theorem C. Moreover, the (DGCS)-phenom enon is equivalent\nto saying that the term λω(1/λ) in (4.3) is optimal.\nLet us assume now that δ >0. Ifσ >1/2, orσ= 1/2 andδis large enough,\nthen it is reasonable to expect that the first (negative) term in the right-hand side of\n(4.1) dominates the second one, and hence E(t)≤ E(0), which is enough to establish\nwell-posedness in Sobolev spaces. Theorem 3.1 confirms this intuition .\nIfσ≤1/2 andc(t) is constant, then (1.8) can be explicitly integrated, obtaining\nthat\nE(t)≤ E(0)exp/parenleftbig\n−2δλ2σt/parenrightbig\n. (4.4)\nIfc(t) isω-continuous and satisfies the strict hyperbolicity assumption (1.4) , then\nwe expect a superposition of the effects of the coefficient, repres ented by (4.3), and the\neffects of the damping, represented by (4.4). We end up with\nE(t)≤M1E(0)exp/parenleftbig\n[M2λω(1/λ)−2δλ2σ]t/parenrightbig\n. (4.5)\nTherefore, it is reasonable to expect that E(t) satisfies an estimate independent of\nλ, which guarantees well-posedness in Sobolev spaces, provided tha tλω(1/λ)≪λ2σ, or\nλω(1/λ)∼λ2σandδis large enough. Theorem 3.2 confirms this intuition. The same\nargument applies if σ= 0 andω(x) =Lx, independently of L(see Remark 3.6).\nOn the contrary, in all other cases the right-hand side of (4.5) dive rges asλ→\n+∞, opening the door to the (DGCS)-phenomenon. We are able to show that it does\nhappen provided that λω(1/λ)≫λ2σ. We refer to the first part of section 6 for further\ncomments.\n5 Proofs of well-posedness and regularity results\nAll proofs of our main results concerning well-posedness and regula rity rely on suitable\nestimates for solutions to the ordinary differential equation (1.8) w ith initial data\nuλ(0) =u0, u′\nλ(0) =u1. (5.1)\nFor the sake of simplicity in the sequel we write u(t) instead of uλ(t).\n165.1 Supercritical dissipation\nLet us consider the case σ≥1/2. The key tool is the following.\nLemma 5.1. Let us consider problem (1.8)–(5.1) under the following ass umptions:\n•the coefficient c: [0,+∞)→Ris measurable and satisfies the degenerate hyper-\nbolicity assumption (1.7),\n•δ,λ,σare positive real numbers such that\n4δ2λ4σ−2≥µ2. (5.2)\nThen the solution u(t)satisfies the following estimates.\n(1) For every t≥0it turns out that\n|u(t)|2≤2\nδ2λ4σu2\n1+3u2\n0, (5.3)\n|u′(t)|2≤/parenleftbigg\n2+µ2\n2\nδ4λ8σ−4/parenrightbigg\nu2\n1+3µ2\n2\n2δ2λ4σ−4u2\n0. (5.4)\n(2) Let us assume in addition that λ≥1andσ≥1/2, and letαandβbe two real\nnumbers satisfying (1.6).\nThen for every t≥0it turns out that\nλ4β|u′(t)|2+λ4α|u(t)|2≤/parenleftbigg\n2+2\nδ2+µ2\n2\nδ4/parenrightbigg\nλ4βu2\n1+3/parenleftbigg\n1+µ2\n2\n2δ2/parenrightbigg\nλ4αu2\n0.(5.5)\n(3) In addition to the assumptions of the statement (2), let u s assume also that there\nexistsr>0satisfying the following three inequalities:\nδλ4σ−2>rµ2,2δr≤1,4δ2λ4σ−2≥(1+2rδ)µ2.(5.6)\nThen for every t≥0it turns out that\nλ4β|u′(t)|2+λ4α|u(t)|2≤/bracketleftbigg\n2/parenleftbigg\n1+2µ2\n2\nδ4+1\nδ2/parenrightbigg\nλ4βu2\n1+3/parenleftbigg\n1+2µ2\n2\nδ2/parenrightbigg\nλ4αu2\n0/bracketrightbigg\n×\n×exp/parenleftbigg\n−2rλ2(1−σ)/integraldisplayt\n0c(s)ds/parenrightbigg\n. (5.7)\n17ProofLet us consider the energy E(t) defined in (1.9). Since\n−3\n4|u′(t)|2−4\n3δ2λ4σ|u(t)|2≤2δλ2σu(t)u′(t)≤ |u′(t)|2+δ2λ4σ|u(t)|2,\nwe easily deduce that\n1\n4|u′(t)|2+2\n3δ2λ4σ|u(t)|2≤E(t)≤2|u′(t)|2+3δ2λ4σ|u(t)|2∀t≥0.(5.8)\nStatement (1) The time-derivative of E(t) is\nE′(t) =−2/parenleftbig\nδλ2σ|u′(t)|2+δλ2σ+2c(t)|u(t)|2+λ2c(t)u(t)u′(t)/parenrightbig\n.(5.9)\nThe right-hand side is a quadratic form in u(t) andu′(t). The coefficient of |u′(t)|2\nis negative. Therefore, this quadratic form is less than or equal to 0 for all values of u(t)\nandu′(t) if and only if\n4δ2λ4σ−2c(t)≥c2(t),\nand this is always true because of (1.7) and (5.2). It follows that E′(t)≤0 for (almost)\neveryt≥0, and hence\nδ2λ4σ|u(t)|2≤E(t)≤E(0)≤2u2\n1+3δ2λ4σu2\n0, (5.10)\nwhich is equivalent to (5.3).\nIn order to estimate u′(t), we rewrite (1.8) in the form\nu′′(t)+2δλ2σu′(t) =−λ2c(t)u(t),\nwhich we interpret as a first order linear equation with constant coe fficients in the\nunknownu′(t), with the right-hand side as a forcing term. Integrating this differ ential\nequation in u′(t), we obtain that\nu′(t) =u1exp/parenleftbig\n−2δλ2σt/parenrightbig\n−/integraldisplayt\n0λ2c(s)u(s)exp/parenleftbig\n−2δλ2σ(t−s)/parenrightbig\nds. (5.11)\nFrom (1.7) and (5.3) it follows that\n|u′(t)| ≤ |u1|+µ2λ2·max\nt∈[0,T]|u(t)|·/integraldisplayt\n0e−2δλ2σ(t−s)ds\n≤ |u1|+µ2λ2\n2δλ2σ/parenleftbigg2\nδ2λ4σu2\n1+3u2\n0/parenrightbigg1/2\n,\nand therefore\n|u′(t)|2≤2|u1|2+µ2\n2λ4\n2δ2λ4σ/parenleftbigg2\nδ2λ4σu2\n1+3u2\n0/parenrightbigg\n,\nwhich is equivalent to (5.4).\n18Statement (2) Exploiting (5.3) and (5.4), with some simple algebra we obtain that\nλ4β|u′(t)|2+λ4α|u(t)|2≤/parenleftbigg\n2+µ2\n2\nδ4·1\nλ4(2σ−1)+2\nδ2·1\nλ4(β+σ−α)/parenrightbigg\nλ4βu2\n1\n+3/parenleftbigg\n1+µ2\n2\n2δ2·1\nλ4(α−β+σ−1)/parenrightbigg\nλ4αu2\n0.\nAll exponents of λ’s in denominators are nonnegative owing to (1.6). Therefore,\nsinceλ≥1, all those fractions can be estimated with 1. This leads to (5.5).\nStatement (3) Let us define C(t) as in (3.9). To begin with, we prove that in this\ncase the function E(t) satisfies the stronger differential inequality\nE′(t)≤ −2rλ2(1−σ)c(t)E(t), (5.12)\nand hence\nE(t)≤E(0)exp/parenleftbig\n−2rλ2(1−σ)C(t)/parenrightbig\n∀t≥0. (5.13)\nComing back to (5.9), inequality (5.12) is equivalent to\nλ2σ/parenleftbig\nδ−rλ2−4σc(t)/parenrightbig\n|u′(t)|2+δλ2σ+2(1−2rδ)c(t)|u(t)|2+λ2(1−2rδ)c(t)u(t)u′(t)≥0.\nAs in the proof of statement (1), we consider the whole left-hand s ide as a quadratic\nform inu(t) andu′(t). Sincec(t)≤µ2, from the first inequality in (5.6) it follows that\nδλ4σ−2>rµ2≥rc(t),\nwhich is equivalent to saying that the coefficient of |u′(t)|2is positive. Therefore, the\nquadratic form is nonnegative for all values of u(t) andu′(t) if and only if\n4λ2σ/parenleftbig\nδ−rλ2−4σc(t)/parenrightbig\n·δλ2σ+2c(t)(1−2rδ)≥λ4c2(t)(1−2rδ)2,\nhence if and only if\n(1−2rδ)c(t)/bracketleftbig\n4δ2λ4σ−2−(1+2rδ)c(t)/bracketrightbig\n≥0,\nand this follows from (1.7) and from the last two inequalities in (5.6).\nNow from (5.13) it follows that\nδ2λ4σ|u(t)|2≤E(t)≤E(0)exp/parenleftbig\n−2rλ2(1−σ)C(t)/parenrightbig\n, (5.14)\nwhich provides an estimate for |u(t)|. In order to estimate u′(t), we write it in the form\n(5.11), and we estimate the two terms separately. The third inequa lity in (5.6) implies\nthat 2δλ4σ−2≥rµ2. SinceC(t)≤µ2t, it follows that\n2δλ2σt≥rλ2−2σµ2t≥rλ2−2σC(t),\n19and hence\n/vextendsingle/vextendsingleu1exp/parenleftbig\n−2δλ2σt/parenrightbig/vextendsingle/vextendsingle≤ |u1|exp/parenleftbig\n−2δλ2σt/parenrightbig\n≤ |u1|exp/parenleftbig\n−rλ2(1−σ)C(t)/parenrightbig\n.(5.15)\nAs for the second terms in (5.11), we exploit (5.14) and we obtain tha t\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0λ2c(s)u(s)exp/parenleftbig\n−2δλ2σ(t−s)/parenrightbig\nds/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤λ2µ2/integraldisplayt\n0|u(s)|exp/parenleftbig\n−2δλ2σ(t−s)/parenrightbig\nds\n≤µ2[E(0)]1/2\nδλ2σ−2exp/parenleftbig\n−2δλ2σt/parenrightbig/integraldisplayt\n0exp/parenleftbig\n−rλ2(1−σ)C(s)+2δλ2σs/parenrightbig\nds.\nFrom the first inequality in (5.6) it follows that\n2δλ2σ−rλ2(1−σ)c(s)≥2δλ2σ−rλ2(1−σ)µ2≥δλ2σ,\nhence\n/integraldisplayt\n0exp/parenleftbig\n−rλ2(1−σ)C(s)+2δλ2σs/parenrightbig\nds\n≤1\nδλ2σ/integraldisplayt\n0/parenleftbig\n2δλ2σ−rλ2(1−σ)c(s)/parenrightbig\nexp/parenleftbig\n2δλ2σs−rλ2(1−σ)C(s)/parenrightbig\nds\n≤1\nδλ2σexp/parenleftbig\n2δλ2σt−rλ2(1−σ)C(t)/parenrightbig\n,\nand therefore\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayt\n0λ2c(s)u(s)exp/parenleftbig\n−2δλ2σ(t−s)/parenrightbig\nds/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤µ2[E(0)]1/2\nδ2λ4σ−2exp/parenleftbig\n−rλ2(1−σ)C(t)/parenrightbig\n.(5.16)\nFrom (5.11), (5.15) and (5.16) we deduce that\n|u′(t)| ≤/parenleftbigg\n|u1|+µ2[E(0)]1/2\nδ2λ4σ−2/parenrightbigg\nexp/parenleftbig\n−rλ2(1−σ)C(t)/parenrightbig\n,\nand hence\n|u′(t)|2≤/parenleftbigg\n2|u1|2+2µ2\n2E(0)\nδ4λ8σ−4/parenrightbigg\nexp/parenleftbig\n−2rλ2(1−σ)C(t)/parenrightbig\n. (5.17)\nFinally, we estimate E(0) as in (5.10). At this point, estimate (5.7) follows from\n(5.17) and (5.14) with some simple algebra (we need to exploit that λ≥1 and assump-\ntion (1.6) exactly as in the proof of statement (2)). /square\n205.1.1 Proof of Theorem 3.1\nLet us fix a real number ν≥1 such that 4 δ2ν4σ−2≥µ2(such a number exists because of\nourassumptions on δandσ). Letusconsiderthecomponents uk(t)ofu(t)corresponding\nto eigenvalues λk≥ν. Sinceλk≥1 and 4δ2λ4σ−2\nk≥µ2, we can apply statement (2) of\nLemma 5.1 to these components. If u0kandu1kdenote the corresponding components\nof initial data, estimate (5.5) read as\nλ4β\nk|u′\nk(t)|2+λ4α\nk|uk(t)|2≤/parenleftbigg\n2+2\nδ2+µ2\n2\nδ4/parenrightbigg\nλ4β\nk|u1,k|2+3/parenleftbigg\n1+µ2\n2\n2δ2/parenrightbigg\nλ4α\nk|u0,k|2.\nSumming over all λk≥νwe obtain exactly (3.3).\nThis proves that uν,+(t) is bounded with values in D(Aα) andu′\nν,+(t) is bounded\nwith values in D(Aβ). The same estimate guarantees the uniform convergence in the\nwhole half-line t≥0 of the series defining Aαuν,+(t) andAβu′\nν,+(t). Since all summands\nare continuous, and the convergence is uniform, the sum is continu ous as well. Since\nlow-frequency components uν,−(t) andu′\nν,−(t) are continuous (see Remark 3.3), (3.2) is\nproved. /square\n5.1.2 Proof of Theorem 3.8\nLet us fix a real number ν≥1 such that 4 δ2ν4σ−2>µ2(such a number exists because of\nour assumptions on δandσ). Then there exists r>0 such that the three inequalities in\n(5.6) hold true for every λ≥ν. Therefore, we can apply statement (3) of Lemma 5.1 to\nall components uk(t) ofu(t) corresponding to eigenvalues λk≥ν. Ifu0kandu1kdenote\nthe corresponding components of initial data, estimate (5.7) read as\n/parenleftBig\nλ4β\nk|u′\nk(t)|2+λ4α\nk|uk(t)|2/parenrightBig\nexp/parenleftbigg\n2rλ2(1−σ)\nk/integraldisplayt\n0c(s)ds/parenrightbigg\n≤K/parenleftBig\nλ4β\nk|u1k|2+λ4α\nk|u0k|2/parenrightBig\nfor everyt≥0, whereKis a suitable constant depending only on µ2andδ. Summing\nover allλk≥νwe obtain exactly (3.11). The continuity of u(t) andu′(t) with values\nin the suitable spaces follows from the uniform convergence of serie s as in the proof of\nTheorem 3.1. /square\n5.2 Subcritical dissipation\nLet us consider the case 0 ≤σ≤1/2. The key tool is the following.\nLemma 5.2. Let us consider problem (1.8)–(5.1) under the following ass umptions:\n•the coefficient c: [0,+∞)→Rsatisfies the strict hyperbolicity assumption (1.4)\nand theω-continuity assumption (2.7) for some continuity modulus ω(x),\n•δ>0,λ>0, andσ≥0are real numbers satisfying (3.6).\n21Then the solution u(t)satisfies the following estimates.\n(1) It turns out that\n|u′(t)|2+2λ2µ1|u(t)|2≤4u2\n1+2/parenleftbig\n3δ2λ4σ+λ2µ2/parenrightbig\nu2\n0∀t≥0.(5.18)\n(2) Let us assume in addition that λ≥1,σ∈[0,1/2], and there exists a constant\nr∈(0,δ)such that\n4(δ−r)(δµ1−rµ2)≥/bracketleftbigg\nλ1−2σω/parenleftbigg1\nλ/parenrightbigg/bracketrightbigg2\n+2δ(1+2r)/bracketleftbigg\nλ1−2σω/parenleftbigg1\nλ/parenrightbigg/bracketrightbigg\n+8rδ3.(5.19)\nThen for every t≥0it turns out that\n|u′(t)|2+2λ2µ1|u(t)|2≤/bracketleftbig\n4u2\n1+2/parenleftbig\n3δ2λ4σ+λ2µ2/parenrightbig\nu2\n0/bracketrightbig\nexp/parenleftbig\n−2rλ2σt/parenrightbig\n.(5.20)\nProofFor everyε>0 we introduce the regularized coefficient\ncε(t) :=1\nε/integraldisplayt+ε\ntc(s)ds∀t≥0.\nIt is easy to see that cε∈C1([0,+∞),R) and satisfies the following estimates:\nµ1≤cε(t)≤µ2∀t≥0, (5.21)\n|c(t)−cε(t)| ≤ω(ε)∀t≥0, (5.22)\n|c′\nε(t)| ≤ω(ε)\nε∀t≥0. (5.23)\nApproximated energy For everyε >0 we consider the approximated hyperbolic\nenergyEε(t) defined in (1.10). Since\n−1\n2|u′(t)|2−2δ2λ4σ|u(t)|2≤2δλ2σu(t)u′(t)≤ |u′(t)|2+δ2λ4σ|u(t)|2,\nwe deduce that\n1\n2|u′(t)|2+µ1λ2|u(t)|2≤Eε(t)≤2|u′(t)|2+(3δ2λ4σ+λ2µ2)|u(t)|2(5.24)\nfor everyε>0 andt≥0. The time-derivative of Eε(t) is\nE′\nε(t) =−2δλ2σ|u′(t)|2−2δλ2σ+2c(t)|u(t)|2\n−2λ2(c(t)−cε(t))u(t)u′(t)+λ2c′\nε(t)|u(t)|2, (5.25)\nhence\nE′\nε(t)≤ −2δλ2σ|u′(t)|2−/parenleftbig\n2δλ2σ+2c(t)−λ2|c′\nε(t)|/parenrightbig\n|u(t)|2\n+2λ2|c(t)−cε(t)|·|u(t)|·|u′(t)|. (5.26)\n22Statement (1) We claim that, for a suitable choice of ε, it turns out that\nE′\nε(t)≤0∀t≥0. (5.27)\nIf we prove this claim, then we apply (5.24) with that particular value o fεand we\nobtain that\n1\n2|u′(t)|2+µ1λ2|u(t)|2≤Eε(t)≤Eε(0)≤2u2\n1+(3δ2λ4σ+λ2µ2)u2\n0,\nwhich is equivalent to (5.18).\nInordertoprove(5.27),weconsiderthewholeright-handsideof( 5.26)asaquadratic\nform in|u(t)|and|u′(t)|. Since the coefficient of |u′(t)|2is negative, this quadratic form\nis less than or equal to 0 for all values of |u(t)|and|u′(t)|if and only if\n2δλ2σ·/parenleftbig\n2δλ2σ+2c(t)−λ2|c′\nε(t)|/parenrightbig\n−λ4|c(t)−cε(t)|2≥0,\nhence if and only if\n4δ2λ4σ−2c(t)≥ |c(t)−cε(t)|2+2δλ2σ−2|c′\nε(t)|. (5.28)\nNow in the left-hand side we estimate c(t) from below with µ1, and we estimate from\nabove the terms in the right-hand side as in (5.22) and (5.23). We obt ain that (5.28)\nholds true whenever\n4δ2µ1≥ω2(ε)\nλ4σ−2+2δω(ε)\nλ2σε.\nThis condition is true when ε:= 1/λthanks to assumption (3.6). This completes\nthe proof of (5.18).\nStatement (2) Let us assume now that λ≥1 and that (5.19) holds true for some\nr∈(0,δ). In this case we claim that, for a suitable choice of ε>0, the stronger estimate\nE′\nε(t)≤ −2rλ2σEε(t)∀t≥0 (5.29)\nholds true, hence\nEε(t)≤Eε(0)exp/parenleftbig\n−2rλ2σt/parenrightbig\n∀t≥0.\nDue to (5.24), this is enough to deduce (5.20). So it remains to prove (5.29).\nOwing to (5.25), inequality (5.29) is equivalent to\n2λ2σ(δ−r)|u′(t)|2+/bracketleftbig\n2λ2σ+2(δc(t)−rcε(t))−λ2c′\nε(t)−4rδ2λ6σ/bracketrightbig\n|u(t)|2\n+2/bracketleftbig\nλ2(c(t)−cε(t))−2rδλ4σ/bracketrightbig\nu(t)u′(t)≥0.\nKeeping (1.4) and (5.21) into account, the last inequality is proved if w e show that\n2λ2σ(δ−r)|u′(t)|2+/bracketleftbig\n2λ2σ+2(δµ1−rµ2)−λ2|c′\nε(t)|−4rδ2λ6σ/bracketrightbig\n|u(t)|2\n23−2/bracketleftbig\nλ2|c(t)−cε(t)|+2rδλ4σ/bracketrightbig\n|u(t)|·|u′(t)| ≥0.\nAs in the proof of the first statement, we consider the whole left-h and side as a\nquadratic form in |u(t)|and|u′(t)|. The coefficient of |u′(t)|is positive because r < δ.\nTherefore, this quadratic form is nonnegative for all values of |u(t)|and|u′(t)|if and\nonly if\n2λ2σ(δ−r)·/bracketleftbig\n2λ2σ+2(δµ1−rµ2)−λ2|c′\nε(t)|−4rδ2λ6σ/bracketrightbig\n≥/bracketleftbig\nλ2|c(t)−cε(t)|+2rδλ4σ/bracketrightbig2.\nNow we rearrange the terms, and we exploit (5.22) and (5.23). We ob tain that the\nlast inequality is proved if we show that\n4(δ−r)(δµ1−rµ2)≥λ2−4σω2(ε)+2δ/parenleftbig\n1+2rελ2σ/parenrightbigω(ε)\nελ2σ+8rδ3\nλ2−4σ.(5.30)\nFinally, we choose ε:= 1/λ, so that (5.30) becomes\n4(δ−r)(δµ1−rµ2)≥/bracketleftbigg\nλ1−2σω/parenleftbigg1\nλ/parenrightbigg/bracketrightbigg2\n+2δ/parenleftbigg\n1+2r\nλ1−2σ/parenrightbigg/bracketleftbigg\nλ1−2σω/parenleftbigg1\nλ/parenrightbigg/bracketrightbigg\n+8rδ3\nλ2−4σ.\nSinceλ≥1 andσ≤1/2, this inequality follows from assumption (5.19). /square\n5.2.1 Proof of Theorem 3.2\nLet us rewrite (3.5) in the form\nΛ∞= limsup\nλ→+∞λ1−2σω/parenleftbigg1\nλ/parenrightbigg\n. (5.31)\nDue to (3.4), there exists ν≥1 such that (3.6) holds true for every λ≥ν. Therefore,\nwe can apply statement (1) of Lemma 5.2 to the components uk(t) ofu(t) corresponding\nto eigenvalues λk≥ν. Ifu0kandu1kdenote the corresponding components of initial\ndata, estimate (5.18) read as\n|u′\nk(t)|2+2λ2\nkµ1|uk(t)|2≤4|u1k|2+2/parenleftbig\n3δ2λ4σ\nk+λ2\nkµ2/parenrightbig\n|u0k|2.\nSinceσ≤1/2 and we chose ν≥1, this implies that\n|u′\nk(t)|2+2λ2\nkµ1|uk(t)|2≤4|u1k|2+2/parenleftbig\n3δ2+µ2/parenrightbig\nλ2\nk|u0k|2.\nSumming over all λk≥νwe obtain exactly (3.7).\nThis proves that uν,+(t) is bounded with values in D(A1/2) andu′\nν,+(t) is bounded\nwith values in H. The continuity of u(t) andu′(t) with values in the same spaces follows\nfrom the uniform convergence of series as in the proof of Theorem 3.1./square\n245.2.2 Proof of Theorem 3.9\nLet us rewrite (3.5) in the form (5.31). Due to (3.4), there exists r >0 andν≥1\nsuch that (5.19) holds true for every λ≥ν. Therefore, we can apply statement (2) of\nLemma 5.2 to the components uk(t) ofu(t) corresponding to eigenvalues λk≥ν. Ifu0k\nandu1kdenote the corresponding components of initial data, estimate (5 .20) reads as\n/parenleftbig\n|u′\nk(t)|2+2λ2\nkµ1|uk(t)|2/parenrightbig\nexp/parenleftbig\n2rλ2σ\nkt/parenrightbig\n≤4|u1k|2+2/parenleftbig\n3δ2λ4σ\nk+λ2\nkµ2/parenrightbig\n|u0k|2.\nSinceσ≤1/2 and we chose ν≥1, this implies that\n/parenleftbig\n|u′\nk(t)|2+2λ2\nkµ1|uk(t)|2/parenrightbig\nexp/parenleftbig\n2rλ2σ\nkt/parenrightbig\n≤4|u1k|2+2/parenleftbig\n3δ2+µ2/parenrightbig\nλ2\nk|u0k|2\nfor everyt≥0. Summing over all λk≥νwe obtain (3.12) with a constant Kdepending\nonly onµ1,µ2, andδ. The continuity of u(t) andu′(t) with values in the suitable spaces\nfollows from the uniform convergence of series as in the proof of Th eorem 3.1. /square\n6 The (DGCS)-phenomenon\nIn this section we prove Theorem 3.10. Let us describe the strateg y before entering into\ndetails. Roughly speaking, what we need is a solution u(t) whose components uk(t) are\nsmall at time t= 0 and huge at time t>0. The starting point is given by the following\nfunctions\nb(ε,λ,t) := (2ελ−δλ2σ)t−εsin(2λt),\nw(ε,λ,t) := sin(λt)exp(b(ε,λ,t)), (6.1)\nγ(ε,λ,t) := 1+δ2\nλ2−4σ−16ε2sin4(λt)−8εsin(2λt). (6.2)\nWith some computations it turns out that\nw′′(ε,λ,t)+2δλ2σw′(ε,λ,t)+λ2γ(ε,λ,t)w(ε,λ,t) = 0 ∀t∈R,\nwhere “primes” denote differentiation with respect to t. As a consequence, if we set\nc(t) :=γ(ε,λ,t) andε:=ω(1/λ), the function u(t) :=w(ε,λ,t) turns out to be a\nsolution to (1.8) which grows as the right-hand side of (4.5). Unfort unately this is not\nenough, because we need to realize a similar growth for countably ma ny components\nwith the same coefficient c(t).\nTo this end, we argue as in [6]. We introduce a suitable decreasing sequ encetk→0+,\nand in the interval [ tk,tk−1] we design the coefficient c(t) so thatuk(tk) is small and\nuk(tk−1) is huge. Then we check that the piecewise defined coefficient c(t) has the\nrequired time-regularity, and that uk(t) remains small for t∈[0,tk] and remains huge\nfort≥tk−1. This completes the proof.\nRoughly speaking, the coefficient c(t) plays on infinitely many time-scales in order\nto “activate” countably many components, but these countably m any actions take place\n25onebyoneindisjointtimeintervals. Ofcoursethismeansthatthelen gthstk−1−tkofthe\n“activationintervals”tendto0as k→+∞. Inordertoobtainenoughgrowth, despiteof\nthe vanishing length of activationintervals, we areforced to assum e thatλω(1/λ)≫λ2σ\nasλ→+∞. In addition, components do not grow exactly as exp( λω(1/λ)t), but just\nmore than exp( ϕ(λ)t) and exp(ψ(λ)t).\nThis is the reason why this strategy does not work when λω(1/λ)∼λ2σandδ\nis small. In this case one would need components growing exactly as ex p(λω(1/λ)t),\nbut this requires activation intervals of non-vanishing length, which are thus forced to\noverlap. In a certain sense, the coefficient c(t) should work once againoninfinitely many\ntime-scales, but now the countably many actions should take place in the same time.\nDefinition of sequences From (3.13) and (3.14) it follows that\nlim\nx→+∞x1−2σω/parenleftbigg1\nx/parenrightbigg\n= +∞, (6.3)\nlim\nx→+∞1\nx1−2σω(1/x)+ϕ(x)\nxω(1/x)+ψ(x)\nxω(1/x)= 0, (6.4)\nand a fortiori\nlim\nx→+∞x1+2σω/parenleftbigg1\nx/parenrightbigg\n= +∞, (6.5)\nlim\nx→+∞x2σ+ϕ(x)+ψ(x)\nxω/parenleftbigg1\nx/parenrightbigg\n= 0. (6.6)\nLet us consider the sequence {λk}, which we assumed to be unbounded. Due to\n(6.5) and (6.4) we can assume, up to passing to a subsequence (not relabeled), that the\nfollowing inequalities hold true for every k≥1:\nλk>4λk−1, (6.7)\nλ1+2σ\nkω/parenleftbigg1\nλk/parenrightbigg\n≥δ4\n210π21\nλ2−8σ\nk−1+4k2\nπ2λ2\nk−1, (6.8)\nλ1+2σ\nkω/parenleftbigg1\nλk/parenrightbigg\n≥4k2\nπ2λ3\nk−1/parenleftbig\nλ2σ\nk−1+ϕ(λk−1)+ψ(λk−1)/parenrightbig\nω/parenleftbigg1\nλk−1/parenrightbigg\n,(6.9)\nλ1+2σ\nkω/parenleftbigg1\nλk/parenrightbigg\n≥λk−1/parenleftbig\nλ2σ\nk−1+ϕ(λk−1)+ψ(λk−1)/parenrightbig\nω/parenleftbigg1\nλk−1/parenrightbigg\n,(6.10)\n1\nλ1−2σ\nkω(1/λk)+ϕ(λk)\nλkω(1/λk)+ψ(λk)\nλkω(1/λk)≤π2\n4k21\nλ2\nk−1. (6.11)\nNow let us set\ntk:=4π\nλk, s k:=π\nλk/floorleftbigg\n2λk\nλk−1/floorrightbigg\n, (6.12)\n26where⌊α⌋denotes the largest integer less than or equal to α, and\nεk:=/braceleftbiggλ2σ\nk+ϕ(λk)+ψ(λk)\nλkω/parenleftbigg1\nλk/parenrightbigg/bracerightbigg1/2\n.\nProperties of the sequences We collect in this section of the proof all the properties\nof the sequences which are needed in the sequel. First of all, it is clear thatλk→+∞,\nhencetk→0 andεk→0 (because of (6.6)). Moreover it turns out that\ntk−1\n4=π\nλk−1≤sk≤2π\nλk−1=tk−1\n2. (6.13)\nKeeping (6.7) into account, it follows that\ntk<sk<tk−1∀k≥1,\nand in particular also sk→0. In addition, it turns out that\nsin(λktk) = sin(λksk) = 0 (6.14)\nand\n|cos(λktk)|=|cos(λksk)|= 1. (6.15)\nSinceσ <1/2,λk→+∞,εk→0,tk→0, keeping (6.3) and (6.4) into account, we\ndeduce that the following seven inequalities are satisfied provided th atkis large enough:\nδ2\nλ2−4σ\nk+16ε2\nk+8εk≤1\n2, (6.16)\nεk≤1\n4, (6.17)\n16πεk+16πδ\nλ1−2σ\nk≤2π, (6.18)\n1\nλ1−2σ\nkω(1/λk)+ϕ(λk)\nλkω(1/λk)+ψ(λk)\nλkω(1/λk)≤1\n52·210·π2, (6.19)\nδ2\n(4π)2−4σ(2tk)1−2σsup/braceleftbiggx1−2σ\nω(x):x∈(0,tk)/bracerightbigg\n≤1\n5, (6.20)\nλ1−2σ\nkω/parenleftbigg1\nλk/parenrightbigg\n≥δ2, (6.21)\n2δ2\nλ2−4σ\nk−1ω(1/λk−1)≤1\n5. (6.22)\n27Letk0∈Nbe a positive integer such that (6.16) through (6.22) hold true for e very\nk≥k0. From (6.21) it follows that\nεkλk≥δλ2σ\nk∀k≥k0. (6.23)\nFrom (6.19) it follows that\n32πεk\nω(1/λk)≤1\n5∀k≥k0. (6.24)\nSincesk≥π/λk−1(see the estimate from below in (6.13)), from (6.8) it follows that\nεkλksk≥δ2\n321\nλ2−4σ\nk−1∀k≥k0, (6.25)\nεkλksk≥2k∀k≥k0, (6.26)\nfrom (6.9) it follows that\nεkλksk≥2kεk−1λk−1∀k≥k0, (6.27)\nand from (6.11) it follows that\nεkλksk≥2k/parenleftbig\nλ2σ\nk+ϕ(λk)+ψ(λk)/parenrightbig\n∀k≥k0. (6.28)\nAs a consequence of (6.26) through (6.28) it turns out that\n2εkλksk≥kεk−1λk−1+2k/parenleftbig\nλ2σ\nk+ϕ(λk)+ψ(λk)/parenrightbig\n+k∀k≥k0.(6.29)\nFinally, from (6.10) it follows that\nεkλk≥εk−1λk−1∀k≥k0. (6.30)\nDefinition of c(t)andu(t) For every k≥1, letℓk:R→Rbe defined by\nℓk(t) :=δ2\ntk−1−sk/parenleftbigg1\nλ2−4σ\nk−1−1\nλ2−4σ\nk/parenrightbigg\n(t−sk)+1+δ2\nλ2−4σ\nk∀t∈R.\nThanks to (6.14), ℓk(t) is the affine function such that\nℓk(sk) =γ(εk,λk,sk) and ℓk(tk−1) =γ(εk−1,λk−1,tk−1).\nLetk0∈Nbe such that (6.16) through (6.22) hold true for every k≥k0. Let us set\nc(t) :=\n\n1 if t≤0,\nγ(εk,λk,t) ift∈[tk,sk] for somek≥k0,\nℓk(t) if t∈[sk,tk−1] for somek≥k0+1,\nγ(εk0,λk0,sk0) ift≥sk0.\n28The following picture describes this definition. The coefficient c(t) is constant for\nt≤0 and fort≥sk0. In the intervals [ tk,sk] it coincides with γ(εk,λk,t), hence it\noscillates, with period of order λ−1\nkand amplitude of order εk, around a value which\ntends to 1. In the intervals [ sk,tk−1] it is just the affine interpolation of the values at\nthe endpoints.\n/Bullet /Bullet /Bullet /Bullet /Bullet /Bullet /Bullet\nsk+2tk+1sk+1tksk tk−1sk−1period∼λ−1\nkperiod∼λ−1\nk−1\n1+δ2\nλ2−4σ\nk∼εk∼εk−1\nFor everyk≥k0, let us consider the solution uk(t) to the Cauchy problem\nu′′\nk(t)+2δλ2σ\nku′\nk(t)+λ2\nkc(t)uk(t) = 0,\nwith “initial” data\nuk(tk) = 0, u′\nk(tk) =λkexp/parenleftbig\n(2εkλk−δλ2σ\nk)tk/parenrightbig\n. (6.31)\nThen we set\nak:=1\nkλkexp(−kϕ(λk)), (6.32)\nand finally\nu(t) :=∞/summationdisplay\nk=k0akuk(t)ek.\nWeclaim that c(t) satisfies (3.15)and(3.16), andthat u(t)satisfies (3.17)and(3.18).\nThe rest of the proof is a verification of these claims.\nEstimate and continuity of c(t) We prove that\n|c(t)−1| ≤1\n2∀t≥0, (6.33)\n29which is equivalent to (3.15), and that c(t) is continuous on the whole real line.\nTo this end, it is enough to check (6.33) in the intervals [ tk,sk], because in the\nintervals [sk,tk−1]the function c(t) isjust aninterpolationofthevalues at theendpoints,\nand it is constant for t≤0 and fort≥sk0.\nIn the intervals [ tk,sk] the function c(t) coincides with γ(εk,λk,t), hence from (6.2)\nit turns out that\n|c(t)−1|=|γ(εk,λk,t)−1| ≤δ2\nλ2−4σ\nk+16ε2\nk+8εk, (6.34)\nso that (6.33) follows immediately from (6.16).\nSince the right-hand side of (6.34) tends to 0 as k→+∞, the same estimate shows\nalso thatc(t)→1 ast→0+, which proves the continuity of c(t) int= 0, the only point\nin which continuity was nontrivial.\nEstimate on c′(t) We prove that\n|c′(t)| ≤32εkλk∀t∈(tk,sk),∀k≥k0, (6.35)\n|c′(t)| ≤32εkλk∀t∈(sk,tk−1),∀k≥k0+1. (6.36)\nIndeed in the interval ( tk,sk) it turns out that\n|c′(t)|=|γ′(εk,λk,t)|=/vextendsingle/vextendsingle−64ε2\nkλksin3(λkt)cos(λkt)−16εkλkcos(2λkt)/vextendsingle/vextendsingle\n≤64ε2\nkλk+16εkλk= 16εkλk(4εk+1),\nso that (6.35) follows from (6.17).\nIn the interval ( sk,tk−1) it turns out that\n|c′(t)|=δ2\ntk−1−sk/parenleftbigg1\nλ2−4σ\nk−1−1\nλ2−4σ\nk/parenrightbigg\n≤δ2\ntk−1−sk·1\nλ2−4σ\nk−1≤δ2\nsk·1\nλ2−4σ\nk−1,\nwhere the last inequality follows from the estimate from above in (6.13 ). At this point\n(6.36) is equivalent to (6.25).\nModulus of continuity of c(t) Let us prove that c(t) satisfies (3.16). Since c(t) is\ncontinuous, and constant for t≤0 andt≥sk0, it is enough to verify its ω-continuity in\n(0,sk0]. In turn, the ω-continuity in (0 ,sk0] is proved if we show that\n|c(ti)−c(tj)| ≤1\n5ω(|ti−tj|)∀i≥k0,∀j≥k0, (6.37)\n|c(a)−c(b)| ≤1\n5ω(|a−b|)∀(a,b)∈[tk,sk]2,∀k≥k0, (6.38)\n30|c(a)−c(b)| ≤1\n5ω(|a−b|)∀(a,b)∈[sk,tk−1]2,∀k≥k0+1. (6.39)\nIndeed, any interval [ s,t]⊆(0,sk0] can be decomposed as the union of at most 5\nintervals whose endpoints fit in one of the 3 possibilities above.\nLet us prove (6.37). From (6.14) it turns out that\n|c(ti)−c(tj)|=δ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nλ2−4σ\ni−1\nλ2−4σ\nj/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤δ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nλ2\ni−1\nλ2\nj/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−2σ\n,\nwhere the inequality follows from the fact that the function x→x1−2σis (1−2σ)-H¨ older\ncontinuous with constant equal to 1. Now from (6.12) it follows that\nδ2/vextendsingle/vextendsingle/vextendsingle/vextendsingle1\nλ2\ni−1\nλ2\nj/vextendsingle/vextendsingle/vextendsingle/vextendsingle1−2σ\n=δ2\n(4π)2−4σ|t2\ni−t2\nj|1−2σ=δ2\n(4π)2−4σ|ti+tj|1−2σ|ti−tj|1−2σ\nω(|ti−tj|)ω(|ti−tj|).\nSince|ti+tj| ≤2tk0and|ti−tj| ≤tk0, we conclude that\n|c(ti)−c(tj)| ≤δ2\n(4π)2−4σ(2tk0)1−2σsup/braceleftbiggx1−2σ\nω(x):x∈(0,tk0)/bracerightbigg\nω(|ti−tj|),\nso that (6.37) follows from (6.20).\nLet us prove (6.38). Since c(t) isπ/λkperiodic in [ tk,sk], for every ( a,b)∈[tk,sk]2\nthere exists ( a1,b1)∈[tk,sk]2such thatc(a) =c(a1),c(b) =c(b1), and|a1−b1| ≤π/λk.\nThus from (6.35) it follows that\n|c(a)−c(b)|=|c(a1)−c(b1)| ≤32εkλk|a1−b1|= 32εkλk|a1−b1|\nω(|a1−b1|)ω(|a1−b1|),\nso that we are left to prove that\n32εkλk|a1−b1|\nω(|a1−b1|)≤1\n5. (6.40)\nDue to (2.6), (2.5), and the fact that |a1−b1| ≤π/λk, it turns out that\n|a1−b1|\nω(|a1−b1|)≤π/λk\nω(π/λk)≤π\nλkω(1/λk),\nso that now (6.40) follows from (6.24).\nLet us prove (6.39). Since c(t) is affine in [ sk,tk−1], for every aandbin this interval\nit turns out that\n|c(a)−c(b)|=δ2\ntk−1−sk/parenleftbigg1\nλ2−4σ\nk−1−1\nλ2−4σ\nk/parenrightbigg\n|a−b|.\n31Sincesk≤tk−1/2, it follows that\n|c(a)−c(b)| ≤2δ2\ntk−11\nλ2−4σ\nk−1·|a−b|=2δ2\ntk−11\nλ2−4σ\nk−1·|a−b|\nω(|a−b|)·ω(|a−b|).\nDue to (2.6), (2.5), and the fact that |a−b| ≤tk−1, it turns out that\n|a−b|\nω(|a−b|)≤tk−1\nω(tk−1)≤tk−1\nω(1/λk−1),\nso that now (6.39) is a simple consequence of (6.22).\nEnergy functions Let us introduce the classic energy functions\nEk(t) :=|u′\nk(t)|2+λ2\nk|uk(t)|2,\nFk(t) :=|u′\nk(t)|2+λ2\nkc(t)|uk(t)|2.\nDue to (3.15), they are equivalent in the sense that\n1\n2Ek(t)≤Fk(t)≤3\n2Ek(t)∀t∈R.\nTherefore, proving (3.17) is equivalent to showing that\n∞/summationdisplay\nk=k0a2\nkEk(0)exp(2rϕ(λk))<+∞ ∀r>0, (6.41)\nwhile proving (3.18) is equivalent to showing that\n∞/summationdisplay\nk=k0a2\nkFk(t)exp(−2Rψ(λk)) = +∞ ∀R>0,∀t>0. (6.42)\nWe are thus left to estimating Ek(0) andFk(t).\nEstimates in [0,tk] We prove that\nEk(0)≤λ2\nkexp(4π)∀k≥k0. (6.43)\nTo this end, we begin by estimating Ek(tk). From (6.31) we obtain that uk(tk) = 0\nand\n|u′\nk(tk)| ≤λkexp(2εkλktk) =λkexp(8πεk),\nso that\nEk(tk)≤λ2\nkexp(16πεk). (6.44)\n32Now the time-derivative of Ek(t) is\nE′\nk(t) =−4δλ2σ\nk|u′\nk(t)|2−2λ2\nk(c(t)−1)u′\nk(t)uk(t)∀t∈R.\nTherefore, from (3.15) it follows that\nE′\nk(t)≥ −4δλ2σ\nkEk(t)−λk|c(t)−1|·2|u′\nk(t)|·λk|uk(t)| ≥ −/parenleftbigg\n4δλ2σ\nk+λk\n2/parenrightbigg\nEk(t)\nfor everyt∈R. Integrating this differential inequality in [0 ,tk] we deduce that\nEk(0)≤Ek(tk)exp/bracketleftbigg/parenleftbigg\n4δλ2σ\nk+λk\n2/parenrightbigg\ntk/bracketrightbigg\n.\nKeeping (6.44) and (6.12) into account, we conclude that\nEk(0)≤λ2\nkexp/parenleftbigg\n16πεk+16πδ\nλ1−2σ\nk+2π/parenrightbigg\n,\nso that (6.43) follows immediately from (6.18).\nEstimates in [tk,sk] In this interval it turns out that uk(t) :=w(εk,λk,t), where\nw(ε,λ,t) is the function defined in (6.1). Keeping (6.14) and (6.15) into accou nt, we\nobtain that uk(sk) = 0 and\n|u′\nk(sk)|=λkexp(b(εk,λk,sk)) =λkexp/parenleftbig\n(2εkλk−δλ2σ\nk)sk/parenrightbig\n.\nTherefore, from (6.23) it follows that\n|u′\nk(sk)| ≥λkexp(εkλksk),\nand hence\nFk(sk) =Ek(sk)≥λ2\nkexp(2εkλksk). (6.45)\nEstimates in [sk,tk−1] We prove that\nFk(tk−1)≥λ2\nkexp(2εkλksk−4δλ2σ\nktk−1). (6.46)\nIndeed the time-derivative of Fk(t) is\nF′\nk(t) =−4δλ2σ\nk|u′\nk(t)|2+λ2\nkc′(t)|uk(t)|2∀t∈(sk,tk−1).\nSincec′(t)>0 in (sk,tk−1), it follows that\nF′\nk(t)≥ −4δλ2σ\nk|u′\nk(t)|2≥ −4δλ2σ\nkFk(t)∀t∈(sk,tk−1),\nand hence\nFk(tk−1)≥Fk(sk)exp/parenleftbig\n−4δλ2σ\nk(tk−1−sk)/parenrightbig\n≥Fk(sk)exp/parenleftbig\n−4δλ2σ\nktk−1/parenrightbig\n.\nKeeping (6.45) into account, we have proved (6.46).\n33Estimates in [tk−1,+∞) We prove that\nFk(t)≥λ2\nkexp/parenleftbig\n2εkλksk−8δλ2σ\nkt−64εk−1λk−1t/parenrightbig\n∀t≥tk−1.(6.47)\nTo this end, let us set\nIk:= [tk−1,+∞)\\k−1/uniondisplay\ni=k0{ti,si}.\nFirst of all, we observe that\n|c′(t)| ≤32εk−1λk−1∀t∈Ik (6.48)\nIndeed we know from (6.35) and (6.36) that\n|c′(t)| ≤32εiλi∀t∈(ti,si)∪(si,ti−1),\nand of course c′(t) = 0 for every t>sk0. Now it is enough to observe that\nIk= (tk0,sk0)∪(sk0,+∞)∪k−1/uniondisplay\ni=k0+1[(ti,si)∪(si,ti−1)],\nand thatεiλiis a nondecreasing sequence because of (6.30).\nNow we observe that the function t→Fk(t) is continuous in [ tk−1,+∞) and differ-\nentiable in Ik, with\nF′\nk(t) =−4δλ2σ\nk|u′\nk(t)|2+λ2\nkc′(t)|uk(t)|2\n≥ −4δλ2σ\nk|u′\nk(t)|2−|c′(t)|\nc(t)·λ2\nkc(t)|uk(t)|2\n≥ −/parenleftbigg\n4δλ2σ\nk+|c′(t)|\nc(t)/parenrightbigg\nFk(t).\nTherefore, from (6.48) and (3.15) it follows that\nF′\nk(t)≥ −/parenleftbig\n4δλ2σ\nk+64εk−1λk−1/parenrightbig\nFk(t)∀t∈Ik,\nand hence\nFk(t)≥Fk(tk−1)exp/bracketleftbig\n−/parenleftbig\n4δλ2σ\nk+64εk−1λk−1/parenrightbig\n(t−tk−1)/bracketrightbig\n≥Fk(tk−1)exp/bracketleftbig\n−/parenleftbig\n4δλ2σ\nk+64εk−1λk−1/parenrightbig\nt/bracketrightbig\nfor everyt≥tk−1. Keeping (6.46) into account, we finally obtain that\nFk(t)≥λ2\nkexp/parenleftbig\n2εkλksk−4δλ2σ\nktk−1−4δλ2σ\nkt−64εk−1λk−1t/parenrightbig\n,\nfrom which (6.47) follows by simply remarking that t≥tk−1.\n34Conclusion We are now ready to verify (6.41) and (6.42). Indeed from (6.32) an d\n(6.43) it turns out that\na2\nkEk(0)exp(2rϕ(λk))≤1\nk2λ2\nkexp(−2kϕ(λk))·λ2\nkexp(4π)·exp(2rϕ(λk))\n=1\nk2exp(4π+2(r−k)ϕ(λk)).\nThe argument of the exponential is less than 4 πwhenkis large enough, and hence\nthe series in (6.41) converges.\nLet us consider now (6.42). For every t>0 it turns out that t≥tk−1whenkis large\nenough. For every such kwe can apply (6.47) and obtain that\na2\nkFk(t)exp(−2Rψ(λk))\n≥1\nk2exp/parenleftbig\n−2kϕ(λk)−2Rψ(λk)+2εkλksk−8δλ2σ\nkt−64εk−1λk−1t/parenrightbig\n.\nKeeping (6.29) into account, it follows that\na2\nkFk(t)exp(−2Rψ(λk))\n≥1\nk2exp/parenleftbig\n(k−64t)εk−1λk−1+2(k−R)ψ(λk)+(2k−8δt)λ2σ\nk+k/parenrightbig\n≥1\nk2exp(k)\nwhenkis large enough. 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Nishihara ; Decay properties of solutions of some quasilinear hyperbolic equa-\ntions with strong damping. Nonlinear Anal. 21(1993), no. 1, 17–21.\n[18]M. Reed, B. Simon ;Methods of Modern Mathematical Physics, I: Functional\nAnalysis. Second edition . Academic Press, New York, 1980.\n[19]Y. Shibata ; Onthe rate of decay of solutions to linear viscoelastic equation. Math.\nMethods Appl. Sci. 23(2000), no. 3, 203–226.\n36"    },    {        "title": "1311.3518v1.The_dimension_of_the_leafwise_reduced_cohomology.pdf",        "content": "arXiv:1311.3518v1  [math.GT]  14 Nov 2013THE DIMENSION OF THE LEAFWISE REDUCED\nCOHOMOLOGY\nJES´US A.´ALVAREZ L ´OPEZ AND GILBERT HECTOR\nAbstract. Geometric conditions are given so that the leafwise reduced co-\nhomology is of infinite dimension, specially for foliations with dense leaves on\nclosed manifolds. The main new definition involved is the int ersection number\nof subfoliations with “appropriate coefficients”. The leafw ise reduced cohomol-\nogy is also described for homogeneous foliations with dense leaves on closed\nnilmanifolds.\n1.Introduction\nLetFbe aC∞foliation on a manifold M. Theleafwise de Rham complex\n(Ω·(F),dF) is the restriction to the leaves of the de Rham complex of M; i.e.,\nΩ(F) is the space of differential forms on the leaves that are C∞on the ambient\nmanifoldM, anddFis the de Rham derivative on the leaves. We use the notation\nΩ(F) =C∞(/logicalandtextT∗F) meaning C∞sections on M. The cohomology H·(F) =\nH·(Ω(F),dF) is called the leafwise cohomology ofF. It is well known that H·(F)\ncanalso be defined as the cohomologyof Mwith coefficients in the sheafofgerms of\nC∞functions which are locally constant on the leaves, but we do not use this. The\n(weak)C∞topology on Ω( F) induces a topology on H·(F), which is non-Hausdorff\nin general [15]. The quotient space of H·(F) over the closure of its trivial subspace\nis called the leafwise reduced cohomology ofF, and denoted by H·(F). Similarly,\nwe can also define Ω·\nc(F),H·\nc(F) andH·\nc(F) by considering compactly supported\nC∞sections of/logicalandtextTF∗.\nFor degree zero we have that H0(F) =H0(F) is the space of C∞functions on\nMthat are constant on each leaf—the so called (smooth) basic functions ; thus\nH0(F)∼=Rif the leaves are dense. Though density of the leaves seems to yield\nstrong restrictions on the leafwise cohomology also for higher degr ee, this cohomol-\nogy may be of infinite dimension when leaves are dense and Mis closed. In fact,\nfor dense linear flows on the two-dimensional torus, we have dim H1(F) = 1 when\nthe slope of the leaves is a diophantine irrational number [18], but dim H1(F) =∞\nif the slope is a Liouville’s irrational number [30]. Nevertheless H1(F)∼=Rin both\ncases. This computation was later generalized to the case of linear f oliations on tori\nof arbitrary dimension [20, 8].\nOther known properties of the leafwise cohomology are the following ones. The\nleafwisecohomologyofdegreeonewithcoefficientsinthenormalbun dleisrelatedto\nthe infinitesimal deformations of the foliation [18]. For p= dimF, the dual space\nHp\nc(F)′is canonically isomorphic to the space of holonomy invariant transver se\ndistributions [15]—recall that for a topological vector space V, the dual space V′is\n1991Mathematics Subject Classification. 57R30.\nPartially supported by Xunta de Galicia (Spain).\n12 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR\nthe space of continuous linear maps V→R.H·(F) is invariant by leaf preserving\nhomotopies, and Mayer-Vietoris arguments can be applied [10], whic h was used to\ncomputeH·(F) for someexamples. For an arbitraryflow Fon the two-torus, it was\nproved that dim H1(F) =∞ifFis not minimal, and dim H1(F) = 1 if and only if\nFisC��conjugate to a Diophantine linear flow [7]. The triviality of H1(F) implies\nthe triviality of the linear holonomy [10], and is equivalent to Thurston’s stability\nif codim F= 1 andMis closed [6]. However, more general relations between the\nleafwise cohomology and the geometry of the foliation remain rather unknown.\nThe above examples of linear foliations on tori could wrongly suggest thatH·(F)\nmaybe offinite dimension if Mis closedand the leavesaredense. In fact, S. Hurder\nand the first author gave examples of foliations with dense leaves on closed Rie-\nmannian manifolds with an infinite dimensional space of leafwise harmon ic forms\nthat areC∞on the ambient manifold [4], and this space is canonically injected in\nthe leafwise reduced cohomology; indeed this injection is an isomorph ism at least\nfor the so called Riemannian foliations [5]. So a natural problem is the fo llowing:\nGive geometric properties characterizing C∞foliations whose leafwise reduced co-\nhomology is of finite dimension; specially for foliations wi th dense leaves on closed\nmanifolds .\nThe aim of this paper is to give an approach to this problem. The first a nd main\ngeometric idea we use is the intersection number of subfoliations with “appropriate\ncoefficients”. To explain it, consider the simplest example where M=T×Lwith\nthefoliation Fwhoseleavesarethe slices {∗}×L,whereT,Lareclosedmanifolds of\ndimensions q,p. Let (Ω·(L),dL) be the de Rhamcomplex of L, and letH·(L),H·(L)\ndenote the homology and cohomology of Lwith real coefficients. Then Ω·(F) is\ntheC∞closure ofC∞(T)⊗Ω·(L), wheredF= 1⊗dL. So\nHk(F) =Hk(F)≡C∞(T)⊗Hk(L)\nbecauseH·(L) is of finite dimension. Assume Lis oriented for simplicity. Then\nrecall that Poincar´ e duality and integration of differential forms e stablish canonical\nisomorphisms\nHk(L)∼=Hq−k(L), Hk(L)′∼=Hk(L),\nsuch that the canonical pairing between Hk(L)⊗Hk(L)′→Rcorresponds to the\nintersection pairing Hp−k(L)⊗Hk(L)→R[32]. Hence\nHk(F)∼=C∞(T)⊗Hp−k(L), (1)\nHk(F)′∼=C∞(T)′⊗Hk(L), (2)\nsuch that the canonical pairing Hk(F)⊗Hk(F)′→Rcorresponds to the product\nof the evaluation of distributions on C∞functions and the intersection pairing.\nNow observe that, according to [32], the right hand side spaces in (1 ) and (2) are\nrespectively generated by elements of the form f⊗[K1] andD⊗[K2], wherefis\naC∞function on T,Dis a distribution on T, andK1,K2⊂Lare closed oriented\nsubmanifolds of dimensions p−k,k. Hence dim Hk(F) =∞is equivalent to the\nexistence of sequences of elements fm⊗[K1] andDn⊗[K2] as above so that K1,K2\nhavenon-trivial intersection number and Dn(fn)/\\e}atio\\slash= 0 if and only if m=n; of course\nthis holds just when Hk(L)/\\e}atio\\slash= 0 andq>0.\nNow consider each element fm⊗[K1] as the family of homology classes\nfm(t)[{t}×K1]∈Hp−k({t}×L), t∈T ,LEAFWISE REDUCED COHOMOLOGY 3\ndetermined by the family of closed oriented submanifolds {t} ×K1of the leaves\nofFand the family of coefficients fm(t). The elements Dn⊗[K2] have a similar\ninterpretationby considering distributions as generalized function s. A key property\nhere is that the families {t}×K1and{t}×K2depend smoothly on t, determining\nC∞subfoliations F1,F2ofF. Other key properties are the C∞dependence of\nthe coefficients fm(t) ont, and the distributional dependence of the generalized\ncoefficients Dn(t) ont. This means that the fmareC∞basic functions of F1\nand theDnare “distributional basic functions” of F2; i.e., theDnare holonomy\ninvariant transverse distributions of F2. It turns out that these key properties are\nenough to generalize the above ideas in a way that can be applied even when the\nleavesaredense, obtainingourfirstmaintheoremthatroughlyass ertsthefollowing:\nFor aC∞oriented foliation Fof dimension p, we have dimHk\nc(F) =∞whenF\nhas oriented subfoliations F1,F2of dimensions k−p,p, and there is a sequence of\nbasic functions fmofF1and a sequence of transverse invariant distributions Dnof\nF2, such that the corresponding “intersection numbers” are no n-trivial if and only if\nm=n—certain simple conditions are also required for the “inter section numbers”\nto be defined . We do not know whether such conditions form a characterization\nof the cases where dim Hk\nc(F) =∞; this depends on whether it is possible to\n“smoothen” the representatives of classes in certain leafwise hom ologies introduced\nin [3]. Indeed the above fmandDnplay the rˆ ole of coefficients in homology,\nassigning a number to each leaf of the subfoliations; the way these n umbers vary\nfrom leaf to leaf is what makes these coefficients appropriate.\nThough these conditions are difficult to check in general, this result h as many\neasy to apply corollaries. For instance, suppose an oriented foliatio nFisRiemann-\nian—in the sense that all of its holonomy transformations are local isom etries for\nsome Riemannian metric on local transversals [29, 25]. Then dim H·\nc(F) =∞if\nFis of positive codimension and some leaf of Fcontains homology classes with\nnon-trivial intersection. These conditions are quite simple to verify . In this case,\nthe infinitely many linearly independent classes obtained in H·\nc(F) can be consid-\nered as “transverse diffusions” of the homology classes in the leaf. This diffusion\nidea is inspired by the unpublished preprints [19, 4]. Indeed [19] is the g erminal\nwork about the relation of the analysis on the leaves and on the ambie nt manifold\nobtained by transverse diffusion.\nOther consequences of the above general theorem hold when Fis asuspension\nfoliation. That is, the ambient manifold of Fis the total space of a fiber bundle\nM→Bwith the leaves transverse to the fibers, and such that the restr iction\nof the bundle projection to each leaf is a regular covering of the bas eB. Now\ndimH·\nc(F) =∞whenBis oriented and has homology classes with non-trivial\nintersection satisfying additional properties with respect to the h olonomy of F. In\nthis case the leaves may not contain homology classes with non-trivia l intersection,\nand thus the idea of “transverse diffusion” of homology classes in th e leaves may\nfail. In fact we shall see that the infinite dimension of H1\nc(F) may be more related\nto the number of ends of the leaves.\nTo explain another theorem of this paper, recall that a foliation Fon a manifold\nMis aLie foliation when it has a complete transversal diffeomorphic to an open\nsubset of a Lie group Gso that holonomy transformationson this transversalcorre-\nspond to restrictions of left translations on G—this type of foliations play a central\nrˆ ole in the study of Riemannian foliations [25]. The Lie algebra gofGis called the4 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR\nstructural Lie algebra ofF; we may also simply say that Fis a Lieg-foliation. In\nthis case, if Mis closed and oriented, and gis compact semisimple, then we obtain\nthat dim H·(F) =∞when some additional hypotheses are satisfied. Again we use\nhomology classes with non-trivial intersection in the hypotheses, b ut now they live\nin the homology of M. The proof of this result is reduced to the case of suspension\nfoliations to apply what we already know. This reduction process con tains rather\ndelicate arguments based on the work [2] of the first author.\nThe above results are negative in the sense that all of them give con ditions for\nthe nonexistence of finite Betti numbers for the reduced leafwise cohomology. In\ncontrast, our final theorem shows that the reduced leafwise coh omology of the so\ncalledhomogeneous foliations with dense leaves in closed nilmanifolds is isomorphic\nto the cohomology of the Lie algebra defining the foliation. This has be en also\nproved by X. Masa with different techniques.\nAcknowledgment . We wish to thank F. Alcalde for many helpful conversations.\nThe first author would like to thank the hospitality of the Institut de Math´ ema-\ntiques et d’Informatique of the University Claude Bernard of Lyon s everal times\nduring the preparation of this work. We would like also to thank the re feree for\nimportant corrections.\n2.Main results\nFor the sake of simplicity, all manifolds, foliations, maps, functions, differential\nforms and actions will be assumed to be C∞from now on, unless the contrary is\nexplicitly stated.\nLetFbe a foliation on a manifold M. For any closed saturated subset S⊂M,\nlet Ω·\nS(F)⊂Ω·(F) be the subcomplex of leafwise differential forms whose support\nhas compact intersection with S. Consider the topology on Ω·\nS(F) determined as\nfollows: A sequence αn∈Ω·\nS(F) converges to zero if it converges to zero in Ω·(F)\nand there is a compact subset K⊂Ssuch thatS∩suppαn⊂Kfor alln. We have\nthe corresponding cohomology H·\nS(F), and reduced cohomology H·\nS(F). With this\nnotations, observe that Ω·(F) = Ω·\n∅(F) and Ω·\nc(F) = Ω·\nM(F) as topological vector\nspaces.\nLetf: (M1,F1)→(M2,F2) be a map of foliated manifolds, and let Si⊂Mi,\ni= 1,2, beclosedsaturatedsubsetssuchthat the restriction f:S1→S2is aproper\nmap. Then f∗(Ω·\nS2(F2))⊂Ω·\nS1(F1), yielding a homomorphism f∗:H·\nS2(F2)→\nH·\nS1(F1). In particular we get f∗:H·\nc(F2)→ H·\nS1(F1) iff:S1→M2is proper.\nThe following is what we need to define the intersection number of sub foliations\nwith “appropriate coefficients”:\n•An oriented foliation Fon a manifold M, and two immersed oriented sub-\nfoliationsιi: (Mi,Fi)→(M,F),i= 1,2.\n•dimF= dimF1+dimF2, and codim F= codim F1.\n•Eachιiistransversely regular in the sense that it defines embeddings of\nsmall enough local transversals of Fiinto local transversals of F; i.e. the\nhomomorphism defined by the differential of ιibetween the normal bundles\nofFiandFis injective on the fibers.\n•A compactly supported basic function fofF1.\n•A holonomy invariant transverse distribution DofF2such that the map\nι2: suppD→Mis proper.LEAFWISE REDUCED COHOMOLOGY 5\n•The restrictions ι1|suppfandι2|suppDintersect transversely inFin the\nsense that, for all leaves LiofFiandLofFsuch thatL1⊂suppf,L2⊂\nsuppDandι1(L1)∪ι2(L2)⊂L, the immersed submanifolds ιi:Li→L\nintersect transversely in L.\nObserve that there are open neighborhoods, N1of suppfandN2of suppD, such\nthat theιi|Niintersect transversely in F. Consider the pull-back diagram\nTσ1− −−− →N1\nσ2/arrowbt/arrowbtι1\nN2ι2− −−− →M .\nHere\nT={(x1,x2)∈N1×N2|ι1(x1) =ι2(x2)},\nand theσiare restrictions of the factor projections. It is easy to check th atι1×ι2:\nN1×N2→M×Mis transverse to the diagonal ∆, and thus Tis a manifold\nwith dimT= codim F2. Moreover the σiare immersions, and σ2is transverse\ntoF2. SoDdefines a distribution on T, which will be denoted by DT. We also\nhave the locally constant intersection function ε:T→ {±1}, whereε(x1,x2) =±1\ndepending on whether the identity\nTιi(xi)F ≡ι1∗Tx1F1⊕ι2∗Tx2F2\nis orientation preserving or orientation reversing. On the other ha nd\n(ι1(suppf)×ι2(suppD))∩∆\niscompactbecauseit isaclosedsubsetofthe compactspace ι1(suppf)×ι1(suppf).\nSo\nsuppσ∗\n1f∩suppDT= (ι1×ι2)−1((ι1(suppf)×ι2(suppD))∩∆)\nis a compact subspace of Tsinceι1×ι2: suppf×suppD→M×Mis a proper\nmap. Thus the following definition makes sense.\nDefinition 2.1. With the above notations, the intersection number of (ι1,f) and\n(ι2,D), denoted by /a\\}b∇acketle{t(ι1,f),(ι2,D)/a\\}b∇acket∇i}ht, is defined as DT(g) for any compactly sup-\nported function gonTwhich is equal to the product εσ∗\n1fon some neighborhood\nof suppσ∗\n1f∩suppDT.\nNow our first main theorem is the following.\nTheorem 2.2. LetFbe an oriented foliation on a manifold M, andιi: (Mi,Fi)→\n(M,F),i= 1,2, transversely regular immersed oriented subfoliations. S uppose\ndimF= dimF1+ dimF2, andcodimF= codim F1. Letfmbe a sequence of\ncompactly supported basic functions of F1, andDna sequence of holonomy invari-\nant transverse distributions of F2such that each restriction ι2: suppDn→Mis\na proper map. Suppose each pair ι1|suppfmandι2|suppDnintersect transversely in\nF, and/a\\}b∇acketle{t(ι1,fm),(ι2,Dn)/a\\}b∇acket∇i}ht /\\e}atio\\slash= 0if and only if m=n. ThendimHk\nc(F) =∞for\nk= dimF2.\nThe following two corollaries are the first type of consequences of T heorem 2.2;\nthe second corollary follows directly from the first one.6 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR\nCorollary 2.3. LetFbe an oriented foliation of codimension q >0,La leaf of\nF, andh:π1(L)→G∞\nqits holonomy representation, where G∞\nqis the group of\ngerms at the origin of local diffeomorphisms of Rqwith the origin as fixed point. Let\nιi:Ki→L,i= 1,2, be smooth immersions of closed oriented manifolds of com-\nplementary dimension and nontrivial intersection. Suppos e there is a Riemannian\nmetric on Rqso that the elements in the image of the composites\n(3) π1(Ki)π1(ιi)− −−− →π1(L)h− −−− →G∞\nq\nare germs of local isometries. Then dimHki\nc(F) =∞forki= dimKi,i= 1,2.\nCorollary 2.4. LetFbe an oriented Riemannian foliation of positive codimensio n.\nSuppose some leaf of Fhas homology classes of complementary degrees, k1andk2,\nwith non-trivial intersection. Then dimHki\nc(F) =∞,i= 1,2.\nBeforestatingthenexttypeofcorollariesofTheorem2.2, recallt hatasuspension\nfoliation Fis given as follows. Let π:L→Bbe a regular covering map of an\noriented manifold, and let Γ be its group of deck transformations. F or any effective\naction of Γ on some manifold T, consider the right diagonal action of Γ on L×T:\n(z,t)γ= (zγ,γ−1t) forγ∈Γ and (z,t)∈L×T. Then Fis the foliation on\nM= (L×T)/Γwhoseleavesaretheprojectionsofthesubmanifolds L×{∗} ⊂L×T.\nThe element in Mdefined by each ( z,t)∈L×Twill be denoted by [ z,t]. The\nmapρ:M→Bgiven byρ([z,t]) =π(z) is a fiber bundle projection with typical\nfiberT. The leaves of Fare transverse to the fibers of ρ, and define coverings of\nB. The leaf that contains [ z,t] can be canonically identified to L/Γt, where Γ tis\nthe isotropy subgroup of Γ at t. This leaf is dense if and only if the Γ-orbit of tis\ndense inT.\nCorollary 2.5. With the above notation, let h:π1(B)→Γbe the surjective\nhomomorphism defined by the regular covering LofB, and letιi:Ki→B,i= 1,2,\nbe immersions of connected oriented manifolds of complemen tary dimension in B.\nSupposeK1is a closed manifold, ι2a proper map, and the homology class defined\nbyι1has non-trivial intersection with the locally finite homolo gy class defined by\nι2. For each i, letΓi⊂Γbe the image of the composite\nπ1(Ki)π1(ιi)− −−− →π1(B)h− −−− →Γ.\nLetfmbe a sequence of compactly supported Γ1-invariant functions on T, andDn\na sequence of Γ2-invariant distributions on Tsuch thatDn(fm)/\\e}atio\\slash= 0if and only if\nm=n. ThendimHk\nc(F) =∞fork= dimK2.\nCorollary 2.6. LetB,L,h,Γ,T,F,Ki,ιiandΓibe as in Corollary 2.5. Let\nµbe aΓ2-invariant measure on T. Suppose the closure of the image of Γ1in the\ntopological group of diffeomorphisms of T(with the weak C∞topology)is a compact\nLie group, and there is an infinite sequence of Γ1-invariant open subsets of Twith\nnon-trivial µ-measure and pairwise disjoint Γ2-saturations. Then dimHk\nc(F) =∞\nfork= dimK2.\nObserve that, in Corollary 2.6, the infinite sequence of Γ 1-invariant open sets\nmay not be Γ 2-invariant, and their Γ 2-saturations may not be Γ 1-invariant.\nCorollary 2.7. LetB,L,h,Γ,TandFbe as in Corollary 2.5. Suppose that there\nis a loopc:S1→Bwith a lift to Lthat joins two distinct points of the end set of L.\nLeta=h([c])∈Γ, where[c]is the element of π1(B)represented by c, and assumeLEAFWISE REDUCED COHOMOLOGY 7\nthat the closure Hof the image of /a\\}b∇acketle{ta/a\\}b∇acket∇i}htin the topological group of diffeomorphisms\nofT(with the weak C∞topology)is a compact Lie group. Suppose also that there\nis an infinite sequence of disjoint non-trivial H-invariant open subsets of T. Then\ndimH1\nc(F) =∞.\nIn Corollaries 2.5, 2.6 and 2.7, if Bis compact, then the leaves of Fcan only\nbe dense when Lhas either one end or a Cantor space of ends, as follows from the\nfollowing.\nProposition 2.8. LetΓbe a finitely generated group with two ends, and C⊂Γ\nan infinite subgroup. Suppose Γacts continuously on some connected T1topological\nspaceX. Then the Γ-orbits are dense in Xif and only if so are the C-orbits.\nNow letFbe a Lie g-foliation on a closed manifold M. The following property\ncharacterizes such a type of foliations [11, 24, 25]. Let /tildewiderMbe the universal covering\nofM,/tildewideFthe lift of Fto/tildewiderM, andGthe simply connected Lie group with Lie\nalgebrag. Then the leaves of /tildewideFare the fibers of a fiber bundle /tildewiderM→G, which\nis equivariant with respect to some homomorphism h:π1(M)→G, where we\nconsider the right action of π1(M) on/tildewiderMby deck transformations and the right\naction ofGon itself by right translations. This hand its image are respectively\ncalled the holonomy homomorphism andholonomy group ofF. Observe that the\nfibers ofDare connected because Gis simply connected (a connected coveringof G\nis given by the quotient of /tildewiderMwhose points are the connected components of these\nfibers).\nTheorem 2.9. With the above notation, suppose that Mis oriented and the struc-\ntural Lie algebra gofFis compact semisimple. Let ιi:Ki→M,i= 1,2,\nbe immersions of closed oriented manifolds of complementar y dimension defining\nhomology classes of Mwith non-trivial intersection. Let Γibe the image of the\ncomposite\nπ1(Ki)π1(ιi)− −−− →π1(M)h− −−− →G.\nSuppose the group generated by Γ1∪Γ2is not dense in G. Letk= dimK2, and\nsuppose either 1≤k≤2orι1is transverse to F. ThendimHk(F) =∞.\nThe following is our final theorem.\nTheorem 2.10. LetHbe a simply connected nilpotent Lie group, K⊂Ha normal\nconnected subgroup, and Γ⊂Ha discrete uniform subgroup whose projection to\nH/Kis dense. Let Fbe the foliation of the closed nilmanifold Γ\\Hdefined as the\nquotient of the foliation on Hwhose leaves are the translates of K. Then there is\na canonical isomorphism H·(F)∼=H·(k), wherekis the Lie algebra of K.\nThe following two examples are of different nature. In both of them t here are\ninfinitely many linearly independent leafwise reduced cohomology class es of degree\none. But these classes are induced by the handles in the leaves in Exa mple 2.11,\nwhereasthey areinduced by the “branches”of the leavesthat de fine a Cantorspace\nof ends in Example 2.12.\nExample 2.11 ([4]).LetLbe aZ-covering of the compact oriented surface of\ngenus two; i.e., Lis a cylinder with infinitely many handles attached to it. Each\nhandle contains two circles defining homology classes with non-trivial intersection.\nHence for any injection of Zinto then-torusRn/Zn, the corresponding suspension8 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR\nfoliation fulfills the hypotheses of Corollary 2.6, and thus has infinite d imensional\nreduced leafwise cohomologyof degree one. We could also use Corolla ry 2.4 instead\nof Corollary 2.6.\nExample 2.12. Let Γ be the free group with two generators, and La Γ-covering\nof the compact orientable surface of genus two. This Lhas a Cantor space of ends.\nHence, for any injective homomorphism of Γ in a compact Lie group, t he reduced\nleafwise cohomology of degree one of the corresponding suspensio n foliation is of\ninfinite dimension by Corollary 2.7.\n3.Leafwise reduced cohomology and subfoliations\nThis section is devoted to the proof of Theorem 2.2. With the notatio ns in-\ntroduced in Section 2, the idea of the proof is the following. The ( ι1,fm) yield\nelementsζm∈ Hr\nc(F) by “leafwise Poincar´ eduality”. On the other hand, the argu-\nmentsin[15]showthateach Dncanbeconsideredasanelementin Hr\nSn(F2)′, where\nSn= suppDn⊂M. Moreover there are homomorphisms ι∗\n2:H·\nc(F)→ H·\nSn(F2)\nsince theι2:Sn→Mare proper maps. Then the result follows by verifying\n/a\\}b∇acketle{t(ι1,fm),(ι2,Dn)/a\\}b∇acket∇i}ht=Dn(ι∗\n2ζm).\nWe first explain the way “leafwise Poincar´ e duality” works. Consider thetrans-\nverse complex Ω·\nc(TrF) introduced in [15], which will be only used for degree zero.\nFor any representative Hof the holonomy pseudogroup of F, defined on some\nmanifoldT, Ω0\nc(TrF) is defined as the quotient of C∞c(T) over the subspace gen-\nerated by the functions of the type φ−h∗φ, whereh∈ Handφ∈C∞c(T) with\nsuppφ⊂domh. As a topological vector space, Ω0\nc(TrF) is independent of chosen\nrepresentative of the holonomy pseudogroup. From the definition it easily follows\nthat the dualspaceΩ0\nc(TrF)′canbe canonicallyidentified tothe spaceofholonomy\ninvariant transverse distributions of F.\nNow consider the representative Hof the holonomy pseudogroup induced by an\nappropriate locally finite covering of Mby foliation patches Ui; that is, iffi:Ui→\nTiis the localquotient map whosefibers arethe plaques in Ui, then appropriateness\nof this covering means that each equality fj=hi,jfionUi∩Ujdetermines a\ndiffeomorphisms hi,j:fi(Ui∩Uj)→fj(Ui∩Uj), and the collection of all of these\ndiffeomorphisms generate the pseudogroup HonT=/unionsqtext\niTi. Fix also a partition\nof unityφisubordinated to the covering Ui. With these data we have a map\nΩp\nc(F)→Ω·\nc(T) given by α/mapsto→/summationtext\ni/integraltext\nfiφiα, wherep= dimFand/integraltext\nfidenotes\nintegration along the fibers of fi. This “integration along the leaves” induces an\nisomorphism Hp(F)∼=Ω0\nc(TrF) of topological vector spaces, which is independent\nof the choice of the Uiandφi[15,§3.3]. So\nHp\nc(F)′≡Hp\nc(F)′∼=Ω0\nc(TrF)′;\ni.e., any holonomy invariant distribution Dcan be canonically considered as an\nelement in Hp\nc(F)′. Moreover Dcan be also considered as an element in Hp\nS(F)′≡\nHp\nS(F)′forS= suppDasfollowsfromthe followingargument. Forany α∈Ωp\nc(F),\nit is easily verified that D/parenleftBig/summationtext\ni/integraltext\nfiφiα/parenrightBig\ndepends only on the restriction of αto any\nneighborhood of the support of DinM. Therefore, if ζ∈Hp\nS(F),α∈Ωp\nS(F)\nis any representative of ζ, andβ∈Ωp\nc(F) has the same restriction as αto some\nneighborhood of S, thenD/parenleftBig/summationtext\ni/integraltext\nfiφiβ/parenrightBig\ndoes not depend on the choices of αand\nβ, and thus this is a good definition of D(ζ).LEAFWISE REDUCED COHOMOLOGY 9\nTheorem 2.2 will follow easily from the following result, which will be prove d in\nSection 4.\nProposition 3.1. LetFbe an oriented foliation on a manifold M. Letι1:\n(M1,F1)→(M,F)be a transversely regular immersed oriented subfoliation w ith\ncodimF= codim F1, andfa compactly supported basic function of F1. Then\nthere is a class ζ∈ Hk\nc(F),k= dimF −dimF1, such that\n(4) /a\\}b∇acketle{t(ι1,f),(ι2,D)/a\\}b∇acket∇i}ht=D(ι∗\n2ζ)\nfor any subfoliation ι2: (M2,F2)→(M,F)and any holonomy invariant transverse\ndistribution DofF2so that the left hand side of (4)is defined. In the right hand\nside of(4),Dis considered as an element of Hk\nS(F2)′forS= suppD, andι∗\n2\ndenotes the homomorphism Hk\nc(F)→ Hk\nS(F2)induced byι2, which is defined since\nι2:S→Mis a proper map.\nWe do not know whether (4) completely determines ζ. If so,ζcould be called\ntheleafwise Poincar´ e dual class of (ι1,f).\nProof of Theorem 2.2. Letζm∈ Hk\nc(F) be the classes defined by the ( ι1,fm) ac-\ncording to Proposition 3.1. If Pn∈ Hk\nc(F)′is given by the composite\nHk\nc(F)ι∗\n2− −−− → Hk\nSn(F2)Dn− −−− →R,\nwe havePn(ζm)/\\e}atio\\slash= 0 if and only if m=nby Proposition 3.1, yielding the linear\nindependence of the ζm. /square\n4.Leafwise Poincar ´e duality\nThis section will be devoted to the proof of Proposition 3.1.\n4.1.On the Thom class of a vector bundle. Thefollowinglemma isatechnical\nstep in the proof of Proposition 3.1, which will be proved in Section 4.2.\nLemma 4.1. LetMbe a manifold and π:E→Man oriented vector bundle.\nIdentifyMto the image of the zero section, whose normal bundle is canon ically\noriented. There is a sequence Φnof representatives of the Thom class of Esuch\nthat, iffis any function on M, V is any neighborhood of MinE,K⊂Mis any\ncompact subset, and φ:V→Eis any map which restricts to the identity on M\nand its differential induces an orientation preserving auto morphism of the normal\nbundle ofM, thenπ−1(K)∩φ−1(suppΦ n)is compact for large enough n, and the\nsequence of functions/integraltext\nπφ∗(π∗fΦn)converges to foverKwith respect to the C∞\ntopology.\nCorollary 4.2. Letπ:E→Mbe an oriented vector bundle, and ι:N→Man\nimmersion. Let πN:ι∗E→Nbe the pull-back vector bundle, and ˜ι:ι∗E→E\nthe canonical homomorphism. Identify MandNto the image of the zero sections\nofEandι∗E, respectively, and consider the induced orientations on th eir normal\nbundles. Let Vbe an open neighborhood of Ninι∗E, andh:V→Ean extension\nofιsuch that the homomorphism between the normal bundles of NandM, defined\nby the differential of h, restricts to orientation preserving isomorphisms betwee n\nthe fibers. Let Φnbe the forms on Egiven by Lemma 4.1, K⊂Na compact\nsubset, and fa function on M. Thenπ−1\nN(K)∩h−1(suppΦ n)is compact for large\nenoughn, and the sequence of functions/integraltext\nπNh∗(π∗fΦn)converge to ι∗foverK\nwith respect to the C∞topology.10 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR\nProof.LetU1,...,U mbe a finite open cover of Ksuch that each ι:Ui→Mis an\nembedding. For each i, there is a compactly supported function fionMwhich is\nsupportedinsometubularneighborhood Wiofι(Ui), andsuchthat f=f1+···+fm\non some neighborhood of ι(K). Then, taking a neighborhood Viof eachUiinVso\nthath:Vi→Eis an embedding, we get\n(5)/integraldisplay\nπNh∗(π∗fΦn) =/summationdisplay\ni/integraldisplay\nπN|π−1\nN(Ui)(h|Vi)∗(π∗fiΦn)\naroundK, yielding the result if each term in the right hand side of (5) converge s\ntoι∗fi. Therefore we can assume ι, ˜ιandhare embeddings.\nWith this assumption, there is an open disk bundle DoverV, and extensions\n˜ι′,h′:D→π−1(V) of ˜ιandh, respectively, which are diffeomorphisms onto open\nsubsets ofE. Letφdenote the composite\n˜ι′(D)(˜ι′)−1\n− −−− →Dh′\n− −−− →π−1(V).\nClearly,φsatisfies the conditions of Lemma 4.1, and we can suppose fis supported\nin ˜ι′(D). So/integraltext\nπφ∗(π∗fΦn) converges to fover any compact subset of Vwith\nrespect to the C∞topology. But\nι∗/integraldisplay\nπφ∗(π∗fΦn) =/integraldisplay\nπN((˜ι′)∗φ∗(π∗fΦn)|V)\n=/integraldisplay\nπNh∗(π∗fΦn),\nand the result follows. /square\nObserve that Lemma 4.1 is a particular case of Corollary 4.2. The coro llary\ncould be proved directly with the arguments of the lemma, but the no tation would\nbecome more complicated.\n4.2.Proof of Lemma 4.1. The following easy observations will be used to prove\nLemma 4.1.\nRemark 1.LetEandFbe vector bundles over the manifolds MandN, respec-\ntively. Suppose f:E→Fis a homomorphism which restricts to isomorphisms\non the fibers, and let g:M→Nbe the map induced by f. Thus the homomor-\nphismE→g∗F, canonically defined by f, is an isomorphism. Therefore there is a\ncomposite of homeomorphisms\nC∞(F)→C∞(g∗F)→C∞(E).\nHere, the first homomorphism is canonically defined by the pull-back d iagram of\ng∗F, and the second one is induced by the inverse of E→g∗F. Ifs/mapsto→s′by the\nabove composite, then s′is determined by f(s′(x)) =s(g(x)) forx∈M.\nRemark 2.SetE=Rn×Rk, and letπi,i= 1,2, denote the factor projections of\nEontoRnandRk, respectively. Let Kbe a compact subset of Rn, andφ:V→W\na diffeomorphism between open neighborhoods of Rn×{0}. Supposeφrestricts to\nthe identity on Rn× {0}. For anyr >0, letBr,Sr⊂Rkrespectively denote the\nEuclidean ball and the Euclidean sphere of radius r centered at the o rigin. Then\nthere is anR>0 and an open neighborhood UofKsuch that, for every x∈UandLEAFWISE REDUCED COHOMOLOGY 11\neveryy∈BR,{x} ×Rkintersects transversely φ−1(Rn× {y}) at just one point.\nMoreover, the map\nσ:U×BR→(U×Rk)∩φ−1(Rn×BR),\ndetermined by\n{σ(x,y)}= ({x}×Rk)∩φ−1(Rn×{y}),\nis a diffeomorphism. Indeed σis smooth because each ( U×Rk)∩φ−1(Rn×{y}) can\nbe given as the graph of a map ψy:U→Rkdepending smoothly on y∈BR, and\nσ(x,y) = (x,ψy(x)). It also has a smooth inverse since ( x,y) = (x,π2φσ(x,y)).\nTherefore, for r≤R,π1: (U×Rk)∩φ−1(Rn×Sr)→Uis a sphere bundle, whose\nfibers are of volume uniformly bounded by Crk−1for someC >0 ifUandRare\nsmall enough.\nTo begin with the proof of Lemma 4.1, fix a Riemannian structure on E, and\nletBr,Sr⊂Erespectively denote the corresponding open disk bundle and spher e\nbundle of radius r. SetS=S1. Letψbe a global angular form of S[9,§11]. (IfE\nis of rankk,ψis a differential form of degree k−1 restricting to unitary volume\nforms on the fibers and so that dψ=−π∗e, whereerepresents the Euler class of\nS.) Letr:E→Rdenote the radius function, and h:E\\M→Sthe deformation\nretraction given by h(v) =v/r(v). For each n, let alsoρnbe a function on [0 ,∞)\nsuch that −1≤ρn≤0,ρ′\nn≥0,ρn≡ −1 on a neighborhood of 0, and ρn≡0 on\n[1/n,∞). Then each\nΦn=d(ρn(r)h∗ψ) =ρ′\nn(r)dr∧h∗ψ−ρn(r)π∗e\nrepresents the Thom class of E[9,§12].\nLocal orthonormal frames canonically define isomorphisms of trivia lity ofE\nwhich restrict to local isomorphisms between restrictions of each Srand trivial\nsphere bundles with typical fiber the Euclidean sphere of radius r. So Remark 2\nand the conditions satisfied by φyield the existence of some R,C >0 and some\nrelatively compact open neighborhood UofKinMso that\n•π−1(U)∩φ−1(BR)⊂V,\n•the map\nφ:π−1(U)∩φ−1(BR)→φπ−1(U)∩BR\nis a diffeomorphism whose differential is of fiberwise uniformly bounded\nnorm, and\n•for 0< r≤R,φ−1(Sr) is transverse to the fibers of πoverUandπ:\nπ−1(U)∩φ−1(Sr)→Uis a sphere bundle whose fibers are of volume\nuniformly bounded by Crk−1.\nTheφ∗Φnalso represent the Thom class of EoverUforn>1/R. Hence\nf−/integraldisplay\nπφ∗(π∗fΦn)\n=/integraldisplay\nπ(π∗f−φ∗π∗f)φ∗Φn\n=/integraldisplay1/n\n0ρ′\nn(r)dr/integraldisplay\nπ|π−1(U)∩φ−1(Sr)(π∗f−φ∗π∗f)φ∗h∗ψ (6)\n−/integraldisplay\nπ(π∗f−φ∗π∗f)ρn(φ∗r)φ∗π∗e. (7)12 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR\nWe have to prove that (6) and (7) converge uniformly to zero on Kasn→ ∞, as\nwell as all of its derivatives of any order.\nTake a Riemannian metric on M, and a splitting TE=V ⊕H, whereVis the\nvertical bundle of πandHthe horizontal bundle of any Riemannian connection.\nThis yields a Riemannian structure on TEdefined in the standard way by using\nthe canonical isomorphisms V∼=π∗EandH∼=π∗TM. We also have TM=H|M,\nH|Sr⊂TSr, and\n(8) TSr= (V ∩TSr)⊕(H|Sr).\nFinally, we can assume\n(9) V ∩φ−1\n∗(H) = 0 over π−1(U)∩φ−1(BR)\nby the properties of φ.\nBy the conditions on φ, the supremum of |π∗f−φ∗π∗f|overπ−1(U)∩φ−1(B1/n)\nconvergestozeroas n→ ∞. Alsothepointwisenormof φ∗π∗eisuniformlybounded\nonπ−1(U)∩φ−1(B1/n), thus (7) converges uniformly to zero as n→ ∞. On the\notherhand, because the fiberwise normof each h∗:TSr→TSisr−1, the pointwise\nnorm ofφ∗h∗ψis uniformly bounded on π−1(U)∩φ−1(Sr) byC1r−k+1withC1>0\nindependent of r≤R. So (6) also converges uniformly to zero on Uasn→ ∞by\nthe estimate on the volume of the fibers of πonπ−1(U)∩φ−1(Sr).\nNow fix vector fields X1,...,X monU. By (8) and (9) the Xihave liftings Yi\nwhich are sections of φ−1\n∗Hoverπ−1(U)∩φ−1(BR). For any subset I⊂ {1,...,m},\nletθIdenote the composite of Lie derivatives θY1···θYlifI={i1,...il}with\ni1< i2<···< il, and letθ∅be the identity homomorphism. Then the order m\nderivativeX1···Xmover (6) and (7) is respectively given by\n(10)/summationdisplay\nI,J/integraldisplay1/n\n0ρ′\nn(r)dr/integraldisplay\nπ|π−1(U)∩φ−1(Sr)θI(π∗f−φ∗π∗f)θJφ∗h∗ψ,\nand\n(11) −/summationdisplay\nI,J/integraldisplay\nπθI(π∗f−φ∗π∗f)ρn(φ∗r)θJφ∗π∗e,\nwhereI,Jruns over the partitions of {1,...,m}. By the properties of Handφ,\nthe supremum of the |θI(π∗f−φ∗π∗f)|onπ−1(U)∩φ−1(B1/n) converges to zero\nasn→ ∞. Hence (11) converges uniformly to zero on Kbecause the pointwise\nnormofthe θJφ∗π∗ecanbe uniformlybounded on π−1(K)∩φ−1(BR). The uniform\nconvergenceof (10)tozerofollowsbyestimatingthepointwisenor moftheθJφ∗h∗ψ\nonπ−1(K)∩φ−1(Sr) byC2r−k+1for someC2>0 independent of r. This in\nturn follows by proving a similar estimate for the pointwise norm of θ′\nJh∗ψon\nφπ−1(K)∩Sr, where the θ′\nJare defined in the same way as the θJby using the\nY′\ni=φ∗Yiinstead of the Yi. To do this, consider the multiplication map µ:\n[0,R]×S→BR. Since\nµ∗: [0,1]×(H|S)⊂T([0,1]×S)→ H\nrestricts to isomorphisms on the fibers, by Remark 1 there are smo oth sections Y′′\ni\nof [0,1]×(H|S) so thatµ∗(Y′′\ni(r,v)) =Y′\ni(rv). Also because the composite\n(0,R]×Sµ− −−− →BR\\Mh− −−− →SLEAFWISE REDUCED COHOMOLOGY 13\nis the second factor projection, µ∗h∗ψis the form canonically defined by ψon\n(0,R]×S, which extends smoothly to [0 ,R]×S. So, ifθ′′\nJis defined in the same\nway as theθJby using the Y′′\niinstead of the Yi, the pointwise norm of the θ′′\nJµ∗h∗ψ\nis uniformly bounded. Then the desired estimation of the pointwise no rm of the\nθ′\nJh∗ψfollows by observing that the fiberwise norm of µ∗:{r}×TS→TSrisr.\n4.3.Proof of Proposition 3.1. Recall that any local diffeomorphism φ:M→N\ninduces a homomorphism of complexes, φ∗: Ωc(M)→Ωc(N), defined as follows.\nFor anyα∈Ωc(M), choose a finite open cover U1,...,U nof suppαsuch that\neach restriction φ:Ui→φ(Ui) is a diffeomorphism. There is a decomposition\nα=α1+···+αnso that supp αi⊂Ui. For eachi, there is a unique βi∈Ωc(N)\nsupported in φ(Ui) such that βi|φ(Ui)corresponds to αi|Uibyφ. Defineφ∗α=\nβ1+···+βn. This definition is easily checked to be independent of the choices\ninvolved and compatible with the differential maps. If φ: (M,F)→(N,G) is\na local diffeomorphism of foliated manifolds, we similarly have a homomor phism\nφ∗: Ωc(F)→Ωc(G) which is compatible with the leafwise de Rham derivative.\nMoreoverφ∗is surjective if so is φ.\nNow Proposition 3.1 can be proved as follows.\nThere is a canonical injection of TF1as vector subbundle of ι∗\n1TF. LetE=\nι∗\n1TF/TF1, andπ:E→M1the bundle projection. Identify M1with the image\nof the zero section of E. Fixing any Riemannian metric on M, there are induced\nRiemannian metrics on the Mi, and an induced Riemannian structure on E. For\neachr >0, letBr⊂Edenote the open disk bundle of radius roverM1. Then\nthere is an R >0 and an open neighborhood Uof the support of finM1such\nthat, ifV=π−1(U)∩BR, the restriction of ι1toUcan be extended to a map of\nfoliated manifolds, ˜ ι1: (V,π∗F1|V)→(M,F), which is defined over each x∈M1\nas a composite of the restriction of the canonical homomorphism\n(ι∗\n1TF/TF1)x→Tι1(x)F/ι1∗TxF1≡(ι1∗TxF1)⊥∩TxF,\nand the exponential map of the leaves of Fdefined on the ball of radius Rcentered\nat zero in (ι1∗TxF1)⊥∩TxF. By elementary properties of the exponential map and\nsinceι1is transversely regular with codim F1= codim F,Rcan be chosen so that\n˜ι1is a local diffeomorphism and ˜ ι∗\n1F=π∗F1|V.\nEis of rankk, and with an induced orientation. The representatives Φ nof its\nThom class, given by Lemma 4.1, can be assumed to be supported in BR. The Φ n\nare of degree k, closed and compactly supported in the vertical direction, i.e. with\ncompactly supported restrictions to the fibers. Moreover all the Φn|BRare pairwise\ncohomologousin the complex of forms in Ω·(BR) which are compactly supported in\nthe vertical direction. On the other hand, fis basic and compactly supported. So\ntheπ∗fΦnrestrict to leafwise closed forms αn∈Ωk\nc(π∗F1|V) which are pairwise\ncohomologous. Thus the ˜ ι1∗αn∈Ωk\nc(F) are leafwise closed and define the same\nclassζ∈ Hk\nc(F).\nLetU1,...,U m, be an open cover of the support of finUsuch that each\nι1:Uj→Mis an embedding, j= 1,...,m. The above Rcan be chosen small\nenough so that the ˜ ι1:Vj=π−1(Uj)∩BR→˜ι1(Vj) are diffeomorphisms. Take a\ndecomposition f=f1+···+fmwith eachfjcompactly supported in Uj, and let\nαn,j∈Ωk\nc(π∗F1|Vj) be the restriction to the leaves of π∗fjΦn. Then, by definition,\n˜ι1∗αn=βn,1+···+βn,m,14 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR\nwhere each βn,j∈Ωk\nc(F) is the extension by zero of the forms in Ωk\nc(F|˜ι1(Vj)) which\ncorrespond to the αn,j|Vjby ˜ι1.\nGivenι2: (M2,F2)→(M,F), we use the same notation as in the preamble of\nDefinition 2.1. We can clearly assume the Ujare contained in N1. Letι=ι1σ1=\nι2σ2:T→M. There is a canonical isomorphism\nι∗TF∼=σ∗\n1TF1⊕σ∗\n2TF2\nbecause the ιi|Niintersect transversely in F. Soσ∗\n1E∼=σ∗\n2TF2canonically. This\nisomorphism will be considered as an identity.\nLetπT:σ∗\n1E→Tbe the pull-back vector bundle projection, and ˜ σ1:σ∗\n1E→E\nthe canonical homomorphism. Identify Tto the image of the zero section of σ∗\n1E.\nFor eachj, take a relatively compact open subset Oj⊂σ−1\n1(Uj) containing the\ncompact set supp σ∗\n1fj∩suppDT. The above Rcan be chosen small enough so that\nσ2:Oj→N2hasan extensionto alocaldiffeomorphism ˜ σ2:π−1\nT(Oj)∩˜σ−1\n1(BR)→\nN2defined as the composite of the restriction of the canonical homom orphism\nσ∗\n1E≡σ∗\n2TF2→TF2, and the exponential map of the leaves of F2defined on the\ntubular neighborhood of radius Rof the zero section in TF2. In this way, ˜ σ2maps\neach fiber of πTinto a leaf of F2. Observe that the diagram\n˜σ−1\n1(BR)∩π−1\nT(Oj)˜σ1− −−− →Vj\n˜σ2/arrowbt/arrowbt˜ι1\nN2ι2− −−− →M\nis obviously non-commutative in general. This is the main technical diffic ulty. To\nsolve it, we have chosen the Φ nso that their supports concentrate around M1and\nsatisfy the needed properties at the limit (Lemma 4.1 and Corollary 4.2 ).\nWe need the observation that\n(12) σ2σ−1\n1(A) = (ι2|N2)−1ι1(A)\nfor any subset A⊂N1, as can be easily checked.\nUsing the compactness of BR∩π−1(suppfj) and since\nsuppfj=/intersectiondisplay\n0<r<RBr∩π−1(suppfj),\nwe easily get\nι1(suppfj) =/intersectiondisplay\n0<r<R˜ι1/parenleftbig\nBr∩π−1(suppfj)/parenrightbig\n.\nTherefore/intersectiondisplay\n0<r<RsuppD∩ι−1\n2˜ι1/parenleftbig\nBr∩π−1(suppfj)/parenrightbig\n= suppD∩ι−1\n2ι1(suppfj)\n= suppD∩σ2σ−1\n1(suppfj)\n=σ2(σ−1\n2(suppD)∩σ−1\n1(suppfj))\n=σ2(suppDT∩suppσ∗\n1fj),\nwhere the second equality follows by (12). Then, since the\nsuppD∩ι−1\n2˜ι1/parenleftbig\nBr∩π−1(suppfj)/parenrightbig\nare compact, and since\n˜σ2/parenleftbig\n˜σ−1\n1(BR)∩π−1\nT(Oj)/parenrightbigLEAFWISE REDUCED COHOMOLOGY 15\nis an open neighborhood of\nσ2(suppDT∩suppσ∗\n1fj),\nthere is an r<Rsuch that\nsuppD∩ι−1\n2˜ι1/parenleftbig\nBr∩π−1(suppfj)/parenrightbig\n⊂˜σ2/parenleftbig\n˜σ−1\n1(BR)∩π−1\nT(Oj)/parenrightbig\n.\nSo\nsuppD∩suppι∗\n2βn,j⊂˜σ2(Wj)\nfor large enough n, where\nWj= ˜σ−1\n2ι−1\n2˜ι1(Vj)∩˜σ−1\n1(BR)∩π−1\nT(Oj).\nWe can assume this holds for every n. Hence there is some ωn,j∈Ωk\nc(F2) which is\nsupported in ˜ σ(Wj) and has the same restriction to some neighborhood of supp D\nasι∗\n2βn,j. IfFπTis the foliation on σ∗\n1Edefined by the fibers of πT, there is some\nγn,j∈Ωk\nc(FπT|Wj) such that (˜ σ2|Wj)∗ωn,j=ι∗\n2βn,j. We get\n(13) D(ι∗\n2ζ) =/summationdisplay\njDT/parenleftBigg/integraldisplay\nπT|Wjγn,j/parenrightBigg\nby definition. Let hj:Wj→Ebe the immersion given by the composite\nWj˜σ2− −−− →ι−1\n2˜ι1(Vj)ι2− −−− →˜ι1(Vj)˜ι−1\n1− −−− →Vj⊂E .\nClearlyhjis an extension of σ1:Oj→N1⊂M1⊂E, andγn,j=h∗\nj(π∗fjΦn)\naroundWj∩π−1\nT(σ−1\n1(suppfj)∩suppDT). Moreover the homomorphism between\nthe normal bundles of OjandM1, defined by the differential of hj, restricts to\nisomorphismsonthefibers. Theseisomorphismsareorientationpre servingonfibers\nover points with ε= 1, and orientation reversing on fibers over points with ε=−1.\nTherefore/integraltext\nπT|Wjγn,jconverges to εσ∗\n1fjonσ−1\n1(suppfj)∩suppDTwith respect\nto theC∞topology by Corollary 4.2. Hence (13) is equal to /a\\}b∇acketle{t(ι1,f),(ι2,D)/a\\}b∇acket∇i}ht, and\nthe proof is complete.\nRemark3.Observe that, in Proposition 3.1, ζhas representativessupported in any\nneighborhood of ι1(M1). Thus, in Theorem 2.2, the linearly independent classes\nζm∈ H·\nc(F) also have representatives supported in any neighborhood of ι1(M1).\n5.Case where the leaves have homology classes with non-trivia l\nintersection\nThis section will be devoted to the proof of Corollary 2.3.\nLetMbetheambientmanifoldof F. Letπi:ι∗\ni(TM/TF)→Kidenotethepull-\nbackvectorbundle projection, andidentify Kitothe imageofitszerosection. Fix a\nRiemannian metric on Mand, for some R>0, letMibe the tubular neighborhood\nof radiusRaroundKiinι∗\ni(TM/TF). SuchRcan be chosen so that the maps\n˜ιi:Mi→Mare well defined as composites of the restrictions of the canonical\nhomomorphisms ι∗\ni(TM/TF)→TM/TF ≡(TF)⊥, and the restriction of the\nexponential map of Mto the tubular neighborhood of radius Rof the zero section\nofTF⊥. ChooseRsmall enough so that ˜ ιi:π−1\ni(xi)∩Mi→˜ιi(π−1\ni(xi)∩Mi)\nis an embedded transversal of Ffor eachiand eachxi∈Mi. Observe that\n˜ι1(π−1\n1(x1)∩M1) = ˜ι2(π−1\n2(x2)∩M2) ifι1(x1) =ι2(x2). The ˜ιiare thus transverse\ntoF, and the Fi= ˜ι∗\niFhave the same codimension as F. Then ˜ιiare transversely16 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR\nregular immersions of foliated manifolds. By deforming the ιiif needed, we can\nsupposethe ιiintersecteachothertransversely,andthusthe˜ ιiintersecttransversely\ninF. Moreover the orientations of the Kiinduce orientations of the Fi.\nEachKiis a closed leaf of Fiwhose holonomy representation is given by the\ncomposite (3). So the holonomy group of Kiis given by germs of local isometries.\nHenceFiisaRiemannianfoliationaround Ki, asfollowseasilyfrom[14, Theorem2\nin Chapter IV]. (See also [12, Theorem 2.29 in Chapter II] or [17].) We ca n assume\nthe whole Fiis Riemannian, which can thus be described as follows [16, 26]. Fix\na transverse Riemannian structure of Fi. LetQibe theO(q)-principal bundle\noverMiof transverse orthonormal frames of Fiwith the transverse Levi-Civita\nconnection, and ˆFithe horizontal lifting of FitoQi[24, 25]. Let Pibe a leaf\nclosure of ˆFioverKi. ThenPiis anHi-principal bundle over Kifor some closed\nsubgroupHi⊂O(q). For the open disk B⊂Rqof radiusRcentered at the origin,\nwe can assume Mi≡(Pi×B)/Hias fiber bundles over Ki, where the Hi-action on\nP2×Bis the diagonal one; i.e. ( z,v)h= (zh,h−1v) for (z,h)∈Pi×Bandh∈Hi.\nMoreover the above identity can be chosen so that Fiis identified to the foliation\nwhose leaves are the projections of products of leaves of ˆFiinPiand points in B.\n(This description is simpler than the one in [16] and [26] because the le af closure\nKiis just a compact leaf.)\nConsider the transverse Riemannian structure of each Fidefined by the Eu-\nclidean metric on Busing the above description. Since the elements in the image\nof the composites (3) are germs of local isometries for the same me tric onRq, the\ncomposite\nB≡π−1\n2(x2)∩M2˜ι2−→˜ι2(π−1\n2(x2)∩M2) = ˜ι1(π−1\n1(x1)∩M1)˜ι−1\n1−→π−1\n1(x1)∩M1≡B\nis an isometry around the origin for all ( x1,x2)∈K1×K2withι1(x1) =ι2(x2). We\ncan assume such composite is an isometry on the whole B, which will be denoted\nbyφx2,x1.\nWith the above description, any compactly supported basic functio nfofF1can\nbe canonically considered as an H1-invariant compactly supported function on B,\nand any compactly supported holonomy invariant transverse distr ibutionDofF2\ncan be canonically considered as a compactly supported H2-invariant distribution\nonB. For suchfandD, we clearly have\n(14) /a\\}b∇acketle{t(˜ι1,f),(˜ι2,D)/a\\}b∇acket∇i}ht=/summationdisplay\nε(x1,x2)D(φ∗\nx2,x1f),\nwhere the sum runs over the pairs ( x1,x2)∈K1×K2withι1(x1) =ι2(x2). Here\nε(x1,x2) =±1 depending on whether the identity\nTιi(xi)B≡ι1∗Tx1K1⊕ι2∗Tx2K2\nisorientationpreservingororientationreversing. Let fmbeasequenceofcompactly\nsupportedO(q)-invariant functions in Bwith integral equal to one and pairwise\ndisjoint supports, and let µmbe the restriction of the Euclidean measure to the\nsupport offm. Then\n/a\\}b∇acketle{t(˜ι1,fm),(˜ι2,µn)/a\\}b∇acket∇i}ht=/a\\}b∇acketle{tι1,ι2/a\\}b∇acket∇i}ht/integraldisplay\nBfmdµn\nby (14), where /a\\}b∇acketle{tι1,ι2/a\\}b∇acket∇i}htis theintersectionnumberof ι1andι2inB. Sodim Hk2\nc(F) =\n∞by Theorem 2.2. Similarly, dim Hk1\nc(F) =∞, which completes the proof.LEAFWISE REDUCED COHOMOLOGY 17\n6.Case of suspension foliations\nProof of Corollary 2.5. Recall the notation used for suspension foliations in the\nstatement of Corollary 2.5, and consider the fiber bundles Mi=ι∗\niMoverKi.\nEach canonical map ˜ ιi:Mi→Mis transverse to F, and let Fi= ˜ι∗\niF. Then ˜ιiare\ntransverselyregularimmersionsoffoliatedmanifolds. Bydeforming theιiifneeded,\nwe can suppose the ιiintersect each other transversely, thus the ˜ ιiintersect each\nother transversely in F. Moreover the orientations of the Kiinduce orientations of\ntheFi.\nThe group of deck transformations of each pull-back covering map ι∗\niL→Kiis\nisomorphic to Γ i, andFiis canonically isomorphic to the corresponding suspension\nfoliation given by the restriction to Γ iof the Γ-action on T. Hence the fmcan be\ncanonically considered as compactly supported basic functions of F1, and theDn\ncan be canonically considered as holonomy invariant transverse dist ributions of F2.\nThe ˜ι2: suppDn→Mare clearly proper, and we easily get\n/a\\}b∇acketle{t(˜ι1,fm),(˜ι2,Dn)/a\\}b∇acket∇i}ht=/a\\}b∇acketle{tι1,ι2/a\\}b∇acket∇i}htDn(fm).\nTherefore the result follows from Theorem 2.2. /square\nProof of Corollary 2.6. LetAnbe a sequence of Γ 1-saturated open subsets of T\nwith non-trivial µ-measure and pairwise disjoint Γ 2-saturations. Clearly, there\nare open sets BnofTwith positive µ-measure and such that Bn⊂An. Since\nthe closure of Γ 1in the group of diffeomorphisms of Tis a compact Lie group,\nthere exists a sequence of non-negative Γ 1-invariant functions fnonTsuch that\nBn⊂suppfn⊂An. Letµnbe the Γ 2-invariant measure on Tdefined as the\nproduct of µand the characteristic function of the closure of the Γ 2-saturation\nof suppfn. Then/integraltext\nTfmdµn/\\e}atio\\slash= 0 if and only if m=n, and the result follows by\nCorollary 2.5. /square\nProof of Corollary 2.7. Since some lift of ctoLjoins two distinct points of its\nend set,Lis disconnected by some codimension one immersed closed submanifold ,\nι:K→L, such that candπιdefine homology classes of Bwith non-trivial\nintersection. Clearly, the composite\nπ1(K)π1(πι)− −−− →π1(B)h− −−− →Γ\nis trivial, and the image of the composite\nπ1(S1)π1(c)− −−− →π1(B)h− −−− →Γ\nis/a\\}b∇acketle{ta/a\\}b∇acket∇i}ht. Take a sequence Anof disjoint non-trivial H-invariant open subsets of T.\nSinceHis an abelian compact Lie group (a torus), there is an H-invariant prob-\nabilistic measure supported in any H-orbit inT. Take thus one of such measures\nµnsupported in each An. Then the result follows from Corollary 2.6 by taking as\nµthe sum of the µn. /square\nTo prove Proposition 2.8, we use the following.\nLemma 6.1. LetΓbe a finitely generated group, and Xa connected T1topological\nspace. For any continuous action of ΓonX, a finite union of orbits is dense if and\nonly if so is each orbit in the union.18 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR\nProof.Takex1,...,x n∈Xsuch that\nX=Γx1∪...∪Γxn=Γx1∪...∪Γxn.\nEach orbit closure Γxican be decomposed as a disjoint union of sets\nLi=/intersectiondisplay\nF(Γ\\F)xi, Ii=Γxi\\Li,\nwhereFruns over the finite subsets of Γ. We have X=L∪I, whereL=/uniontextn\ni=1Li\nandI=/uniontextn\ni=1Ii. Moreover, since Lis saturated we have L∩I=∅. SoI=∅\nbecauseXisT1and connected. (If we had I/\\e}atio\\slash=∅, for anyy∈I,{y}would be\nclosed inXbecauseXisT1. But since Lis closed and I=X\\Lis discrete, {y}\nwould be also open in X. ThusXwould not be connected.) Therefore X=Land\nLi=Γxifor eachi. But each Liis closed in X, andLi∩Lj/\\e}atio\\slash=∅impliesLi=Lj,\nobtainingX=Lifor everyiby the connectedness of X. /square\nProof of Proposition 2.8. Clearly,ifthe C-orbitsaredensein X,soaretheΓ-orbits.\nReciprocally, suppose the Γ-orbits are dense. By a theorem of Sta llings [31],\nthere is a finite normal subgroup F⊂Γ such that Γ 1= Γ/Fis isomorphic either\ntoZor to the diedric group Z2∗Z2. The action of Γ on Xdefines an action of\nΓ1on the connected T1spaceX1=X/Fwith dense orbits. Since Cis infinite, so\nis its projection C1to Γ1, and any infinite subgroup of such Γ 1is of finite index.\nThereforeanyΓ 1-orbitinX1isafiniteunionof C1-orbits,andthusthe C1-orbitsare\ndense inX1by Lemma 6.1. This implies the density of the CF-orbits inXbecause\nthe canonical projection of XontoX1is open and continuous. But any CF-orbit\nis a finite union of C-orbits. Hence the C-orbits are dense by Lemma 6.1. /square\n7.Case of Lie foliations with compact semisimple structural L ie\nalgebra\nTheorem 2.9 will be proved in this section (Corollaries 7.16 and 7.17).\n7.1.Construction of a spectral sequence for an arbitrary Lie fol iation on\na closed manifold. LetFbe a Lie foliationwith dense leaveson a closedmanifold\nM. Letgbe the structural Lie algebra of F, andGthe simply connected Lie group\nwith Lie algebra g. Letπ:/tildewiderM→Mbe the universal covering map. Then the leaves\nof/tildewideF=π∗Fare the fibers of a fiber bundle D:/tildewiderM→G. It will be convenient\nto consider the right action of π1(M) on/tildewiderMby deck transformations and the left\naction ofGon itself by left translations. Thus Dis anti-equivariant with respect to\nthe holonomy homomorphism h:π1(M)→G; i.e.,D(˜xγ) =h(γ)−1D(˜x) [11]. The\ndensity ofthe leavesimplies the density of Γ = h(π1(M)) inG. The homomorphism\nhdefines an action of π1(M) onGby left translations, yielding the corresponding\nsuspension foliation GonN=/parenleftBig\n/tildewiderM×G/parenrightBig\n/π1(M) (defined as in Section 6). Gis a\nLie foliation with the same transverse structure as F, given by ( G,Γ).\nThe section (id ,D) :/tildewiderM→/tildewiderM×Gisπ1(M)-equivariant:\n(id,D)(˜xγ) = (˜xγ,D(˜xγ)) = (˜xγ,γ−1D(˜x)) = (˜x,D(˜x))γ .\nThus (id,D) defines a section s:M→N, andNis trivial as principal G-bundle\noverM. Clearlysis transverse to G, ands∗G=F.LEAFWISE REDUCED COHOMOLOGY 19\nLetD:/tildewiderM×G→Gbe defined by D(˜x,g) =g−1D(˜x). SuchDisπ1(M)-\ninvariant:\nD((˜x,g)a) =D(˜xa,h(a)−1g) =g−1h(a)D(˜xa) =g−1D(˜x).\nSoDdefines a map DN:N→G. ClearlyDNs= const e, whereeis the identity\nelement inG. Moreover DNisG-anti-equivariant:\nDN([˜x,g]g′) =DN([˜x,gg′]) = (gg′)−1D(˜x) = (g′)−1DN([˜x,g]).\nThereforeDNis the composite of the second factor projection of the trivializatio n\nofN→Mdefined bysand the inversion map on G.\nLet/tildewideFalso denote the foliation on Ndefined by the lifting of Fto all the leaves\nofG./tildewideFis a subfoliation of Gwhose leaves are the intersections of the leaves of G\nwith all the translations of s(M).\nLetν⊂TGbe aG-invariant subbundle so that TG=ν⊕T/tildewideF. We get\n/logicalanddisplay\nTG∗=/logicalanddisplay\nν∗⊗/logicalanddisplay\nT/tildewideF∗,\nand thus there is a bigrading of Ω = Ω( G) defined by\nΩu,v=C∞/parenleftBiggu/logicalanddisplay\nν∗⊗v/logicalanddisplay\nT/tildewideF∗/parenrightBigg\n, u,v∈Z.\nFor simplicity, dGwill be denoted by d. There is a decomposition of das sum of\nbihomogeneous components, d=d0,1+d1,0+d2,−1, where each double subindex\ndenotes the corresponding bidegree. From d2= 0 we get\n(15) d2\n0,1=d2\n2,−1=d0,1d1,0+d1,0d0,1= 0,\n(16) d1,0d2,−1+d2,−1d1,0=d2\n1,0+d0,1d2,−1+d2,−1d0,1= 0.\nThe decreasing filtration of (Ω ,d) by the differential ideals\n(17) Fl= Ωl,·∧Ω,\ndepends only on/parenleftBig\nG,/tildewideF/parenrightBig\n; it could be defined without using ν. So we get a spectral\nsequence (Ei,di) which converges to H·(G). As for the spectral sequence of a\nfoliation (see e.g. [1]), in this case there are canonical identities\n(18) ( E0,d0)≡(Ω,d0,1),(E1,d1)≡(H(Ω,d0,1),d1,0∗).\nTheC∞topology on the space of differential forms induces a topology on ea chEi\nwhich is not Hausdorff in general.\nAt eachz∈Nwe have\nDN∗:νz∼=− −−− →TDN(z)G.\nSo for each X∈gthere is a well defined vector field Xν∈C∞(ν) which isDN-\nprojectable and such that DN∗Xν=X. SuchXνisG-invariant since Xν\nzg∈νzg\nand\nDN∗(Xν\nzg) =g−1DN∗Xν\nz=g−1XDN(z)=Xg−1DN(z)=XDN(zg).\nLetθXandiXrespectively denote the Lie derivative and interior product on Ω\nwith respect to Xν. (We are considering θXandiXas operators on the leaves of G,\nbut preserving smoothness on N.) By comparing bidegrees in the usual formulas20 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR\nthat relate Lie derivatives, interior products and the de Rham deriv ative, we easily\nget\nd0,1iX+iXd0,1= 0,\n(θX)0,0d0,1=d0,1(θX)0,0,\ni[X,Y]= (θX)0,0iY−iY(θX)0,0,\n(θX)0,0=d1,0iX+iXd1,0,\n(θ[X,Y])0,0= (θXθY−θYθX)0,0−d0,1iΞ(X∧Y)−iΞ(X∧Y)d0,1,\nwhere Ξ :/logicalandtext2g→C∞/parenleftBig\nT/tildewideF/parenrightBig\nis given by\nΞ(X∧Y) = [Xν,Yν]−[X,Y]ν.\nTherefore we get the operation ( g,i1,θ1,E1,d1), wherei1X≡iX∗andθ1X≡\n(θX)0,0∗according to (18), and the algebraic connection D∗\nN:g∗→E1,0\n1⊂Ω1,0\n[13]. Then\nEu,v\n2∼=Hu(g;θ1:g→End(E0,v\n1)).\nLetφ:N×g→Nbe defined by φ(z,X) =Xν\n1(z), whereXν\ntdenotes the uni-\nparametric group of transformations defined by Xν, considered as group of trans-\nformations of the leaves of Gpreserving smoothness on N. Then the following\ndiagram is commutative\nN×gφ− −−− →N\nDN×exp/arrowbt/arrowbtDN\nG×G− −−− →G,\nwhere the lowest map denotes the operation on G. (This follows because Xt=\nRexp(tX)for allX∈g.)\n7.2.Tensor product decomposition of E2whengis compact semisimple.\nFrom now on suppose gis compact semisimple, and thus Gis compact [28].\nTheorem 7.1. With the above notations,\nEu,v\n2∼=Hu(g)⊗E0,v\n2=Hu(g)⊗(E0,v\n1)θ1=0.\nThe resultfollowswith the sametypeofargumentsasin thosegivenin Sections2\nand 3 of [2] to prove Theorem 3.5 in [2]. We will indicate the main steps in th e\nproof because some of them will be needed later.\nConsider the canonical biinvariant metric on G[28, Chapter 6], and let C⊂G\nandC∗⊂gbe the cut locus and tangential cut locus corresponding to the iden tity\nelemente∈G. LetB∗be the radial domain in gbounded by C∗, and letB=\nexp(B∗). From the general properties of the cut locus we have C=∂B=G\\B,\nexp :B∗→Bis a diffeomorphism, CandC∗have Lebesgue measure zero, and B∗\nis compact (since so is G) [22, 21]. Consider the compact space\nF={(X,Y,Z)∈B∗3: exp(X) exp(Y) = exp(Z)} ⊂g3,\nand for each X∈B∗the compact slice\nFX={(Y,Z)∈g2: (X,Y,Z)∈F} ⊂g2.\nSmoothness on FandFXwill refer to the smoothness obtained by considering\nthese spaces as subspaces of g3andg2, respectively.LEAFWISE REDUCED COHOMOLOGY 21\nLetι:g2→g2be the involution ( Y,Z)/mapsto→(Z,Y). Fora= exp(X) we also have\nthe smooth map JX:B∩L−1\naB→FXgiven by JX(g) = (log(g),log(ag)), where\nlog = exp−1:B→B∗. LetWX=JX(B∩L−1\naB)⊂FX.\nLemma 7.2 ([2, Proposition 2.2]) .We have:\n(i)WXis open inFXandJX:B∩L−1\naB→WXis a diffeomorphism.\n(ii)ι(FX) =F−X, and the diagram\nB∩L−1\naBJX− −−− →FX\nLa/arrowbt/arrowbtι\nB∩LaBJ−X− −−− →F−X\nis commutative.\nForX,Y∈B∗letWX,Y=JX(B∩L−1\naB∩L−1\nbB)⊂FX, wherea= exp(X)\nandb= exp(Y). We have the diffeomorphism JX,Y=JYJ−1\nX:WX,Y→WY,X.\nLet ∆ be the unique biinvariant volume form on Gsuch that/integraltext\nG∆ = 1, which\ndefines a Haar measure µonG. Then for each X∈B∗letµXbe the Borel measure\nonFX, concentrated on WX, where it corresponds to µbyJX.\nCorollary 7.3 ([2, Proposition 2.3]) .We have:\n(i)µX(FX) =µX(WX) =µX(WX,Y) =µ(B∩L−1\naB∩L−1\nbB)\n=µ(B∩L−1\naB) =µ(G) = 1\n(ii)µXcorresponds to µ−Xbyι:FX→F−X.\n(iii)µXcorresponds to µYbyJX,Y:WX,Y→WY,X.\nLetI= [0,1], and define continuous maps σ,η:F×I→Gby setting\nσ(ξ,t) = exp(tZ),\nη(ξ,t) =/braceleftBigg\nexp(2tX) if t∈I1= [0,1/2]\nexp(X) exp((2t−1)Y) ift∈I2= [1/2,1],\nwhereξ= (X,Y,Z)∈F. The map σis smooth, and so are the restrictions of ηto\neachF×Ii(i= 1,2).\nLemma 7.4 ([2, page 178]) .There is a finite open cover Q1,...,Q kofF, and\ncontinuous maps Hj:Qj×I×I→Gwith smooth restrictions to each Qj×Ii×I,\ni= 1,2,j= 1,...,k, so that\nHj(·,·,0) =σ|Qj×I,Hj(·,·,1) =η|Qj×I,\nHj(ξ,0,s) =efor alls∈Iandξ∈Qj,\nHj(ξ,1,s) = exp(Z)for alls∈Iandξ= (X,Y,Z)∈Qj.\nLemma 7.5. For eachj= 1,...,kthere exists a unique continuous map\nHj:N×Qj×I×I→N\nwith smooth restrictions to each N×Qj×Ii×I,i= 1,2, such that\n(i)DNHj(z,ξ,t,s) =DN(z)Hj(ξ,t,s),\n(ii)Hj(z,ξ,0,s) =z,\n(iii) (d/dt)Hj(z,ξ,t,s)∈νfort/\\e}atio\\slash= 1/2.\nMoreover for ξ= (X,Y,Z)∈Qjwe have22 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR\n(iv)Hj(·,ξ,1,0) =φZ,\n(v)Hj(·,ξ,1,1) =φYφX\n(vi)Hj(z,ξ,1,s)∈D−1\nN(D(z) exp(Z))for allz∈Nand alls∈I.\nProof.It is completely similar to the proofs of Lemmas 3.1 and 3.2 in [2]. /square\nTherefore, for all ξ= (X,Y,Z)∈Qj,Hj(·,ξ,1,·) :N×I→Nis an/tildewideF-integrable\nhomotopy of φZtoφYφX[10]. Hence the corresponding homotopy operator in Ω\npreservesthefiltration, andthusits(0 ,−1)-bihomogeneouscomponent kj,ξ: Ω→Ω\nsatisfies\n(φ∗\nXφ∗\nY−φ∗\nZ)0,0=d0,1kj,ξ+kj,ξd0,1.\nDefine the operators ρ,λ: Ω→Ω by setting\nρ(α) =/integraldisplay\nB∗φ∗\nXα∆∗(X), λ(α) =/integraldisplay\nB∗ΦXα∆∗(X),\nwhere ∆∗= exp∗∆ and Φ Xis the homogeneous operator of degree −1 on Ω associ-\natedtothehomotopy φtX(t∈I) [9]. Theoperators ρandλarelinearhomogeneous\nof degrees 0 and −1, respectively, satisfying ρ−id =dλ+λd. Moreover, since φtX\npreserves the pair of foliations/parenleftBig\nG,/tildewideF/parenrightBig\n(becauseXνis an infinitesimal transforma-\ntion of/parenleftBig\nG,/tildewideF/parenrightBig\n), ΦXreduces the filtration at most by a unit. Therefore the bihomo-\ngeneous operators ρ1≡ρ0,0∗andλ1≡λ−1,0∗onE1satisfyρ1−id =d1λ1+λ1d1.\nForα∈Ω andX∈B∗, by Lemma 7.2 and Corollary 7.3 we have\nφ∗\nXρ(α) =/integraldisplay\nFXφ∗\nXφ∗\nYαdµX(Y,Z),\nρ(α) =/integraldisplay\nWX,−Xφ∗\nYαdµX(Y,Z)\n=/integraldisplay\nW−X,Xφ∗\nYαdµ−X(Y,Z)\n=/integraldisplay\nF−Xφ∗\nYαdµ−X(Y,Z)\n=/integraldisplay\nFXφ∗\nZαdµX(Y,Z).\nSo\n(19) ( φ∗\nXρ−ρ)α=/integraldisplay\nFX(φ∗\nXφ∗\nY−φ∗\nZ)αdµX(Y,Z).\nTake a smooth partition of unity f1,...,f kofFsubordinated to the open cover\nQ1,...,Q k. Then the fj(X,·,·) form a partition of unity of FXsubordinated to\nthe open cover given by the slices\nQj,X={(Y,Z)∈g2: (X,Y,Z)∈Qj}.\nLet ΨX: Ω→Ω be the (0 ,−1)-bihomogeneous linear operator given by\nΨXα=k/summationdisplay\nj=1/integraldisplay\nQj,Xkj,ξαfj(ξ)dµX(Y,Z),\nwhereξ= (X,Y,Z) for each (Y,Z)∈Qj,X. From (19) we get\n(20) ( φ∗\nXρ−ρ)0,0=d0,1ΨX+ΨXd0,1.LEAFWISE REDUCED COHOMOLOGY 23\nLemma 7.6. ΨXαdepends continuously on X∈B∗for eachα∈Ωfixed.\nProof.It is completely analogous to the proof of Lemma 3.3 in [2]. /square\nLemma 7.7. Forα∈Ω,X∈gandt∈Rwe have\nφ∗\ntXα=α+/integraldisplayt\n0φ∗\nsXθXαds=α+θX/integraldisplayt\n0φ∗\nsXαds.\nProof.It is completely analogous to the proof of Lemma 3.4 in [2]. /square\nLemma 7.8. ρ1(E1) = (E1)θ1=0, and\nρ1∗:E2∼=− −−− →H((E1)θ1=0).\nProof.First, we shall prove that ρ1(E1)⊂(E1)θ1=0. Take any α∈ker(d0,1)\ndefining [α]∈E1. If [α]∈ρ1(E1), we can suppose α=ρ0,0βfor someβ∈ker(d0,1).\nThen\n(21) ( φ∗\nX)0,0α−α=d0,1ΨXβfor allX∈B∗\nby (20). Thus Lemmas 7.6 and 7.7 yield\n(θX)0,0α=d0,1/parenleftbigg\nΨXβ−(θX)0,0/integraldisplay1\n0ΨsXβds/parenrightbigg\nas in [2, page 181]. Therefore ρ1([α])∈(E1)θ1=0.\nLetι: (E1)θ1=0→E1be the inclusion map. If [ α]∈(E1)θ1=0, since (θX)0,0\ndepends linearly on X∈g, there is a linear map X/mapsto→βXofgto Ω so that\n(θX)0,0α=d0,1βXfor allX∈g. Thus by Lemma 7.7 we get\nρ0,0α=α+d0,1/integraldisplay\nB∗/integraldisplay1\n0(φ∗\nsX)0,0βXds∆∗(X),\nyieldingρ1ι= id. In particular ρ1(E1) = (E1)θ1=0. We also have ιρ1−id =\nd1λ1+λ1d1, and the result follows. /square\nEnd of the proof of Theorem 7.1. SinceGiscompact,therepresentation θgissemisim-\nple [13, Sections 4.4 and 5.12]. So\nH((E1)θ1=0)∼=H(g)⊗(E0,·\n1)θ1=0\nby [13, Theorem V in Section 4.11, and Section 5.26]. The result now follo ws from\nLemma 7.8. /square\n7.3.Relation between H·(F)andE2.\nTheorem 7.9. With the above notations, H·(F)∼=E0,·\n2.\nTo begin with the proof of Theorem 7.9, the section s:M→Ndefines a\nhomomorphism ( s∗)1:E0,·\n1→H·(F) sinces∗d0,1=dFs∗. By restricting ( s∗)1, we\nget (s∗)2:E0,·\n2= (E0,·\n1)θ1=0→H·(F). We will prove that ( s∗)2is an isomorphism.\nFor anyX∈gsetsX=φXs:M→N, which is an embedding, but not a\nsection ofπNin general. Nevertheless sX(M) =s(M) exp(X). Analogously to s,\nthe mapsXalso defines ( s∗\nX)1:E0,·\n1→H·(F). LetUXbe the neighborhood of\nsX(M) given by\nUX=/uniondisplay\nY∈B∗φYsX(M) =sX(M)B=D−1\nN(exp(X)B).24 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR\nFor eachX∈gand eachx∈M,sXdefines an isomorphism\nsX∗:TxF∼=− −−− →TsX(x)/tildewideF.\nSo\ns∗\nX:/logicalandtextTsX(x)/tildewideF∗ ∼=− −−− →/logicalandtextTxF∗.\nFor eachω∈Ω(F), letωXbe the unique smooth section of/logicalandtextT/tildewideF∗oversX(M)\nsuch thats∗\nXωX=ω. DefineTXω∈Ω0,·(G|UX) by setting\n(TXω)(φYsX(x)) = (φ∗\n−Y)0,0ωX(sX(x))\nforY∈B∗andx∈M. This is well defined since ( x,Y)/mapsto→φYsX(x) is a dif-\nfeomorphism of M×B∗ontoUX. Moreover d0,1TX=TXdFsinced0,1≡d/tildewideFon\nΩ0,·≡Ω·/parenleftBig\n/tildewideF/parenrightBig\n, and (φYsX)∗/tildewideF=Ffor allX,Y∈g. Therefore TXdefines a map\nTX∗:H·(F)→E0,·\n1(G|UX).\nThe inclusion map ιX:UX→Ninduces (ι∗\nX)1:E0,·\n1→E0,·\n1(G|UX).\nLemma 7.10. Ifζ∈(E0,·\n1)θ1=0, thenTX∗(s∗\nX)1ζ= (ι∗\nX)1ζ.\nProof.By Lemma 7.8 we have ρ1ζ=ζ. We thus can choose forms α,γ∈Ω0,·such\nthatd0,1α= 0,ζ= [α], andα=ρ0,0α+d0,1γ. Then (21) yields\n(φ∗\nY)0,0(α−d0,1γ)−(α−d0,1γ) =d0,1ΨYα\nfor anyY∈B∗. So\n(22) ( φ∗\nY)0,0α−α=d0,1(ΨYα+(φ∗\nY)0,0γ−γ).\nClearly (s∗\nXα)X=α|sX(M). Hence\n(TXs∗\nXα)(φYsX(x)) = (φ∗\n−Y)0,0(α(sX(x)))\n= (α+d0,1(Ψ−Yα+(φ∗\n−Y)0,0γ−γ))(φYsX(x))\nby (22). But since each φYsX(M) is/tildewideF-saturated, d0,1≡d/tildewideFcommutes with the\nrestriction to each φYsX(M). Therefore we get\nTXs∗\nXα=α+d0,1ηX\nonUX, whereηXis the (0,·)-form onUXdefined by\nηX(φYsX(x)) = (Ψ −Yα+(φ∗\n−Y)0,0γ−γ)(φYsX(x)),\nwhich finishes the proof. /square\nSinceGis compact, there is a finite sequence 0 = X1,X2,...,X lof elements of\ngsuch that\nG=B∪exp(X2)B∪···∪exp(Xl)B .\nLetUj=UXjTj=TXj,sj=sXjandιj=ιXjforj= 1,...,l. ThenN=\nU1∪···∪Ul. Leth1,...,h lbe a smooth partition of unity of Gsubordinated to the\nopen cover exp( X1)B,...,exp(Xl)Bso thath1(e) = 1. Then D∗\nNh1,...,D∗\nNhlis\na partition of unity of N subordinated to U1,...,U l.\nForω∈Ω(F), defineTω∈Ω0,·by setting\nTω=l/summationdisplay\nj=1D∗\nNhjTjω.LEAFWISE REDUCED COHOMOLOGY 25\nSince each D∗\nNhjis constant along the leaves of /tildewideF, we getd0,1T=TdF.SoT\ndefines a map T∗:H·(F)→E0,·\n1.\nLemma 7.11. Ifζ∈(E0,·\n1)θ1=0, thenT∗(s∗)1ζ=ζ.\nProof.For eachX∈g, let (φ∗\nX)1:E1→E1be the homomorphism defined by φ∗\nX\n((φ∗\nX)1≡(φ∗\nX)0,0∗). SincesX=φXs, by (21) we have\n(s∗\nX)1ζ=s∗\n1(φ∗\nX)1ζ=s∗\n1ζ .\nTherefore, by Lemma 7.10,\n(ι∗\nj)1ζ=Tj∗(s∗\nj)1ζ=Tj∗(s∗)1ζ\nforj= 1,...,l. So, ifζ= [α] forα∈Ω0,·withd0,1α= 0, there is some βj∈Ω0,·\nfor eachjsuch thatα−Tjs∗α=d0,1βjoverUj. Let\nβ=l/summationdisplay\nj=1D∗\nNhjβj∈Ω0,·.\nSince eachD∗\nNhjis constant on the leaves of /tildewideFandd0,1≡d/tildewideF, we get\nd0,1β=l/summationdisplay\nj=1D∗\nNhjd0,1βj\n=l/summationdisplay\nj=1D∗\nNhj(α−Tjs∗α)\n=α−Ts∗α,\nand the proof is complete. /square\nLemma 7.12. (s∗)2:E0,·\n2→H·(F)is surjective.\nProof.Take anyω∈Ω(F) withdFω= 0, and take any function f≥0 compactly\nsupported in Bsuch that/integraltext\nBf(g)∆(g) = 1. Then α=D∗\nNfT1ωis a (0,·)-form\ncompactly supported in U1and satisfying d0,1α= 0. Soαdefines a class ζ∈E0,·\n1.\nWe shall prove that ( s∗)1ρ1ζ= [ω].\nForx∈MandY∈B∗we have\nα(φYs(x)) =f(exp(Y))(φ∗\n−Y)0,0(ωX1(s(x))).\nSo\n((φ∗\nY)0,0α)(s(x)) =f(exp(Y))ωX1(s(x)),\nyielding\n(ρ0,0α)(s(x)) =/integraldisplay\nB∗((φ∗\nY)0,0α)(s(x))∆∗(Y)\n=ωX1(s(x))/integraldisplay\nB∗f(exp(Y))∆∗(Y)\n=ωX1(s(x))/integraldisplay\nGf(g)∆(g)\n=ωX1(s(x)).\nTherefores∗ρ0,0α=s∗ωX1=ω, and the proof follows. /square\nCorollary 7.13. T∗(H·(F))⊂E0,·\n2.26 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR\nProof.It follows directly from Lemmas 7.11 and 7.12. /square\nEnd of the proof of Theorem 7.9. By Corollary 7.13 we can consider T∗:H·(F)→\nE0,·\n2. By Lemma 7.11 we have T∗(s∗)2= id. On the other hand, ( s∗)2T∗= id\nbecause (D∗\nNh1)(s(x)) = 1 for all x∈Msinceh1(e) = 1. So ( s∗)2is an isomor-\nphism. /square\nCorollary 7.14. H1(F)∼=H1(G)andH1(F)∼=H1(G).\nProof.Theorem 7.1 yields E2,0\n2∼=H2(g)⊗E0,0\n2= 0 since gis compact semisimple.\nSoE0,1\n2=E0,1\n∞∼=H1(G) canonically. Then H1(F)∼=H1(G) as topological vector\nspaces by Theorem 7.9, obtaining also H1(F)∼=H1(G). /square\nCorollary 7.15. H2(F)andH2(F)are of finite dimension if and only if so are\nH2(G)andH2(G), respectively.\nProof.The leaves of Gare dense since so are the leaves of F. ThusH0(G)∼=R,\nyieldingE·,0\n2∼=H·(g)byTheorem7.1. Ontheotherhand, H1(g) =H2(g) = 0since\ngis compact semisimple [28]. So E1,·\ni=E2,·\ni= 0 for 2 ≤i≤ ∞by Theorem 7.1.\nHenceE0,2\n3=E0,2\n2∼=H2(F) (using Theorem 7.9), and E3,0\n3=E3,0\n2∼=H3(g).\nTherefore, since\nE0,2\n∞=E0,2\n4= ker(d3:E0,2\n3→E3,0\n3),\nH2(G)∼=E0,2\n∞can be identified to the kernel of some continuous homomorphism of\nH2(F) toH3(g), and the result follows. /square\nCorollary 7.16. SupposeMis oriented. Let ιi:Ki→M,i= 1,2, be smooth\nimmersions of closed oriented manifolds of complementary d imension which define\nhomology classes of Mwith non-trivial intersection. Let Γibe the image of the\ncomposite\nπ1(Ki)π1(ιi)− −−− →π1(M)h− −−− →G.\nSuppose the group generated by Γ1∪Γ2is not dense in G. If1≤k= dimK2≤2,\nthendimHk(F) =∞.\nProof.The result follows directly applying Corollaries 7.14, 7.15, and 2.6 to G./square\nCorollary 7.17. SupposeMis oriented. Let ιi:Ki→M,i= 1,2, be smooth\nimmersions of closed oriented manifolds of complementary d imension which define\nhomology classes of Mwith non-trivial intersection. Let Γibe the image of the\ncomposite\nπ1(Ki)π1(ιi)− −−− →π1(M)h− −−− →G.\nSuppose the group generated by Γ1∪Γ2is not dense in G. Ifι1is transverse to F,\nthendimHk(F) =∞fork= dimK2.\nProof.By Corollary 7.16, we can assume k >2. LetFlH·(G) andFlH·(G),l=\n0,1,2,..., be the filtrations of H·(G) andH·(G) induced by (17). We have\nH·(G)/F1H·(G)∼=E0,·\n∞⊂E0,·\n2∼=H·(F),\nwhere both isomorphisms preserve the topologies, and E0,·\n∞is a closed subspace of\nE0,·\n2. (The last isomorphism follows from Theorem 7.9.) So H·(G)/F1H·(G) can be\ninjected into H·(F), and it is enough to prove that Hk(G)/F1Hk(G) is of infinite\ndimension.LEAFWISE REDUCED COHOMOLOGY 27\nThis is a special case of the setting of Theorem 2.2 and Corollaries 2.5 a nd 2.6.\nThe proofs of those results yield linearly independent classes ζm∈ Hk(G). In this\ncase, we shall prove that the ζmare also linearly independent modulo F1Hk(G).\nConsider the pull-back bundles ι∗\niNoverKi. The canonical maps ˜ ιi:ι∗\niN→N\nareimmersionstransverseto G, whichthuscanbeconsideredastransverselyregular\nimmersionsof ( ι∗\niN,Gi) into (N,G), where Gi= ˜ι∗\niG. We can assume the ιiintersect\neach other transversely, and thus the ˜ ιiintersect transversely in G.\nLet/tildewideH·⊂H·(G) and/tildewideH·⊂ H·(G) be the subspaces given by the classes that\nhave representatives supported in π−1\nN(U) for any open subset U⊂Mcontaining\nι1(K1). SetF1/tildewideH·=/tildewideH·∩F1H·(G) andF1/tildewideH·=/tildewideH·∩F1H·(G). Sinceζm∈/tildewideHkby\nRemark3, it is enoughto provethat the ζmarelinearly independent modulo F1/tildewideHk.\nHence, according to the proof of Theorem 2.2, it is enough to prove thatι2can be\nchosen so that ˜ ι∗\n2/parenleftBig\nF1/tildewideHk/parenrightBig\n= 0 where ˜ ι∗\n2:H·(G)→ H·(G2). In fact we shall prove\nthe stronger property that the choice of ι2can be made so that ˜ ι∗\n2/parenleftBig\nF1/tildewideHk/parenrightBig\n= 0 for\n˜ι∗\n2:H·(G)→H·(G2).\nSinceι1is transverseto F, we can choose ι2such that, for some open subset U⊂\nMcontainingι1(K1), each connected component of ι2(K2)∩Uis contained in some\nleaf ofF. So, for every leaf L2ofG2, the connected components of ˜ ι2(L2)∩π−1\nN(U)\nare contained in leaves of /tildewideF, yielding ˜ι∗\n2α= 0 over˜ι−1\n2π−1\nN(U) for anyα∈F1Ω·(G).\nMoreoverUandι2can be chosen so that the connected components of ι−1\n2(U) are\ncontractible;thus˜ ι−1\n2π−1\nN(U)≡ι−1\n2(U)×Gcanonically,wheretheslices ι−1\n2(U)×{∗}\nare the leaves of the restriction G2,UofG2to ˜ι−1\n2π−1\nN(U). HenceHl(G2,U) = 0 for\nl>0. Finally, the abovechoices can be made so that, for some open sub setV⊂M,\nwe haveι1(K1)∩V=∅,U∪V=M, and each connected component of ι−1\n2(U∩V)\nis contractible. Thus, as above, Hl(G2,U∩V) = 0 forl >0, where G2,U∩Vis the\nrestriction of G2to ˜ι−1\n2π−1\nN(U∩V). Therefore, by using the Mayer-Vietoris type\nspectral sequence (cf. [10])\n··· →Hl−1(G2,U∩V)→Hl(G2)→Hl(G2,U)⊕Hl(G2,V)→Hl(G2,U∩V)→ ···\nand sincek>2, we get\n(23) Hk(G2)∼=Hk(G2,V)\nby the restriction homomorphism.\nNowanyξ∈F1/tildewideHkcanbedefinedbyaleafwiseclosedform α∈Ωk(G) supported\ninM\\Vwithα+dGβ∈F1Ωk(G) for some β∈Ωk−1(G). Then ˜ι∗\n2(α+dGβ) is\nsupported in ˜ ι−1\n2π−1\nN(V), where it is the G2-leafwise derivative of ˜ ι∗\n2β. So ˜ι∗\n2ξis\nmapped to zeroin Hk(G2,V), and thus ˜ ι∗\n2ξ= 0 by (23), which finishes the proof. /square\n8.Case of foliations on nilmanifolds Γ\\Hdefined by normal\nsubgroups of H\nThe goal of this section is to prove Theorem 2.10. It will be done by ind uction,\nwhich needs leafwise reduced cohomology with coefficients in a vector bundle with\na flatF-partial connection. Thus we shall prove a more general theorem by taking\narbitrary coefficients.\nForafoliation Fonamanifold Mandavectorbundle VoverM, aflatF-partial\nconnection on Vcan be defined as a flat connection on the restriction of Vto the28 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR\nleaves whose local coefficients are smooth on each foliation chart of FonM. So the\ncorresponding de Rham derivative dFwith coefficients in Vpreserves smoothness\nonM; i.e.dFpreservesΩ( F,V) =C∞(/logicalandtextTF∗⊗V). ThenH·(F,V) canbe defined\nin the same way as H·(F) by using (Ω( F,V),dF) instead of (Ω( F),dF).\nConsider the following particular case. Let Hbe a simply connected nilpotent\nLie group, K⊂Ha normal connected subgroup, and Γ ⊂Ha discrete uniform\nsubgroup whose projection to H/Kis dense. Then let Fbe the foliation on the\nnilmanifold M= Γ\\Hdefined as the quotient of the foliation /tildewideFonHwhose leaves\nare the translates of K. In this case, Mis closed and the leaves of Fare dense.\nLet/tildewideVbe anH×K-vector bundle over Hfor the left action of H×KonHgiven\nby (h,k)h′=hh′k−1, (h,k)∈H×Kandh′∈H. We also consider the induced\nleft actions of HandKonH. The space of H-invariant sections of /tildewideVwill be\ndenoted by C∞/parenleftBig\n/tildewideV/parenrightBig\nH, and the subspaces of invariant sections will be denoted in\na similar way for other actions. Suppose /tildewideVis endowed with an H×K-invariant\nflat/tildewideF-partial connection, and let Vbe the induced vector bundle on Mwith the\ninduced flat F-partial connection. The structure of H×K-vector bundle on V\ncanonically defines an action of konC∞/parenleftBig\n/tildewideV/parenrightBig\nH, wherekis the Lie algebra of K.\nMoreover the induced differential map on/logicalandtextk∗⊗C∞/parenleftBig\n/tildewideV/parenrightBig\nHcorresponds to d/tildewideFby\nthe canonical injection of this space in Ω( F,V).\nTheorem 8.1. With the above notations, H·(F,V)∼=H·/parenleftBig\nk,C∞/parenleftBig\n/tildewideV/parenrightBig\nH/parenrightBig\n.\nThe result will follow by induction on the codimension qofF.\nForq= 0 andVthe trivial line bundle, this is just a well known theorem of\nK. Nomizu [27]. If q= 0 andVis arbitrary, the result still follows with the obvious\nadaptation of the arguments in [27].\nSupposeq >0 and the result is true for foliations of codimension less than q.\nThe proof has two cases.\nCase1.AssumeK∩Γ = 1. ThegroupΓisnilpotentsincesois H, thusthecenterof\nΓ is non-trivial. Let abe a non-trivial element in the center of Γ. By the universal\nproperty of Mal’cev’s completion [23], there exists a one dimensional c onnected\nsubgroupLof the center of Hcontaining /a\\}b∇acketle{ta/a\\}b∇acket∇i}htas a discrete uniform subgroup. L\nis isomorphic to RsinceHis simply connected. Let H1=H/L, and Γ 1= Γ//a\\}b∇acketle{ta/a\\}b∇acket∇i}ht.\nClearly Γ 1is canonically injected in H1as a discrete uniform subgroup. We get\nL∩K= 1 because /a\\}b∇acketle{ta/a\\}b∇acket∇i}ht∩K= 1, and thus there is a canonical injection of KintoH1\nas a normal subgroup, defining a foliation F1on the nilmanifold M1= Γ1\\H1.F1\nis a foliationofthe type considered in the statement ofthis theorem , ofcodimension\nq−1, but observe that the canonical injection of KintoH1may not have trivial\nintersection with Γ 1. The projection H//a\\}b∇acketle{ta/a\\}b∇acket∇i}ht →H1is canonically an S1-principal\nbundle (considering S1≡L//a\\}b∇acketle{ta/a\\}b∇acket∇i}ht), so the induced map π:M→M1is also an\nS1-principal bundle in a canonical way. Then Vcanonically is an S1-vector bundle\nso that the partial connection is invariant, and thus induces the ve ctor bundle\nV1=V/S1overM1with the corresponding flat F1-partial connection. The lifting\nofV1toH1is/tildewideV1=/tildewideV/L, which satisfies the same properties as /tildewideVwith respect to\nK1instead ofK.LEAFWISE REDUCED COHOMOLOGY 29\nFor eachx∈M1and eachm∈Z, define\nCm,x={f∈C∞(π−1(x),C) :f(yθ) =f(y)e2πmθi\nfor ally∈π−1(x) and allθ∈S1≡R/Z}.\nIt is easy to see that\nCm=/unionsqdisplay\nx∈M1Cm,x\nis a one-dimensional C-vectorbundle over M1in a canonical way. For m∈Z, define\nalso\nΩ(F,V⊗C)m={α∈Ω(F,V⊗C) :α(yθ) =α(y)e2πmθi\nfor ally∈π−1(x) and allθ∈S1},\nandsimilarlydefine C∞/parenleftBig/parenleftBig\n/tildewideV//a\\}b∇acketle{ta/a\\}b∇acket∇i}ht/parenrightBig\n⊗C/parenrightBigm\nconsideringthe S1-principalbundle H//a\\}b∇acketle{ta/a\\}b∇acket∇i}ht →\nH1. By the Fourier series expression for functions on S1, we get that Ω( F,V⊗C)\nis theC∞closure of /circleplusdisplay\nm∈ZΩ(F,V⊗C)m.\nIt can be easily seen that there is a canonical isomorphism\n(24) Ω( F1,V1⊗Cm)∼=Ω(F,V⊗C)m\ndefined byπ∗and the canonical identity\nC∞(Cm)≡C∞(M,C)m.\nSinceFis preserved by the S1-action onM,dFpreserves each Ω( F,V⊗C)mand\ncorresponds to dF1by (24). By induction\nH·(F1,V1⊗Cm)∼=H·/parenleftbigg\nk,C∞/parenleftBig\n/tildewideV1⊗˜Cm/parenrightBig\nH1/parenrightbigg\n.\nBut\nC∞/parenleftBig\n/tildewideV1⊗/tildewideCm/parenrightBig\nH1∼=C∞/parenleftBig/parenleftBig\n/tildewideV//a\\}b∇acketle{ta/a\\}b∇acket∇i}ht/parenrightBig\n⊗C/parenrightBigm\nH//angbracketlefta/angbracketright\ncanonically, which is obviously trivial if m/\\e}atio\\slash= 0. ButC0is the trivial complex line\nbundle, so\nH·(F,V⊗C)∼=H·(F1,V1⊗C0)\n∼=H·/parenleftbigg\nk,C∞/parenleftBig\n/tildewideV1⊗C/parenrightBig\nH1/parenrightbigg\n∼=H·/parenleftBig\nk,C∞/parenleftBig\n/tildewideV⊗C/parenrightBig\nH/parenrightBig\n.\nCase2.In the general case, let G=H/Kand Γ 1the projection of Γ to G. We\nuse Mal’cev’s construction for the pair ( G,Γ1). It yields a simply connected nilpo-\ntent Lie group H1containing Γ 1as a discrete uniform subgroup, and a surjective\nhomomorphism D1:H1→Gwhich is the identity on Γ 1. The kernel K1ofD1\ndefines a foliation Gof codimension qon the nilmanifold M1= Γ1\\H1, and we have\nK1∩Γ1= 1. SoGis the type of foliation we have considered in Case 1.\nGis the classifying foliation for foliations with transverse structure g iven by\n(G,Γ1). So there is a smooth map f:M→M1which is transverse to Gand so\nthatF=f∗G. In this particular case, fcan be constructed in the following way.\nBy the universal property of Mal’cev’s construction, the surject ive homomorphism30 J.A. ´ALVAREZ L ´OPEZ AND G. HECTOR\nof Γ to Γ 1can be uniquely extended to a surjective homomorphism ˜f:H→H1,\nwhich defines a map f:M→M1. We have D1˜f=D. SoKis projected onto\nK1, and thus F=f∗F1. Moreover fis a locally trivial bundle with fiber the\nnilmanifold P/(P∩Γ), wherePis the kernel of ˜f.\nFix a vector subbundle ν⊂TFwhich is complementary to the subbundle\nτ⊂TFof vectors that are tangent to the fibers of f. Then we get a canoni-\ncal isomorphism/logicalanddisplay\nTF∗⊗V∼=/logicalanddisplay\nν∗⊗/logicalanddisplay\nτ∗⊗V ,\nyielding a bigrading of Ω( F,V) given by\nΩu,v(F,V) =C∞/parenleftBiggu/logicalanddisplay\nν∗⊗v/logicalanddisplay\nτ∗⊗V/parenrightBigg\n.\nConsider the filtration of Ω( F,V) given by the differential subspaces\nFkΩ(F,V) =/circleplusdisplay\nu≥kΩu,·(F,V),\nwhich depend only on FandV; in fact they could be defined without using ν. This\nfiltration induces a spectral sequence ( Ei,di) converging to H·(F,V), whose terms\n(E0,d0) and (E1,d1) can be described as follows. The derivative dFdecomposes\nas sum of bihomogeneous operators dF,0,1,dF,1,0anddF,2,−1, where each double\nsubindex indicates the corresponding bidegree. These operators satisfy identities\nwhich are similar to those in (15) and (16), yielding\n(E0,d0)≡(Ω(F,V),dF,0,1),\n(E1,d1)≡(H(Ω(F,V),dF,0,1),dF,1,0∗).\nLetk1be the Lie algebra of K1. EachX∈k1canonically defines a vector field\nX1onM1which is tangent to the leaves of F1. LetXνbe the unique vector field\nonMwhich is a section of νand projects to X1. Forα∈Ω0,v(F) ands∈C∞(V),\ndefineθX(α⊗s) to be the (0 ,·)-component of\nθXνα⊗s+α⊗∇Xνs,\nwhere∇denotes the flat F-partial connection of V. It can be easily checked that\nθXdF,0,1=dF,0,1θX. SoθXdefines an operator, also denoted by θX, onE0,·\n1. In\nthis way, we get a representation θofk1onE0,·\n1, and a canonical isomorphism\nEu,v\n2∼=Hu(k1,θ).\nDefine\nV1,y=H·/parenleftbig\nf−1(y),V|f−1(y)/parenrightbig\n, y∈M1,\nV1=/unionsqdisplay\ny∈M1V1,y,\nand let/tildewideV1be the lifting of V1toH1. It is easy to see that /tildewideV1canonically is a\nH1×K1-vector bundle over the H1×K1-manifoldH1with anH1×K1-invariant\nflat/tildewideF1-partial connection. (The fibers of /tildewideV1are of finite dimension since the fibers\noffare compact.) It is also easily seen that there is a canonical isomorph ism\nC∞(V1)∼=E0,·\n1. Moreover the representation of k1onE0,·\n1corresponds to the\nrepresentation of k1onC∞/parenleftBig\n/tildewideV1/parenrightBig\ndefined by the flat partial connection of /tildewideV1. So\nEu,·\n2∼=Hu/parenleftBig\nk1,C∞/parenleftBig\n/tildewideV1/parenrightBig/parenrightBig\n∼=Hu(F1,V1).LEAFWISE REDUCED COHOMOLOGY 31\nLetEibe the quotient of Eiover the closure 0iof its trivial subspace. Then\nEu,·\n2∼=Hu(F1,V1)∼=Hu/parenleftbigg\nk1,C∞/parenleftBig\n/tildewideV1/parenrightBig\nH1/parenrightbigg\nby Case 1.\nIf the above filtration is restricted to the space of differential for ms in Ω( F,V)\nwhose lifting to HisH-left invariant, we get a spectral sequence ( Ei,di) converging\ntoH·/parenleftBig\nk1,C∞/parenleftBig\n/tildewideV/parenrightBig\nH/parenrightBig\n, and there is a canonical homomorphism ( Ei,di)→(Ei,di)\nof spectral sequences. Analogously, we have a canonical isomorp hism\nEu,·\n2∼=Hu(k1,C∞(V1)H1).\nSo the composite E2→E2→ E2is an isomorphism, and thus E2∼=E2⊕02as\ndifferential complexes. Then E3∼=E3⊕H(02,d2), yielding H(02,d2)∼=03, and\nthe above decomposition is of differential complexes. We get E4∼=E4⊕H(03,d3).\nContinuing with these arguments, we finally obtain Ei∼=Ei⊕0ias topological\ndifferential complexes for i≥2, and thus\nH·(F,V)∼=E∞∼=E∞⊕0∞.\nHence\nH·(F,V)∼=E∞∼=H·(k,C∞(V)H)\nas desired.\nRemark 4.For general Lie foliations with dense leaves and nilpotent structura l Lie\nalgebra,theclassifyingfoliationsareofthetypeconsideredinTheo rem2.10. Onthe\none hand, if the ambient manifold is closed and the classifying map can b e chosen\nto be a fiber bundle, then a spectral sequence argument shows th at the leafwise\nreduced cohomology is of finite dimension. 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Spectral theory of foliation geometric oper ators. Preprint, 1992.\n[20] A. El Kacimi-Alaoui and A. Tihami. Cohomologie bigradu ´ ee de certains feuilletages. Bull.\nSoc. Math. Belge , s´ erie B-38, 1986.\n[21] W. Klingenberg. Riemannian Geometry , volume 1 of De Gruyter Studies in Mathematics .\nDe Gruyter, Berlin, 1982.\n[22] S. Kobayashi. On conjugate and cut loci. In Studies in Global Geometry and Analysis , pages\n96–122. Math. Assoc. of Amer., 1967.\n[23] A.I. Mal’cev. On a class of homogeneous spaces. Transl. Amer. Math. Soc. , 39:276–307, 1951.\n[24] P. Molino. G´ eom´ etrie globale des feuilletages riema nniens.Nederl. Akad. Wetensch. Indag.\nMath., 44:45–76, 1982.\n[25] P. Molino. Riemannian Foliations , volume 73 of Progress in Math. Birkh¨ auser Boston Inc.,\nBoston, MA, 1988.\n[26] P. Molino and M. Pierrot. Th´ eor´ emes des slice et holon omie des feuilletages Riemanniens.\nAnnales Inst. Fourier, Grenoble , 37:207–223, 1987.\n[27] K. Nomizu. 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Helv. ,\n28:17–86, 1954.\nDepartamento de Xeometr ´ıa e Topolox ´ıa, Facultade de Matem ´aticas, Universidade\nde Santiago de Compostela, 15782 Santiago de Compostela, Sp ain\nE-mail address :jesus.alvarez@usc.es\nInstitut Girard Desargues, UPRESA 5028, 43, boulevard du 11 No vembre 1918, Uni-\nversit´e Claude Bernard-Lyon I, 69622 Villeurbanne Cedex, France\nE-mail address :hector@geometrie.univ-lyon1.fr"    },    {        "title": "2210.16931v1.Intrinsic_polynomial_squeezing_for_Balakrishnan_Taylor_beam_models.pdf",        "content": "arXiv:2210.16931v1  [math.AP]  30 Oct 2022Intrinsic polynomial squeezing for\nBalakrishnan-Taylor beam models\nE. H. Gomes Tavares, M. A. Jorge Silva, V. Narciso and A. Vicen te\nAbstract We explore the energy decay properties related to a model in e xtensible\nbeams with the so-called energy damping . We investigate the influence of the non-\nloncal damping coefficient in the stability of the model. We pr ove, for the first time,\nthat the corresponding energy functional is squeezed by pol ynomial-like functions\ninvolving the power of the damping coefficient, which arises i ntrinsically from the\nBalakrishnan-Taylor beam models. As a consequence, it is sh own that such models\nwith nonlocal energy damping are never exponentially stabl e in its essence.\n1 Introduction\nIn 1989 Balakrishnan and Taylor [3] derived some prototypes of vibrating exten-\nsible beams with the so-called energy damping . Accordingly, the following one\ndimensional beam equation is proposed\n/u1D715/u1D461/u1D461/u1D462−2/u1D701√\n/u1D706/u1D715/u1D465/u1D465/u1D462+/u1D706/u1D715/u1D465/u1D465/u1D465/u1D465/u1D462−/u1D6FC/bracketleftbigg/uni222B.dsp/u1D43F\n−/u1D43F/parenleftbig/u1D706|/u1D715/u1D465/u1D465/u1D462|2+ |/u1D715/u1D461/u1D462|2/parenrightbig/u1D451/u1D465/bracketrightbigg/u1D45E\n/u1D715/u1D465/u1D465/u1D461/u1D462=0,(1)\nwhere/u1D462=/u1D462(/u1D465,/u1D461)represents the transversal deflection of a beam with length 2 /u1D43F >0\nin the rest position, /u1D6FC > 0 is a damping coefficient, /u1D701is a constant appearing\nin Krylov-Bogoliubov’s approximation, /u1D706 > 0 is related to mode frequency and\nspectral density of external forces, and /u1D45E=2(/u1D45B+/u1D6FD) +1 with/u1D45B∈Nand 0≤/u1D6FD<1\n2.\nE. H. Gomes Tavares\nState University of Londrina, 86057-970, Londrina, PR, Bra zil,\ne-mail:eduardogomes7107@gmail.com\nM. A. Jorge Silva\nState University of Londrina, 86057-970, Londrina, PR, Bra zil.\ne-mail:marcioajs@uel.br\nV. Narciso\nState University of Mato Grosso do Sul, 79804-970, Dourados , MS, Brazil.\ne-mail:vnarciso@uems.br\nA. Vicente\nWestern Paraná State University, 85819-110, Cascavel, PR, Brazil.\ne-mail:andre.vicente@unioeste.br\n12 E. H. Gomes Tavares, M. A. Jorge Silva, V. Narciso and A. Vice nte\nWe still refer to [3, Sect. 4] for several other beam equation s taking into account\nnonlocal energy damping coefficients, as well as [2, 4, 6, 7, 12 , 17, 18] for associated\nmodels. A normalized /u1D45B-dimensional equation corresponding to (1) can be seen as\nfollows\n/u1D715/u1D461/u1D461/u1D462−/u1D705Δ/u1D462+Δ2/u1D462−/u1D6FC/bracketleftbigg/uni222B.dsp\nΩ/parenleftBig\n|Δ/u1D462|2+ |/u1D715/u1D461/u1D462|2/parenrightBig\n/u1D451/u1D465/bracketrightbigg/u1D45E\nΔ/u1D715/u1D461/u1D462=0, (2)\nwhere we denote /u1D706=1 and/u1D705=2/u1D701;Ωmay represent an open bounded of R/u1D45B;\nand the symbols ΔandΔ2stand for the usual Laplacian and Bi-harmonic operators,\nrespectively. Additionally, in order to see the problem wit hin the frictional context\nof dampers, we rely on materials whose viscosity can be essen tially seen as friction\nbetween moving solids. In this way, besides reflecting on a mo re challenging model\n(at least) from the stability point of view, one may metaphys ically supersede the\nviscous damping in (2) by a nonlocal frictional one so that we cast the model\n/u1D715/u1D461/u1D461/u1D462−/u1D705Δ/u1D462+Δ2/u1D462+/u1D6FC/bracketleftbigg/uni222B.dsp\nΩ/parenleftBig\n|Δ/u1D462|2+ |/u1D715/u1D461/u1D462|2/parenrightBig\n/u1D451/u1D465/bracketrightbigg/u1D45E\n/u1D715/u1D461/u1D462=0. (3)\nThe main goal of this paper is to explore the influence of the no nloncal damping\ncoefficient in the stability of problem (3). Unlike the existi ng literature on extensible\nbeams with full viscous or frictional damping, we are going t o see for the first time\nthat the feature of the energy damping coefficient\nE/u1D45E(/u1D461):=E/u1D45E(/u1D462,/u1D462/u1D461)(/u1D461)=/bracketleftbigg/uni222B.dsp\nΩ/parenleftBig\n|Δ/u1D462(/u1D461)|2+ |/u1D715/u1D461/u1D462(/u1D461)|2/parenrightBig\n/u1D451/u1D465/bracketrightbigg/u1D45E\n, /u1D45E > 0, (4)\nnot only prevents exponential decay, but also gives us a poly nomial range in terms\nof/u1D45Ewhose energy is squeezed and goes to zero polynomially when t ime goes to\ninfinity. More precisely, by noting that the corresponding e nergy functional is given\nby\n/u1D438/u1D705(/u1D461):=/u1D438/u1D705(/u1D462,/u1D462/u1D461)(/u1D461)=/uni222B.dsp\nΩ/parenleftBig\n|Δ/u1D462(/u1D461)|2+ |/u1D715/u1D461/u1D462(/u1D461)|2+/u1D705|∇/u1D462(/u1D461)|2/parenrightBig\n/u1D451/u1D465, /u1D705≥0,(5)\nthen it belongs to an area of variation between upper and lowe r polynomial limits as\nfollows\n/u1D4500/u1D461−1\n/u1D45E/lessorsimilar/u1D438/u1D705(/u1D461)/lessorsimilar/u1D4360/u1D461−1\n/u1D45E, /u1D461→ +∞, (6)\nfor some constants 0 < /u1D450 0≤/u1D4360depending on the initial energy /u1D438/u1D705(0), /u1D705≥0.\nIndeed, such a claim corresponds to an intrinsic polynomial range of (uniform)\nstability and will follow as a consequence of a more general r esult that is rigorous\nstated in Theorem 2. See also Corollary 1. In particular, we c an conclude that (3) is\nnot exponentially stable when dealing with weak initial dat a, that is, with solution in\nthe standard energy space. See Corollary 2.\nIn conclusion, Theorem 2 truly reveals the stability of the a ssociated energy\n/u1D438/u1D705(/u1D461), which leads us to the concrete conclusions provided by Coro llaries 1-2, being\npioneering results on the subject. Due to technicalities in the well-posedness process,Intrinsic polynomial squeezing for Balakrishnan-Taylor b eam models 3\nwe shall work with /u1D45E≥1/2. In Section 2 we prepare all notations and initial results.\nThen, all precise details on the stability results shall be g iven in Section 3.\n1.1 Previous literature, comparisons and highlights\nIn what follows, we are going to highlight that our approach a nd results are different\nor else provide generalized results, besides keeping more p hysical consistency in\nworking exactly with (4) instead of modified versions of it. I ndeed, there are at least\nthree mathematical ways of attacking the energy damping coe fficient (4) along the\nequation (3) (or (2)), namely:\n1. Keeping the potential energy in (4), but neglecting the ki netic one;\n2. Keeping the kinetic energy in (4), but neglecting the pote ntial one;\n3. Keeping both potential and kinetic energies, but conside ring them under the action\nof a strictly (or not) positive function /u1D440(·)as a non-degenerate (or possibility\ndegenerate) damping coefficient.\nIn the first case, equation (3) becomes to\n/u1D715/u1D461/u1D461/u1D462−/u1D705Δ/u1D462+Δ2/u1D462+/u1D6FC/bracketleftbigg/uni222B.dsp\nΩ|Δ/u1D462|2/u1D451/u1D465/bracketrightbigg/u1D45E\n/u1D715/u1D461/u1D462=0 inΩ× (0,∞). (7)\nThis is, for sure, the most challenging case once the damping coefficient becomes\nnow to a real degenerate coefficient. In [5, Theorem 3.1], work ing on a bounded\ndomainΩwith clamped boundary condition, it is proved the following with/u1D45E=1\nin (7): for every/u1D445 >0, there exist constants /u1D436/u1D445=/u1D436(/u1D445)>0and/u1D6FE/u1D445=/u1D6FE(/u1D445)>0\ndepending on /u1D445such that\n/u1D438/u1D705(/u1D461) ≤/u1D436/u1D445/u1D438/u1D705(0)/u1D452−/u1D6FE/u1D445/u1D461, /u1D461 > 0, (8)\nonly holds for every regular solution /u1D462of(3)with initial data (/u1D4620,/u1D4621)satisfying\n/ba∇dbl(/u1D4620,/u1D4621)/ba∇dbl(/u1D43B4(Ω)∩/u1D43B2\n0(Ω))×/u1D43B2\n0(Ω)≤/u1D445. (9)\nWe stress that (8) only represents a local stability result since it holds on every\nball with radius /u1D445 > 0 in the strong topology (/u1D43B4(Ω) ∩/u1D43B2\n0(Ω)) ×/u1D43B2\n0(Ω),but\nthey are not independent of the initial data. Moreover, as ob served by the authors\nin [5], the drawback of (8)-(9) is that it could not be proved i n the weak topology\n/u1D43B2\n0(Ω) ×/u1D43F2(Ω), even taking initial data uniformly bounded in /u1D43B2\n0(Ω) ×/u1D43F2(Ω).\nAlthough we recognized that our results for (3) can not be fai rly compared to\nsuch a result, we do can conclude by means of the upper and lowe r polynomial\nbounds (6) that the estimate (8) will never be reached for wea k initial data given in\n/u1D43B2\n0(Ω) ×/u1D43F2(Ω). Therefore, our results act as complementary conclusions t o [5] by\nclarifying such drawback raised therein, and yet giving a di fferent point of view of\nstability by means of (6) and its consequences concerning pr oblem (3).4 E. H. Gomes Tavares, M. A. Jorge Silva, V. Narciso and A. Vice nte\nIn the second case, equation (3) falls into\n/u1D715/u1D461/u1D461/u1D462−/u1D705Δ/u1D462+Δ2/u1D462+/u1D6FC/bracketleftbigg/uni222B.dsp\nΩ|/u1D715/u1D461/u1D462|2/u1D451/u1D465/bracketrightbigg/u1D45E\n/u1D715/u1D461/u1D462=0 inΩ× (0,∞). (10)\nUnlike the first case, here we have an easier setting because t he kinetic damping\ncoefficient provides a kind of monotonous (polynomial) dampi ng whose computa-\ntions to achieve (6) remain unchanged (and with less calcula tions). This means that\nall results highlighted previously still hold for this part icular case. In addition, they\nclarify what is precisely the stability result related to pr oblems addressed in [19, 20],\nwhich in turn represent particular models of abstract dampi ng given by [1, Section\n8]. In other words, in terms of stability, our methodology pr ovides a way to show\nthe existence of absorbing sets with polynomial rate (and no t faster than polynomial\nrate depending on /u1D45E) when dealing with generalized problems relate to (10), sub ject\nthat is not addressed in [19, 20].\nFinally, in the third case let us see equations (2)-(3) as fol lows\n/u1D715/u1D461/u1D461/u1D462−/u1D705Δ/u1D462+Δ2/u1D462+/u1D440/parenleftbigg/uni222B.dsp\nΩ/parenleftBig\n|Δ/u1D462|2+ |/u1D715/u1D461/u1D462|2/parenrightBig\n/u1D451/u1D465/parenrightbigg\n/u1D434/u1D715/u1D461/u1D462=0 inΩ× (0,∞),(11)\nwhere operator /u1D434represents the Laplacian operator /u1D434=−Δor else the identity one\n/u1D434=/u1D43C. Thus, here we clearly have two subcases, namely, when /u1D440(·) ≥ 0 is a non-\ndegenerate or possibly degenerate function. For instance, when/u1D440(/u1D460)=/u1D6FC/u1D460/u1D45E, /u1D460≥0,\nand/u1D434=−Δ, then we go back to problem (2). For this (degenerate) nonloc al strong\ndamping situation with /u1D45E≥1, it is considered in [11, Theorem 3.1] an upper\npolynomial stability for the corresponding energy, which a lso involves a standard\nnonlinear source term. Nonetheless, we call the attention t o the following prediction\nresult provided in [11, Theorem 4.1] for (2) addressed on a bo unded domain Ωwith\nclamped boundary condition and /u1D45E≥1:By taking finite initial energy 0</u1D438/u1D705(0)<\n∞, then/u1D438/u1D705(/u1D461)given in (5)satisfies\n/u1D438/u1D705(/u1D461) ≤3/u1D438/u1D705(0)/u1D452−/u1D6FF∫/u1D461\n0/ba∇dbl/u1D462(/u1D460)/ba∇dbl2/u1D45E/u1D451/u1D460, /u1D461 > 0, (12)\nwhere/u1D6FF=/u1D6FF(1\n/u1D438/u1D705(0))>0is a constant proportional to 1//u1D438/u1D705(0).\nAlthough the estimate (12) provides a new result with an exponential face , it\ndoes not mean any kind of stability result. Indeed, it is only a peculiar estimate\nindicating that prevents exponential decay patterns as rem arked in [11, Section 4].\nIn addition, it is worth pointing out that our computations t o reach the stability result\nfor problem (3) can be easily adjusted to (2), even for /u1D45E≥1/2 thanks to a inequality\nprovided in [1, Lemma 2.2]. Therefore, through the polynomi al range (6) we provide\nhere a much more accurate stability result than the estimate expressed by (12), by\nconcluding indeed that both problems (2) and (3) are never ex ponentially stable in\nthe topology of the energy space.\nOn the other hand, in the non-degenerate case /u1D440(/u1D460)>0, /u1D460≥0, but still taking\n/u1D434=−Δ, a generalized version of (11) has been recently approached by [16] in a\ncontext of strong attractors , that is, the existence of attractors in the topology ofIntrinsic polynomial squeezing for Balakrishnan-Taylor b eam models 5\nmore regular space than the weak phase space. In this occasio n, the/u1D4361-regularity\nfor/u1D440 > 0 brings out the non-degeneracy of the damping coefficient, wh ich in turn\nallowed them to reach interesting results on well-posednes s, regularity and long-time\nbehavior of solutions over more regular spaces. Such assump tion of positiveness\nfor the damping coefficient has been also addressed by other au thors for related\nproblems, see e.g. [8, 9, 10]. From our point of view, in spite of representing a nice\ncase, the latter does not portray the current situation of th is paper so that we do not\nprovide more detailed comparisons with such a non-degenera te problems, but we\nrefer to [5, 8, 9, 10, 11, 16] for a nice survey on this kind of no n-degenerate damping\ncoefficients. Additionally, we note that the suitable case of non-degenerate damping\ncoefficient/u1D440(/u1D460)>0, /u1D460≥0, and/u1D434=/u1D43Cin (11) has not been considered in the\nliterature so far and shall be concerned in another work by th e authors in the future.\nAt light of the above statements, one sees e.g. when /u1D440(/u1D460)=/u1D6FC/u1D460/u1D45E, /u1D460≥0,and\n/u1D434=/u1D43C, then problem (11) falls into (3), being a problem not yet add ressed in the\nliterature that brings out a new branch of studies for such a n onlocal (possibly\ndegenerate) damped problems, and also justifies all new stab ility results previously\nspecified.\n2 The problem and well-posedness\nLet us consider again the beam model with energy damping\n/u1D715/u1D461/u1D461/u1D462+Δ2/u1D462−/u1D705Δ/u1D462+/u1D6FC/bracketleftbigg/uni222B.dsp\nΩ/parenleftBig\n|/u1D715/u1D461/u1D462|2+ |Δ/u1D462|2/parenrightBig\n/u1D451/u1D466/bracketrightbigg/u1D45E\n/u1D715/u1D461/u1D462=0 inΩ× (0,∞),(13)\nwith clamped boundary condition\n/u1D462=/u1D715/u1D462\n/u1D715/u1D708=0 on/u1D715Ω× [0,∞), (14)\nand initial data\n/u1D462(/u1D465,0)=/u1D4620(/u1D465), /u1D715/u1D461/u1D462(/u1D465,0)=/u1D4621(/u1D465), /u1D465∈Ω. (15)\nTo address problem (13)-(15), we introduce the Hilbert phas e space (still called\nenergy space )\nH:=/u1D43B2\n0(Ω) ×/u1D43F2(Ω),\nequipped with the inner product/angbracketleftbig\n/u1D4671,/u1D4672/angbracketrightbig\nH:=/angbracketleftbig\nΔ/u1D4621,Δ/u1D4622/angbracketrightbig\n+/angbracketleftbig\n/u1D463.alt1,/u1D463.alt2/angbracketrightbig\nfor/u1D467/u1D456=(/u1D462/u1D456,/u1D463.alt/u1D456) ∈\nH, /u1D456=1,2,and norm /ba∇dbl/u1D467/ba∇dblH=/parenleftbig/ba∇dblΔ/u1D462/ba∇dbl2+ /ba∇dbl/u1D463.alt/ba∇dbl2/parenrightbig1/2,for/u1D467=(/u1D462,/u1D463.alt) ∈ H,where\n/an}b∇acke⊔le{⊔/u1D462,/u1D463.alt/an}b∇acke⊔∇i}h⊔:=/uni222B.dsp\nΩ/u1D462/u1D463.alt/u1D451/u1D465 ,/ba∇dbl/u1D462/ba∇dbl2:=/an}b∇acke⊔le{⊔/u1D462,/u1D462/an}b∇acke⊔∇i}h⊔and/ba∇dbl/u1D467/ba∇dbl2\nH:=/an}b∇acke⊔le{⊔/u1D467,/u1D467/an}b∇acke⊔∇i}h⊔H.\nIn order to stablish the well-posedness of (13)-(15), we defi ne the vector-valued\nfunction/u1D467(/u1D461):=(/u1D462(/u1D461),/u1D463.alt(/u1D461)),/u1D461≥0,with/u1D463.alt=/u1D715/u1D461/u1D462. Then we can rewrite system\n(13)-(15) as the following first order abstract problem6 E. H. Gomes Tavares, M. A. Jorge Silva, V. Narciso and A. Vice nte\n/braceleftBigg/u1D715/u1D461/u1D467=A/u1D467+M(/u1D467), /u1D461 > 0,\n/u1D467(0)=(/u1D4620,/u1D4621):=/u1D4670,(16)\nwhereA:D(A) ⊂ H → H is the linear operator given by\nA/u1D467=(/u1D463.alt,−Δ2/u1D462),D(A) :=/u1D43B4(Ω) ∩/u1D43B2\n0(Ω), (17)\nandM:H → H is the nonlinear operator\nM(/u1D467)=(0,/u1D705Δ/u1D462−/u1D6FC/ba∇dbl/u1D467/ba∇dbl2/u1D45E\nH/u1D463.alt), /u1D467=(/u1D462,/u1D463.alt) ∈ H. (18)\nTherefore, the existence and uniqueness of solution to the s ystem (13)-(15) relies\non the study of problem (16). Accordingly, we have the follow ing well-posedness\nresult.\nTheorem 1. Let/u1D705,/u1D6FC≥0and/u1D45E≥1\n2be given constants. If /u1D4670∈ H, then (16)has a\nunique mild solution /u1D467in the class/u1D467∈/u1D436([0,∞),H).\nIn addition, if /u1D4670∈ D(A) , then/u1D467is a regular solution lying in the class\n/u1D467∈/u1D436([0,∞),D(A)) ∩/u1D4361([0,∞),H).\nProof. To show the local version of the first statement, it is enough t o prove that A\ngiven in (17) is the infinitesimal generator of a /u1D4360-semigroup of contractions /u1D452A/u1D461\n(which is very standard) and Mset in (18) is locally Lipschitz on Hwhich will be\ndone next. Indeed, let /u1D45F >0 and/u1D4671,/u1D4672∈ Hsuch that max {/ba∇dbl/u1D4671/ba∇dblH,/ba∇dbl/u1D4672/ba∇dblH} ≤/u1D45F. We\nnote that\n/bardblex/bardblex/bardblex/ba∇dbl/u1D4671/ba∇dbl2/u1D45E\nH/u1D463.alt1− /ba∇dbl/u1D4672/ba∇dbl2/u1D45E\nH/u1D463.alt2/bardblex/bardblex/bardblex≤/bracketleftBig\n/ba∇dbl/u1D4671/ba∇dbl2/u1D45E\nH+ /ba∇dbl/u1D4672/ba∇dbl2/u1D45E\nH/bracketrightBig\n/ba∇dbl/u1D463.alt1−/u1D463.alt2/ba∇dbl+/barex/barex/barex/ba∇dbl/u1D4671/ba∇dbl2/u1D45E\nH− /ba∇dbl/u1D4672/ba∇dbl2/u1D45E\nH/barex/barex/barex/ba∇dbl/u1D463.alt1+/u1D463.alt2/ba∇dbl.\n(19)\nThe first term on the right side of (19) can be estimated by\n/bracketleftBig\n/ba∇dbl/u1D4671/ba∇dbl2/u1D45E\nH+ /ba∇dbl/u1D4672/ba∇dbl2/u1D45E\nH/bracketrightBig\n/ba∇dbl/u1D463.alt1−/u1D463.alt2/ba∇dbl ≤2/u1D45F2/u1D45E/ba∇dbl/u1D4671−/u1D4672/ba∇dblH.\nNow, from a suitable inequality provided in [1] /one.supwe estimate the second term as\nfollows /barex/barex/barex/ba∇dbl/u1D4671/ba∇dbl2/u1D45E\nH− /ba∇dbl/u1D4672/ba∇dbl2/u1D45E\nH/barex/barex/barex/ba∇dbl/u1D463.alt1+/u1D463.alt2/ba∇dbl ≤4/u1D45E/u1D45F2/u1D45E/ba∇dbl/u1D4671−/u1D4672/ba∇dblH.\nPlugging the two last estimates in (19), we obtain\n/bardblex/bardblex/bardblex/ba∇dbl/u1D4671/ba∇dbl2/u1D45E\nH/u1D463.alt1− /ba∇dbl/u1D4672/ba∇dbl2/u1D45E\nH/u1D463.alt2/bardblex/bardblex/bardblex\nH≤2(2/u1D45E+1)/u1D45F2/u1D45E/ba∇dbl/u1D4671−/u1D4672/ba∇dblH.\nThus,\n/one.supSee [1, Lemma 2.2]: Let/u1D44Bbe a normed space with norm /ba∇dbl · /ba∇dbl/u1D44B. Then, for any /u1D460≥1we have\n/barex/barex/ba∇dbl/u1D462/ba∇dbl/u1D460\n/u1D44B− /ba∇dbl/u1D463.alt/ba∇dbl/u1D460\n/u1D44B/barex/barex≤/u1D460max{/ba∇dbl/u1D462/ba∇dbl/u1D44B,/ba∇dbl/u1D463.alt/ba∇dbl/u1D44B}/u1D460−1/ba∇dbl/u1D462−/u1D463.alt/ba∇dbl/u1D44B,∀/u1D462,/u1D463.alt∈/u1D44B. (20)Intrinsic polynomial squeezing for Balakrishnan-Taylor b eam models 7\n/ba∇dblM(/u1D4671) −M(/u1D4672)/ba∇dblH≤/parenleftBig\n/u1D705+2(2/u1D45E+1)/u1D6FC/u1D45F2/u1D45E/parenrightBig\n/ba∇dbl/u1D4671−/u1D4672/ba∇dblH,\nandMis locally Lipschitz in H.\nHence, according to Pazy [15, Chapter 6], if /u1D4670∈ H(/u1D4670∈/u1D437(A)), there exists a\ntime/u1D461max∈ (0,+∞]such that (16) has a unique mild (regular) solution\n/u1D467∈/u1D436([0,/u1D461max),H) (/u1D467∈/u1D436([0,/u1D461max),/u1D437(A)) ∩/u1D4361([0,/u1D461max),H)).\nMoreover, such time /u1D461maxsatisfies either the conditions /u1D461max=+∞or else/u1D461max<+∞\nwith\nlim\n/u1D461→/u1D461−max/ba∇dbl/u1D467(/u1D461)/ba∇dblH=+∞. (21)\nIn order to show that /u1D461max=+∞, we consider /u1D4670∈/u1D437(A)and the corresponding\nregular solution /u1D467of (16). Taking the inner product in Hof (16) with /u1D467, we obtain\n1\n2/u1D451\n/u1D451/u1D461/bracketleftbig\n/ba∇dbl/u1D467(/u1D461)/ba∇dbl2\nH+/u1D705/ba∇dbl∇/u1D462(/u1D461)/ba∇dbl2/bracketrightbig\n+/u1D6FC/ba∇dbl/u1D467(/u1D461)/ba∇dbl2/u1D45E\nH/ba∇dbl/u1D715/u1D461/u1D462(/u1D461)/ba∇dbl2=0/u1D461∈ [0,/u1D461max).(22)\nIntegrating (22) over (0,/u1D461), /u1D461∈ [0,/u1D461max), we get\n/ba∇dbl/u1D467(/u1D461)/ba∇dblH≤ (1+/u1D450′/u1D705)1/2/ba∇dbl/u1D4670/ba∇dblH, /u1D461∈ [0,/u1D461max).\nHere, the constant /u1D450′>0 comes from the embedding /u1D43B2\n0(Ω)↩→/u1D43B1\n0(Ω). The\nlast estimate contradicts (21). Hence, /u1D461/u1D45A/u1D44E/u1D465=+∞. Using a limit process, one can\nconclude the same result for mild solutions.\nThe proof of Theorem 1 is then complete.\n3 Lower-upper polynomial energy’s bounds\nBy means of the notations introduced in Section 2, we recall t hat the energy functional\ncorresponding to problem (13)-(15) can be expressed by\n/u1D438/u1D705(/u1D461)=1\n2/bracketleftbig\n/ba∇dbl(/u1D462(/u1D461),/u1D715/u1D461/u1D462(/u1D461))/ba∇dbl2\nH+/u1D705/ba∇dbl∇/u1D462(/u1D461)/ba∇dbl2/bracketrightbig\n, /u1D461≥0. (23)\nOur main stability result reveals that /u1D438/u1D705(/u1D461)is squeezed by decreasing polynomial\nfunctions as follows.\nTheorem 2. Under the assumptions of Theorem 1, there exists an increasing function\nJ:R+→R+such that the energy /u1D438/u1D705(/u1D461)satisfies\n/bracketleftbig\n2/u1D45E+1/u1D6FC/u1D45E/u1D461+/bracketleftbig\n/u1D438/u1D705(0)/bracketrightbig−/u1D45E/bracketrightbig−1//u1D45E≤/u1D438/u1D705(/u1D461) ≤/bracketleftbigg/u1D45E\nJ (/u1D438/u1D705(0))(/u1D461−1)++/bracketleftbig\n/u1D438/u1D705(0)/bracketrightbig−/u1D45E/bracketrightbigg−1//u1D45E\n,(24)\nfor all/u1D461 >0, where we use the standard notation /u1D460+:=(/u1D460+ |/u1D460|)/2.\nProof. Taking the scalar product in /u1D43F2(Ω)of (13) with /u1D715/u1D461/u1D462, we obtain8 E. H. Gomes Tavares, M. A. Jorge Silva, V. Narciso and A. Vice nte\n/u1D451\n/u1D451/u1D461/u1D438/u1D705(/u1D461)=−/u1D6FC||(/u1D462(/u1D461),/u1D715/u1D461/u1D462(/u1D461))||2/u1D45E\nH/ba∇dbl/u1D715/u1D461/u1D462(/u1D461)/ba∇dbl2, /u1D461 > 0. (25)\nLet us prove the lower and upper estimates in (24) in the seque l.\nLower bound. We first note that\n||(/u1D462(/u1D461),/u1D715/u1D461/u1D462(/u1D461))||2/u1D45E\nH/ba∇dbl/u1D715/u1D461/u1D462(/u1D461)/ba∇dbl2≤2/u1D45E+1[/u1D438/u1D705(/u1D461)]/u1D45E+1,\nand replacing it in (25), we get\n/u1D451\n/u1D451/u1D461/u1D438/u1D705(/u1D461) ≥ − 2/u1D45E+1/u1D6FC[/u1D438/u1D705(/u1D461)]/u1D45E+1, /u1D461 > 0. (26)\nThus, integrating (26) and proceeding a straightforward co mputation, we reach the\nfirst inequality in (24).\nUpper bound. Now, we are going to prove the second inequality of (24). To do so,\nwe provide some proper estimates and then apply a Nakao’s res ult (cf. [13, 14]).\nWe start by noting that\n||(/u1D462(/u1D461),/u1D715/u1D461/u1D462(/u1D461)||2/u1D45E\nH/ba∇dbl/u1D715/u1D461/u1D462(/u1D461)/ba∇dbl2≥ /ba∇dbl/u1D715/u1D461/u1D462(/u1D461)/ba∇dbl2(/u1D45E+1), (27)\nand replacing (27) in (25), we get\n/u1D451\n/u1D451/u1D461/u1D438/u1D705(/u1D461) +/u1D6FC/ba∇dbl/u1D715/u1D461/u1D462(/u1D461)/ba∇dbl2(/u1D45E+1)≤0, /u1D461 > 0, (28)\nwhich implies that /u1D438/u1D705(/u1D461)is non-increasing with /u1D438/u1D705(/u1D461) ≤/u1D438/u1D705(0)for every/u1D461 >0.\nAlso, integrating (28) from /u1D461to/u1D461+1, we obtain\n/u1D6FC/uni222B.dsp/u1D461+1\n/u1D461/ba∇dbl/u1D715/u1D461/u1D462(/u1D460)/ba∇dbl2(/u1D45E+1)/u1D451/u1D460≤/u1D438/u1D705(/u1D461) −/u1D438/u1D705(/u1D461+1):=[/u1D437(/u1D461)]2. (29)\nUsing Hölder’s inequality with/u1D45E\n/u1D45E+1+1\n/u1D45E+1=1 and (29), we infer\n/uni222B.dsp/u1D461+1\n/u1D461/ba∇dbl/u1D715/u1D461/u1D462(/u1D460)/ba∇dbl2/u1D451/u1D460≤1\n/u1D6FC1\n/u1D45E+1[/u1D437(/u1D461)]2\n/u1D45E+1. (30)\nFrom the Mean Value Theorem for integrals, there exist /u1D4611∈ [/u1D461,/u1D461+1\n4]and/u1D4612∈\n[/u1D461+3\n4,/u1D461+1]such that\n/ba∇dbl/u1D715/u1D461/u1D462(/u1D461/u1D456)/ba∇dbl2≤4/uni222B.dsp/u1D461+1\n/u1D461/ba∇dbl/u1D715/u1D461/u1D462(/u1D460)/ba∇dbl2/u1D451/u1D460≤4\n/u1D6FC1\n/u1D45E+1[/u1D437(/u1D461)]2\n/u1D45E+1, /u1D456=1,2. (31)\nOn the other hand, taking the scalar product in /u1D43F2(Ω)of (13) with /u1D462and inte-\ngrating the result over [/u1D4611,/u1D4612], we haveIntrinsic polynomial squeezing for Balakrishnan-Taylor b eam models 9\n/uni222B.dsp/u1D4612\n/u1D4611/u1D438/u1D705(/u1D460)/u1D451/u1D460=/uni222B.dsp/u1D4612\n/u1D4611/ba∇dbl/u1D715/u1D461/u1D462(/u1D460)/ba∇dbl2/u1D451/u1D460+1\n2[(/u1D715/u1D461/u1D462(/u1D4611),/u1D462(/u1D4611)) − (/u1D715/u1D461/u1D462(/u1D4612),/u1D462(/u1D4612))]\n−/u1D6FC\n2/uni222B.dsp/u1D4612\n/u1D4611||(/u1D462(/u1D460),/u1D715/u1D461/u1D462(/u1D460))||2/u1D45E\nH(/u1D715/u1D461/u1D462(/u1D460),/u1D462(/u1D460))/u1D451/u1D460. (32)\nLet us estimate the terms in the right side of (32). Firstly, w e note that through\nHölder’s inequality, (31) and Young’s inequality, we obtai n\n|(/u1D715/u1D461/u1D462(/u1D4611),/u1D462(/u1D4611)) − (/u1D715/u1D461/u1D462(/u1D4612),/u1D462(/u1D4612))| ≤/u1D4512/summationdisplay.1\n/u1D456=1/ba∇dbl/u1D715/u1D461/u1D462(/u1D461/u1D456)/ba∇dbl/ba∇dblΔ/u1D462(/u1D461/u1D456)/ba∇dbl\n≤8/u1D451\n/u1D6FC1\n2(/u1D45E+1)[/u1D437(/u1D461)]1\n/u1D45E+1sup\n/u1D4611≤/u1D460≤/u1D4612[/u1D438/u1D705(/u1D460)]1/2\n≤128/u1D4512\n/u1D6FC1\n/u1D45E+1[/u1D437(/u1D461)]2\n/u1D45E+1+1\n8sup\n/u1D4611≤/u1D460≤/u1D4612/u1D438/u1D705(/u1D460),\nwhere the constant /u1D451 >0 comes from the embedding /u1D43B2\n0(Ω)↩→/u1D43F2(Ω). Addition-\nally, using that /u1D438/u1D705(/u1D461) ≤/u1D438/u1D705(0), we have\n/ba∇dbl(/u1D462(/u1D461),/u1D715/u1D461/u1D462(/u1D461))/ba∇dbl2/u1D45E\nH≤2/u1D45E[/u1D438/u1D705(/u1D461)]/u1D45E≤2/u1D45E[/u1D438/u1D705(0)]/u1D45E.\nFrom this and (30) we also get\n/barex/barex/barex/barex/uni222B.dsp/u1D4612\n/u1D4611||(/u1D462(/u1D460),/u1D715/u1D461/u1D462(/u1D460))||2/u1D45E\nH(/u1D715/u1D461/u1D462(/u1D460),/u1D462(/u1D460))/u1D451/u1D460/barex/barex/barex/barex≤22/u1D45E+3/u1D4512[/u1D438/u1D705(0)]2/u1D45E\n/u1D6FC−/u1D45E\n/u1D45E+1[/u1D437(/u1D461)]2\n/u1D45E+1\n+1\n8/u1D6FCsup\n/u1D4611≤/u1D460≤/u1D4612/u1D438/u1D705(/u1D460).\nRegarding again (30) and replacing the above estimates in (3 2), we obtain\n/uni222B.dsp/u1D4612\n/u1D4611/u1D438/u1D705(/u1D460)/u1D451/u1D460≤ K (/u1D438/u1D705(0)) [/u1D437(/u1D461)]2\n/u1D45E+1+1\n8sup\n/u1D4611≤/u1D460≤/u1D4612/u1D438/u1D705(/u1D460), (33)\nwhere we set the function Kas\nK(/u1D460):=/bracketleftbigg64/u1D4512+1\n/u1D6FC1\n/u1D45E+1+2(/u1D45E+1)/u1D4512/u1D6FC2/u1D45E+1\n/u1D45E+1/u1D4602/u1D45E/bracketrightbigg\n>0.\nUsing once more the Mean Value Theorem for integrals and the f act that/u1D438/u1D705(/u1D461)is\nnon-increasing, there exists /u1D701∈ [/u1D4611,/u1D4612]such that\n/uni222B.dsp/u1D4612\n/u1D4611/u1D438/u1D705(/u1D460)/u1D451/u1D460=/u1D438/u1D705(/u1D701)(/u1D4612−/u1D4611) ≥1\n2/u1D438/u1D705(/u1D461+1),\nand then10 E. H. Gomes Tavares, M. A. Jorge Silva, V. Narciso and A. Vic ente\nsup\n/u1D461≤/u1D460≤/u1D461+1/u1D438/u1D705(/u1D460)=/u1D438/u1D705(/u1D461)=/u1D438/u1D705(/u1D461+1) + [/u1D437(/u1D461)]2≤2/uni222B.dsp/u1D4612\n/u1D4611/u1D438/u1D705(/u1D460)/u1D451/u1D460+ [/u1D437(/u1D461)]2.\nThus, from this and (33), we arrive at\nsup\n/u1D461≤/u1D460≤/u1D461+1/u1D438/u1D705(/u1D460) ≤ [/u1D437(/u1D461)]2+2/uni222B.dsp/u1D4612\n/u1D4611/u1D438/u1D705(/u1D460)/u1D451/u1D460\n≤ [/u1D437(/u1D461)]2+2K (/u1D438/u1D705(0)) [/u1D437(/u1D461)]2\n/u1D45E+1+1\n4sup\n/u1D461≤/u1D460≤/u1D461+1/u1D438/u1D705(/u1D460),\nand since 0<2\n/u1D45E+1≤2, we obtain\nsup\n/u1D461≤/u1D460≤/u1D461+1/u1D438/u1D705(/u1D460) ≤4\n3[/u1D437(/u1D461)]2\n/u1D45E+1/bracketleftBig\n[/u1D437(/u1D461)]2/u1D45E\n/u1D45E+1+2K (/u1D438/u1D705(0))/bracketrightBig\n. (34)\nObserving that [/u1D437(/u1D461)]2/u1D45E\n/u1D45E+1≤ [/u1D438/u1D705(/u1D461) +/u1D438/u1D705(/u1D461+1)]/u1D45E\n/u1D45E+1≤2/u1D45E\n/u1D45E+1[/u1D438/u1D705(0)]/u1D45E\n/u1D45E+1,and de-\nnoting by\nJ(/u1D460):=/parenleftbigg4\n3/parenrightbigg/u1D45E+1/bracketleftBig\n(2/u1D460)/u1D45E\n/u1D45E+1+2K(/u1D460)/bracketrightBig/u1D45E+1\n>0, (35)\nand also recalling the definition of [/u1D437(/u1D461)]2in (29), we obtain from (34) that\nsup\n/u1D461≤/u1D460≤/u1D461+1[/u1D438/u1D705(/u1D460)]/u1D45E+1≤ J (/u1D438/u1D705(0)) [/u1D438/u1D705(/u1D461) −/u1D438/u1D705(/u1D461+1)].\nHence, applying e.g. Lemma 2.1 of [14] with /u1D438/u1D705=/u1D719,J (/u1D438/u1D705(0))=/u1D4360,and/u1D43E=0,\nwe conclude /u1D438/u1D705(/u1D461) ≤/bracketleftbigg\n/u1D45E\nJ(/u1D438/u1D705(0))(/u1D461−1)++1/bracketleftbig\n/u1D438/u1D705(0)/bracketrightbig/u1D45E/bracketrightbigg−1//u1D45E\n,which ends the proof of\nthe second inequality in (24).\nThe proof of Theorem 2 is therefore complete.\nRemark 1. It is worth point out that we always have\n/bracketleftbig\n22/u1D45E+1/u1D6FC/u1D45E/u1D461+/bracketleftbig\n/u1D438/u1D705(0)/bracketrightbig−/u1D45E/bracketrightbig−1//u1D45E≤/bracketleftbigg/u1D45E\nJ (/u1D438/u1D705(0))(/u1D461−1)++/bracketleftbig\n/u1D438/u1D705(0)/bracketrightbig−/u1D45E/bracketrightbigg−1//u1D45E\n,(36)\nso that it makes sense to express /u1D438/u1D705(/u1D461)between the inequalities in (24). Indeed, from\nthe definition Jin (35) one easily sees that J (/u1D438/u1D705(0)) ≥1\n22/u1D45E+1/u1D6FC,from where one\nconcludes (36) promptly.\nCorollary 1. (Polynomial Range of Decay). Under the assumptions of Theorem 2,\nthe energy functional /u1D438/u1D705(/u1D461)defined in (23)decays squeezed as follows\n/u1D4500/u1D461−1\n/u1D45E/lessorsimilar/u1D438/u1D705(/u1D461)/lessorsimilar/u1D4360/u1D461−1\n/u1D45Eas/u1D461→ +∞, (37)\nfor some constants 0</u1D4500≤/u1D4360depending on the initial energy /u1D438/u1D705(0).Intrinsic polynomial squeezing for Balakrishnan-Taylor b eam models 11\nIn other words, /u1D438/u1D705(/u1D461)decays polynomially at rate /u1D461−1//u1D45E(/u1D45E≥1/2) as/u1D461→ +∞ .\n⊓ ⊔\nCorollary 2. (Non-Exponential Stability). Under the assumptions of Theorem 2,\nthe energy/u1D438/u1D705(/u1D461)set in (23)never decays exponentially as /u1D452−/u1D44E/u1D461(/u1D44E >0) as/u1D461→ +∞ .\n⊓ ⊔\nReferences\n1. F. Aloui, I. Ben Hassen, A. Haraux, Compactness of traject ories to some nonlinear second\norder evolution equations and applications, J. Math. Pures Appl. 100 (2013), no. 3, 295-326.\n2. A. V. Balakrishnan, A theory of nonlinear damping in flexib le structures. Stabilization of\nflexible structures, p. 1-12, 1988.\n3. A. V. Balakrishnan and L. W. Taylor, Distributed paramete r nonlinear damping models for\nflight structures, in: Proceedings Daming 89, Flight Dynami cs Lab and Air Force Wright\nAeronautical Labs, WPAFB, 1989.\n4. R. W. Bass and D. Zes, Spillover, Nonlinearity, and flexibl e structures, in The Fourth NASA\nWorkshop on Computational Control of Flexible Aerospace Sy stems, NASA Conference\nPublication 10065 ed. L.W.Taylor (1991) 1-14.\n5. M. M. Cavalcanti, V. N. Domingos Cavalcanti, M. A. Jorge Si lva, V. Narciso, Stability for\nextensible beams with a single degenerate nonlocal damping of Balakrishnan-Taylor type, J.\nof Differential Equations, 290 (2021), 197-222.\n6. E. H. Dowell, Aeroelasticity of plates and shells, Gronin ger, NL, Noordhoff Int. Publishing\nCo. (1975).\n7. T. J. Hughes and J. E. Marsden, Mathematical foundation of elasticity, Englewood C. Prentice-\nHall, 1983.\n8. M. A. Jorge Silva and V. Narciso, Long-time behavior for a p late equation with nonlocal weak\ndamping, Differential Integral Equations 27 (2014), no. 9-1 0, 931-948.\n9. M. A. Jorge Silva and V. Narciso, Attractors and their prop erties for a class of nonlocal\nextensible beams, Discrete Contin. Dyn. Syst. 35 (2015) no. 3, 985-1008.\n10. M. A. Jorge Silva and V. Narciso, Long-time dynamics for a class of extensible beams with\nnonlocal nonlinear damping, Evol. Equ. Control Theory 6 (20 17), no. 3, 437-470.\n11. M. A. Jorge Silva, V. Narciso and A. Vicente, On a beam mode l related to flight structures\nwith nonlocal energy damping, Discrete and Continuous Dyna mical Systems Series B. 24\n(2019), 3281-3298.\n12. C. Mu, J. Ma, On a system of nonlinear wave equations with B alakrishnan-Taylor damping,\nZ. Angew. Math. Phys. 65 (2014) 91-113.\n13. M. Nakao, Convergence of solutions of the wave equation w ith a nonlinear dissipative term\nto the steady state, Mem. Fac. Sci. Kyushu Univ. Ser. A, 30 (19 76), 257-265.\n14. M. Nakao, A difference inequality and its application to n onlinear evolution equations, J.\nMath. Soc. Japan 30 4 (1978) 747-762.\n15. A. Pazy, Semigroups of linear operators and application s to partial differential equations, vol.\n44, Springer-Verland,1983.\n16. Y. Sun, Z. Yang, Strong attractors and their robustness f or an extensible beam\nmodel with energy damping, Discrete and Continuous Dynamic al Systems, 2021. (doi:\n10.3934/dcdsb.2021175)\n17. Y. You, Inertial manifolds and stabilization of nonline ar beam equations with Balakrishnan-\nTaylor damping, Abstr. Appl. Anal. 1(1) (1996) 83-102.\n18. W. Zhang, Nonlinear Damping Model: Response to Random Ex citation in: 5th Annual NASA\nSpacecraft Control Laboratory Experiment (SCOLE) Worksho p (1988) 27-38.12 E. H. Gomes Tavares, M. A. Jorge Silva, V. Narciso and A. Vic ente\n19. C. Zhao, C. Zhao, C. Zhong, The global attractor for a clas s of extensible beams with nonlocal\nweak damping, Discrete Contin. Dyn. Syst. - B, 25 (2020), 935 -955.\n20. C. Zhao, S. Ma, C. Zhong, Long-time behavior for a class of extensible beams with nonlocal\nweak damping and critical nonlinearity Journal of Mathemat ical Physics 61 (2020), p. 032701."    },    {        "title": "1911.12786v1.Transport_properties_of_spin_superfluids__comparing_easy_plane_ferro__and_antiferromagnets.pdf",        "content": "Transport properties of spin superfluids—comparing easy-plane ferro- and\nantiferromagnets\nMartin Evers and Ulrich Nowak\nFachbereich Physik, Universität Konstanz, 78457 Konstanz, Germany\n(Dated: December 2, 2019)\nWe present a study on spin-superfluid transport based on an atomistic, classical spin model. Easy-\nplane ferro- as well as antiferromagnets are considered, which allows for a direct comparison of these\ntwo material classes based on the same model assumptions. We find a spin-superfluid transport\nwhich is robust against variations of the boundary conditions, thermal fluctuations, and dissipation\nmodeled via Gilbert damping. Though the spin accumulations is smaller for antiferromagnets the\nrange of the spin-superfluid transport turns out to be identical for ferro- and antiferromagnets. Fi-\nnally, we calculate and explore the role of the driving frequency and especially the critical frequency,\nwhere phase slips occur and the spin accumulation breaks down.\nI. INTRODUCTION\nSpin transport in magnetic insulators [1, 2] has been\nintensively studied beacause of the fundamental interest\nin the various physical phenomena that occur in these\nmaterials and because of their potential for future appli-\ncations. Magnetic insulators do not exhibit Joule heat-\ning [3] as no electron transport is involved and many of\nthese are oxides with exceptionally low magnetic damp-\ning [4], which hopefully allows for energy efficient trans-\nport properties. It has even been shown that the realiza-\ntion of logic elements is possible [5], such that devices are\ncompatible and integratable with CMOS technology [6].\nStudies on transport in this material class focuses mostly\non transport of magnons [7], i.e. quanta of spin waves—\nthe elementary excitations of the magnetic ground state.\nAs magnons are quasi particles, their number is not con-\nserved and each magnon mode shows an exponential de-\ncay upon transport through the system on a length scale\nξcalled magnon propagation length [8–13]. This is even\ntrue at zero temperature and in a clean system without\nany disorder due to the coupling of the magnons to elec-\ntronic and phononic degrees of freedom, a fact which is\ndescribed phenomenologically via Gilbert damping in the\nequation of motion as will be explained below.\nIn contrast to this damped magnonic transport, a pro-\nposal for spin transport was made that carries the name\nspin superfluidity. The original idea is in fact quite old\n[14, 15] and rests on a similarity of the magnetic or-\nder parameter—either the magnetization of a ferromag-\nnet or the Néel vector of an antiferromagnet—compared\nto the order parameter of superfluidity—the macroscopic\nwave function—as it occurs for He-4 below the lambda\ntransition. For instance, in easy-plane ferromagnets the\nmagnetizationfeaturesaspontaneouslybrokenrotational\nsymmetry in the easy plane ( SO(2)symmetry) that is\nequivalent to the spontaneously broken gauge invariance\nof the macroscopic wave function ( U(1)symmetry). This\nsymmetry leads in both cases to currents that are sta-\nble against small deviations—the supercurrents. [16] One\nstriking difference of spin-superfluid transport to spin-\nwave transport is its distance dependence: for spin su-perfluidity it is expected to be non-exponential, pushing\nthe limit of the range of magnonic transport.\nThe first experimental realizations of a spin superfluid\nwas achieved in a system of nuclear spins of He-3 atoms\n[17]—a model system which is not in a solid state. Only\nrecently the physics of spin superfluidity has drawn again\nattention for the case of solid magnets [18–23], including\na proposed dissipationless transport in metallic magnets\n[18]. However, König et al. neglected spin-orbit inter-\naction in their model for the electrons, which is one of\nthe reasons for Gilbert damping in magnets [24]. But ev-\nery known material exhibits spin-orbit interaction—since\nspinandangularmomentumofanatomareneverexactly\nzero—and therefore also magnetic damping, even if it is\nsmall. Consequently, spinsuperfluidsdoalwaysshowdis-\nsipation in contrast to their conventional counterparts.\nRecent theoretical work has focused on insulators\nrather than metals, usually based of phenomenological\nmodelsincluding theLandau-Lifshitz-Gilbertequationof\nmotion for both ferro- and antiferromagnets. [16, 19, 20]\nThe experimental detection of spin superfluidity in solid-\nstate magnets has been reported for magnon condensates\n[25], where the origin of the spin-superfluid order param-\neter is different to the cases described above, and also\nin antiferromagnetic solids [23]. However, the interpre-\ntation of the experimental findings is still controversially\ndiscussed [16, 26–28].\nIn the following, we will investigate and compare spin\nsuperfluidityinferro-andantiferromagneticmodels. The\ngeometry of our model resembles that of an experimen-\ntal non-local spin-transport investigation as sketched in\nfig. 1. In the corresponding experiments [29] at one side\n(here on the left) a spin current is injected into the mag-\nnet viathe spin-Halleffect causedby an electricalcurrent\nthrough an attached heavy-metal stripe. The resulting\nspin current is detected using the inverse spin-Hall ef-\nfect at another position (here the right-hand side). In\nour model we avoid the details of the excitation mech-\nanism and model the effect of the injected spin current\nby an appropriate boundary condition that triggers the\ndynamics of the spin systems that we investigate. This\nis done from the perspective of an atomistic, classicalarXiv:1911.12786v1  [cond-mat.mes-hall]  28 Nov 20192\nFigure 1. Basic concept of non-local spin transport as in an\nexperimental setup: heavy metal stripes are attached to the\nmagnet to inject a spin current via the spin-Hall effect (here\non the left hand side). The spin current in a certain distance\n(here at the right end) is detected via inverse spin-Hall effect.\nspin model, which has some advantages: the approach is\nnot restricted to small deviations from the ground state,\nfinite temperatures can be investigated and our calcu-\nlations are not limited to the steady state only. Fur-\nthermore, we are able to compare ferro- and antiferro-\nmagnetic systems. Their behavior turns out to be very\nsimilar, except for the resulting spin accumulation that is\nmuchlowerforthelatter. However,fromanexperimental\npoint of view antiferromagnets are much more promising,\nsince these are not prone to a breakdown of spin super-\nfluidity as a consequence of dipolar interactions, which is\nhard to avoid in ferromagnets. [22]\nII. ATOMISTIC SPIN MODEL\nWe consider the following classical, atomistic spin\nmodel of Heisenberg type [30], comprising Nnormal-\nized magnetic moments Sl=µl/µSon regular lattice\nsitesrl. We assume a simple cubic lattice with lattice\nconstanta. The Hamiltonian for these moments, in the\nfollowing called “spins”, is given by\nH=−J\n2/summationdisplay\n/angbracketleftn,m/angbracketrightSn·Sm−dz/summationdisplay\nn(Sn\nz)2,(1)\ntaking into account Heisenberg exchange interaction of\nnearest neighbors quantified by the exchange constant\nJ, where each spin has Nnbnearest neighbors. Further-\nmore, a uniaxial anisotropy with respect to the zdirec-\ntion with anisotropy constant dzis included. In this work\nwe consider the easy-plane case dz<0, where the mag-\nnets ground state readsgSl=±(cos(gϕ),sin(gϕ),0)with\nsome arbitrary, but uniform anglegϕ∈[0,2π](SO(2)\nsymmetry) and an alternating sign ±in case of antifer-\nromagnetic order ( J <0).\nThe time evolution of the spins Slis governed by\nthe stochastic Landau-Lifshitz-Gilbert (LLG) equationof motion [31–33]\ndSl\ndt=−γ\nµS(1 +α2)/bracketleftbig\nSl×/parenleftbig\nHl+αSl×Hl/parenrightbig/bracketrightbig\n(2)\nHl=−∂H\n∂Sl+ξl\n/angbracketleftbig\nξl\nβ(t)/angbracketrightbig\n= 0,/angbracketleftBig\nξl\nβ(t)ξl/prime\nη(t/prime)/angbracketrightBig\n=δll/primeδβηδ(t−t/prime)2µSαkBT\nγ\ndescribing the motion of a spin in its effective field Hl,\nwhereγisthegyromagneticratio, αtheGilbertdamping\nconstant,kBthe Boltzmann constant and Tthe absolute\ntemperature. The properties of the thermal noise ξlare\nchosen such that the dissipation-fluctuation theorem is\nsatisfied [34]. The material parameters define our sys-\ntem of units,|J|for the energy, tJ:=µS/γ|J|for the\ntime,afor the distance. Numerically the LLG equation\nis solved either by the classical Runge-Kutta method in\ncase of zero temperature, or at finite temperature using\nstochastic Heun’s method. At zero temperature the dis-\nsipated power per spin due to Gilbert damping follows\ndirectly from the time evolution of the spins Sl(t)[35]:\nPdiss=1\nNdH\ndt=1\nN/summationdisplay\nn∂H\n∂Sn/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\neff.field·∂Sn\n∂t/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nLLG.(3)\nWe study a magnetic wire extended along xdirection.\nThe system size for our numerical simulations is given\nbyN=Nx×Ny×Nzspins along x-,y- andzdirec-\ntion, where Nx/greatermuchNy,Nz. For transverse directions we\nuse periodic boundary conditions if not noted otherwise.\nBoundary spins at x=Nxa(the right-hand side) are\ndenotedSl/vextendsingle/vextendsingle\nrightand at this side an open boundary con-\ndition is applied, Sl/vextendsingle/vextendsingle\nright= 0. At the opposite side, at\nx= 0, we use a time-dependent boundary condition,\nSl/vextendsingle/vextendsingle\nleft=±(cos(ω0t),sin(ω0t),0), (4)\nin form of an in-plane precession with frequency ω0that\ninjects a spin current from this side. The alternating sign\n(±) is used only for antiferromagnetic systems, according\nto the sublattices with antiparallel spin orientation.\nThe use of this boundary condition creates an ex-\ncitation with well-defined frequency ω0. Alternatively,\nwe also assumed an externally given spin accumulation\nµ=µezat the left-hand side that causes additional\ntorques on the spins and drives them out of equilibrium,\nwhich directly maps an experimental implementation us-\ning a spin-Hall-generated spin accumulation to the model\nutilized here. This method has been used for instance in\n[22]. In appendix B we calculate how this spin accumula-\ntion maps to the excitation frequency ω0and we further-\nmore confirmed numerically that both mechanisms lead\nto the same response for ferro- and antiferromagnets.\nAlthough an atomistic picture—comprising discrete\ndegrees of freedom—is studied numerically, the micro-\nmagnetic approximation is of particular value for analyt-\nical considerations of ferromagnets. This approximation3\nassumes that spatial variations of magnetic structures\nare small compared to the atomic distance a. In this\ncase differences can be approximated as derivatives and\nthe spins form a continuous field S(r,t). It is handy to\nuse cylindrical coordinates\nS=/parenleftBig/radicalbig\n1−S2zcosϕ,/radicalbig\n1−S2zsinϕ, Sz/parenrightBig\n,where definitions Sz(rl) :=Sl\nzandϕ(rl) :=ϕllink the\natomistic picture to the micromagnetics. Note that for\na spin superfluid Szis considered as the spin-superfluid\ndensity and ϕits phase. The use of the micromagnetic\napproximationforferromagnetsallowstoreformulatethe\nLLGequationintermsofdifferentialequationsfor Szand\nϕthat read\nµS\nγ˙ϕ=Ja2/bracketleftBigg\n1\n1−S2z∆Sz+Sz|∇Sz|2\n(1−S2z)2+Sz|∇ϕ|2/bracketrightBigg\n+ 2dzSz−αµS\nγ˙Sz\n1−S2z(5)\nµS\nγ˙Sz=−Ja2/bracketleftbig/parenleftbig\n1−S2\nz/parenrightbig\n∆ϕ−2Sz∇Sz·∇ϕ/bracketrightbig\n+α/parenleftbig\n1−S2\nz/parenrightbigµS\nγ˙ϕ. (6)\nThese two equations are strictly equivalent to the LLG\nequation eq. (2) for zero temperature with the only as-\nsumption of the micromagnetic approximation. If one\nexpands these equations in lowest order in ∇ϕ,∆ϕ,∇Sz,\nand∆Szfor an easy-plane magnet, which implies espe-\ncially assuming|Sz|/lessmuch1, but keeping|∇ϕ|2, one ends up\nwith\nµS\nγ˙ϕ=Ja2∆Sz+Ja2Sz|∇ϕ|2+ 2dzSz−αµS\nγ˙Sz(7)\nµS\nγ˙Sz=−Ja2∆ϕ+αµS\nγ˙ϕ. (8)\nImportantly, keeping the |∇ϕ|2term is actually required\nif the damping takes relatively high values, a fact which\nwe checked numerically. Furthermore, these equations\nare very similar to others already reported in [19, 21], but\nnot exactly equivalent. Ref. [19] uses more approxima-\ntions, especially neglecting the |∇ϕ|2-term, and ref. [21]\nconsiders a different starting point, namely a quantum\ntheory at low temperatures, where this term has a dif-\nferentSz-dependence. Because of this difference, the re-\nsult from [21] does not exactly match our numerical re-\nsults of the atomistic spin model, nor does it match the\nclassical micromagnetic theory. Hence, we use eqs. (7)\nand (8) that do describe the atomistic spin simulations\nwell. However, eqs. (7) and (8) can be solved in steady\nstate for a special case: a ferromagnet that is of length L\nalongxdirection and exhibits translational invariance in\ny- andzdirection as carried out in appendix A. Steady\nstate means a coherent precession of all spins with a fre-\nquency ˙ϕ=ω0and a stationary profile Sz(x). This so-\nlution of eqs. (7) and (8) reads:\nsϕ(x,t) =α\n2µSω0\nγJ(x−L)2\na2+ω0t+ϕ0(9)\nsSz(x) =sSz(L)\n1 +µ2\nSω2\n0\n2γ2Jdzα2/parenleftbigx−L\na/parenrightbig2, (10)\nwith a spin accumulation at the right end of the sys-\ntem (atx=Nxa=:L) ofsSz(L) =µSω0/2γdz, avalue which is independent of L—one of the striking fea-\ntures of spin superfluidity. Another feature is the mono-\ntone increase of ϕwhich implies the formation of an in-\nplane spin spiral with winding number Nw, which reads\n2πNw=/integraltext\ndϕ=ϕ(L)−ϕ(0). Note furthermore, that an\nopen boundary condition at the right end is an assump-\ntion that leads to solutions eqs. (9) and (10), correspond-\ning to a Neumann condition ∇ϕ/vextendsingle/vextendsingle\nright= 0, which must be\njustified as a realistic choice.\nFor the numerical study of eq. (2) we assume an open\nboundary at the right end. Equation (10) assumes the\nsame and results in a finite Szatx= 0, which contra-\ndicts the numerical driving boundary at this side, eq. (4),\nthat forces Sz(x=0) = 0. Furthermore, in an experiment\nan open boundary at the right end might not be feasi-\nble because of outflowing spin currents, for example into\nan attached heavy metal. Thus, the real behavior at\nthe boundaries for sure deviates from the ideal solution\neq. (10) and raises the question how strong that devia-\ntion is and in how far the boundary conditions influence\nthe overall bulk behavior of the spin transport. This\nis examined numerically from the full model eq. (2) by\nvarying the boundary conditions on the left and right.\nOne example of the variations we tested is an absorbing\nboundary condition on the right, modeling an outflow-\ning spin current by an enhanced damping. As result we\nobserve the profile Sl\nzto show only little change in that\ncase compared to an open boundary and also that in all\ncases the numerical profiles well follow eq. (10) (see in\nthe following fig. 2 a) as example). Other variations of\nthe boundary condition which we tested have also hardly\nany impact on the magnets overall response.\nIII. EASY-PLANE FERROMAGNET\nIn a first step of the numerical investigation, we con-\nsider a collinear ferromagnet as most simple case, with\nparameters J > 0for the ferromagnetic state and dz=\n−0.01Jas in-plane anisotropy. Let us describe the phe-4\n0 1000 2000 3000 4000 5000012345610-3\n010203040506070\n0 0.5 1 1.5 2 2.5\n10-300.010.020.03\n5 6\n10-40.0240.026\nFigure 2. Spin superfluidity in a 1D ferromagnet at T= 0in the steady state: a)depicts the spin accumulation Szand the\nin-plane angle ϕforω0tJ=−2×10−4; numerical data (blue and red symbols) follow perfectly the theoretical curve eqs. (9)\nand (10) (black, dashed lines), except for the vicinity of the left boundary. This is an artifact of the boundary condition, eq. (4),\nused for the numerics. b)shows the spin accumulation at the right end of the system SN\nzversus driving frequency ω0; for small\ndriving frequencies up to a critical value ωcritthe numerical data follow the analytical curvesSz(L); for larger frequencies the\nspin accumulation breaks down, deviating form the theoretical curve, due to phase slips and spin wave excitations.\nnomenology of the spin superfluid in a 1D system of size\nNx×Ny×Nz= 5000×1×1at temperature T= 0.\nThis model is equivalent to a 3D system with transla-\ntional invariance in y- andzdirection. Furthermore, we\nsetα= 0.05andω0tJ=−2×10−4.\nStarting from a uniform ferromagnet as initial condi-\ntion, the boundary spin starts to rotate and due to ex-\nchange the next spin will follow this rotation and ac-\ncordingly drive its neighbor and so on. But since a spin\ncannot immediately follow the dynamics of its neighbor,\nthere is a certain phase difference Dϕbetween the spins,\ni.e., the neighbor to the right is lagging behind. In the\nmicromagnetic approximation this effect is described by\na phase gradient ∇ϕ≈Dϕ/a. The rotation of the spins\nspeeds up, until it reaches the final precession frequency,\ngiven by the driving frequency ω0. At the same time\ntheout-of-planecomponent Sz—thespinaccumulation—\nincreases until it reaches a steady state profile. The time\nscale of this transient phase for reaching a steady state\ncan be quantified: ˙ϕ(t)andSz(t)follow a limited expo-\nnential growth on a characteristic time τt≈5×105tJ\nfor the parameters used here. τtscales positively with\nsystem size Nxand damping α.\nEventually, the numerical time evolution reaches a\nsteady state as shown in fig. 2 a). This steady state\nverifies the analytical solution eqs. (9) and (10) in bulk\nwith a deviation only at the left boundary as anticipated\nand described above. Note that the finite spin accumu-\nlationSzas a consequence of this type of dynamics has\nimportantfeatures: itisalong-rangespintransportsince\nit decays non-exponential and it would allow to measure\nspin transport by means of the inverse spin-Hall effect.\nFurthermore, it could also be addressed, for instance, by\nmagneto-optical measurements—if sensitive to the out-\nof-plane magnetization for a geometry as studied here.\nFor a further investigation, we vary the frequency ω0\nand find two different regimes, one for sufficiently smallω0, where the system is able to follow the excitation\nwithout disturbance, and one for large ω0where the sys-\ntems response deviates from the theoretical expectation.\nThesetworegimes, whichwewillcalllinearandnonlinear\nregime in the following, are sharply separated by a crit-\nical frequency ωcrit. The existence of these two regimes\ncan be seen from the data depicted in fig. 2 b). Here, as\na measure, we consider the spin accumulation of the last\nspinSN\nzat the right end of the system. Below ωcritwe\nfind just the analytical valuesSz(L), see eq. (10), which\nscales linearly with ω0. Atωcritthis behavior breaks\ndown and the spin accumulation SN\nzdecreases with in-\ncreasing pumping frequency. This breakdown can be un-\nderstood in terms of the phase gradient ∇ϕwhich scales\nlinearly with the driving frequency ω0, see eq. (9). How-\never, one can expect a maximum phase gradient ∇ϕfor a\nspin-superfluid state given by the Landau criterion [36]:\nif the phase gradient exceeds locally a critical value, it\nis energetically favorable for the spins at this position to\nrotate out of the x-yplane and return to the plane by\nunwinding the spiral. Hence, the winding number Nw\ndecreases by one—an effect which is called a phase slip.\nThe Landau criterion for the stability of a spin superfluid\nwith respect to phase slips reads [36]\n|∇ϕ|</radicalbigg\n−2dz\nJa2. (11)\nNote that this relation is not exact as a uniform Szis\nassumed for its derivation. Nevertheless do we observe\nthese phase slips numerically. In the linear regime the\nwinding number is constant in the steady state, whereas\nin the nonlinear regime it relaxes by one at a regular\nrate Γpsas shown in fig. 3 at the example of ω0tJ=\n−6.5×10−4.\nTheω0dependence of the phase-slip rate Γpsis de-\npicted in fig. 4. Each phase slip is accompanied by the\nexcitation of a broad spin-wave spectrum on top of the5\n4 4.005 4.01 4.015 4.02\n10748.54949.550\n00.511.510-7\nFigure 3. Winding number Nwand dissipated power Pdiss\nin the steady state of a driven ferromagnet for ω0tJ=\n−6.5×10−4, well in the nonlinear regime. At a rate of Γps\nphase slips relax the winding number of the in-plane spiral.\nFor each such event the dissipated power spikes.\nspin superfluid. These spin waves are visible as oscil-\nlations ofSzaround the spin-superfluid magnitude and,\nhence, there is strictly speaking no steady state any more\nas the phase slips and the spin-wave excitation are def-\ninitely time dependent. In particular, for systems with\nlow Gilbert damping this dynamics leads to determinis-\nticchaos, thoughthe spin-superfluidbackground remains\nvisible. These findings have some severe implications as\nthere is a maximum spin accumulation, which is achieved\nright at the edge between the linear and the nonlinear\nregime. Furthermore, driving the system in the nonlin-\near regime means also to waste energy to the phase slips\nand the excitation of incoherent spin waves.\nFor the parameters here the critical frequency takes\nthe valueωcrittJ≈−5.15×10−4, which is determined\nfrom fig. 2 b). We also tested different parameters, vary-\ningαandL(data not shown), and find that ωcritscales\nnegatively with αandL. Our numerical result can be\ncompared to the analytical prediction above, eq. (11).\nFrom eq. (9) follows that the maximum phase gradient\nis given by∇sϕ(0) =αµSω0L/γJa2, which, inserted into\neq. (11), implies\n|ωcrit|=γJa\nαµSL/radicalbigg\n−2dz\nJ. (12)\nFor our parameter set this takes value 4×10−41/tJ,\nwhich is slightly lower compared to the numerical value\nabove. This discrepancy is probably due to the fact that\neq. (11) ignores the spatial dependence of Sz. Further-\nmore, a test for very low damping α= 10−4showed that\neq. (11) is even more inaccurate in that case.\nAnother important quantity is the dissipated power\ngiven by eq. (3), which takes negative sign as it lowers\nthe total energy. Figure 4 depicts its dependence on ω0.\nBelowthecriticalfrequency, inthesteadystate, itistime\nindependent as the dynamics is completely stationary.\nIn this regime it scales quadratically with the excitation\nfrequency,Pdiss∝ω2\n0, a result which has already been\n10-410-310-210-1010-910-810-7\n10-610-410-2Figure 4. Time-averaged dissipated power /angbracketleftPdiss/angbracketrightand phase-\nslip rate Γpsin the steady state versus driving frequency\nω0comparing ferro- (FM) and antiferromagnets (AFM). The\nperpendicular dash-dotted lines mark ωcritfor the FM and\nthe AFM, where the latter takes on the higher value. For\nω0< ω critthe dissipated power scales as /angbracketleftPdiss/angbracketright∝ω2\n0and is\nidentical for FMs and AFMs. Above ωcritthe increase slows\ndown and the curve flattens for very high ω0. In this regime,\n/angbracketleftPdiss/angbracketrightis higher for AFMs as compared to FMs. For the\nphase-slip rate we find Γps= 0for|ω0|<|ωcrit|and similar,\nincreasing values for the FM and the AFM above ωcrit. The\ndeviation between FM and AFM near ωcritis due to the dif-\nferent critical frequency, i.e. the data almost coincide when\nplotted versus ω0−ωcrit.\nreported before [37]. This behavior changes above ωcrit.\nThetimeevolutionofthedissipationinthisregimeshows\nthat the phase slips notably contribute to the dissipated\npower, i.e. for each phase slip Pdissspikes as shown in\nfig. 3. Because of this time-dependence of Pdiss, we have\nto consider an average over time for the evaluation of the\ndissipated power. Still, the dissipated power increases\nfurtherwith ω0butlessthanlinearandthecurvenotably\nflattens.\nSo far, our results were obtained from zero-\ntemperature simulations. In the following we address\nthe robustness of spin-superfluid transport at finite tem-\nperature. For this, we consider a finite cross section\nNy×Nz>1andNx= 2000 and vary the tempera-\nture. An average over Navrealizations of thermal noise\nis carried out and, furthermore, data are averaged over\nthecrosssectioninordertoreducethenoise. Thespecific\nchoice of parameters in provided is table I.\nFigure 5 presents the numerical results for the exam-\nple ofkBT/J = 10−2forSzandϕ. The spin-superfluid\ntransport remains in tact but, in particular, the spin ac-\ncumulation Szshows strong thermal fluctuations despite\nthe averages taken over the cross section and the Navre-\nalizations. However, on average the spin accumulation\nclearly deviates from its equilibrium value, which is zero.\nTo quantify the influence of the temperature we calculate\nthe spatial average over the xdirection/angbracketleftSz/angbracketrightxand com-\nparethistothezero-temperaturevalue, givenbyeq.(10).\nThe results are included in Table I. Furthermore, the in-\nplane angle/angbracketleftϕ/angbracketrightNavshows only little fluctuations and its6\n00.020.04\n0 500 1000 1500 200001020\nFigure 5. Spin superfluidity in a ferromagnet at finite tem-\nperaturekBT/J = 10−2and forω0tJ=−2×10−4: shown is\nthe spin accumulation Szand the in-plane angle ϕ. Blue lines\nrepresent the numerical data, black dash-dotted lines the an-\nalytical solution at zero-temperature. The spin accumulation\nis subjected to strong thermal fluctuations but still has a fi-\nniteaveragevalue /angbracketleftSz/angbracketrightx=/summationtext\nnSn\nz/Nx, depictedasreddashed\nline. Its value is only slightly lower than the zero-temperature\nvalue. Thermal fluctuations are much less pronounced for the\nin-plane angle.\nTable I. Averaged spin accumulation of a ferromagnet driven\nwithω0tJ=−2×10−4for different temperatures. The cor-\nresponding zero-temperature value is /angbracketleftSz/angbracketright= 0.01, from which\nno significant deviation is observed.\nkBT/JNx×NyNav/angbracketleftSz/angbracketrightx\n10−44×4 38 0.010\n10−24×4 15 0.009\n0.05 8×8 5 0.010\n0.10 8×8 4 0.011\n0.20 14×145 0.012\nspatial profile shows hardly any deviation from the zero-\ntemperature behavior, given by eq. (9). Overall, we find\nno significant difference to the zero temperature case.\nWe also checked whether phase slips due to thermal ac-\ntivation can be observed, but from the available data\nwe could not observe a single one with the conclusion\nthat ΓpstJ<4×10−5. Hence, spin superfluidity is very\nrobust against thermal fluctuations, even though these\nfluctuations are a problem in our simulations in terms of\nthe signal-to-noise ratio.\nIV. EASY-PLANE ANTIFERROMAGNETS\nFor antiferromagnets, the magnetic unit cells comprise\ntwoatoms—denotedAandBinthefollowing—thatform\ntwo sublattices. We write all properties using this label-\ning so thatASlandBSlare spins of the corresponding\nsublattices. In the ground state both sublattices have\nopposite orientation,ASl=−BSl. The field equations,\neqs. (5) and (6), do not hold as these require a small\nin-plane angle difference between two neighboring spins\nDϕ, which is obviously not true in this case. However,it is reasonable to define phase differences and gradients\nwithin each sublattice, i.e.ADϕas phase difference be-\ntween a spin of sublattice A and its next-nearest neigh-\nbor, which is the nearest neighbor within sublattice A.\nAccordingly,BDϕdefines the phase difference of sublat-\nticeB. Assuming sufficiently weak excitation, spatial\nvariationswithineachsublatticearesmallsuchthatami-\ncromagnetic approximation inside the sublattices reads\n∇A,Bϕ≈A,BDϕ/2a. Interestingly, numerical results re-\nveal that the antiferromagnetic system in bulk fulfills\nfield equation (8), applied separately to each sublattice.\nThe other eq. (7) is not valid, as has been reported before\n[20] for a phenomenological model for antiferromagnets.\nConsequently, the antiferromagnet is expected to exhibit\nthe same in-plane angleA,Bϕ(up to phase difference of π\nbetweensublattices)asaferromagnetwithcorresponding\nparameters, but not the same spin accumulationA,BSz.\nBefore we discuss the numerical results in detail, let\nus first introduce two differences to the ferromagnet that\nare essential for understanding the following results: the\nroleofexchangeand(interlinkedwiththis)thetransverse\ngeometry. Just as in a ferromagnet, a spin-superfluid dy-\nnamics imposes a finite spin accumulationA,BSzwhich,\nremarkably, carries the same sign for both sublattices\nleading to a small out-of-plane magnetization. But this is\nof course antagonized by the antiferromagnetic exchange\nthat favors antiparallel orientation of all components be-\ntween sublattices. Consequently, the exchange interac-\ntions must lower the spin accumulation Sztremendously\nas compared to the ferromagnet (compare fig. 6 a) and\nfig. 2 a)). This also implies that the behavior of Szis\ndetermined by the number of nearest neighbors Nnbof\na spin as more neighbors imply stronger exchange cou-\npling. Consequently, a 1D spin chain is less prone to this\nexchange reduction than a 3D system. We checked this\nnumerically by comparing 1D, 2D and 3D models and,\nindeed, the spin accumulation of the spin superfluid Sz\nscales linearly with Nnb.\nThere is another implication: at a boundary the num-\nberofneighborsislocallyreduced—andthereforetheim-\nportance of the exchange—, resulting in deviations of the\nsublattice componentsA,BSz, see fig. 6 a) for a 1D setup\n(the effect is less pronounced in 3D). This 1D setup owns\nonly boundaries along the xdirection and the question\nwhether for finite cross section Ny×Nz>1these devi-\nations aty- andzboundaries significantly influence the\nbulk behavior has also been tested numerically. Fortu-\nnately, deviations at transversal boundaries quickly fall\noff with distance to the boundary over a few lattice con-\nstants. The bulk then behaves qualitatively and quan-\ntitatively just as a 1D system, except for the reduced\nspin accumulation due to the number of neighbors as\ndiscussed above. The study of 1D systems is preferable\nto keep computational costs feasible.\nWe turn now to the presentation of the numerical data\nfor a 1D system. The model parameters are the same as\ngiven above for the ferromagnet, except for the exchange\nconstant which is now negative. Similarly to the ferro-7\n050100024610-5\n10002000300040004900 5000\n0 0.5 1 1.5 2 2.5\n10-300.511.510-4\n5.566.5\n10-41.31.41.510-4\nFigure 6. Spin superfluidity in antiferromagnetic spin chains: a)the spin accumulation in the stead state resolved for the two\nsublattices A and B. In the bulk both take the same value, leading to a finite total spin accumulation, which is two orders of\nmagnitude lower as compared to the ferromagnet. At the boundaries the profiles show deviations from bulk behavior because\nof the broken exchange right at the boundary. b) the spin accumulation at the right end of the system as function of driving\nfrequencyω0; as for the ferromagnet there are two regimes separated by a critical frequency ωcrit.\nmagnet, the system reaches a steady state after a tran-\nsientphasecharacterizedbyalimitedexponentialgrowth\non a time scale τt, which is roughly the same as for the\nferromagnet. In the steady state the sublattice-resolved\nin-plane anglesA,Bϕboth follow exactly the same profile\nas the ferromagnet, i.e. eq. (9), but with a phase differ-\nence ofπbetween the two sublattices because of the an-\ntiferromagnetic order (data for the antiferromagnet not\nshown).\nThe spin accumulation deviates from the behavior of\na ferromagnet as depicted in fig. 6 a). The bulk profiles\n(away from boundaries at x= 0andx=Nxa) are iden-\ntical in the two sublattices,ASz=BSz. Hence, a measur-\nable spin accumulation is present, but it is two orders\nof magnitude lower than in comparable ferromagnetic\ncases. This is the aforementioned exchange reduction.\nIf we consider the spin accumulation Szin bulk, in the\ndata in fig. 6 a) hardly a space dependence is observed in\ncontrast to the ferromagnet, where Sl\nzhas a finite slope.\nThe antiferromagnet exhibits this in the same way, but\nit is also much smaller and the profile becomes roughly\nconstant. Contrary to the ferromagnet, there are distur-\nbances at the boundaries in the profile of Szwhich we\nalready discussed before.\nDriving the antiferromagnet with the time-dependent\nboundary condition eq. (4) at frequency ω0leads to the\nvery same two different regimes as for ferromagnets, a\nlinear regime up to a critical frequency ωcritand above—\nin the nonlinear regime—phase slips occur. These phase\nslips reduce the winding number, lead to the excitation\nof spin waves, and a further increase of the spin accu-\nmulation is not possible. We quantify this behavior in a\nsimilar way as for the ferromagnet. It is, however, not\npossible to use the spin accumulation of the last spin\nSN\nzas a measure because of the deviating profile at the\nboundary. Instead, we take the spin accumulation at the\nend of the bulk in form of a spatial average over the spins\nin the range xl/a∈[4900,4920],Send\nz:=/angbracketleftbig\nSl\nz/angbracketrightbig\n[4900,4920].This range is chosen such that it is sufficiently separated\nfrom the boundary. The data for the ω0dependence of\nthespinaccumulationareshowninfig.6, panelb): These\nshow that critical frequencies takes roughly same values\nfor ferro- and antiferromagnets, a result which has been\ntested and confirmed for another parameter set with dif-\nferentNx,α, anddz. For the data set shown here the\nvalue isωcrittJ≈−5.75×10−4. However, the decrease\nof the spin accumulation Send\nzwith increasing driving fre-\nquencyω0in the nonlinear regime is less pronounced for\nantiferromagnets. We also calculated the ω0dependence\nof the time-averaged dissipated power /angbracketleftPdiss/angbracketrightand of the\nphase-slip rate Γps, both shown in fig. 4. Similar to other\nfeatures these properties behave for the antiferromagnets\nvery much as for ferromagnets: below ωcritthe dissipated\npower shows exactly the same dependence and above it\nis dominated by phase slips. However, a difference is that\naboveωcritthe dissipated power increases faster with ω0.\nOne reason for this might be the dynamics of spin waves\nthat very much differ between ferro- and antiferromag-\nnets. The phase-slip rate differs slightly, however, this\nseems to be solely due to the fact that ωcritdiffers for\nferro- and antiferromagnets. When Γpsis plotted versus\nω0−ωcrit, both curves match almost.\nThe next step is to consider finite temperature. Again\nthis requires a finite cross section for which we use\nNx×Ny×Nz= 2000×4×4and we test two temper-\natures,kBT/J = 10−2andkBT/J = 10−4. As before,\nthe magnetic response is very similar to that of a ferro-\nmagnet: the in-plane angles follow the zero-temperature\nprofiles, as well as does the average spin accumulation\nfor the lower of the two temperatures. The only major\ndifference is the ratio of the spin-superfluid spin accumu-\nlation to the thermal fluctuations, which is two orders of\nmagnitude smaller as a result of the lower spin-superfluid\nsignal and an equal strength of the fluctuations. For the\nhigher temperature, this even leads to an average Szthat\nis essentially zero. This does not mean that there is no8\nspin-superfluid spin accumulation, but rather that the\navailable numerical data are not sufficient to resolve it\nand more averaging is needed. Note that the in-plane\nangle is not affected by this—it is as robust against the\nfluctuations just as for the ferromagnet.\nV. DISCUSSION AND CONCLUSION\nOur comparative study addresses spin superfluidity in\nferro- and antiferromagnets, where one should bear in\nmind that the former are less promising for spin super-\nfluidity as the latter because of the negative influence of\nthe stray field [22]. Nevertheless, the former can help to\nunderstand the behavior of the latter, which we utilize\nin this work. One of the striking features of spin super-\nfluidity is the transport range that leads to a spin ac-\ncumulation at the end of the system Sz(L)(see eqs. (9)\nand (10)) that does depend on the system length L—\na completely different situation compared to spin-wave\ntransport where the intensity decays exponentially with\nthe distance. However, this non-exponential decay does\nnot imply the possibility of an infinite transport range\nsince with increasing system size the critical frequency\nlowers until no undisturbed spin superfluid is possible\nanymore.\nWe present a full analytical solution for the steady\nstate of the ferromagnet, which slightly deviates from the\nanalytical theory reported before [19, 21]. This theory\nis tested numerically by the full atomistic model, which\nallows to test the robustness of the spin-superfluid trans-\nport against varying boundary conditions, against high\nexcitation frequencies and finite temperature. We show\nthat this kind of transport is remarkably robust: bound-\nary conditions and also elevated temperature hardly\nhamper the magnets spin-superfluid response.\nFurthermore, we identify the critical frequency ωcrit—\na manifestation of the Landau criterion—as the limiting\nfactor for the range of this transport. Above this critical\nfrequency phase slips occur, which also sets a limit to\nthe spin accumulation that can be achieved. In ref. [38]\nanother limitation on the spin current of such a spin su-\nperfluid is discussed, which rests on the fact that |Sz|\nis bounded above. But the estimated values would re-\nquire an out-of-plane component that takes quite large\nvalues|Sz|>0.1, which our simulations reveal to be\nhardly possible even for low damping. This is in particu-lar true for the case for antiferromagnets and, therefore,\nwe conclude that the critical frequency—and therefore\nthe phase slips—is a more relevant limitation on spin su-\nperfluid transport.\nThe direct comparison of antiferromagnets to ferro-\nmagnets shows that both exhibit the very same behavior:\nDriven by an in-plane rotation, both form an in-plane\nspin spiral that exhibits exactly the same behavior, in-\ncluding a spin accumulation in form of an out-of-plane\nmagnetization. Antiferromagnets show in principle the\nsame transport range as ferromagnets with a spin accu-\nmulation at the end of the system independent of the\nsystem length, provided the excitation frequency ω0is\nkept constant ( ω0itself depends on the magnets geome-\ntry in experimental setups, see eq. (B12)). Furthermore,\nthe critical frequency takes very similar value for the two\ntypes of magnets. This general accordance of spin super-\nfluidity for both types of magnets is in contrast to spin-\nwavetransportthatisknowntobedifferentforferro-and\nantiferromagnets[39]. Yetthereisamajordeviation: the\nantiferromagnetic exchange lowers tremendously the spin\naccumulation.\nOurstudyalsocoversanexaminationofthedissipation\nofaspinsuperfluidandoftheeffectoffinitetemperature.\nWe proof the principle robustness of spin superfluidity\nagainst thermal fluctuations, i.e. that quite high temper-\natures are required before thermal phase slips start to\nhamper the transport. 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Sonin, Advances in Physics 59, 181 (2010),\nhttps://doi.org/10.1080/00018731003739943.\n[37] H. Ochoa, R. Zarzuela, and Y. Tserkovnyak, Phys. Rev.\nB98, 054424 (2018).\n[38] Y. Tserkovnyak and M. Kläui, Phys. Rev. Lett. 119,\n187705 (2017).\n[39] F. Keffer, H. Kaplan, and Y. Yafet, Amer-\nican Journal of Physics 21, 250 (1953),\nhttps://doi.org/10.1119/1.1933416.\n[40] A. Manchon, J. Železný, I. M. Miron, T. Jungwirth,\nJ. Sinova, A. Thiaville, K. Garello, and P. Gambardella,\nRev. Mod. Phys. 91, 035004 (2019).\nAppendix A: Analytical theory for a 1D ferromagnet\nThe ferromagnet in the micromagnetic approximation\nunder the assumption of small out-of-plane component,\n|Sz|/lessmuch1, is described by the LLG equation in cylindrical\ncoordinates, eqs. (7) and (8). Assuming translational\ninvariancealong y-andzdirectionleadstoa1Dproblem:\nµS\nγ˙ϕ=Ja2∂2\nxSz+Ja2Sz(∂xϕ)2+ 2dzSz−αµS\nγ˙Sz\n(A1)\nµS\nγ˙Sz=−Ja2∂2\nxϕ+αµS\nγ˙ϕ. (A2)\nSteady state means ˙ϕ=ω0and ˙Sz= 0. This allows to\nintegrate the latter equation,\nsϕ(x,t) =α\n2µSω0\nγJ/parenleftbiggx−L\na/parenrightbigg2\n+ω0t+ϕ0,(A3)\nwhere the first integration constant follows from the Neu-\nmann boundary condition at the right end, ∂xϕ(L) = 0\n(no outflow of spin current), and the second one satisfies\nthe condition ˙ϕ=ω0and allows for an arbitrary phase\nϕ0. This is inserted in the first equation, which then\nreads\n−Ja2∂2\nxSz=−µSω0\nγ+µ2\nSω2\n0\nγ2J/parenleftbigg\nαx−L\na/parenrightbigg2\nSz+ 2dzSz.\n(A4)\nWe argue that the second-derivative term can be ne-\nglected−Ja2∂2\nxSz≈0. This is justified in a twofold\nmanner: first we compared the relevance of all terms\nin that equation numerically by calculating those three\nterms from simulations of the full atomistic spin model,\neq. (2). Indeed the result is that in steady state the\nsecond-derivative term is several orders of magnitude\nsmaller compared to the other two. The second reason\nfollows a-posteriori from the calculated solution and is10\nspin injector (using SHE)\nspins not subjected\nto SHEspins driven\nby SHEspins not subjected\nto SHE\nFigure 7. 1D setup for calculation of the excitation frequency\nω0of a magnet driven by a spin injector utilizing the spin-Hall\neffecttoexertexternaltorquesonthespins. Thesetorquesare\napplied in the region [l1,l2]and vanish outside. Furthermore,\nthe Gilbert damping in [l1,l2]is enhanced by αd. The ground\nstateSis in-plane, the spin accumulation µperpendicular.\nexplainedbelow. From −Ja2∂2\nxSz≈0followsthesteady-\nstate solution for Sz:\nsSz=µSω0\n2γdz\n1 +µ2\nSω2\n0\n2γ2Jdz/parenleftbig\nαx−L\na/parenrightbig2. (A5)\nThis solution does not fulfill eq. (A4), however, we can\ninsert it and calculate the deviation by calculating\n∂2\nxsSz=−2µSω0\nγJα2\na2sS2\nz+ 4/parenleftbiggµSω0\nγJ/parenrightbigg2α4(x−L)2\na4sS3\nz\n=O/parenleftbig\nS2\nz/parenrightbig\n.\nThis allows the conclusion that the correction by taking\nthe second derivative into account is of higher order in\nSzand neglecting this is consistent with the original as-\nsumption|Sz|/lessmuch1. Hence, eqs. (A3) and (A5) form the\nanalytical solution for a 1D setup.\nAppendix B: Frequency of a spin superfluid\nThe usual excitation of a spin current in a magnet\nrests on a spin accumulation µat an interface between\nthe magnet and a heavy metal, which is created by an\nelectrical current. Normally for that the spin-Hall effect\nis utilized. The aim of this appendix is to calculate the\nresulting excitation frequency ω0of a spin superfluid.\nWe assume here that the spin accumulation is per-\npendicular to the magnets ground state, i.e. µ∝ez.\nConsequently, there is an additional damping-like torque\n[22, 40] in the LLG equation (here written as viscousdamping):\n˙Sl=−γ\nµSSl×Hl+αlSl×˙Sl+α/prime\nlSl×/parenleftbigg\nSl×µl\n~/parenrightbigg\n.\n(B1)\nA subsetVdof the total volume of the magnet is driven,\ni.e. subjected to the additional torques and the driving\nalso creates an enhanced damping α/prime\nlwithinVd:\nµl=/braceleftbigg\nµdezforrl∈Vd\n0else(B2)\nαl=α0+α/prime\nlwithα/prime\nl=/braceleftbigg\nαdforrl∈Vd\n0else.(B3)\nα0is the intrinsic Gilbert damping of the magnet.\nTo proceed we consider the LLG equation in the fol-\nlowing form, resolved for the time derivative:\n˙Sl=−γ\nµS(1 +α2\nl)Sl×/parenleftbig\nHl+αlSl×Hl/parenrightbig\n+Tl\n1Sl×Al+Tl\n2Sl×/parenleftbig\nSl×Al/parenrightbig\n.(B4)\nTl\n1andTl\n2parameterize arbitrary additional torques with\nrespecttoanaxis Alandforthespecificchoice Al=µl/~,\nTl\n1=αlα/prime\nl/(1+α2\nl)andTl\n2=−α/prime\nl/(1+α2\nl)eq. (B4) is equiva-\nlent to eq. (B1). However, for the sake of generality we\nconsider for the calculation eq. (B4). Assuming Al∝ez\nand using cylindrical coordinates and again the micro-\nmagnetic approximation, this form of the LLG reads\nµS\nγ˙ϕ=Ja2Sz|∇ϕ|2+ 2dzSz−αµS\nγ˙Sz\n−µS\nγAz(T1+αT2) (B5)\nµS\nγ˙Sz=−Ja2∆ϕ+αµS\nγ˙ϕ+µS\nγAz(αT1−T2),(B6)\nan extension of eqs. (7) and (8). In the same spirit as\nin appendix A we can solve these equations in one di-\nmension in steady-state (assuming ˙Sz= 0and ˙ϕ=ω0),\nwhere the geometry depicted in fig. 7 is assumed. We ap-\nply the external spin accumulation in the interval [l1,l2],\nwhereas the total magnet expands over [0,L]. Therefore,\nT1,2(x) =/braceleftbigg\nTd\n1,2forx∈[l1,l2]\n0else\nA(x) =/braceleftbigg\nAd\nzezforx∈[l1,l2]\n0else.11\nIn the 1D setup eq. (B6) reads\n∂2\nxϕ=α(x)µS\nγJa2ω0+µS\nγJa2Az(x) [α(x)T1(x)−T2(x)]\n=\n\n=:¯ω0/bracehtipdownleft/bracehtipupright/bracehtipupleft/bracehtipdownright\nα0µS\nγJa2ω0 forx∈[0,l1]\n(α0+αd)µS\nγJa2ω0\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=:¯ω/prime\n0+µS\nγJa2Ad\nz/bracketleftbig\n(α0+αd)Td\n1−Td\n2/bracketrightbig\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft /bracehtipupright\n=:tforx∈[l1,l2]\nα0µS\nγJa2ω0 forx∈[l2,L], (B7)\nwhich can be integrated. There are six boundary conditions to consider, each one at the left and right end of the\nmagnet, where we assume a Neumann condition ∂xϕ(0) =∂xϕ(L) = 0, i.e. no outflow of spin currents. Furthermore, ϕ\nand∂xϕmust be continuous at l1andl2, delivering four internal boundary conditions. But there is another condition,\na gauge condition for ϕ, which allows to add an arbitrary constant phase to ϕ(x)without altering the physics. (In\npractice this gauge phase depends on the prehistory of the magnet, i.e. on how it had reached its steady state, and also\nwhich exact instant in time is considered.) As gauge we use ϕ(0) = 0. Altogether there are 6 integration constants\nand the unknown frequency ω0in combination with 6 boundary conditions and a gauge, such that the problem has a\nunique solution.\nAs result we obtain\nϕ=\n\n1\n2¯ω0x2forx∈[0,l1]\n1\n2(¯ω/prime\n0+t)x2+ (¯ω0−¯ω/prime\n0−t)l1x+1\n2(¯ω/prime\n0−¯ω0+t)l2\n1forx∈[l1,l2]\n1\n2¯ω0x2+ (¯ω/prime\n0−¯ω0+t)/bracketleftbig\n(l2−l1)x+1\n2(l2\n1−l2\n2)/bracketrightbig\nforx∈[l2,L](B8)\nSz=µS\nγω0+Az(x) [T1(x) +α(x)T2(x)]\nJa2(∂xϕ)2+ 2dz(B9)\nand, importantly, we also gain\nω0=−Ad\nz/bracketleftbig\n(α0+αd)Td\n1−Td\n2/bracketrightbig\n(l2−l1)\nα0L+αd(l2−l1). (B10)\nThis holds true for arbitrary torques taking form\neq. (B4). If the specific case of the spin injector utiliz-\ning the spin-Hall effect is considered, then inserting the\nparameters T1,T2andAreads\nω0=−µd\n~αd\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n=:τ·l2−l1\nα0L+αd(l2−l1).(B11)\nThe former factor τis the strength of the spin-Hall effect\non the magnet [40]:\nτ=γ\nMs~\n2eηϑSHjel1\nd,\nwith spin transparency of the interface η, spin-Hall an-\ngleϑSH, saturation magnetization Msand thickness d\nof the magnet. jelis the electric current density. The\nlatter factor in eq. (B11) is a geometric factor that is ba-\nsically the ratio between the driven volume l2−l1and the\ntotal volume L, weighted with the total damping of the\nmagnet, where the Gilbert-damping enhancement can beexpressed as [22]\nαd=g⊥~2\n2e2γ\nMsd,\nwith transverse spin mixing conductance g⊥of the in-\nterface. This rigorous derivation holds only true for 1D\nferromagnets, however, the natural extension to 2D and\n3D is given by\nω0=−τ·Vd\nα0V+αdVd, (B12)\nwhereVis the magnets total and Vdthe driven volume.\nThe validity of this expression has been checked numeri-\ncally for 1D and 2D systems using various geometries by\ninvestigating the full atomistic LLG eq. (B4). As a result\nwe obtain very good agreement with the analytical calcu-\nlation except for two cases. First, when the assumption\n|Sz|/lessmuch 1is violated and second if the setup is not ef-\nfectively one dimensional, i.e. if the system is not driven\nover the entire transverse width. However, such a mis-\nmatch in usually small for realistic experimental setups.\nWe furthermore did not only simulate ferromagnets, but12\nalso antiferromagnets with same parameters except for\nthe sign of J. These simulations result in exactly the\nsame frequencies ω0as the corresponding ferromagnetsand thus eqs. (B10) to (B12) are also valid for antiferro-\nmagnets, even though note that the resulting spin accu-\nmulation deviates."    },    {        "title": "1907.04499v1.Determination_of_the_damping_co_efficient_of_electrons_in_optically_transparent_glasses_at_the_true_resonance_frequency_in_the_ultraviolet_from_an_analysis_of_the_Lorentz_Maxwell_model_of_dispersion.pdf",        "content": "Determination of the damping coe\u000ecient of\nelectrons in optically transparent glasses at the\ntrue resonance frequency in the ultraviolet from\nan analysis of the Lorentz-Maxwell model of\ndispersion\nSurajit Chakrabarti\n(Ramakrishna Mission Vidyamandira)\nHowrah, India\nThe Lorentz-Maxwell model of dispersion of light has been analyzed in this paper\nto determine the true resonance frequency in the ultraviolet for the electrons in\noptically transparent glasses and the damping coe\u000ecient at this frequency. For\nthis we needed the refractive indices of glass in the optical frequency range. We\nargue that the true resonance condition in the absorption region prevails when\nthe frequency at which the absorption coe\u000ecient is maximum is the same as the\nfrequency at which the average energy per cycle of the electrons is also a max-\nimum. We have simultaneously solved the two equations obtained from the two\nmaxima conditions numerically to arrive at a unique solution for the true resonance\nfrequency and the damping coe\u000ecient at this frequency. Assuming the damping\ncoe\u000ecient to be constant over a small frequency range in the absorption region,\nwe have determined the frequencies at which the extinction coe\u000ecient and the re-\n\rectance are maxima. These frequencies match very well with the published data\nfor silica glasses available from the literature.\n1arXiv:1907.04499v1  [physics.optics]  10 Jul 20191 Introduction\nThe Lorentz-Maxwell model of dispersion of electromagnetic waves in matter is\nvery successful in describing the properties of matter under the action of electro-\nmagnetic waves over its whole spectrum where the wavelength is large compared to\nthe interatomic distances. The model is generally studied in the optical frequency\nrange where only the oscillation of electrons bound to atoms and molecules is rel-\nevant for the study of dispersion. Two important parameters of the model namely\nthe natural oscillation frequency and the plasma frequency of the electrons in a\ndielectric medium like glass can be easily determined from the refractive indices\nof a glass prism measured in the optical band [1] where glass is transparent. In\na condensed system like glass one has to include the e\u000bect of the local \feld on\nthe electrons apart from the \feld of the incident wave. This leads to another fre-\nquency which is conventionally known as the resonance frequency and is related to\nthe plasma and the natural oscillation frequencies of the electron [2]. Though it is\ncalled the resonance frequency, there is no proof that the absorption coe\u000ecient is\nmaximum at this frequency.\nIn order to study the absorption of EM waves in matter, a phenomenologi-\ncal variable called the damping coe\u000ecient is introduced in the Lorentz-Maxwell\nmodel. Glass is opaque in the ultraviolet indicating that it has a strong absorption\nthere. In scienti\fc literature, there are innumerable experimental works which have\nstudied the interaction of silica glasses with electromagnetic waves over its whole\nspectrum. A summary of these works can be found in Kitamura et al. [3]. From\nthe experimental data on the extinction coe\u000ecient for silica glass in the ultraviolet,\nwe can \fnd the frequency at which this coe\u000ecient is maximum. However, as far as\nwe are aware, there has been no theoretical study so far which has determined this\nfrequency by an analysis of the Lorentz-Maxwell model of dispersion. The main\nproblem with the theoretical analysis is the fact that it has not been possible so\nfar to determine the value of the damping coe\u000ecient theoretically.\n2In this work we have determined the damping coe\u000ecient at the true resonance\nfrequency which we de\fne to be the frequency at which the absorption coe\u000ecient\nfor the energy of the electromagnetic \feld in the medium is maximum. We have\ndone this theoretically by taking the natural oscillation frequency and the plasma\nfrequency determined from the refractive indices of glass in the optical region as two\nknown parameters of the Lorentz-Maxwell model. We have formed two algebraic\nequations containing the true resonance frequency and the damping coe\u000ecient\nas two unknown variables. We have solved these two equations simultaneously by\nnumerical method to \fnd a unique solution for the two variables. With the value of\nthe damping coe\u000ecient known, we have explored the anomalous dispersion region\nin the ultraviolet for glass.\nIt is well known that the Kramers-Kronig relations [3] allow us to determine\nthe imaginary part of the dielectric constant from an integral of the real part over\nthe whole range of frequencies and vice versa. The theory is based on a very\ngeneral causality argument and a linear response of the medium to an external\nperturbation. We have, on the other hand, determined the damping coe\u000ecient of\nthe Lorentz-Maxwell model of dispersion starting from the refractive indices in the\noptical region corresponding to the real value of the dielectric constant. From this\nwe have extracted the information about the absorptive region in the ultraviolet\ncorresponding to the imaginary part of the dielectric constant.\nIn section 2 we give the outline of the Lorentz-Maxwell model. In section 3\nwe o\u000ber our physical argument for the method adopted to determine the damping\ncoe\u000ecient and the true resonance frequency. The next four sections are just an\nexecution of these ideas. We conclude with a summary of the work.\n32 Lorentz-Maxwell model of dispersion\nIn the Lorentz model [4] of dispersion of light in a dense medium like a solid or\nliquid, electrons execute forced simple harmonic oscillations with damping in the\ncombined \feld of the incident electromagnetic wave of frequency !and the local\n\feld. The local \feld arises as a result of the interaction of the electron with the\n\felds of other atoms close by. Without any loss of generality we can assume that\nthe direction in which the electron is oscillating is the ydirection. We can write\nthe equation of motion as\ny+\r_y+!2\n0y=qE0\n0\nme\u0000i!t: (1)\nwhereE0\n0is the amplitude of the e\u000bective \feld acting on the electrons. Here !0is\nthe natural oscillation frequency and \ris the damping coe\u000ecient of the electron.\nIn the steady state the electron will oscillate at a frequency !of the incident wave\nthough shifted in phase. E0\n0is related to the amplitude of the \feld ( Ei0) outside\nfrom where it is incident on the medium as\nE0\n0=1 +\u001f\n3\n1 +D\u001fEi0: (2)\nThe\u001f\n3term in equation (2) arises as a result of the e\u000bect of local \feld in the\nLorentz-Lorenz theory of dielectric polarizability valid for an isotropic medium [5]\nwhere\u001fis the electric susceptibility. Dis the depolarisation factor, a dimensionless\nnumber of the order of unity [6]. The dielectric function of the medium is given by\n\u000f= 1 +\u001f: (3)\nUsing Maxwell's phenomenological relation \u000f=n2\ncwherencis the complex refrac-\ntive index and the Lorentz-Lorenz equation [5], we arrive at the following equation\nfor a number of resonance regions [7,8].\nn2\nc\u00001\nn2\nc+ 2=Nq2\n3\u000f0mX\njfj\n!2\n0j\u0000!2\u0000i\rj!: (4)\n4Herefjis the fraction of electrons that have a natural oscillation frequency !0j\nand damping constant \rjwith \u0006fj= 1.Nis the density of electrons taking part\nin dispersion. It is a common practice to assume a single dominant absorption\nfrequency which is true in many practical cases and which makes the analysis\nsimpler [9]. With this assumption fj= 1 and equation (4) can be written as\nn2\nc= 1 +!2\np\n!2\nn\u0000!2\u0000i\r!(5)\nwhere the plasma frequency !pis given by\n!2\np=Nq2\n\u000f0m(6)\nand we de\fne\n!2\nn=!2\n0\u0000!2\np\n3: (7)\nIn scienti\fc literature [2], !0is known as the natural oscillation frequency of the\nelectrons and !nis known conventionally as the resonance frequency. So far, au-\nthors have used some chosen values of the damping coe\u000ecient \rand the plasma\nfrequency which mimic the absorptive properties of dielectric materials, in order\nto carry out model analysis [9]. We have actually determined the damping coe\u000e-\ncient from a prior knowledge of the natural oscillation frequency and the plasma\nfrequency of a glass medium.\nIn the optical limit where the absorption in glass is negligible we take \r= 0.\nIn this limit the refractive index is real and equation (5) reduces to\nn2= 1 +!2\np\n!2\nn\u0000!2: (8)\nwhich is essentially the Sellmeier's formula [7] for dispersion in the frequency do-\nmain with one absorption band. If we have a set of measurements of refractive\nindices of a glass prism for several optical wavelengths, we can determine !nand\n!pusing equation (8) [1].The resonance wavelength which falls in the ultraviolet\nregion, has been determined in a similar work [7]. Once !nand!pare known, !0\ncan be determined using equation (7).\n5In the absorptive region the dielectric function picks up an imaginary part\ngiven by\nn2\nc=\u000f=\u000f1+i\u000f2: (9)\nThe refractive index ( n) and the extinction coe\u000ecient ( \u0014) known as optical con-\nstants are written as\nnc=n+i\u0014 (10)\nwhere\u0014represents the attenuation factor of the amplitude of the electromagnetic\nwave in an absorptive medium. Using the last two equations and equation (5) we\nobtain for the real and imaginary parts of the complex dielectric function [10],\n\u000f1=n2\u0000\u00142= 1 +!2\np(!2\nn\u0000!2)\n(!2\nn\u0000!2)2+\r2!2(11)\nand\n\u000f2= 2n\u0014=!2\np\r!\n(!2\nn\u0000!2)2+\r2!2: (12)\nWe can express n2and\u00142as functions of frequency !using equations (11) and\n(12). The details and the \fnal expressions have been shown in Appendix A.\nThe absorption coe\u000ecient of the incident EM wave \u000bis given by [7,10],\n\u000b=2!\nc\u0014 (13)\nwherecis the speed of light in vacuum. \u000bgives the attenuation coe\u000ecient of\nthe intensity of the incident wave. Intensity is the rate of \row of energy per\nunit area normal to a surface. \u000bwill be a maximum at the frequency at which\nthe absorption of energy by the electrons from the EM \feld is maximum. This\ngives the condition of resonance. It is a general practice to consider !nde\fned in\nequation (7) as the resonance frequency though there is no proof that the energy\nabsorption is maximum at this frequency. So we do not assume a priori !nto be\nthe resonance frequency. In the next section we will describe our strategy to \fnd\nthe true resonance frequency and in the results section we will see that the true\nresonance frequency is di\u000berent from both !0and!nand lies between them. There\n6is no real reason to call !nthe resonance frequency. We treat the true resonance\nfrequency as an unknown variable to be found from our analysis.\nThe damping coe\u000ecient \ris introduced in the Lorentz model to explain absorp-\ntion. We model \rsuch that it is zero in the optical band and upto the frequency\n!n. In the absorption band we assume that \ris constant from frequency !nto\n!0. Above!0,\rfalls down and rises again to another constant value of \rin the\nnext resonance region if the material under study has one. With this model for \r\nin mind we can extrapolate equation (8) to \fnd !nand!p. In the next section we\nwill explain how to get the constant value of \rin the absorption region and the\ntrue resonance frequency.\nEven if the system under study may have several absorption bands, we can\nstudy it with the assumption of a single resonance region. The optical waves os-\ncillate the outermost electrons of atoms and molecules having the lowest natural\nfrequencies and as a result we get the phenomenon of refraction. With an anal-\nysis of the refractive indices in the optical region under this assumption of single\nresonance, we are most likely to \fnd information about the absorption band with\nthe lowest natural oscillation frequency in the ultraviolet closest to the optical\nband. This will of course depend on the strength of the resonance. The justi\fca-\ntion of the single resonance calculations with the chosen model for \rcan be found\nfrom the results of our theoretical calculations which will be found to match the\nexperimental results very well.\n73 Physical argument for the method adopted to\ndetermine the damping coe\u000ecient at the true\nresonance frequency\nFrom various experiments on the absorption of EM waves in matter, we know\nthat the absorption coe\u000ecient ( \u000b) attains a maximum value at a characteristic\nfrequency. We try to \fnd this frequency where \u000bis maximum. We di\u000berentiate\n\u000bwith respect to frequency and equating the derivative to zero get one equation.\nHowever, we have two unknown variables in the theory - the damping coe\u000ecient\nand the true resonance frequency. We look for a second equation.\nThe incident EM wave interacts with the electrons bound to the atoms and\nmolecules.The electrons execute a forced simple harmonic oscillation with damp-\ning. The total energy of the electron is time dependent, as the electron is being\nperturbed by a time dependent harmonic force.The average energy of the electron\nper cycle can be worked out easily [11]. We \fnd the frequency at which this av-\nerage energy per cycle is maximum. This leads to another equation involving the\ntwo unknown variables. When the frequency at which \u000bis maximum is the same\nas the frequency at which the average energy per cycle of the electron is also a\nmaximum, the electromagnetic wave will share its energy most with the electrons\nand will be attenuated most. This will constitute the true condition of resonance.\nBy solving the two equations simultaneously using numerical method, we \fnd both\nthe variables. We call the characteristic frequency, the true resonance frequency\n!tand the damping coe\u000ecient at the true resonance frequency \rt.\nHeitler [12] has proposed a quantum theory of the phenomenon of damping.\nAccording to this theory the damping coe\u000ecient is dependent on frequency though\nof a very slowly varying nature near resonances. This gives support to our earlier\nassumption that the damping coe\u000ecient is a constant within a small frequency\nrange about the resonance frequency. However, it can be taken as zero in the\n8optical band where glass is transparent and absorption is negligible.\n4 Condition for the maximum of the absorption\ncoe\u000ecient as a function of frequency\nOur aim in this section is to \fnd the frequency at which \u000bis maximum. We\n\frst di\u000berentiate \u000bwith respect to !assuming\rconstant. In order to \fnd the\nderivative of \u000bwe \frst di\u000berentiate equations (11) and (12) with respect to !. We\n\fnd two algebraic equations involvingdn\nd!andd\u0014\nd!. By eliminatingdn\nd!from the two\nequations, we get the expression ford\u0014\nd!and henced\u000b\nd!using equation (13). We have\nshown the di\u000berentiations in Appendix B. Eliminatingdn\nd!between equations (B.2)\nand (B.3) we get\n2d\u0014\nd!(n+\u00142\nn) =A\u0000B\nC(14)\nwhere\nA=!2\np(!2\nn\u0000!2)2[\r\u00002\u0014\nn!] (15)\nand\nB=!2\np!\r[!\r2\u00004(!2\nn\u0000!2)!\u00002\u0014\nn!2\nn\r] (16)\nand\nC= [(!2\nn\u0000!2)2+\r2!2]2: (17)\nFrom this we get\nd\u000b\nd!=\u0014\nc[2 +n\n\u0014!\nn2+\u00142A\u0000B\nC]: (18)\nIf\u000bis maximum thend\u000b\nd!should be zero. So we write at the maximum\n!(A\u0000B)\nC=\u00002\u0014\nn(n2+\u00142): (19)\nIt is to be noted that two sides of equation (19) are dimensionless and they will be\ncompared later numerically to \fnd the solution for the true resonance frequency\nand the damping coe\u000ecient.\n95 Condition for the maximum of the average en-\nergy per cycle of the electron as a function of\nfrequency\nIn the steady state the electron will oscillate at a frequency !as given by the\nsteady state solution of equation (1) and the total energy of the system averaged\nover a period is given by [11],\nE(!) =1\n4(qE0\n0)2\nm(!2+!2\n0)\n[(!2\n0\u0000!2)2+ (!\r)2]=1\n4(qE0\n0)2\nmg(!) (20)\nwhere\ng(!) =(!2+!2\n0)\n[(!2\n0\u0000!2)2+ (!\r)2]: (21)\nEquation (2) shows the relationship between the incident electric \feld and the \feld\nacting on an electron. With the variation of frequency in the ultraviolet we can\nimagine that the amplitude of the incident \feld is kept constant. However, the\namplitudeE0\n0is dependent on \u001fwhich is frequency dependent. Lorentz theory is\nbased on the assumption that the response \u001fof the medium to the external \feld\nis small [13]. In equation (2), \u001fappears both in the numerator as well as in the\ndenominator. With the depolarization factor Dpositive, any variation of \u001fin the\nnumerator will be o\u000bset to some extent by the variation in \u001fin the denominator.\nSo we neglect the variation of the term E0\n0with frequency and assume it to be\nconstant. To \fnd the derivative of the average energy per cycle E(!), it is su\u000ecient\nto \fnd the derivative of the function g(!) given by equation (21) with respect to\n!. Equating the derivative to zero, we \fnd the condition at which the average\nenergy per cycle is maximum. It turns out that the frequency is given by\n!=!0[r\n4\u0000(\r\n!0)2\u00001]1\n2: (22)\nIf the incident electromagnetic wave can oscillate the bound electrons steadily at\nfrequency!given by the last equation, then the wave has to deliver maximum\nenergy per cycle and its absorption will be maximum.\n10It is clear from equation (22) that for real values of !we should have the ratio\nf=\r\n!0<p\n3. By trial we take several positive values of fupto its maximum of\np\n3. For each value of fwe \fnd the values of \rand!using equation (22) with the\nknown value of !0. With these values we determine the refractive index nand the\nextinction coe\u000ecient \u0014in the resonance region using equations (A.3) and (A.4)\nrespectively. We put these values in equation (19) and try to see for which value\noffthis equation is satis\fed. From fwhich satis\fes equation (19) we calculate \rt\nand!tusing equation (22).\n6 Determination of the damping coe\u000ecient and\nthe true resonance frequency\nThe results of an experiment performed with a prism made of \rint glass have been\nreported by Chakrabarti [1]. In this experiment1\nn2\u00001has been plotted against the\ninverse wavelength squared at optical range following equation (8). From this plot\nwe have determined the values of !n,!p:The value of !0has been estimated using\nequation (7). The errors in these frequencies are less than 1% :Refractive indices as\na function of wavelengths have been shown in table 1 and the parameters needed\nfor this work have been shown in table 2.\nIt has been shown in the discussion following equation (22) that the maximum\nvalue off=\r\n!0isp\n3. So we take trial values of flike 0.1, 0.2 upto 1.7. For all these\nvalues offwe determine \rand then!using equation (22). We then determine n\nand\u0014using equations (A.3) and (A.4) respectively and calculate A,B,Caccording\nto equations (15),(16),(17) respectively. With these values we try to see whether\nthey satisfy equation (19) or not. The algorithm for this is as follows: suppose\nwe call the left side of equation (19), Y1 and the right side, Y2. Now we form\na parameter, Y=Y1\u0000Y2. For all values of ffrom 0.1 to 1.7 we calculate Y\nand \fnd between which two values of f, the sign of Ychanges. In our case there\n11was only one sign change in Ybetweenfvalues 0.6 and 0.7. Now we check this\nregion more closely, that is between 0.60 to 0.70 at an interval of 0.01. We \fnd\nthat forf= 0:65,Y=\u00000:001 and for f= 0:66,Y= 0:053. So there is a zero\ncrossing between fvalues 0.65 and 0.66. Proceeding similarly we \fnally \fnd that\nforf= 0:6501,Y=\u00000:0005 and for f= 0:6502,Y= 0:000055. So the solution\nlies between 0.6501 and 0.6502. We take the solution as f= 0:65015 with an error\nof 0:00005 which is just \u00060:008%. From this value of fwe get\rtsince we know\nthe value of !0. We get the value of !tby putting the value of fin equation (22).\nThese values are shown in table 3. The values of the damping coe\u000ecient and the\ntrue resonance frequency turn out to be\n\rt= 11:6\u00021015rad/s\nwith an error of only 0 :76% and\n!t= 16:8\u00021015rad/s\nwith an error similar to the error in !0.!tlies between !nand!0. This solution\nis unique.\nWe \fnd that !tdi\u000bers signi\fcantly from !n. We have presented the parameters\nat true resonance frequency in table 4. The maximum value of the absorption\ncoe\u000ecient\u000bat the true resonance frequency has come out to be 8 :18\u0002107m\u00001\ncorresponding to an attenuation length 12 :2 nm.\nOnce we determine the damping coe\u000ecient, we can determine the absorption\ncoe\u000b\fcient \u000bin a small frequency range about !twhere we assume \ris constant\nand equal to \rt. In \fgure 1 we plot \u000bas a function of x=!\n!0betweenx=\n!n\n!0= 0:81 to 1. We assume that within the frequency range !nto!0,\rremains\nconstant. We clearly \fnd that \u000battains a maximum value at x=!t\n!0= 0:94:\nJackson [14] has given the frequency range for absorption coe\u000ecient \u000bin the\nultraviolet as well as the plasma frequency of water. In the ultraviolet, we are\nconcerned with electronic oscillations as the most important component responsible\n12for dispersion. So the properties of glass will not be too di\u000berent from water at\nthese frequencies. This work [14] shows that the maximum value of \u000bis around\n108m\u00001which is approximately the same as the maximum value that we have\nobtained. The frequency at which the maximum occurs is also the same in order\nof magnitude as !t:The full width at half maximum of the absorption curve can\nbe read o\u000b approximately. It is determined to be approximately 15 \u00021015rad/s\nwhich is of the same order of magnitude that we have obtained for the value of \rt.\nThe damping coe\u000ecient determined in this work is rather large.This broad ab-\nsorption in the ultraviolet is due to the outer electrons in the atoms and molecules\nof the solid which take part in the process of dispersion. The outer electrons are\na\u000bected by the collisions and the electric \felds of the neighbouring atoms. Con-\nsequently an extensive region of continuous absorption is obtained in solids and\nliquids [8,15]. So this large value of \rtis expected. This value of \rtis valid only in\nthe resonance region.\n7 Some other results\nWe assume the damping coe\u000b\fcient \rtis constant within the small frequency range\nfrom!nto!0in which!tlies. We now \fnd the frequency !\u0014at which the extinction\ncoe\u000ecient\u0014is maximum, assuming !\u0014is close to!t. For\u0014to be maximum,d\u0014\nd!\nshould be zero and from equation (14) this should occur when A=B:Here we scan\na parameter!\n!0from 0.1 to 1.0. For each value of !and known \rtwe calculate\nnand\u0014using equations (A.3) and (A.4) respectively. We then check whether\nthe condition A=Bis satis\fed. Once again we \fnd a unique solution with\n!\u0014\n!0= 0:84843:This corresponds to an angular frequency !\u0014= 15:1\u00021015rad/s.\nFrom this we get\n\u0017\u0014= 2:40\u00021015Hz\n13corresponding to a wavelength\n\u0015\u0014= 0:125\u0016m:\nIn \fgure 2 we have plotted \u0014determined by our theoretical analysis, as a function of\n!\n!0:This \fgure clearly shows that \u0014has attained a maximum value. The maximum\nvalue of\u0014that we have obtained at !=!\u0014is 0.769. This is of the same order\nof magnitude as the maximum value shown in \fgure 2 of Kitamura et al.[3] which\ngives the maximum of \u0014in the ultraviolet at a wavelength very near to 0 :12\u0016m.\nThis is very close to the wavelength \u0015\u0014that we have obtained. In the last column\nof table 3 we show the value of !\u0014.\u0015\u0014that we have determined may be seen from\n\fgure 1 of [3] to be the nearest to the optical wavelengths on long wavelength side\nas we claimed in last paragraph of section 2. The data in \fgure 1 of [3] may be\nshowing other resonance/s at shorter wavelengths or larger frequencies.\nIn \fgure 3 we show a plot of re\rectance Ras a function of frequency in the\nabsorption region where Ris the normal re\rectivity de\fned in [3] as\nR=(n\u00001)2+\u00142\n(n+ 1)2+\u00142: (23)\nWe \fnd that this function has a peak at!\n!0= 0:908 with the maximum value of R=\n0:118. The peak position corresponds to != 16:2\u00021015rad/s:The re\rectance peak\nposition corresponds to an energy 10.6 eV. Experimentally, silica glasses show the\nlowest frequency re\rectance peak in the ultraviolet region at 10.2 eV [16] which is\nvery close to what we have found theoretically. This is once again as we claimed in\nthe last paragraph of section 2. Sigel has shown in \fgure 3 [16] that the re\rectance\npeak position is the same for two glassy materials though their re\rectance values\nat this frequency are di\u000berent. Similarly the actual values of refractive indices and\nextinction coe\u000ecients at the same frequency can vary signi\fcantly due to glass\nmanufacturing process [3]. Data for these coe\u000ecients in the absorption region for\n\rint glasses were not available in the literature. However, the order of magnitude\nof these coe\u000ecients is similar to that of other silica glasses. Since we are getting\n14the peak positions of the extinction coe\u000ecient and the re\rectance data very close\nto the published data for silica glasses, we conclude that the value of the damping\ncoe\u000ecient obtained by us is correct.\nWe have shown in table 5 the values of n,\u0014,\u000b, the absorption length1\n\u000band\ng(!) at frequencies close to !t. They have been calculated at three frequencies !0,\n!\u0014and!nfor the same \rt. Comparing tables 4 and 5 we \fnd that \u000bandg(!) are\nindeed maxima at !t. The anomalous nature of variation of the refractive index is\nevident from the values of nat frequencies around !t.\nIn section 4 we determined the condition for \u000bto be maximum. In a similar\nway we can \fnd the condition for the refractive index nto be an extremum by\nequating its derivative to zero. This derivative can be obtained by eliminatingd\u0014\nd!\nfrom equations (B.2) and (B.3). Interestingly, this condition gives two solutions\nfor the frequency. The refractive index is maximum at one and minimum at the\nother frequency. We \fnd that the Lorentz-Maxwell model of dispersion reproduces\nall the features of anomalous dispersion in the absorption region as observed in\nactual experiments [3].\n8 Conclusions\nThe Lorentz-Maxwell model of dispersion of electromagnetic waves in matter has\nbeen studied in this paper with an analysis of the phenomenon of absorption in\nthe ultraviolet in dielectrics like \rint glass. We have shown that if we know the\nrefractive indices of glass fairly accurately in the optical frequencies, we can explore\nthe anomalous dispersion region in the ultraviolet quantitatively. The key \fnding\nof this work is that the damping coe\u000ecient of the model can be determined by\na simple argument. We also determine the frequency at which the absorption\ncoe\u000ecient is maximum. We call this the true resonance frequency. In the optical\nregion where glass is transparent, the damping coe\u000ecient can be assumed to be\n15zero. In the absorptive part the damping coe\u000ecient has been taken to be a constant\nwithin a short range of frequencies. The value of the damping coe\u000ecient matches\nin order of magnitude with the experimental width of the absorption coe\u000ecient\ndata for water available from literature.\nOnce the damping coe\u000ecient is determined, we can \fnd the frequencies at\nwhich the extinction coe\u000ecient and the re\rectance are maxima. These frequencies\nmatch very well with the experimental data available in the literature for silica\nglasses. This indirectly shows that the value of \rdetermined by us is correct. Our\nassumption of a single resonance should give us the information of the absorption\nband closest to the optical frequencies. We actually observe this by comparing the\npeak positions of the extinction coe\u000ecient and re\rectance data obtained by us with\nthat found from literature. Refractive indices estimated at di\u000berent frequencies\nclose to the true resonance frequency in the absorption region reveal the anomalous\nnature of dispersion. All the features of dispersion by a dielectric like glass in the\nultraviolet absorption region have been reproduced from our theoretical analysis\nof the Lorentz-Maxwell model.\n9 Acknowledgment\nThe author would like to thank Prof. Jayanta Kumar Bhattacharjee for some\nhelpful discussions. Thanks are due to Prof. Debashis Mukherjee for making some\nhelpful comments on the paper.\n16Appendix A\nEquations (11) and (12) can be easily inverted and we get [2],\nn2=1\n2[(\u000f2\n1+\u000f2\n2)1\n2+\u000f1] (A.1)\nand\n\u00142=1\n2[(\u000f2\n1+\u000f2\n2)1\n2\u0000\u000f1]: (A.2)\nWe express \u000f1and\u000f2as functions of frequency using equations (11) and (12)\nrespectively. After a fairly straightforward algebra we arrive at the \fnal expressions\nforn2and\u00142as functions of frequency !.\nn2=[!4\np\r2!2+ (!04+\r2!2+!2\np!02)2]1\n2+ (!04+\r2!2+!2\np!02)\n2(!04+\r2!2)(A.3)\n\u00142=[!4\np\r2!2+ (!04+\r2!2+!2\np!02)2]1\n2\u0000(!04+\r2!2+!2\np!02)\n2(!04+\r2!2)(A.4)\nwhere\n!02=!2\nn\u0000!2: (A.5)\nThese equations are exact and will be used for determining nand\u0014in the resonance\nregion in the ultraviolet.\nIt can be easily checked that in the limit \rtending to zero, \u0014becomes zero\nat all frequencies and vice versa. Thus the medium is transparent in the optical\nfrequencies as it should. In this limit the refractive index nsatis\fes a relation\nwhich has been used in the \frst place to get the parameters !nand!p[1].\nAppendix B\nWe \frst \fnd the derivative of1\n(!2n\u0000!2)2+\r2!2with respect to !and get\nd\nd![1\n(!2\nn\u0000!2)2+\r2!2] =2![2(!2\nn\u0000!2)\u0000\r2]\n[(!2\nn\u0000!2)2+\r2!2]2(B.1)\n17Di\u000berentiating equation (11) with respect to !we get\nndn\nd!\u0000\u0014d\u0014\nd!=\u0000!!2\np[(!2\nn\u0000!2)2+\r2!2] +!!2\np(!2\nn\u0000!2)[2(!2\nn\u0000!2)\u0000\r2]\n[(!2\nn\u0000!2)2+\r2!2]2\n=!!2\np[(!2\nn\u0000!2)2\u0000!2\nn\r2]\n[(!2\nn\u0000!2)2+\r2!2]2(B.2)\nSimilarly di\u000berentiating equation (12) with respect to !we get\n2\u0014dn\nd!+ 2nd\u0014\nd!=!2\np\r[(!2\nn\u0000!2)2+\r2!2] + 2!2!2\np\r[2(!2\nn\u0000!2)\u0000\r2]\n[(!2\nn\u0000!2)2+\r2!2]2\n=!2\np\r(!2\nn\u0000!2)[(!2\nn\u0000!2) + 4!2]\u0000!2!2\np\r3\n[(!2\nn\u0000!2)2+\r2!2]2:(B.3)\n18REFERENCES\n1. Chakrabarti S 2006 Phys. Educ. 23167- 175 (New Delhi, India: South\nAsian Publishers PVT LTD)\n2. Christy R W 1972 Am.J.Phys. 401403-1419\n3. Kitamura R, Pilon L and Jonasz M 2007 Applied Optics 46(33) 8118-8133\n4. Almog I F, Bradley M S and Bulovic V 2011 The Lorentz Oscillator and its\nApplications ( MIT OpenCourseWare, MIT6-007S11/lorentz)\n5. Born M, Wolf E 1980 Principles of Optics ( New York:Pergamon Press,\nSixth.Edn.) p 85\n6. Kittel C 1976 Introduction To Solid State Physics (New Delhi:WileyEastern\nLimited, 5th Edn.) p 405\n7. Hecht E 2002 Optics ( Delhi,India:Pearson Education, 4th Edn.) p 71,\n85,70,128\n8. Feynman R P, Leighton R B, Sands M 2003 The Feynman Lectures on\nPhysics, 2nd Volume (New Delhi,India:Narosa Publishing House) p 1211,1210\n9. Oughstun K E, Cartwright N A 2003 Optics Express 11(13) 1541-1546\n10. Tanner D B 2013 Optical e\u000bects in solids (www.phys.u\r.edu/tanner/notes.pdf\n)\n11. Kleppner D, Kolenkow R J 1973 An Introduction To Mechanics (New Delhi\n:Tata Mcgraw Hill ) p 426\n12. Heitler W 1954 The quantum theory of radiation (Oxford University Press,\n3rd. Edn.) p 163\n1913. Seitz F 1940 The Modern Theory of Solids (McGraw-Hill ,International Series\nIn Pure And Applied Physics) p 629\n14. Jackson J D 1999 Classical Electrodynamics 3rd edn. (John Wiley & Sons,\nInc) p 314\n15. Jenkins F A , White H E 1957 Fundamentals of Optics (Mcgraw-Hill book\ncompany,Inc, 3rd Edn.) p 486\n16. Sigel G H Jr 1973/74 Journal of Non-Crystalline Solids 13372-398\n20Table 1: Refractive indices as a function of wavelengths for the \rint glass\nprism [1]\nwavelength refractive index\n\u0015(nm) n\n706.544 1.6087\n667.815 1.6108\n587.574 1.6167\n504.774 1.6259\n501.567 1.6264\n492.193 1.6277\n471.314 1.6311\n447.148 1.6358\n438.793 1.6377\n21Table 2: Parameters obtained from \ftting of data of refractive indices to\nLorentz model[1]\n!n N ! p !0\nrad/s m\u00003rad/s rad/s\n14:5\u000210151:02\u0002102918:0\u0002101517:8\u00021015\nTable 3: Table for \rt,!tand!\u0014\nf=\rt\n!0\rt !t !\u0014\nrad/s rad/s rad/s\n0.65015 11 :6\u0002101516:8\u0002101515:1\u00021015\nTable 4: Parameters at the true resonance frequency\nfrequency n \u0014 \u000b1\n\u000bg(!) =4mE(!)\n(qE0\n0)2\n! m\u00001\u0016m s2=rad2\n!t 0.995 0.729 8 :18\u00021070.0122 0:153\u000210\u000031\nTable 5: Values of some parameters in the ultraviolet region\nfrequency n \u0014 \u000b1\n\u000bg(!) =4mE(!)\n(qE0\n0)2\n! m\u00001\u0016m s2=rad2\n!0 0.906 0.678 8 :06\u00021070.0124 0:149\u000210\u000031\n!\u0014 1.18 0.769 7 :76\u00021070.0129 0:141\u000210\u000031\n!n 1.26 0.763 7 :39\u00021070.0135 0:133\u000210\u000031\n22 7.2 7.4 7.6 7.8 8 8.2 8.4\n 0.8  0.85  0.9  0.95  1  1.05α(107 m-1)\nω/ω0Figure 1: Distribution of the absorption coe\u000ecient \u000bin the absorption region\n23 0.7 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78\n 0.75  0.8  0.85  0.9  0.95  1κ\nω/ω0Figure 2: Distribution of the extinction coe\u000ecient \u0014in the absorption region\n24 0.115 0.1155 0.116 0.1165 0.117 0.1175 0.118 0.1185\n 0.8 0.82 0.84 0.86 0.88  0.9 0.92 0.94 0.96 0.98  1 1.02R\nω/ω0Figure 3: Distribution of Re\rectance Rin the absorption region\n25"    },    {        "title": "2112.06941v2.Cosmic_ray_streaming_in_the_turbulent_interstellar_medium.pdf",        "content": "DRAFT VERSION JANUARY 24, 2022\nTypeset using L ATEXpreprint2 style in AASTeX63\nCosmic ray streaming in the turbulent interstellar medium\nSIYAO XU1AND ALEX LAZARIAN2, 3\n1Institute for Advanced Study, 1 Einstein Drive, Princeton, NJ 08540, USA; sxu@ias.edua\n2Department of Astronomy, University of Wisconsin, 475 North Charter Street, Madison, WI 53706, USA; lazarian@astro.wisc.edu\n3Centro de Investigaci ´on en Astronom ´ıa, Universidad Bernardo O’Higgins, Santiago, General Gana 1760, 8370993,Chile\nABSTRACT\nWe study the streaming instability of GeV \u0000100GeV cosmic rays (CRs) and its damping in the turbulent\ninterstellar medium (ISM). We find that the damping of streaming instability is dominated by ion-neutral col-\nlisional damping in weakly ionized molecular clouds, turbulent damping in the highly ionized warm medium,\nand nonlinear Landau damping in the Galactic halo. Only in the Galactic halo, is the streaming speed of CRs\nclose to the Alfv ´en speed. Alfv ´enic turbulence plays an important role in both suppressing the streaming in-\nstability and regulating the diffusion of streaming CRs via magnetic field line tangling, with the effective mean\nfree path of streaming CRs in the observer frame determined by the Alfv ´enic scale in super-Alfv ´enic turbu-\nlence. The resulting diffusion coefficient is sensitive to Alfv ´en Mach number, which has a large range of values\nin the multi-phase ISM. Super-Alfv ´enic turbulence contributes to additional confinement of streaming CRs,\nirrespective of the dominant damping mechanism.\n1.INTRODUCTION\nThe resonant streaming instability (Wentzel 1974; Kul-\nsrud & Pearce 1969; Wentzel 1969; Skilling 1971) is im-\nportant for confining cosmic rays (CRs) with energies up to\n\u0018100GeV in the Galaxy (Farmer & Goldreich 2004). It\nhas many astrophysical implications on, e.g., shock accel-\neration (Bell 1978), heating of intracluster media (Guo &\nOh 2008; Brunetti & Jones 2014), launching galactic winds\n(Ipavich 1975; Wiener et al. 2017; Mao & Ostriker 2018;\nHolguin et al. 2019; Quataert et al. 2021), transport of CRs\nin starburst galaxies (Krumholz et al. 2020) and around CR\nsources (Marcowith et al. 2021), and explaining PAMELA\nand AMS-02 observations at Earth (Blasi et al. 2012; Amato\n& Casanova 2021).\nThe self-generated Alfv ´en waves by CRs via the streaming\ninstability are subject to various damping effects, including\nion-neutral collisional damping in a partially ionized medium\n(Kulsrud & Pearce 1969; Plotnikov et al. 2021; Armillotta\net al. 2021), nonlinear Landau damping in a collisionless\nmedium (Kulsrud 2005), as well as turbulent damping by\nbackground Alfv ´enic turbulence (Lazarian 2016). Unlike\nother damping mechanisms depending on plasma conditions,\nturbulent damping depends on properties of magnetohydro-\ndynamic (MHD) turbulence. Measurements in different in-\nterstellar phases reveal a large range of turbulence parame-\nters, e.g., Alfv ´en Mach number MAthat characterizes the\nmagnetization level of turbulence (Lazarian et al. 2018; Hu\net al. 2019).\nBased on the theoretical understanding of MHD turbulence\ndeveloped since Goldreich & Sridhar (1995) and Lazarian &\nVishniac (1999), Farmer & Goldreich (2004) first formulated\naHubble Fellowthe turbulent damping rate for trans-Alfv ´enic (MA= 1) tur-\nbulence. Lazarian (2016) further provided a detailed analysis\non turbulent damping in both super-Alfv ´enic (MA>1) and\nsub-Alfv ´enic (MA<1) turbulence. When the growth of\nstreaming instability is limited by turbulent damping, the re-\nsulting streaming speed of CRs can deviate from the Alfv ´en\nspeed and is sensitive to turbulence parameters. In addition,\ndue to the magnetic field line tangling in super-Alfv ´enic tur-\nbulence, CRs streaming along turbulent magnetic fields have\nan effective mean free path determined by the Alfv ´enic scale\nlA=LM\u00003\nA(Lazarian 2006; Brunetti & Lazarian 2007),\nwhereLis the injection scale of turbulence, and an isotropic\ndistribution on scales larger than lA. The above effect on the\nspatial diffusion of streaming CRs has not been addressed in\nprevious studies.\nIn this work, we focus on the effect of Alfv ´enic turbulence\non the streaming speed and diffusive propagation of stream-\ning CRs in the energy range GeV \u0000100GeV in different tur-\nbulence regimes. We also examine the relative importance\nbetween turbulent damping and other damping mechanisms\nof streaming instability in various interstellar phases. In par-\nticular, in a partially ionized medium, as MHD turbulence\nis also subject to ion-neutral collisional damping (Xu et al.\n2015, 2016; Xu & Lazarian 2017a), the relative importance\nbetween turbulent damping and ion-neutral collisional damp-\ning of CR-driven Alfv ´en waves depends on the ionization\nfraction and the coupling state between ions and neutrals in\ndifferent ranges of length scales.\nThe paper is organized as follows. The description on\nstreaming instability and different damping effects is pre-\nsented in Section 2. In Section 3, we compare turbulent\ndamping and ion-neutral collisional damping in both weakly\nand highly ionized media, and we derive the correspond-\ning streaming speed and diffusion coefficient in different\nregimes. The comparison between turbulent damping andarXiv:2112.06941v2  [astro-ph.HE]  20 Jan 20222\nnonlinear Landau damping in the Galactic halo is carried out\nin Section 4. Discussion and our summary are in Section 5\nand Section 6, respectivley.\n2.GROWTH AND DAMPING OF CR-DRIVEN ALFV ´EN\nWA VES\n2.1. Growth of Alfv ´en waves\nThe same resonance condition, \u0015\u0018rL, applies to both\ngyroresonant scattering of CRs by Alfv ´en waves and genera-\ntion of Alfv ´en waves via the CR resonant streaming instabil-\nity, where\u0015is the wavelength of Alfv ´en waves, and rLis the\nLarmor radius of CRs. For CRs streaming from a source to\na sink, when their bulk drift velocity, i.e., streaming velocity\nvD, is larger than the Alfv ´en speedVA, the Alfv ´en waves ex-\ncited by streaming CRs become unstable. The wave growth\nrate is (Kulsrud & Pearce 1969)\n\u0000CR= \n 0nCR(>rL)\nn\u0010vD\nVA\u00001\u0011\n; (1)\nwhen neutrals and ions are strongly coupled together with\nthe Alfv ´en wave frequency \u0018r\u00001\nLVAmuch smaller than the\nneutral-ion collisional frequency \u0017ni=\rd\u001aiin a weakly ion-\nized medium or the ion-neutral collisional frequency \u0017in=\n\rd\u001anin a highly ionized medium. Here \rdis the drag coeffi-\ncient (Shu 1992), \u001aiand\u001anare the ion and neutral mass den-\nsities, \n0=eB0=(mc)is the nonrelativistic gyrofrequency,\neandmare the proton electric charge and mass, cis the light\nspeed,nCR(> rL)is the number density of CRs with the\nLarmor radius larger than rL\u0018\u0015,nis the total number den-\nsity of gas,vD\u0000VAis the drift velocity in the wave frame,\nVA=B0=p4\u0019\u001a,B0is the mean magnetic field strength,\nand\u001a=\u001ai+\u001anis the total mass density.\nWhen neutrals and ions are weakly coupled with\nr\u00001\nLVAi> \u0017inin a partially ionized medium, where VAi=\nB0=p4\u0019\u001aiis the Alfv ´en speed in ions, or in a fully ionized\nmedium, the growth rate is\n\u0000CR= \n 0nCR(>rL)\nni\u0010vD\nVAi\u00001\u0011\n: (2)\nHereniis the ion number density.\nThe CR-generated Alfv ´en waves in turn scatter the CRs.\nThe quasilinear gyroresonant scattering of CRs in the wave\nframe regulates vD\u0000VA(i). In a steady state, the ampli-\ntude of CR-driven Alfv ´en waves is stabilized by the balance\nbetween \u0000CRand the damping rate of Alfv ´en waves. The\npitch-angle scattering corresponding to this wave amplitude\nis also in balance with the net streaming (Kulsrud 2005). The\nnet drift velocity in the wave frame in a steady state is (Kul-\nsrud 2005; Wiener et al. 2013)\nvD\u0000VA(i)=1\n3vrL\nHB2\n0\n\u000eB(rL)2; (3)\nwherev\u0018cfor relativistic CRs, His the distance from the\nsource to the sink, and \u000eB(rL)2=B2\n0is the relative magnetic\nfluctuation energy of the resonant Alfv ´en waves.The damping of streaming instability depends on both\nproperties of the background MHD turbulence and plasma\nconditions of the surrounding medium. Next we will discuss\ndifferent damping mechanisms.\n2.2. Turbulent damping\nTurbulent damping was first mentioned in Yan & Lazar-\nian (2002) and later studied in detail by Farmer & Goldre-\nich (2004) for trans-Alfv ´enic turbulence and Lazarian (2016)\nin various turbulence regimes for a more general astrophysi-\ncal application. For strong MHD turbulence with the critical\nbalance (Goldreich & Sridhar 1995) between the turbulent\nmotion in the direction perpendicular to the local magnetic\nfield and the wave-like motion along the local magnetic field\n(Lazarian & Vishniac 1999), i.e.,\nx?\nux=xk\nVA; (4)\nwherex?andxkare the length scales of a turbulent eddy\nperpendicular and parallel to the local magnetic field, and\nux=Vst(x?=Lst)1\n3 (5)\nis the turbulent velocity at x?. The corresponding turbulent\ncascading rate, i.e., eddy turnover rate, is\nuxx\u00001\n?=VstL\u00001\n3\nstx\u00002\n3\n?: (6)\nHere\nVst=VA; Lst=lA=LM\u00003\nA; (7)\nfor super-Alfv ´enic turbulence with the Alfv ´en Mach number\nMA=VL=VA>1,lAis the Alfv ´enic scale, and\nVst=VLMA; Lst=ltran=LM2\nA; (8)\nfor sub-Alfv ´enic turbulence with MA<1, whereVLis the\nturbulent velocity at the injection scale Lof turbulence.\nWe follow the analysis in Lazarian (2016) to derive the tur-\nbulent damping rate. The CR-driven Alfv ´en waves propagate\nalong the local magnetic field. For the Alfv ´en waves with the\nwavelength \u0015, the distortion by the turbulent motion at the\nresonant perpendicular scale x?is most efficient. \u0015andx?\nare related by\nx?\nVA=\u0015\nux: (9)\nThe scaling relations in Eqs. (4) and (9) are illustrated in Fig.\n1, and they give\n\u0015=ux\nVAx?=u2\nx\nV2\nAxk: (10)\nBy inserting Eq. (5) into Eq. (9), one finds\nx?=\u00153\n4\u0010VA\nVst\u00113\n4L1\n4\nst: (11)3\nFigure 1. Sketch of the relation between xkandx?for strong\nanisotropic MHD turbulence and the relation between x?and\u0015for\nturbulent damping of CR-driven Alfv ´en waves.\nThe turbulent damping rate is determined by the eddy\nturnover rate at x?(Eqs. (6) and (11)),\n\u0000st=ux\nx?=V\u00001\n2\nAV3\n2\nstL\u00001\n2\nst\u0015\u00001\n2: (12)\nNote thatx?should lie within the range of strong MHD tur-\nbulence, i.e., [xmin;?;Lst], wherexmin;?is the perpendicular\ndamping scale of MHD turbulence and determined by micro-\nscopic plasma effects. The corresponding range of rL\u0018\u0015is\n(Eq. (11)),\nVst\nVAL\u00001\n3\nstx4\n3\nmin;?<rL<Vst\nVALst: (13)\nEqs. (12) and (13) become (Eq. (7))\n\u0000st=VAL\u00001\n2M3\n2\nA\u0015\u00001\n2=VLL\u00001\n2M1\n2\nA\u0015\u00001\n2; (14)\nand\nl\u00001\n3\nAx4\n3\nmin;?<rL<lA; (15)\nfor super-Alfv ´enic turbulence, and (Eq. (8))\n\u0000st=VAL\u00001\n2M2\nA\u0015\u00001\n2=VLL\u00001\n2MA\u0015\u00001\n2; (16)\nand\nL\u00001\n3M4\n3\nAx4\n3\nmin;?<rL<LM4\nA; (17)\nfor sub-Alfv ´enic turbulence. We see that \u0000stincreases with\nMA. Naturally, a larger amplitude of turbulence can result\nin a more efficient turbulent damping. For the same reason,\n\u0000stof sub-Alfv ´enic turbulence is smaller than that of super-\nAlfv ´enic turbulence under the same physical condition.\n2.3. Ion-neutral collisional damping in a partially ionized\nmedium\nAlfv ´en waves propagating in the partially ionized interstel-\nlar medium (ISM) with a wide range of ionization fractions,\ne.g., from weakly ionized molecular clouds (MCs) to highlyionized warm phases, are subject to the damping effect due\nto the collisional friction between ions and neutrals.\nIn a weakly ionized medium with \u0017ni< \u0017in, when ions\nand neutrals are strongly coupled together with the wave fre-\nquency!=VAkk< \u0017ni, the ion-neutral collisional (IN)\ndamping rate is (Piddington 1956; Kulsrud & Pearce 1969)\n\u0000IN=\u0018nV2\nAk2\nk\n2\u0017ni; (18)\nwherekkis the wavevector component parallel to the mag-\nnetic field, and \u0018n=\u001an=\u001a. When neutrals and ions are de-\ncoupled from each other, i.e., in the weak coupling regime\nwith!=VAikk>\u0017in, there is\n\u0000IN=\u0017in\n2: (19)\nMHD turbulent cascade in a weakly ionized medium is\nalso subject to IN damping (Xu et al. 2015, 2016; Xu &\nLazarian 2017a). We consider that the driving of turbulence\noccurs in the strong coupling regime. MHD turbulence is\ndamped when \u0000INin Eq. (18) equalizes with the turbulent\ncascading rate ukk?, whereukis the turbulent velocity at\nwavenumber k, andk?is the wavevector component perpen-\ndicular to the magnetic field. For strong MHD turbulence, k?\nandkkare related by the critical balance relation (see Section\n2.2)\nk?uk=kkVA: (20)\nThe corresponding IN damping scale of MHD turbulence is\n(Xu et al. 2015, 2016)\nxmin;?=\u00102\u0017ni\n\u0018n\u0011\u00003\n2L\u00001\n2\nstV3\n2\nst; (21)\nwhich gives the smallest perpendicular scale of MHD turbu-\nlent cascade. It becomes\nxmin;?=\u00102\u0017ni\n\u0018n\u0011\u00003\n2L\u00001\n2V3\n2\nL (22)\nfor super-Alfv ´enic turbulence, and\nxmin;?=\u00102\u0017ni\n\u0018n\u0011\u00003\n2L\u00001\n2V3\n2\nLM1\n2\nA (23)\nfor sub-Alfv ´enic turbulence. With\nukk?=VAkk<\u0017ni<\u0017in; (24)\nand\n\u0018nV2\nAk2\nk\n2\u0017ni<\u0018n\u0017ni\n2<\u0017ni\n2<\u0017in\n2; (25)\nstrong MHD turbulence injected in the strong coupling\nregime cannot cascade into the weak coupling regime, and\n\u0000INof Alfv ´en waves in the weak coupling regime is larger\nthan\u0000INand the eddy turnover rate of MHD turbulence in\nthe strong coupling regime (Xu et al. 2016).4\nIn a highly ionized medium with \u0017in< \u0017ni, in the strong\ncoupling regime with VAkk<\u0017in,\u0000INis given by Eq. (18).\nWhen ions are decoupled from neutrals with VAikk> \u0017in,\nthere is (Xu et al. 2016)\n\u0000IN=\u0017ni\u001fV2\nAik2\nk\n2\u0002\n(1 +\u001f)2\u00172\nni+V2\nAik2\nk\u0003; (26)\nwhere\u001f=\u001an=\u001ai. When neutrals and ions are decoupled\nfrom each other with VAikk>\u0017ni, the above expression can\nbe reduced to Eq. (19). As ukk?=VAkk(orVAikk)>\u0000IN\nin both strong and weak coupling regimes, MHD turbulence\nin a highly ionized medium is not damped by IN damping.\nBriefly, IN damping is sensitive to the ionization fraction,\nand the damping effect in a weakly ionized medium is much\nstronger than that in a highly ionized medium.\n2.4. Nonlinear Landau damping\nIn the fully ionized gaseous Galactic halo or corona\n(Spitzer 1990; McKee 1993), Alfv ´en waves are subject to\nnonlinear Landau (NL) damping due to the resonant interac-\ntions of thermal ions with the beat waves produced by cou-\nples of Alfven waves (Lee & V ¨olk 1973; Kulsrud 1978). The\ndamping rate is (Kulsrud 1978)\n\u0000NL=1\n2\u0010\u0019\n2\u00111\n2vth\nc\u000eB(rL)2\nB2\n0\n; (27)\nwhere \n =eB0=(\rmc)\u0018c=rLis the gyrofrequency of rel-\nativistic CRs with the Lorentz factor \r,vth=p\nkBTi=mi\nis the average thermal ion speed, kBis the Boltzmann con-\nstant,Tiis ion temperature, and miis ion mass. Unlike \u0000st\nand\u0000IN,\u0000NLdepends on the amplitude of CR-generated\nAlfv ´en waves.\n3.TURBULENT DAMPING VS. IN DAMPING\nDepending on the driving condition of MHD turbulence\nand the plasma condition in different interstellar phases,\nthe dominant damping mechanism of streaming instability\nvaries. We first compare turbulent damping with IN damp-\ning in weakly and highly ionized media, and then compare\nturbulent damping with NL damping in a fully ionized hot\nmedium (see Section 4). As the streaming instability and\nwave damping together determine vD, a proper description\nof the damping effect in different regimes is important for\ndetermining the diffusion coefficient of CRs and understand-\ning their confinement in the Galaxy.\n3.1. Dominant damping mechanism in different regimes\n(1) Weakly ionized medium. We first consider the case\nwhen both MHD turbulence and CR-driven Alfv ´en waves are\nin the strong coupling regime, i.e., r\u00001\nLVA< \u0017ni. If the\nturbulent damping is the dominant damping mechanism, we\nshould have\n(i):\u0000st(x?)>\u0000IN(xk); (28)\nso that MHD turbulence is not damped at x?, and\n(ii):\u0000st(x?)>\u0000IN(rL): (29)We easily see\nrL<x?<xk (30)\nbased on the relation in Eq. (10), meaning\n\u0000IN(rL)>\u0000IN(xk): (31)\nTherefore, if condition (ii) is satisfied, then condition (i) is\nnaturally satisfied.\nAs an example, using the following parameters, we have\nVA\nrL\u0017ni\n=0:07\u0010B0\n1\u0016G\u00112\u0010nH\n100cm\u00003\u0011\u00003\n2\u0010ne=nH\n0:1\u0011\u00001\u0010ECR\n10GeV\u0011\u00001\n<1;\n(32)\nwherene=nHis the ionization fraction, neandnHare num-\nber densities of electrons and atomic hydrogen, mi=mn=\nmH,mnis neutral mass, mHis hydrogen atomic mass,\n\rd= 5:5\u00021014cm3g\u00001s\u00001(Shu 1992), and ECRis the\nenergy of CR protons. The values used here do not represent\nthe typical conditions of MCs, but are still considered as a\npossibility given the large variety of interstellar conditions.\nCondition (ii) in Eq. (29) can be rewritten as (Eqs. (14) and\n(18))\nMA>\u0010\u0018n\n2\u0017niVAL1\n2r\u00003\n2\nL\u00112\n3\n= 2\u0010B0\n1\u0016G\u00115\n3\u0010nH\n100cm\u00003\u0011\u00001\u0010ne=nH\n0:1\u0011\u00002\n3\n\u0010L\n0:1pc\u00111\n3\u0010ECR\n10GeV\u0011\u00001(33)\nfor super-Alfv ´enic turbulence driven on small length scales,\ne.g., near supernova shocks when the shock and shock pre-\ncursor interact with interstellar or circumstellar density inho-\nmogeneities (e.g., Xu & Lazarian 2017b, 2021). We note that\nthe outer scale of this turbulence is determined by the size of\ndensity clumps. For instance, the typical size of ubiquitous\nHI clouds in the ISM is 0:1pc (Inoue et al. 2009). As this\nscale is much larger than rLof low-energy CRs considered\nhere, the CR-induced Alfv ´en waves are subject to turbulent\ndamping in this scenario.\nWith the above parameters used, in Fig. 2(a), the shaded\narea shows the ranges of MAandne=nHfor turbulent damp-\ning to dominate over IN damping. The solid line represents\nMA=\u0010\u0018n\n2\u0017niVAL1\n2r\u00003\n2\nL\u00112\n3; (34)\nbelow which, IN damping dominates over turbulent damping.\nIn the area above the solid line, as MHD turbulence is also\nsubject to IN damping, to ensure that the condition in Eq.\n(15) is also satisfied, other constraints on MAindicated in\nFig. 2(a) are\nMA<\u00102\u0017ni\n\u0018nL\nVA\u00111\n3; (35)5\ncorresponding to (Eq. (22))\nxmin;?<lA; (36)\nMA<h\u00102\u0017ni\n\u0018n\u00112\nV\u00002\nALrLi1\n3; (37)\ncorresponding to (Eqs. (11) and (22))\n\u0000st(x?)>\u0000IN(xk); (38)\nand\nMA<\u0010L\nrL\u00111\n3; (39)\ncorresponding to\nrL<lA: (40)\nIn addition, the vertical dashed line indicates the ne=nH\nvalue corresponding to r\u00001\nLVA=\u0017ni. Toward a larger\nne=nH, the Alfv ´en waves are in the strong coupling regime.\nIn Fig. 2(b), using the MAvalue given by Eq. (34), we\nillustrate the relation between different length scales. For the\nregime of interest, we have\nrL;min<VA\n\u0017ni<rL<x?<xk<lA; (41)\nwhererL;min=l\u00001\n3\nAx4\n3\nmin;?is given in Eq. (15).\nIn typical MC conditions, we find that CR-driven Alfv ´en\nwaves are in the weak coupling regime with\nVAi\nrL\u0017in\u00192\u0002103\u0010B0\n10\u0016G\u00112\u0010nH\n100cm\u00003\u0011\u00003\n2\u0010ne=nH\n10\u00004\u0011\u00001\n2\n\u0010ECR\n1GeV\u0011\u00001\n\u001d1:\n(42)\nFor MHD turbulence injected at a large scale in the strong\ncoupling regime, there is always (see Section 2.3)\n\u0000st(x?)<\u0000IN(rL) =\u0017in\n2: (43)\nTherefore, the damping of CR-driven Alfv ´en waves in MCs\npredominantly comes from ion-neutral collisions.\n(2) Highly ionized medium. At a high ionization fraction,\nwhen both MHD turbulence and CR-generated Alfv ´en waves\nare in the strong coupling regime, i.e., r\u00001\nLVA<\u0017in, similar\nto the analysis for the strong coupling regime in a weakly ion-\nized medium, \u0000st(x?)should be compared with \u0000IN(rL)to\ndetermine the relative importance between the two damping\neffects. When MHD turbulence at x?is in the strong cou-\npling regime, but CR-generated Alfv ´en waves are in the weak\ncoupling regime and also have r\u00001\nLVAi> \u0017ni, IN damping\nis more important than turbulent damping. When MHD tur-\nbulence atx?is also in the weak coupling regime, there is\nalways\n\u0000st(x?)>\u0000IN(rL) =\u0017in\n2; (44)\nand MHD turbulence dominates the wave damping.By using the typical parameters of the warm ionized\nmedium (WIM) (Reynolds 1992), we find that CR-generated\nAlfv ´en waves are in the weak coupling regime and further\nhave\nVAi\nrL\u0017ni\n=7:6\u0002103\u0010B0\n1\u0016G\u00112\u0010ni\n0:1cm\u00003\u0011\u00003\n2\u0010ECR\n1GeV\u0011\u00001\n\u001d1:\n(45)\nAs discussed above, under the condition\n\u0000st(x?)\n\u0000IN(rL)=\u0000st(x?)\n\u0017in\n2>1; (46)\nturbulent damping dominates over IN damping. The above\ncondition can be rewritten as (Eq. (14))\nMA>\u0010\u0017in\n2V\u00001\nAiL1\n2r1\n2\nL\u00112\n3\n= 0:2\u0010B0\n1\u0016G\u0011\u00001\u0010ni\n0:1cm\u00003\u00111\n3\u0010nn\n0:01cm\u00003\u00112\n3\n\u0010L\n100pc\u00111\n3\u0010ECR\n1GeV\u00111\n3(47)\nfor super-Alfv ´enic turbulence, which is naturally satisfied,\nand (Eq. (16))\nMA>\u0010\u0017in\n2V\u00001\nAiL1\n2r1\n2\nL\u00111\n2\n= 0:3\u0010B0\n1\u0016G\u0011\u00003\n4\u0010ni\n0:1cm\u00003\u00111\n4\u0010nn\n0:01cm\u00003\u00111\n2\n\u0010L\n100pc\u00111\n4\u0010ECR\n1GeV\u00111\n4(48)\nfor sub-Alfv ´enic turbulence, where nnis the neutral num-\nber density, and L\u0018100pc is the typical injection scale of\ninterstellar turbulence driven by supernova explosions. We\nnote thatVA\u0019VAican be used for estimating MAof MHD\nturbulence injected in the strong coupling regime in a highly\nionized medium. As the above constraints on MAcan be eas-\nily satisfied in the WIM, turbulent damping is likely to be the\ndominant damping effect for CR-generated Alfv ´en waves in\nthe WIM.\n3.2.vDin different regimes\nKnowing the dominant damping mechanism in different\ncoupling regimes and at different ionization fractions, we can\nfurther determine vDat the balance between wave growth\nand damping.\n(1) Weakly ionized medium. In the strong coupling\nregime, when MHD turbulence dominates the wave damping,\nat the balance between growth and damping rates of Alfv ´en\nwaves (Eqs. (1) and (12)), we find\nvD\nVA= 1 + \n\u00001\n0\u0010nCR(>rL)\nn\u0011\u00001\nV\u00001\n2\nAV3\n2\nstL\u00001\n2\nstr\u00001\n2\nL;(49)6\n10-210-1ne/nH100101102MAst (x) = IN (rL)st(x) = IN(x||)xmin, = lArL = lAst(x) > IN(rL)\n(a)\n10-210-1ne/nH1081010101210141016length scales [cm]lAVA/nirL,minrLxx|| (b)\nFigure 2. (a) Ranges of MAandne=nHfor turbulent damping to dominate over IN damping (shaded area above the solid line) and for IN\ndamping to dominate over turbulent damping (below the solid line) in a weakly ionized medium, where super-Alfv ´enic turbulence driven on\na small length scale ( 0:1pc) is considered. Other limits on MAare indicated by other lines as explained in the text. (b) Relation between\ndifferent length scales, where the MAvalue corresponding to the solid line in (a) is used. The vertical dashed lines in (a) and (b) correspond to\nr\u00001\nLVA=\u0017niwithne=nH= 0:007.\nwhich is (Eq. (7))\nvD\nVA\u00191 + 1:7\u0002105\u0010B0\n1\u0016G\u0011\u00001\u0010nH\n100cm\u00003\u00115\n4\n\u0010L\n0:1pc\u0011\u00001\n2\u0010VL\n1km s\u00001\u00113\n2\u0010ECR\n10GeV\u00111:1(50)\nfor super-Alfv ´enic turbulence, where we adopt the integral\nnumber density of CRs near the Sun (Wentzel 1974)\nnCR(>rL) = 2\u000210\u000010\r\u00001:6cm\u00003: (51)\nWhen ion-neutral collisions dominate the wave damping,\nthere is (Eqs. (1) and (18))\nvD\nVA= 1 + \n\u00001\n0\u0010nCR(>rL)\nn\u0011\u00001\u0018nV2\nAr\u00002\nL\n2\u0017ni\n\u00191 + 4:9\u0002104\u0010B0\n1\u0016G\u00113\u0010nH\n100cm\u00003\u0011\u00001\n\u0010ne=nH\n0:1\u0011\u00001\u0010ECR\n10GeV\u0011\u00000:4\n:(52)\nWe see that with the parameters adopted here, in the strong\ncoupling regime there is vD\u001dVAdue to the strong damping\nof CR-generated Alfv ´en waves irrespective of the dominant\ndamping mechanism.\nIn a typical MC environment, when CR-driven Alfv ´en\nwaves are in the weak coupling regime and mainly subjectto IN damping, we have (Eqs. (2) and (19))\nvD\nVAi= 1 +\u0017in\n2\n\u00001\n0\u0010nCR(>rL)\nni\u0011\u00001\n\u00191 + 26:5\u0010B0\n10\u0016G\u0011\u00001\u0010nH\n100cm\u00003\u00112\n\u0010ne=nH\n10\u00004\u0011\u0010ECR\n1GeV\u00111:6\n:(53)\nWe see that vDis significantly larger than VAidue to the\ndamping effect.\n(2) Highly ionized medium. In the WIM, CR-driven\nAlfv ´en waves are in the weak coupling regime and mainly\nsubject to turbulent damping. \u0000CR= \u0000stgives (Eqs. (2),\n(7), (8), (12))\nvD\nVAi=1 + \n\u00001\n0\u0010nCR(>rL)\nni\u0011\u00001\nV\u00001\n2\nAiV3\n2\nstL\u00001\n2\nstr\u00001\n2\nL\n\u00191 + 3:2\u0010B0\n1\u0016G\u0011\u00001\u0010ni\n0:1cm\u00003\u00115\n4\u0010ECR\n1GeV\u00111:1\n\u0010VL\n10km s\u00001\u00113\n2\u0010L\n100pc\u0011\u00001\n2(54)\nfor super-Alfv ´enic turbulence, and\nvD\nVAi\u00191 + 0:9\u0010B0\n1\u0016G\u0011\u00003\n2\u0010ni\n0:1cm\u00003\u00113\n2\u0010ECR\n1GeV\u00111:1\n\u0010VL\n5km s\u00001\u00112\u0010L\n100pc\u0011\u00001\n2\n(55)\nfor sub-Alfv ´enic turbulence, where we consider VA\u0019VAi\nandni\u0019nH, andVL\u001810km s\u00001is the typical turbu-\nlent velocity for supernova-driven turbulence (Chamandy &7\nShukurov 2020). The second term in Eqs. (54) and (55) be-\ncomes the dominant term at higher CR energies, and vDis\nenergy dependent. The larger vDin Eq. (54) is caused by\nthe stronger turbulent damping in super-Alfv ´enic turbulence\nthan in sub-Alfv ´enic turbulence (see Section 2.2).\n3.3. Diffusion coefficient in different regimes\nThe diffusion coefficient Dof streaming CRs depends on\nbothvDand the characteristic scale of turbulent magnetic\nfields. In super-Alfv ´enic turbulence, lAis the characteris-\ntic tangling scale of turbulent magnetic fields, at which the\nturbulent and magnetic energies are in equipartition. Over\nlAthe field line changes its orientation in a random walk\nmanner. Therefore, lAis the effective mean free path of\nCRs streaming along turbulent magnetic field lines (Brunetti\n& Lazarian 2007). In sub-Alfv ´enic turbulence, magnetic\nfields are weakly perturbed with an insignificant change of\nmagnetic field orientation on all length scales. So the mag-\nnetic field structure cannot provide additional confinement\nfor streaming CRs. In this case, streaming CRs do not have a\ndiffusive propagation in the observer frame, but we still intro-\nduce a diffusion coefficient to quantify the CR confinement\nand adopt the CR gradient scale length Hfor calculating D.\nIn a weakly ionized medium, e.g., MCs, by using Eq. (53)\nand considering super-Alfv ´enic turbulence, we have\nD=vDlA=VAivD\nVAiLM\u00003\nA\n\u00191:8\u00021028cm2s\u00001\u0010nH\n100cm\u00003\u00113\n2\u0010ne=nH\n10\u00004\u00111\n2\n\u0010ECR\n1GeV\u00111:6\u0010L\n10pc\u0011\nM\u00003\nA;(56)\nwhere the factor M\u00003\nAcan be much smaller than unity. Here\nwe consider L\u001810pc for turbulence in MCs, and we note\nthat as MHD turbulence is in the strong coupling regime, VA\nshould be used when calculating MA. AsD/M\u00003\nA, a slow\ndiffusion with a small Dis expected at a large MA.\nIn a highly ionized medium, e.g., the WIM, we have\nD=vDlA (57)\nfor super-Alfv ´enic turbulence, and\nD=vDH (58)\nfor sub-Alfv ´enic turbulence, where vDis given in Eq. (54)\nand Eq. (55), respectively, and H\u00181kpc as the scale height\nof the WIM. In Fig. 3, we present Das a function of ECR\nfor both super- and sub-Alfv ´enic turbulence with MA= 1:4\nand0:7in the WIM. The smaller Din super-Alfv ´enic turbu-\nlence is caused by the tangling of turbulent magnetic fields.\nWe seeD/E1:1\nCRin both turbulence regimes. This steep en-\nergy scaling can be important for explaining the CR spectrum\nobserved at Earth below \u0018100 GeV (Blasi et al. 2012).\n4.TURBULENT DAMPING VS. NL DAMPING\n100101102ECR [GeV]1026102710281029D [cm2 s-1]MA = 0.7MA = 1.4 ECR1.1Figure 3. Diffusion coefficient vs. ECRin super- and sub-Alfv ´enic\nturbulence in the WIM.\nIn the Galactic halo, if NL damping is the dominant damp-\ning mechanism of CR-generated Alfv ´en waves, at the bal-\nance\n\u0000CR= \u0000NL; (59)\nby combining Eqs. (2), (3), and (27), one can obtain\nvD=VAi+pc\n3H\u00001\n2\"\n2\n3r\n2\n\u0019\n0\nvthnCR(>rL)\nni1\nVAi#\u00001\n2\n;\n(60)\nand\n\u000eB(rL)2\nB2\n0=\"\n2\n3r\n2\n\u0019\n0\nvthnCR(>rL)\nnic\nVAir2\nL\nH#1\n2\n:(61)\nInserting Eq. (61) into Eq. (27) yields\n\u0000NL=\u0010\u0019\n2\u00111\n4\u00101\n6\u00111\n2v1\n2\nth\"\n\n0nCR(>rL)\nnic\nVAi1\nH#1\n2\n;\n(62)\nwhich becomes smaller at a larger Hwith a smaller CR gra-\ndient. To be consistent with our assumption that NL damping\ndominates over turbulent damping, the condition (Eqs. (12)\nand (62))\n\u0000NL\n\u0000st=\"\u0010\u0019\n2\u00111\n21\n6Lst\nHnCR(>rL)\nnivth\nVstc\nVst\n0rL\nVst#1\n2\n>1\n(63)\nshould be satisfied.\nIf turbulent damping dominates over NL damping, the bal-\nance (Eqs. (2) and (12))\n\u0000CR= \u0000st (64)\ngives (see also Eq. (54))\nvD=VAi+ \n\u00001\n0\u0010nCR(>rL)\nni\u0011\u00001\nV1\n2\nAiV3\n2\nstL\u00001\n2\nstr\u00001\n2\nL:(65)8\nThen inserting the above expression into Eq. (3) gives\n\u000eB(rL)2\nB2\n0=c\n3H\n0nCR(>rL)\nniV\u00001\n2\nAiV\u00003\n2\nstL1\n2\nstr3\n2\nL:(66)\nMoreover, \u0000NLcorresponding to the above relative magnetic\nfluctuation energy is (Eqs. (27) and (66))\n\u0000NL=1\n6\u0010\u0019\n2\u00111\n2c\nH\n0vthnCR(>rL)\nniV\u00001\n2\nAiV\u00003\n2\nstr1\n2\nLL1\n2\nst:\n(67)\nUnder the assumption of dominant turbulent damping, there\nshould be\n\u0000NL\n\u0000st=\u0010\u0019\n2\u00111\n21\n6Lst\nHnCR(>rL)\nnivth\nVstc\nVst\n0rL\nVst<1:\n(68)\nComparing the conditions in Eqs. (63) and (68), we see\nthat at \u0000NL= \u0000st, there is\n\u0010\u0019\n2\u00111\n21\n6Lst\nHnCR(>rL)\nnivth\nVstc\nVst\n0rL\nVst= 1: (69)\nUsing the typical parameters in the Galactic halo (Farmer\n& Goldreich 2004), we find that the turbulence in this low-\ndensity environment has\nMA\u00190:1\u0010VL\n10km s\u00001\u0011\u0010B0\n1\u0016G\u0011\u00001\u0010ni\n10\u00003cm\u00003\u00111\n2:(70)\nFor sub-Alfv ´enic turbulence, Eq. (69) can be rewritten as\nMA=h\u0010\u0019\n2\u00111\n21\n6L\nHnCR(>rL)\nnivth\nVAic\nVAi\n0rL\nVAii1\n4;(71)\nwhich is shown as the solid line in Fig. 4. Other parameters\nareTi= 106K,L= 100 pc, andECR= 1 GeV . The area\nabove and below the solid line corresponds to the parame-\nter space for turbulent damping and NL damping to be the\ndominant damping mechanism, respectively. When turbulent\ndamping is dominant, another constraint on MAis (Eq. (17))\nMA>\u0010rL\nL\u00111\n4\u00190:008; (72)\nwhich is naturally satisfied in this situation.\nGiven the small MAof MHD turbulence in the Galactic\nhalo, the wave damping is more likely to be dominated by\nNL damping. Using Eq. (60), we find\nvD\nVAi\u00191+0:02\u0010H\n5kpc\u0011\u00001\n2\u0010B0\n1\u0016G\u0011\u00001\u0010ni\n10\u00003cm\u00003\u00113\n4\n\u0010Ti\n106K\u00111\n4\u0010ECR\n1GeV\u00110:8\n:\n(73)\nvDis very close to VAi, indicative of the insignificant wave\ndamping in the Galactic halo. Therefore, GeV CRs can be\nconfined due to streaming instability. For CRs with ECR<\n12345H [kpc]00.20.40.60.81MAFigure 4. Comparison between turbulent damping and NL damping\nfor Alfv ´en waves generated by GeV CRs in the Galactic halo. The\nshaded area shows the ranges of MAandHfor turbulent damping\nto dominate over NL damping. The solid line represents the relation\nin Eq. (71).\n100GeV , there is approximately\nD\u0019VAiH\n= 1:1\u00021029cm2s\u00001\u0010B0\n1\u0016G\u0011\u0010ni\n10\u00003cm\u00003\u0011\u00001\n2\u0010H\n5kpc\u0011\n;\n(74)\nwhich is energy independent.\n5.DISCUSSION\nEffect of MHD turbulence on diffusion of streaming CRs.\nIn cases when MHD turbulence dominates the damping of\nstreaming instability, e.g., in the WIM, MHD turbulence is\nimportant for setting vD. Super-Alfv ´enic turbulence also\nprovides additional confinement to streaming CRs due to the\nfield line tangling at lA, irrespective of the dominant damping\nmechanism. In addition, the non-resonant mirroring interac-\ntion of CRs with slow and fast modes in MHD turbulence\ncan also suppress the diffusion of CRs in the vicinity of CR\nsources (Lazarian & Xu 2021; Xu 2021). The relative im-\nportance between mirroring and streaming instability in af-\nfecting CR diffusion near CR sources will be investigated in\nour future work. In this work we do not consider gyroreso-9\nnant scattering and resonance-broadened transit time damp-\ning (TTD) by fast modes of MHD turbulence (Xu & Lazarian\n2018, 2020), as fast modes are damped at a large scale due to\nIN damping in a weakly ionized medium (Xu et al. 2016) and\ntheir energy fraction is small at a small MA(Hu et al. 2021).\nMoreover, the energy scaling of diffusion coefficient corre-\nsponding to scattering by fast modes is incompatible with\nAMS-02 observations at CR energies .103GeV (Kempski\n& Quataert 2021).\nCutoff range of Alfv ´en waves in a weakly ionized medium.\nIn a weakly ionized medium, within the cutoff range of kk\nthere is no propagation of Alfv ´en waves due to the severe IN\ndamping (Kulsrud & Pearce 1969). The boundary [k+\nc;k;k\u0000\nc;k]\nof the cutoff range is set by != \u0000INin both strong and\nweak coupling regimes (Xu et al. 2015),\nk+\nc;k=2\u0017ni\nVA\u0018n; k\u0000\nc;k=\u0017in\n2VAi: (75)\nIf CR-driven Alfv ´en waves fall in the cutoff range with\nk+\nc;k<r\u00001\nL<k\u0000\nc;k; (76)\nthe streaming instability cannot occur. We note that for GeV\nCRs in a typical MC environment, the CR-driven Alfv ´en\nwaves are in the weak coupling regime with r\u00001\nL\u001dk\u0000\nc;k(see\nEq. (42)).\nMicroscopic vs. macroscopic diffusion. By “microscopic\ndiffusion”, we refer to the diffusion in the wave frame caused\nby the gyroresonant scattering of CRs by CR-amplified\nAlfv ´en waves, while the “macroscopic diffusion” in the ob-\nserver frame is a result of both streaming of CRs and tangling\nof turbulent magnetic fields on scales much larger than rL.\nThe former was included in calculating the total diffusion co-\nefficient in earlier studies, e.g., Hopkins et al. (2021). In this\nwork we only consider the latter as it can be directly com-\npared with observations in the observer frame, which was\nalso adopted in Krumholz et al. (2020) for studying CR trans-\nport in starburst galaxies.\nCoupling between CRs and gas. In the Galactic disk with\nsuper-Alfv ´enic turbulence (Hu et al. 2019), due to the strong\nIN damping and turbulent damping, CRs have fast streaming\nand do not suffer significant energy loss via wave genera-\ntion. The coupling between CRs and gas is caused by field\nline tangling. This coupling can result in additional pres-\nsure support and suppression of star formation. By contrast,\nin the Galactic halo with sub-Alfv ´enic turbulence, due to the\nweak NL damping, CRs are well self-confined and coupled to\nthe gas via streaming instability, and thus effectively transfer\nmomentum to the gas. Both wave damping and turbulent tan-\ngling can significantly affect the transport of streaming CRs\nand their coupling with the gas, and thus are important for\nstudying CR-driven galactic winds.\nTurbulence and CRs at phase transition. In the multi-phase\nISM, the transition from hot/warm to cold gas can be driven\nby, e.g., passage of shock waves (Inutsuka et al. 2015). The\nphase transition induces various instabilities and turbulence.The turbulent mixing layers at the interfaces between dif-\nferent gas phases have been recently studied in detail by Ji\net al. (2019). As the transport of CRs is sensitive to the\nturbulent magnetic field structure, when CRs interact with\nthe shock-compressed magnetic field, they can be reflected\noff the shock surface (Xu & Lazarian 2021) and undergo an\nabrupt change of trajectory.\n6.SUMMARY\nWe study the damping of streaming instability of GeV-100\nGeV CRs and the resulting diffusion coefficients in different\nMHD turbulence regimes and interstellar phases.\nIn a partially ionized medium, both CR-generated Alfv ´en\nwaves and MHD turbulence are subject to IN damping. The\ndamping rate depends on the ionization fraction and the cou-\npling state between ions and neutrals. In both weakly ion-\nized MCs and highly ionized WIM, CR-generated Alfv ´en\nwaves are in the weak coupling regime. In a weakly ionized\nmedium, IN damping is strong and dominates the damping\nof both MHD turbulence and CR-amplified Alfv ´en waves. In\na highly ionized medium, IN damping is so weak that MHD\nturbulence injected in the strong coupling regime can cas-\ncade into the weak coupling regime and dominates the wave\ndamping.\nBoth IN damping in MCs and turbulent damping in the\nWIM act to suppress the streaming instability, leading to a\nstreaming speed of CRs larger than the Alfv ´en speed. The\nresulting diffusion coefficient is thus dependent on CR en-\nergies. The steep energy scaling of diffusion coefficient in\nthe WIM (see Fig. 3) is important for explaining the CR\nspectrum observed at Earth, as the turbulence properties mea-\nsured in the nearby ( .1kpc) ISM (Armstrong et al. 1995)\nare similar to that in the WIM (Chepurnov & Lazarian 2010).\nWe find that MHD turbulence not only can affect the CR\nstreaming speed by turbulent damping but also causes the\ndiffusive propagation of streaming CRs by the field line tan-\ngling. The latter effect was not considered in most earlier\nstudies on CR streaming. Because of the field line tangling\nin super-Alfv ´enic turbulence at lA=LM\u00003\nA, CRs streaming\nalong turbulent field lines have an effective mean free path\ngiven bylA. At a large MAin, e.g., cold interstellar phases,\na significant reduction of the diffusion coefficient by M\u00003\nA\nis expected. The slow diffusion of streaming CRs in star-\nforming MCs can have an important influence on the Galac-\ntic disk structure and star formation (Semenov et al. 2021).\nIn the multi-phase ISM with a large variety of MAof inter-\nstellar turbulence, measuring MAwith new techniques (e.g.,\nLazarian et al. 2018; Xu & Hu 2021) is necessary for realistic\nmodeling of diffusion coefficients of streaming CRs.\nIn the diffuse Galactic halo, MHD turbulence is sub-\nAlfv ´enic with a small MA, and NL damping is a more impor-\ntant mechanism for damping CR-generated Alfv ´en waves.\nThis finding is different from that in Lazarian (2016). The\nresulting streaming speed is basically given by Alfv ´en speed,\nand CRs are confined mainly due to streaming instability.\nIn the WIM and Galactic halo, the global pressure gradient\nformed by streaming CRs plays an important dynamical role10\nin driving galactic outflows and affecting galaxy evolution\n(Padovani et al. 2020).\nIn addition to the interstellar turbulence injected on \u0018\n10\u0000100pc, we also considered a special case with small-\nscale (.0:1pc) preshock turbulence in supernova rem-\nnants, which is driven by the interaction between the CR\nprecursor and upstream density inhomogeneities. 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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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An identification algorithm is p roposed based on the\nfact that the output of system happens to be decomposed into a p roduct of an exponential func-\ntion and a periodic function. The former contains information of the damping coefficient, while\nthe latter does not. The convergence and error analysis are also d eveloped. Three examples,\nnamely an anti-stable wave equation with boundary anti-damping, th e Schr¨ odinger equation\nwith internal anti-damping, and two connected strings with middle jo int anti-damping, are in-\nvestigated and demonstrated by numerical simulations to show the effectiveness of the proposed\nalgorithm.\nKeywords: Identification; damping coefficient; anti-stable PDEs; anti-damping coefficient.\nAMS subject classifications: 35K05, 35R30, 65M32, 65N21, 15A22.\n1 Introduction\nLetHbe a Hilbert space with the inner product /an}bracketle{t·,·/an}bracketri}htand inner product induced norm /bardbl·/bardbl, and\nletY=R(orC). Consider the dynamic system in H:\n/braceleftBigg\n˙x(t) =A(q)x(t), x(0) =x0,\ny(t) =Cx(t)+d(t),(1.1)\n∗Corresponding author. Email: bzguo@iss.ac.cn\n1whereA(q) :D(A(q))⊂H→His the system operator depending on the coefficient q, which is\nassumed to be a generator of C0-semigroup Tq= (Tq(t))t∈R+onH,C:H→Yis the admissible\nobservation operator for Tq([20]),x0∈His the initial value, and d(t) is the external disturbance.\nVarious PDE control systems with damping mechanism can be fo rmulated into system (1.1),\nwhereqis the damping coefficient. For a physical system, if the dampi ng is produced by material\nitself that dissipates the energy stored in system, then the system keeps stable. The identification\nof damping coefficient has been well considered for distribut ed parameter systems like Kelvin-Voigt\nviscoelastic damping coefficient in Euler-Bernoulli beam in vestigated in [4], and a more general\ntheoretical framework for various classes of parameter est imation problems presented in [5]. In\nthese works, the inverse problems are formulated as least sq uare problems and are solved by finite\ndimensionalization. For more revelent works, we can refer t o the monograph [6]. Sometimes,\nhowever, the source of instability may arise from the negati ve damping. One example is the\nthermoacoustic instability in duct combustion dynamics an d the other is the stick-slip instability\nphenomenon in deep oil drilling, see, for instance, [7] and t he references therein. In such cases,\nthe negative damping will result in all the eigenvalues loca ted in the right-half complex plane, and\nthe open-loop plant is hence “anti-stable” (exponentially stable in negative time) and the qin such\nkind of system is said to be the anti-damping coefficient.\nAwidely investigated probleminrecent years isstabilizat ion foranti-stable systemsbyimposing\nfeedback controls. A breakthrough on stabilization for an a nti-stable wave equation was first\nreached in [19] where a backstepping transformation is prop osed to design the boundary state\nfeedback control. By the backstepping method, [11] general izes [19] to two connected anti-stable\nstrings with joint anti-damping. Very recently, [12, 13] in vestigate stabilization for anti-stable\nwave equation subject to external disturbance coming throu gh the boundary input, where the\nsliding mode control and active disturbance rejection cont rol technology are employed. It is worth\npointing out that in all aforementioned works, the anti-dam ping coefficients are always supposed\nto be known.\nOn the other hand, a few stabilization results for anti-stab le systems with unknown anti-\ndamping coefficients are also available. In [16], a full state feedback adaptive control is designed for\nan anti-stable wave equation. By converting thewave equati on into acascade of two delay elements,\nan adaptive output feedback control and parameter estimato r are designed in [7]. Unfortunately,\nno convergence of the parameter update law is provided in the se works.\nIt can be seen in [7, 16] that it is the uncertainty of the anti- damping coefficient that leads to\ncomplicated design for adaptive control and parameter upda te law. This comes naturally with the\nidentification of unknown anti-damping coefficient. To the be st of our knowledge, there are few\nstudies on this regard. Our focus in the present paper is on si multaneous identification for both\nanti-damping (or damping) coefficient and initial value for s ystem (1.1), where the coefficient qis\nassumed to be in a prior parameter set Q= [q,q] (qorqmay be infinity) and the initial value is\nsupposed to be nonzero.\nWe proceed as follows. In Section 2, we propose an algorithm t o identify simultaneously the\n2coefficient and initial value through the measured observati on. The system may not suffer from\ndisturbance or it may suffer from general bounded disturbance . In Section 3, a wave equation\nwith anti-damping term in the boundary is discussed. A Schr¨ odinger equation with internal anti-\ndamping term is investigated in Section 4. Section 5 is devot ed to coupled strings with middle joint\nanti-damping. In all these sections, numerical simulation s are presented to verify the performance\nof the proposed algorithms. Some concluding remarks are pre sented in Section 6.\n2 Identification algorithm\nBefore giving the main results, we introduce the following w ell known Ingham’s theorem [14, 15, 23]\nas Lemma 2.1.\nLemma 2.1. Assume that the strictly increasing sequence {ωk}k∈Zof real numbers satisfies the\ngap condtion\nωk+1−ωk≥γfor allk∈Z, (2.1)\nfor someγ >0. Then, for all T >2π/γ, there exist two positive constants C1andC2, depending\nonly onγandT, such that\nC1/summationdisplay\nk∈Z|ak|2≤/integraldisplayT\n0/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nk∈Zakeiωkt/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndt≤C2/summationdisplay\nk∈Z|ak|2, (2.2)\nfor every complex sequence (ak)k∈Z∈ℓ2, where\nC1=2T\nπ/parenleftbigg\n1−4π2\nT2γ2/parenrightbigg\n, C2=8T\nπ/parenleftbigg\n1+4π2\nT2γ2/parenrightbigg\n. (2.3)\nTo begin with, we suppose that there is no external disturban ce in system (1.1), that is,\n/braceleftBigg\n˙x(t) =A(q)x(t), x(0) =x0,\ny(t) =Cx(t).(2.4)\nThe succeeding Theorem 2.1 indicates that identification of the coefficient qand initial value x0\ncan be achieved exactly simultaneously without error for A(q) with some structure.\nTheorem 2.1. LetA(q)in system (2.4) generate a C0-semigroup Tq= (Tq(t))t∈R+and suppose\nthatA(q)andCsatisfy the following conditions:\n(i).A(q)has a compact resolvent and all its eigenvalues {λn}n∈N(or{λn}n∈Z) admit the\nfollowing expansion:\nλn=f(q)+iµn,···<µn<µn+1<···, (2.5)\nwheref:Q→Ris invertible, µnis independent of q, and there exists an L>0such that\nµnL\n2π∈Zfor alln∈N. (2.6)\n(ii). The corresponding eigenvectors {φn}n∈Nform a Riesz basis for H.\n3(iii). There exist two positive numbers κandKsuch thatκ≤ |κn| ≤Kfor alln∈N, where\nκn:=Cφn, n∈N. (2.7)\nThen both coefficient qand initial value x0can be uniquely determined by the output y(t),t∈[0,T],\nwhereT >2L. Precisely,\nq=f−1/parenleftBigg\n1\nLln/bardbly/bardblL2(T1,T2)\n/bardbly/bardblL2(T1−L,T2−L)/parenrightBigg\n, (2.8)\nfor anyL<T1<T2−L, and\nx0=1\nL/summationdisplay\nn∈N1\nκn/parenleftbigg/integraldisplayL\n0y(t)e−λntdt/parenrightbigg\nφn. (2.9)\nProof.Since{φn}n∈Nforms a Riesz basis for H, there exists a sequence {ψn}n∈Nof eigenvectors\nofA(q)∗, which is biorthogonal to {φn}n∈N, that is, /an}bracketle{tφn,ψm/an}bracketri}ht=δnm. In this way, we can express\nthe initial value x0∈Hasx0=/summationtext\nn∈N/an}bracketle{tx0,ψn/an}bracketri}htφn,and the solution of system (2.4) as\nx(t) =Tq(t)x0=/summationdisplay\nn∈Neλnt/an}bracketle{tx0,ψn/an}bracketri}htφn. (2.10)\nBy (2.6), there exists an increasing sequence {Kn} ⊂Zsuch that\nµn=2πKn\nL, n∈N, (2.11)\nwhich implies that {µn}satisfies the following gap condition\nµn+1−µn=2π(Kn+1−Kn)\nL≥2π\nL/definesγ. (2.12)\nIn addition, by the assumption that {φn}forms a Riesz basis for Hand|Cφn|is uniformly bounded\nwith respect to n, it follows from Proposition 2 of [8] or Theorem 2 of [9] that Cis admissible for\nTq. So generally, we have\ny(t) =Cx(t) =/summationdisplay\nn∈Neλnt/an}bracketle{tx0,ψn/an}bracketri}htCφn=ef(q)t/summationdisplay\nn∈Neiµnt/an}bracketle{tx0,ψn/an}bracketri}htCφn/definesef(q)tPL(t),(2.13)\nwherePL(t) =/summationtext\nn∈Neiµnt/an}bracketle{tx0,ψn/an}bracketri}htCφnis well-defined and it follows from (2.11) that PL(t) is a\nY-valued function of period L. For anyT2−L>T1>L,\n/integraldisplayT2\nT1|y(t)|2dt=e2f(q)L/integraldisplayT2−L\nT1−L|y(t)|2dt, (2.14)\nthat is,\n/bardbly/bardblL2(T1,T2)=ef(q)L/bardbly/bardblL2(T1−L,T2−L). (2.15)\nTo obtain (2.8), we need to show that /bardbly/bardblL2(T1,T2)/ne}ationslash= 0 forT2−T1> L. Actually, it follows from\n(2.13) that\n/bardbly/bardbl2\nL2(T1,T2)=/integraldisplayT2\nT1/vextendsingle/vextendsingle/vextendsingleef(q)tPL(t)/vextendsingle/vextendsingle/vextendsingle2\ndt≥C3/integraldisplayT2\nT1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nn∈Neiµnt/an}bracketle{tx0,ψn/an}bracketri}htCφn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndt, (2.16)\n4whereC3= min/braceleftbig\ne2T1f(q),e2T2f(q)/bracerightbig\n>0. By Lemma 2.1 and the gap condition (2.12), it follows\nthat forT2−T1>2π\nγ=L,\n/integraldisplayT2\nT1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nn∈Neiµnt/an}bracketle{tx0,ψn/an}bracketri}htCφn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndt≥C1κ2/summationdisplay\nn∈N|/an}bracketle{tx0,ψn/an}bracketri}ht|2, (2.17)\nwhere\nC1=2(T2−T1)\nπ/parenleftbigg\n1−L2\n(T2−T1)2/parenrightbigg\n>0 forT2−T1>L.\nThe inequality (2.16) together with (2.17) gives\n/bardbly/bardbl2\nL2(T1,T2)≥C1C3κ2/summationdisplay\nn∈N|/an}bracketle{tx0,ψn/an}bracketri}ht|2. (2.18)\nNotice that {φn}n∈Nforms a Riesz basis for Hand so does {ψn}n∈NforH, there are two positive\nnumbersM1andM2such that\nM1/summationdisplay\nn∈N|/an}bracketle{tx0,ψn/an}bracketri}ht|2≤ /bardblx0/bardbl2≤M2/summationdisplay\nn∈N|/an}bracketle{tx0,ψn/an}bracketri}ht|2. (2.19)\nCombining (2.18) with (2.19) yields\n/bardbly/bardblL2(T1,T2)≥C/bardblx0/bardbl>0, (2.20)\nwhereC=κ/radicalBig\nC1C3\nM2>0. The identity (2.8) then follows from (2.15).\nThe inequality (2.20) means that system (2.4) is exactly obs ervable for T2−T1> L. So the\ninitial value x0can be uniquely determined by the output y(t),t∈[T1,T2]. We show next how to\nreconstruct the initial value from the output.\nActually, it follows from (2.11) that\n1\nL/integraldisplayL\n0ei(µm−µn)tdt=δnm, (2.21)\nHence,/integraldisplayL\n0y(t)e−λntdt=/integraldisplayL\n0/parenleftBigg/summationdisplay\nm∈Nei(µm−µn)t/an}bracketle{tx0,ψm/an}bracketri}htCφm/parenrightBigg\ndt=κnL·/an}bracketle{tx0,ψn/an}bracketri}ht,(2.22)\nTherefore the initial value x0can be reconstructed by\nx0=/summationdisplay\nn∈N/an}bracketle{tx0,ψn/an}bracketri}htφn=1\nL/summationdisplay\nn∈N1\nκn/parenleftbigg/integraldisplayL\n0y(t)e−λntdt/parenrightbigg\nφn. (2.23)\nThis completes the proof of the theorem.\nRemark 2.1. Clearly, (2.8) and (2.9) provide an algorithm to reconstruc tqandx0from the\noutput. It seems that the condition (2.6) is restrictive but it is satisfied by some physical systems\ndiscussed in Sections 3-5. Condition (2.6) is only for ident ification of q. For identification of initial\nvalue only, this condition can be removed. From numerical st andpoint, the function PL(t) in (2.13)\n5can be approximated by the finite series in (2.13) with the firs tNterms for sufficiently large N.\nHence condition (2.6) can be relaxed in numerical algorithm to be\nC1.There exists an Lsuch that: everyµnL\n2πis equal to (or close to) some integer for n∈\n{1,2,···,N},for some sufficiently large N.\nObviously, the relaxed condition C1 can still ensure that PL(t) is close to a function of period\nL. In this case, some points µnmay be very close to each other and the corresponding Riesz ba sis\nproperty of the family of divided differences of exponentials eiµntdeveloped in [1, Section II.4] and\n[2, 3] can be used. For the third condition, |Cφn| ≤Kimplies that Cis admissible for Tqwhich\nensures that the output belongs to L2\nloc(0,∞;Y), and|Cφn| ≥κimplies that system (2.4) is exactly\nobservable which ensures the unique determination of the in itial value. It is easily seen from (2.15)\nthat the coefficient qcan always be identified as long as /bardbly/bardblL2(T1,T2)/ne}ationslash= 0 for some time interval\n[T1,T2], which shows that the identifiability of coefficient qdoes not rely on the exact observability\nyet approximate observability.\nRemark 2.2. The condition T2−T1>Lin Theorem 2.1 is only used in application of Ingham’s\ninequalityin(2.17)toensurethat /bardbly/bardblL2(T1,T2)/ne}ationslash= 0. Inpracticalapplications, however, thiscondition\nis not always necessary. Actually, any L<T1<T2is applicable in (2.8) as long as /bardbly/bardblL2(T1,T2)/ne}ationslash= 0.\nSimilar remark also applies for Theorem 2.2 below.\nRemark 2.3. It should be noted that for identification of damping coefficie nt in [4, 5, 6], the\ndistributed observations are always required. In Theorem 2 .1, however, we use only boundary\nmeasurement and our identification algorithm utilizes phys ics of the system that the anti-damping\ncoefficient can make the measurement have an exponential term in (2.13). We should also point\nout that identification of the damping or anti-damping coeffic ientqin Theorem 2.1 does not rely on\nthe knowledge of the initial value. Actually, after qbeing estimated, there are various methods for\ninitial value reconstruction, see, e.g. [18, 21] and the ref erences therein. The idea of the algorithm\nfor reconstruction of the initial value here is borrowed fro m the Riesz basis approach proposed in\n[21].\nNow we come to the system with external disturbance which is i nevitable in many situations.\nSuppose that system (1.1) is corrupted by an unknown general bounded disturbance d(t) in obser-\nvation. It should be noted that system (1.1) is supposed to be anti-stable in Theorem 2.2 below\nwhereas in Theorem 2.1, there is no constraint on the stabili ty of system.\nTheorem 2.2. Suppose that system (1.1) is anti-stable and all the conditi ons in Theorem 2.1 are\nsatisfied. If the inverse of f(q)is continuous and the disturbance d(t)is bounded, i.e. |d(t)| ≤M\nfor someM >0and allt≥0, then for any T2−L>T1>L,\nlim\nT1→+∞qT1=q,lim\nT1→+∞/bardblˆx0T1−x0/bardbl= 0, (2.24)\nwhere\nqT1=f−1/parenleftBigg\n1\nLln/bardbly/bardblL2(T1,T2)\n/bardbly/bardblL2(T1−L,T2−L)/parenrightBigg\n, (2.25)\n6and\nˆx0T1=1\nL/summationdisplay\nn∈N1\nκn/parenleftbigg/integraldisplayT1+L\nT1y(t)e−λntdt/parenrightbigg\nφn, T1≥0. (2.26)\nMoreover, for sufficiently large T1, the errors |f(qT1)−f(q)|and/bardblˆx0T1−x0/bardblsatisfy\n|f(qT1)−f(q)|<4\nLM√T2−T1\n/bardbly/bardblL2(T1−L,T2−L)−M√T2−T1, (2.27)\nand\n/bardblˆx0T1−x0/bardbl ≤CM\nκ√\nLe−f(q)T1for someC >0. (2.28)\nProof.Introduce\nye(t) =CTq(t)x0=y(t)−d(t) =ef(q)tPL(t), (2.29)\nwherePL(t) is defined in (2.13). We first show that\nlim\nT1→+∞/bardblye/bardblL2(T1,T2)= +∞. (2.30)\nSince system (1.1) is anti-stable, the real part of the eigen valuesf(q)>0. It then follows from\n(2.29) that\n/bardblye/bardbl2\nL2(T1,T2)=/integraldisplayT2\nT1/vextendsingle/vextendsingle/vextendsingleef(q)tPL(t)/vextendsingle/vextendsingle/vextendsingle2\ndt≥e2f(q)T1/integraldisplayT2\nT1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/summationdisplay\nn∈Neiµnt/an}bracketle{tx0,ψn/an}bracketri}htCφn/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\ndt. (2.31)\nUsing the same arguments as (2.16)-(2.20) in the proof of The orem 2.1, we have\n/bardblye/bardblL2(T1,T2)≥Cef(q)T1/bardblx0/bardbl, (2.32)\nwhereC=κ/radicalBig\nC1\nM2>0. Sincef(q)>0, x0/ne}ationslash= 0,(2.30) holds. Therefore for sufficiently large T1,\n/bardbly/bardblL2(T1,T2)\n/bardbly/bardblL2(T1−L,T2−L)=/bardblye+d/bardblL2(T1,T2)\n/bardblye+d/bardblL2(T1−L,T2−L)≤/bardblye/bardblL2(T1,T2)+/bardbld/bardblL2(T1,T2)\n/bardblye/bardblL2(T1−L,T2−L)−/bardbld/bardblL2(T1−L,T2−L).(2.33)\nSince|d(t)| ≤M, for any finite time interval I,\n/bardbld/bardblL2(I)=/parenleftbigg/integraldisplay\nI|d(t)|2dt/parenrightbigg1\n2\n≤M/radicalbig\n|I|, (2.34)\nwhere|I|represents the length of the time interval I. Hence\n/bardbly/bardblL2(T1,T2)\n/bardbly/bardblL2(T1−L,T2−L)≤eLf(q)+ε(T1,T2)\n1−ε(T1,T2), (2.35)\nwhere\nε(T1,T2) =M√T2−T1\n/bardblye/bardblL2(T1−L,T2−L). (2.36)\nSimilarly,\n/bardbly/bardblL2(T1,T2)\n/bardbly/bardblL2(T1−L,T2−L)≥eLf(q)−ε(T1,T2)\n1+ε(T1,T2). (2.37)\n7It is clear from (2.30) and (2.36) that lim T1→+∞ε(T1,T2) = 0. This together with (2.35) and (2.37)\ngives\nlim\nT1→+∞/bardbly/bardblL2(T1,T2)\n/bardbly/bardblL2(T1−L,T2−L)=eLf(q). (2.38)\nSincef−1(q) is continuous,\nlim\nT1→+∞qT1=f−1/parenleftBigg\n1\nLln lim\nT1→+∞/bardbly/bardblL2(T1,T2)\n/bardbly/bardblL2(T1−L,T2−L)/parenrightBigg\n=q.\nWe next show convergence of the initial value. Similarly wit h the arguments (2.21)-(2.23) in\nthe proof of Theorem 2.1, we have\nx0=1\nL/summationdisplay\nn∈N1\nκn/parenleftbigg/integraldisplayT1+L\nT1ye(t)e−λntdt/parenrightbigg\nφn,∀T1≥0.\nIt then follows from (2.26) that for arbitrary T1≥0,\nˆx0T1−x0=1\nL/summationdisplay\nn∈N1\nκn/parenleftbigg/integraldisplayT1+L\nT1d(t)e−λntdt/parenrightbigg\nφn. (2.39)\nIn view of the Riesz basis property of {φn}, it follows that\n/bardblˆx0T1−x0/bardbl2=/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble1\nL/summationdisplay\nn∈N1\nκn/parenleftbigg/integraldisplayT1+L\nT1d(t)e−λntdt/parenrightbigg\nφn/vextenddouble/vextenddouble/vextenddouble/vextenddouble/vextenddouble2\n≤M2\nL2κ2e−2f(q)T1/summationdisplay\nn∈N/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayL\n0/parenleftBig\nd(t+T1)e−f(q)t/parenrightBig\ne−iµntdt/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n,(2.40)\nwhereM2>0 is introduced in (2.19). To estimate the last series in (2.4 0), we need the Riesz\nbasis (sequence) property of the exponential system Λ :=/braceleftbig\nfn=eiµnt/bracerightbig\nn∈N. There are two cases\naccording to the relation between the sets {Kn}n∈Nintroduced in (2.11) and integers Z:\nCase 1: {Kn}n∈N=Z, that is, Λ =/braceleftBig\nei2nπ\nLt/bracerightBig\nn∈Z. In this case, since/braceleftbig\neint/bracerightbig\nn∈Zforms a Riesz\nbasis forL2[−π,π], Λ forms a Riesz basis for L2[−L\n2,L\n2].\nCase 2: {Kn}n∈N/subsetnoteql Z. In this case, it is noted that the exponential system/braceleftbig\neiµnt/bracerightbig\nn∈Nforms\na Riesz sequence in L2[−L\n2,L\n2].\nIn each case above, by properties of Riesz basis and Riesz seq uence (see, e.g., [23, p. 32-35,\np.154]), there exists a positive constant C4>0 such that\n/summationdisplay\nn∈N|(g,fn)|2≤C4/bardblg/bardbl2\nL2[−L\n2,L\n2], (2.41)\nfor allg∈L2[−L\n2,L\n2], where ( ·,·) denotes the inner product in L2[−L\n2,L\n2].\nWe return to the estimation of /bardblˆx0T1−x0/bardbl. By variable substitution of t=L\n2−sin (2.40),\ntogether with (2.41), we have\n/bardblˆx0T1−x0/bardbl2≤M2\nL2κ2e−2f(q)T1/summationdisplay\nn∈N/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplayL\n2\n−L\n2/bracketleftbigg\nd/parenleftbigg\nT1+L\n2−s/parenrightbigg\nef(q)(s−L\n2)e−iµnL\n2/bracketrightbigg\neiµnsds/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\n≤M2C4\nL2κ2e−2f(q)T1/integraldisplayL\n2\n−L\n2/vextendsingle/vextendsingle/vextendsingle/vextendsingled/parenleftbigg\nT1+L\n2−s/parenrightbigg\nef(q)(s−L\n2)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2\nds\n≤M2M2C4\nLκ2e−2f(q)T1.\n8Therefore,\n/bardblˆx0T1−x0/bardbl ≤/radicalbigg\nM2C4\nLM\nκe−f(q)T1, (2.42)\nwhich implies that /bardblˆx0T1−x0/bardblwill tend to zero as T1→+∞forf(q)>0. The inequality (2.28)\nwith the positive number C=√M2C4is also concluded.\nFinally, we estimate |f(qT1)−f(q)|. SettingT1large enough so that ε(T1,T2)<1, it follows\nfrom (2.25) and (2.35) that\nLf(qT1)≤lneLf(q)+ε(T1,T2)\n1−ε(T1,T2)<Lf(q)+2ε(T1,T2)\n1−ε(T1,T2). (2.43)\nSimilarly, for T1large enough so that ε(T1,T2)≤1\n4, it follows from (2.25) and (2.37) that\nLf(qT1)>Lf(q)−4ε(T1,T2)\n1+ε(T1,T2). (2.44)\nCombining (2.43) and (2.44), and setting T1large enough so that ε(T1,T2)≤1\n4, we have\n|f(qT1)−f(q)|<4ε(T1,T2)\nL.\nThe error estimation (2.27) comes from the fact\nε(T1,T2)≤M√T2−T1\n/bardbly/bardblL2(T1−L,T2−L)−M√T2−T1. (2.45)\nWe thus complete the proof of the theorem.\nRemark 2.4. Theorem 2.2 shows that when system (1.1) is anti-stable, the nqT1defined in (2.25)\ncan be regarded as an approximation of the coefficient qwhenT1is sufficiently large. Roughly\nspeaking, the ε(T1,T2) defined in (2.36) reflects the ratio of the energy, in L2norm, of the distur-\nbanced(t) which is an unwanted signal, with the energy of the real outp ut signalye(t). We may\nregard 1/ε(T1,T2) as signal-to-noise ratio (SNR) which is well known in signa l analysis. Theorem\n2.2 indicates that qT1defined in (2.25) is an approximation of the coefficient qwhen SNR is large\nenough. However, if system (1.1) is stable, i.e.f(q)<0, similar analysis shows that the output\nwill be exponentially decaying oscillation, which implies that the unknown disturbance will account\nfor a large proportion in observation and the SNR can not be to o large. In this case, it is difficult\nto extract enough useful information from the corrupted obs ervation as that with large SNR.\nRemark 2.5. Theanti-stability assumptionin Theorem 2.2 is almost nece ssary since otherwise, we\nmay have the case of y(t) =Cx(t)+d(t)≡0 for which we cannot obtain anything for identification.\nRemark 2.6. It is well known that the inverse problems are usually ill-po sed in the sense of\nHadamard, that is, arbitrarily small error in the measureme nt data may lead to large error in\nsolution. Theorem 2.2 shows that if system (1.1) is anti-sta ble, our algorithm is robust against\nbounded unknown disturbance in measurement data. Actually , similar to the analysis in Theorem\n2.2, it can be shown that when system (1.1) is not anti-stable , the algorithm in Theorem 2.1 is also\nnumerically stable in the presence of small perturbations i n the measurement data, as long as the\nperturbation is relatively small in comparison to the outpu t. Some numerical simulations validate\nthis also in Example 3.1 in Section 3.\n93 Application to wave equation\nIn this section, we apply the algorithm proposed in previous section to identification of the anti-\ndamping coefficient and initial values for a one-dimensional vibrating string equation described by\n([7, 16])\n\nutt=uxx, 0<x<1, t>0,\nu(0,t) = 0, ux(1,t) =qut(1,t), t≥0,\ny(t) =ux(0,t)+d(t), t ≥0,\nu(x,0) =u0(x), ut(x,0) =u1(x),0≤x≤1,(3.1)\nwherexdenotes theposition, tthetime, 0 <q/ne}ationslash= 1theunknownanti-dampingcoefficient, u0(x) and\nu1(x) the unknown initial displacement and initial velocity, re spectively, and y(t) is the boundary\nmeasured output corrupted by the disturbance d(t).\nLetH=H1\nE(0,1)×L2(0,1), whereH1\nE(0,1) ={f∈H1(0,1)|f(0) = 0}, equipped with the\ninner product /an}bracketle{t·,·/an}bracketri}htand the inner product induced norm\n/bardbl(f,g)/bardbl2=/integraldisplay1\n0/bracketleftbig\n|f′(x)|2+|g(x)|2/bracketrightbig\ndx.\nDefine the system operator A:D(A)(⊂ H)→ Has\n/braceleftBigg\nA(f,g) = (g,f′′),\nD(A) =/braceleftbig\n(f,g)∈H2(0,1)×H1\nE(0,1)|f(0) = 0,f′(1) =qg(1)/bracerightbig\n,(3.2)\nand the observation operator CfromHtoCas\nC(f,g) =f′(0),(f,g)∈D(A). (3.3)\nIt is indicated in [21] that the operator Agenerates a C0-group on H.\nLemma 3.1. [21] LetAbe defined by (3.2) and let q/ne}ationslash= 1. Then the spectrum of Aconsists of all\nisolated eigenvalues given by\nλn=1\n2ln1+q\nq−1+inπ, n∈Z, if q> 1, (3.4)\nor\nλn=1\n2ln1+q\n1−q+i2n+1\n2π, n∈Z, if0<q<1, (3.5)\nand the corresponding eigenfunctions Φn(x)are given by\nΦn(x) =/parenleftbiggsinhλnx\nλn,sinhλnx/parenrightbigg\n,∀n∈Z. (3.6)\nMoreover, {Φn(x)}n∈Zforms a Riesz basis for H.\nLemma 3.2. [21] Let Abe defined by (3.2) and let q/ne}ationslash= 1. Then the adjoint operator A∗ofAis\ngiven by\n/braceleftBigg\nA∗(v,h) =−(h,v′′),\nD(A∗) =/braceleftbig\n(v,h)∈H2(0,1)×H1(0,1)|v(0) = 0,v′(1)+qh(1) = 0/bracerightbig\n,(3.7)\n10andσ(A∗) =σ(A). The eigenvector Ψn(x)ofA∗corresponding to λnis given by\nΨn(x) =/parenleftbiggsinhλnx\nλn,−sinhλnx/parenrightbigg\n,∀n∈Z. (3.8)\nIt is easy to verify that for any n,m∈Z,/an}bracketle{tΦn,Ψm/an}bracketri}ht=δnm.System (3.1) can be written as the\nfollowing evolutionary equation in H:\ndX(t)\ndt=AX(t), t>0, X(0) = (u0,u1), (3.9)\nwhereX(t) = (u(·,t),ut(·,t)), and the solution of (3.9) is given by\nX(t) =/summationdisplay\nn∈Zeλnt/an}bracketle{tX(0),Ψn/an}bracketri}htΦn. (3.10)\nThus\ny(t) =/summationdisplay\nn∈Zeλnt/an}bracketle{tX(0),Ψn/an}bracketri}ht+d(t). (3.11)\nIt can be seen from Lemma 3.1 that when q= 1, the real part of the eigenvalues is + ∞, while\nfor 0<q/ne}ationslash= 1, the real part is finite positive. Hence, we suppose 1 /∈Qas usual (see, e.g., [7, 16]),\nwhereQ= [q,q] is the prior parameter set.\nWe takeq∈Q= (1,+∞) as an example to illustrate how to apply the algorithms prop osed in\nprevious section to simultaneous identification for the ant i-damping coefficient qand initial values.\nThe following Corollaries 3.1-3.2 are the direct consequen ces of Theorem 2.1 and Theorem 2.2,\nrespectively, by noticing that for system (3.1), the releva nt function and parameters now are\nf(q) =1\n2lnq+1\nq−1, µn=nπ, L= 2, κn= 1. (3.12)\nCorollary 3.1. Suppose that d(t) = 0in system (3.1). Then both the coefficient qand initial values\nu0(x)andu1(x)can be uniquely determined by the output y(t),t∈[0,T], whereT >4. Specifically,\nqcan be recovered exactly from\nq=/bardbly/bardblL2(T1,T2)+/bardbly/bardblL2(T1−2,T2−2)\n/bardbly/bardblL2(T1,T2)−/bardbly/bardblL2(T1−2,T2−2),2≤T1<T2−2, (3.13)\nand the initial values u0(x)andu1(x)can be reconstructed from\nu0(x) =1\n2/summationdisplay\nn∈Z/parenleftbigg/integraldisplay2\n0y(t)e−λntdt/parenrightbiggsinhλnx\nλn, u1(x) =1\n2/summationdisplay\nn∈Z/parenleftbigg/integraldisplay2\n0y(t)e−λntdt/parenrightbigg\nsinhλnx.(3.14)\nNotice that in (3.14), the observation interval [0 ,2] is the minimal time interval for observation\nto identify the initial values for any identification algori thm.\nCorollary 3.2. Suppose that q∈Q= (1,+∞)in system (3.1) and the disturbance is bounded, i.e.\n|d(t)| ≤Mfor someM >0and allt≥0. Then for any T2−2>T1≥2,\nlim\nT1→+∞qT1=q,lim\nT1→+∞/bardbl(ˆu0T1,ˆu1T1)−(u0,u1)/bardbl= 0, (3.15)\n11where\nqT1=/bardbly/bardblL2(T1,T2)+/bardbly/bardblL2(T1−2,T2−2)\n/bardbly/bardblL2(T1,T2)−/bardbly/bardblL2(T1−2,T2−2), (3.16)\nand\nˆu0T1(x) =1\n2/summationdisplay\nn∈Z/parenleftbigg/integraldisplayT1+2\nT1y(t)e−λntdt/parenrightbiggsinhλnx\nλn,\nˆu1T1(x) =1\n2/summationdisplay\nn∈Z/parenleftbigg/integraldisplayT1+2\nT1y(t)e−λntdt/parenrightbigg\nsinhλnx.(3.17)\nTo end this section, we present some numerical simulations f or system (3.1) to illustrate the\nperformance of the algorithm.\nExample 3.1. The observation with random noises when system (1.1) is stabl e.\nA simple spectral analysis together with Theorem 2.1 shows t hat Corollary 3.1 is also valid for\nq∈Q= (−∞,−1). In this example, the damping coefficient qand initial values u0(x),u1(x) are\nchosen as\nq=−3, u0(x) =−3sinπx, u 1(x) =πcosπx. (3.18)\nIn this case, the output can be obtained from (3.11) (with d(t) = 0), where the infinite series is\napproximated by a finite one, that is, {n∈Z}is replaced by {n∈Z|−5000≤n≤5000}. Some\nrandom noises are added to the measurement data and we use the se data to test the algorithm\nproposed in Corollary 3.1.\nLetT1= 2,T2= 2.5. Then the damping coefficient qcan be recovered from (3.13), and the\ninitial values u0(x) andu1(x) can be reconstructed from (3.14). Table 1 lists the numeric al results\nfor the damping coefficients (the second column in Table 1) and Figure 1(a)-1(c) for the initial\nvalues in various cases of noise levels. In Table 1, the absol ute errors of the real damping coefficient\nand the recovered ones, and the L2-norm of the differences between the exact initial values and t he\nreconstructed ones are also shown.\nIt is worth pointing out that in reconstruction of the initia l values from (3.14), the infinite series\nis approximated by a finite one once again, that is, {n∈Z}is replaced by {n∈Z| |n| ≤1000},\nwhich accounts for the zero value of the reconstructed initi al velocity at the left end. This is also\nthe reason that the errors of the initial velocity (the last c olumn in Table 1) are relatively large\neven if there is no random noise in the measured data.\nTable 1: Absolute errors with different noise levels\nNoise Level Recovered qErrors forqErrors foru0(x) Errors for u1(x)\n0 -3.0000 9.3259E-15 1.1744E-08 2.2215E-01\n1% -2.9994 6.2498E-04 1.1618E-03 2.2748E-01\n3% -2.9979 2.0904E-03 3.5604E-03 2.6662E-01\n120 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í3í2.5í2í1.5í1í0.50\nxinitial displacementreal u0(x)\nrecovered\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í4í3í2í101234\nxinitial velocityreal u1(x)\nrecovered\n(a) without random noise0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í3.5í3í2.5í2í1.5í1í0.50\nxinitial displacementreal u0(x)\nrecovered\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í4í3í2í101234\nxinitial velocityreal u1(x)\nrecovered\n(b) with 1% random error\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í3.5í3í2.5í2í1.5í1í0.50\nxinitial displacementreal u0(x)\nrecovered\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í4í3í2í101234\nxinitial velocityreal u1(x)\nrecovered\n(c) with 3% random error\nFigure 1: The initial values: initial displacement (upper) and initial velocity (lower)\nExample 3.2. The observation with general bounded disturbance when syste m (1.1) is anti-stable.\nThe anti-damping coefficient and initial values are chosen as\nq= 3, u0(x) = 3sinπx, u 1(x) =πcosπx. (3.19)\nand the observation is corrupted by the bounded disturbance :\nd(t) = 2sin1\n1+t+3cos10t. (3.20)\nThe relevant parameters in Corollary 3.2 are chosen to be T2=T1+3, and let T1be different values\nincreasing from 2 to 10. The corresponding anti-damping coe fficientsqT1recovered from (3.16) are\ndepicted in Figure 2. It is seen that qT1converges to the real value q= 3 asT1increases. Setting\nT1= 0,3,7 in (3.17) and reconstructing the initial values produce re sults in Figure 2 from which\nwe can see that the reconstructed initial values become clos er to the real ones as T1increases.\n132 3 4 5 6 7 8 9 102.9533.053.13.153.2\nT1Real and recovered antidamping coefficient\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 101234\nxInitial dispacement u0(x)\nreal u0(x)\nT1=0\nT1=3\nT1=7\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í505\nxInitial velocity u1(x)\nreal u1(x)\nT1=0\nT1=3\nT1=7\nFigure 2: anti-damping coefficient qand initial values u0, u1\n4 Application to Schr¨ odinger equation\nIn this section, we consider a quantum system described by th e following Schr¨ odinger equation:\n\n\nut=−iuxx+qu, 0<x<1, t>0,\nux(0,t) = 0, u(1,t) = 0, t≥0,\ny(t) =u(0,t)+d(t), t ≥0,\nu(x,0) =u0(x), 0≤x≤1.(4.1)\nwhereu(x,t) is the complex-valued state, iis the imaginary unit, and the potential q>0 andu0(x)\nare the unknown anti-damping coefficient and initial value, r espectively.\nLetH=L2(0,1) be equipped with the usual inner product /an}bracketle{t·,·/an}bracketri}htand the inner product induced\nnorm/bardbl·/bardbl.Introduce the operator Adefined by\n/braceleftBigg\nAφ=−iφ′′+qφ,\nD(A) =/braceleftbig\nφ∈H2(0,1)|φ′(0) =φ(1) = 0/bracerightbig\n.(4.2)\nA straightforward verification shows that such defined Agenerates a C0-semigroup on H.\nLemma 4.1. [17] Let Abe defined by (4.2). Then the spectrum of Aconsists of all isolated\neigenvalues given by\nλn=q+i/parenleftbigg\nn−1\n2/parenrightbigg2\nπ2, n∈N, (4.3)\n14and the corresponding eigenfunctions φn(x)are given by\nφn(x) =√\n2cos/parenleftbigg\nn−1\n2/parenrightbigg\nπx, n∈N. (4.4)\nIn addition, {φn(x)}n∈Nforms an orthonormal basis for H.\nSystem (4.1) can be rewritten as the following evolutionary equation in H:\ndX(t)\ndt=AX(t), t>0, X(0) =u0, (4.5)\nand the solution of (4.5) is given by\nX(t) =/summationdisplay\nn∈Neλnt/an}bracketle{tX(0),φn/an}bracketri}htφn. (4.6)\nThus\ny(t) =√\n2/summationdisplay\nn∈Neλnt/an}bracketle{tu0,φn/an}bracketri}ht+d(t). (4.7)\nThe relevant function and parameters in Theorems 2.1-2.2 fo r system (4.1) are\nf(q) =q, µn=/parenleftbigg\nn−1\n2/parenrightbigg2\nπ2, L=8\nπ, κn=√\n2.\nParallel toSection 3, wehavetwocorollaries correspondin gtotheexact observation andobservation\nwith general bounded disturbance, respectively, for syste m (4.1). Here we only list the latter one\nand the former is omitted.\nCorollary 4.1. Suppose that q∈Q= (0,+∞)in system (4.1) and the disturbance is bounded, i.e.\n|d(t)| ≤Mfor someM >0and allt≥0. Then for any T2−8\nπ>T1>8\nπ,\nlim\nT1→+∞qT1=q,lim\nT1→+∞/bardblˆu0T1−u0/bardbl= 0, (4.8)\nwhere\nqT1=π\n8ln/bardbly/bardblL2(T1,T2)\n/bardbly/bardblL2(T1−8\nπ,T2−8\nπ),8\nπ<T1<T2−8\nπ, (4.9)\nand\nˆu0T1(x) =π\n8/summationdisplay\nn∈Z/parenleftBigg/integraldisplayT1+8\nπ\nT1y(t)e−λntdt/parenrightBigg\ncos/parenleftbigg\nn−1\n2/parenrightbigg\nπx. (4.10)\nWe also give a numerical simulation to test the algorithm pro posed in Corollary 4.1 for system\n(4.1), where the anti-damping coefficient qand initial value u0(x) are chosen as\nq= 0.7, u0(x) = sinπx+icosπx, (4.11)\nand the observation is corrupted by the disturbance\nd(t) = 2sint\n10+t+3icos20t. (4.12)\n15The observation can be obtained from (4.7) by a finite series a pproximation, that is, {n∈N}\nis replaced by {n∈N|n≤5000}. The relevant parameters in Corollary 4.1 are chosen to be\nT2=T1+1, andT1increasing from 2 .55 to 10. The corresponding anti-damping coefficients qT1\nrecovered from (4.9) are shown in Figure 3. It is obvious that qT1is convergent to the real value\nq= 0.7 asT1increases. Setting T1= 0,3,7, respectively, in (4.10), the reconstructed initial valu es\nare shown in Figure 3 from which it is seen that the errors betw een the reconstructed initial values\nand the real ones become smaller as T1increases.\n3 4 5 6 7 8 9 100.60.650.7\nT1Real and recovered antidamping coefficient\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í0.500.511.5\nxReal part of initial value\nRe(u0)\nT1=0\nT1=3\nT1=7\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í1.5í1í0.500.511.5\nxImaginary part of initial value\nIm(u0)\nT1=0\nT1=3\nT1=7\nFigure 3: coefficient (upper), real part (middle) and imagina ry part (lower) of initial value u0(x)\n5 Application to coupled strings equation\nIn this section, we consider the following two connected ant i-stable strings with joint anti-damping\ndescribed by \n\nutt(x,t) =uxx(x,t),x∈(0,1\n2)∪(1\n2,1), t>0,\nu/parenleftBig\n1\n2−,t/parenrightBig\n=u/parenleftBig\n1\n2+,t/parenrightBig\n, t≥0,\nux/parenleftBig\n1\n2−,t/parenrightBig\n−ux/parenleftBig\n1\n2+,t/parenrightBig\n=qut(1,t), t≥0,\nu(0,t) =ux(1,t) = 0, t≥0,\nu(x,0) =u0(x),ut(x,0) =u1(x),0≤x≤1,\ny(t) =ux(0,t)+d(t), t≥0,(5.1)\nwhereq >0,q/ne}ationslash= 2 is the unknown anti-damping constant. System (5.1) model s two connected\nstrings with joint vertical force anti-damping, see [10, 11 , 22] for more details.\n16LetH=H1\nE(0,1)×L2(0,1) be equipped with the inner product /an}bracketle{t·,·/an}bracketri}htand its induced norm\n/bardbl(u,v)/bardbl2=/integraldisplay1\n0/bracketleftbig\n|u′(x)|2+|v(x)|2/bracketrightbig\ndx,\nwhereH1\nE(0,1) =/braceleftbig\nu|u∈H1(0,1),u(0) = 0/bracerightbig\n. Then system (5.1) can be rewritten as an evolution-\nary equation in Has follows:\nd\ndtX(t) =AX(t), (5.2)\nwhereX(t) = (u(·,t),ut(·,t))∈ HandAis defined by\nA(u,v) = (v(x),u′′(x)), (5.3)\nwith the domain\nD(A) =\n\n(u,v)∈H1(0,1)×H1\nE(0,1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleu(0) =u′(1) = 0, u|[0,1\n2]∈H2(0,1\n2),\nu|[1\n2,1]∈H2(1\n2,1), u′(1\n2−)−u′(1\n2+) =qv(1\n2),\n\n,(5.4)\nwhereu|[a,b]denotes the function u(x) confined to [ a,b].\nWe assume without loss of generality that the prior paramete r set forqisQ= (2,+∞) since\nthe case for Q= (0,2) is very similar.\nLemma 5.1. [22] LetAbe defined by (5.3)-(5.4) and q∈Q= (2,+∞). ThenA−1is compact on\nHand the eigenvalues of Aare algebraically simple and separated, given by\nλn=1\n2lnq+2\nq−2+inπ, n∈Z. (5.5)\nThe corresponding eigenfunctions Φn(x)are given by\nΦn(x) = (φn(x),λnφn(x)),∀n∈Z, (5.6)\nwhere\nφn(x) =\n\n√\n2\nλncoshλn\n2sinhλnx, 0<x<1\n2,\n√\n2\nλnsinhλn\n2coshλn(1−x),1\n2<x<1.\nand{Φn(x)}n∈Zforms a Riesz basis for H. In addition, Agenerates a C0-semigroup on H.\nLemma 5.2. [22] LetAbe defined by (5.3)-(5.4) and q∈Q= (2,+∞). Then the adjoint operator\nA∗ofAis given by\nA∗(u,v) =−(v,u′′), (5.7)\nwith the domain\nD(A∗) =\n\n(u,v)∈H1(0,1)×H1\nE(0,1)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleu(0) =u′(1) = 0, u|[0,1\n2]∈H2(0,1\n2),\nu|[1\n2,1]∈H2(1\n2,1), u′(1\n2−)−u′(1\n2+) =−qv(1\n2)\n\n,\nandσ(A∗) =σ(A). The eigenfunctions Ψn(x)ofA∗corresponding to λnare given by\nΨn(x) =/parenleftbig\nφn(x),−λnφn(x)/parenrightbig\n,∀n∈Z. (5.8)\n17A direct calculation shows that {Ψn(x)}is biorthogonal to {Φn(x)}. Hence, the solution of\n(5.2) can be expressed as\nX(t) =/summationdisplay\nn∈Zeλnt/an}bracketle{tX(0),Ψn/an}bracketri}htΦn. (5.9)\nDefine the observation operator CfromHtoCto be\nC(f,g) =f′(0),(f,g)∈D(A). (5.10)\nThen\ny(t) =/summationdisplay\nn∈Zeλnt/an}bracketle{tX(0),Ψn/an}bracketri}htCΦn+d(t). (5.11)\nThe succeeding Corollary 5.1 is a direct consequence of Theo rem 2.2 by noticing that for system\n(5.1), the relevant function and parameters now are\nf(q) =1\n2lnq+2\nq−2, µn=nπ, L= 2, κn=√\n2coshλn\n2,\nand a simple calculation shows that\n√\n2\n2/bracketleftBigg/parenleftbiggq+2\nq−2/parenrightbigg1\n4\n−/parenleftbiggq−2\nq+2/parenrightbigg1\n4/bracketrightBigg\n≤ |κn| ≤√\n2\n2/bracketleftBigg/parenleftbiggq+2\nq−2/parenrightbigg1\n4\n+/parenleftbiggq−2\nq+2/parenrightbigg1\n4/bracketrightBigg\n,∀n∈Z.\nThe corollaries corresponding to Theorem 2.1 are omitted he re.\nCorollary 5.1. Suppose that q∈Q= (2,+∞)in system (5.1) and the disturbance is bounded, i.e.\n|d(t)| ≤Mfor someM >0and allt≥0. Then for any T2−2>T1≥2,\nlim\nT1→+∞qT1=q,lim\nT1→+∞/bardbl(ˆu0T1,ˆu1T1)−(u0,u1)/bardbl= 0,\nwhere\nqT1=2/parenleftbig\n/bardbly/bardblL2(T1,T2)+/bardbly/bardblL2(T1−2,T2−2)/parenrightbig\n/bardbly/bardblL2(T1,T2)−/bardbly/bardblL2(T1−2,T2−2), (5.12)\nand\nˆu0T1(x) =\n\n1\n2/summationdisplay\nn∈Z/parenleftbigg/integraldisplayT1+2\nT1y(t)e−λntdt/parenrightbiggsinhλnx\nλn, 0<x<1\n2,\n1\n2/summationdisplay\nn∈Z/parenleftbigg/integraldisplayT1+2\nT1y(t)e−λntdt/parenrightbigg\ntanhλn\n2coshλn(1−x)\nλn,1\n2<x<1.\nˆu1T1(x) =\n\n1\n2/summationdisplay\nn∈Z/parenleftbigg/integraldisplayT1+2\nT1y(t)e−λntdt/parenrightbigg\nsinhλnx, 0<x<1\n2,\n1\n2/summationdisplay\nn∈Z/parenleftbigg/integraldisplayT1+2\nT1y(t)e−λntdt/parenrightbigg\ntanhλn\n2coshλn(1−x),1\n2<x<1.(5.13)\nAs before, we present some numerical simulations for system (5.1) to showcase the effectiveness\nof the algorithm proposed in Corollary 5.1. The anti-dampin g coefficient qand initial values u0(x)\nandu1(x) are chosen to be\nq= 3, u0(x) = sinx, u1(x) = cosx, (5.14)\n18and the observation is corrupted by the disturbance\nd(t) = sint2\n10+t+cos10t. (5.15)\nTheobservation is obtained from(5.11) by afiniteseries app roximation, that is, {n∈N}is replaced\nby{n∈N| |n| ≤5000}. The relevant parameters in Corollary 5.1 are chosen to be T2=T1+1, and\nT1increases from 2 to 8. The corresponding anti-damping coeffic ientsqT1recovered from (5.12)\nare plotted in Figure 4. It can be seen that qT1converges to the real value q= 3 asT1increases.\nLetT1= 0,3,7 in (5.13), and the reconstructed initial values are shown i n Figure 4, from which\nit is seen that the reconstructed initial values become clos er to the real ones as T1increases.\n2 3 4 5 6 7 82.9533.053.13.15\nT1Real and recovered antidamping coefficient\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 100.20.40.60.81\nxInitial dispacement u0(x)\nreal u0(x)\nT1=0\nT1=3\nT1=7\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1í10123\nxInitial velocity u1(x)\nreal u1(x)\nT1=0\nT1=3\nT1=7\nFigure 4: anti-damping coefficient q(upper), initial values u0(middle) and u1(lower)\n6 Concluding remarks\nIn this paper, we propose an algorithm to reconstruct simult aneously an anti-damping coefficient\nand an initial value for some anti-stable PDEs. When the meas ured output is exact, the recovered\nvalues are exact whereas if the measured output suffers from bo unded unknown disturbance, the\napproximated values of the anti-damping coefficient and init ial value can also be obtained. Some\nnumericalexamplesarecarriedouttovalidatetheeffectiven ess ofthealgorithm. Itisverypromising\nto apply the algorithm presented here to stabilization of an ti-stable systems with an unknown anti-\ndamping coefficient.\n19Acknowledgements\nThis work was supported by the National Natural Science Foun dation of China and the National\nResearch Foundation of South Africa.\nReferences\n[1] S.A. Avdonin and S.A. Ivanov, Families of Exponentials: The Method of Moments in Control-\nlability Problems for Distributed Parameter Systems , Cambridge University Press, 1995.\n[2] S.A. Avdonin and W. Moran, Ingham-type inequalities and Riesz bases of divided differences,\nInt. J. Appl. Math. Comput. Sci. , 11(2001), 803–820.\n[3] S.A. Avdonin and S.A. Ivanov, Riesz bases of exponential s and divided differences, St. Peters-\nburg Math. J. , 13(2002), 339–351.\n[4] H.T. Banks and I.G. Rosen, Computational methods for the identification of spatially varying\nstiffness and damping in beams, Control Theory Adv. Tech. , 3(1987), 1–32.\n[5] H.T. Banks and K. Ito, A unified framework for approximati on in inverse problems for dis-\ntributed parameter systems, Control Theory Adv. Tech. , 4(1988), 73–90.\n[6] H.T. Banks and K. Kunisch, Estimation Techniques for Distributed Parameter System ,\nBirkh¨ auser, Boston, 1989.\n[7] D. Bresch-Pietri and M. Krstic, Output-feedback adapti ve control of a wave PDE with bound-\nary anti-damping, Automatica , 50(2014), 1407–1415.\n[8] B.Z. Guo and Y.H. Luo, Controllability and stability of a second-order hyperbolic system with\ncollocated sensor/actuator, Systems Control Lett. , 46(2002), 45–65.\n[9] B.Z. Guo and Y.H. Luo, Riesz basis property of a second-or der hyperbolic system with collo-\ncated scalar input-output, IEEE Trans. Autom. Control , 47(2002), 693–698.\n[10] B.Z. Guo and W.D. Zhu, On the energy decay of two coupled s trings through a joint damper,\nJ. Sound Vib. , 203(1997), 447–455.\n[11] B.Z. Guo and F.F. Jin, Arbitrary decay rate for two conne cted strings with joint anti-damping\nby boundary output feedback, Automatica , 46(2010), 1203–1209.\n[12] B.Z. Guo and F.F. Jin, Sliding mode and active disturban ce rejection control to stabilization\nof one-dimensional anti-stable wave equations subject to d isturbance in boundary input, IEEE\nTrans. Autom. Control , 58(2013), 1269–1274.\n[13] B.Z. Guo and F.F. Jin, Output feedback stabilization fo r one-dimensional wave equation sub-\nject to boundary disturbance, IEEE Trans. Autom. Control , 60(2015), 824–830.\n20[14] A.E. Ingham, Sometrigonometrical inequalities with a pplications in thetheory of series, Math.\nZ., 41(1936), 367–379.\n[15] V. Komornik and P. Loreti, Fourier Series in Control Theory , Springer-Verlag, New York,\n2005.\n[16] M. Krstic, Adaptive control of an anti-stable wave PDE, Dyn. Contin. Discrete Impuls. Syst.\nSer. A. Math. Anal. , 17(2010), 853–882.\n[17] M. Krstic, B.Z. Guo, and A. Smyshlyaev, Boundary contro llers and observers for the linearized\nSchr¨ odinger equation, SIAM J. Control Optim. , 49(2011), 1479–1497.\n[18] K. Ramdani, M. Tucsnak, and G. Weiss, Recovering the ini tial state of an infinite-dimensional\nsystem using observers, Automatica, 46(2010), 1616–1625.\n[19] A. Smyshlyaev and M. Krstic, Boundary control of an anti -stable wave equation with anti-\ndamping on the uncontrolled boundary, Systems Control Lett. , 58(2009), 617–623.\n[20] G. Weiss, Admissible observation operators for linear semigroups, Israel J. Math. , 65(1989),\n17-43.\n[21] G.Q. Xu, State reconstruction of a distributed paramet er system with exact observability, J.\nMath. Anal. Appl. , 409(2014), 168–179.\n[22] G.Q. Xu and B.Z. Guo, Riesz basis property of evolution e quations in Hilbert spaces and\napplication to a coupled string equation, SIAM J. Control Optim. , 42(2003), 966–984.\n[23] R. M. Young, An Introduction to Nonharmonic Fourier Series , Academic Press, New York,\n1980.\n21"    },    {        "title": "1606.09326v2.Skyrmion_dynamics_in_a_chiral_magnet_driven_by_periodically_varying_spin_currents.pdf",        "content": "arXiv:1606.09326v2  [cond-mat.mes-hall]  6 Dec 2016Skyrmion dynamics in a chiral magnet driven by periodically\nvarying spin currents\nRui Zhu*and Yin-Yan Zhang\nDepartment of Physics, South China University of Technolog y,\nGuangzhou 510641, People’s Republic of China\nAbstract\nIn this work, we investigated the spin dynamics in a slab of ch iral magnets induced by an\nalternating (ac) spin current. Periodic trajectories of th e skyrmion in real space are discovered\nunder the ac current as a result of the Magnus and viscous forc es, which originate from the Gilbert\ndamping, the spin transfer torque, and the β-nonadiabatic torque effects. The results are obtained\nby numerically solving the Landau-Lifshitz-Gilbert equat ion and can be explained by the Thiele\nequation characterizing the skyrmion core motion.\nPACS numbers: 75.78.-n, 72.25.-b, 71.70.-d\n*Corresponding author. Electronic address: rzhu@scut.edu.cn\n1I. INTRODUCTION\nThe skyrmion spin texture is a kind of topologically-nontrivial magnet ic vortex formed\nmost typically in the bulk chiral magnets (CMs) and magnetic thin films1–3. In CMs it is\nbelieved that the spin-orbit coupling induced Dzyaloshinskii-Moriya int eraction (DMI) gov-\nerns the spin twisting1. Recently the magnetic skyrmion structure attracts intensive fo cus,\nboth in the fundamental theoretic aspect and in its potential applic ation in the information\ntechnology4–7. In the magnetic skyrmion state the emergent electrodynamic effe ct originates\nfrom its nontrivial spin topology and gives rise to the topological Hall effect and a remark-\nable current-driven spin transfer torque effect1,8–14. The so-called skyrmionics makes use\nof the skyrmion as a memory unit favored by its topologically protect ed long lifetime and\nultralow driving current, which is five or six orders smaller than that f or driving a magnetic\ndomain wall1,15.\nAlthough the current-driven spin dynamics in the CMs with DMI has be en intensively\nstudied recently, less work of an alternating current (ac) driven s kyrmion dynamics was\nreported. The skyrmion-motion-induced ac current generation h as been predicted, which\nshares the reversed effect of our consideration16. In this work, we investigated the ac-\nspin-current driven skyrmion dynamics with the DMI, Gilbert damping17, adiabatic and\nnonadiabatic spin torques, and different current profiles taken int o account. Our proposition\nis inspired by the following several aspects. Firstly, it is both theore tically and technically\ninteresting to know the behavior of a skyrmion when an external ac current is applied.\nSecondly, ahigh-speedlow-powermodulationofaskyrmionisfavora bleforpotentialmemory\nprocessing. Lastly but not least, we noticed the mathematical sign ificance of the solution of\nthe Landau-Lifshitz-Gilbert (LLG) equation of a collinear magnet wit h periodically varying\nspin-currents applied, in which chaos is observed18,19.\nThe topological property of a spin texture can be described by the surface integral of the\nsolid angle of the unitary spin-field vector n(r). The skyrmion number is so defined as S=\n1\n4π/integraltext\nn·/parenleftBig\n∂n\n∂x×∂n\n∂y/parenrightBig\nd2rcounting how many times the spin field wraps the unit sphere. More\nspecific topological properties of a skyrmion can be considered by a nalyzing its radial and\nwhirling symmetric pattern1,20–22. In the continuum field theory, as a result of topological\nprotection, the skyrmion cannot be generated from a topologically trivial magnetic state\nsuch as a ferromagnet or a helimagnet by variation without a topolog ically nontrivial force\n2such as a spatially nontrivial spin current11,12, geometrical constriction13, domain wall pair\nsource5,6, the edge spin configuration22, etc., and vise versa. It is predicted by simulation\nthat the skyrmion can be generated from a quasi-ferromagnetic a nd helimagnetic state by\nexternal Lorentzian and radial spin current12and that transformation is possible between\ndifferent topologically-nontrivial states such as that between the domain-wall pair and the\nskyrmion5,6. The local current flowing from the scanning tunneling microscope t o generate\nthe skyrmion in experiment can be approximated by a radial spin curr ent, which imbues\nnontrivial topology into the helimagnet11. Also an artificial magnetic skyrmion can be\ntailored by an external magnetic field with nontrivial geometric distr ibution23. When the\nboundary geometry of the material is tailored such as by a notch in a long plate, a skyrmion\ncan be generated by a collinear spin current13. In this case, the nontrivial constriction\ntopology contributes to the formation of the skyrmion. The unifor m current can move and\nrotate a skyrmion without changing its topology15,24. In this work we will show that these\ntopological behaviors of the skyrmion are retained in the spin dynam ics driving by an ac\nspin current.\nAlmost all kinds of ferromagnetic and vortex spin dynamics can be de scribed by the LLG\nequation. The behavior of the LLG equation is of importance in both t he physical and\nmathematical sciences18. It has been shown by previous works that the spin torque effect\ndriven by a periodic varying spin current can be described as well by t he LLG equation\nwith the original time-independent current replaced by the time-de pendent current in the\nspin torque term18,19. Although chaotic behaviors are predicted in the spatially-uniform a c\nspin-current driven collinear ferromagnetic spin structure18,19, which is well described by the\nsingle-spin LLG equation, no similar phenomenon is reported in a spatia lly-nonuniform spin\nlattice, the latter of which can be attributed to the relaxation proc esses of the inter-site\nscattering. Even if some sort of chaotic behavior occurs after a lo ng time of evolution, it\nis workable to restore the original state by applying a magnetic field a fter some time. The\ninfluence of it on the skyrmionics exploitation is not large. In this work , we use a matrix-\nbased fourth-order Runge-Kutta method to solve the LLG equat ion with both the adiabatic\nand nonadiabatic spin torques taken into account. Analytical solut ion of the generalized\nThiele equation1,13,24reproduces our numerical results.\n3II. THEORETIC FORMALISM\nWe consider a thin slab of CM modulated by a constant magnetic field an d an ac spin\ncurrent. The strong DMI makes the material a skyrmion-host. In the continuum approxi-\nmation, the Hamiltonian of the localized magnetic spin in a CM can be desc ribed as1,12,13\nH=−J/summationtext\nrMr·/parenleftbig\nMr+ex+Mr+ey/parenrightbig\n−D/summationtext\nr/parenleftbig\nMr×Mr+ex·ex+Mr×Mr+ey·ey/parenrightbig\n−B·/summationtext\nrMr,(1)\nwithJandDthe ferromagnetic and Dzyaloshinskii-Moriya (DM) exchange energ ies, re-\nspectively. The dimensionless local magnetic moments Mrare defined as Mr≡ −Sr//planckover2pi1,\nwhereSris the local spin at rand/planckover2pi1is the plank constant divided by 2 π. We assume that\nthe length of the vector |Mr|=Mis fixed, therefore Mr=Mn(r) withn(r) the unitary\nspin field vector. The unit-cell dimension is taken to be unity. An exte rnal magnetic field\nBis applied perpendicular to the slab plane to stabilize the skyrmion confi guration. The\nBohr magneton µBis absorbed into Bto have it in the unit of energy. The typical DMI\nD= 0.18Jis used throughout this work13. This DM exchange strength corresponds to the\ncritical magnetic fields Bc1= 0.0075Jbetween the helical and skyrmion-crystal phases and\nBc2= 0.0252Jbetween the skyrmion-crystal and ferromagnetic phases, resp ectively. We\nadoptB= (0,0,0.01J) in our numerical considerations with J= 1 meV.\nTheextendedformoftheLLGequationthattakesintoaccountth eDMIandtheadiabatic\nand nonadiabatic spin torque effects can be expressed in the followin g formula1,12,13,25\ndMr\ndt=−γMr×Beff\nr+α\nMMr×dMr\ndt+pa3\n2eM[j(r,t)·∇]Mr\n−pa3β\n2eM2{Mr×[j(r,t)·∇]Mr}.(2)\nBy assuming that the energy of a magnet with the local magnetizatio nMrin a spatially\nvarying magnetic field Beff\nrhas the form of H=−γ/planckover2pi1/summationtext\nrMr·Beff\nr, we have\nBeff\nr=−1\n/planckover2pi1γ∂H\n∂Mr, (3)\nand therefore the first term in the right hand side of Eq. (2) is\n−γMr×Beff\nr=−J\n/planckover2pi1Mr×/parenleftbig\nMr+ex+Mr+ey+Mr−ex+Mr−ey/parenrightbig\n−1\n/planckover2pi1(Mr×B)\n−D\n/planckover2pi1Mr×\n/parenleftbig\nMr−ey,z−Mr+ey,z/parenrightbig\nex+(Mr+ex,z−Mr−ex,z)ey\n+/parenleftbig\nMr+ey,x−Mr+ex,y−Mr−ey,x+Mr−ex,y/parenrightbig\nez\n.(4)\n4The second to the last terms of Eq. (2) sequentially correspond to the effect of the Gilbert\ndamping, the time-dependent spin current j(t) =jesin(ωt)-induced adiabatic and nonadia-\nbatic spin torques, respectively. pmeasures the polarization of the conduction electrons, eis\nthe positive electron charge, and ais the average in-plane lattice constant of the CM. In our\nconsiderations the frequency of the ac spin current ωis small enough in comparison of the\nmagnetization evolution rate. Therefore, the spin torques can be satisfactorily described by\nusing the time-dependent current in thestandard torque expres sion, which has been justified\nby previous studies18,19. Here, the unit of time is set to be t0=/planckover2pi1/J≈6.6×10−13s. A\nphenomenologically expectedvalueof α= 0.1isusedinafterwardsnumerical considerations.\nBy looking deep into Eqs. (2) and (4), we can make some predictions o f the behavior of\nthe local magnetization. We know that the effect of the magnetic fie ld together with the\nGilbert term is to precess the magnetic spin into the direction of the e xternal field. The first\nterm in the right hand side of Eq. (4) is that the effective magnetic fie ld is in the direction\nof neighboring spins. Therefore the evolution tends to form a ferr omagnet. This contributes\nto the centripetal force of the magnetization in the direction of Mr×Mr′, which results in\nthe precession of one around the other. The effective field in the DM term in Eq. (4) is\n−∇×Mrwith unitary lattice constants. The integral counterpart of the c url is/contintegraltext\nMr′·dl.\nWhentheneighboringspinsformaring, theenergyisthelowest, hen cegeneratingaspiraling\nforce to the CM. It helps our understanding if we analogize all the ot her terms in the right\nhand side of Eq. (2) to the effect of a magnetic field. The local “magn etic field” of the\nphenomenological Gilbert damping force is proportional to −dMr/dtin the standard linear-\nresponse damping form, proportional to the velocity and pointing o ppositely to it. While\nthe local spin is precessing, the direction of Mr×dMr/dtpoints to the precession axis of\nMradding a force swaying to that axis. The last two terms are the effec t of the current-\ninduced spin torques. For convenience of interpretation, we discu ss the case that j(r,t) is\nspatially-uniform and along the x-direction. Then [ j(r,t)·∇]Mr=jx(t)∂Mr/∂x. In the\ncase of the adiabatic torque, this term adds a velocity to Mrmaking it sway to the direction\nofMr+ex, andMr+extoMr+2ex, and etc. if jx(t) is positive. Therefore, the complete spin\ntexture moves along the direction of the external spin current like a relay race no matter\nit is a skyrmion or a domain wall. In a periodic magnetic structure such a s a ferromagnet\nand a helimagnet the “relay race” goes back to itself and hence no sp in structure movement\noccurs. Following this physical picture, the local “magnetic field” of the nonadiabatic spin\n5torque is along the direction of jx(t)∂Mr/∂x. It exerts a velocity perpendicular to that\noriginates from the adiabatic spin torque. Its result is the motion of the spin texture in the\ndirection perpendicular to the spin current. We have already analyz ed the mechanisms of\ntheLLGequation termby term. However, they affectsthe system collaboratively. While the\nadiabatic spin torque moves the skyrmion along the spin current, th e Gilbert damping force\ncontributes a velocity in the direction of Mr×dMr/dtandtherefore the effect is a transverse\nmotion of the skyrmion, which is the so-called Hall-like motion12. Also it is noticeable that\nthe transverse velocity resulting from the Gilbert damping and the n onadiabatic spin torque\nis opposite to each other. In real situations, both αandβare much less than 1. The\nadiabatic spin torque makes the main contribution to the motion of th e skyrmion. And\nwhen the two transverse force is equal, the motion of the skyrmion is straightly along the\ndirection of the spin current. Therefore, periodic trajectories o f the skyrmion in real space\ncan be predicted under the influence of a spatially uniform ac spin cur rent.\nThe previous discussions are well expressed in the Thiele equation de scribing the motion\nof the center of mass of a skyrmion as1,13,24,26.\nG×[−j(t)−vd]+κ[−βj(t)−αvd]−∇U(r) =0, (5)\nwherevd=dR/dt=/parenleftBig\n˙X,˙Y/parenrightBig\nwithR= (X,Y) the center of mass coordinates, κis a\ndimensionless constant of the order of unity, and G= 2πSezis the gyrovector with ezin the\ndirection perpendicular to the CM plane. The minus sign before j(t) is because of that the\ndirection of the motion of conduction electrons is opposite to that o f the current. The Thiele\nequation (5) describes5coaction of the Magnus force Fg=G×[−j(t)−vd], the viscous\nforceFv=κ[−βj(t)−αvd], and the confining force Fp=−∇U(r). In our considerations,\nperiodic boundary conditions are used to justify an infinite two-dime nsional model. The\napplied magnetic field is spatially uniform and the impurity effect is neglec ted. Therefore\n∇U≈0. The analytical result of Eq. (5) assuming S=−1 andj(t) =jesin(ωt)excan be\nobtained as \n\nX=αβκ2+4π2\n(α2κ2+4π2)ωjecos(ωt),\nY=2πκ(β−α)\n(α2κ2+4π2)ωjecos(ωt).(6)\nSince the spin current is time dependent, FgandFvinstantaneously change their direc-\ntion with the motion of the skyrmion core and simultaneously react on the motion of the\nskyrmion, giving rise to the trigonometric trajectory of the skyrm ion shown in Eq. (6),\n6which agrees with the simulation results. Because the skyrmion vort ex moves in the relay\nfashion under the effect of the spin torque shown by the LLG equat ion, there is a π/2 phase\nlag between its core motion and the sinusoidally varying spin current.\nIII. NUMERICAL RESULTS AND INTERPRETATIONS\nBy multiplying ˜ α−1with\n˜α= 1−α\n0−(Mr)z(Mr)y\n(Mr)z0−(Mr)x\n−(Mr)y(Mr)x0\n, (7)\nfrom the left to Eq. (2), the matrix-based Runge-Kutta method is developed. In Figs. 1 to\n3, numerical results of our simulations are given. We set M= 1,p= 0.5, anda= 4˚A. The\nintegral step h= 0.1t0is used and its convergence is justified by comparison with the result s\nofh= 0.01t0. WithD= 0.18J, the natural helimagnet wavevector Q= 2π/λ=D/J\nwith the diameter of the skyrmion λ=D/J≈35 in the unit of a. A 30×30 square\nlattice is considered which approximately sustains a single skyrmion. P eriodic boundary\ncondition is used to allow the considered patch to fit into an infinite plan e. While part of the\nskyrmion moves out of the slab, complementary part enters from t he outside as the natural\nground state of a CM is the skyrmion crystal. We use the theoretica lly perfect skyrmion\nprofilen(r) = [cosΦ( ϕ)sinΘ(r),sinΦ(ϕ)sinΘ(r),cosΘ(r)] with Θ( r) =π(1−r/λ) and\nΦ(ϕ) =ϕin the polar coordinates as the initial state and it would change into a n atural\nskyrmion in less than one current period. The skyrmion number for t his state S=−1. The\nspatially-uniform ac spin current is applied in the x-direction as j(t) =jesin(ωt)exwithin\nthe CM plane.\nVariationofthe skyrmion number intimedriven bythe acspincurrent is shown inFig. 1.\nItcanbeseen thatcosinusoidal variationof Soriginatesfromthesinusoidal j(t)withexactly\nthe same period. Fig. 2 shows the snapshots of the spin profile at th e bottoms and peeks of\nthecosinusoidal variationof SandFig. 3shows thetrajectoriesof thecenter oftheskyrmion\n(see Ref. 27 for Supplementary Movie). The skyrmion number is a de monstration of the\nmotion pattern of the skyrmion. While the skyrmion moves to one side of the CM slab, only\npart of a skyrmion is within the view and hence the skyrmion number is r educed. Previous\nauthors have found that the velocity of the skyrmion increases line arly with the increase of\n7the current amplitude and that thedynamical threshold current t o move a skyrmion isin the\nsameorderofthatneededforadomainwall1. Herewehavereobtainedthetwo points. Itcan\nbeseeninFig. 1(a)thatthepeakhight oftheskyrmionnumber incre ases withtheamplitude\nof the current density and it becomes almost invisible when jeis as small as 1010Am−2. In\nFig. 1(b), the evolutions of Sfor different ac periods are shown. The frequency of the ac\ncurrent is in the order of GHz, which is sufficiently adiabatic as the rat e of the spin dynamics\nis in the order of 10−12s. We can see that the periodic pattern of Sis better kept with larger\namplitudes for smaller ac frequencies. It shows that the phenomen on is a good adiabatic\none. Within our numerical capacity, it can be predicted that strong cosinusoidal variation\ncan occur at MHz or smaller ac frequencies, which promises experime ntal realization.\nThe variation of Sis the result of the motion of the skyrmion. The periodic translation o f\nskyrmion is the result of the coaction of the instantaneous Magnus and viscous forces. The\nspin current gives rise to the drift velocity of the spin texture. As a combined result of the\nGilbert damping, the DMI, the adiabatic and nonadiabatic spin torque s, the skyrmion Hall\neffect, namely, the transverse motion of the skyrmion perpendicu lar to the spin current, is\nobserved in topologically-nontrivial spin textures. As shown in Fig. 2 , in spite of its motion,\nthe topological properties of the skyrmion are conserved becaus e the initial skyrmion state\nand the natural skyrmion ground state share similar topology and n o topology-breaking\nsource such as an in-plane magnetic field is present. When part of th e skyrmion moves out\nof the CM slab, only the remaining part contributes to the skyrmion n umber and hence Sis\ndecreased. The cosinusoidal variation of Sdirectly reflects the oscillating trajectory of the\nskyrmion Shown in Figs. 2 and 3. We can see that the skyrmion change s from the initial\nartificial skyrmion state into the natural skyrmion state with S=−1 conserved, as shown\nin Fig. 2(a) and (b). At the times of integer periods the skyrmion is at the center of the CM\nslab and at the times of half-integer periods the skyrmion moves to t he left side as shown\nin Fig. 2 (c) to (f).\nAs predicted by the Thiele equation, the trajectory of the skyrmio n follows a cosinusoidal\npattern expressed in Eq. (6). It is interesting that the trajecto ry of the skyrmion results\nfrom the competition between the drift motion of the skyrmion and t he skyrmion Hall effect\nunder the influence of the adiabatic and nonadiabatic spin torque eff ects. The adiabatic spin\ntorque effect exerts a force to align the spin at each site to its + x-direction neighbor while\nj(t) is in the exdirection, which results in the motion of the spin pattern to the + xdirection\n8in a relay fashion. The Gilbert damping effect and the nonadiabatic spin torque add a\ntransverse velocity to the moving skyrmion perpendicular to its orig inal velocity. These two\nforces are in opposite directions when αandβare both positive. Therefore the transverse\nmotion is determined by the sign and relative strength of these two e ffects. From Eq. (6)\nwe can see that when β−α >0 the skyrmion’s y-direction motion is in a cosinusoidal form\nand when β−α <0 it is in a negative cosinusoidal form. For the x-direction motion of the\nskyrmion, the direction is the same in the two cases and the magnitud e is slightly smaller for\nthe latter because |4π2| ≫ |αβκ2|holds for all physical parameter settings. And physically\nit is because the x-direction motion of the skyrmion is mainly determined by the adiabatic\nspin torque, which is the prerequisite for any motion of the skyrmion .\nOur simulation results of the skyrmion trajectories for β= 0.5α,α, and 2αwith fixed\nα= 0.1 are shown in Fig. 3. Good agreement with the prediction by the Thiele equation\nis obtained. In the three cases, Xevolves cosinusoidally with the initial position ( X,Y) =\n(15,15) at the center of the CM slab. For β= 0.5α,Yevolves minus-cosinusoidally; for\nβ=α,Yis constant at 15; for β= 2α,Yevolves cosinusoidally. As the difference between\nβandαis small in Fig. 3 (a) and (c), the cosinusoidal pattern shrinks into a s tep jump.\nBesides the oscillation, a tiny linear velocity of the skyrmion can be see n in Fig. 3 (a) and\n(c). And the directions of this velocity are different in the two cases . We attribute this\nlinear velocity to the whirling of the skyrmion from the artificial initial p rofile to the natural\nprofile sustained by the real CM. Because at this whirling step, the G ilbert damping and\nthe adiabatic and nonadiabatic torques are already in effect, the init ial linear velocities are\ndifferent in the two cases.\nIV. CONCLUSIONS\nIn this work, we have investigated the dynamics of the skyrmion in a C M driven by\nperiodically varying spin currents by replacing the static current in t he LLG equation by\nan adiabatic time-dependent current. Oscillating trajectories of t he skyrmion are found\nby numerical simulations, which are in good agreement with the analyt ical solution of the\nThiele equation. In the paper, physical behaviors of the general L LG equation with the\nGilbert damping and the adiabatic and nonadiabatic spin torques coex istent are elucidated.\nEspecially, the effect of the nonadiabatic spin torque is interpreted both physically and\n9numerically.\nV. AUTHOR CONTRIBUTION STATEMENT\nR.Z. wrote the program and the paper. Y.Y.Z. made the simulation.\nVI. ACKNOWLEDGEMENTS\nR.Z. would like to thank Pak Ming Hui for stimulation and encouragemen t of the work.\nThis project was supported by the National Natural Science Foun dation of China (No.\n11004063) and the Fundamental Research Funds for the Centra l Universities, SCUT (No.\n2014ZG0044).\n101N. Nagaosa and Y. Tokura, Nat. Nanotechnol. 8, 899 (2013).\n2S. M¨ uhlbauer, B. Binz, F. Jonietz, C. Pfleiderer, A. Rosch, A . Neubauer, R. Georgii, and P.\nB¨ oni, Science 323, 915 (2009).\n3S.Heinze, K. V. Bergmann, M. Menzel, J. Brede, A. Kubetzka, R . Wiesendanger, G. Bihlmayer,\nand S. Bl¨ ugel, Nat. Phys. 7, 713 (2011).\n4X. Zhang, M. Ezawa, and Y. Zhou, Sci. Rep. 5, 9400 (2015).\n5X. Xing, P. W. T. Pong, and Y. Zhou, Phys. Rev. B 94, 054408 (2016).\n6Y. Zhou and M. Ezawa, Nat. Commun. 5, 4652 (2014).\n7J. Sampaio, V. Cros, S. Rohart, A. Thiaville, and A. Fert, Nat . Nanotechnol. 8, 839 (2013).\n8K. Hamamoto, M. Ezawa, and N. Nagaosa, Phys. Rev. B 92, 115417 (2015).\n9A. Neubauer, C. 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Lett. 30, 230 (1972).\n27Supplementary Movie.\n12/s48 /s53/s48/s48 /s49/s48/s48/s48 /s49/s53/s48/s48 /s50/s48/s48/s48/s45/s49/s46/s48/s48/s45/s48/s46/s57/s53/s45/s48/s46/s57/s48/s45/s48/s46/s56/s53/s45/s49/s46/s48/s48/s45/s48/s46/s57/s53/s45/s48/s46/s57/s48/s45/s48/s46/s56/s53/s45/s48/s46/s56/s48/s45/s48/s46/s55/s53\n/s116/s83/s32/s106\n/s101/s61/s48\n/s32/s84/s61/s52/s48/s48\n/s32/s84/s61/s53/s48/s48\n/s32/s84/s61/s54/s48/s48/s40/s98/s41/s83/s32/s106\n/s101/s61/s49/s48/s49/s48\n/s65/s109/s45/s50\n/s32/s106\n/s101/s61/s49/s48/s49/s49\n/s65/s109/s45/s50\n/s32/s106\n/s101/s61/s50 /s49/s48/s49/s49\n/s65/s109/s45/s50/s40/s97/s41/s32\nFIG. 1: Variation of the skyrmion number S in time (a) for differ ent current amplitudes and (b)\nfor different ac current frequencies. The time tand ac spin current period Tare in the unit of\nt0≈6.6×10−13s.β= 0.05. In panel (a), T= 500. In panel (b), je= 1011Am−2.\n13(0,−0.91763)\n  \n(a)\n−1−0.500.51(750,−0.77054)\n(b)(1000,−0.97276)\n(c)\n(1250,−0.76819)\n(d)(1500,−0.97019)\n(e)(1750,−0.76716)\n(f)\nFIG. 2: Snapshots of the dynamical spin configurations at the bottoms and peaks of the skyrmion\nnumber shown in Fig. 1. The in-plane components of the magnet ic moments are represented by\narrows and their z-components are represented by the color plot. The paramete rs areje= 2×1011\nAm−2,T= 500, and β= 0.05. On the top of each panel are the ( t,S) values.\n14/s57/s49/s48/s49/s49/s49/s50/s49/s51/s49/s52/s49/s53/s49/s54\n/s57/s49/s48/s49/s49/s49/s50/s49/s51/s49/s52/s49/s53\n/s48 /s53/s48/s48 /s49/s48/s48/s48 /s49/s53/s48/s48 /s50/s48/s48/s48/s57/s49/s48/s49/s49/s49/s50/s49/s51/s49/s52/s49/s53\n/s32/s88\n/s32/s89/s40/s97/s41/s32 /s61/s48/s46/s48/s53\n/s40/s98/s41/s32 /s61/s48/s46/s49\n/s116/s40/s99/s41/s32 /s61/s48/s46/s50\nFIG. 3: Variation of the skyrmion center coordinates ( X,Y) in time (a) for β= 0.05, (b) for\nβ= 0.1, and (c) for β= 0.2. Other parameters are the same as Fig. 2.\n15"    },    {        "title": "1303.1192v1.Angle_Dependent_Spin_Wave_Resonance_Spectroscopy_of__Ga_Mn_As_Films.pdf",        "content": "arXiv:1303.1192v1  [cond-mat.mtrl-sci]  5 Mar 2013Angle-Dependent Spin-Wave Resonance Spectroscopy of (Ga, Mn)As Films\nL. Dreher,1,∗C. Bihler,1E. Peiner,2A. Waag,2W. Schoch,3W. Limmer,3S.T.B. Goennenwein,4and M.S. Brandt1\n1Walter Schottky Institut, Technische Universit¨ at M¨ unch en, Am Coulombwall 4, 85748 Garching, Germany\n2Institut f¨ ur Halbleitertechnik, Technische Universit¨ a t Braunschweig,\nHans-Sommer-Straße 66, 38023 Braunschweig, Germany\n3Institut f¨ ur Quantenmaterie, Universit¨ at Ulm, 89069 Ulm , Germany\n4Walther-Meißner-Institut, Bayerische Akademie der Wisse nschaften,\nWalther-Meißner-Straße 8, 85748 Garching, Germany\n(Dated: October 31, 2018)\nA modeling approach for standing spin-wave resonances base d on a finite-difference formulation\nof the Landau-Lifshitz-Gilbert equation is presented. In c ontrast to a previous study [Bihler et al.,\nPhys. Rev. B 79, 045205 (2009)], this formalism accounts for elliptical ma gnetization precession and\nmagnetic properties arbitrarily varying across the layer t hickness, including the magnetic anisotropy\nparameters, the exchange stiffness, the Gilbert damping, an d the saturation magnetization. To\ndemonstrate the usefulness of our modeling approach, we exp erimentally study a set of (Ga,Mn)As\nsamples grown by low-temperature molecular-beam epitaxy b y means of angle-dependent stand-\ning spin-wave resonance spectroscopy and electrochemical capacitance-voltage measurements. By\napplying our modeling approach, the angle dependence of the spin-wave resonance data can be re-\nproduced in a simulation with one set of simulation paramete rs for all external field orientations.\nWe find that the approximately linear gradient in the out-of- plane magnetic anisotropy is related\nto a linear gradient in the hole concentrations of the sample s.\nPACS numbers: 75.50.Pp, 76.50.+g, 75.70.-i, 75.30.Ds\nKeywords: (Ga,Mn)As; spin wave resonance; magnetic anisot ropy\nI. INTRODUCTION\nDue to their particular magnetic properties, in-\ncluding magnetic anisotropy,1–3anisotropic magneto-\nresistance4,5and magneto-thermopower,6in the past\nyears ferromagnetic semiconductors have continued to\nbe of great scientific interest in exploring new physics\nand conceptual spintronic devices.7–11The most promi-\nnentferromagneticsemiconductoris(Ga,Mn)As, wherea\nsmall percentage of Mn atoms on Ga sites introduces lo-\ncalizedmagneticmomentsaswellasitinerantholeswhich\nmediate the ferromagneticinteractionofthe Mn spins ( p-\ndexchange interaction).12Both theoretical and experi-\nmental studies have shown that the magnetic anisotropy,\ni.e., the dependence of the free energy of the ferromagnet\non the magnetization orientation, depends on the elas-\ntic strain and the hole concentration in the (Ga,Mn)As\nlayer,12,13openingupseveralpathwaystomanipulatethe\nmagnetic anisotropy of (Ga,Mn)As.14–16\nA common spectroscopic method to probe the mag-\nnetic anisotropy of ferromagnets and in particular\n(Ga,Mn)As, is angle-dependent ferromagnetic resonance\n(FMR),17–23where FMR spectra are taken as a func-\ntion of the orientation of the external magnetic field. If\nthe magnetic properties of the ferromagnet are homo-\ngeneous, a zero wave vector ( k= 0) mode of collec-\ntively, uniformally precessing magnetic moments couples\nto the microwavemagnetic field, e.g., in a microwavecav-\nity, allowing for a detection of the magnetization preces-\nsion. The resonance field of this mode, referred to as\nuniform resonance magnetic field, depends on the em-\nployedmicrowavefrequency and the magnetic anisotropyparameters. Thus, by recording FMR spectra at differ-\nent orientations of the external field with respect to the\ncrystal axes, the anisotropy parameters can be deduced\nfrom the experiment. However, if the magnetic prop-\nerties of a ferromagnetic layer are non-homogeneous or\nthe spins at the surface and interface of the layer are\npinned, non-propagating modes with k/negationslash= 0, referred to\nas standing spin-wave resonances (SWR), can be excited\nby the cavity field and thus be detected in an FMR ex-\nperiment. On one hand this can hamper the derivation\nof anisotropy parameters, on the other hand a detailed\nanalysis of these modes can elucidate the anisotropy pro-\nfile of the layer and the nature of spin pinning condi-\ntions. Furthermore, the excitation of spin waves is of\ntopical interest in combination with spin-pumping,24–27\ni.e., the generation of pure spin currents by a precessing\nmagnetization.28–30In this context, the exact knowledge\nof the magnetization precession amplitude as a function\nof the position coordinate within the ferromagnet is of\nparticular importance.24\nSeveral publications report on SWR modes in\n(Ga,Mn)As with a mode spacing deviating from what is\nexpected according to the Kittel model for magnetically\nhomogeneous films with pinned spins at the surface.31–36\nThese results have been attributed to an out-of-plane\nanisotropyfieldlinearly31,36orquadraticallyvarying33–35\nas a function of the depth into the layer, as well as to\nspecific spin pinning conditions at the surface and at the\ninterface to the substrate.35While most of these studies\nhave focused on the spacings of the resonancefields when\nmodeling SWR measurements, in Ref. 36 a more sophis-\nticated approach, based on a normal mode analysis,37,38\nwas employed to model resonance fields as well as rela-2\ntive mode intensities for the external field oriented along\nhigh-symmetry directions, assuming a circularly precess-\ning magnetization.\nIn this work, we present a more general modeling ap-\nproach for SWR, based on a finite-difference formulation\nof the Landau-Lifshitz-Gilbert (LLG) equation. This ap-\nproach holds for any orientation of the external mag-\nnetic field and accounts for elliptical magnetization pre-\ncession [Sec. II]. It allows for a simulation of arbitrar-\nily varying profiles of the magnetic properties across the\nthickness of the film, including vatiations of the mag-\nnetic anisotropy parameters, the exchange stiffness, and\nthe Gilbert damping parameter. As the result ofthe sim-\nulation, we obtain the Polder susceptibility tensor as a\nfunction of the depth within the ferromagnet. Based on\nthis result, the absorbedpowerupon spin waveresonance\nandthe magnetizationprecessionamplitude asafunction\nof the depth can be calculated for any orientation of the\nexternal magnetic field.\nWe apply our modeling approach to a set of four\n(Ga,Mn)As samples epitaxially grown with different\nV/III flux ratios [Sec. III], motivated by the obser-\nvation that V/III flux ratios of /lessorsimilar3 lead to a gra-\ndient in the hole concentration p[Ref. 39], which in\nturn is expected to cause non-homogeneous magnetic\nanisotropyparameters.31,36Electrochemicalcapacitance-\nvoltage (ECV) measurements revealed a nearly linear\ngradient in pacross the thickness of the layers investi-\ngated. To show that our modeling approach is capa-\nble of simulating SWR spectra for arbitrary magnetic\nfield orientations, angle-dependent SWR data were taken\nand compared with the model using one set of magnetic\nparameters for each sample, revealing gradients in the\nuniform resonance magnetic fields. We discuss the in-\nfluence of the gradient in pon the observed uniform\nresonance field gradients as well as possible influences\nof strain and saturation magnetization gradients on the\nobserved out-of-plane anisotropy profile. It should be\nemphasized, however, that the objective of this work is\nto show the usefulness of our modeling approach, while\na detailed investigation of the origin of the gradient in\nthe out-of-plane magnetic anisotropy profile and there-\nfore a detailed understanding of the particular materials\nphysics of (Ga,Mn)As is beyond the scope of this study.\nFinally, we summarize our results and discuss further po-\ntential applications of this work [Sec. IV].\nII. THEORETICAL CONSIDERATIONS\nIn this section, we provide the theoretical framework\nnecessary to describe the full angle dependence of the\nspin-wave resonance spectra presented in Sec. III. Refer-\nring to the coordinate system depicted in Fig. 1, we start\nfrom the canonical expression for the free enthalpy den-\nsity (normalized to the saturation magnetization M) forφ0Θ0\n123m2\nm1m3≈1\nm\nx||[100]y||[010]z||[001]\nSubstrateFerromag net\nFIG. 1: (color online) Relation between the two coordinate\nsystems employed. The ( x,y,z) frame of reference is spanned\nby the cubic crystal axes, while the (1 ,2,3) coordinate sys-\ntem is determined by the equilibrium orientation of the mag-\nnetization (3-direction) and two transverse directions, t he 2-\ndirection being parallel to the film plane; the latter system is\nzandµ0Hdependent, as described in the text.\na tetragonally distorted (Ga,Mn)As film13,20,40,41\nG= const −µ0H·m+B001m2\nz+B4⊥m4\nz\n+B4/bardbl(m4\nx+m4\ny)+1\n2B1¯10(mx−my)2.(1)\nHere,µ0His a static external magnetic field, B001\nis a uniaxial out-of-plane anisotropy parameter, re-\nflecting shape and second-order crystalline anisotropy,13\nB4⊥,B4/bardbl, andB1¯10are fourth-order crystalline and\nsecond-order uniaxial in-plane anisotropy parameters,\nrespectively;1mx,my,mzdenote the components of the\nnormalized magnetization vector m(z) =M(z)/M(z)\nalong the cubic axes [100], [010], and [001], respectively.\nWe assume the magnetic properties of the layer to be ho-\nmogeneouslaterally(within the xyplane) and inhomoge-\nneous vertically (along z); the anisotropy parameters in\nEq. (1) and the magnetization are consequently a func-\ntion of the spatial variable z. To obtain the anisotropy\nparameters from Eq. (1) in units of energy density, it\nwould therefore be required to know the zdependence\nand the absolute value of M.\nThe minimum of Eq. (1) determines the equilibrium\norientation of the magnetization, given by the angles\nθ0=θ0(z) andφ0=φ0(z), cf. Fig. 1. To describe the\nmagnetization dynamics, we introduce a new frame of\nreference (1 ,2,3) shown in Fig. 1, in which the equilib-\nrium orientation of the magnetization m0coincides with\nthe axis 3. For small perturbations, the magnetization\nprecesses around its equilibrium with finite transverse\ncomponents of the magnetization mi(i= 1,2) as illus-\ntrated in the inset in Fig. 1. The transformation between\nthe two coordinate systems is given in the Appendix A\nby Eqs. (A1) and (A2). We write for the (normalized)\nmagnetization3\nm=\n0\n0\n1\n\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nm0+\nm1\nm2\n0\n+O(m2\n1,m2\n2).(2)\nThe evolution ofthe magnetizationunder the influence\nof an effective magnetic field µ0Heffis described by the\nLLG equation42,43\n∂tm=−γm×µ0Heff+αm×∂tm,(3)\nwhereγis the gyromagnetic ratio and αa phenomeno-\nlogical damping parameter. The effective magnetic field\nis given by36\nµ0Heff=−∇mG+Ds\nM∇2M+µ0h(t),(4)\nwhere∇m= (∂m1,∂m2,∂m3) isthe vectordifferentialop-\neratorwith respect to the componentsof m,Ds= 2A/M\nis the exchange stiffness with the exchange constant A,\n∇2is the spatial differential operator ∇2=∂2\nx+∂2\ny+∂2\nz,\nandh(t) =h0e−iωtis the externally applied microwave\nmagnetic field with the angular frequency ω;h(t) is ori-\nented perpendicularly to µ0H. Since the magnetic prop-\nerties are independent of xandy, Eq. (3) simplifies to\n∂tm=−γm×[−∇mG+Dsm′′+µ0h(t)]+αm×∂tm,\n(5)\nwithm′′=∂2\nzm, neglecting terms of the order of m2\ni(for\ni= 1,2). By definition of the (1 ,2,3) coordinate system,\nthe only non-vanishing component of ∇mGin the equi-\nlibrium is along the 3-direction. For small deviations of\nmfrom the equilibrium we find44\n∇mG=\nG11m1+G21m2\nG12m1+G22m2\nG3\n, (6)\nwhere we have introduced the abbreviations Gi=\n∂miG|m=m0andGij=∂mi∂mjG|m=m0; the explicit ex-\npressions for these derivatives are given in the Appendix\nA.\nInthefollowing,wecalculatethetransversemagnetiza-\ntion components assuming a harmonic time dependence\nmi=mi,0e−iωt. The linearized LLG equation, consider-\ning only the transverse components, reads as\n/parenleftbigg\nH11H12\nH21H22/parenrightbigg/parenleftbigg\nm1\nm2/parenrightbigg\n−Ds/parenleftbigg\nm′′\n1\nm′′\n2/parenrightbigg\n=µ0/parenleftbigg\nh1\nh2/parenrightbigg\n,(7)\nwherewe haveintroducedthe abbreviations H11=G11−\nG3−iαω/γ,H12=H∗\n21=G12+iω/γ, andH22=G22−\nG3−iαω/γ. We have dropped all terms which are non-\nlinear inmiand products of miwith the driving field.\nResonant uniform precession of the magnetization\n(m′′\ni= 0) occurs at the so called uniform resonance fieldµ0Huni(z), which is found by solving the homogeneous\n(h= 0) equation\nH11(z)H22(z)−H12(z)H21(z) = 0\n⇔(G11−G3)(G22−G3)−G2\n12=/parenleftbiggω\nγ/parenrightbigg2\n(8)\nforµ0H, neglecting the Gilbert damping ( α= 0). Equa-\ntion(8)canbeusedtoderiveanisotropyparametersfrom\nangle-dependent FMR spectra. As extensively discussed\nby Baselgia et al.44, using Eq. (8) is equivalent to using\nthe method of Smit and Beljers, which employs second\nderivatives of the free enthalpy with respect to the spher-\nical coordinates.41,45,46\nTo illustrate the role of the uniform resonance field in\nthe context of spin-wave resonances, we consider the spe-\ncial case where magnetization is aligned along the [001]\ncrystal axis ( θ0= 0), before we deal with the general\ncase of arbitrary field orientations. Neglecting the uniax-\nial in-plane anisotropy( B1¯10= 0) since this anisotropyis\ntypically weaker than all other anisotropies,13,41we find\nG3=−µ0H+2B001+4B4⊥andG11=G22=G12= 0,\nresulting in the uniform resonance field\nµ0H001\nuni(z) =ω/γ+2B001(z)+4B4⊥(z).(9)\nTo find the eigenmodes of the system, we consider the\nunperturbed and undamped case, i.e., α= 0 and h= 0\nin Eq. (7). With m2=im1= ˜mwe find the spin-wave\nequation\nDs˜m′′+µ0H001\nuni(z)˜m=µ0H˜m (10)\nin agreementwith Ref. 36. The relationofthe anisotropy\nparameters defined in Ref. 36 to the ones used here\nis given by B001=K100\neff/M+B1¯10,B1¯10=−K011\nu/M,\n2B4⊥=−K⊥\nc1/M, and 2B4/bardbl=−K/bardbl\nc1/M. Equation (10)\nis mathematically equivalent to the one-dimensional\ntime-independent Schr¨ odinger equation, where the\nuniform resonance field corresponds to the potential,\n˜mto the wave function, µ0Hto the energy, and Dsis\nproportional to the inverse mass. To calculate the actual\nprecession amplitude of the magnetization, the coupling\nof the eigenmodes of Eq. (10) to the driving field is\nrelevant, which is proportional to the net magnetic\nmoment of the mode.36,38In analogy to a particle in a\nbox, the geometry of the uniform resonance field as well\nas the boundary conditions determine the resonance\nfields and the spatial form of the precession amplitude.\nFor the remainder of this work, we assume the spins\nto exhibit natural freedom at the boundaries of the\nfilm, i.e.,∂z˜m= ˜m′= 0 at the interfaces,36,47since\nthese boundary conditions have been shown to describe\nthe out-of-plane SWR data of similar samples well.36\nTo graphically illustrate the influence of the uniform\nresonance field on the SWR modes, we consider in Fig. 2\na ferromagnetic layer with a thickness of 50 nm with\nconstant magnetic properties across the layer (a) and\nwith a linearly varying uniform resonance field (c); in4\n(a) (b)\n(c) (d) m (arb. u.)~ m (arb. u.)~\nFIG. 2: Simulation to demonstrate the influence of the uni-\nform resonance field µ0H001\nunion theSWRmodes for m0||[001],\nassuming circular precession. In (a), µ0H001\nuniis set to be con-\nstant across the layer, while in (c) it varies linearly (blue ,\ndashed lines), in analogy to a square potential and a trian-\ngular potential, respectively. The dotted black lines are t he\nresonance fields, calculated assuming boundary conditions of\nnatural freedom, see text. The solid red lines show the eigen -\nmodes of the system, i.e., the precession amplitude ˜ mof the\nmagnetization; for each mode the dotted line corresponds to\n˜m= 0. As can be seen in (a), for a constant uniform reso-\nnancefieldthefirstmodeoccursattheuniformresonance field\nand exhibits a constant precession amplitude across the lay er,\ni.e., an FMR mode. The second and third mode (higher-order\nmodes are not shown) exhibit a non-uniform magnetization\nprofile. In order to couple to the driving field the modes need\nto have a finite net magnetic moment. As can be seen in\n(a), the positive and negative areas of the second and third\nmode are equal, thus these modes are not visible in the SWR\nspectrum (b). This is in contrast to the case of the linearly\nvarying uniform resonance field (c) where the mode profile is\ngiven by Airy functions, which have a nonzero net magnetic\nmoment also for the second and third mode, resulting in a\nfinite SWR intensity of these modes (d). The spectra in (b)\nand (d) were calculated by integrating over the eigenmodes\n˜mand convoluting the square of the result with Lorentzians.\nboth cases we assume Ds= 13 Tnm−2, a similar value\nas obtained in previous studies.36For these conditions,\nwe numerically solve Eq. (10) by the finite difference\nmethod described in the Appendix B1, in orderto obtain\nthe resonance fields (eigenvalues) and the zdependence\nof the transverse magnetic moments (eigenfunctions).\nTo which amount a mode couples to the driving field is\ndetermined by the net magnetic moment of the mode,\nwhich is found by integrating ˜ m(z) over the thickness of\nthe film. For the magnetically homogeneous layer, the\nonly mode that couples to the driving field is the uniformprecession mode at µ0H001\nuni, since modes of higher order\nhave a zero net magnetic moment [Fig. 2 (a)], resulting\nin one resonance at the uniform resonance field, cf. Fig. 2\n(b). For the non-uniform layer, with µ0H001\nuni(z) linearly\nvarying across the film, the mode profile is given by Airy\nfunctions31,36,38and various non-uniform modes couple\nto the driving field, resulting in several spin-wave reso-\nnances with their amplitude proportional to the square\nof the net magnetic moment36,38of the corresponding\nmode, cf. Fig. 2 (c) and (d).\nWe now turn to the general case of arbitrary field ori-\nentations. Due to the magnetic anisotropy profile, the\nmagnetizationorientationisaprioriunknownandafunc-\ntion ofzandµ0H. Furthermore, the assumption of a\ncircularly precessing magnetization is not generally jus-\ntified. To solve Eq. (7) for arbitrary field orientations,\nwe employ a finite difference method as outlined in the\nAppendix B2. BysolvingEq.(7), weobtainthe zdepen-\ndent generalized Polder susceptibility tensor ¯ χ(µ0H,z),\nwhich relates the transverse magnetization components\nMi(z) =M(z)mi(z) with the components of the driving\nfield by\n/parenleftbigg\nM1\nM2/parenrightbigg\n= ¯χ(µ0H,z)/parenleftbigg\nh1\nh2/parenrightbigg\n. (11)\nInamicrowaveabsorptionmeasurement, thecomponents\nMiwhich are out-of-phase with the driving field are de-\ntected. The absorbed power density is related to the\nimaginary part of ¯ χ(µ0H,z) and can be calculated by48\nP=ωµ0\n2z0Im/braceleftbigg/integraldisplay0\n−z0/bracketleftbigg/parenleftbigh∗\n1,h∗\n2/parenrightbig\n¯χ(µ0H,z)/parenleftbigg\nh1\nh2/parenrightbigg/bracketrightbigg\ndz/bracerightbigg\n,\n(12)\nwherez0is the thickness ofthe ferromagneticlayer. Note\nthat the position coordinate zis negative in the film,\ncf. Fig. 1.\nTo obtain an impression of how gradients in differ-\nent anisotropy parameters influence the SWR spectra,\nwe plot in Fig. 3 simulated SWR spectra together with\nthe magnetization precession cone as a function of depth\nin the ferromagnetic layer. We assume a constant sat-\nuration magnetization (its value is not relevant for the\noutcome of the simulation), a constant exchange stiff-\nnessDs= 35 Tnm2unless otherwise specified, α= 0.09,\nandB001= 90 mT, B4||=−50 mT,B4⊥= 15 mT.\nIn Fig. 3 (a), we assume B001to vary across the layer\nthickness according to B001(z) =B001−b001×zwith\nb001=−0.8 mT/nm. Figure 3 (a i) shows the simu-\nlated SWR spectra calculated by taking the first deriva-\ntive of Eq. (12) with respect to µ0Hfor different angles\nψdefined in the inset in Fig. 3 (c iv). We observe sev-\neral SWR modes for µ0H||[001] which become less as\nµ0His tilted away from [001]. At ψ= 40◦only one\nmode is visible while for ψ= 0◦we again observe mul-\ntiple SWR modes. This observation can be understood\nby considering the uniform resonance fields as a func-\ntion of the depth for these orientations. In Fig. 3 (a5\n90\n03060\n90\n03060\n90\n03060\nSWR Intensi ty (arb . units)ψ (deg.) ψ (deg.) ψ (deg.)\n200 600 400\nµ0H (mT)ψµ0H\n[110][001](a i) (a ii) (a iii) (a iv)\n(b i) (b ii) (b iii) (b iv)\n(c i) (cii) (ciii) (c iv)Im(m1m2-m1m2) (arb. u.) * *\nFIG. 3: Atlas illustrating the influence of gradients in the a nisotropy parameters on SWR spectra. In (a) all anisotropy\nparameters are kept constant with the values given in the tex t, exceptB001which is varied linearly. Correspondingly, in (b)\nand (c)B4⊥andB4||were varied linearly, respectively. Panels (i) show the firs t derivative of simulations using Eq. (12) with\nrespect toµ0Hand panels (ii)-(iv) show the precession cone Im( m∗\n1m2−m1m∗\n2) in a color plot together with the uniform\nresonance field µ0Huni(z) (dashed blue lines) at three different external field orient ations; the black dotted lines indicate the\nresonance field positions of the modes. Panel (a i) additiona lly shows the influence of a linear gradient in the exchange st iffness\nparameter on the spin-wave spectra, see text for further det ails and discussion.\nii)-(a iv), we show the uniform resonance field (dashed\nblue line) for ψ= 0◦,ψ= 30◦, andψ= 90◦, respec-\ntively, together with the magnetization precession cone\nIm(m∗\n1m2−m1m∗\n2) in a contour plot as a function of\ndepth andµ0H. Atψ= 90◦, the uniform resonance field\nvaries strongly across the film, which can be understood\nby considering Eq. (9). This results in several spin wave\nmodes with their resonance fields indicated by dotted\nlines.\nFor other field orientations, the formula for the uni-\nform resonance field can also be derived but results in a\nlonger, more complex equation than Eq. (9). Important\nin this context is that positive values of B001lead to an\nincrease(decrease) ofthe resonancefield for the magneti-\nzation oriented perpendicular (parallel) to the film plane,\naccountingfor the reversedsign ofthe slopesof µ0Huniin\nFig. 3 (a ii) and (a iv). Consequently, in between those\ntwo extreme cases µ0Hunimust be constant across the\nlayer for some field orientation, in our case for ψ= 30◦,\nresulting in a single SWR mode, cf. Fig. 3 (a i) and (a\niii). In addition to the SWR simulations with constantDs, weplotinFig.3(ai)simulatedSWRspectrawith Ds\nvarying linearly across the film with Ds= 35−65 Tnm2\n(blue, dotted lines) and Ds= 35−5 Tnm2(green, dotted\nlines). A decreasing Dsleads to a decreasing spacing in\nthe modes and vice versa for an increasing Dsas can be\nseen, e.g., for µ0H||[001].\nIn Fig. 3 (b), we consider the case where all magnetic\nparameters are constant with the values given above, ex-\nceptB4⊥(z) =B4⊥−b4⊥×zwithb4⊥=−0.4 mT/nm.\nAs evident from Eq. (9), this results in the same slope of\nµ0Huniforψ= 90◦as in the case above where we varied\nB001only, cf. Fig. 3 (a iv) and (b iv). In contrast to the\ncasedepictedin(a), however,herefor ψ= 0◦theuniform\nresonancefield is constant. This can be understood when\nevaluating the parametersthat enter in the calculation of\nthe uniform resonance field [Eq. (8)]. If mis in the film\nplane, none of the parameters in Eqs. (A4)-(A6) depends\nonB4⊥, resulting in a constant uniform resonance field\nforψ= 0◦. Asmis tilted away from the film plane,\nB4⊥enters in some of the terms Eqs. (A4)-(A6). As a\nconsequence, µ0Hunivaries, first such that it increases6\n[cf. Fig. 3 (b iii)] and finally, such that it decreases as a\nfunction of depth [cf. Fig. (b iv)].\nFinally, we discuss the case where all parameters are\nconstant except B4||(z) =B4||−b4||×zwithb4||=\n−0.4 mT/nm [Fig. 3 (c)]. Here, µ0Huniis constant for\nψ= 90◦aspredictedbyEq.(9). As mistiltedawayfrom\n[001] a varying B4||leads to a varying uniform resonance\nfield as shown in Fig. 3 (c ii) and (c iii). Here, a sign\nreversal of the slope as it was the case in Fig. 3 (a) and\n(b) does not take place and multiple resonances occur,\nstarting from ψ= 60◦[Fig. 3 (c i)].\nIII. EXPERIMENTAL RESULTS AND\nDISCUSSION\n(Ga,Mn)As samples with a nominal Mn concentration\nof≈4% were grown on (001)-oriented GaAs substrates\nby low-temperature molecular-beam epitaxy at a sub-\nstrate temperature of 220◦C using V/III flux ratios of\n1.1, 1.3, 1.5, and 3.5, referred to as sample A, B, C, and\nD, respectively. The layer thickness was 210-280 nm as\ndetermined from the ECV measurements, cf. Fig. 4. For\nsamples with V/III flux ratios /lessorsimilar3 a gradient in the hole\nconcentration has been reported,39hence this set of sam-\nples was chosen to study the influence of a gradient in p\non the out-of-plane magnetic anisotropy. Further details\non the sample growth can be found in Refs. 39 and 41.\nThe hole concentration profile of the as-grown\n(Ga,Mn)As layers were determined by ECV profiling us-\ning a BioRad PN4400 profiler with a 250 ml aqueous\nsolution of 2.0 g NaOH+9.3 g EDTA as the electrolyte.\nFor further details on the ECV analysis see Ref. 39. The\nresults of the ECV measurements for the layers investi-\ngated areshown in Fig. 4(a). Except for the sample with\nV/III=3.5, they reveala nearlylinearly varying hole con-\ncentration across the layer thickness with different slopes\nand with the absolute value of the hole concentration\nat the surface of the layer varying by about 20%. The\nprofiles are reproducible within an uncertainty of about\n15%.\nTo investigate the magnetic anisotropy profiles of the\nsamples, weperformedcavity-basedFMRmeasurements,\nusing a Bruker ESP300 spectrometer operating at a mi-\ncrowave frequency of 9.265 GHz ( X-band) with a mi-\ncrowave power of 2 mW at T= 5 K; we used magnetic\nfield modulation at a frequency of 100 kHz and an ampli-\ntude of3.2mT. Since wearemainlyinterested in the out-\nof-plane magnetic anisotropy, we recorded spectra for ex-\nternalmagneticfieldorientationswithinthe crystalplane\nspanned by the [110] and [001] crystal axes in 5◦steps,\ncf. the inset in Fig. 5. For each orientation, the field was\nramped to 1 T in order to saturate the magnetization\nand then swept from 650 mT to 250 mT; the spectra for\nthe samples investigated are shown in Fig. 5.\nWe start by discussing qualitative differences in the\nspectra. The samples A and B exhibit several pro-\nnounced resonances for the external field oriented along(a)\n(b)\nFIG.4: (a)Theholeconcentration inthedifferent(Ga,Mn)As\nsamples is shown as a function of the depth within the layers\nas determined by ECV profiling. (b) The uniform resonance\nfieldsµ0H001\nuni(z) for the four samples obtained from the sim-\nulations for the out-of-plane orientation of the external fi eld\n(ψ= 90◦) as a function of the depth.\n[001], which we attribute to standing spin-wave reso-\nnances [Fig. 5 (a) and (b)]. For these samples, the [001]\ndirection is the magnetically hardest axis since at this\norientation the resonance field of the fundamental spin-\nwavemode is largerthan at all other orientations. As the\nexternal field is rotated into the film plane, the resonance\nposition of this mode gradually shifts to lower field val-\nues as expected for a pronounced out-of-plane hard axis.\nIn contrast, the samples C and D exhibit the largest res-\nonance fields for a field orientation of 50-60◦[Fig. 5 (c)\nand (d)] pointing to an interplay of second- and fourth-\norder out-of-plane anisotropy with different signs of the\ncorresponding anisotropy parameters. These samples ex-\nhibit spin-wave resonances as well, however they are less\npronounced than for samples A and B.\nTo quantitatively model the spin-wave spectra we nu-\nmerically solve for each magnetic field orientation the\nspin-wave equation (7) by the finite difference method as\noutlined in the Appendix B2. Although this method al-\nlows for the modeling of the SWR for arbitrary profiles\nof the anisotropy parameters, the exchange stiffness, the\nGilbert damping parameter, and the saturation magne-\ntization, we assume the parameters to vary linearly as\na function of z. This approach is motivated by the lin-\near gradient in the hole concentration, which in first ap-\nproximation is assumed to cause a linear gradient in the7\nψ[001]\n[110]µ0Hext\nSimulationExperiment(a) (b)\n(d) (c)\nV/III=1.5 V/III=3.5V/III=1.3 V/III=1.1\nFIG. 5: The spin-wave resonance data (dotted, blue lines) ar e shown together with simulations (red, solid lines) using t he\nnumerical procedure described in the text and in the Appendi x B2. The data were obtained as a function of the external\nmagnetic field orientation and magnitude for samples with a V /III flux ratio of (a) 1.1, (b) 1.3, (c) 1.5, and (d) 3.5. The\nrotation angle ψis defined in the inset and the parameters used for the simulat ions are summarized in Tab. I.8\nanisotropy parameters, resulting in the spin-wave reso-\nnances observed in the samples.31,36In Tab. I, we have\nsummarized the parameters used in the simulation for\nthe different samples. The parameters in capital let-\nters denote the value at the surface of the sample while\nthe ones in lower-case letters denote the slope of this\nparameter; e.g., the zdependence of the second-order,\nuniaxial out-of-plane anisotropy parameter is given by\nB001(z) =B001−b001×z. The layer thickness used for\nthe simulationcanbe inferredfromFig. 4(a) andwasde-\ntermined from the ECV data under the assumption that\nat the position where the hole concentration rapidly de-\ncreases the magnetic properties of the layer abruptly un-\ndergo a transition from ferromagnetic to paramagnetic.\nFor the simulations, we divided each film into n= 100\nlayers with constant magnetic properties within each\nlayer. For the gyromagnetic ratio we used γ=gµB//planckover2pi1\nwithg= 2.21\nAs result of the simulation we obtain the Polder sus-\nceptibility tensor ¯ χ(µ0H,z) and the transverse magneti-\nzationcomponentsasafunctionof zandµ0H. Addition-\nally, weobtainthe zdependenceoftheuniformresonance\nfield by solving Eq. (8) for each field orientation. In an\nSWR absorption experiment with magnetic field mod-\nulation, the obtained signal is proportional to the first\nderivative of the absorbed microwave power with respect\nto the magnetic field. Thus, we calculate the absorbed\npowerusingthesimulatedsusceptibilityandEq.(12) and\nnumerically differentiate the result in order to compare\nthe simulated SWR spectra with the experiment. Addi-\ntionally, we use a global scaling factor, accounting, e.g.,\nfor the modulation amplitude, which is the same for all\nfield orientations, and we multiply all the simulated data\nwith this factor. In Fig. 5, we plot the experimental data\ntogether with the simulations using the parameters given\nin Tab. I, demonstrating that a reasonableagreementbe-\ntweentheoryandexperimentcanbefoundwithonesetof\nsimulation parameters for all magnetic field orientations\nfor each sample.\nWe will now exemplarily discuss the angle dependence\nof the SWR spectrum of sample A shown in Fig. 5(a)\nbased on the uniform resonance field and the resulting\nmagnetizationmodeprofileobtainedfromthesimulation.\nTo this end, we plot in Fig. 6 (a)-(c) the magnetization\nprecession amplitude Im( m∗\n1m2−m1m∗\n2) for selected ex-\nternal field orientations as a function of depth and ex-\nternal magnetic field in a contour plot, together with the\ncorresponding uniform resonance field. In Fig. 6 (d)-(f),\nwe show for each external field orientation a magnifica-\ntion of the corresponding SWR spectrum together with\nthesimulation. Notethatincontrasttothenormal-mode\napproach (Appendix B1) used to calculate the modes in\nFig. 2, where the coupling of each mode to the cavity\nfield has to be found by integration, the approach elabo-\nrated in the Appendix B2 directly yields the transverse\nmagnetization components, already accounting for the\ncoupling efficiency and the linewidth. Further, the ap-\nproach presented in the Appendix B2 is also valid whenthe differencein the resonancefields oftwomodesis com-\nparable with or smaller than their linewidth, in contrast\nto the normal-mode approach38.\nIf the external field is parallel to the surface normal\n(ψ= 90◦) the uniform resonance field varies by about\n350 mT across the film thickness [cf. the dashed line\nin Fig. 6 (a)], resulting in several well-resolved stand-\ning spin-wave modes. The spin-wave resonance fields are\nplotted as dotted lines in Fig. 6 (a); since the spacing of\nthe resonance fields is larger than the SWR linewidth,\nthe modes are clearly resolved, cf. Fig. 6 (a) and (d).\nIn the simulation two regions with different b001values\nwereused in orderto reproducethe spacingofthe higher-\norder spin-wave modes found in the experiment. Using\nthe same slope as in the first 100 nm for the entire layer\nwould lead to a smaller spacing between the third and\nhigher order modes. Instead of defining two regions with\ndifferent slopes b001, a gradient in the exchange stiffness\nwith positiveslopecouldalsobe usedto modelthe exper-\nimentally found mode spacing as discussed in the context\nof Fig. 3. Since the exchange interaction in (Ga,Mn)As is\nmediated by holes12andpdecreases across the layer, we\nrefrainfrom modeling ourresultswith a positivegradient\ninDs. Further, the results in Ref. 36 rather point to a\nnegative gradient in Dsin a similar sample. However, a\ndecreasing Mn concentration as a function of the depth\ncould lead to an increase of Ds.34\nFinally, we note, since B1¯10= 0 in the simulations,\nthe magnetization precesses circularly for ψ= 90◦and\nthus Im(m∗\n1m2−m1m∗\n2) = 2sin2τ,49with the precession\ncone angle τ. For all other orientations, mprecesses\nelliptically which is accounted for in our simulations. In\nthe simulations of the precession amplitudes, we have\nassumed an externally applied microwave magnetic field\nwithµ0h= 0.1 mT.\nAt an external field orientation of ψ= 50◦the uni-\nform resonance field is nearly constant across the layer,\nand consequently only one SWR mode is observed with\nan almost uniform magnetization precession across the\nlayer, cf. Fig. 6 (b). The precession amplitude is a mea-\nsureforthe SWR intensity. While the fundamental mode\natψ= 90◦exhibits a larger precession cone at the in-\nterface, it rapidly decays as a function of the depth, in\ncontrast to the nearly uniform precession amplitude for\nψ= 50◦. Since the entire layer contributes to the power\nabsorption, consequently, the SWR mode at ψ= 50◦is\nmore intense than the fundamental mode for ψ= 90◦,\nwhich is indeed observed in the experiment [cf. Fig. 6 (d)\nand (e)].\nFor the magnetic field within the film plane [ ψ= 0◦,\ncf.Fig.6(c)], the uniformresonancefieldagainvarieslin-\nearly across the film, however in a less pronounced way\nthan for the out-of-plane field orientation and with an\nopposite sign of the slope. The sign reversal of the slope\ncan be understood in terms of the uniaxial out-of-plane\nanisotropy parameter B001: positive values of these pa-\nrameters lead to an increase (decrease) of the resonance\nfield for the magnetization oriented perpendicular (par-9\nTABLE I: Simulation parameters and their zdependence of the samples under study as obtained by fitting t he simulations to\nthe SWR measurements. For the anisotropy parameters the cap ital letters denote the value at the surface of the film and the\nlower case letters the slope as described in the text. For sam ple A, the first value of b001was used for the first 100 nm and the\nsecond one for the remaining layer. In addition to the anisot ropy parameters, the saturation magnetization is also assu med to\nvary linearly across the layer, while its absolute value is u nknown and not important for the SWR simulations.\nSample V/III B001 b001 B4/bardblb4/bardblB4⊥b4⊥Dsα∂M(z)\n∂zM(0)\n(mT) (mT\nnm) (mT) (mT\nnm) (mT) (mT\nnm) (Tnm2) (1\nµm)\nA 1.1 90 -0.1, -0.3 -50 0.05 25 -0.3 35 0.09 -3\nB 1.3 130 -0.5 -50 0 0 0 20 0.06 -4\nC 1.5 75 -0.4 -55 -0.04 -15 0 40 0.11 -4\nD 3.5 91 -0.3 -55 -0.04 -15 0 20 0.09 -3\nallel) to the film plane, accounting for the slopes of the\nuniform resonance fields in Fig. 6. Since the gradient in\nthe uniform resonance field is less pronounced for ψ= 0◦\nthan forψ= 90◦, the spin-wave modes are not resolved\nforψ= 0◦, since their spacing is smaller than the SWR\nlinewidth, leading to one rather broad line [cf. Fig. 6 (c)\nand (f)]. A steeper gradient in B4||in combination with\na different Gilbert damping (or with an additional inho-\nmogeneous damping parameter) and amplitude scaling\nfactor, could improve the agreement of simulation and\nexperiment in the in-plane configuration, as discussed\nlater. A detailed study of the in-plane anisotropy pro-\nfile is however beyond the scope of this work. Given that\nthe presented simulations were obtained with one set of\nparameters, the agreement of theory and experiment is\nreasonably good also for the in-plane configuration, since\nsalient features of the SWR lineshape are reproduced in\nthe simulation.\nHaving discussed the angle-dependence of the SWR\nspectra, we turn to the zdependence of the out-of-plane\nanisotropy of sample A. Our simulations reveal that it\nis governed by the zdependence of both B001(z) and\nB4⊥(z). Assuming only a gradient in B001results in a\nreasonable agreement of theory and experiment for the\nexternal field oriented along [001] and [110], but fails to\nreproduce the spectra observed for the intermediate field\norientations, e.g., ψ= 50◦. This is illustrated by the\ndashed black line in Fig. 6 (e), which represent simu-\nlations with a constant B4⊥(z) forψ= 50◦. As can\nbe seen, this simulation produces several spin-wave res-\nonances, whereas in the experiment only one resonance\nis present, which is better reproduced by the simulation\nwith bothB001(z) andB4⊥(z) varying across the layer.\nWe will now discuss the anisotropy parameters of all\nsamples. In contrast to sample A, the out-of-plane\nanisotropy profile of all other samples appears to be gov-\nerned by a gradient in B001(z). As already discussed\nqualitatively, the hard axis of the samples is determined\nby an interplay of B001andB4⊥. For sample A and B\nB4⊥is positive and zero, leading to an out-of-plane hard\naxis. Incontrast,sampleCandDexhibitanegative B4⊥,\nleading to a hard axis between out-of-plane and in-plane.\nTheB4||parameter is negative and of similar magnitude\nfor all samples.Since the out-of-plane anisotropy profile of sample A\nis governed by B001(z) andB4⊥(z), a comparison of the\nout-of-plane anisotropy profile between all samples based\non anisotropy parameters is difficult. We therefore com-\npare the uniform resonance fields, where both anisotropy\nparameters enter. As evident from Fig. 6, the strongest\ninfluence of the magnetic inhomogeneity of the layers on\nthe uniform resonance fields is observed for the exter-\nnal field along [001]. To compare the hole concentration\nprofile in Fig. 4 (a) with the anisotropy profile, we there-\nfore plot in Fig. 4 (b) the zdependence of the uniform\nresonance field µ0H001\nunifor this field orientation. The\nfigure demonstrates that the gradient in µ0H001\nuniis cor-\nrelated with the gradient in p. For the sample with the\nstrongest gradient in pthe gradient in µ0H001\nuniis also\nmost distinct while the samples with a weaker gradient\ninpexhibit a less pronounced gradient in µ0H001\nuni. How-\never, for sample D, exhibiting a nearly constant p, we\nstillobservestandingspinwaveresonancesfor µ0H||[001]\n[Fig. 5 (d)], reflected in a slight gradient of µ0H001\nuni. This\nobservation suggests that aditionally other mechanisms\nlead to a variation of the anisotropy profile. One possi-\nbilitywouldbeagradientintheelasticstrainofthelayer,\ndue to a non-homogeneous incorporation of Mn atoms in\nthe lattice. However, x-ray diffraction measurements of\nthis sample, in combination with a numerical simulation\nbased on dynamic scattering theory, reveal a variation of\nthe vertical strain ∆ εzzas small as 3 ×10−5across the\nlayer. According to the measurements in Ref. 13, such a\nvariation in strain would lead to a variation of the B001\nparameter by a few mT only, insufficient to account for\nthe variation of µ0Huniby almost 100 mT across the\nlayer. A more likely explanation seems to be a varia-\ntion of the saturation magnetization, which should also\ninfluence the anisotropy parameters. In the simulation,\na non-homogeneous saturation was assumed, potentially\nexplaining also the observed gradient in the anisotropy\nparameters and therefore in the uniform resonance field.\nIn contrast to the out-of-plane anisotropy parameters,\nB4||was found to depend only weakly on z, for all sam-\nples except sample B where it was constant. Addition-\nally,B1¯10, typically of the order of a few mT,13might\nhave an influence and interplay with B4||in determining\nthe in-plane anisotropy. We here however focus on the10\n0°\n(b) (c)\n(a)Im(m1m2-m1m2) (10-5) * *\n0 1.2Im(m1m2-m1m2) (10-5) * *\n0 0.3Im(m1m2-m1m2) (10-5) * *\n0 0.53\n(d) (e) (f)001arb.(a)\nFIG. 6: Simulated magnetization mode profile and uniform res onance field of sample A. The contour plots show the magneti-\nzation precession amplitude Im( m∗\n1m2−m1m∗\n2) as a function of the position within the film and the external magnetic field\nfor the external field aligned (a) along [001], (b) at an angle of 50◦with respect to [110] (cf. the inset in Fig. 5) and (c) along\n[110]. The blue, dashed lines in (a)-(c) show the uniform res onance field, obtained by numerically solving Eq. (8) for eac h given\nfield orientation. The dotted black lines in (a) indicate the resonance magnetic fields. In (d)-(f), a magnification of the data\n(blue dotted lines) and simulation (red solid lines) from Fi g. 5 (a) is shown using the same scale for all orientations. In (e), a\nsimulation with a different set of parameters is shown for com parison (black, dashed line), see text.11\nout-of-plane anisotropy and therefore neglect B1¯10in our\nsimulations. An in-plane rotation of the external field\nwould be required for a more accurate measurement of\nB4||andB1¯10, but is outside the scope of this work.\nAccording to the valence-band model in Ref. 12, an\noscillatory behavior of the magnetic anisotropy parame-\nters is expected as a function of p. Therefore, depend-\ning on the absolute value of p, different values for, e.g.,\n∂B001/∂pare expected. In particular, there are regions\nwhere a anisotropy parameter might be nearly indepen-\ndent ofpand other regions with a very steep pdepen-\ndence. Since the absolute value of pis unknown, a quan-\ntitative discussion of the pdependence of the obtained\nanisotropy parameters based on the model in Ref. 12\nis not possible. In addition to p, thep-dexchange\nintegral,12which mayalsovary asa function ofthe depth\nin a non-homogneous film, also influences the anisotropy\nparameters,12further complicating a quantitative analy-\nsis.\nFor all samples, we used a constant exchange stiffness\nDsin our modeling. As alluded to above, there is some\nambiguity in this assumption, since the exchange stiff-\nness and the gradient in the anisotropy both influence\nthe mode spacing. For simplicity, however, we intended\nto keep as many simulation parameters as possible con-\nstant. The absolute values obtained for the exchange\nstiffness agree within a factor of 2 with the ones obtained\nin previous experiments36,50but are a factor of 2-4 larger\nthan theoretically predicted.51For the reasons discussed\nabove,thereisalargeuncertaintyalsointhederivationof\nthe absolute value of Dsfrom standing spin-wave modes\nin layers with a gradient in the magnetic anisotropy con-\nstants.\nIn order to use one parameter set for all field-\norientations, the Gilbert damping parameter was as-\nsumed to be isotropic in the simulations. The modeling\nof the SWR data could be further improved by assum-\ning a non-isotropic damping, its value being larger for\nµ0H||[110] than for µ0H||[001] [cf. Fig. 5]. This how-\never, only improves the result when assuming a field\norientation-dependent scaling factor for the amplitude,\nwhich could be motivated, e.g., by the assumption that\nthe microwave magnetic field present at the sample po-\nsition depends on the sample orientation within the\ncavity. The absolute values of αdetermined here are\ncomparable with the ones obtained by ultra-fast opti-\ncal experiments,52but are larger than the typical α=\n0.01...0.03 values found by frequency-dependent FMR\nstudies.53,54As already alluded to, inhomogeneous line-\nbroadening mechanisms may play a dominant role,54in\nparticular for as-grown samples.55We therefore assume\nthat the values for αobtained in this study overesti-\nmate the actual intrinsic Gilbert damping. A frequency-\ndependent SWR study would be required to determine\nthe intrinsic α. Such a study could possibly also reveal a\np-dependent αas theoretically predicted.55In our study,\nassuming a zdependent αdid not improve the agree-\nment between simulation and experiment, corroboratingthe conjecture that inhomogeneous broadening mecha-\nnisms dominate the linewidth and therefore obscure a\npossiblezdependence of α.\nIV. SUMMARY\nWehavepresentedafinitedifference-typemodelingap-\nproach for standing spin-wave resonances based on a nu-\nmerical solution of the LLG equation. With this generic\nformalism, SWR spectra can be simulated accounting for\nelliptical magnetization precession, for arbitrary orienta-\ntionsofthe externalmagneticfield, andforarbitrarypro-\nfiles of all magnetic properties, including anisotropy pa-\nrameters, exchange stiffness, Gilbert damping, and sat-\nuration magnetization. The approach is applicable not\nonly to (Ga,Mn)As but to all ferromagnets.\nFour(Ga,Mn)Assamples, epitaxiallygrownwithV/III\nflux ratios of 1.1, 1.3, 1.5, and 3.5 were investigated by\nECV and spin-wave resonance spectroscopy, revealing a\ncorrelation of a linear gradient in the hole concentration\nwith the occurrence of standing spin wave resonances, in\nparticularfortheexternalfieldorientedout-of-plane. Us-\ning the presented modeling approach, the SWR spectra\ncould be reproduced in a simulation with one parameter\nset for all external field orientations. The simulation re-\nsults demonstrate that the profileof the out-of-planeuni-\nformresonancefieldiscorrelatedwiththeholeconcentra-\ntion profile. However, our measurements and simulations\nshow,that anon-uniformholeconcentrationprofileisnot\nthe only cause that leads to the observed non-uniform\nmagnetic anisotropy; possibly, a variation in the satura-\ntion magnetization also influences the anisotropy param-\neters. To gain a quantitative understanding of this issue,\nmore samples with known hole concentrations would be\nrequired, where both the absolute values and the profiles\nofpare varied. Such a study was, however, outside the\nscope of this work.\nBesides the modeling of SWR intensities and\nlinewidths, the presented formalism yields the magne-\ntization precession amplitude as a function of the po-\nsition within the ferromagnet. It can therefore be used\nto investigate spin-pumping intensities in (Ga,Mn)As/Pt\nbilayers.27The spin-pumping signal, detected as a volt-\nage across the Pt layer, should be proportional to\nthe magnetization precession cone in the vicinity of\nthe (Ga,Mn)As/Pt interface. By measuring the spin-\npumping signal as well as the SWR intensities of\n(Ga,Mn)As/Pt and by using our modeling approach, it\nshould be possible to investigate to which extent a mag-\nnetization mode which is localized at a certain posi-\ntion within the (Ga,Mn)As layer contributes to the spin-\npumping signal.12\nAcknowledgments\nThis work was supported by the Deutsche Forschungs-\nGemeinschaft via Grant No. SFB 631 C3 (Walter Schot-\ntky Institut) and Grant No. Li 988/4 (Universit¨ at Ulm).\nAppendix A: Coordinate Transformation and Free\nEnthalpy derivatives\nThe transformation between the crystallographiccoor-\ndinate system ( x,y,z) and the equilibrium system (1,2,3)\nis given by\n\nmx\nmy\nmz\n=T\nm1\nm2\nm3\n, (A1)\nwith\nT=\ncosθ0cosφ0−sinφ0sinθ0cosφ0\ncosθ0sinφ0cosφ0sinθ0sinφ0\n−sinθ00 cos θ0\n.(A2)\nThe derivatives of the free enthalpy density Eq. (6) with\nrespect to the magnetization components are\nG3=∂m3G|m=m0=−µ0H3+2B001cos2θ0\n+B1¯10(sinθ0cosφ0−sinθ0sinφ0)2\n+ 4B4⊥cos4θ0\n+ 4B4/bardblsin4θ0(cos4φ0+sin4φ0) (A3)\nG21=G12=∂m1∂m2G|m=m0\n= cosθ0(1−2cos2φ0)[B1¯10\n+ 12B4/bardblsin2θ0cosφ0sinφ0] (A4)\nG11=∂m1∂m1G|m=m0= 2B001sin2θ0\n+ 12cos2θ0sin2θ0[B4⊥\n+B4/bardbl(cos4φ0+sin4φ0)]\n+B1¯10cos2θ0(cosφ0−sinφ0)2(A5)\nG22=∂m2∂m2G|m=m0= 2B1¯10(sinφ0+cosφ0)2\n+ 24B4/bardblsin2θ0cos2φ0sin2φ0. (A6)Appendix B: Finite Difference Method\nIn this Appendix, we describe how the spin-waveequa-\ntion can be numerically solved by the finite difference\nmethod. We start with the simple case of a circulary pre-\ncessing magnetization, neglecting Gilbert damping and\nthe driving field (Sec. B1). Then we turn to the gen-\neral case, where the magnetization precesses elliptically\nand the Gilbert damping as well as the driving field are\nincluded (Sec. B2).\n1. The One-Dimensional, Homogeneous,\nUndamped Case\nHere, we describe how the resonance fields and the\nspin-wave modes can be found, assuming a circularly\nprecessing magnetization m2=im1= ˜m, a constant ex-\nchangestiffness, and a zindependent equilibrium magne-\ntization. This case has been considered in Ref. 36 using a\nsemi-analytical approach to solve the spin-wave equation\nEq. (10). The approach considered here, is slightly more\ngeneral, as it is straight forward to determine resonance\nfields and eigenmodes of the system for an arbitrary z\ndependence of the uniform resonance field. To solve\nEq. (10), we divide the ferromagnetic film into a finite\nnumbernof layers with equal thickness land constant\nmagnetic properties within each of these layers. The z\ndependence of ˜ mandµ0H001\nuniis thus given by an index\nj= 1...n. Within each of these layers the uniform reso-\nnance field and ˜ m(z) are thus constant and given by the\nvaluesµ0H001,j\nuni=:Kjand ˜mj, respectively. The second\nderivative of ˜ mis approximated by\n˜m′′(z=j·l)≈˜mj−1−2˜mj+ ˜mj+1\nl2.(B1)\nConsequently, Eq. (10) is converted to the homogeneous\nequation system\n\n.........\n... Kj−1+2d−d 0...\n...−d Kj+2d−d ...\n...0 −d Kj+1+2d ...\n.........\n\n...\n˜mj−1\n˜mj\n˜mj+1\n...\n=µ0H\n...\n˜mj−1\n˜mj\n˜mj+1\n...\n, (B2)\nwith the abbreviation d=−Ds/l2. The boundary condi-\ntion of natural freedom36(von Neumann boundary con-dition) reads as ˜ m0= ˜m1and ˜mn−1= ˜mnand can be\nincorporated in Eq. (B2). Since the matrix on the left13\nhand side of Eq. (B2) is sparse, it can be efficiently diag-\nonalized numerically, yielding the resonancefields (eigen-\nvalues) and the corresponding modes (eigenvectors). Af-\nter diagonalizingthe matrix, the relevantresonancefields\nare found by sorting the eigenvalues and considering only\nthe modes with positive resonance fields, corresponding\nto the bound states in the particle-in-a-box analogon.\nThe SWR amplitude of each mode is proportional to its\nnet magnetic moment; thus, the amplitudes can be found\nby integrating the (normalized) eigenmodes. The mode\nprofile, the resonance fields, and the SWR intensities are\nillustrated in Fig. 2 for a constant and a linearly varying\nuniform resonance field. The finite linewidth of the SWR\nmodes can be accounted for by assuming a Lorentzian\nlineshape for each mode with a certain linewidth and\nwith the resonance fields and intensities calculated as\ndescribed above36. Note that this approach to derive\nresonance fields and intensities is only valid if the mode\nseparation is large compared with the linewidth of the\nmodes; this restriction does not apply to the model pre-\nsented in the Appendix B2.\n2. The General Case\nTosolveEq.(7)forarbitrary µ0Handarbitrarilyvary-\ning magnetic properties, we again divide the ferromag-netic film into a finite number nof layers with equal\nthicknessland constant magnetic properties within each\nof these layers. In contrast to the case in the Appendix\nB1, where only the uniform resonance field was varied\nacross the layer, here potentially all magnetic proper-\nties entering Eq. (7) can be assumed to be zdependent.\nAdditionally, the components of the driving field µ0hi\n(i= 1,2), can also vary as a function of z, since the\n(1,2,3) frame of reference is zdependent and thus the\nprojections of the driving field have to be calculated for\neach layer. The zdependence of the components mi\n(i= 1,2), of the parameters H11,H12,H21,H22(de-\nfined in Sec. II) and the exchange stiffness is thus given\nby the index j= 0...n; the second derivative of each of\nthe components miis approximated as in Eq. (B1).\nThe linearized LLG equation Eq. (7), is thus converted\ninto the inhomogeneous equation system\n\n..................\n...Hj−1\n11−2dj−1Hj−1\n12dj−10 0 0 ...\n... Hj−1\n21Hj−1\n22−2dj−10dj−10 0 ...\n... dj0Hj\n11−2djHj\n12dj0...\n... 0 djHj\n21Hj\n22−2dj0 dj...\n... 0 0 dj+10Hj+1\n11−2dj+1Hj+1\n12...\n... 0 0 0 dj+1Hj+1\n21Hj+1\n22−2dj+1...\n..................\n\n...\nmj−1\n1\nmj−1\n2\nmj\n1\nmj\n2\nmj+1\n1\nmj+1\n2...\n=µ0\n...\nhj−1\n1\nhj−1\n2\nhj\n1\nhj\n2\nhj+1\n1\nhj+1\n2...\n,(B3)\nwith the abbreviation dj=−Dj\ns/l2. 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Technologieentwicklung, 07745 Jena, Germany\n4)Nanosystems Initiative Munich, 80799 Munich, Germany\n5)Institut f ur Festk orperphysik, Technische Universit at Dresden, 01062 Dresden,\nGermany\n6)Center for Transport and Devices of Emergent Materials, Technische Universit at Dresden, 01062 Dresden,\nGermany\n(Dated: 8 December 2016)\nThe magnetostatic mode (MSM) spectrum of a 300 \u0016m diameter single crystalline sphere of yttrium iron\ngarnet is investigated using broadband ferromagnetic resonance (FMR). The individual MSMs are identi\fed\nvia their characteristic dispersion relations and the corresponding mode number tuples ( nmr) are assigned.\nTaking FMR data over a broad frequency and magnetic \feld range allows to analyze both the Gilbert\ndamping parameter \u000band the inhomogeneous line broadening contribution to the total linewidth of the\nMSMs separately. The linewidth analysis shows that all MSMs share the same Gilbert damping parameter\n\u000b= 2:7(5)\u000210\u00005irrespective of their mode index. In contrast, the inhomogeneous line broadening shows a\npronounced mode dependence. This observation is modeled in terms of two-magnon scattering processes of\nthe MSMs into the spin-wave manifold, mediated by surface and volume defects.\nThe ferrimagnetic insulator yttrium iron garnet (YIG)\nhas numerous applications in technology and funda-\nmental research due to its low intrinsic Gilbert damp-\ning and large spin-wave propagation length.1It is used\nas prototypical material in various experiments in spin\nelectronics2{4and spin caloritronics5,6and is indispens-\nable for microwave technology.\nRecently, YIG spheres attracted attention in the\n\feld of quantum information technology.7{15For exam-\nple, strong coupling between magnons and photons in\nYIG/cavity hybrid systems can be employed for the up-\nand down-conversion of quantum signals between mi-\ncrowave and optical frequencies, enabling a long-range\ntransmission of quantum information between microwave\nquantum circuits.14{16Here, the damping of the mag-\nnetic excitation plays a crucial role, since it limits the\ntime-scale in which energy and information is exchanged\nand stored in the magnon-photon hybrid system.\nOne type of magnetic excitations in YIG spheres17{19\nare magnetostatic modes (MSMs) which resemble stand-\ning spin-wave patterns within the sphere. Although the\nlinewidth of MSMs in YIG spheres has been studied at\n\fxed frequencies in the past,20{22the respective contri-\nbutions of intrinsic Gilbert damping and inhomogeneous\nline broadening23to the total linewidth have not yet been\ninvestigated. In particular, it is not evident from the\nliterature, whether di\u000berent MSMs feature the same or\ndi\u000berent Gilbert damping.24,25\nHere, we report on the study of dynamic properties of\nmultiple MSMs for a 300 \u0016m diameter YIG sphere us-\ning broadband ferromagnetic resonance. The frequency\na)Electronic mail: stefan.klingler@wmi.badw.deand magnetic \feld resolved FMR data allows to separate\nGilbert damping and inhomogeneous line broadening of\nthe MSMs. One and the same Gilbert damping parame-\nter\u000b= 2:7(5)\u000210\u00005is found for all MSMs, independent\nof their particular mode index. However, the inhomoge-\nneous line broadening markedly di\u000bers between the ob-\nserved MSMs. This \fnding is attributed to two-magnon\nscattering processes of the MSMs into the spin-wave man-\nifold, mediated by surface and volume defects.\nThe MSM pro\fles and eigenfrequencies of a magnetic\nsphere can be calculated in the magnetostatic approx-\nimationr\u0002H= 0,17{19using the Landau-Lifshitz-\nGilbert equation (LLG).26,27The resonance frequencies\n\n of the MSMs are obtained by solving the characteristic\nequation:17{19\nn+ 1 +\u00180dPm\nn(\u00180)=d\u00180\nPmn(\u00180)\u0006m\u0017= 0; (1)\nwhere\u00182\n0= 1 + 1=\u0014,\u0014= \n H=\u0000\n\n2\nH\u0000\n2\u0001\n,\u0017=\n\n=\u0000\n\n2\nH\u0000\n2\u0001\n, \nH=\u00160Hi=\u00160Msand \n =!=\r\u0016 0Ms.\nHere,\r=gJ\u0016B=~is the gyromagnetic ratio, gJis the\nLand\u0013 eg-factor,\u0016Bis the Bohr magneton, ~is the reduced\nPlanck constant, \u00160is the vacuum permeability and Ms\nis the saturation magnetization. The angular frequency\nof the applied microwave \feld is denoted as != 2\u0019f.\nThe internal \feld is given by Hi=H0+Hani+Hdemag ,\nwhereH0is the applied static magnet \feld, Haniis the\nanisotropy \feld, and Hdemag =\u0000Ms=3 is the demagneti-\nzation \feld of a sphere.\nThe mode pro\fles of the MSMs have the form of asso-\nciated Legendre polynomials Pm\nn, where the localization\nof the MSMs at the surface is related to the mode index\nn2N.21The indexjmj\u0014ncorresponds to an angular-\nmomentum quantum number of the MSM,28where thearXiv:1612.02360v1  [cond-mat.mtrl-sci]  7 Dec 20162\nbar above the mode index mis used for indices m < 0.\nThe index r\u00150 enumerates the solutions of the char-\nacteristic equation (1) for given nandmfor increasing\nfrequencies.18,29In total, each MSM is uniquely identi-\n\fed by the index tuple ( nmr). For more information and\nplots of the MSM mode patterns, the review of Ref. 19\nis recommended.\nThe Gilbert damping parameter phenomenologically\naccounts for the viscous (linearly frequency-dependent)\nrelaxation of magnetic excitations. Assuming a domi-\nnant Gilbert-type damping for all MSM modes, the full\nlinewidth at half maximum (FWHM) \u0001 f(nmr)of a MSM\nresonance line at frequency f(nmr)\nres is given by:30\n\u0001f(nmr)= 2\u000bf(nmr)\nres + \u0001f(nmr)\n0: (2)\nHere, \u0001f0denotes the inhomogeneous line broadening\ncontributions to the total linewidth. For a two-magnon\nscattering process mediated by volume and surface de-\nfects the latter can be written as:21\n\u0001f(nmr)\n0 = \u0001fm-mF(nmr)+ \u0001f0\n0: (3)\nHere, \u0001fm-maccounts for the two-magnon scattering pro-\ncess of the MSMs into the spin-wave manifold.21,22,31The\nfactorF(nmr)represents the ratio of the linewidth of a\nparticular MSM with respect to the uniform precessing\n(110)-mode.21,22,32,33It therefore accounts for the surface\nsensitivity of the speci\fc mode compared to the (110)-\nmode. The two-magnon scattering processes can be sup-\npressed if a perfectly polished YIG sphere is used, due to\nthe vanishing ability of the system to transfer linear and\nangular momentum from and to the lattice.21The term\n\u0001f0\n0represents a constant contribution to the linewidth\nin which all other frequency-independent broadening ef-\nfects are absorbed. The complete scattering theory used\nin this letter is presented in Ref. 21.\nFig. 1 (a) shows a sketch of the measurement setup.\nThe YIG sphere with a diameter of d= 300\u0016m is placed\nin a disk shaped Vespel sample holder (diameter 6 mm,\nnot shown), which has a centered hole with a diameter\nof 350\u0016m. The sphere in the sample holder is exposed\nto a static magnetic \feld in order to align the easy [111]-\ndirection of the YIG crystal parallel to the \feld direc-\ntion. The orientation of the sphere is subsequently \fxed\nusing photoresist and the alignment is con\frmed by Laue\ndi\u000braction.\nThe oriented YIG sphere is placed on a 50 \n impedance\nmatched coplanar waveguide (CPW) structure. The\nsphere is placed in the middle of the w= 300\u0016m wide\ncenter conductor, with the YIG [110]-axis aligned par-\nallel to the long axis of the center conductor of the\nCPW. Additionally, a pressed crumb of Diphenylpicryl-\nhydrazyl (DPPH) is glued on the center conductor, where\nthe distance between the YIG sphere and the DPPH is\nl\u00191 cm. DPPH is a spin marker with a g-factor34of\ngDPPH = 2:0036(3). The measurement of its resonance\nfrequency\nfDPPH =gDPPH\u0016B\n2\u0019~\u00160HDPPH\n0 (4)\nP1 P2 z, [111] y, [110] \nx, h xVNAelectro magnet top view (a)  side view \nYIG \nDPPH CPW \nH0\nIm ∆S 21   ,Re ∆S 21 (a.u.)  \n-10 0 10\nf-f res  (MHz)(b) (530)-mode  \nH0w\nlaa/2 P1 \nP2 hrf FIG. 1. (a) The CPW with the YIG sphere and the DPPH\nis positioned in the homogeneous \feld of an electromagnet.\nThe CPW is connected to port 1 (P1) and port 2 (P2) of a\nvector network analyzer (VNA). The YIG sphere is placed\non top of the center conductor of the CPW with its [111]-\naxis parallel to the applied magnetic \feld H0inz-direction.\n(b) Typical normalized transmission spectrum of the (530)-\nmode at\u00160H0= 0:8 T (symbols) including a \ft to Eq. (5)\n(lines).\nprovides an independent magnetic \feld reference at the\nsample position, in addition to Hall probe measurements.\nThe static magnetic \feld calculated from the DPPH\nresonance frequency is denoted as HDPPH\n0 . The stray\n\feld originating from the YIG sphere at the location of\nthe DPPH creates a systematic measurement error of\n\u000e\u00160Hstray\u001440\u0016T, as estimated using a dipole approxi-\nmation.\nFor the broadband FMR experiments, the CPW is po-\nsitioned between the pole shoes of an electromagnet with\na maximum \feld strength of j\u00160H0j\u00142:25 T. The pole\nshoe diameter is a= 6 cm, while the pole shoe sepa-\nration isa=2, to ensure a su\u000ecient homogeneity of the\napplied magnetic \felds. The measured radial \feld gra-\ndient creates a systematic \feld measurement error of\n\u000e\u00160Hdisp= 0:3 mT forl= 1 cm displacement from the\ncenter axis.\nThe CPW is connected to port 1 (P1) and port 2 (P2)\nof a vector network analyzer (VNA) and the complex\nscattering parameter S21is recorded as a function of H0\nandf\u001426:5 GHz. The applied microwave power is -\n20 dBm to avoid non-linear e\u000bects causing additional line\nbroadening. The microwave current \rowing along the\ncenter conductor generates a microwave magnetic \feld\npredominately in the x-direction at the location of the\nYIG sphere. This results in an oscillating torque on\nthe magnetization, which is aligned in parallel to the z-\ndirection by the external static \feld H0. Forf=f(nmr)\nres ,\nthe excited resonant precession of the magnetization re-\nsults in an absorption of microwave power.\nIn order to eliminate the e\u000bect of the frequency depen-\ndent background transmission of the CPW, the following\nmeasurement protocol is applied: First, S21is measured\nfor \fxedH0in a frequency range fDPPH\u00061 GHz. Second,\nS21is measured for the same frequency range at a slightly3\nFIG. 2. (a) Normalized transmission magnitude j\u0001S21j\nplotted versus applied magnetic \feld \u00160H0and microwave\nfrequencyfrelative to the DPPH resonance fDPPH . The\ncontrast between the dashed lines is stretched for better vis-\nibility. (b) Calculated and measured dispersions of various\nMSMs (lines and open circles, respectively).\nlarger magnetic \feld H0+ \u0001H0, with\u00160\u0001H0= 100 mT.\nSince for this \feld no YIG and DPPH resonances are\npresent in the observed frequency range, the latter mea-\nsurement contains the pure background transmission.\nThird, the normalized transmission spectra is obtained\nas \u0001S21=S21(H0)=S21(H0+ \u0001H0), which corrects the\nmagnitude and the phase of the signal. This procedure is\nrepeated for all applied magnetic \felds. The transmitted\nmagnitude around the resonance can be expressed as:30\n\u0001S21(f) =A+Bf+Z\n\u0010\nf(nmr)\nres\u00112\n\u0000if2\u0000if\u0001f(nmr):(5)\nHere,Ais a complex o\u000bset parameter, Bis a complex lin-\near background and Zis a complex scaling parameter.35\nFig. 1 (b) exemplary shows the real and imaginary part of\n\u0001S21for the (530)-mode at \u00160H0= 0:8 T. In addition, a\n\ft of Eq. 5 to the data is shown, which adequately models\nthe shape of the resonances.\nFig. 2 (a) shows the normalized transmitted magnitude\nj\u0001S21jas a function of H0andf\u0000fDPPH on a linear\ncolor-coded scale. The frequency axis is chosen relative\nto the DPPH resonance frequency, so that all modes with\na linear dispersion f(nmr)\nres/H0appear as straight lines,whereas modes with a non-linear dispersion are curved.\nNote, that the \feld values displayed on the y-axis repre-\nsent the magnetic \feld strength measured with the Hall\nprobe.\nThe di\u000berent modes appearing in the color plot in\nFig. 2 (a) can be identi\fed in a straightforward manner.\nAt \frst, all visible resonances are \ftted using Eq. (5)\nin order to extract f(nmr)\nres and \u0001f(nmr). Furthermore,\nthe DPPH resonance line is identi\fed as straight line at\nf\u0000fDPPH = 0 MHz and the resonance \felds HDPPH\n0 are\ncalculated using Eq. (4).\nSecond, the straight lines at about f\u0000fDPPH\u0019\n\u000060 MHz and f\u0000fDPPH\u0019\u0000740 MHz are identi\fed as\nthe (110)- and (210)-mode, respectively. A simultaneous\n\ft of the dispersion relations18\nf(110)\nres =gYIG\u0016B\n2\u0019~\u00160(H0+Hani) (6)\nand\nf(210)\nres =gYIG\u0016B\n2\u0019~\u00160\u0012\nH0+Hani\u00002\n15Ms\u0013\n(7)\nto the measured values of f(110)\nres,f(210)\nres and\u00160HDPPH\n0\nyieldsgYIG = 2:0054(3),\u00160Ms= 176:0(4) mT and\n\u00160Hani=\u00002:5(4) mT. The error of gYIGis given by the\nsystematic error introduced by the \feld normalization\nusinggDPPH . The errors in \u00160Haniand\u00160Msare given\nby\u000e\u00160Hdisp+\u000e\u00160Hstray. All values are in good agree-\nment with previously reported material parameters36{40\nfor YIG (gYIG = 2:005(2),\u00160Hani=\u00005:7 mT and\n\u00160Ms= 180 mT) and, hence, justify the (110)- and (210)-\nmode assignments.\nThird, the complete MSM manifold is computed using\nthe extracted material parameters. The mode numbers\nof the remaining modes are determined from the charac-\nteristic dispersions. Fig. 2 (b) shows the dispersions of\nthe identi\fed modes as function of f(nmr)\nres\u0000fDPPH and\nHDPPH\n0 , with very good agreement of theory (lines) and\nexperiment (circles). Slight deviations between model\npredictions and data might be attributed to a non-perfect\nspherical shape of the sample, which would change the\nboundary conditions for the magnetization dynamic in\nthe YIG spheroid, and thus the dispersion relations.\nIn Fig. 3 (a) the linewidth \u0001 f(nmr)of each MSM is\nplotted versus its resonance frequency f(nmr)\nres . The o\u000bset\n\u0001f(nmr)\n0 is magni\fed by a factor of 5 to emphasize the\ndi\u000berences in the inhomogeneous line broadening. Indi-\nvidual \fts of all \u0001 f(nmr)to Eq. (2) yield identical slopes\nfor all modes within a small scatter, which is also evident\nfrom the linewidth data in Fig. 3 (a). Hence, the Gilbert\ndamping parameter and inhomogeneous line broadening\nare obtained from a simultaneous \ft of Eq. (2) to the\nextracted data points. Here, \u000bis a shared \ft parameter\nfor all MSMs, but the inhomogeneous line broadening\n\u0001f(nmr)\n0 is \ftted separately for each mode. To avoid\n\ftting errors, the linewidths data are disregarded when\na mode anti-crossing is observed, since this results in a4\n5 10 15 20 25 ∆f (nmr) (MHz) \nfres       (GHz) (a) \n0(110)\n(440)\n(531)\n(530)\n(511)\n(631)\n(502)\n(nmr) 246810 Offset x5 \n(b) \n0.00.51.0 1.5 2.0 ∆f0(nmr) (MHz) \n-500 0 500 -250 250 ∆f00=0.3 MHz (110) \n(440) (531) (530) (511) \n(631) \n(502) Measurement \nTheory \n fres      - fDPPH  (MHz) (nmr) \nFIG. 3. (a) Linewidth vs. resonance frequency of the\nmeasured MSMs. The Gilbert damping of all MSMs is\n\u000b= 2:7(5)\u000210\u00005as evident from the same slope of all\ncurves. The inhomogeneous line broadening is di\u000berent for\neach MSM. Note that the data points are plotted with an o\u000b-\nset proportional to the inhomogeneous line broadening. (b)\nInhomogeneous line broadening as a function of f\u0000fDPPH .\npronounced increase in linewidth.41As evident from the\nsolid \ft curves in Fig. 3 (a) the evolution of the linewidth\nwith resonance frequency of all measured MSMs can be\nwell described with a shared Gilbert damping parameter\nof\u000b= 2:7(5)\u000210\u00005, independent of the mode num-\nber and the mode intensity. The latter strongly sug-\ngests a negligible e\u000bect of radiative damping on the mea-\nsured linewidths.42The error in \u000bis given by the scat-\nter of\u000bfrom the independent \fts. Other groups report\nGilbert damping parameters for YIG \flms43{49larger\nthan\u000b= 6:15\u000210\u00005, whereas for bulk YIG37,49,50values\nof\u000b= 4\u000210\u00005are found. Hence, the Gilbert damp-\ning parameter obtained here is the smallest experimen-\ntal value reported so far. The results are in agreement\nwith the notion, that the Gilbert damping parameter is a\nbulk property which only depends on intrinsic damping\ne\u000bects. However, the inhomogeneous line broadening is\nindeed di\u000berent for the various MSMs.\nFig. 3 (b) shows the extracted values for the inhomo-\ngeneous line broadening (\flled dots) as a function of\nf(nmr)\nres\u0000fDPPH . The error bars indicate the variation of\nthe inhomogeneous line broadening between global andindividual \fts. In order to show the approximate posi-\ntion of the modes in comparison to Fig. 2, the x-scale is\ncalculated for a magnetic \feld strength of \u00160H= 0:5 T.\nAdditionally, the linewidths \u0001 f(nmr)\n0 for all modes are\ncalculated using the two-magnon scattering theory, given\nin Eq. (4) of Ref. 21 (open circles). For the calculations of\nthe linewidths, a pit radius R= 350 nm and a constant\nlinewidth contribution of \u0001 f0\n0= 30 kHz was assumed.\nSince the calculated \u0001 fm-mare slightly frequency depen-\ndent, the average linewidth values for the measured \feld\nand frequency range are used and the standard deviation\nis indicated by the error bars of the open symbols. For\nmost MSMs the variation is smaller than 10 kHz. Never-\ntheless, the (440)-mode should show a prominent peak in\nthe linewidth measurement at about f(440)\nres = 10 GHz in\nFig. 3 (a),21which is however not observed in the experi-\nmental data. Additionally, the (110)-MSM shows a much\nlarger linewidth than expected from the calculations. In\na perfect sphere the (110)-mode is degenerate with the\n(430)-mode,18but in a real sphere this degeneracy might\nbe lifted. If the di\u000berence of the (110)- and (430)-mode\nfrequencies is smaller than the linewidth of the measured\nresonance, an additional inhomogeneous line broadening\nis expected. Indeed, a careful analysis of the (110)-MSM\nline shape reveals a second resonance line in very close\nvicinity to the (110)-mode, yielding an arti\fcial inhomo-\ngeneous line broadening of this mode. Besides these two\nMSMs, an excellent quantitative agreement between the\ntwo-magnon scattering model and experiment is found.\nIn conclusion, broadband ferromagnetic resonance ex-\nperiments on magnetostatic modes in a YIG sphere are\npresented and various magnetostatic modes are identi-\n\fed. The linewidth analysis of the data allows to distin-\nguish between the Gilbert damping and inhomogeneous\nline broadening. A very small Gilbert damping parame-\nter of\u000b= 2:7(5)\u000210\u00005is found for all MSMs, indepen-\ndent of their mode indices. 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R oschmann, IEEE Transactions on Magnetics 17, 2973 (1981)."    },    {        "title": "1712.07323v1.Unifying_ultrafast_demagnetization_and_intrinsic_Gilbert_damping_in_Co_Ni_bilayers_with_electronic_relaxation_near_the_Fermi_surface.pdf",        "content": " 1 Unifying ultrafast demagnetization and intrinsic Gilbert damping in Co/Ni \nbilayers with electronic relaxation near the Fermi surface  \nWei Zhang, Wei He*, Xiang -Qun Zhang, and Zhao -Hua Cheng*  \nState Key Laboratory of Magnetism and Beijing National Laboratory for \nCondensed Matter Physics, Institute of Physics, Chinese Academy of \nSciences, Beijing 100190, P. R. China  \nJiao Teng  \nDepartment of Materials Physics and Chemistry, University of Sci ence and \nTechnology Beijing, Beijing 100083, P. R. China  \nManfred Fä hnle  \nMax Planck Institute for Intelligent Systems, Heisenbergstra e 3, 70569 \nStuttgart, Germany  \nAbstract  \nThe ability to controllably manipulate the laser -induced ultrafast magnetic \ndynamics  is a prerequisite for future high speed spintronic devices. The optimization \nof devices requires the controllability of the ultrafast demagnetization time, , and \nintrinsic Gilbert damping, . In previous attempts to establish the relationship \nbetween \nM  and \nrint  , the rare -earth doping of a permalloy film with two different \ndemagnetization mechanism is not a suitable candidate. Here, we  choose Co/Ni \nbilayers to  investigate the relations between and  by means of ti me-resolved \nmagneto -optical Kerr effect (TRMOKE) via adjusting the thickness of the Ni layers, \nand obtain an approximately  proportional  relation between these two parameters.   \nM\nintr\nM\nintr 2 The remarkable agreement between TRMOKE experiment and the prediction of \nbreathi ng Fermi -surface model confirms that a large Elliott -Yafet spin -mixing \nparameter \n2b  is relevant to the strong spin -orbital coupling at the Co/Ni interface. \nMore importantly, a proportional relation between \nM  and \nintr  in such metallic \nfilms or heterostructures  with electronic relaxation near Fermi surface suggests the \nlocal spin -flip scattering domains the mechanism of ultrafast demagnetization, \notherwise the spin -current mechanism domains. It is a n effective method to \ndistinguish the dominant contributions to ultrafast magnetic quenching in metallic \nheterostructures by investigating both the ultrafast demagnetization time and Gilbert \ndamping simultaneously. Our work can open a novel avenue to manip ulate the  \nmagnitude and efficiency of Terahertz  emission in metallic heterostructures such as \nthe perpendicular magnetic anisotropic Ta/Pt/ Co/Ni /Pt/Ta  multi layers,  and then it has \nan immediate implication of the design of high frequency spintronic devices . \n \nPACS numbers: 75.78.Jp, 75.40.Gb, 76.50.+g, 78.47.+p  \n*Correspondence and requests for materials  should be addressed to Z.H.C \n(zhcheng@iphy.ac.cn ) or W.H. ( hewei@iphy.ac.cn ) \n   3 Since the pioneering work on ultrafast demagnetization of Ni thin film after \nfemtosecond laser irradiation was demonstrated in 1996 by Beaurepaire et al1, the \nquest for ultrafast modification of the magnetic moments has triggered a new field of \nresearch : Femtomagnetism . It leads to the dawn of a new ear for breaking the ultimate \nphysical limit for the speed of magnetic switching and manipulation, which are \nrelevant to current and future information storage. In the past two decades, the \nultrafast dynamics  in hundreds of femtoseconds have been probed with the \nfemtosecond laser pulse using magneto -optical Kerr1 or Faraday effect2, or other \ntime-resolved techniques such as the high -harmonic generation (HHG) of extreme \nultraviolet(XUV) radiation3, magnetic cir cular dichroism4, or spin resolved two -photo \nphotoemission5.  \nNevertheless, the microscopic mechanism underlying ultrafast quenching of \nmagnetization remains elusive. Various mechanisms including electron -phonon \nmediated spin -flip scattering6-9, electron -electron scattering10,11, electron -magnon \nscattering12,13, direct angular momentum transfer from photon to electron mediated by \nspin-orbit coupling14,15, coherent interaction among spins electrons and photons16, \nwere proposed to explain the ultraf ast spin dynamics.  In addition, since Malinowski17 \net al first proposed that the laser excited spin current transport could increase and \nspeed up the magnetic quenching in metallic heterostructures, the laser -induced \nsuper -diffusive spin current was raise d to play an important role in determining the \nultrafast demagnetization in metallic films or heterostructures18-22. However, the \nrecent demonstration23 shows that the unpolarized hot electrons transport can  4 demagnetize a ferromagnet, indicating the local spin angular momentum dissipation is \nunavoidable even when super -diffusive spin transport domains in the metallic \nheterostructures. Moreover, even in th e similar samples, the local spin -flip scattering \nand nonlocal spin transport mechanism were proposed respectively by different \nexperimental tools19, 24 to explain the ultrafast demagnetization . It is harmful for \nclarifying the underlying ultrafast demagne tization mechanism in such metallic \nheterostructures. Therefore, an effective method to distinguish  the two dominant \ncontribution s to ultrafast  demagnetization in metallic heterostructures is highly \ndesirable19,23,24. Here, we propose that investigating bo th the ultrafast demagnetization \ntime and Gilbert damping25 simultaneously is a candidate method, although the \nrelationship between the two parameters has never been unified successfully so far \nbetween the experiments and theoretical predictions.    \nAn inv erse relation between and was first derived by Koopmans et al. \nfrom a quantum -mechanical calculation on the basis of the Elliot t-Yalfet (EY) \nspin-flip scattering model6. Later, the attempted experiments have ever been carried \nout to demonstrate the predict ion in rare -earth -doped permalloys26,27 and amorphous \nTbFeCo films28. In this case, t he localized 4f electrons  rather than itinerant 5 d6s \nelectrons domain most of the large magnetic moment  in rare -earth elements. Because \nthe 4 f electrons are far from the Fermi level, their ultrafast demagnetization processes \nare medicated by 5 d6s electrons after laser pulse excitation7. The indirect excitation \nleads to the so called type_II ultrafast demagnetization behavior in rare -earth elemen ts, \nwhich is much slower than that of itinerant electrons. Therefore, it is not unexpected \nM\nintr 5 that the ultrafast demagnetization time \nM  of permalloys increases with the doping \ncontents of rare -earth elements increasing. Meanwhile , it happ ened  that the Gilbert \ndamping constant of permalloys is  also increased by doping 4 f elements, which \nmainly comes from the so called “slow relaxing impurities mechanism”29. Therefore, \nby introducing the extra mechanism unavoidablely ，a trivial consequence wa s \nobtained that the ultrafast demagnetization time increases as the Gilbert damping  \n\n increases in rare -earth -doped permalloys26. In hindsight, from this experiment, one \ncan not confirm the relation between ultrafast demagnetization time  and Gilbert \ndamping  \ndue to the defects of the experimental design. A genuine relation between \nultrafast demagnetization time and Gilbert damping should be explored in a clean \nsystem without extra demagnetization mechanism. So far, the explicit relationship \nbetween the two parameters has never been unified successfully between the \nexperiments and theoretical predictions. Our work in Co/Ni bilayers with the electrons \nrelaxing at the Fermi surface can fill in the blank.   \nIn the cas e of pure 3 d itinerant electrons  relaxing near the Fermi surface after the \nlaser excitation , both ultrafast demagnetization  and Gilbert damping are determined \nby spin -flip scattering of itinerant electrons at quasi -particles or impurities . Based on \nthe breathing Fermi -surface model of Gilbert damping and on the EY relation for the \nspin-relaxation time, a proportional relation between and was derived by \nFä hnle et al30,31 for the materials with conductivity -like damping. And an inverse \nrelation was also d erived which is similar with that proposed by B. Koopmans et al \nwhen the resistivity -type damping domains in the materials.  Although the predicted \nM\nM\nM\nintr 6 single numerical values of intr/M are in good agreement with the experimental ones \nfor Fe, Ni, or Co, for a confirmation of the explicit relation between and  one \nhas to vary the values on the two parameters systematically for one system, as we do \nit in our paper by changing the thickness of the films.    \nCo/Ni bilayers with a stack of Ta (3 nm)/Pt (2 nm)/Co ( 0.8 nm)/Ni ( dNi nm)/Pt (1 \nnm)/Ta (3 nm) were grown on glass substrates by DC magnetron sputtering32, 33. The \nthickness of Ni layer changes from dNi = 0.4 nm to dNi = 2.0 nm. T heir static \nproperties have been shown in the Part Ⅰof the Supplementary Materials34. Both\nand for Co/Ni bilayer systems have been achieved by using time -resolved \nmagneto -optical Kerr effect (TRMOKE) technique21, 35. The reasons for selecting the \nCo/Ni bilayers are three -fold. First, Co/Ni bilayers with perpendicular magnetic \nanisotropy  (PMA) are one of candidates for perpendicular magnetic recording (PMR) \nmedia and spintronic devices36-39. Second, the electrons in both Co and Ni are \nitinerant near the Fermi surface and they have the same order of magnitude of \ndemagnetization time7,10. Without rare earth element doping in 3 d metals, one can \nexclude the possibility of an extra slow demagnetization accompanied by doping with \n4f rare-earth metals. Third, both and in Co/Ni bilayers can be tuned by \nchanging the Ni thickness. Therefore, Co/Ni bilayers provide an ideal system to \ninvestigate the relation between and . A nearly p roportional  relationship \nbetween and was evident  in Co/Ni bilayers, suggesting that the \nconductivity -like damping30, 31 plays a dominant role. It is distinct i n physics with \nprevious experiments26 where  the seemingly similar results have been obtained via \nM\nintr\nM\nintr\nM\nintr\nM\nintr\nM\nintr 7 introducing extra slow demagnetization mechanism. Moreover, we discussed the \norigin of Gilbert damping, analyzed its influence on the relation between \nM  and \nintr\n and proposed a new approach to distinguish the intrinsic spin -flip and extrinsic \nspin current mechanism for ultrafast demagnetization in metallic heterostructures. The \nfinding for this unification can provid e the possibility for manipulating the \nlaser -induced ultrafast demagnetization via Gilbert damping in high frequency or \nultrafast spintronic devices  such as the Terahertz emitters .  \nFig. 1(a) shows time -resolved MOKE signals40 for films with various Ni lay er \nthickness measured with an external field  Oe. The quantitative values of \nintrinsic Gilbert damping constant41-44 in Fig. 1(b) can be obtained by eliminating the \nextrinsic contributions (See the Supplementary Materials [34],  PartⅡ for details). It \nwas observed that intr decreases with increasing Ni layer thickness. On the one hand, \nprevious investigations39, 45 have been reported that the large PMA origins from the \nstrong spin -orbit coupling effect at Co/Ni interface. A thickness modification in Co/Ni \nbilayer can change the competition between interface and volume effect, and \nconsequently the PMA. When we plot the intrinsic Gilbert damping constant as a \nfunction of effective anisotropy field  in Fig. 4 in the Part Ⅱ in Supplementary \nMaterial (See the Supplementary Materials [34],  PartⅡ for details), a proportional \nrelation was confirmed in our Co/Ni bilayer system, which demonstrates that \nspin-orbit coupling contributes to both Gilbert damping and PMA  (Also, for the \nachievement of effec tive anisotropy field, please see the Supplementary Materials [34]  \nPartⅡfor details ). On the other hand, the interface between Ni and Pt maybe also \n4000H 8 modified via changing Ni layer thickness. Because the Gilbert damping increases \nlinearly when the Ni layer b ecomes thinner, it seems that the spin current dissipation \nis involved partly. A similar trend was observed in a Pt/CoFeB/Pt system46, in which a \npure non -local spin pumping effect domains the Gilbert damping. Therefore, the total \nGilbert damping equals to  α=𝛼𝑖𝑛𝑡𝑟 +𝛼𝑠𝑝 , in which 𝛼𝑠𝑝  represents the \ncontributions from spin current. Due to the low spin diffusion length of Pt, the \nmagnetization precession in Ni layer entering the Pt layer would be absorbed \ncompletely like in the system of Py/Pt and Py/Pd47 and so on. H owever，we have to \naddress that, i n the case of the variation of ferromagnetic layer thickness, the amount \nof spin current pumped out of ferromagnet is determined entirely by the parameter of \ninterfacial mixing conductance  𝐺𝑒𝑓𝑓𝑚𝑖𝑥 48,49.  It is a constant value once the normal \nmetal thickness is fixed , although the Gilbert damping in thinner magnetic layer is \nenhanced. Therefore, given the spin current contributes partly to the Gilbert damping \nat present, the spin angular momentum transferring from Ni layer to Pt layer would be  \nthe same  for various Ni lay er thickness.       \n  The central strategy  of our study is to establish a direct correlation between \nultrafast demagnetization time and the intrinsic Gilbert damping constant. The \nintrinsic Gilbert damping constant was extracted from magnetization precessi on in \nhundreds of ps timescale. The laser -induced ultrafast demagnetization dynamics has \nbeen measured carefully within time delay of 2.5  ps at a step of 15  fs and low laser \nfluence of 1 was used. Fig. 2 (a) shows the TRMOKE signals of the ultrafast \ndemagn etization evolution after optical excitation. A rapid decrease of magnetization \n2/cmmJ 9 takes place on the sub -picosecond timescale followed by a pronounced recovery. As \ncan be seen in this figure, the ultrafast demagnetization rate is different by changing \nthe Ni  thickness.  \nTo identify the effect of the heat transport across the film thickness on \ndemagnetization time, a numerical simulation50 was carried out to demonstrate that \nthe demagnetization time  variation induced with the thicknesses ranged from 1.2 nm \nto 2.8 nm  is so small that can be ignored  (See the Supplementary Materials [34], Part \nⅢ for details), although a relatively large error of  could be resulted in when the \nsample thickness spans very large. According to the simulation results, the heat \ntransport not only affects  the rate of ultrafast magnetization loss but also the \nmaximum magnetic quenching. So, in experiment we obtain the ultrafast \ndemagnetization time for various samples with almost the sa me maximum quenching \nof 9% to suppress the influence of heat transport7, 21, 51 -54 as well as the non local spin \ncurrent effect17. The temporal evolution of magnetization in sub -picosecond time \nscale was fitted by the analytic solution based on the  phenome nological  three \ntemperature model (3TM)1, 17:  \n \n(1)  \nwhere  presents the convolution product with the Gaussian laser pulse \nprofile, whose full width at half maximum (FWHM) is . A temporal stretching of \nthe laser pulse was introduced by the excited hot ele ctrons55, which is the trigger for \nthe observed ultrafast demagnetization. In the fitting procedure, the demagnetization \nM\n),()()()(\n1)(\n321 1 2\n5.0\n01\nGt\nM EEt\nM EM EtGtAteAAeAA\ntA\nMtMM M \n\n \n\n\n\n\n\n\n\n\n \n),(GtG\nG 10 time  we cared can be influenced by the value of , which is inter -dependence \nwith   within the three temperature model. As is shown in  Table 1  in the \nSupplementary Material34 Part Ⅳ,  was fixed at 330 fs for various samples to \neliminate its relevance with . The time variable in eq. (5) corresponds to  \n, with the free fit parameter characterizing the onset of the \ndemagnetization dynamic s of the actual data trace, which is fixed as 100 fs for various \nsamples.  is a step function,  is the Dirac delta function and  are \nthe fitting constants. The two critical time parameters  are the ultrafast \ndemagnetization time and magnetization recov ery time, respectively. The well fitted \ncurves by 3TM are also shown as the solid lines in Fig. 3(a) from which the ultrafast \ndemagnetization time  and the magnetization recovery time  were evaluated. \nWithin 3TM model, the magnetization recovery process is affected by , \ncharactering the electron -phonon relaxation, and , representing heat transport \ntimescale through the substrates as well as demagnetization time . In the fitting \nprocedure by 3TM model, we assigned a fixed value to  and varies slightly to \nexclude the heat transport effect through thickness. Via changing the single \nparameter ， , we can accurately reproduce the experimental results for various \nsamples. And the heat transport across the thickness domains within 3TM model \ncharacterized by the  parameter of , which is shown in  Table. 1 in  Part Ⅳ of \nSupplementary Material34 as around 2 ps. It is about three times bigger than  \nindicating that we are not mixing the heat transport and the electron -phonon \nrelaxation56. Only in this case, are both th e values of and  genuine. The value \nM\nG\nM\nG\nM\n0 expt tt\n0t\n)(t\n)(t\n3 2 1,,AAA\nE M,\nM\nE\nE\n0\nM\nE\n0\nM\n0\nE\nE\nM 11 of  indicates that the heat was transferred through the substrate in less than 3 ps in \nthis paper, rather than what was observed by F. Busse et al57 where the heat was \ntrapped laterally in the Gaussian profile up to 1  ns. Therefore, the lateral heat \ntransport effect can be ignored, and hencely the modification of precessional \ndynamics here. As illustrated in Fig. 2(b), it can be clearly seen that decreases with \nincreasing dNi.  \n      \nBy replotting Fig. 1(b) and Fig. 2(b), an approximately proportional \nrelationship between and intr was confirmed by our experimental results \n(Fig. 2(c)). This relationship between intr and is consistent well with the \ntheoretical prediction  based on the breathing Fermi -surface model30,31,58 \nfor materials with conductivity -like damping contributions. On the basis of the \nbreathing Fermi -surface model, the Elliott -Yafet spin -mixing parameter 𝑏2 in Co/Ni \nbilayers can be estimated from the theoretical equation30, 31 shown as the red solid l ine \nin Fig. 2(c): \n                                              (2) \nwhere the quantity contains the derivatives of the single -electron energies with respect \nto the orientation e of the magnetization M=Me. p is a material -specific parameter \nwhich should be close to 4. If we use =   from ab initio density \nfunctional electron theory calculation for fcc bulk Ni31, the experimental value of \nElliott -Yafet spin -mixing parameter 𝑏2 = 0.28 can be estimated in Co/Ni bilayers, \nwhich is far larger than that of Co or Ni. The significant enhancement of spin -mixing \n0\nM\nM\nM\nint Mr\n2pbFM\nelM\nelF\nJ231087.1 12 parameters is related to the strong spin -orbital coupling at the Co/Ni interface since b2 \nis proportional to 2 in first -order  perturbation theory, where  is the coefficient of the \nspin-orbit coupling. A detailed ab initio calculation  for Elliott -Yafet spin -mixing \nparameter in Co/Ni bilayers is highly desirable.  For a derivation of eq. (2) it must be \nassumed that the same types of spin-flip scattering processes are relevant for the \nultrafast  demagnetization and for the damping. The assumption does not say anything  \nabout these  detailed  types. It has been shown in Ref. 9 that mere electron -phonon  \nscatterings cannot explain the expe rimentally observed demagnetization  quantitatively. \nIn reality there are also contributions from  electron -electron scatterings11, \nelectron -magnon scatterings12 and from a combination of electron -phonon and \nelectron -magnon scatterings13. Because both for de magnetization and for damping ，\nthe spin  angular momentum  has to be transferred from the electronic spin system to \nthe lattice,  there is no reason why different types of theses spin -flip scatterings  should \nbe relevant for the two situations. Therefore , the Elliott -Yafet  relation, eq. (2) should \nbe applicable for our system. It would not be  valid if non -local spin -diffusion \nprocesses would contribute a lot to  demagnetization. Examples are a superdiffusive \nspin current in the  direction perpendicular to th e film plane, or a lateral diffusion out \nof the spot irradiated by the laser pulse and investigated by the TRMOKE.  However, \nwe definitely found the validity of the Elliott -Yafet relation,  and this shows that \nnonlocal spin -diffusion processes are so small t hat can be neglected in our \nexperiment.  \nDespite this , previous demonstrations17,19-21 show that the ultrafast spin current  13 caused by the transport of spin -majority and spin -minority electrons in the antiparallel \n(AP) state of magnetic multilayers  after the  laser pulse  accelerates the ultrafast \ndemagnetization. Similarly, as is indicated in Fig. 1(b), with the assistance of interface \nbetween FM (Ni) and NM (Pt), the spin current induced by the flow of spin -up and \nspin-down electrons in opposite directions59 may contribute partly to the Gilbert \ndamping in Pt/Co/Ni/Pt mulitilayers.  The femtosecond laser induced spin current lives \nvery shortly which is in sub -picosecond timescale, while the duration of spin current \ntriggered by spin precession is in the timescal e of nanosecond. The difference of the \nduration of the spin current is just related to the timescale of the perturbation of the \nsystem. One has to note that spin currents at the femtosecond time scale gives rise to a \nlowering of the demanetization time17, while spin pumping induced spin current gives \nrise to the enhancement of Gilbert damping and thus a lowering of the relaxation time. \nTherefore, when spin current contributes largely to both ultrafast demagnetization and \nspin precession dynamics, an inverse  relationship between ultrafast demagnetization \ntime and Gilbert damping could be expected. That is, t he more spin current \ntransferred from ferromagnetic layer to normal metal, the faster ultrafast \ndemagnetization should be. Therefore, a t present paper, to  explain the experimental \nresults the local Ellio tt-Yafet scattering theory suffices. And , the non -local spin \ncurrent effect can be ignored, although it contributes partly to the fitted value of \nspin-mixing parameter  𝑏2 . The discussions here inspire us t o continuously  clarify \nthe various relationships between ultrafast demagnetization time and Gilbert dam ping \ncoming from different microscopic mechanisms, which is helpful for understanding  14 the underlying physics of ultrafast spin dynamics as well as the ap plication of ultrafast \nspin current triggered by ultrashort laser60, 61. For instance, recently, the researchers \nare seeking for the potential candidates as the Terahertz waves emitters including the \nmetallic heterostructures. Previous demonstrations show that the magnitude and \nefficiency of Terahertz signals in these multilayers are determined by Gilbert \ndamping60. The investigations of the relationship between Gilbert damping and \nultrafast demagnetization time will open up a new avenue to tailor the Terah ertz \nemission.  \nMeanwhile , the dominant contribution to ultrafast demagnetization in metallic \nheterostructures, either from the localized spin -flip scattering or non -local spin \ntransport, has been a controversial issue for a long time23. Here, a new approa ch, by \nestablishing the relation between the demagnetization time and Gilbert damping, is \nproposed to distinguish the two mechanisms . The proportional relationship indicates \nthe localized spin -flip scattering mechanism domains, otherwise the nonlocal spin \ncurrent domains.          \nIn conclusion, the fast and ultrafast dynamic properties of Ta(3 nm)/Pt(2 \nnm)/Co(0.8 nm)/Ni( dNi nm)/Pt(1 nm)/Ta(3 nm) bilayers with  the electrons  relaxing \nnear the Fermi surface have been investigated by using TRMOKE pump -probe \ntechnique. An genuine  proportional relationship , contrast to previous trivial \nconsequence induced by impurities mechanism,  between ultrafast demagnetization \ntime and Gilbert damping constant is confirmed fr om experimental results. The \nestimated value of spin -mixing parameter on the basis of breathing Fermi -surface  15 model is far larger than that of Co or Ni , which is  originated from the strong \nspin-orbital coupling at the interface. More importantly, distingui shing the dominant \nmechanism underlying ultrafast demagnetization in metallic heterstructures has been a \ntough task for a long time. Here, an effective method by unification of the ultrafast \ndemagnetization time and Gilbert damping is proposed to solve thi s task, namely that, \na proportional relation  between the two parameters indicates the local spin flip \nscattering mechanism domains, otherwise the non local spin current effect domains.   \n   16 Acknowledgments  \nThis work was supported by the National Basic Research Program of China (973 \nprogram, Grant Nos. 2015CB921403 and 2016YFA0300701), the National Natural \nSciences Foundation of China (51427801, 11374350, and 11274361). The authors \nthank Hai -Feng Du, Da -Li Su n and Qing -feng Zhan for critical reading and \nconstructive suggestions for the manuscript.  The authors are indebted to B. Koopmans \nand M. Haag for helpful discussions.  \nAuthor Contributions  \nZ.H.C. supervised project. Z.H.C. and W.Z conceived and designed th e \nexperiments. W.Z. and W.H. performed the polar Kerr loops and TRMOKE \nmeasurement. 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Lett. 101, 072401 (2012).  \n61. T. Seifert, S. Jaiswal, U. Martens, J. Hannegan, L. Braun, P. Maldonado, F. \nFreimuth, A. Kronenberg, J. Henrizi, I. Radu, E. Beaurepaire, Y. Mokrousov , P. \nM. Oppeneer , M. Jourdan , G. Jakob , D. Turchinovich , L. M. M. Hayde n , M. \nWolf , M. Mü nzenberg , M. Klä ui , and T. Kampfrath ,  Nat. Photonics. 10, 483 \n(2016).   23  \n \n \nFigure caption:  \nFIG. 1 Spin precession.  (a)TRMOKE signals of Co/Ni bilayers with dNi=0.4-2.0 nm \nin applied field H = 4000 Oe. (b) Intrinsic Gilbert damping constant as a function of \ndNi. \nFIG. 2 Ultrafast demagnetization.  (a) Ultrafast demagnetization curves with various \nNi layer thickness. (b) Ultrafast demagnetization time as a function of Ni layer \nthickness. (c) Ult rafast demagnetization time as a function of Gilbert damping \nconstant. The red full line indicates theoretical fitting .  \n   24  \n \n \n \n \n                  \n \n \n \n \n \n \n \nFig. 1 (Color Online)  Spin precession.  (a)TRMOKE signals of Co/Ni bilayers \nwith dNi=0.4-2.0 nm in applied field H = 4000 Oe. (b) Intrinsic Gilbert damping \nconstant as a function of dNi. \n  \n 25  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig.2 (Color Online)  Ultrafast demagnetization.  (a) Ultrafast demagnetization \ncurves with various Ni layer thickness. (b) Ultrafast demagnetization time as a \nfunction of Ni layer thickness. (c) Ultrafast demagnetization time as a function of \nGilbert damping constant. The red full line indicates theoret ical fitting.  \n \n \n 26  \nSupplementary Information  \n \nUnifying ultrafast demagnetization and intrinsic Gilbert damping in Co/Ni \nbilayers with electronic relaxation near the Fermi surface  \n \n \n \nPartⅠ \n \n \nThe measurements of static properties for Ta (3 nm)/Pt (2 nm)/Co (0.8 nm)/Ni \n(dNi nm)/Pt (1 nm)/Ta (3 nm) . \n \nFig. 1(a) shows the polar magneto -optical Kerr signal measured at room \ntemperature with maximum applied field of 300 Oe. The static polar Kerr loops of \nCo/Ni bilayers were acquired using a laser diode with a wavel ength of 650 nm.  All \nsamples show very square loops  with a remanence ratio of about 100% , indicating t he \nwell-established perpendicular magnetization anisotropy ( PMA) of the samples. The \nmeasured coercivity Hc decreases with dNi from 103Oe for dNi = 0.4 nm to 37Oe for \ndNi =2.0 nm (Fig. 1(b)). The decrease of coercivity implies that the PMA decreases \nwith the thickness of Ni.  \n \n  27  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig.1  Static magnetic properties of of Ta (3 nm)/Pt (2 nm)/Co (0.8 nm)/Ni (d Ni \nnm)/Pt (1 nm)/Ta (3 nm) bilayers.  (a) Polar -MOKE loops with various thickness of \nNi layer d Ni. (b) Coercivity Hc and effective anisotropy field as a function of Ni \nlayer thickness d Ni. \n \neff\nKH\n 28  \n \n \n \n \nPartⅡ \n \nThe measurements of spin dynamics for Co/Ni bilayers in ns timescales and the \nanalysis of extrinsic contributions to spin precession  \n \nIn this part, we show the details of spin precession experiment.  For example, Fig. \n2(a) illustrates the scheme for laser -induced magnetization precession. The direction \nof applied field is fixed at .  \nThe typical time -resolved magnetization dynamics with various applied fields for \nTa(3 nm)/Pt(2 nm)/Co(0.8 nm)/Ni(0.8 nm)/Pt(1 nm)/Ta(3 nm) shown in Fig. 2(b) can \nbe best fitted by using the damped harmonic function added to an \nexponential -decaying background1:  \n                (1) \nwhere A and B are the background magnitudes, and  is the background recovery rate. \nC, , f and  are magnetization precession amplitude, relaxation time, frequency and \nphase, respectively. From the f itting curves shown in Fig. 2(b) as the solid lines, the \nvalues of precession frequency f and relaxation time  are extracted. Since the applied \nfields are large enough, we can obtain the Gilbert damping constant using the \nfollowing relationship2  \n                                                (2).  \n80H\n( ) exp( ) exp( )sin(2 )tM t A B t C ft         \n\n1)2( f 29 In the case of films with a relatively low Gilbert damping3-7 as well as thickness \nlarger than the optical penetration depth8, ultrafast laser may generate non -uniform \nspin waves and  affect the relation ship between demagnetization and Gilbert damping \nas extrinsic contributions . In order to check the contribution of non -uniform modes, \nwe performed a fast Fourier transform shown in Fig. 2(c). Only the uniform \nprecession mode was excited at present Co/Ni bi layers with perpendicular magnetic \nanisotropy.     \nBoth  and f as a function of H are plotted in Fig.3. Since the overall damping \nconstant consists of intrinsic damping and extrinsic damping whereby the second one \narises from inhomogeneities in the sample , the Gilbert damping constant decreases \nmonotonously to a constant value as the applied field increases (Fig. 3(a)).  In the low \nexternal fields range, the inhomogeneously distributed anisotropy may lead to higher \n values. Fortunately, the sufficient high  field we used can suppress the extrinsic \ncontributions to the magnetization precession, because for high fields the \nmagnetization dynamics is mainly determined by the external field9. In addition, \nbecause of the interaction between femtosecond laser sourc e and the thin films, the \nlateral heat distribution across the film plane has to be considered as another  \ncandidate contributions to affect the processional dynamics. As is shown by F. Busse \net al6, the heat was trapped as the Gaussian distribution across the film plane of \nCoFeB up to 1 ns due to the use of regenerative amplifier. It can enhance the laser \npower largely while the pump laser spot kept as large as around 90  μm. This \nfacilitates the occurrence of the temperature profile, and consequently the sp in-waves  30 in the range of laser spot size. However, in the absent of regenerative amplifier at \npresent, the laser spot is so small as less than 10 1,10 that one can excite the \nnonequilibrium state of the samples. And the laser fluence used here is around 1\n, which is far weaker than that used in previous report6. Although smaller \nlaser spot seems easier to trigger the nonuniform spin waves, the very low laser power \nwe used here can suppress the influence of lateral heat distribution on the relaxation \ntime o f spin dynamics at present.  M oreover, the absence of non -uniform spin wave \ndemonstrated in Fig. 2(c) in the pump laser spot confirms that the lateral heat \ntransport can be neglected here. In fact, it is found in the main text, within the three \ntemperature  model  (3TM model) describing the ultrafast demagnetization dynamics, \nthat the heat induced by laser pulse mainly transports along the thickness direction to \nsubstrate in less than a few picoseconds. The observation of pronounced \nmagnetization recovery aft er ultrafast demagnetization can exclude the possibility of \nlateral heat trap.     \n   In order to avoid the effect of extrinsic damping constant, the intrinsic \ndamping constants were obtained by fitting the overall damping factor as the function \nof applied  fields with the expression  shown as the red line in Fig. 3(a) : \n                                            (3) \nwhere  and are the intrinsic and extrinsic parts of the damping factor, \nrespectively. The intrinsic part is independent of the external field  or precession \nfrequency, while the extrinsic part is field -dependent.  \nm\n2/cmmJ\n0/\nint 1HH\nrae \nintr\n0/\n1HHe 31 The experimental f-H relation in Fig. 3(b) can be fitted by analytic Kittel formula \nderived from LLG equation2: \n                                                (4) \nwhere , . The \nequilibrium angle of magnetization  was calculated from the relationship\n. The direction of applied field is fixed at . In the \nabove equations, and are the effective perpendicular magnetization \nanisotropy and gyromagnetic ratio, respectively, wher e , . In \nour calculation, the Lande factor  was set to 2.2 as the bulk Co value2. is the \nonly adjustable parameter. The variation of effective field with the thickness of Ni \nlayer was also plotted in Fig. 1(b).  When we plot the intrinsic Gilbert damping \nconstant as a function of effective anisotropy field  in Fig. 4, a proportional relation \nwas confirmed in our Co/Ni bilayer system, which demonstrates that spin -orbit \ncoupling contributes to both Gilbert damping and PMA .   \n \n \n \n \n \n \n \n2 12HH f\n 2\n1 cos ) cos(eff\nK H H H H  \n  2cos ) cos(2eff\nK H H H H  \n\n) sin(22sin    H eff\nKHH\n80H\neff\nKH\n\nseff eff\nKMKH2\n2Bg\nh\ng\neff\nKH 32  \n \n \n \n \n \n \n \n \n \n \nThe numerical simulation for ultrafast demagnetization  \n \n \n \n \n \nFig. 2   (a) Scheme of TRMOKE. (b): TRMOKE signals with various applied \nfield for Ta (3 nm)/Pt (2 nm)/Co (0.8 nm)/Ni (0.8 nm)/Pt (1 nm)/Ta (3 nm) \nbilayers. (c): Fast Fourier transform ation s ignals.  \n \n \n \n \n \n \n 33  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 3  Gilbert damping and precession frequency. Field dependence of overall \ndamping constant (a) and precession frequency (b) of Co/Ni bilayers with\n \n \nnm dnm dNi Co 8.0 ,8.0  \n 34  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 4  Dependence of intrinsic Gilbert damping constant on the effective \nanisotropy field.  \n \n \n \n \n \n \n \n 35  \n \n \n \nPartⅢ \n \nNumerical simulation for the effect of heat transport across the film thickness \non the ultrafast demagnetization time  \n \n \nTo estimate the evolution of heat  transport profile in time, w e carried out a \nnumerical simulation based on M3TM11 model, in which the heat transport12 was \ndominated by electrons and a temperature gradient across the film thickness was \nintroduced.  It is divided in thin slabs in the direct ion normal to the film plane, and the \nslabs is 0.1 nm thick. For each slab, the evolution of the electron and phonon \ntemperatures \neT  and \npT  are determined by a  set of coupled differential equations :13  \n)),)(coth()( 1()()()()),( )(()()),( )(( ))( ()()(\nzTmTzmTzTzRmdtzdmzTzTgdtz dTCzTzTg zTdtzdTzT\nec\ncpp e epp\npe p ep ez ze\ne\n    \n                (5) \n \nWhere \nsMMm ,\n)()(\n0zTzT\npe 4, \n228\nD atB atcBep sf\nEVTkgaR ,with \nat the atomic \nmagnetic moment in units of Bohr magneton \nB , \natV  the atomic volume, and \nDE is \nthe Debye energy.  \neC and \npC  are the heat capacities of the e and p systems \nrespectively. \n)(zTez is the electron temperature gradient normal to the film . \nBk is  36 the Boltzmann constant. \n0k is the material dependent electronic thermal conductivity. \nepg\nis the e -p coupling constant and determines  the decay of the electronic \ntemperature until equilibrium is  reached14. \nsfa represents the spin -flip probability11. \nThe equations of motion for each slab thus describe heating of the electron system by \na Gaussian laser pulse, heat diffusi on by electrons to neighboring slabs, e -p \nequilibration, and finally the evolution of the magnetization due to e -p spin -flip \nscattering. In the simulation, the total magneto -optical signal was obtained by the \ncalculation of \ndzztzm t ) exp(),( )(  .  \nThe electronic system after the action of the laser pulse is in a strongly  \nnon-equilibrium situation. Nevertheless, one can describe the electron  system by use \nof an electron temperature. The reason is that the laser  photons excite electrons, but \nthese excite d electrons thermalize more or  less instantly due to very rapid and \nfrequent electron -electron  scatterings via their Coulomb interactions. This is the \nassumption of the  accepted Elliott -Yafet scenario  which describes the  effect of the \nlaser pulse directly after the action of the laser pulse.   \nFig.4(a) shows the simulated ultrafast demagnetization curves for various film \nthicknesses. We can clearly observe that the evolution of magnetization curves looks \nalmost identical for various film thicknesses , indicating that the effect of heat \ntransport on the demagnetization time can be neglected. Despite this, for the \nremagnetization part, a deviation from the experimental curves occurs. This is mainly \nbecause that the heat diffusion can almost be neglected d uring the ultrafast \ndemagnetization timescale, but starts playing an increasing role from ps timescale  37 onwards. The similar phenomenon was reported previously by B. Koopmans et al. \nFortunately, what we should be focused on here is in the ultrafast demagnet ization \ntimescale, in which the effect of heat transport can be neglected.  In fact, as is shown \nin Fig. 4(b), less than 10 fs variation was induced with the thicknesses ranged from \n1.2 nm to 2.8 nm . The parameters used in the simulation is given in Table.1 .   \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  38  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 4(a) Dependence of demagnetization as a function of delay time after pulsed laser \nheating at \n0t  (b) Maximum demagnetization and demagnetization time  \nversus the sample thickness.  \n \n \n \n 39  \n \n \n \n \n \nTable 1: Parameters used in the M3TM12,13,15. \n \nParameters  Value  Units  \n\n 5400  \n) /(23KmJ \npC\n \n61033.2  \n) /(3KmJ  \nepg\n \n181005.4  \n) /(3sKmJ  \nDE\n 0.036  \neV  \nat\n 0.62  \ncT\n 630 \nK  \n0\n 90.7 \n) /(smKJ \nsfa\n 0.185  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  40  \n \n \n \n \n \n \nPart  Ⅳ \n \nTable . 1 Values of the main fit parameters of ultrafast demagnetizations curves \nfor various thicknesses of the samples.  \n \n \n \n \n \n \n \n \nReferences :  dNi (nm)       \n0.4 200 860 2.3 330 100 \n0.8 170 860 2.1 330 100 \n1.0 150 860 2.0 330 100 \n1.5 120 860 2.3 330 100 \n2.0 90 860 2.0 330 100 \n)(fsM\n)fsE（\n)(0ps\n)(fsG\n)(0fst 41 1、W. 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Walowski, D. Watanabe, T. Kubota, X. Zhang, \nH. Naganuma, M. Oogane, Y. Ando, and T. Miyazaki, Phys. Rev. Lett. 106, 117201 \n(2011)  \n10、W. He, T. Zhu, X. -Q. Zhang, H. -T. Yang, and Z. -H. Cheng, Sci. Rep. 3, 2883 \n(2013).   42 11、B. Koopmans,  G. Malinowski, F. Dalla Longa, D. Steiauf, M. Fä hnle, T. Roth, M. \nCinchetti and M. Aeschlimann, Nat. Mater . 9, 259265 (2010).  \n12、K. C. Kuiper, G. Malinowski, F. Dalla Longa, and B. Koopmans, J. Appl. Phys \n109, 07D316 (2011).  \n13、K. C. Kuiper, T. Roth, A. J. Schellekens, O. Schmitt, B. Koopmans,  M. Cinchetti, \nand M. Aeschlimann, Appl. Phys. Lett. 105, 202402 (2014).  \n14、L.I. Berger, Optical properties of selected inorganica and organic solids, in \nHandbook of Chemistry and Physic s, 88th Edition, p. 12 -144 - 12-159. 3.1.5, 3.4  \n15、U. A. Macizo, Modeling of ultrafast laser -induced magnetization dynamics within \nthe Landau -Lifshitz -Bloch approach , PhD Thesis 2012. P.78  \n \n \n "    },    {        "title": "1502.04695v2.Role_of_nonlinear_anisotropic_damping_in_the_magnetization_dynamics_of_topological_solitons.pdf",        "content": "Role of nonlinear anisotropic damping in the magnetization dynamics of topological solitons\nJoo-V on Kim\u0003\nInstitut d’Electronique Fondamentale, Univ. Paris-Sud, 91405 Orsay, France and\nCNRS, UMR 8622, 91405 Orsay, France\n(Dated: May 31, 2021)\nThe consequences of nonlinear anisotropic damping, driven by the presence of Rashba spin-orbit coupling\nin thin ferromagnetic metals, are examined for the dynamics of topological magnetic solitons such as domain\nwalls, vortices, and skyrmions. The damping is found to a \u000bect Bloch and N ´eel walls di \u000berently in the steady\nstate regime below Walker breakdown and leads to a monotonic increase in the wall velocity above this transition\nfor large values of the Rashba coe \u000ecient. For vortices and skyrmions, a generalization of the damping tensor\nwithin the Thiele formalism is presented. It is found that chiral components of the damping a \u000bect vortex- and\nhedgehog-like skyrmions in di \u000berent ways, but the dominant e \u000bect is an overall increase in the viscous-like\ndamping.\nPACS numbers: 75.60.Ch, 75.70.Kw, 75.75.-c, 75.78.Fg\nI. INTRODUCTION\nDissipation in magnetization dynamics is a longstanding\nproblem in magnetism [1–3]. For strong ferromagnets such as\ncobalt, iron, nickel, and their alloys, a widely used theoretical\napproach to describe damping involves a local viscous form\ndue to Gilbert for the Landau-Lifshitz equation of motion,\n@m\n@t=\u0000\r0m\u0002He\u000b+\u000b0m\u0002@m\n@t; (1)\nwhich appears as the second term on the right hand side, pro-\nportional to the damping constant \u000b0. This equation describes\nthe damped magnetization precession about a local e \u000bective\nfieldHe\u000b=\u0000(1=\u00160Ms)\u000eU=\u000em, which is given by a variational\nderivative of the magnetic energy Uwith respect to the mag-\nnetization field described by the unit vector m, with\r0=\u00160\r\nbeing the gyromagnetic constant and Msis the saturation mag-\nnetization. Despite the multitude of physical processes that\nunderlie dissipation in such materials, such as the scattering\nof magnons with electrons, phonons, and other magnons, the\nform in Eq. (1) has proven to be remarkably useful for describ-\ning a wide range of dynamical phenomena from ferromagnetic\nresonance to domain wall motion.\nOne feature of the dissipative dynamics described in Eq. (1)\nis that it is local, i.e., the damping torque only depends on the\nlocal magnetization and its time dependence. With the ad-\nvent of magnetic heterostructures, however, this restriction of\nlocality has been shown to be inadequate for systems such\nas metallic multilayers in which nonlocal processes can be\nimportant [4]. A striking example involves spin pumping,\nwhich describes how spin angular momentum can be dissi-\npated in adjacent magnetic or normal metal layers through the\nabsorption of spin currents generated by a precessing magne-\ntization [5, 6]. Early experimental observations of this phe-\nnomena involved iron films sandwiched by silver layers [7]\nand permalloy films in close proximity with strong spin-orbit\nnormal metals such as palladium and platinum [8, 9], where\n\u0003joo-von.kim@u-psud.frferromagnetic resonance line widths were shown to depend\nstrong on the composition and thickness of the adjacent lay-\ners. Such observations also spurred other studies involving\nferromagnetic multilayers separated by normal metal spacers,\nwhere spin pumping e \u000bects can lead to a dynamic coupling\nbetween the magnetization in di \u000berent layers [10, 11]. In\nthe context of damping, such dynamic coupling was shown to\ngive rise to a configuration dependent damping in spin-valve\nstructures [12, 13].\nA generalization of the spin-pumping picture in the context\nof dissipation was given by Zhang and Zhang, who proposed\nthat spin currents generated within the ferromagnetic material\nitself can lead to an additional contribution to the damping,\nprovided that large magnetization gradients are present [14].\nThis theory is based on an sdmodel in which the local mo-\nments (4 d) are exchange coupled to the delocalized conduc-\ntion electrons (3 s), which are treated as a free electron gas.\nThe spin current “pumped” at one point in the material by\nthe precessing local moments are dissipated at another if the\ncurrent encounters strong spatial variations in the magneti-\nzation such as domain walls or vortices – a mechanism that\ncan be thought of as the reciprocal process of current-induced\nspin torques in magnetic textures [15–18]. For this reason,\nthe mechanism is referred to as “feedback” damping since the\npumped spin currents generated feed back into the magnetiza-\ntion dynamics in the form of a dissipative torque. This addi-\ntional contribution is predicted to be both nonlinear and non-\nlocal, and can have profound consequences for the dynamics\nof topological solitons such as domain walls and vortices as a\nresult of the spatial gradients involved. Indeed, recent experi-\nments on vortex wall motion in permalloy stripes indicate that\nsuch nonlinear contributions can be significant and be of the\nsame order of magnitude as the usual Gilbert damping char-\nacterized by \u000b0in Eq. (1) [19].\nAn extension to this feedback damping idea was proposed\nrecently by Kim and coworkers, who considered spin pump-\ning involving a conduction electron system with a Rashba\nspin-orbit coupling (RSOC) [20]. By building upon the\nZhang-Zhang formalism, it was shown that the feedback\ndamping can be expressed as a generalization of the Landau-arXiv:1502.04695v2  [cond-mat.mtrl-sci]  4 Jun 20152\nLifshitz equation [14, 20],\n@m\n@t=\u0000\r0m\u0002He\u000b+m\u0002D LL(m)\u0001@m\n@t; (2)\nwhere the 3\u00023 matrixDLLrepresents the generalized damping\ntensor, which can be expressed as [20]\nDi j\nLL=\u000b0\u000ei j+\u0011X\nk(Fki+˜\u000bR\u000f3ki)\u0010\nFk j+˜\u000bR\u000f3k j\u0011\n:(3)\nHere,\u000b0is the usual Gilbert damping constant, \u0011=\ng\u0016B~G0=(4e2Ms) is a constant related to the conductivity G0\nof the spin bands [14], Fki=(@m=@xk)iare components of\nthe spatial magnetization gradient, ˜ \u000bR=2\u000bRme=~2is the\nscaled Rashba coe \u000ecient,\u000fi jkis the Levi-Civita symbol, and\nthe indices ( i jk) represent the components ( xyz) in Cartesian\ncoordinates. In addition to the nonlinearity present in the\nZhang-Zhang picture, the inclusion of the \u000bRterm results\nin an anisotropic contribution that is related to the underly-\ning symmetry of the Rashba interaction. Numerical estimates\nbased on realistic parameters suggest that the Rashba con-\ntribution can be much larger than the nonlinear contribution\n\u0011alone [20], which may have wide implications for soliton\ndynamics in ultrathin ferromagnetic films with perpendicular\nmagnetic anisotropy, such as Pt /Co material systems, in which\nlarge spin-orbit e \u000bects are known to be present.\nIn this article, we explore theoretically the consequences\nof the nonlinear anisotropic damping given in Eq. (3) on the\ndynamics of topological magnetic solitons, namely domain\nwalls, vortices, and skyrmions, in which spatial gradients can\ninvolve 180\u000erotation of the magnetization vector over length\nscales of 10 nm. In particular, we examine the role of chiral-\nity in the Rashba-induced contributions to the damping, which\nare found to a \u000bect chiral solitons in di \u000berent ways. This ar-\nticle is organized as follows. In Section II, we discuss the\ne\u000bects of nonlinear anisotropic damping on the dynamics of\nBloch and N ´eel domain walls, where the latter is stabilized\nby the Dzyaloshinskii-Moriya interaction. In Section III, we\nexamine the consequences of this damping for vortices and\nskyrmions, and we derive a generalization to the damping\ndyadic appearing in the Thiele equation of motion. Finally,\nwe present some discussion and concluding remarks in Sec-\ntion IV.\nII. BLOCH AND N ´EEL DOMAIN WALLS\nThe focus of this section are domain walls in ultrathin\nfilms with perpendicular magnetic anisotropy. Consider a\n180\u000edomain wall representing a boundary separating two\noppositely magnetized domains along the xaxis, with zbe-\ning the uniaxial anisotropy axis that is perpendicular to the\nfilm plane. We assume that the magnetization remains uni-\nform along the yaxis. The unit magnetization vector m(x;t)\ncan be parametrized in spherical coordinates ( \u0012;\u001e), such that\nm=(sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012). With this definition, thespherical angles for the domain wall profile can be written as\n\u0012(x;t)=2 tan\u00001exp \n\u0006x\u0000X0(t)\n\u0001!\n;\n\u001e(x;t)=\u001e0(t); (4)\nwhere X0(t) denotes the position of the domain wall, \u0001 =pA=K0represents the wall width parameter that depends on\nthe exchange constant Aand the e \u000bective uniaxial anisotropy\nK0, and the azimuthal angle \u001e0(t) is a dynamic variable but\nspatially uniform. The anisotropy constant, K0=Ku\u0000\n\u00160M2\ns=2, involves the di \u000berence between the magnetocrys-\ntalline ( Ku) and shape anisotropies relevant for an ultrathin\nfilm. In this coordinate system, a static Bloch wall is given by\n\u001e0=\u0006\u0019=2, while a static N ´eel wall is given by \u001e0=0;\u0019. A\npositive sign in the argument of the exponential function for\n\u0012in Eq. (4) describes an up-to-down domain wall profile go-\ning along the +xdirection, while a negative sign represents a\ndown-to-up wall.\nTo determine the role of the nonlinear anisotropic damping\nterm in Eq. (3) on the wall dynamics, it is convenient to com-\npute the dissipation function W(˙X0;˙\u001e0) for the wall variables,\nwhere the notation ˙X0\u0011@tX0, etc., denotes a time derivative.\nThe dissipation function per unit surface area is given by\nW\u0010˙X0;˙\u001e0\u0011\n=Ms\n2\rZ1\n\u00001dx˙miDi j\nLL(m) ˙mj; (5)\nwhere mi=mi\u0002x\u0000X0(t);\u001e0(t)\u0003and the Einstein summation\nconvention is assumed. By using the domain wall ansatz\n(4), the integral in Eq. (5) can be evaluated exactly to give\nW=W0+WNL, where W0represents the usual (linear) Gilbert\ndamping,\nW0=\u000b0Ms\u0001\n\r0BBBB@˙X2\n0\n\u00012+˙\u001e2\n01CCCCA; (6)\nwhile WNLis the additional contribution from the nonlinear\nanisotropic damping,\nWNL=Ms\u0001\n\r2666641\n3\u000b3sin2\u001e0(t)˙X2\n0\n\u00012\n+ 2\n3\u000b1\u0006\u0019\n2\u000b2cos\u001e0(t)+\u000b3cos2\u001e0(t)!\n˙\u001e2\n0#\n;(7)\nwhere\u000b1\u0011\u0011=\u00012,\u000b2\u0011\u0011˜\u000bR=\u0001, and\u000b3\u0011\u0011˜\u000b2\nRare dimen-\nsionless nonlinear damping constants. In contrast to the linear\ncase, the nonlinear anisotropic dissipation function exhibits a\nconfiguration-dependent dissipation rate where the prefactors\nof the ˙X2\n0and˙\u001e2\n0terms depend explicitly on \u001e0(t).\nIn addition to the nonlinearity a chiral damping term, pro-\nportional to \u000b2, appears as a result of the Rashba interaction\nand is linear in the Rashba coe \u000ecient\u000bR. The sign of this\nterm depends on the sign chosen for the polar angle \u0012in the\nwall profile (4). To illustrate the chiral nature of this term, we\nconsider small fluctuations about the static configuration by\nwriting\u001e0(t)=\u001e0+\u000e\u001e(t), where\u000e\u001e(t)\u001c\u0019is a small angle.\nThis approximation is useful for the steady state regime below3\nWalker breakdown. For up-to-down Bloch walls ( \u001e0=\u0006\u0019=2),\nthe nonlinear part of the dissipation function to first order in\n\u000e\u001e(t) becomes\nWNL;Bloch\u0019Ms\u0001\n\r266664\u000b3\n3˙X2\n0\n\u00012+ 2\u000b1\n3+Cx\u0019\u000b2\n2\u000e\u001e(t)!\n˙\u001e2\n0377775:(8)\nThe quantity Ci=\u00061 is a component of the chirality vec-\ntor [21],\nC=1\n\u0019Z1\n\u00001dxm\u0002@xm; (9)\nwhich characterizes the handedness of the domain wall. For\na right-handed Bloch wall, \u001e0=\u0000\u0019=2 and the only nonva-\nnishing component is Cx=1, while for a left-handed wall\n(\u001e0=\u0000\u0019=2) the corresponding value is Cx=\u00001. Thus, the\nterm proportional to \u000b2depends explicitly on the wall chiral-\nity. Similarly for up-to-down N ´eel walls, the same lineariza-\ntion about the static wall profile leads to\nWNL;Neel\u0019Ms\u0001\n\r 2\u000b1\n3+Cy\u0019\u000b2\n2+\u000b3!\n˙\u001e2\n0; (10)\nwhere Cy=1 for a right-handed N ´eel wall (\u001e0=0) and\nCy=\u00001 for its left-handed counterpart ( \u001e0=\u0019). Since the\nfluctuation\u000e\u001e(t) is taken to be small, the chiral damping term\nis more pronounced for N ´eel walls in the steady-state velocity\nregime since it does not depend on the fluctuation amplitude\n\u000e\u001e(t) as in the case of Bloch walls.\nTo better appreciate the magnitude of the chirality-\ndependent damping term, it is instructive to estimate numer-\nically the relative magnitudes of the nonlinear damping con-\nstants\u000b1;\u000b2;\u000b3. Following [Ref. 20], we assume \u0011=0:2 nm2\nand\u000bR=10\u000010eV m. If we suppose \u0001 = 10 nm, which is\nconsistent with anisotropy values measured in ultrathin films\nwith perpendicular anisotropy [22], the damping constants can\nbe evaluated to be \u000b1=0:002,\u000b2=0:052, and\u000b3=1:37.\nSince\u000b0varies between 0.01–0.02 [23] and 0.1–0.3 [24] de-\npending on the material system, the chiral term \u000b2is compa-\nrable to Gilbert damping in magnitude, but remains almost an\norder of magnitude smaller than the nonlinear component \u000b3\nthat provides the dominant contribution to the overall damp-\ning.\nThe full equations of motion for the domain wall dynam-\nics can be obtained using a Lagrangian formalism that ac-\ncounts for the dissipation given by W[25, 26]. For the sake\nof simplicity, we will focus on wall motion driven by mag-\nnetic fields alone, where a spatially-uniform magnetic field\nHzis applied along the +zdirection. In addition, we include\nthe Dzyaloshinskii-Moriya interaction appropriate for the ge-\nometry considered [27, 28] when considering the dynamics\nof N ´eel walls. From the Euler-Lagrange equations with the\nRayleigh dissipation function,\nd\ndt@L\n@˙X0\u0000@L\n@X0+@W\n@˙X0=0; (11)\nwith an analogous expression for \u001e0, the equations of motion\nfor the wall coordinates are found to be\n˙\u001e0+\u0012\n\u000b0+\u000b3\n3sin2\u001e0\u0013˙X0\n\u0001=\r0Hz; (12)\n0 5 10 15 20 \nm0Hz (mT)010 20 30 Wall Velocity (m/s) \n0 0.1 0.2 0.3 0.4 \naR (eV nm)0.9 0.95 1vw / v w,0\naR (eV nm)aR = 0  0.05 eV nm 0.1 eV nm 0.15 eV nm (a)\n(b) (c)\n0 0.1 0.2 0.3 0.4 0.15 0.2 0.25 df w / pFIG. 1. (Color online) Bloch wall dynamics. (a) Steady-state domain\nwall velocity,h˙X0i, as a function of perpendicular applied magnetic\nfield,\u00160Hz, for several values of the Rashba coe \u000ecient,\u000bR. The\nhorizontal dashed line indicates the Walker velocity and the arrows\nindicate the Walker transition. (b) The ratio between the Walker ve-\nlocity, vW, to its linear damping value, vW;0, as a function of \u000bR. (c)\nDeviation in the wall angle from rest at the Walker velocity, \u000e\u001eW, as\na function of \u000bR\n˙X0\n\u0001\u0000 \n\u000b0+2\u000b1\n3+\u0019\u000b2\n2cos\u001e0+\u000b3cos2\u001e0!\n˙\u001e0\n=\u0000\r0 \u0019\n2Dex\n\u00160Ms\u0001+2K?\n\u00160Mscos\u001e0!\nsin\u001e0;(13)\nwhere Dexis the Dzyaloshinskii-Moriya constant [28] and\nK?represents a hard-axis anisotropy that results from vol-\nume dipolar charges. The Dzyaloshinskii-Moriya interaction\n(DMI) is present in ultrathin films in contact with a strong\nspin-orbit coupling material [29, 30] and favors a N ´eel-type\nwall profile [31, 32]. The DMI itself can appear as a con-\nsequence of the Rashba interaction and therefore its inclu-\nsion here is consistent with the nonlinear anisotropic damping\nterms used [20, 33, 34].\nResults from numerical integration of these equations of\nmotion for Bloch and N ´eel walls are presented in Figs. 1 and\n2. We used parameters consistent with ultrathin films with\nperpendicular anisotropy, namely \u000b0=0:1,Ms=1 MA /m,\n\u0001 = 10 nm, and K?=\u00160NxM2\ns=2 with the demagnetiza-\ntion factor Nx=0:02 [28]. To study the dynamics of the\nDzyaloshinskii (N ´eel) wall we assumed a value of Dex=1\nmJ/m2, which is much stronger than the volume dipolar in-\nteraction represented by K?and is of the same order of mag-\nnitude as values determined by Brillouin light spectroscopy\nin Pt/Co/Al2O3films [35]. As in the discussion on numeri-\ncal estimates above, we assumed \u0011=0:2 nm2but considered\nseveral di \u000berent values for the Rashba coe \u000ecient\u000bR. The\nsteady-state domain wall velocity, h˙X0i, was computed as a\nfunction of the perpendicular applied magnetic field, Hz. In4\nthe precessional regime above Walker breakdown in which\n\u001e0(t) becomes a periodic function in time, h˙X0iis computed\nby averaging the wall displacement over few hundred periods\nof precession.\nFor the Bloch case [Fig. 1(a)], the Walker field is observed\nto increase with the Rashba coe \u000ecient, which is consistent\nwith the overall increase in damping experienced by the do-\nmain wall. However, there are two features that di \u000ber qual-\nitatively from the behavior with linear damping. First, the\nWalker velocity is not attained for finite \u000bR, where the peak\nvelocity at the Walker transition is below the value reached\nfor\u000bR=0. This is shown in more detail in Fig. 1(b), where\nthe ratio between the Walker velocity, vW, and its linear damp-\ning value, vW;0, is shown as a function of \u000bR. The Walker limit\nis set by the largest extent to which the wall angle \u001e0can de-\nviate from its equilibrium value, \u001e0;eq. By assuming ˙\u001e=0\nin the linear regime, we can determine this limit by rearrang-\ning Eqs. 12 and 13 to obtain the following relationship for the\nBloch wall,\n2Hz\nNxMs=\u0000\u0012\n\u000b0+\u000b3\n3sin2\u001e0\u0013\nsin 2\u001e0: (14)\nThe angle\u001e0=\u001eWfor which the right hand side of this\nequation is an extremum determines the Walker limit. In\nFig. 1(c), we present this limit in terms of the deviation an-\ngle,\u000e\u001eW\u0011j\u001eW\u0000\u001e0;eqj, which is shown as a function of \u000bR.\nAs the Rashba parameter is increased, the maximum wall tilt\npossible in the linear regime decreases from the linear damp-\ning value of \u0019=4, which results in an overall reduction in the\nWalker velocity. Second, the field dependence of the wall ve-\nlocity below Walker breakdown is nonlinear and exhibits a\nslight convex curvature, which becomes more pronounced as\n\u000bRincreases. This curvature can be understood by examining\nthe wall mobility under fields, which can be deduced from Eq.\n(12) by setting ˙\u001e=0,\n˙X0=\r0\u0001\n\u000b0+(\u000b3=3)sin2\u001e0Hz: (15)\nSince the angle \u001e0for Bloch walls varies from its rest value of\n\u001e0;eq=\u0006\u0019=2 at zero field to \u001eWat the Walker field, the sin2\u001e0\nterm in the denominator decreases from its maximum value of\nsin2\u001e0;eq=1 at rest with increasing applied field and therefore\nan increase in the mobility is seen as Hzincreases, resulting\nin the convex shape of the velocity versus field relation below\nWalker breakdown.\nIt is interesting to note that the nonlinear damping terms\na\u000bect the Dzyaloshinskii (N ´eel) wall motion di \u000berently. In\ncontrast to the Bloch case, the Walker velocity for increasing\n\u000bRslightly exceeds the linear damping value, which can be\nseen by the arrows marking the Walker transition in Fig. 2(a)\nand in detail in Fig. 2(b). In addition, the field dependence of\nthe velocity exhibits a concave curvature below breakdown,\nwhich can also be understood from Eq. (15) by considering\nthat\u001e0instead deviates from the rest value of \u001e0;eq=0 or\u0019\nat zero field. As for the Bloch wall case, the deviation angle\nat breakdown is determined by the value of \u001e0that gives an\n0 50 100 150 200 0100 200 300 Wall Velocity (m/s) \n0 0.1 0.2 0.3 0.4 11.0008 aR = 0  0.05 eV nm 0.1 eV nm 0.15 eV nm vw / v w,0\naR (eV nm) aR (eV nm)m0Hz (mT)(a)\n(b)(c)\n0 0.1 0.2 0.3 0.4 0.5 0.51 0.52 0.53 df w / pFIG. 2. (Color online) Dzyaloshinskii (N ´eel) wall dynamics. (a)\nSteady-state domain wall velocity, h˙X0i, as a function of perpendic-\nular applied magnetic field, \u00160Hz, for several values of the Rashba\ncoe\u000ecient,\u000bR. The horizontal dashed line indicates the Walker ve-\nlocity and the arrows indicate the Walker transition. (b) The ratio\nbetween the Walker velocity, vW, to its linear damping value, vW;0, as\na function of \u000bR. (c) The wall angle at the Walker velocity, \u001eW, as a\nfunction of \u000bR\nextremum for the right hand side of\n2Hz\nNxMs=\u0000\u0012\n\u000b0+\u000b3\n3sin2\u001e0\u0013 \u0019Dex\n2K?\u0001cos\u001e0+sin 2\u001e0!\n;(16)\nand is also seen to decrease with increasing Rashba coe \u000e-\ncient [Fig. 2(c)]. In contrast to the Bloch wall case, how-\never, changes in \u001eWhave a comparatively modest e \u000bect on the\nWalker velocity. The shape of the velocity versus field curve\nis consistent with experimental reports of field-driven domain\nwall motion in the Pt /Co (0.6 nm) /Al2O3system [36], which\npossess a large DMI value [35] and harbors N ´eel-type domain\nwall profiles at equilibrium [37].\nAs the preceding discussion shows, the di \u000berences in the\nfield dependence of the wall velocity for the two profiles are\na result of the DMI, rather than the chiral damping term that\nis proportional to \u000b2. This was verified by setting \u000b2=0\nfor the N ´eel wall case with D,0, which did not modify\nthe overall behavior of the field dependence of the velocity. In\nthe one-dimensional approximation for the wall dynamics, the\nDMI enters the equations of motion like an e \u000bective magnetic\nfield along the xaxis, which stabilizes the wall structure by\nminimizing deviations in the wall angle \u001e0(t).\nIII. VORTICES AND SKYRMIONS\nThe focus of this section is on the dissipative dynamics\nof two-dimensional topological solitons such as vortices and\nskyrmions. The equilibrium magnetization profile for these5\nmicromagnetic objects are described by a nonlinear di \u000ber-\nential equation similar to the sine-Gordon equation, where\nthe dispersive exchange interaction is compensated by dipo-\nlar interactions for vortices [38, 39] and an additional uniax-\nial anisotropy for skyrmions [40]. The topology of vortices\nand skyrmions can be characterized by the skyrmion winding\nnumber Q,\nQ=1\n4\u0019\"\ndxdy m\u0001\u0010\n@xm\u0002@ym\u0011\n: (17)\nWhile the skyrmion number for vortices ( Q=\u00061=2) and\nskyrmions ( Q=\u00061) are di \u000berent, their dynamics are quali-\ntatively similar and can be described using the same formal-\nism. For this reason, vortices and skyrmions will be treated\non equal footing in what follows and distinctions between the\ntwo will only be drawn on the numerical values of the damp-\ning parameters.\nA key approximation used for describing vortex or\nskyrmion dynamics is the rigid core assumption, where it is\nassumed that the spin structure of the soliton remains unper-\nturbed from its equilibrium state during motion. Within this\napproximation, the dynamics is given entirely by the position\nof the core in the film plane, X0(t)=[X0(t);Y0(t)], which al-\nlows the unit magnetization vector to be parametrized as\n\u0012(x;y;t)=\u00120[kx\u0000X0(t)k];\n\u001e(x;y;t)=qtan\u00001\"y\u0000Y0(t)\nx\u0000X0(t)#\n+c\u0019\n2; (18)\nwhere qis a topological charge and cis the chirality. An il-\nlustration of the magnetization field given by the azimuthal\nangle\u001e(x;y) is presented in Fig. 3. q=1 corresponds to a\nvortex or skyrmion, while q=\u00001 represents the antivortex or\nantiskyrmion.\nThe dynamics of a vortex or skyrmion in the rigid core ap-\nproximation is given by the Thiele equation,\nG\u0002˙X0+DT\u0001˙X0=\u0000@U\n@X0; (19)\nwhere\nG=Msd\n\r\"\ndxdy sin(\u0012)(r\u0012\u0002r\u001e) (20)\nis the gyrovector and U(X0) is the e \u000bective potential that is ob-\ntained from the magnetic Hamiltonian by integrating out the\nspatial dependence of the magnetization. The damping dyadic\nin the Thiele equation, DT, can be obtained from the dissipa-\ntion function in the rigid core approximation, W(˙X0), which is\ndefined in the same way as in Eq. (5) but with the ansatz given\nin Eq. (18). For this system, it is more convenient to eval-\nuate the dyadic by performing the integration over all space\nafter taking derivatives with respect to the core velocity. In\nother words, the dyadic can be obtained using the Lagrangian\nformulation by recognizing that\nDT\u0001˙X0=Msd\n2\r\"\ndxdy@\n@˙X0\u0010\n˙miDi j\nLL(m) ˙mj\u0011\n: (21)\nc = 0\nq = +1\nq = –1c = 1 c = 2 c = 3 (a)\n(b) (c)\n1\n– 1 0FIG. 3. (Color online) In-plane magnetization fields for vortices and\nskyrmions. (a) Vector fields given by \u001e(x;y) in (18) for di \u000berent\nvalues of qandc. (b) V ortex and (c) skyrmion for spin structure with\nc=1;q=1, where the arrows indicate the in-plane components\n(mx;y) and the color code gives the perpendicular component of the\nmagnetization ( mz).\nBy using polar coordinates for the spatial coordinates, ( x;y)=\n(rcos';rsin'), assuming translational invariance in the film\nplane, and integrating over ', the damping dyadic is found to\nbe\nDT=Msd\n\r \n(\u000b0D0+\u000b1D1+\u000b3D3)I+\u000b2D2\"\na110\n0a22#!\n;\n(22)\nwhereIis the 2\u00022 identity matrix and the dimensionless\ndamping constants are defined as \u000b1\u0011\u0011=r2\nc,\u000b2\u0011\u0011˜\u000bR=rc, and\n\u000b3\u0011\u0011˜\u000b2\nR, in analogy with the domain wall case where the\ncore radius rcplays the role here as the characteristic length\nscale. The coe \u000ecients Didepend on the core profile and are\ngiven by\nD0=\u0019Z1\n0dr \nr(@r\u00120)2+sin2\u00120\nr!\n; (23)\nD1=2\u0019r2\ncZ1\n0dr1\nr(@r\u00120)2sin2\u00120; (24)\nD2=2\u0019rcZ1\n0dr1\nr(@r\u00120)sin\u00120(r(@r\u00120)cos\u00120+sin\u00120);\n(25)\nD3=\u0019Z1\n0dr \nr(@r\u00120)2cos2\u00120+sin2\u00120\nr!\n; (26)\nwhere the expression for D0is a known result but D1;D2and\nD3are new terms that arise from the nonlinear anisotropic\ndamping due to RSOC.\nThe coe \u000ecients a11anda22are configuration-dependent\nand represent the chiral component of the Rashba-induced\ndamping. For vortex-type spin textures ( c=1;3 and q=1),6\nTABLE I. Coe \u000ecients a11anda22of the chiral damping term in\nEq. (22) for di \u000berent vortex /skyrmion charges qand chirality c.\nq=1 q=\u00001\nc 0 1 2 3 0 1 2 3\na11 1 0\u00001 0\u00001\u00001 1 1\na22 1 0\u00001 0 1\u00001\u00001 1\na11=a22=0, which indicates that the \u000b2term plays no role\nfor such configurations. This is consistent with the result for\nBloch domain walls discussed previously, since the vortex-\ntype texture [Fig. 3(b)], particularly the vortex-type skyrmion\n[Fig. 3(c)], can be thought of as being analogous to a spin\nstructure generated by a 2 \u0019revolution of a Bloch wall about\nan axis perpendicular to the film plane. The rigid core approx-\nimation implies that fluctuations about the ground state are ne-\nglected, which is akin to setting \u000e\u001e(t)=0 in Eq. (8). As such,\nno contribution from \u000b2is expected for vortex-type textures.\nOn the other hand, a finite contribution appears for hedgehog-\ntype vortices and skyrmions ( q=1), where a11=a22=1\nforc=0 and a11=a22=\u00001 for c=2. This can be un-\nderstood with the same argument by noting that hedgehog-\ntype textures can be generated by revolving N ´eel-type domain\nwalls. A summary of these coe \u000ecients is given in Table I.\nFor antivortices ( q=\u00001), it is found that the coe \u000ecients\naiiare nonzero for all winding numbers considered. We can\nunderstand this qualitatively by examining how the magneti-\nzation varies across the core along two orthogonal directions.\nFor example, for c=0, the variation along the xandyaxes\nacross the core are akin to two N ´eel-type walls of di \u000berent\nchiralities, which results in nonvanishing contributions to a11\nanda22but with opposite sign. The sign of these coe \u000ecients\ndepends on how these axes are oriented in the film plane, as\nwitnessed by the di \u000berent chiralities cin Fig. 3. Such damping\ndynamics is therefore strongly anisotropic, which may have\ninteresting consequences on the rotational motion of vortex-\nantivortex dipoles, for example, where the antivortex configu-\nration oscillates between the di \u000berent cvalues in time [41].\nFor vortex structures, we can provide numerical estimates\nof the di \u000berent damping contributions \u000biDiby using the Usov\nansatz for the vortex core magnetization,\ncos\u00120=8>>>><>>>>:r2\nc\u0000r2\nr2c+r2r\u0014rc\n0 r>rc: (27)\nLetLrepresent the lateral system size. The coe \u000ecients Diare\nthen found to be D0=\u0019[2+ln(L=rc)],D1=D2=14\u0019=3,\nandD3=\u0019[4=3+ln(L=rc)]. We note that for D0andD3,\nthe system size Land core radius rcappear as cuto \u000bs for the\ndivergent 1=rterm in the integral. By assuming parameters of\n\u000b0=0:1,\u0011=0:05 nm2, and\u000bR=0:1 eV nm, along with\ntypical scales of rc=10 nm and L=1\u0016m, the damping\nterms can be evaluated numerically to be \u000b0D0\u00192:1,\u000b1D1\u0019\n0:0073,\u000b2D2\u00190:19, and\u000b3D3\u00196:4. As for the domain\nwalls, the Rashba term \u000b3D3is the dominant contribution and\nis of the same order of magnitude as the linear damping term,while the chiral term \u000b2D2is an order of magnitude smaller\nand the nonlinear term \u000b1D1is negligible in comparison.\nFor skyrmion configurations, a similar ansatz can be used\nfor the core magnetization,\ncos\u0012\u00120\n2\u0013\n=8>>>><>>>>:r2\nc\u0000r2\nr2c+r2r\u0014rc\n0 r>rc: (28)\nWe note that this di \u000bers from the “linear” profiles discussed\nelsewhere [40], but the numerical di \u000berences are small and do\nnot influence the qualitative features of the dynamics. The ad-\nvantage of the ansatz in Eq. (28) is that the integrals for Di\nhave simple analytical expressions. Because spatial variations\nin the magnetization for a skyrmion are localized only to the\ncore, in contrast to the circulating in-plane moments of vor-\ntices that extend across the entire system, the damping con-\nstants Dihave no explicit dependence on the system size. By\nusing Eq. (28), we find D0=D3=16\u0019=3,D1=496\u0019=15, and\nD2=52\u0019=5. By using the same values of \u000b0,\u0011, and\u000bRas for\nthe vortices in the preceding paragraph, we find \u000b0D0\u00191:7,\n\u000b1D1\u00190:052,\u000b2D2\u00190:43, and\u000b3D3\u00193:3.\nIV . DISCUSSION AND CONCLUDING REMARKS\nA clear consequence of the nonlinear anisotropic damp-\ning introduced in Eq. (3) is that it provides a mechanism by\nwhich the overall damping constant, as extracted from domain\nwall experiments, for example, can di \u000ber from the value ob-\ntained using linear response methods such as ferromagnetic\nresonance [19]. However, the Rashba term can also a \u000bect the\nferromagnetic linewidth in a nontrivial way. To see this, we\nconsider the e \u000bect of the damping by evaluating the dissipa-\ntion function associated with a spin wave propagating in the\nplane of a perpendicularly magnetized system with an ampli-\ntude b(t) and wave vector kjj. The spin wave can be expressed\nasm=\u0002b(t) cos( kjj\u0001rjj);b(t) sin(kjj\u0001rjj);1\u0003, which results in a\ndissipation function per unit volume of\nWsw=Ms\n2\r˙b(t)2\u0010\n\u000b0+\u000b3+\u0011b(t)2kkjjk2\u0011\n; (29)\nwhere a term proportional to the chiral part \u0011˜\u000bRspatially aver-\nages out to zero. The Rashba contribution \u000b3\u0011\u0011˜\u000b2\nRleads to\nan overall increase in the damping for linear excitations and\nplays the same role as the usual Gilbert term \u000b0in this ap-\nproximation, which allows us to assimilate the two terms as\nan e\u000bective FMR damping constant, \u000bFMR\u0019\u000b0+\u000b3. On\nthe other hand, the nonlinear feedback term proportional to\n\u0011is only important for large spin wave amplitudes and de-\npends quadratically on the wave vector. This is consistent\nwith recent experiments on permalloy films (in the absence of\nRSOC) in which the linear Gilbert damping was recovered in\nferromagnetic resonance while nonlinear contributions were\nonly seen for domain wall motion [19]. This result also sug-\ngests that the large damping constant in ultrathin Pt /Co/Al2O3\nfilms as determined by similar time-resolved magneto-optical7\nmicroscopy experiments, where it is found that \u000bFMR=0:1–\n0:3 [24], may partly be due to the RSOC mechanism described\nhere (although dissipation resulting from spin pumping into\nthe platinum underlayer is also likely to be important [42]).\nIncidentally, the nonlinear term \u0011b(t)2may provide a physi-\ncal basis for the phenomenological nonlinear damping model\nproposed in the context of spin-torque nano-oscillators [43].\nFor vortices and skyrmions, the increase in the overall\ndamping due to the Rashba term \u000b3can have important con-\nsequences for their dynamics. The gyrotropic response to any\nforce, as described by the Thiele equation in Eq. (19), depends\non the overall strength of the damping term. This response\ncan be characterized by a deflection angle, \u0012H, that describes\nthe degree to which the resulting displacement is noncollinear\nwith an applied force. This is analogous to a Hall e \u000bect. By\nneglecting the chiral term \u000b2D2, the deflection or Hall angle\ncan be deduced from Eq. (19) to be\ntan\u0012H=G0\n\u000b0D0+\u000b1D1+\u000b3D3; (30)\nwhere G0=2\u0019for vortices and G0=4\u0019for skyrmions. Con-\nsider the skyrmion profile and the magnetic parameters dis-\ncussed in Section III. With only the linear Gilbert damping\nterm (\u000b0D0) the Hall angle is found to be \u0012H=82:3\u000e, which\nunderlies the largely gyrotropic nature of the dynamics. If\nthe full nonlinear damping is taken into account [Eq. (30)],\nwe find\u0012H=68:3\u000e, which represents a significant reduction\nin the Hall e \u000bect and a greater Newtonian response to an ap-\nplied force. Aside from a quantitative increase in the overalldamping, the presence of the nonlinear terms can therefore af-\nfect the dynamics qualitatively. Such considerations may be\nimportant for interpreting current-driven skyrmion dynamics\nin racetrack geometries, where the interplay between edge re-\npulsion and spin torques is crucial for determining skyrmion\ntrajectories [44, 45].\nFinally, we conclude by commenting on the relevance of\nthe chiral-dependent component of the damping term, \u000b2. It\nhas been shown theoretically that the Rashba spin-orbit cou-\npling leading to Eq. (3) also gives rise to an e \u000bective chiral\ninteraction of the Dzyaloshinskii-Moriya form [34]. This in-\nteraction is equivalent to the interface-driven form considered\nearlier, which favors monochiral N ´eel wall structures in ul-\ntrathin films with perpendicular magnetic anisotropy. Within\nthis picture, a su \u000eciently strong Rashba interaction should\nonly favor domain wall or skyrmion spin textures with one\ngiven chirality as determined by the induced Dzyaloshinskii-\nMoriya interaction. So while some non-negligible di \u000berences\nin the chiral damping between vortices and skyrmions of dif-\nferent chiralities were found, probing the dynamics of solitons\nwith distinct chiralities may be very di \u000ecult to achieve exper-\nimentally in material systems of interest.\nACKNOWLEDGMENTS\nThe author acknowledges fruitful discussions with P. Borys,\nJ.-Y . Chauleau, and F. Garcia-Sanchez. This work was par-\ntially supported by the Agence Nationale de la Recherche un-\nder Contract No. ANR-11-BS10-003 (NanoSWITI) and No.\nANR-14-CE26-0012 (ULTRASKY).\n[1] L. Landau and E. Lifshitz, Phys. Z. Sowjet 8, 153 (1935).\n[2] M. Sparks, Ferromagnetic-relaxation theory (McGraw-Hill,\nNew York, 1964).\n[3] T. Gilbert, IEEE. Trans. Magn. 40, 3443 (2004).\n[4] Y . 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Fert, Nat.\nNanotech. 8, 839 (2013)."    },    {        "title": "1806.03172v1.Brownian_motion_of_magnetic_domain_walls_and_skyrmions__and_their_diffusion_constants.pdf",        "content": "Brownian motion of magnetic domain walls and skyrmions,\nand their di\u000busion constants\nJacques Miltat,\u0003Stanislas Rohart, and Andr\u0013 e Thiaville\nLaboratoire de Physique des Solides, Universit\u0013 e Paris-Sud,\nUniversit\u0013 e Paris-Saclay, CNRS, UMR 8502, F-91405 Orsay Cedex, France\n(Dated: October 8, 2018)\nExtended numerical simulations enable to ascertain the di\u000busive behavior at \fnite temperatures\nof chiral walls and skyrmions in ultra-thin model Co layers exhibiting symmetric - Heisenberg - as\nwell as antisymmetric - Dzyaloshinskii-Moriya - exchange interactions. The Brownian motion of\nwalls and skyrmions is shown to obey markedly di\u000berent di\u000busion laws as a function of the damping\nparameter. Topology related skyrmion di\u000busion suppression with vanishing damping parameter,\nalbeit already documented, is shown to be restricted to ultra-small skyrmion sizes or, equivalently,\nto ultra-low damping coe\u000ecients, possibly hampering observation.\nI. INTRODUCTION\nThe prospect of ultra-small stable information bits in\nmagnetic layers in presence of the Dzyaloshinskii-Moriya\n(DM) interaction [1] combined to the expectation of their\nminute current propagation [2], notably under spin-orbit\ntorques [3], builds up a new paradigm in information\ntechnology. In stacks associating a metal with strong\nspin-orbit interactions e.g. Pt and a ferromagnetic metal\nsuch as Co, that may host isolated skyrmions, large do-\nmain wall velocities have also been forecast [4] and ob-\nserved [5]. The DM interaction induces chiral magnetiza-\ntion textures, walls or skyrmions, that prove little prone\nto transformations of their internal structure, hence their\nextended stability and mobility.\nIn order, however, to achieve low propagation cur-\nrents, steps will need to be taken towards a reduction\nof wall- or skyrmion-pinning. Recent experimental stud-\nies indicate that skyrmions fail to propagate for cur-\nrents below a threshold roughly equal to 2 1011Am\u00002for\n[Pt/Co/Ta]nand [Pt/CoFeB/MgO]nmultilayers [6], or\n2:5 1011Am\u00002for [Pt/(Ni/Co/Ni)/Au/(Ni/Co/Ni)/Pt]\nsymmetrical bilayers [7]. Only in one seldom instance did\nthe threshold current fall down to about 2 :5 1010Am\u00002\nfor a [Ta/CoFeB/TaO] stack, still probably, however, one\norder of magnitude higher than currents referred to in\nsimulation work applying to perfect samples [8].\nIn a wall within a Co stripe 50 nm wide, 3 nm thick, the\nnumber of spins remains large, typically 216for a 5 nm\nwide wall. A skyrmion within a Co monolayer (ML) over\nPt or Ir, on the other hand, contains a mere 250 spins,\nsay 28. Assuming that a sizeable reduction of pinning\nmight somehow be achieved, then a tiny structure such\nas a skyrmion is anticipated to become sensitive, if not\nextremely sensitive, to thermal \ructuations.\nIn this work, we show, on the basis of extended nu-\nmerical simulations, that both chiral walls and skyrmions\nwithin ferromagnets obey a di\u000busion law in their Brow-\nnian motion at \fnite temperature [9, 10]. The di\u000busion\n\u0003jacques.miltat@u-psud.fr\nx z q \nwS tS L \na) b) q FIG. 1. a) Wall within a narrow stripe: wSis the stripe width,\ntSits thickness. The stripe element length Lis solely de\fned\nfor computational purposes. qis the wall displacement; b)\nsnapshot of the magnetization distribution: color coding after\nmx. The wall region mx\u00191 appears red. Thermal \ructua-\ntions are visible within domains: T= 25 K,wS= 100 nm,\ntS= 0:6 nm,\u000b= 0:5.\nlaw is shown to be valid over a broad range of damp-\ning parameter values. The thermal di\u000busion of domain\nwalls seems to have attracted very little attention, ex-\ncept for walls in 1D, double potential, structurally un-\nstable, lattices [11], a source of direct inspiration for\nthe title of this contribution. Chiral magnetic domain\nwalls are found below to behave classically with a mobil-\nity inversely proportional to the damping parameter. As\nshown earlier [12, 13], such is not the case for skyrmions,\na behavior shared by magnetic vortices [14]. Vortices and\nskyrmions in ferromagnetic materials are both charac-\nterized by a de\fnite topological signature. In contradis-\ntinction, skyrmions in antiferromagnetic compounds are\ncharacterized by opposite sign spin textures on each sub-\nlattice, with, as a result, a classical, wall-like, dependence\nof their di\u000busion constant [15]. Lastly, ferrimagnets do\ndisplay reduced skyrmion Hall angles [16], most likely\nconducive to modi\fed di\u000busion properties.arXiv:1806.03172v1  [cond-mat.mes-hall]  8 Jun 20182\nII. DOMAIN WALL DIFFUSION\nWe examine here, within the micromagnetic frame-\nwork, the Langevin dynamics of an isolated domain wall\nwithin a ferromagnetic stripe with thickness tS, width\nwSand \fnite length L(see Fig. 1). The wall is located\nat mid-position along the stripe at time t= 0. Thermal\nnoise is introduced via a stochastic \feld ~HRduncorrelated\nin space, time and component-wise, with zero mean and\nvariance\u0011proportional to the Gilbert damping parame-\nter\u000band temperature T[17] :\nh~HRdi=~0\nhHi\nRd(~ r;t)Hj\nRd(~r0;t0)i=\u0011\u000eij\u000e(~ r\u0000~r0)\u000e(t\u0000t0)\n\u0011=2kBT\n\r0\u00160MS\u000b(1)\nwhere,kBis Boltzmann constant, \u00160and\r0are the vac-\nuum permability and gyromagnetic ratio, respectively,\nMSthe saturation magnetization. Written as such, the\nfunctions\u000e(~ r\u0000~r0) and\u000e(t\u0000t0) have the dimension of\nreciprocal volume and time, respectively. Applied to nu-\nmerical simulations, the variance of the stochastic \feld\nbecomes\u0011=2kBT\n\r0\u00160MSVdt\u000b, whereVis the computation\ncell volume and dtthe integration time step.\nA. Simulation results\nThe full set of numerical simulations has been per-\nformed by means of an in-house code ported to graph-\nical processing units (GPU's). Double precision has\nbeen used throughout and the GPU-speci\fc version of\nthe \"Mersenne twister\" [18] served as a source of long-\nsequence pseudo-random numbers generator.\nMaterial parameters have been chosen such as to mimic\na 3-ML Co layer (thickness tS= 0:6 nm) on top of Pt\nwith an exchange constant equal to A= 10\u000011J/m, a\nMs= 1:09 106A/m saturation magnetization, a Ku=\n1:25 106J/m3uniaxial anisotropy constant allowing for\na perpendicular easy magnetization axis within domains,\nand a moderate-to-high DM interaction (DMI) constant\nDDM= 2 mJ/m2. In order to temper the neglect of short\nwavelength excitations [19], the cell size has been kept\ndown toLx=Ly= 1 nm, whilst Lz=tS= 0:6 nm.\nThe stripe length has been kept \fxed at L= 1\u0016m,\na value compatible with wall excursions within the ex-\nplored temperature range. The latter has, for reasons\nto be made clear later, been restricted to \u00191=3 of the\npresumed Curie temperature for this model Co layer. Fi-\nnally, the integration time constant, also the \ructuating\n\feld refresh time constant, has been set to dt= 25 fs.\nAs shown by the snapshot displayed in Fig. 1b, the wall\nmay acquire some (moderate) curvature and/or slanting\nduring its Brownian motion. Because wall di\u000busion is\ntreated here as a 1D problem, the wall position qis de-\n0510152025\n487488489Time [ns]Wall Position [nm]ΔtqFIG. 2. Excerpt of a wall trace displaying wall position \ructu-\nations vstime:T= 77 K,\u000b= 0:5,wS= 100 nm,tS= 0:6 nm.\nqis the wall displacement during time interval \u0001 t.\n\fned as the average position owing to :\nq=L\nNxNyPNx\ni=1PNy\nj=1mz(i;j)\n[hmziL\u0000hmziR](2)\nwhere,iandjare the computation cell indices, Nxand\nNythe number of cells along the length and the width\nof the stripe, respectively, hmziLis the \ructuations aver-\naged value of the zmagnetization component far left of\nthe domain wall,hmziRthe average value of mzfar right.\nRegardless of sign, hmziRandhmziLare expected to be\nequal in the absence of any Hz\feld.\nFig. 2 displays the position as a function of time of a\nwall within a wS= 100 nm wide stripe immersed in a\nT= 77 K temperature bath. A 2 ns physical time win-\ndow has been extracted from a simulation set to run for\n1:5\u0016s. The \fgure shows short term wall position \ruc-\ntuations superimposed onto longer time di\u000busion. Ac-\ncording to Einstein's theory of Brownian motion [9], the\nprobability P(x;t) of \fnding a particle at position xat\ntimetobeys the classical di\u000busion equation @tP(x;t) =\nD@2\nx2P(x;t) with, as a solution, a normal (gaussian) dis-\ntributionP(x;t) = 1=p\n4\u0019Dtexp(\u0000x2=4Dt), whereDis\nthe di\u000busion constant.\nSo does the raw probability of \fnding a (sti\u000b) wall in\na narrow stripe at position qafter a time interval \u0001 t, as\nshown in Fig. 3 (see Fig. 2 for variable de\fnition). It\nought to be mentioned that the average wall displace-\nmenthq(\u0001t)iis always equal to 0, with an excellent ac-\ncuracy, provided the overall computation time is large\nenough. The \ft to a normal distribution proves rather\nsatisfactory, with, however, as seen in Fig. 3, a slightly\nincreasing skewness in the distributions as a function of\nincreasing \u0001 t. Skewness, however, 1) remains moderate3\n05 1031 1041.5 1042 104\n-30-20-100102030Δt = 0.2 nsN\nq - <q>  [nm]05 1031 1041.5 1042 104\n-30-20-100102030Δt = 0.5 ns\nq - <q>  [nm]N\n05 1031 1041.5 1042 104\n-30-20-100102030Δt = 1.0 ns\nq - <q>  [nm]N\n05 1031 1041.5 1042 104\n-30-20-100102030q - <q>  [nm]Δt = 2.0 nsN\nFIG. 3. Wall within stripe: event statistics with time interval\n\u0001tas a parameter; \u000b= 0:5,wS= 100 nm, tS= 0:6 nm,\nT= 25K. The continuous blue lines are \fts to a gaussian\ndistribution, the variance of which increases with \u0001 t.\nup to \u0001tvalues typically equal to 5 \u000010 ns, 2) is seen to\nreverse sign with time interval (compare Fig. 3b and c),\nexcluding intrinsic biasing. The distributions standard\ndeviation is clearly seen to increase with increasing \u0001 t.\nAlternatively, one may represent the variance hq2i\n(hqi= 0) as a function of the time interval \u0001 t: if di\u000bu-\nsion applies, then a linear dependence is expected, with\na 2Dslope for a one-dimensional di\u000busion. Fig.4a shows,\nfor various temperatures, that a linear law is indeed ob-\nserved. Lastly, as shown in Fig.4b, the di\u000busion constant\nincreases linearly with increasing temperature. The er-\nror bars measuring the departure from strict linearity in\nFig.4a remain limited in extent. For the stripe width\nand damping parameter considered here ( wS= 100 nm,\n\u000b= 0:5), the ratio of di\u000busion constant to temperature\nis found to amount to D=T= 0:187 nm2ns\u00001K\u00001.\nB. Wall di\u000busion constant (analytical)\nThiele's equation [20] states that a magnetic texture\nmoves at constant velocity ~ vprovided the equilibrium of\n3 forces be satis\fed:\n~G\u0002~ v+\u000bD~ v=~F (3)\nwhere,~Fis the applied force, ~FG=~G\u0002~ vis the gyrotropic\nforce,~Gthe gyrovector, ~FD=\u000bD~ vthe dissipation force,\nDthe dissipation dyadic.\nFor the DMI hardened N\u0013 eel wall considered here : ~G=\n050100150200250300\n012345Δt [ns]< q2 > [nm2]\n25° K50° K77° K120° K150° K\na)0102030\n050100150T [K]D  [nm2 ns-1]\nb)FIG. 4. a) Variance hq2i(nm2) of the wall displacement vs\ntime interval \u0001 twith temperature Tas a parameter. Thick\nlines represent a linear \ft to data; b) Di\u000busion constant D\nas a function of temperature (square full symbols). Dis pro-\nportional to the slope of the hq2ivs\u0001tcurves in Fig.4a (see\ntext for details). The error bars are deduced from the slopes\nof straight lines through the origin that encompass all data\npoints in Fig.4a for a given temperature and the \ft time\nbracket, 1\u00005 ns. For the sake of legibility, the error bars\nhave been moved-up by 2 :5 units. Continuous line: linear\n\ft through the origin. The dashed line is the analytical ex-\npectation in the \"low\" noise limit. \u000b= 0:5,wS= 100 nm,\ntS= 0:6 nm.\n~0. For a 1D wall, the Thiele equation simply reads :\n\u000bDxxvx=Fx (4)\nwhere,Dxx=\u00160MS\n\r0R\nV(@~ m\n@x)2d3r.\nThe calculation proceeds in two steps, \frst evaluate\nthe force, hence, according to Eqn.4, the velocity auto-\ncorrelation functions, then integrate vstime in order to\nderivehq2i. The force, per de\fnition, is equal to minus\nthe partial derivative of the energy Ew.r.t. the displace-\nmentq, namelyFx=\u0000@E\n@q=\u0000\u00160MSR\nV@~ m\n@x\u0001~H d3r.\nFormally,\nhFx(t)Fx(t0)i= (\u00160MS)2\u0002 (5)*Z\nV@~ m(~ r;t)\n@x\u0001~H(~ r;t)d3rZ\nV@~ m(~r0;t0)\n@x\u0001~H(~r0;t0)d3r0+\nAs noticed earlier [14], since the random \feld noise is\n\"multiplicative\" [17], moving the magnetization vector\nout of the average brackets is, strictly speaking, not al-\nlowed, unless considering the magnetization vector to\nonly marginally di\u000ber from its orientation and modulus\nin the absence of \ructuations (the so-called \"low\" noise\nlimit [14]):\nhFx(t)Fx(t0)i= (\u00160MS)2\u0002 (6)\nZ\nVX\ni;j\"\n@mi(~ r;t)\n@x@mj(~r0;t0)\n@xD\nHi(~ r;t)Hj(~r0;t0)E#\nd3r d3r0\nIf due account is being taken of the fully uncorrelated4\ncharacter of the thermal \feld (Eqn.1), the force auto-\ncorrelation function becomes:\nhFx(t)Fx(t0)i= 2\u000bkBTDxx\u000e(t\u0000t0) (7)\nThe velocity auto-correlation function follows from\nEqn.4. Lastly, time integration ( q(t) =Rt\n0vx(t0)dt0)\nyields :\nhq2(t)i= 2Dt;D=kBT\n\u000bDxx(8)\nIn order to relate the di\u000busion constant to a more directly\nrecognizable wall mobility, Dxxmay be expanded as :\nDxx=\u00160MS\n\r02wStS\n\u0001T(9)\nwhere, \u0001Thas been called the Thiele wall width (implic-\nitly de\fned in [21]). Dmay thus be expressed as :\nD=kBT\n2\u00160MS1\nwStS\r0\u0001T\n\u000b(10)\nthus, proportional to the wall mobility \r0\u0001T=\u000b.\nA directly comparable result may be obtained after\nconstructing a full Langevin equation from the ( q;\u001e)\nequations of domain wall motion (Slonczewski's equa-\ntions [22]), where \u001eis the azimuthal magnetization angle\nin the wall mid-plane. In this context, the wall mobility\nis\u0016W=\r0\u0001=\u000b, where \u0001 is the usual wall width, inci-\ndentally equal to the Thiele wall width in the case of a\npure Bloch wall. The Langevin equation [10] here reads:\nmD\n2wStSd2\nq2\u000b\ndt2+1\n22\u00160Ms\n\u0016WwStSd\nq2\u000b\ndt=kBT(11)\nwhere,mDis D oring's wall mass density (kg =m2):\nmD=\u0000\n1 +\u000b2\u0001\u0012\r0\n2\u00160Ms\u0013\u000021\n\u0019jDDMj(12)\nan expression valid in the limit jDDMj\u001dKE\u000b=\nKu\u00001\n2\u00160M2\ns. Note that the DMI constant DDMex-\nplicitly enters the expression of the wall mass, as a con-\nsequence of the wall structure sti\u000bening by DMI. In the\nstationary regime, hq2iis proportional to time tand the\nwall di\u000busion constant exactly matches Eqn.10, after sub-\nstitution of \u0001 Tby \u0001. Finally, the characteristic time for\nthe establishment of stationary motion is:\nt0=mD1\n2\u00160Ms\r0\u0001\n\u000b(13)\nFor the parameters of our model 3-ML Co layer on top\nof Pt, D oring's mass density is equal to \u00183 10\u00008kg=m2\nfor\u000b= 0:5, and the characteristic time amounts to\nt0'25 ps. Still for \u000b= 0:5,wS= 100 nm and\ntS= 0:6 nm,D=Tamounts to 0 :153 nm2ns\u00001K\u00001for\n\u0001T= 4:13 nm, i.e. the value computed from a properly\nconverged wall pro\fle at T= 0. The relative di\u000berence\n0255075\n050100150T [K]a)wS = 25 nmwS = 50 nmwS = 100 nmD  [nm2 ns-1]\n00.250.50.751\n00.010.020.030.040.051/wS [nm-1]D /T  [nm2 ns-1 K-1]\nb)FIG. 5. a) Di\u000busion constant Das a function of temperature\nwith the stripe width wSas a parameter (full symbols); b)\nD=Tas a function of the inverse of the stripe width. \u000b= 0:5,\ntS= 0:6 nm, throughout. Solid blue lines: linear \ft through\nthe origin, dashed line: analytical expectation.\nbetween simulation and theoretical values is found to be\nof the order of\u001920%.\nOwing to Eqn.10, Dis expected to prove inversely pro-\nportional to both the stripe width wSand the Gilbert\ndamping parameter \u000b, a behavior con\frmed by simula-\ntions. Fig.5a displays the computed values of the dif-\nfusion coe\u000ecient as a function of temperature with the\nstripe width as a parameter, whilst Fig.5b states the lin-\near behavior ofDvswS\u00001. The slope proves, however,\nsome 13:5% higher than anticipated from Eqn.10. Lastly,\nthe 1=\u000bdependence is veri\fed in Fig.6 showing the com-\nputed variation of Dvstemperature with \u000bas a param-\neter for a narrow stripe ( wS= 25 nm) as well as the\ncorresponding \u000bdependence ofD=T. The dotted line\nrepresents Eqn.10 without any adjusting parameter. The\nrelative di\u000berence between simulation data and theoret-\nical expectation is beyond, say \u000b= 0:25, seen to grow\nwith increasing \u000bbut also appears to be smaller for a\nnarrow stripe as compared to wider tracks.\nAltogether, simulation results only moderately depart\nfrom theoretical predictions. The Brownian motion of\na DMI-sti\u000bened wall in a track clearly proves di\u000busive.\nThe di\u000busion constant is classically proportional to the\nwall mobility and inversely proportional to the damping\nparameter. Unsurprisingly, the smaller the track width,\nthe larger the di\u000busion constant. In order to provide an\norder of magnitude, the di\u000busion induced displacement\nexpectation,p\n2D\u0001t, for a wall sitting in a 100 nm-wide,\npinning-free, track for 25 ns at T= 300 K proves essen-\ntially equal to\u0006the stripe width.\nIII. SKYRMION DIFFUSION\nOutstanding observations, by means of Spin Polarized\nScanning Tunneling Microscopy, have revealed the exis-\ntence of isolated, nanometer size, skyrmions in ultra-thin5\n0255075100125150175200\n020406080100α = 0.125α = 0.25α = 0.5α = 0.075α = 0.05\nα = 0.8a)T [K]D  [nm2 ns-1]\n0246810\n-20-1001020\n00.20.40.60.81D /T  [nm2 ns-1 K-1](%)\nαb)wS = 25 nmwS = 50 nmwS = 100 nm\nFIG. 6. a) Di\u000busion constant Das a function of temperature\nwith the damping constant \u000bas a parameter ( wS= 25 nm,\ntS= 0:6 nm). Solid blue lines: linear \ft through the ori-\ngin; b)D=T(large semi-open symbols) as a function of \u000bfor\nwS= 25 nm and tS= 0:6 nm; dotted blue curve: analyt-\nical expectation. Full symbols: relative di\u000berence between\ncomputational and analytical results (%).\nFIG. 7. a) Snapshot of a skyrmion immersed in a 12.5 K\ntemperature bath ( \u000b= 0:5), together with the underlying\nlattice. Red cells: sz\u0019+1, blue cells: sz\u0019\u00001. The white\ncross indicates the barycenter of lattice site positions satisfy-\ningsz\u00150:5.\n\flms such as a PdFe bilayer on an Ir(1111) single crystal\nsubstrate [23] [24]. We analyse below the thermal motion\nof skyrmions in a model system made of a Co ML on top\nof Pt(111). We deal with skyrmions with a diameter of\nabout 2:5 nm containing at T= 0 about 250 spins.\nA. Simulation results\nIn order to monitor the Brownian motion of an iso-\nlated skyrmion, rather than micromagnetics, it is pre-\nferred to simulate the thermal agitation of classical\nspins,~ s(jsj= 1), on a triangular lattice. Lat-\ntice e\u000bects and frequency cuto\u000bs in thermal excitations\nare thus avoided. Such simulations have already been\nused e.g. for the determination of the barrier to col-\nlapse of an isolated skyrmion [25, 26]. The parame-\nFIG. 8. Example of skyrmion trajectory. Distances in atomic\nunits (1 at:u:= 2:51\u0017A). The trajectory started at the origin\nof coordinates at time t= 0 and stopped at the cross location\nat physical time t\u0019100 ns.T= 25 K,\u000b= 1.\nters are: lattice constant a= 2:51\u0017A, magnetic mo-\nment\u0016At= 2:1\u0016B/atom, Heisenberg exchange nearest\nneighbor constant J= 29 meV/bond, Dzyaloshinskii-\nMoriya exchange d=\u00001:5 meV/bond, magnetocrys-\ntalline anisotropy 0 :4 meV/atom. The stochastic \feld\nis still de\fned by Eqn.1 after substitution of the prod-\nuctMSVby the magnetic moment per atom. The code\nfeatures full magnetostatic (dipole-dipole) interactions.\nFast Fourier Transforms implementation ensues from the\ndecomposition of the triangular lattice into two rectangu-\nlar sublattices, at the expense of a multiplication of the\nnumber of dipole-dipole interaction coe\u000ecients. Lastly,\nthe base time step, also the stochastic \feld refresh time,\nhas been given a low value in view of the small atomic\nvolume, namely dt= 2:5 fs for\u000b\u00150:1,dt= 1 fs below.\nTime steps that small may be deemed little compatible\nwith the white thermal noise hypothesis [17]. They are in\nfact dictated by the requirement for numerical stability,\nprimarily w.r.t. exchange interactions.\nFig.7 is a snapshot of an isolated skyrmion in the\nmodel Co ML with a temperature raised to 12 :5 K. The\nskyrmion is at the center of a 200 at. u.- i.e. \u001950 nm-size\nsquare computation window, that contains 46400 spins\nand is allowed to move with the di\u000busing skyrmion. Do-\ning so alleviates the computation load without restricting\nthe path followed by the skyrmion. Free boundary con-\nditions (BC's) apply. The window, however, proves su\u000e-\nciently large to render the con\fning potential created by\nBC's ine\u000bective. The skyrmion position as a function of\ntime is de\fned simply as the (iso)barycenter of the con-\ntiguous lattice site positions x(k),y(k), wheresz\u00150:5:\nqSk\nx=1\nKKX\nk=1x(k) ;qSk\ny=1\nKKX\nk=1y(k) (14)6\n01 1042 1043 1044 1045 1046 1047 1048 104\n01 1042 1043 1044 1045 1046 1047 104\n-50050NΔt = 0.2 ns\nqx,y - <qx,y>  [at. u.]01 1042 1043 1044 1045 1046 1047 1048 104\n01 1042 1043 1044 1045 1046 1047 104\n-50050Δt = 0.5 nsN\nqx,y - <qx,y>  [at. u.]\n01 1042 1043 1044 1045 1046 1047 1048 104\n01 1042 1043 1044 1045 1046 1047 104\n-50050NΔt = 1.0 ns\nqx,y - <qx,y>  [at. u.]01 1042 1043 1044 1045 1046 1047 1048 104\n01 1042 1043 1044 1045 1046 1047 104\n-50050N\nqx,y - <qx,y>  [at. u.]Δt = 2.0 ns\nFIG. 9. Skyrmion: event statistics with time interval \u0001 t\nas a parameter for the displacement components qx(black\nfull symbols) and qy(red open symbols), labeled qx;yin the\n\fgures. In each panel, the curves have been o\u000bset vertically\nfor legibility. Solid lines: \ft to a gaussian distribution. \u000b=\n0:25,T= 25 K\nwhere,kis the lattice site index, Kthe number of lattice\nsites satisfying the above condition. Such a de\fnition\nproves robust vsthermal disorder such as displayed in\nFig. 7. Similarly to the case of wall di\u000busion, we analyze\n\frst the distributions of the displacement components\nqx;qy. The event statistics for each value of the time\ninterval is clearly gaussian (see Fig.9). However, the noise\nin the distributions appears larger when compared to the\nwall case. It also increases faster with \u0001 t. On the other\nhand, the raw probabilities for hq2\nxiandhq2\nyibarely di\u000ber\nas anticipated from a random process. The behavior of\nhq2i(q2=q2\nx+q2\ny)vs\u0001tis displayed in Fig.10a.\nThe range of accessible temperatures is governed by\nthe thermal stability of the tiny skyrmion within a Co\nML: with a lifetime of '1\u0016s at 77 K [25{28], tem-\nperatures have been con\fned to a \u001450 K range. When\ncompared to the wall case (Fig.4a), the linear dependence\nofhq2iwith respect to \u0001 tappears less satisfactory, al-\nthough, over all cases examined, the curves do not display\na single curvature, but rather meander gently around a\nstraight line. The slope is de\fned as the slope of the\nlinear regression either for time intervals between 0 :25\nand 2:5 ns (thick line segments in Fig.10a) or for the full\nrange 0 to 5 ns (dashed lines). Then, the ratio of the\ndi\u000busion constant to temperature, D=T, for an isolated\nskyrmion within the model Co ML considered here is\nequal to 0:250 and 0:249 nm2ns\u00001K\u00001, respectively, for\n\u000b= 0:5 (see Fig.10b). The di\u000berence proves marginal.\nLastly, error bars appear even narrower than in the wall\n01000200030004000\n01234550° K25° K12.5° K4.2° K< q2 > [at.u.2]\nΔt [ns]a)01020\n0255075T [K]b)D  [nm2 ns-1]FIG. 10. a) Variance (at :u:2) of the skyrmion displacement\nhq2ivstime interval \u0001 twith temperature Tas a parameter.\nThick and dashed lines represent a linear \ft to data with\ndi\u000berent time coverage, namely [0 :25\u00002:5 ns] and [0\u00005 ns];\nb) Di\u000busion constant Das a function of temperature for a\n[0:25\u00002:5 ns]- (open symbols) and [0 \u00005 ns]- (full symbols)\nlinear \ft. Solid blue line: linear \ft through the origin. Dashed\nline: analytical expectation in the \"low\" noise limit. In order\nto ensure legibility, the error bars as de\fned in the caption\nof Fig.4 and pertaining to the [0 :25\u00002:5 ns] \ft time bracket\nhave been moved-up by one unit. \u000b= 0:5.\ncase.\nB. Skyrmion di\u000busion constant (analytical)\nThe gyrovector ~Gin Thiele's equation (Eqn.3) has in\nthe case of a skyrmion or a vortex, and in many other\ninstances such as lines within walls, a single non-zero\ncomponent, here Gz. Thiele's equation, in components\nform, reads:\n\u0000Gzvy+\u000b[Dxxvx+Dxyvy] =Fx\n+Gzvx+\u000b[Dyxvx+Dyyvy] =Fy(15)\nBecause of the revolution symmetry of a skyrmion at rest,\nDxyorDyxmay safely be neglected and Dyy=Dxx.\nAccordingly, the velocities may be expressed as:\nvx=\u000bDFx+GFy\nG2+ (\u000bD)2;vy=\u000bDFy\u0000GFx\nG2+ (\u000bD)2(16)\nwhere,G=Gz,D=Dxx=Dyy.\nSimilarly to the stochastic \feld, the force components\nare necessarily uncorrelated. The velocity autocorrela-\ntion functions may now be obtained following the same\nlines as in the wall case, yielding, in the low noise ap-\nproximation:\nhvx(t)vx(t0)i=hvy(t)vy(t0)i= 2kBT\u000bD\nG2+ (\u000bD)2\u000e(t\u0000t0)\n(17)7\n00.050.10.150.20.250.30.35\n-20-1001020304050\n0246810α(%)D /T  [nm2 ns-1 K-1]\nFIG. 11. Computed values of D=Tvs\u000b(large open symbols);\nblack line: guide to the eye; blue (resp. red) solid curves: an-\nalytical values with [ \r0SAt=\u00160\u0016At]D= 4\u0019(resp. 14:5). The\nblue curve thus corresponds to the Belavin-Polyakov pro\fle\nlimit. The relative di\u000berence between simulation and theory\nis indicated by small full symbols (% : right scale).\nThe average values of the displacements squared, hq2\nxi\nandhq2\nyifollow from time integration:\n\nq2\nx(t)\u000b\n=\nq2\ny(t)\u000b\n= 2kBT\u000bD\nG2+ (\u000bD)2t (18)\nAs shown previously [12, 13], the di\u000busion constant for a\nskyrmion thus reads:\nD=kBT\u000bD\nG2+ (\u000bD)2(19)\nThe following relations do apply:\n\nq2\nx(t)\u000b\n=\nq2\ny(t)\u000b\n= 2Dt\n\nq2(t)\u000b\n=\nq2\nx(t) +q2\ny(t)\u000b\n= 4Dt(20)\nRelation (19) implies a peculiar damping constant de-\npendence with, assuming for the time being DandGto\nhave comparable values, a gradual drop to zero of the\ndi\u000busion constant with decreasing \u000b(\u000b\u00141), termed\n\"di\u000busion suppression by G\" by C. Sch utte et al. [12].\nDi\u000busion suppression is actually not a complete surprise\nsince, for electrons in a magnetic \feld, a similar e\u000bect is\nleading to the classical magnetoresistance. A similar de-\npendenceD(\u000b) is also expected for a vortex. Boundary\nconditions, however, add complexity to vortex di\u000busion.\nWhat nevertheless remains, is a linear dependence of D\nvs\u000b[14], namely, di\u000busion suppression.\nThe classical expressions for GzandDxxvalid for a\nmagnetization continuum need to be adapted when deal-\ning with discrete spins. We obtain:\nGz=\u00160\u0016At\n\r0X\nk[~ s(k)\u0001[@x~ s(k)\u0002@y~ s(k)]]\nDxx=\u00160\u0016At\n\r0X\nkh\n[@x~ s(k)]2i (21)where,\u0016Atis the moment per atom.\nThe dimensionless product\r0SAt\n\u00160\u0016AtGz(Eqn.21), where\nSAtis the surface per atom, amounts to 4 \u0019, irrespec-\ntive of the skyrmion size in a perfect material at T= 0.\nStated otherwise, the skyrmion number is 1 [29]. In\nthe Belavin-Polyakov pro\fle limit [30], the dimention-\nless product\r0SAt\n\u00160\u0016AtDxx(Eqn.21) also amounts to 4 \u0019. In\nthis limit,Dis proportional to \u000b=(1+\u000b2).Dxxincreases\nwith skyrmion radius beyond the Belavin-Polyakov pro-\n\fle limit (see supplementary material in [7]). For a\nskyrmion at rest in the model Co ML considered here,\nD=Dxx\u001914:5\u00160\u0016At=(\r0SAt). For that value of\nDxx, and for the parameters used in the simulations,\nD=T, the ratio of the theoretical skyrmion di\u000busion con-\nstant to temperature, is equal 0 :234 nm2ns\u00001K\u00001, for\n\u000b= 0:5 (SAt=a2p\n3=2), to be compared to the 0 :250\nvalue extracted from simulations. More generally, Fig.11\ncompares numerical D=Tvalues calculated for a broad\nspectrum of \u000bvalues with theoretical expectations for\nD= 14:5\u00160\u0016At=(\r0SAt) and in the Belavin-Poliakov\nlimit. The average di\u000berence between analytical and sim-\nulation results is, in the \u000b= (0;1) interval, seen to be of\nthe order of'15%.\nIV. DISCUSSION\nIn the present study of thermal di\u000busion characteris-\ntics, satisfactory agreement between simulations and the-\nory has been attained for DMI sti\u000bened magnetic tex-\ntures, be it walls in narrow tracks or skyrmions. The\n\u000bdependence of the di\u000busion constants has been thor-\noughly investigated, with, as a result, a con\frmation of\nBrownian motion suppression in the presence of a non-\nzero gyrovector or, equivalently, a topological signature.\nThe theory starts with the Thiele relation applying to\na texture moving under rigid translation at constant ve-\nlocity. Furthermore, the chosen values of the components\nof the dissipation dyadic, are those valid for textures at\nrest, atT= 0. The\u000bdependence of the di\u000busion con-\nstants clearly survives these approximations. And, yet, a\nwall within a narrow stripe or a skyrmion in an ultra-thin\nmagnetic layer are deformable textures, as obvious from\nFigs.1,7. Simulations, on the other hand, rely on the\npioneering analysis of Brownian motion, here meaning\nmagnetization/spin orientation \ructuations [17], within\na particle small enough to prove uniformly magnetized\nand then extend the analysis to ultra-small computation\ncell volumes down to the single spin. Both approaches\nrely on the hypothesis of a white -uncorrelated- noise at\n\fnite temperature.\nThe discussion of results is organized in two parts. In\nthe \frst, results are analyzed in terms of a sole action of\nstructure plasticity on the diagonal elements of the dis-\nsipation dyadic. In the second, we envisage, without fur-\nther justi\fcation, how the present results are amended if,\nin the di\u000busion constants of walls and skyrmions (Eqns.8\nand 19), the gyrotropic and dissipation terms are re-8\n01 10-102 10-10\n20406080100f (GHz)S\n 12.5 K  25 K  T = 50 K \na)0123\n010203040506070T(K)< rEq > [nm]\nb)\nFIG. 12. a) Power spectrum Sof the time series rEq(t) for\nthree temperatures. The hatched area corresponds to the fre-\nquency range where a signature of the fundamental skyrmion\nbreathing mode is anticipated to be observed ( \u001939:3 GHz,\nin the present case); b) Equivalent skyrmion radius hrEqias\na function of temperature. Error bars correspond to \u00061\u001b\nof the gaussian distribution, itself a function of temperature.\n\u000b= 0:5, throughout.\nplaced by their time average as deduced from simulations.\nA. Size e\u000bects\nThe integral de\fnition of wall position adopted in this\nwork (Eqn.2) allows for a 1D treatment of wall di\u000busion,\nthus ignoring any di\u000busion characteristics potentially as-\nsociated with wall swelling, tilting, curving or meander-\ning. Additional information is, however, available in the\ncase of skyrmions. We concentrate here on the number,\nn, of spins within the skyrmion satisfying the condition\nsz\u00150:5, and its \ructuations as a function of time. The\nsurface of the skyrmion is nSAtand its equivalent radius,\nrEq, is de\fned by r2\nEq=nSAt=\u0019. The skyrmion radius\nrEqis found to \ructuate with time around its average\nvalue, according to a gaussian distribution that depends\non temperature, but becomes independent of the autocor-\nrelation time interval beyond \u001925 ps. The power spec-\ntrum of the time series rEq(t), shown in Fig.12a, excludes\nthe existence of a signi\fcant power surge around the\nfundamental breathing mode frequency of the skyrmion\n(\u001939:3 GHz for the present model Co ML) [31]. The\nskyrmion radius as de\fned from the discrete ndistribu-\ntion is thus subject to white noise. The average radius\nhrEqi, on the other hand, varies signi\fcantly with tem-\nperature, increasing from \u00191:6 nm to 2:4 nm when the\ntemperature is increased from 4 :2 K to 50 K (Fig.12b)\nand the diagonal element of the dissipation dyadic is ex-\npected to increase with increasing skyrmion radius [3, 7].\nOwing to relations (19,21), the maximum of D(\u000b) is\nfound for\u000b=Gz=Dxx=G=D . For\u000b < G=D , resp.\n\u000b > G=D ,Dincreases, resp. decreases, with D, hence\nthe relative positions of the blue and black continuous\ncurves in Fig.11. At maximum, Dis independent of D\nand amounts to kBT\r0SAt\n\u00160\u0016At1\n2G=kBT\r0SAt\n\u00160\u0016At1\n8\u0019. It ensues\nz\t\r \nf\t\r α\t\n\r\nR/Δ\t\n\ra) b) 00.20.40.60.81\n01020304050R /!\"#D / #\" < 0#D / #\" > 0FIG. 13. Di\u000busion suppression: a) general shape of function\nf(\u000b;R= \u0001) with 0 < \u000b < 1, 1< R= \u0001<50; b) crest line\nseparating the region of di\u000busion suppression ( @D=@\u000b > 0)\nfrom region @D=@\u000b< 0.\nthat the discrepancy between numerical and analytical\nDvalues around \u000b= 1 may not be relaxed by a sole\nvariation of D. On the other hand, allowing Dto increase\nwith skyrmion radius, itself a function of temperature,\nleads to an increase (decrease) of the di\u000busion coe\u000ecient\nfor\u000b<G=D (\u000b>G=D ).\nLikely more important is the reduction, as a function\nof skyrmion size, of the \u000bwindow where di\u000busion sup-\npression is expected. If including the ( R=\u0001 + \u0001=R) de-\npendence of Dxx(see supplementary material in [7]; \u0001 is\nthe wall width and Rthe skyrmion radius), the skyrmion\ndi\u000busion constant may be expressed as:\nD=kBT\r0SAt\n\u00160\u0016At1\n8\u0019f\u0012\n\u000b;R\n\u0001\u0013\n\u0011=R\n\u0001;\u0018=1\n2\u00121 +\u00112\n\u0011\u0013\n;f(\u000b;\u0011) =2\u000b\u0018\n1 + (\u000b\u0018)2(22)\nThe general shape of function f(\u000b;R= \u0001) is shown in\nFig.13a. The maximum of f(\u000b;R= \u0001) is equal to 1 for\nall values of \u000bandR=\u0001. The crest line R\u000b= \u0001 is\nseen to divide the parameter space into two regions (see\nFig.13b), a region close to the axes where @D=@\u000b > 0,\ni.e. the region of di\u000busion suppression, from the much\nwider region where @D=@\u000b < 0, that is, the region of\nwall-like behavior for skyrmion di\u000busion. Clearly, the \u000b\nwindow for di\u000busion suppression decreases dramatically\nwith increasing skyrmion size R=\u0001. A \frst observation\nof skyrmion Brownian motion at a video recording time\nscale (25 ms) may be found in the Supplementary Ma-\nterial of Ref.[32]. Skyrmions are here unusually large\nand most likely escape the di\u000busion suppression window\n(\u000b<0:02 forR=\u0001 = 50). Combining skyrmion thermal\nstability with general observability and damping parame-\nter tailoring may, as a matter of fact, well prove extremely\nchallenging for the observation of topology related di\u000bu-\nsion suppression.9\n0.750.80.850.90.951\n101214161820\n020406080100120140160T (K)< DVF >< mz / mz(T=0) >< sz / sz(T=0) >1 ML3 ML : 0.6 nm\nFIG. 14. Average reduced zmagnetization or spin component\nas a function of temperature (left scale) and time averaged\nvalue of the sole vector function, hDVFi, within the diagonal\nelement of the dissipation tensor in the skyrmion case (right\nscale). These results prove independent of the damping pa-\nrameter provided the time step in the integration of the LLG\nequation be suitably chosen.\nB. Time averaging\nOne certainly expects from the simulation model a fair\nprediction of the average magnetization hMziorhSzivs\ntemperature T, at least for temperatures substantially\nlower than the Curie temperature TC. Fig.14 shows the\nvariation ofhMzi=Mz(T= 0) orhSzi=Sz(T= 0) with\ntemperature for the two model magnetic layers of this\nwork. Although simulation results do not compare unfa-\nvorably with published experimental data [33{35], where,\ntypically, the Curie temperature amounts to \u0019150Kfor\n1 ML, and proves larger than 300 Kfor thicknesses above\n2 ML, a more detailed analysis, potentially including dis-\norder, ought to be performed.\nhGzi=\u00160\u0016Athszi\n\r0hX\nk[~ s(k)\u0001[@x~ s(k)\u0002@y~ s(k)]]i\n=\u00160\u0016Athszi\n\r0SAthGVF\nzi\nhDxxi=\u00160\u0016Athszi\n\r0hX\nk[@x~ s(k)]2i\n=\u00160\u0016Athszi\n\r0SAthDVF\nxxi(23)\nLet us now, without further justi\fcation, substitute in\nthe expression of the skyrmion di\u000busion coe\u000ecient time\naveraged values of GandD, owing to relations (23).\nKeeping in mind the geometrical meaning of GVF\nz, the\ndimensionless vector function in G,hGziis anticipated\nto be a sole function of hszi. Inversely, DVF\nxx, the (di-\nmensionless) vector function in hDxxi, a de\fnite posi-\ntive quantity, steadily increases with thermal disorder.\nIt is even found to be proportional to temperature (notshown). Its time averaged value for the sole skyrmion\nmay only be obtained by subtraction of values computed\nin the presence and absence of the skyrmion.\nFor the skyrmion in our model Co monolayer, hDVF\nxxi\nis found to increase moderately with temperature (see\nFig.14), a result also anticipated from an increase with\ntemperature of the skyrmion radius. Besides, both hGzi\nandhDxxiare expected to decrease with temperature\ndue to their proportionality to hszi.hDxxiis thus sub-\nject to two competing e\u000bects of temperature T. Present\nevidence, however, points at a dominating in\ruence of\nhsz(T)i.\nV. SUMMARY AND OUTLOOK\nSummarizing, it has been shown that the Brownian\nmotion of chiral walls and skyrmions in DMI materials\nobeys di\u000busion equations with markedly di\u000berent damp-\ning parameter ( \u000b) dependence. Although not a new re-\nsult, skyrmions Brownian motion suppression with de-\ncreasing\u000b(\u000b<G=D ) is substantiated by a wide explo-\nration of the damping parameter space. The observation\nof this astonishing topological property might, however,\nbe hampered by the restriction to ultra-small skyrmion\nsizes or ultra-low \u000bvalues. The discrepancy (up to 20%)\nbetween simulation results and theoretical expectations\ncould be reduced by the introduction of time averaged\nvalues for the gyrotropic and dissipation contributions\nto the analytical di\u000busion coe\u000ecients in the \"low\" noise\nlimit, at the expense of a tiny upwards curvature in the\nD(T) curves. A strong theoretical justi\fcation for doing\nso remains, however, lacking at this stage.\nIn this work, the sample has been assumed to be per-\nfect, i.e. devoid of spatial variations of the magnetic\nproperties, even though the lifting of such a restriction is\nanticipated to prove mandatory for a proper description\nof experiments. Di\u000busion in the presence of disorder has\nbeen theoretically studied for a number of disorder and\nrandom walk types [36, 37]. Generally, disorder changes\nthe linear growth with time of the position variance into\na power law, a behavior called superdi\u000busion if the ex-\nponent is larger than 1 and subdi\u000busion if smaller. For\ninstance, if the skyrmion motion in a disordered system\nmay be mapped onto a 2D random walk with an onsite\nresidence time \u001c, probability/\u001c\u0000(1+\u0016)(\u0016<1), then the\ndi\u000busion exponent will be \u0016, meaning subdi\u000busion. 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P. 231, Campus Plaine, B-1050 Brussels, Belgium\nzDepartamento de F ´ısica, Facultad de Ciencias, Universidad Nacional Aut ´onoma de M ´exico,\n04510 M ´exico D.F., Mexico\nE-mail: thomas.gilbert@ulb.ac.be, dps@fciencias.unam.mx\nAbstract. We calculate the diffusion coefficients of persistent random walks on lattices,\nwhere the direction of a walker at a given step depends on the memory of a certain number of\nprevious steps. In particular, we describe a simple method which enables us to obtain explicit\nexpressions for the diffusion coefficients of walks with two-step memory on different classes\nof one-, two- and higher-dimensional lattices.\nSubmitted to: J. Phys. A: Math. Theor.\n1. Introduction\nRandom walks are widely used throughout physics as a model for systems in which the state\nof the system can be viewed as evolving in a stochastic way from one time step to the next.\nTheir properties have been extensively explored and the techniques to study them are well\ndeveloped [1]. In particular, at a large scale, random walks behave diffusively.\nThe most-studied case is that of random walkers which have no memory of their past\nhistory. Many physical applications, however, call for a model in which the choice of possible\ndirections for the walker’s next step are given by probabilities which are influenced by the\npath it took prior to making that choice, so that its jumps are correlated; this is often called a\npersistent orcorrelated random walk [2].\nA walk with zero memory corresponds to the Bernoulli process of the usual uncorrelated\nrandom walk. Walks with single-step memories are the most commonly studied cases of\npersistent random walks, where the random walker determines the direction it takes at a given\nstep in terms of the direction taken on the immediately preceding step [1]. Such walks – not\nnecessarily restricted to lattices, as will be the case here – were first discussed in the context\nof Brownian motion [3] and fluid dynamics [4], and have since found many applications\nin the physics literature, most prominently in polymer conformation theory [5] and tracer\ndiffusion in metals [6], but also in relation to the telegrapher’s equation in the context of\nthermodynamics [7]. Previous works dealing with random walks on lattices with higher-order\nmemory effects include that of Montroll [8], with applications to models of polymers, and\nBender and Richmond [9].\nThe state of the walker is thus specified by two variables, its location and the direction it\ntook at the preceding step. The statistical properties of such persistent walks can be describedarXiv:0908.1271v1  [cond-mat.stat-mech]  10 Aug 2009Diffusion coefficients for persistent random walks 2\nby simple Markov chains and have already been thoroughly investigated in the literature; see\nin particular ref. [10]. We will only provide a short review of results relevant to our purposes,\nwith specific emphasis on the diffusive properties.\nThe statistics of random walks with multi-step memory can in principle be analysed in\nterms of Markov chains, in a similar way to their single-step memory counterpart. However,\nthe number of states of these chains grows exponentially with the number of steps accounted\nfor. This is the source of great technical difficulties, which are present already at the level of\ntwo-step processes.\nOf specific interest to us are random walks with two-step memory. Among the class of\npersistent walks under consideration, these are the simplest case beyond those with single-\nstep memory, and are therefore relevant to problems dealing with the persistence of motion\nof tracer particles where the single-step-memory approximation breaks down. An example\nwhere this occurs is given in recent work by the present authors, on diffusion in a class of\nperiodic billiard tables [11].\nThe paper is organised as follows. The general framework of walks on lattices is briefly\nreviewed in section 2, where we provide the expression of the diffusion coefficient of such\nwalks in terms of the velocity auto-correlations. Successive approximation schemes for\nthe computation of these auto-correlation functions are presented in sections 3, 4, and 5,\npertaining to the number of steps of memory of the walkers, respectively 0, 1, and 2. Specific\nexamples are discussed, namely the one-dimensional lattice and the two-dimensional square,\nhoneycomb and triangular lattices, and their diffusion coefficients are computed. Some of\nthe details of the computations presented in section 5 are deferred to appendices Appendix A\nand Appendix B. Section 6 provides an alternative derivation of the diffusion coefficients of\ntwo-step memory persistent walks with special left–right symmetries. Conclusions are drawn\nin section 7.\n2. Diffusion on a lattice\nWe consider the motion of independent tracer particles undergoing random walks on a regular\nlatticeL. Their trajectories are specified by their initial position r0at time t=0, and the\nsequencefv0;:::;vngof the successive values vi2Vriof their direction vectors at positions\nri, whereVridenotes the space of direction vectors allowed at site ri, which point to the lattice\nsites adjacent to ri. Here we consider dynamics in discrete time, so that the time sequences\nare simply assumed to be incremented by identical time steps tas the tracers move from site\nto site. In the sequel we will loosely refer to the direction vectors as velocity vectors; they are\nin fact dimensionless unit vectors.\nExamples of such motions are random walks on one- and two-dimensional lattices such\nas honeycomb, square and triangular lattices, but also include persistent random walks where\nmemory effects must be accounted for, i.e. when the probability of occurrence of vndepends\non the past history vn\u00001;vn\u00002;:::.\nThe quantity we will be concerned with is the diffusion coefficient Dof such persistent\nprocesses, which measures the linear growth in time of the mean-squared displacement of\nwalkers. This can be written in terms of velocity auto-correlations using the Taylor–Green–\nKubo expression:\nD=`2\n2dt\"\n1+2 lim\nK!¥K\nå\nn=1hv0\u0001vni#\n; (2.1)\nwhere ddenotes the dimensionality of the lattice L, and `is the lattice spacing. TheDiffusion coefficients for persistent random walks 3\n(dimensionless) velocity auto-correlations are computed as averages h\u0001iover the equilibrium\ndistribution m, so that the problem reduces to computing\nhv0\u0001vni=å\nv0;:::;vnv0\u0001vnm(fv0;:::;vng): (2.2)\nAs reviewed below, this can be easily carried out in the simple examples of random walks\nwith zero- and single-step memories. The main achievement of this paper is to describe the\ncomputation of the velocity auto-correlations of random walks with two-step memory. All\nthese cases involve factorisations of the measure m(fv0;:::;vng)by products of probability\nmeasures which depend on a number of velocity vectors, equal to the number of steps of\nmemory of the walkers. These measures will be denoted by pthroughout the paper.\nThe next three sections, sections 3, 4 and 5, are devoted to the computation of\nthe diffusion coefficient (2.1) for random walks with zero-, one- and two-step memories,\nrespectively.\n3. No-Memory Approximation (NMA)\nIn the simplest case, the walkers have no memory of their history as they proceed to their\nnext position. This gives a Bernoulli process for the velocity trials, for which the probability\nmeasure factorises:\nm(fv0;:::;vng) =n\nÕ\ni=0p(vi): (3.1)\nGiven that the lattice is isotropic and that pis uniform, the velocity auto-correlation (2.2)\nvanishes:\nhv0\u0001vni=dn;0: (3.2)\nThe diffusion coefficient of the random walk without memory is then given by\nDNMA=`2\n2dt: (3.3)\n4. One-Step Memory Approximation (1-SMA)\nWe now assume that the velocity vectors obey a Markov process for which vntakes on\ndifferent values according to the velocity at the previous step vn\u00001. We may then write\nm(fv0;:::;vng) =n\nÕ\ni=1P(vijvi\u00001)p(v0): (4.1)\nHere, P(bja)denotes the 1-step conditional probability that the walker moves in a direction\nb,given that it had direction aat the previous step.\nWe denote by zthe coordination number of the lattice, i.e. the number of neighbouring\nsites accessible from each site, and we denote by Rthe rotation operation which takes a vector\nvthrough all the lattice directions v;Rv;:::;Rz\u00001v. In general, the set of allowed orientations\nofvdepends on the lattice site, such as in the two-dimensional honeycomb lattice. We denote\nbyTthe symmetry operator that maps a cell to its neighbors, which corresponds simply to\nthe identity for square lattices and to a reflection for the honeycomb lattice.Diffusion coefficients for persistent random walks 4\nThe idea of our calculation is to express each velocity vector vkin terms of the first one,\nv0, asvk=RikTkv0, where iklies between 0 and z\u00001. Substituting this into the expression\nfor the velocity auto-correlation hv0\u0001vni, equation (2.2), we obtain\nå\nv0;:::;vnv0\u0001vnn\nÕ\ni=1P(vijvi\u00001)p(v0) =z\nå\ni0;:::;in=1v0\u0001RinTnv0min;in\u00001\u0001\u0001\u0001mi1;i0pi0; (4.2)\nwhere\nmin;in\u00001\u0011P(RinTnv0jRin\u00001Tn\u00001v0) (4.3)\nare the elements of the stochastic matrix Mof the Markov chain associated to the persistent\nrandom walk, and pi\u0011p(i)are the elements of its invariant (equilibrium) distribution, denoted\nP, evaluated with a velocity in the ith lattice direction. The invariance of Pis expressed as\nåjmi;jpj=pi. These notations will be used throughout this article.\nThe terms involving Min (4.2) constitute the matrix product of ncopies of M.\nFurthermore, since the invariant distribution is uniform over the zpossible lattice directions,\nwe can choose an arbitrary direction for v0, and hence write\nhv0\u0001vni=v0\u0001Tnv0m(n)\n1;1+v0\u0001RTnv0m(n)\n2;1+\u0001\u0001\u0001+v0\u0001Rz\u00001Tnv0m(n)\nz;1;(4.4)\nwhere m(n)\ni;jdenote the elements of Mn.\nUnder special symmetry assumptions to be discussed in the examples below, one has\nhv0\u0001vni=hcosqin; (4.5)\nwherehcosqidenotes the average angle between two successive velocity vectors. It is then\na general, well-known, result for such symmetric persistent random walks with single-step\nmemory [2] that their diffusion coefficients have the form\nD1SMA =DNMA1+hcosqi\n1\u0000hcosqi(symmetricwalks ): (4.6)\nThe actual value of the diffusion coefficient depends on the probabilities P(RjTvjv),\nwhich are parameters of the model. Specific applications of equation (4.6) are given in the\nexamples below, such as shown in figure 1. To simplify the notation, we denote the conditional\nprobabilities of these walks by Pj, where j=0;:::; z\u00001 corresponds to the relative angle\n2pj=zof the direction that the walker takes with respect to its previous step (up to a reflection\nin the case of the honeycomb lattice). These conventions are shown in figure 2.\n(a)\n (b)\nFigure 1. Examples of walks on (a) square and (b) honeycomb lattices. Note the inversion of\nthe allowed directions at neighbouring sites on the honeycomb lattice.Diffusion coefficients for persistent random walks 5\n(a)\n0 1 (b)\n01\n2\n(c)\n0\n321 (d)\n0\n5 432 1\nFigure 2. The possible directions of motion at a given step for different lattices, relative to the\nincoming direction which is shown by the arrow, are labelled from 0 to z\u00001, corresponding to\nthe angle 2 pj=zthat the lattice direction jmakes with respect to the reference direction, up to\na reflection inversion in the case of the honeycomb lattice. (a) One-dimensional lattice ( z=2);\n(b) honeycomb lattice ( z=3); (c) square lattice ( z=4); (d) triangular lattice ( z=6) .\n4.1. One-dimensional lattice\nThe simplest case is that of a regular one-dimensional lattice. In this case, each site is\nequivalent, and so Tis the identity. Each velocity vector vkhas only two possible values,\nvand\u0000v, so that Ris a reflection. We denote by P0\u0011P(vjv)the probability that the random\nwalker continues in the same direction at the next step, and by P1\u0011P(\u0000vjv)the probability\nthat it reverses direction.\nIt is easy to check that the velocity auto-correlation (4.4) yields (4.5):\nhv0\u0001vni= (P0\u0000P1)n=hcosqin; (4.7)\nand so the diffusion coefficient is given by\nD1SMA =DNMAP0\nP1: (4.8)\n4.2. Two-dimensional square lattice\nOn a two-dimensional square lattice, vkcan take four possible values, and each lattice site is\nagain equivalent. Thus Tis the identity and Rcan be taken as an anticlockwise rotation by\nangle p=2. We denote by P0\u0011P(vjv)the probability that the particle proceeds in the same\ndirection as on its previous step, by P1\u0011P(Rvjv)the probability that the particle turns to the\nleft relative to its previous direction, by P2\u0011P(R2vjv)the probability that it turns around,\nand by P3\u0011P(R3vjv)the probability that it turns right.Diffusion coefficients for persistent random walks 6\nFrom (4.4), the velocity auto-correlation hv0\u0001vniis given by\nhv0\u0001vni=m(n)\n1;1\u0000m(n)\n1;3: (4.9)\nThe transition matrix Mgiven by (4.3) is thus the following cyclic matrix:\nM=0\nBB@P0P1P2P3\nP3P0P1P2\nP2P3P0P1\nP1P2P3P01\nCCA: (4.10)\nTo calculate the elements m(n)\ni;jof the powers Mn, it is possible to compute the eigenvalues\nand eigenvectors of Mand then decompose it as M=Q\u0001L\u0001Q\u00001, where Lis the diagonal matrix\nof the eigenvalues of MandQthe matrix of its eigenvectors. This procedure is, however, not\nnecessary here, since we only require the combination of the m(n)\ni;jwhich appears in (4.9). To\nproceeed, we label the distinct entries of Mnasa(n)\ni, using the fact that Mnis also cyclic if M\nis:\nMn=0\nBBB@a(n)\n1a(n)\n2a(n)\n3a(n)\n4\na(n)\n4a(n)\n1a(n)\n2a(n)\n3\na(n)\n3a(n)\n4a(n)\n1a(n)\n2\na(n)\n2a(n)\n3a(n)\n4a(n)\n11\nCCCA: (4.11)\nWriting Mn=MMn\u00001, we can exploit the particular structure of the matrix to reduce it from\na(4\u00024)-matrix to a (2\u00022)-matrix, by considering the following differences: \na(n)\n1\u0000a(n)\n3\na(n)\n2\u0000a(n)\n4!\n=\u0012\nP0\u0000P2P3\u0000P1\nP1\u0000P3P0\u0000P2\u0013 \na(n\u00001)\n1\u0000a(n\u00001)\n3\na(n\u00001)\n2\u0000a(n\u00001)\n4!\n=\u0012\nP0\u0000P2P3\u0000P1\nP1\u0000P3P0\u0000P2\u0013n\u00001\u0012\nP0\u0000P2\nP1\u0000P3\u0013\n: (4.12)\nThe velocity auto-correlation (4.9) is thus given by\nhv0\u0001vni=a(n)\n1\u0000a(n)\n3\n=\u0000\n1 0\u0001\u0012\nP0\u0000P2P3\u0000P1\nP1\u0000P3P0\u0000P2\u0013n\u00001\u0012\nP0\u0000P2\nP1\u0000P3\u0013\n: (4.13)\nSumming the previous expression over all n, and using the fact that å¥\nn=0An= (I\u0000A)\u00001\nfor a matrix A, where Iis the identity matrix, we obtain the diffusion coefficient (2.1) as\nD1SMA =DNMA\"\n1+2\u0010\n1 0\u0011\u0012\n1\u0000P0+P2P1\u0000P3\nP3\u0000P11\u0000P0+P2\u0013\u00001\u0012\nP0\u0000P2\nP1\u0000P3\u0013#\n: (4.14)\nFor a symmetric process in which P1=P3, which is often imposed by a symmetry of the\nphysical system, equation (4.5) holds, and the diffusion coefficient takes the form (4.6), viz.\nD1SMA =DNMA1+P0\u0000P2\n1\u0000P0+P2: (4.15)\nIf, however, P16=P3, then equation (4.6) is no longer valid. Instead, we have the more\ncomplicated expression\nD1SMA =DNMA1\u0000(P0\u0000P2)2\u0000(P1\u0000P3)2\n(1\u0000P0+P2)2+(P1\u0000P3)2: (4.16)\nSuch asymmetric walks are the lattice equivalent of the continuous-space models of persistent\nrandom walks with chirality considered in [12].Diffusion coefficients for persistent random walks 7\n4.3. Two-dimensional honeycomb lattice\nOn the two-dimensional honeycomb lattice, shown in figure 1(b), each site has z=3\nneighbours. Here, Ris taken to be a clockwise rotation‡ by angle 2 p=3 and the arrangement\nof neighbours differs by a reflection T. We denote by P0\u0011P(\u0000vjv)the probability that the\nparticle turns around, P1\u0011P(\u0000R2vjv)that it turns left relative to its previous direction, and\nP2\u0011P(\u0000Rvjv)that it turns right. The transition matrix Mis thus\nM=0\n@P0P1P2\nP2P0P1\nP1P2P01\nA: (4.17)\nProceeding as with the square lattice, we let\nMn\u00110\nB@a(n)\n1a(n)\n2a(n)\n3\na(n)\n3a(n)\n1a(n)\n2\na(n)\n2a(n)\n3a(n)\n11\nCA; (4.18)\nand obtain the matrix equation0\nB@\u0000a(n)\n1+1\n2[a(n)\n2+a(n)\n3]\n\u0000a(n)\n2+1\n2[a(n)\n1+a(n)\n3]\n\u0000a(n)\n3+1\n2[a(n)\n1+a(n)\n2]1\nCA=0\n@P0P1P2\nP2P0P1\nP1P2P01\nA0\nB@\u0000a(n\u00001)\n1+1\n2[a(n\u00001)\n2+a(n\u00001)\n3]\n\u0000a(n\u00001)\n2+1\n2[a(n\u00001)\n1+a(n\u00001)\n3]\n\u0000a(n\u00001)\n3+1\n2[a(n\u00001)\n1+a(n\u00001)\n2]1\nCA;\n=1\n20\n@P0P1P2\nP2P0P1\nP1P2P01\nAn\u000010\n@1\u00003P0\n1\u00003P2\n1\u00003P11\nA: (4.19)\nThe velocity auto-correlation (4.4) is thus\nhv0\u0001vni=\u0000a(n)\n1+1\n2[a(n)\n2+a(n)\n3];\n=1\n2\u0010\n1 0 0\u00110\n@P0P1P2\nP2P0P1\nP1P2P01\nAn\u000010\n@1\u00003P0\n1\u00003P2\n1\u00003P11\nA: (4.20)\nHence the diffusion coefficient (2.1) is\nD1SMA =DNMA2\n641+\u0010\n1 0 0\u00110\n@1\u0000P0\u0000P1\u0000P2\n\u0000P21\u0000P0\u0000P1\n\u0000P1\u0000P21\u0000P01\nA\u000010\n@1\u00003P0\n1\u00003P2\n1\u00003P11\nA3\n75: (4.21)\nAgain, in the case of an isotropic process for which P2=P1\u0011Ps(‘symmetric’), equation\n(4.5) holds, and substituting Ps= (1\u0000P0)=2 gives the following expression for the diffusion\ncoefficient:\nD1SMA =DNMA1\u0000P0+Ps\n1+P0\u0000Ps=DNMA3(1\u0000P0)\n1+3P0: (4.22)\nFor an asymmetric process for which P16=P2, defining the symmetric and antisymmetric parts\nPs\u0011(P1+P2)=2 and Pa\u0011(P1\u0000P2)=2, we instead obtain\nD1SMA =DNMA1\u00003P2\na\u0000(P0\u0000Ps)2\n3P2a+(1+P0\u0000Ps)2;\n=DNMA3(1\u0000P0)(1+3P0)\u000012P2\na\n(1+3P0)2+12P2a: (4.23)\n‡ We take a clockwise rotation as opposed to an anticlockwise one in the other examples so that the lattice directions\nare still labelled anticlockwise.Diffusion coefficients for persistent random walks 8\n4.4. Two-dimensional triangular lattice\nPersistent walks on a triangular lattice were made popular by Fink and Mao [13] in connection\nto tie knots. Here, each site has z=6 neighbours, so that Ris an anti-clockwise rotation\nby angle p=3. Following our convention [see figure 2 (d)], we denote by P0\u0011P(vjv)the\nprobability that the particle moves forward, P1\u0011P(Rvjv)that it moves in the forward left\ndirection relative to its previous direction, and similarly for P2,P3,P4, and P5.\nProceeding along the lines of the previous subsection, we let\nMn\u00110\nBBBB@a(n)\n1a(n)\n2\u0001\u0001\u0001a(n)\n6\na(n)\n6a(n)\n1\u0001\u0001\u0001a(n)\n5............\na(n)\n2a(n)\n3\u0001\u0001\u0001a(n)\n11\nCCCCA; (4.24)\nin terms of which the velocity auto-correlation (4.4) is given by\nhv0\u0001vni=a(n)\n1+1\n2[a(n)\n2+a(n)\n6\u0000a(n)\n3\u0000a(n)\n5]\u0000a(n)\n4: (4.25)\nA computation similar to that of equation (4.19) yields\nhv0\u0001vni=\u0010\n1 0\u0001\u0001\u00010\u00110\nBBB@P0P1\u0001\u0001\u0001P5\nP5P0\u0001\u0001\u0001P4\n............\nP1P2\u0001\u0001\u0001P01\nCCCAn\u000010\nBBB@P0\u0000P3+1\n2(P1\u0000P2\u0000P4+P5)\nP1\u0000P4+1\n2(P2\u0000P3\u0000P5+P0)\n...\nP5\u0000P2+1\n2(P0\u0000P1\u0000P3+P4)1\nCCCA: (4.26)\nThe diffusion coefficient (2.1) is then\nD1SMA =DNMA2\n66641+\u0000\n1 0\u0001\u0001\u00010\u00010\nBBB@1\u0000P0\u0000P1\u0001\u0001\u0001 \u0000 P5\n\u0000P51\u0000P0\u0001\u0001\u0001 \u0000 P4\n............\n\u0000P1\u0000P2\u0001\u0001\u00011\u0000P01\nCCCA\u00001\n\u00020\nBBB@P0\u0000P3+1\n2(P1\u0000P2\u0000P4+P5)\nP1\u0000P4+1\n2(P2\u0000P3\u0000P5+P0)\n...\nP5\u0000P2+1\n2(P0\u0000P1\u0000P3+P4)1\nCCCA3\n7775: (4.27)\n4.5. d-dimensional hypercubic lattice\nThe case of a hypercubic lattice in arbitrary dimension dwith coordination number z=2d\ncan also be treated, provided that the same probability Psis assigned to scattering along\nall new directions which are perpendicular to the previous direction of motion. We then\nhave P0+Pz=2+2(d\u00001)Ps=1, and the invariant distribution of velocities p(v) =1=(2d)is\nuniform. We then recover the expression (4.15) for the diffusion coefficient in this case.\n5. Two-Step Memory Approximation (2-SMA)\nWe now turn to the main contribution of this paper, namely the calculation of the diffusion\ncoefficient for persistent random walks with 2-step memory, for several two-dimensional\nlattices.Diffusion coefficients for persistent random walks 9\nWe thus assume that the velocity vectors obey a random process for which the probability\nofvntakes on different values according to the velocities at the two previous steps, vn\u00001and\nvn\u00002, so that we may write\nm(fv0;:::;vng) =n\nÕ\ni=2P(vijvi\u00001;vi\u00002)p(v0;v1): (5.1)\nThe velocity auto-correlation (2.2) function is then\nhv0\u0001vni=å\nfvn;:::;v0gv0\u0001vnn\nÕ\ni=2P(vijvi\u00001;vi\u00002)p(v0;v1): (5.2)\nThe calculation of these correlations proceeds in the most straightforward way by\ntransposing the calculation leading to equation (2.2) to the two-step probability transitions\nas characterizing the probability transitions of a two-dimensional Markov chain. Considering\na lattice with coordination number z, the state of the Markov chain is a normalised vector of\ndimension z2. The time evolution is specified by the (z2\u0002z2)stochastic matrix Mwith entries\nmi=(i1\u00001)z+i2;j=(j1\u00001)z+j2\u0011di2;j1P(Ri1T2vjRj1Tv;Rj2v); (5.3)\nwhere i1,i2,i1, and i2take values between 1 and z. Denoting by Pthe invariant distribution of\nthis Markov chain, i.e. the z2-dimensional vector with components pisuch that åz2\nj=1mi;jpj=\npi, equation (5.2) becomes\nhv0\u0001vni=z2\nå\ni0;in=1J(n)\nin;i0m(n\u00001)\nin;i0pi0; (5.4)\nwhere J(n)is the (z2\u0002z2)matrix with elements\nJ(n)\ni=(i1\u00001)z+i2;j=(j1\u00001)z+j2\u0011ei1\u0001Tnej2; (5.5)\nandekdenotes the unit vector along the kth lattice direction.\nUsing the symmetries of the problem and writing Pj k=P(RkTRjTvjRjTv;v)for the\nconditional probability of turning successively by angles 2 pj=zand 2 pk=zwith respect to the\ncurrent direction (with reflection by Twhere needed), we define f\u0011exp(2ip=z), where i\ndenotes the imaginary unit, i=p\u00001, and show, through the examples below, that equation\n(5.4) reduces to the general expression\nhv0\u0001vni=snz\n2\u0010\n1\u0001\u0001\u00011\u00112\n66640\nBBB@P00 P10\u0001\u0001\u0001 Pz\u00001;0\nfP01 fP11\u0001\u0001\u0001 fPz\u00001;1\n............\nfz\u00001P0;z\u00001fz\u00001P1;z\u00001\u0001\u0001\u0001fz\u00001Pz\u00001;11\nCCCAn\u00001\nDiag(1;f;:::;fz\u00001)\n+0\nBBB@P00 P10\u0001\u0001\u0001 Pz\u00001;0\nf\u00001P01 f\u00001P11\u0001\u0001\u0001f\u00001Pz\u00001;1\n............\nf1\u0000zP0;z\u00001f1\u0000zP1;z\u00001\u0001\u0001\u0001f1\u0000zPz\u00001;11\nCCCAn\u00001\nDiag(1;f\u00001;:::;f1\u0000z)3\n77750\nB@p1\n...\npz1\nCA;\n(5.6)\nwhere Diag (1;f;:::;fz\u00001)denotes the matrix with elements listed on the main diagonal and\n0 elsewhere. Here, sis a sign factor which is \u00001 for the honeycomb lattice and +1 for the\nother lattices, and reflects the action of T. Note that the second term in the summation is theDiffusion coefficients for persistent random walks 10\ncomplex conjugate of the first, so that the result is real. The diffusion coefficient for persistent\nrandom walks with 2-step memory is therefore\nD2SMA\nDNMA=1+sz\u0010\n1\u0001\u0001\u00011\u0011\n\u0002 (5.7)\n2\n66640\nBBB@1\u0000P00\u0000P10\u0001\u0001\u0001 \u0000 Pz\u00001;0\n\u0000fP01 1\u0000fP11\u0001\u0001\u0001 \u0000 fPz\u00001;1\n............\n\u0000fz\u00001P0;z\u00001\u0000fz\u00001P1;z\u00001\u0001\u0001\u00011\u0000fz\u00001Pz\u00001;11\nCCCA\u00001\nDiag(1;f;:::;fz\u00001)\n+0\nBBB@1\u0000P00\u0000P10\u0001\u0001\u0001 \u0000 Pz\u00001;0\n\u0000f\u00001P01\u0000f\u00001P11\u0001\u0001\u0001 \u0000 f\u00001Pz\u00001;1\n............\n\u0000f1\u0000zP0;z\u00001\u0000f1\u0000zP1;z\u00001\u0001\u0001\u00011\u0000f1\u0000zPz\u00001;11\nCCCA\u00001\nDiag(1;f\u00001;:::;f1\u0000z)3\n77750\nB@p1\n...\npz1\nCA:\n5.1. One-dimensional lattice\nWe first consider the simplest case, namely the one-dimensional lattice. The stochastic matrix\nMfrom equation (5.3) is the 4 \u00024 matrix\nM=0\nBB@P(vjv;v)P(vjv;\u0000v) 0 0\n0 0 P(\u0000vjv;\u0000v)P(\u0000vjv;v)\nP(\u0000vjv;v)P(\u0000vjv;\u0000v) 0 0\n0 0 P(vjv;\u0000v)P(vjv;v)1\nCCA;\n=0\nBB@P00P100 0\n0 0 P11P01\nP01P110 0\n0 0 P10P001\nCCA; (5.8)\nwhere P00+P01=1 and P10+P11=1.\nConsidering equation (5.4), we compute the invariant distribution of M, which is the\nvector Pwhose components correspond to the four states p(v;v);p(\u0000v;v);p(v;\u0000v)and\np(\u0000v;\u0000v). Given that we must have p(\u0000v;\u0000v) = p(v;v)and p(v;\u0000v) = p(\u0000v;v), the\nequilibrium distribution is obtained as the solution of the system of equations\np(v;v) =P00p(v;v)+P10p(\u0000v;v);\np(v;v)+p(\u0000v;v) =1\n2;(5.9)\ngiving\np1=p4=p(v;v) =P10\n2[1\u0000P00+P10];\np2=p3=p(\u0000v;v) =1\u0000P00\n2[1\u0000P00+P10]:(5.10)\nThe matrix J(n), equation (5.5), is here the same for all n, and has the expression\nJ=0\nBB@1\u00001 1\u00001\n1\u00001 1\u00001\n\u00001 1\u00001 1\n\u00001 1\u00001 11\nCCA: (5.11)Diffusion coefficients for persistent random walks 11\nThe velocity auto-correlation (5.4) is thus\nhv0\u0001vni=2h\u0010\nm(n\u00001)\n1;1\u0000m(n\u00001)\n1;4+m(n\u00001)\n3;4\u0000m(n\u00001)\n3;1\u0011\np1\n+\u0010\nm(n\u00001)\n1;3\u0000m(n\u00001)\n1;2+m(n\u00001)\n3;2\u0000m(n\u00001)\n3;3\u0011\np2i\n;\n=2\u0010\n1 1\u0011 \nm(n\u00001)\n1;1\u0000m(n\u00001)\n1;4m(n\u00001)\n1;3\u0000m(n\u00001)\n1;2\nm(n\u00001)\n3;4\u0000m(n\u00001)\n3;1m(n\u00001)\n3;2\u0000m(n\u00001)\n3;3!\u0012\np1\np2\u0013\n: (5.12)\nSince Mn, the nth power of M, has the symmetries of M, its entries m(n)\ni;jare such that\nm(n)\n1;1=m(n)\n4;4\u0011a(n)\n1; m(n)\n2;1=m(n)\n3;4\u0011b(n)\n4;\nm(n)\n1;2=m(n)\n4;3\u0011a(n)\n2; m(n)\n2;2=m(n)\n3;3\u0011b(n)\n3;\nm(n)\n1;3=m(n)\n4;2\u0011a(n)\n3; m(n)\n2;3=m(n)\n3;2\u0011b(n)\n2;\nm(n)\n1;4=m(n)\n4;1\u0011a(n)\n4; m(n)\n2;4=m(n)\n3;1\u0011b(n)\n1:(5.13)\nWriting Mn=MMn\u00001, we obtain two separate sets of equations for 2 \u00022 matrices, one\ninvolving a(n)\n1;a(n)\n4andb(n)\n1;b(n)\n4, and the other involving a(n)\n2;a(n)\n3andb(n)\n2;b(n)\n3:\n \na(n)\n1a(n)\n4\nb(n)\n1b(n)\n4!\n=\u0012\nP00P10\nP01P11\u0013 \na(n\u00001)\n1a(n\u00001)\n4\nb(n\u00001)\n4b(n\u00001)\n1!\n;\n \na(n)\n2a(n)\n3\nb(n)\n2b(n)\n3!\n=\u0012\nP00P10\nP01P11\u0013 \na(n\u00001)\n2a(n\u00001)\n3\nb(n\u00001)\n3b(n\u00001)\n2!\n:(5.14)\nNote that these equations do not have a simple recursive form, since the elements of the\nmatrices on the two sides do not appear in the same places. However, taking the differences\na(n)\n1\u0000a(n)\n4,a(n)\n3\u0000a(n)\n2,b(n)\n4\u0000b(n)\n1, and b(n)\n2\u0000b(n)\n3, we obtain the recursive system\n \na(n)\n1\u0000a(n)\n4a(n)\n3\u0000a(n)\n2\nb(n)\n4\u0000b(n)\n1b(n)\n2\u0000b(n)\n3!\n=\u0012\nP00 P10\n\u0000P01\u0000P11\u0013 \na(n\u00001)\n1\u0000a(n\u00001)\n4a(n\u00001)\n3\u0000a(n\u00001)\n2\nb(n\u00001)\n4\u0000b(n\u00001)\n1b(n\u00001)\n2\u0000b(n\u00001)\n3!\n;\n=\u0012\nP00 P10\n\u0000P01\u0000P11\u0013n\u00001 \na(1)\n1\u0000a(1)\n4a(1)\n3\u0000a(1)\n2\nb(1)\n4\u0000b(1)\n1b(1)\n2\u0000b(1)\n3!\n;\n=\u0012\nP00 P10\n\u0000P01\u0000P11\u0013n\u00001\u0012\nP00\u0000P10\n\u0000P01P11\u0013\n;\n=\u0012\nP00 P10\n\u0000P01\u0000P11\u0013n\u0012\n1 0\n0\u00001\u0013\n: (5.15)\nPlugging this equation into equation (5.12), we obtain\nhv0\u0001vni=2\u0010\n1 1\u0011\u0012\nP00 P10\n\u0000P01\u0000P11\u0013n\u00001\u0012\np1\n\u0000p2\u0013\n: (5.16)\nThis is equation (5.6), where f=exp(2ip=2) =\u00001. The diffusion coefficient is therefore\ngiven by equation (5.7):\nD2SMA\nDNMA=1+4\u0010\n1 1\u0011\u0012\n1\u0000P00\u0000P10\nP01 1+P11\u0013\u00001\u0012\np1\n\u0000p2\u0013\n;Diffusion coefficients for persistent random walks 12\n=DNMAP10\n[1\u0000P00][1+P00\u0000P10]\n[1\u0000P00+P10]: (5.17)\nIt is a function of the two parameters P00andP10; when these are equal, the process reduces\nto a walk with single-step memory, and the diffusion coefficient to that of the single-step\nmemory approximation (4.8), as it should.\n5.2. Two-dimensional honeycomb lattice\nFor the two-dimensional honeycomb lattice, the stochastic matrix Mof equation (5.3) is a\n(9\u00029)matrix with the following non-zero entries:\nm1;1=m5;5=m9;9=P(vj\u0000v;v) =P00;\nm1;2=m5;6=m9;7=P(R2vj\u0000R2v;v) =P10;\nm1;3=m5;4=m9;8=P(Rvj\u0000Rv;v) =P20;\nm4;1=m8;5=m3;9=P(Rvj\u0000v;v) =P02;\nm4;2=m8;6=m3;7=P(vj\u0000R2v;v) =P12;\nm4;3=m8;4=m3;8=P(R2vj\u0000Rv;v) =P22;\nm7;1=m2;5=m6;9=P(R2vj\u0000v;v) =P01;\nm7;2=m2;6=m6;7=P(Rvj\u0000R2v;v) =P11;\nm7;3=m2;4=m6;8=P(vj\u0000Rv;v) =P21:(5.18)\nGiven the three constraints\nP00+P01+P02=1;\nP10+P11+P12=1;\nP20+P21+P22=1;(5.19)\nthe actual number of independent parameters is six.\nThe invariant distribution Pwith components pican be written in terms of the three\nprobabilities p(v;\u0000v),p(v;\u0000Rv),p(v;\u0000R2v),\np1=p5=p9=p(v;\u0000v);\np2=p6=p7=p(v;\u0000Rv);\np3=p4=p8=p(v;\u0000R2v);(5.20)\nwhich are solutions of the system of equations:\np(v;\u0000v) =P00p(v;\u0000v)+P10p(v;\u0000R2v)+P20p(v;\u0000Rv); (5.21)\np(v;\u0000R2v) =P01p(v;\u0000v)+P11p(v;\u0000R2v)+P21p(v;\u0000Rv); (5.22)\np(v;\u0000v)+p(v;\u0000Rv)+p(v;\u0000R2v) =1\n3: (5.23)\nThe matrix (5.5) has the block structure\nJ(n)= (\u00001)n0\n@B1B1B1\nB2B2B2\nB3B3B31\nA; (5.24)\nwhere\nB1=0\n@\u00001 1=2 1=2\n\u00001 1=2 1=2\n\u00001 1=2 1=21\nA;B2=0\n@1=2\u00001 1=2\n1=2\u00001 1=2\n1=2\u00001 1=21\nA;B3=0\n@1=2 1=2\u00001\n1=2 1=2\u00001\n1=2 1=2\u000011\nA: (5.25)Diffusion coefficients for persistent random walks 13\nSubstituting these expressions into equation (5.4), we find\nhvn+1\u0001v0i=3(\u00001)n+1(h\nm(n)\n1;1\u00001\n2m(n)\n1;5\u00001\n2m(n)\n1;9\u00001\n2m(n)\n4;1+m(n)\n4;5\u00001\n2m(n)\n4;9\u00001\n2m(n)\n7;1\n\u00001\n2m(n)\n7;5+m(n)\n7;9i\np1+h\n\u00001\n2m(n)\n1;2\u00001\n2m(n)\n1;6+m(n)\n1;7+m(m;n)\n4;2\u00001\n2m(n)\n4;6\n\u00001\n2m(n)\n4;7\u00001\n2m(n)\n7;2+m(n)\n7;6\u00001\n2m(n)\n7;7i\np2+h\n\u00001\n2m(n)\n1;3+m(n)\n1;4\u00001\n2m(n)\n1;8\n\u00001\n2m(n)\n4;3\u00001\n2m(n)\n4;4+m(n)\n4;8+m(n)\n7;3\u00001\n2m(n)\n7;4\u00001\n2m(n)\n7;8i\np3)\n;\n=3(\u00001)n+1\u0010\n1 1 1\u0011\n\u0002 (5.26)\n0\nB@m(n)\n1;1\u00001\n2m(n)\n1;5\u00001\n2m(n)\n1;9\u00001\n2m(n)\n1;2\u00001\n2m(n)\n1;6+m(n)\n1;7\u00001\n2m(n)\n1;3+m(n)\n1;4\u00001\n2m(n)\n1;8\n\u00001\n2m(n)\n7;1\u00001\n2m(n)\n7;5+m(n)\n7;9\u00001\n2m(n)\n7;2+m(n)\n7;6\u00001\n2m(n)\n7;7m(n)\n7;3\u00001\n2m(n)\n7;4\u00001\n2m(n)\n7;8\n\u00001\n2m(n)\n4;1+m(n)\n4;5\u00001\n2m(n)\n4;9m(n)\n4;2\u00001\n2m(n)\n4;6\u00001\n2m(n)\n4;7\u00001\n2m(n)\n4;3\u00001\n2m(n)\n4;4+m(n)\n4;81\nCA0\n@p1\np2\np31\nA:\nProceeding along the lines of the computation presented in subsection 5.1, we obtain a\nset of recursive matrix equations (A.9) involving the coefficients of Mn. We refer the reader\nto Appendix A for the details of this derivation.\nWe note that the coefficients which appear in equation (5.26) satisfy identities such as,\nfor instance,\nm(n)\n1;1\u00001\n2m(n)\n1;5\u00001\n2m(n)\n1;9=1\n2h\nm(n)\n1;1+e2ip=3m(n)\n1;5+e\u00002ip=3m(n)\n1;9i\n+1\n2h\nm(n)\n1;1+e\u00002ip=3m(n)\n1;5+e2ip=3m(n)\n1;9i\n:(5.27)\nThus, letting f=exp(2ip=3), we can combine the results of equation (A.9) with equation\n(5.26) to find:\nhvn\u0001v0i= (\u00001)n3\n2\u0010\n1 1 1\u00112\n640\n@P00 P10 P20\nfP01fP11fP21\nf2P02f2P12f2P221\nAn\u000010\n@1 0 0\n0f0\n0 0 f21\nA\n+0\n@P00 P10 P20\nf2P01f2P11f2P21\nfP02fP12fP221\nAn\u000010\n@1 0 0\n0f20\n0 0 f1\nA3\n750\n@p1\np2\np31\nA: (5.28)\nThis is equation (5.6). The diffusion coefficient (2.1) is therefore given by (5.7), which is here\nD2SMA\nDNMA=1\u00003\u0010\n1 1 1\u00112\n640\n@1+P00 P10 P20\nfP011+fP11 fP21\nf2P02 f2P121+f2P221\nA\u000010\n@1 0 0\n0f0\n0 0 f21\nA\n+0\n@1+P00 P10 P20\nf2P011+f2P11f2P21\nfP02 fP12 1+fP221\nA\u000010\n@1 0 0\n0f20\n0 0 f1\nA3\n750\n@p1\np2\np31\nA:(5.29)\nGiven a symmetric process for which left and right probabilities are equal, but the\nprobability of a left–left turn is different than that of a right–left turn, we let\nP02=P01\u0011Pbs=1\u0000P00\n2;\nP10=P20\u0011Psb;P11=P22\u0011Pss;P12=P21=1\u0000Psb\u0000Pss:(5.30)Diffusion coefficients for persistent random walks 14\nCarrying out the matrix inversions in (5.29), we find the diffusion coefficient,\nD2SMA =DNMA3(1\u0000P00)(1+P00\u0000Psb)(2\u0000Psb\u00002Pss)\n(1\u0000P00+Psb)[Psb(7+P00\u00008Pss)+2(1+P00)Pss\u00004P2\nsb]: (5.31)\nIf we further assume complete left–right symmetry and identify the probabilities of left–left\nturns and left–right turns, thus letting\nP12=P21=P11=P22=1\u0000Psb\n2; (5.32)\nequation (5.31) simplifies to\nD2SMA =DNMA3(1\u0000P00)(1+P00\u0000Psb)\n(1\u0000P00+Psb)(1+P00+2Psb): (5.33)\n5.3. Two-dimensional square lattice\nFor the two-dimensional square lattice, the matrix Mof equation (5.3) is the (16\u000216)matrix\nwith the following non-zero entries:\nm1;1=m6;6=m11;11=m16;16=P(vjv;v) =P00;\nm1;2=m6;7=m11;12=m16;13=P(R3vjR3v;v) =P10;\nm1;3=m6;8=m11;9=m16;14=P(R2vjR2v;v) =P20;\nm1;4=m6;5=m11;10=m16;15=P(RvjRv;v) =P30;\nm5;1=m10;6=m15;11=m4;16=P(Rvjv;v) =P03;\nm5;2=m10;7=m15;12=m4;13=P(vjR3v;v) =P13;\nm5;3=m10;8=m15;9=m4;14=P(R3vjR2v;v) =P23;\nm5;4=m10;5=m15;10=m4;15=P(R2vjRv;v) =P33;\nm9;1=m14;6=m3;11=m8;16=P(R2vjv;v) =P02;\nm9;2=m14;7=m3;12=m8;13=P(RvjR3v;v) =P12;\nm9;3=m14;8=m3;9=m8;14=P(vjR2v;v) =P22;\nm9;4=m14;5=m3;10=m8;15=P(R3vjRv;v) =P32;\nm13;1=m2;6=m7;11=m12;16=P(R3vjv;v) =P01;\nm13;2=m2;7=m7;12=m12;13=P(R2vjR3v;v) =P11;\nm13;3=m2;8=m7;9=m12;14=P(RvjR2v;v) =P21;\nm13;4=m2;5=m7;10=m12;15=P(vjRv;v) =P31:\nThe matrix (5.5) is here\nJ(n)\u0011J=0\nBB@B1B1B1B1\nB2B2B2B2\n\u0000B1\u0000B1\u0000B1\u0000B1\n\u0000B2\u0000B2\u0000B2\u0000B21\nCCA; (5.34)\nwhere\nB1=0\nBB@1 0\u00001 0\n1 0\u00001 0\n1 0\u00001 0\n1 0\u00001 01\nCCA;B2=0\nBB@0 1 0\u00001\n0 1 0\u00001\n0 1 0\u00001\n0 1 0\u000011\nCCA; (5.35)Diffusion coefficients for persistent random walks 15\nThe invariant distribution Pwill not be written explicitly here. Due to symmetry, only 4\nof the 16 components are distinct. These four components are most simply computed as the\ninvariant vector of the following (4\u00024)matrix:0\nBB@P00P10P20P30\nP01P11P21P31\nP02P12P22P32\nP03P13P23P331\nCCA0\nBB@p1\np2\np3\np41\nCCA=0\nBB@p1\np2\np3\np41\nCCA; (5.36)\nnormalised so that p1+p2+p3+p4=1=4.\nIn terms of these quantities, the velocity auto-correlations are found to be\nhv0\u0001vn+1i=4h\nm(n)\n1;1\u0000m(n)\n1;11+m(n)\n5;6\u0000m(n)\n5;16\u0000m(n)\n9;1+m(n)\n9;11\u0000m(n)\n13;6+m(n)\n13;16i\np1\n+4h\n\u0000m(n)\n1;7+m(n)\n1;13+m(n)\n5;2\u0000m(n)\n5;12+m(n)\n9;7\u0000m(n)\n9;13\u0000m(n)\n13;2+m(n)\n13;12i\np2\n+4h\n\u0000m(n)\n1;3+m(n)\n1;9\u0000m(n)\n5;8+m(n)\n5;14+m(n)\n9;3\u0000m(n)\n9;9+m(n)\n13;8\u0000m(n)\n13;14i\np3\n+4h\nm(n)\n1;5\u0000m(n)\n1;15\u0000m(n)\n5;4+m(n)\n5;10\u0000m(n)\n9;5+m(n)\n9;15+m(n)\n13;4\u0000m(n)\n13;10i\np4;\n=4\u0010\n1 1 1 1\u00110\nBBBB@m(n)\n1;1\u0000m(n)\n1;11m(n)\n1;13\u0000m(n)\n1;7m(n)\n1;9\u0000m(n)\n1;3m(n)\n1;5\u0000m(n)\n1;15\nm(n)\n13;16\u0000m(n)\n13;6m(n)\n13;12\u0000m(n)\n13;2m(n)\n13;8\u0000m(n)\n13;14m(n)\n13;4\u0000m(n)\n13;10\nm(n)\n9;11\u0000m(n)\n9;1m(n)\n9;7\u0000m(n)\n9;13m(n)\n9;3\u0000m(n)\n9;9m(n)\n9;15\u0000m(n)\n9;5\nm(n)\n5;6\u0000m(n)\n5;16m(n)\n5;2\u0000m(n)\n5;12m(n)\n5;14\u0000m(n)\n5;8m(n)\n5;10\u0000m(n)\n5;41\nCCCCA0\nBB@p1\np2\np3\np41\nCCA:\nUsing the results of Appendix B, this expression reduces to\nhv0\u0001vni=2\u0010\n1 1 1 1\u00112\n6640\nBB@P00 P10 P20 P30\niP01 iP11 iP21 iP31\n\u0000P02\u0000P12\u0000P22\u0000P32\n\u0000iP03\u0000iP13\u0000iP23\u0000iP331\nCCAn\u000010\nBB@1 0 0 0\n0i0 0\n0 0\u00001 0\n0 0 0\u0000i1\nCCA(5.37)\n+0\nBB@P00 P10 P20 P30\n\u0000iP01\u0000iP11\u0000iP21\u0000iP31\n\u00001P02\u00001P12\u00001P22\u00001P32\niP03 iP13 iP23 iP331\nCCAn\u000010\nBB@1 0 0 0\n0\u0000i0 0\n0 0\u00001 0\n0 0 0\u0000i1\nCCA3\n7750\nBB@p1\np2\np3\np41\nCCA;\nwhich is equation (5.6). Substituting this result into equation (2.1), we recover the diffuson\ncoefficient given by equation (5.7) :\nD2SMA\nDNMA=1+4\u0010\n1 1 1 1\u0011\n\u0002 (5.38)\n2\n6640\nBB@1\u0000P00\u0000P10\u0000P20\u0000P30\n\u0000iP011\u0000iP11\u0000iP21\u0000iP31\nP02 P12 1+P22 P32\niP03 iP13 iP23 1+iP331\nCCA\u000010\nBB@1 0 0 0\n0i0 0\n0 0\u00001 0\n0 0 0\u0000i1\nCCA\n+0\nBB@1\u0000P00\u0000P10\u0000P20\u0000P30\niP01 1+iP11 iP21 iP31\nP02 P12 1+P22P32\n\u0000iP03\u0000iP13\u0000iP23\u0000iP331\nCCA\u000010\nBB@1 0 0 0\n0\u0000i0 0\n0 0\u00001 0\n0 0 0 i1\nCCA3\n7750\nBB@p1\np2\np3\np41\nCCA:\nFor a symmetric walk, we substitute\nP03=P01=1\u0000P00\u0000P02\n2;P23=P21=1\u0000P20\u0000P22\n2; (5.39)Diffusion coefficients for persistent random walks 16\nand write\nP33=P11\u0011Pss;P10=P30\u0011Psf;\nP12=P32\u0011Psb;P31=P13=1\u0000Psf\u0000Pss\u0000Psb;(5.40)\nin terms of which the diffusion coefficient is found to be\nD2SMA =DNMA(\n2\u00004P00+2P2\n00\u00003Psb+6P00Psb\u00003P2\n00Psb+P2\nsb\u00002P00P2\nsb+P2\n00P2\nsb\n+3Psf\u00006P02Psf\u00002P00Psf+2P02P00Psf\u0000P2\n00Psf+2PsbPsf+3P02PsbPsf\n+3P00PsbPsf\u0000P02P00PsbPsf+P2\n00PsbPsf\u00002P2\nsbPsf\u00002P00P2\nsbPsf+P2\nsf\n+P02P2\nsf+3P00P2\nsf\u0000P02P00P2\nsf\u00004PsbP2\nsf+2P02PsbP2\nsf\u00002P00PsbP2\nsf\u00002P3\nsf\n+2P02P3\nsf\u00002h\n\u0000(\u00001+P00)2(\u00001+Psb)+[1+P02(\u00003+P00)\u00003P00\n+2(1+P00)Psb]Psf\u00002(\u00001+P02)P2\nsfi\nPss+P2\n20(P02\u0000Psb)[P02(\u00002+Psb\n+Psf+2Pss)\u00002Psb(\u00001+Psb+Psf+2Pss)]+P20[(1+P00)Psb\u0000P02Psf]\n\u0002[3Psb+Psf+2Pss+P02(\u00002+Psb+Psf+2Pss)\n\u00002Psb(Psb+Psf+2Pss)]\u0000P2\n22h\n2\u0000Psb+3Psf\u00002Pss\n\u0000P2\n00(\u00002+Psb+Psf+2Pss)\u0000Psf[Psb+2PsbPsf+2Pss\n\u0000Psf(1\u00002Psf\u00004Pss)]+P00[Psb(2+3Psf)\u00004(1\u0000Pss)\n+Psf(\u00002+3Psf+6Pss)]i\n\u0000P22n\nP2\nsb[(\u00001+P00)2\u00002(1+P00)Psf]\n+P02Psf[\u00006+Psf+6Pss\u0000P00(\u00002+Psf+2Pss)+2Psf(Psf+2Pss)]+\nPsbh\nPsf(3+3P02\u00002Psf+2P02Psf\u00004Pss)+2(\u00001+Pss)\n+P2\n00(\u00002+Psf+2Pss)\u0000P00[\u00004+P02Psf+2P2\nsf+4(1+Psf)Pss]i\n+P20h\nP02[4\u00002Psb+2Psf\u00004Pss+2P00(\u00002+Psb+Psf+2Pss)\n\u00003Psf(Psb+Psf+2Pss)]+Psb[\u00004+Psb\u00003Psf+2Pss\n+4Psf(Psb+Psf+2Pss)\u0000P00(\u00004+3Psb+3Psf+6Pss)]io)\n,(\n[1+Psb\u0000P00(1+Psb)+P20(\u0000P02+Psb)+P22(\u00001+P00\u0000Psf)\n+Psf+P02Psf]h\n2(\u00001+P00)P2\nsb\u00002P02P2\nsf+Psf[1+P22\u0000P00\u0000P22P00\n+P02(2+P20\u00004Pss)]+2[1+P22+P20P02\u0000(1+P22)P00]Pss\n+Psb[3+P22+P20P02\u0000P22P00\u00002(1+P02)Psf\u00004Pss\n+P00(\u00003+2Psf+4Pss)]i)\n: (5.41)\nIf we further assume complete left–right symmetry, so that\nP33=P11=P31=P13=1\u0000Psf\u0000Psb\n2; (5.42)Diffusion coefficients for persistent random walks 17\nthen the diffusion coefficient becomes\nDs\n2SMA =DNMAh\n\u00001+P2\n22+2P22P20P02+P2\n20P2\n02+2P00\u00002P2\n22P00\u00002P22P20P02P00\n\u0000P2\n00+P2\n22P2\n00+Psb\u0000P22Psb\u0000P20Psb\u00003P22P20Psb+P20P02Psb\n\u0000P2\n20P02Psb\u00002P00Psb+2P22P00Psb\u0000P20P00Psb+P22P20P00Psb\n+P20P02P00Psb+P2\n00Psb\u0000P22P2\n00Psb\u00003Psf+3P2\n22Psf+3P02Psf\n\u00003P02Psf+P20P02Psf+P22P20P02Psf\u0000P20P2\n02Psf+P00Psf\u0000P2\n22P00Psf\n\u0000P02P00Psf+P22P02P00Psfi\n=h\n\u00001+P2\n22+2P22P20P02+P2\n20P2\n02+2P00\n\u00002P2\n22P00\u00002P22P20P02P00\u0000P2\n00+P2\n22P2\n00\u0000Psb\u0000P22Psb\u0000P20Psb\n\u0000P22P20Psb\u0000P20P02Psb\u0000P2\n20P02Psb+2P00Psb+2P22P00Psb+P20P00Psb\n+P22P20P00Psb+P20P02P00Psb\u0000P2\n00Psb\u0000P22P2\n00Psb\u0000Psf+P2\n22Psf\n\u0000P02Psf\u0000P22P02Psf\u0000P20P02Psf+P22P20P02Psf\u0000P20P2\n02Psf+P00Psf\n\u0000P2\n22P00Psf+P02P00Psf+P22P02P00Psfi\n: (5.43)\nIt can be checked that this equation boils down to the expression (4.15) in the single-step\nmemory approximation.\nThe validity of equations (5.17), (5.29) and (5.38) has been checked by comparison\nwith diffusion coefficients calculated from direct numerical simulations of the corresponding\npersistent random walk processes.\n5.4. Triangular lattice\nConsider finally the triangular lattice with 6-fold symmetry. We denote the relative directions\nby numbers from 0 to 5, following the conventions shown in figure 2(d).\nThe symmetry of Mis similar to the previous subsections, so that the structure of the\nproblem is by now clear. Having identified the matrix Jand invariant measure Pin equation\n(5.4), the velocity auto-correlation is found to be\nhv0\u0001vn+1i=3h\n2m(n)\n1;1+m(n)\n1;8\u0000m(n)\n1;15\u00002m(n)\n1;22\u0000m(n)\n1;29+m(n)\n1;36+m(n)\n7;1+2m(n)\n7;8\n+m(n)\n7;15\u0000m(n)\n7;22\u00002m(n)\n7;29\u0000m(n)\n7;36\u0000m(n)\n13;1+m(n)\n13;8+2m(n)\n13;15+m(n)\n13;22\n\u0000m(n)\n13;29\u00002m(n)\n13;36\u00002m(n)\n19;1\u0000m(n)\n19;8+m(n)\n19;15+2m(n)\n19;22+m(n)\n19;29\u0000m(n)\n19;36\n\u0000m(n)\n25;1\u00002m(n)\n25;8\u0000m(n)\n25;15+m(n)\n25;22+2m(n)\n25;29+m(n)\n25;36+m(n)\n31;1\u0000m(n)\n31;8\n\u00002m(n)\n31;15\u0000m(n)\n31;22+m(n)\n31;29+2m(n)\n31;36i\np1\n+3h\nm(n)\n1;2\u0000m(n)\n1;9\u00002m(n)\n1;16\u0000m(n)\n1;23+m(n)\n1;30+2m(n)\n1;31+2m(n)\n7;2+m(n)\n7;9\n\u0000m(n)\n7;16\u00002m(n)\n7;23\u0000m(n)\n7;30+m(n)\n7;31+m(n)\n13;2+2m(n)\n13;9+m(n)\n13;16\u0000m(n)\n13;23\n\u00002m(n)\n13;30\u0000m(n)\n13;31\u0000m(n)\n19;2+m(n)\n19;9+2m(n)\n19;16+m(n)\n19;23\u0000m(n)\n19;30\u00002m(n)\n19;31\n\u00002m(n)\n25;2\u0000m(n)\n25;9+m(n)\n25;16+2m(n)\n25;23+m(n)\n25;30\u0000m(n)\n25;31\u0000m(n)\n31;2\u00002m(n)\n31;9\n\u0000m(n)\n31;16+m(n)\n31;23+2m(n)\n31;30+m(n)\n31;31i\np2\n+3h\n\u0000m(n)\n1;3\u00002m(n)\n1;10\u0000m(n)\n1;17+m(n)\n1;24+2m(n)\n1;25+m(n)\n1;32+m(n)\n7;3\u0000m(n)\n7;10Diffusion coefficients for persistent random walks 18\n\u00002m(n)\n7;17\u0000m(n)\n7;24+m(n)\n7;25+2m(n)\n7;32+2m(n)\n13;3+m(n)\n13;10\u0000m(n)\n13;17\u00002m(n)\n13;24\n\u0000m(n)\n13;25+m(n)\n13;32+m(n)\n19;3+2m(n)\n19;10+m(n)\n19;17\u0000m(n)\n19;24\u00002m(n)\n19;25\u0000m(n)\n19;32\n\u0000m(n)\n25;3+m(n)\n25;10+2m(n)\n25;17+m(n)\n25;24\u0000m(n)\n25;25\u00002m(n)\n25;32\u00002m(n)\n31;3\u0000m(n)\n31;10\n+m(n)\n31;17+2m(n)\n31;24+m(n)\n31;25\u0000m(n)\n31;32i\np3\n+3h\n\u00002m(n)\n1;4\u0000m(n)\n1;11+m(n)\n1;18+2m(n)\n1;19+m(n)\n1;26\u0000m(n)\n1;33\u0000m(n)\n7;4\u00002m(n)\n7;11\n\u0000m(n)\n7;18+m(n)\n7;19+2m(n)\n7;26+m(n)\n7;33+m(n)\n13;4\u0000m(n)\n13;11\u00002m(n)\n13;18\u0000m(n)\n13;19\n+m(n)\n13;26+2m(n)\n13;33+2m(n)\n19;4+m(n)\n19;11\u0000m(n)\n19;18\u00002m(n)\n19;19\u0000m(n)\n19;26+m(n)\n19;33\n+m(n)\n25;4+2m(n)\n25;11+m(n)\n25;18\u0000m(n)\n25;19\u00002m(n)\n25;26\u0000m(n)\n25;33\u0000m(n)\n31;4+m(n)\n31;11\n+2m(n)\n31;18+m(n)\n31;19\u0000m(n)\n31;26\u00002m(n)\n31;33i\np4\n+3h\n\u0000m(n)\n1;5+m(n)\n1;12+2m(n)\n1;13+m(n)\n1;20\u0000m(n)\n1;27\u00002m(n)\n1;34\u00002m(n)\n7;5\u0000m(n)\n7;12\n+m(n)\n7;13+2m(n)\n7;20+m(n)\n7;27\u0000m(n)\n7;34\u0000m(n)\n13;5\u00002m(n)\n13;12\u0000m(n)\n13;13+m(n)\n13;20\n+2m(n)\n13;27+m(n)\n13;34+m(n)\n19;5\u0000m(n)\n19;12\u00002m(n)\n19;13\u0000m(n)\n19;20+m(n)\n19;27+2m(n)\n19;34\n+2m(n)\n25;5+m(n)\n25;12\u0000m(n)\n25;13\u00002m(n)\n25;20\u0000m(n)\n25;27+m(n)\n25;34+m(n)\n31;5+2m(n)\n31;12\n+m(n)\n31;13\u0000m(n)\n31;20\u00002m(n)\n31;27\u0000m(n)\n31;34i\np5\n+3h\nm(n)\n1;6+2m(n)\n1;7+m(n)\n1;14\u0000m(n)\n1;21\u00002m(n)\n1;28\u0000m(n)\n1;35\u0000m(n)\n7;6+m(n)\n7;7\n+2m(n)\n7;14+m(n)\n7;21\u0000m(n)\n7;28\u00002m(n)\n7;35\u00002m(n)\n13;6\u0000m(n)\n13;7+m(n)\n13;14+2m(n)\n13;21\n+m(n)\n13;28\u0000m(n)\n13;35\u0000m(n)\n19;6\u00002m(n)\n19;7\u0000m(n)\n19;14+m(n)\n19;21+2m(n)\n19;28+m(n)\n19;35\n+m(n)\n25;6\u0000m(n)\n25;7\u00002m(n)\n25;14\u0000m(n)\n25;21+m(n)\n25;28+2m(n)\n25;35+2m(n)\n31;6+m(n)\n31;7\n\u0000m(n)\n31;14\u00002m(n)\n31;21\u0000m(n)\n31;28+m(n)\n31;35i\np6 (5.44)\nLetting f=exp(2pi=6), we find\nhv0\u0001vn+1i=3\u0010\n1 1 1 1 1 1\u0011\n\u0002 (5.45)\n2\n66666640\nBBBBBB@P00 P10 P20 P30 P40 P50\nfP01 fP11 fP21 fP31 fP41 fP51\nf2P02 f2P12 f2P22 f2P32 f2P42 f2P52\n\u0000P03\u0000P13\u0000P23\u0000P33\u0000P33\u0000P53\nf\u00002P04f\u00002P14f\u00002P24f\u00002P34f\u00002P44f\u00002P54\nf\u00001P05f\u00001P15f\u00001P25f\u00001P35f\u00001P45f\u00001P551\nCCCCCCAn0\nBBBBBB@1 0 0 0 0 0\n0f0 0 0 0\n0 0 f20 0 0\n0 0 0\u00001 0 0\n0 0 0 0 f\u000020\n0 0 0 0 0 f\u000011\nCCCCCCA\n+0\nBBBBBB@P00 P10 P20 P30 P40 P50\nf\u00001P01f\u00001P11f\u00001P21f\u00001P31f\u00001P41f\u00001P51\nf\u00002P02f\u00002P12f\u00002P22f\u00002P32f\u00002P42f\u00002P52\n\u0000P03\u0000P13\u0000P23\u0000P33\u0000P33\u0000P53\nf2P04 f2P14 f2P24f2P34 f2P44 f2P54\nfP05 fP15 fP25fP35 fP45 fP551\nCCCCCCAn0\nBBBBBB@1 0 0 0 0 0\n0f\u000010 0 0 0\n0 0 f\u000020 0 0\n0 0 0\u00001 0 0\n0 0 0 0 f20\n0 0 0 0 0 f1\nCCCCCCA3\n77777750\nBBBBBB@p1\np2\np3\np4\np5\np61\nCCCCCCA;\nwhich is again equation (5.6). The expression of the diffusion coefficient is thus given byDiffusion coefficients for persistent random walks 19\n(5.7).\n6. Two-Step Memory Approximation revisited\nAs seen in the previous section, the symbolic computation of (5.4) quickly becomes tricky.\nHowever, an alternative to the above scheme can be found, provided that the walk has special\nsymmetries. Returning to (5.2), we write\nhv0\u0001vni=å\nv0å\ni1;:::;inv0\u0001Si1;:::;inv0P(Si1;:::;inv0jSi1;:::;in\u00001v0;Si1;:::;in\u00002v0)\n\u0002\u0001\u0001\u0001\u0002 P(Si1;i2v0jSi1v0;v0)p(v0;Si1v0); (6.1)\nwhere we introduce the compact notation Si\u0011RiT, and sequences in the exponent denote\nmultiple composition: Si1;:::;in\u0011Sin\u000eSin\u00001\u000e\u0001\u0001\u0001\u000e Si1, where each iktakes values between 1\nandz. Note that in general Sj\u000eSi6=Si+jwhen Tis non-trivial. The transition probabilities\nP(Sj2v0jSj1v0;Sj0v0)can be seen as matrix elements ˜Qj2\u0000j1;j1\u0000j0, so that (6.1) may be\nrewritten as\nhv0\u0001vni=å\nv0å\ni1;:::;inv0\u0001Si1;:::;inv0˜Qin;in\u00001\u0001\u0001\u0001˜Qi2;i1p(v0;Si1v0): (6.2)\nWe would like to rewrite this expression as a matrix product. However, this is in general\nnot possible, and further approximations are needed. Thus, assuming that the scalar product\nvn\u0001v0factorises as\nSi1;:::;inv0\u0001v0=n\nÕ\nk=1Sikv0\u0001v0; (6.3)\nand defining\nQi;j\u0011˜Qi;jSjv0\u0001v0=P(Sj;iv0jSjv0;v0)Sjv0\u0001v0; (6.4)\nequation (6.2) becomes\nhv0\u0001vni=å\nv0å\ni1;:::;inv0\u0001Sinv0Qin;in\u00001\u0001\u0001\u0001Qi2;i1p(v0;Si1v0);\n=zå\ni1;inV†\ninQn\u00001\nin;i1pi1; (6.5)\nwhere we have introduced the vectors Vin\u0011v0\u0001Sinv0andpi1\u0011p(v0;Si1v0). As can be seen,\nequation (6.5) has an appropriate matrix form and can easily be resummed over nto compute\nthe diffusion coefficient (2.1).\nSince Qis a z\u0002zmatrix, equation (6.5) is much easier to evaluate than (5.4). The\ntrouble is that equation (6.3) is in general incorrect, and turns out to be strictly valid only\nfor one-dimensional walks. Nonetheless, it may also be applied to higher-dimensional walks\nsatisfying special symmetry conditions. We consider the different geometries separately in the\nfollowing and discuss the conditions under which equation (6.5) can be applied. For higher-\ndimensional lattices, we recover by this simpler method the results obtained earlier under the\nrelevant symmetry assumptions.\n6.1. One-dimensional lattice\nThe result (5.17) follows from equation (6.5). Indeed, Rin;:::;i1v0\u0001v0=Rin+\u0001\u0001\u0001+i1v0\u0001v0=\u00061\naccording to the parity of in+\u0001\u0001\u0001+i1, and since this is also a property of the product\nÕn\nk=1Rikv0\u0001v0, we see that equation (6.3) is valid.Diffusion coefficients for persistent random walks 20\nThe vector pion the right-hand side of equation (6.5) is\np1=p(\u0000v;v) =1\u0000P00\n2(1\u0000P00+P10);\np2=p(v;v) =P10\n2(1\u0000P00+P10):(6.6)\nThe vector Vi, on the other hand, has components\nV1=\u00001;\nV2=1:(6.7)\nThe matrix elements Qi;jare defined according to equation (6.4),\nQ=\u0012\n\u0000P11P01\n\u0000P10P00\u0013\n=\u0012\nP10\u00001 1\u0000P00\n\u0000P10 P00\u0013\n: (6.8)\nConsidering equation (2.1) and plugging the above expressions into equation (6.5), we\nhave\nD2SMA =DNMAn\n1+4V†[I2\u0000Q]\u00001po\n; (6.9)\nand we recover equation (5.17).\n6.2. Two-dimensional honeycomb lattice\nConsider equation (6.3) in the case of a honeycomb lattice. The operation Sivis a clockwise\nrotation of vby angle\u0000p=3 ifi=1,p=3 ifi=2, or pifi=3. The operation Si1;:::;invis thus\na rotation by angle [2(i1+:::+in)\u00003n]p=3, and the scalar product\nSi1;:::;inv\u0001v=cos[2(i1+:::+in)\u00003n]p=3: (6.10)\nThis expression is, however, in general different from the product\nSinv\u0001v\u0001\u0001\u0001Si1v\u0001v=n\nÕ\nk=1cos[2(ik)\u00003]p=3: (6.11)\nThis is so, for instance, when n=2 and i1=i2=1, for which (6.10) yields \u00001=2, whereas\n(6.11) yields 1 =4.\nThere is however a special case under which the product structure that we seek can be\nretrieved, as follows. There are a priori nine transition probabilities P(Sj;ivjSjv;v). There\nare, however, a number of left–right symmetries in the system which reduce the number of\nindependent transition probabilities to three:\nP(S1;1vjS1v;v);P(S3;3vjS3v;v);P(S1;2vjS1v;v): (6.12)\nIn the event that the two probabilities P(S1;2vjS1v;v)andP(S1;1vjS1v;v)are equal,\nP(S1;2vjS1v;v) =P(S1;1vjS1v;v)\u0011Pss; (6.13)\nwhich is to say that forward–left and right scatterings are treated as identical events, and the\nnumber of independent parameters reduces to two, which we take to be P00andPss.Diffusion coefficients for persistent random walks 21\nIn this case, the expression of the diffusion coefficient can be obtained in a way similar\nto equation (5.17) for the one-dimensional lattice. This is so because\nSi1:::;in\u00001;1v\u0001v=cosn\n[2(i1+:::+in\u00001+1)\u00003n]p\n3o\n=cosp\n3cosn\n[2(i1+:::+in\u00001)\u00003(n\u00001)]p\n3o\n+sinp\n3sinn\n[2(i1+:::+in\u00001)\u00003(n\u00001)]p\n3o\n; (6.14)\nSi1:::;in\u00001;2v\u0001v=cosn\n[2(i1+:::+in\u00001+2)\u00003n]p\n3o\n=cosp\n3cosn\n[2(i1+:::+in\u00001)\u00003(n\u00001)]p\n3o\n\u0000sinp\n3sinn\n[2(i1+:::+in\u00001)\u00003(n\u00001)]p\n3o\n; (6.15)\nSi1:::;in\u00001;3v\u0001v=cosn\n[2(i1+:::+in\u00001+3)\u00003n]p\n3o\n=\u0000cosn\n[2(i1+:::+in\u00001)\u00003(n\u00001)]p\n3o\n: (6.16)\nThus, given the symmetry between forward–left and right scatterings, the two sine\ncontributions in equations (6.14) and (6.15) cancel, whereas the cosines add up to 1:\nSi1:::;in\u00001;1v\u0001v+Si1:::;in\u00001;2v\u0001v=Si1:::;in\u00001v\u0001v;\n=Si1:::;in\u00001v\u0001v(S1v\u0001v+S2v\u0001v);(6.17)\nSi1:::;in\u00001;3v\u0001v=\u0000Si1:::;in\u00001v\u0001v=Si1:::;in\u00001v\u0001vS3v\u0001v: (6.18)\nWe therefore retrieve an effective product structure, as in equation (6.3), and can compute the\ndiffusion coefficient using equation (6.5), with\nP=1\n3(2\u00002Pss\u0000P00)\u00121\n2(1\u0000P00)\n1\u00002Pss\u0013\n; (6.19)\nV=\u0012\n1\n\u00001\u0013\n; (6.20)\nand\nQ=\u0012\nPss 1=2P00\u00001=2\n1\u00002Pss\u0000P00\u0013\n: (6.21)\nWe obtain the expression of the diffusion coefficient for the symmetric [in the sense of\nequation (6.13)] two-step memory approximation on the honeycomb lattice:\nDs\n2SMA =DNMA3(1\u0000P00)\n(3\u00004Pss+P00)(2Pss+P00)\n(2\u00002Pss\u0000P00): (6.22)\nThis is equation (5.33).\n6.3. Two-dimensional square lattice\nFor the two-dimensional square lattice, recall that Tis the identity and the operation Siv=Riv\nis a anticlockwise rotation of vby angle ip=2, with i=0;:::; 3. The operation Si1;:::;invis thusDiffusion coefficients for persistent random walks 22\na rotation by angle (i1+\u0001\u0001\u0001+in)p=2. Equations similar to (6.14)–(6.16) hold§:\nRi1:::;inv\u0001v=cosh\n(i1+\u0001\u0001\u0001+in)p\n2i\n;\n=cosinp\n2cosh\n(i1+\u0001\u0001\u0001+in\u00001)p\n2i\n\u0000sininp\n2sinh\n(i1+\u0001\u0001\u0001+in\u00001)p\n2i\n;\n=Ri1:::;in\u00001v\u0001vRinv\u0001v\n\u0000(din;1\u0000din;3)sinh\n(i1+\u0001\u0001\u0001+in\u00001)p\n2i\n: (6.23)\nThe last term drops out provided\nP(Ri+1vjRiv;v) =P(Ri+3vjRiv;v): (6.24)\nUnder the additional assumption that\nP(R1+ivjR1v;v) =P(R3+ivjR3v;v); (6.25)\nwe retrieve an effective factorisation similar to that postulated in (6.3), and we can then use\n(6.5) to obtain the corresponding diffusion coefficient. We refer to equations (6.24) and (6.25)\nas defining a complete left–right symmetry. In this case, the invariant distribution is the\nsolution of\np(v;Riv) =å\njP(Ri+jvjRjv;v)p(Riv;Rjv); (6.26)\nå\nip(v;Riv) =1\n4: (6.27)\nWe solve these equations for p(v;v)andp(v;\u0000v), identifying p(v;R1v)andp(v;R3v), and\ndefine\nV=\u0012\n\u00001\n1\u0013\n;P=\u0012\np(v;\u0000v)\np(v;v)\u0013\n; (6.28)\nwith the transition matrix\nQ=\u0012\n\u0000P(vj\u0000v;v)P(\u0000vjv;v)\n\u0000P(\u0000vj\u0000v;v)P(vjv;v)\u0013\n=\u0012\n\u0000P22P02\n\u0000P20P00\u0013\n: (6.29)\nThe resulting expression of the diffusion coefficient is identical to (5.43).\nThe validity of this expression extends to d-dimensional orthogonal lattices under the\nsymmetry assumptions (6.24)–(6.25).\n7. Conclusions\nWe have shown that it is possible to find exact results for the diffusion coefficient of persistent\nrandom walks with two-step memory on regular lattices, by finding the matrix elements which\ngive the velocity auto-correlation function and then resumming then.\n§ Note that, in general, we have the decomposition\nRi1:::;inv\u0001v=å\nw1;:::;wn2f0;1gsgn(w1;:::;wn)Ri1v\u0001vw1\u0001\u0001\u0001Rinv\u0001vwn;\nwhere we introduced the notation vw=vifw=0 and vw=v?ifw=1, and the function sgn (w1;:::;wn) =\u00061,\ndepending on the sequence w1;:::;wn. Equation (6.5) would then be replaced by a more complicated expression\ninvolving the mixed products of two matrices P(Rj+ivjRjv;v)Rjv\u0001vandP(Rj+ivjRjv;v)Rjv\u0001v?.Diffusion coefficients for persistent random walks 23\nWe have applied the results obtained here to approximate the diffusion coefficients of\ncertain periodic billiard tables in [11].\nThe extension to lattice random walks with longer memory is possible, albeit difficult for\nobvious technical reasons. Finally, we remark that the extension to lattices in three dimensions\nis not direct, since in that case, additional information must be specified in order to uniquely\ndefine relative directions [12].\nAcknowledgments\nThe authors thank Hern ´an Larralde for helpful discussions. This research benefitted from the\njoint support of FNRS (Belgium) and CONACYT (Mexico) through a bilateral collaboration\nproject. The work of TG is financially supported by the Belgian Federal Government under\nthe Inter-university Attraction Pole project NOSY P06/02. TG is financially supported by the\nFonds de la Recherche Scientifique F.R.S.-FNRS. DPS acknowledges financial support from\nDGAPA-UNAM project IN105209, and the hospitality of the Universit ´e Libre de Bruxelles,\nwhere most of this work was carried out. TG acknowledges the hospitality of the Weizmann\nInstitute of Science, where part of this work was completed.\nAppendix A. 2SMA on a honeycomb lattice\nIn analogy to equation (5.13), we may write\nMn\u00110\nBBBBBBBBBBBBBBB@a(n)\n1a(n)\n2a(n)\n3a(n)\n4a(n)\n5a(n)\n6a(n)\n7a(n)\n8a(n)\n9\nc(n)\n9c(n)\n7c(n)\n8c(n)\n3c(n)\n1c(n)\n2c(n)\n6c(n)\n4c(n)\n5\nb(n)\n5b(n)\n6b(n)\n4b(n)\n8b(n)\n9b(n)\n7b(n)\n2b(n)\n3b(n)\n1\nb(n)\n1b(n)\n2b(n)\n3b(n)\n4b(n)\n5b(n)\n6b(n)\n7b(n)\n8b(n)\n9\na(n)\n9a(n)\n7a(n)\n8a(n)\n3a(n)\n1a(n)\n2a(n)\n6a(n)\n4a(n)\n5\nc(n)\n5c(n)\n6c(n)\n4c(n)\n8c(n)\n9c(n)\n7c(n)\n2c(n)\n3c(n)\n1\nc(n)\n1c(n)\n2c(n)\n3c(n)\n4c(n)\n5c(n)\n6c(n)\n7c(n)\n8c(n)\n9\nb(n)\n9b(n)\n7b(n)\n8b(n)\n3b(n)\n1b(n)\n2b(n)\n6b(n)\n4b(n)\n5\na(n)\n5a(n)\n6a(n)\n4a(n)\n8a(n)\n9a(n)\n7a(n)\n2a(n)\n3a(n)\n11\nCCCCCCCCCCCCCCCA: (A.1)\nWe have the three sets of equations\n0\nB@a(n)\n1a(n)\n5a(n)\n9\nc(n)\n1c(n)\n5c(n)\n9\nb(n)\n1b(n)\n5b(n)\n91\nCA=0\n@P00P10P20\nP01P11P21\nP02P12P221\nA0\nB@a(n\u00001)\n1a(n\u00001)\n5a(n\u00001)\n9\nc(n\u00001)\n9c(n\u00001)\n1c(n\u00001)\n5\nb(n\u00001)\n5b(n\u00001)\n9b(n\u00001)\n11\nCA; (A.2)\n0\nB@a(n)\n2a(n)\n6a(n)\n7\nc(n)\n2c(n)\n6c(n)\n7\nb(n)\n2b(n)\n6b(n)\n71\nCA=0\n@P00P10P20\nP01P11P21\nP02P12P221\nA0\nB@a(n\u00001)\n2a(n\u00001)\n6a(n\u00001)\n7\nc(n\u00001)\n7c(n\u00001)\n2c(n\u00001)\n6\nb(n\u00001)\n6b(n\u00001)\n7b(n\u00001)\n21\nCA; (A.3)\n0\nB@a(n)\n3a(n)\n4a(n)\n8\nc(n)\n3c(n)\n4c(n)\n8\nb(n)\n3b(n)\n4b(n)\n81\nCA=0\n@P00P10P20\nP01P11P21\nP02P12P221\nA0\nB@a(n\u00001)\n3a(n\u00001)\n4a(n\u00001)\n8\nc(n\u00001)\n8c(n\u00001)\n3c(n\u00001)\n4\nb(n\u00001)\n4b(n\u00001)\n8b(n\u00001)\n31\nCA: (A.4)\nProceeding with our analogy, we seek linear combinations of the elements in the rows of the\nmatrices on the left-hand side of the above equations, so as to obtain a single matrix equationDiffusion coefficients for persistent random walks 24\nsimilar to equation (5.15). Considering the elements in equation (A.2), we write\n0\nB@a(n)\n1+fa(n)\n5+f2a(n)\n9\nfc(n)\n1+f2c(n)\n5+c(n)\n9\nf2b(n)\n1+b(n)\n5+fb(n)\n91\nCA=\n0\n@P00 P10 P20\nfP01fP11fP21\nf2P02f2P12f2P221\nA0\nB@a(n\u00001)\n1+fa(n\u00001)\n5+f2a(n\u00001)\n9\nfc(n\u00001)\n1+f2c(n\u00001)\n5+c(n\u00001)\n9\nf2b(n\u00001)\n1+b(n\u00001)\n5+fb(n\u00001)\n91\nCA: (A.5)\nComparing with equation (A.2), we infer\nf3=1,8\n><\n>:f=1;\nf=exp(2ip=3) =\u00001\u0000ip\n3\n2;\nf=exp(\u00002ip=3) =\u00001+ip\n3\n2:(A.6)\nApplying the same procedure to equations (A.3)–(A.4), we obtain\n0\nB@fa(n)\n2+f2a(n)\n6+a(n)\n7\nf2c(n)\n2+c(n)\n6+fc(n)\n7\nb(n)\n2+fb(n)\n6+f2b(n)\n71\nCA=\n0\n@P00 P10 P20\nfP01fP11fP21\nf2P02f2P12f2P221\nA0\nB@fa(n\u00001)\n2+f2a(n\u00001)\n6+a(n\u00001)\n7\nf2c(n\u00001)\n2+c(n\u00001)\n6+fc(n\u00001)\n7\nb(n\u00001)\n2+fb(n\u00001)\n6+f2b(n\u00001)\n71\nCA: (A.7)\n0\nB@f2a(n)\n3+a(n)\n4+fa(n)\n8\nc(n)\n3+fc(n)\n4+f2c(n)\n8\nfb(n)\n3+f2b(n)\n4+b(n)\n81\nCA=\n0\n@P00 P10 P20\nfP01fP11fP21\nf2P02f2P12f2P221\nA0\nB@f2a(n\u00001)\n3+a(n\u00001)\n4+fa(n\u00001)\n8\nc(n\u00001)\n3+fc(n\u00001)\n4+f2c(n\u00001)\n8\nfb(n\u00001)\n3+f2b(n\u00001)\n4+b(n\u00001)\n81\nCA: (A.8)\nThe system of equations (A.5), (A.7), (A.8) reduces to the single recursive matrix\nequation\n0\nB@a(n)\n1+fa(n)\n5+f2a(n)\n9fa(n)\n2+f2a(n)\n6+a(n)\n7f2a(n)\n3+a(n)\n4+fa(n)\n8\nfc(n)\n1+f2c(n)\n5+c(n)\n9f2c(n)\n2+c(n)\n6+fc(n)\n7c(n)\n3+fc(n)\n4+f2c(n)\n8\nf2b(n)\n1+b(n)\n5+fb(n)\n9b(n)\n2+fb(n)\n6+f2b(n)\n7fb(n)\n3+f2b(n)\n4+b(n)\n81\nCA\n=0\n@P00 P10 P20\nfP01fP11fP21\nf2P02f2P12f2P221\nA\n\u00020\nB@a(n\u00001)\n1+fa(n\u00001)\n5+f2a(n\u00001)\n9:::f2a(n\u00001)\n3+a(n\u00001)\n4+fa(n\u00001)\n8\nfc(n\u00001)\n1+f2c(n\u00001)\n5+c(n\u00001)\n9:::c(n\u00001)\n3+fc(n\u00001)\n4+f2c(n\u00001)\n8\nf2b(n\u00001)\n1+b(n\u00001)\n5+fb(n\u00001)\n9:::fb(n\u00001)\n3+f2b(n\u00001)\n4+b(n\u00001)\n81\nCA;\n=0\n@P00 P10 P20\nfP01fP11fP21\nf2P02f2P12f2P221\nAn\u000010\n@P00 fP10f2P20\nfP01f2P11P21\nf2P02P12 fP221\nA;Diffusion coefficients for persistent random walks 25\n=0\n@P00 P10 P20\nfP01fP11fP21\nf2P02f2P12f2P221\nAn0\n@1 0 0\n0f0\n0 0 f21\nA: (A.9)\nAppendix B. 2SMA on a square lattice\nIn analogy to equations (A.2)–(A.4), there are 64 different entries among the 256 elements of\nMn, which can be obtained through the set of equations\n0\nBBB@a(n)\n1a(n)\n6a(n)\n11a(n)\n16\nd(n)\n1d(n)\n6d(n)\n11d(n)\n16\nc(n)\n1c(n)\n6c(n)\n11c(n)\n16\nb(n)\n1b(n)\n6b(n)\n11b(n)\n161\nCCCA=0\nBB@P00P10P20P30\nP01P11P21P31\nP02P12P22P32\nP03P13P23P331\nCCA0\nBBB@a(n\u00001)\n1a(n\u00001)\n6a(n\u00001)\n11a(n\u00001)\n16\nd(n\u00001)\n16d(n\u00001)\n1d(n\u00001)\n6d(n\u00001)\n11\nc(n\u00001)\n11c(n\u00001)\n16c(n\u00001)\n1c(n\u00001)\n6\nb(n\u00001)\n6b(n\u00001)\n11b(n\u00001)\n16b(n\u00001)\n11\nCCCA;(B.1)\n0\nBBB@a(n)\n2a(n)\n7a(n)\n12a(n)\n13\nd(n)\n2d(n)\n7d(n)\n12d(n)\n13\nc(n)\n2c(n)\n7c(n)\n12c(n)\n13\nb(n)\n2b(n)\n7b(n)\n12b(n)\n131\nCCCA=0\nBB@P00P10P20P30\nP01P11P21P31\nP02P12P22P32\nP03P13P23P331\nCCA0\nBBB@a(n\u00001)\n2a(n\u00001)\n7a(n\u00001)\n12a(n\u00001)\n13\nd(n\u00001)\n13d(n\u00001)\n2d(n\u00001)\n7d(n\u00001)\n12\nc(n\u00001)\n12c(n\u00001)\n13c(n\u00001)\n2c(n\u00001)\n7\nb(n\u00001)\n7b(n\u00001)\n12b(n\u00001)\n13b(n\u00001)\n21\nCCCA;(B.2)\n0\nBBB@a(n)\n3a(n)\n8a(n)\n9a(n)\n14\nd(n)\n3d(n)\n8d(n)\n9d(n)\n14\nc(n)\n3c(n)\n8c(n)\n9c(n)\n14\nb(n)\n3b(n)\n8b(n)\n9b(n)\n141\nCCCA=0\nBB@P00P10P20P30\nP01P11P21P31\nP02P12P22P32\nP03P13P23P331\nCCA0\nBBB@a(n\u00001)\n3a(n\u00001)\n8a(n\u00001)\n9a(n\u00001)\n14\nd(n\u00001)\n14d(n\u00001)\n3d(n\u00001)\n8d(n\u00001)\n9\nc(n\u00001)\n9c(n\u00001)\n14c(n\u00001)\n3c(n\u00001)\n8\nb(n\u00001)\n8b(n\u00001)\n9b(n\u00001)\n14b(n\u00001)\n31\nCCCA;(B.3)\n0\nBBB@a(n)\n4a(n)\n5a(n)\n10a(n)\n15\nd(n)\n4d(n)\n5d(n)\n10d(n)\n15\nc(n)\n4c(n)\n5c(n)\n10c(n)\n15\nb(n)\n4b(n)\n5b(n)\n10b(n)\n151\nCCCA=0\nBB@P00P10P20P30\nP01P11P21P31\nP02P12P22P32\nP03P13P23P331\nCCA0\nBBB@a(n\u00001)\n4a(n\u00001)\n5a(n\u00001)\n10a(n\u00001)\n15\nd(n\u00001)\n15d(n\u00001)\n4d(n\u00001)\n5d(n\u00001)\n10\nc(n\u00001)\n10c(n\u00001)\n15c(n\u00001)\n4c(n\u00001)\n5\nb(n\u00001)\n5b(n\u00001)\n10b(n\u00001)\n15b(n\u00001)\n41\nCCCA:(B.4)\nCombining these quantities, we let\nA(n)\n1(k)\u0011a(n)\n1+fka(n)\n6+f2\nka(n)\n11+f3\nka(n)\n16;\nA(n)\n5(k)\u0011f3\nka(n)\n4+a(n)\n5+fka(n)\n10+f2\nka(n)\n15;\nA(n)\n9(k)\u0011f2\nka(n)\n3+f3\nka(n)\n8+a(n)\n9+fka(n)\n14;\nA(n)\n13(k)\u0011fka(n)\n2+f2\nka(n)\n7+f3\nka(n)\n12+a(n)\n13;\nB(n)\n2(k)\u0011b(n)\n2+fkb(n)\n7+f2\nkb(n)\n12+f3\nkb(n)\n13;\nB(n)\n6(k)\u0011f3\nkb(n)\n1+b(n)\n6+fkb(n)\n11+f2\nkb(n)\n16;\nB(n)\n10(k)\u0011f2\nkb(n)\n4+f3\nkb(n)\n5+b(n)\n10+fkb(n)\n15;\nB(n)\n14(k)\u0011fkb(n)\n3+f2\nkb(n)\n8+f3\nkb(n)\n9+b(n)\n14;\nC(n)\n3(k)\u0011c(n)\n3+fkc(n)\n8+f2\nkc(n)\n9+f3\nkc(n)\n14;\nC(n)\n7(k)\u0011f3\nkc(n)\n2+c(n)\n7+fkc(n)\n12+f2\nkc(n)\n13;\nC(n)\n11(k)\u0011f2\nkc(n)\n1+f3\nkc(n)\n6+c(n)\n11+fkc(n)\n16;\nC(n)\n15(k)\u0011fkc(n)\n4+f2\nkc(n)\n5+f3\nkc(n)\n10+c(n)\n15;Diffusion coefficients for persistent random walks 26\nD(n)\n4(k)\u0011d(n)\n4+fkd(n)\n5+f2\nkd(n)\n10+f3\nkd(n)\n15;\nD(n)\n8(k)\u0011f3\nkd(n)\n3+d(n)\n8+fkd(n)\n9+f2\nkd(n)\n14;\nD(n)\n12(k)\u0011f2\nkd(n)\n2+f3\nkd(n)\n7+d(n)\n12+fkd(n)\n13;\nD(n)\n16(k)\u0011fkd(n)\n1+f2\nkd(n)\n6+f3\nkd(n)\n11+d(n)\n16;\nin terms of which we have\n0\nBBB@A(n)\n1(k)A(n)\n13(k)A(n)\n9(k)A(n)\n5(k)\nD(n)\n16(k)D(n)\n12(k)D(n)\n8(k)D(n)\n4(k)\nC(n)\n11(k)C(n)\n7(k)C(n)\n3(k)C(n)\n15(k)\nB(n)\n6(k)B(n)\n2(k)B(n)\n14(k)B(n)\n10(k)1\nCCCA\n=0\nBB@P00 P10 P20 P30\nfkP01fkP11fkP21fkP31\nf2\nkP02f2\nkP12f2\nkP22f2\nkP32\nf3\nkP03f3\nkP13f3\nkP23f3\nkP331\nCCA0\nBBB@A(n\u00001)\n1(k)A(n\u00001)\n13(k)A(n\u00001)\n9(k)A(n\u00001)\n5(k)\nD(n\u00001)\n16(k)D(n\u00001)\n12(k)D(n\u00001)\n8(k)D(n\u00001)\n4(k)\nC(n\u00001)\n11(k)C(n\u00001)\n7(k)C(n\u00001)\n3(k)C(n\u00001)\n15(k)\nB(n\u00001)\n6(k)B(n\u00001)\n2(k)B(n\u00001)\n14(k)B(n\u00001)\n10(k)1\nCCCA;\n=0\nBB@P00 P10 P20 P30\nfkP01fkP11fkP21fkP31\nf2\nkP02f2\nkP12f2\nkP22f2\nkP32\nf3\nkP03f3\nkP13f3\nkP23f3\nkP331\nCCAn\u000010\nBBB@A(1)\n1(k)A(1)\n13(k)A(1)\n9(k)A(1)\n5(k)\nD(1)\n16(k)D(1)\n12(k)D(1)\n8(k)D(1)\n4(k)\nC(1)\n11(k)C(1)\n7(k)C(1)\n3(k)C(1)\n15(k)\nB(1)\n6(k)B(1)\n2(k)B(1)\n14(k)B(1)\n10(k)1\nCCCA;\n=0\nBB@P00 P10 P20 P30\nfkP01fkP11fkP21fkP31\nf2\nkP02f2\nkP12f2\nkP22f2\nkP32\nf3\nkP03f3\nkP13f3\nkP23f3\nkP331\nCCAn\u000010\nBB@P00fkP10f2\nkP20f3\nkP30\nfkP01f2\nkP11f3\nkP21P31\nf2\nkP02f3\nkP12P22fkP32\nf3\nkP03P13fkP23f2\nkP331\nCCA;\n=0\nBB@P00 P10 P20 P30\nfkP01fkP11fkP21fkP31\nf2\nkP02f2\nkP12f2\nkP22f2\nkP32\nf3\nkP03f3\nkP13f3\nkP23f3\nkP331\nCCAn0\nBB@1 0 0 0\n0fk0 0\n0 0 f2\nk0\n0 0 0 f3\nk1\nCCA; (B.5)\nprovided\nf4\nk=1,fk=exp(ikp=2);k=0;1;2;3; (B.6)\nWe have the following identities,\n2[a(n)\n1\u0000a(n)\n11] =A(n)\n1(1)+A(n)\n1(3);\n2[a(n)\n5\u0000a(n)\n15] =A(n)\n5(1)+A(n)\n5(3);\n2[a(n)\n9\u0000a(n)\n3] =A(n)\n9(1)+A(n)\n9(3);\n2[a(n)\n13\u0000a(n)\n7] =A(n)\n13(1)+A(n)\n13(3);\n2[b(n)\n2\u0000b(n)\n12] =B(n)\n2(1)+B(n)\n2(3);\n2[b(n)\n6\u0000b(n)\n16] =B(n)\n6(1)+B(n)\n6(3);\n2[b(n)\n10\u0000b(n)\n4] =B(n)\n10(1)+B(n)\n10(3);\n2[b(n)\n14\u0000b(n)\n8] =B(n)\n14(1)+B(n)\n14(3);\n2[c(n)\n3\u0000c(n)\n9] =C(n)\n3(1)+C(n)\n3(3);Diffusion coefficients for persistent random walks 27\n2[c(n)\n7\u0000c(n)\n13] =C(n)\n7(1)+C(n)\n7(3);\n2[c(n)\n11\u0000c(n)\n1] =C(n)\n11(1)+C(n)\n11(3);\n2[c(n)\n15\u0000c(n)\n5] =C(n)\n15(1)+C(n)\n15(3);\n2[d(n)\n4\u0000d(n)\n10] =D(n)\n4(1)+D(n)\n4(3);\n2[d(n)\n8\u0000d(n)\n14] =D(n)\n8(1)+D(n)\n8(3);\n2[d(n)\n12\u0000d(n)\n2] =D(n)\n12(1)+D(n)\n12(3);\n2[d(n)\n16\u0000d(n)\n6] =D(n)\n16(1)+D(n)\n16(3);\nwhich, using equation (B.5) yield the velocity auto-correlation (5.37).\nReferences\n[1] Weiss G H 1994, Aspects and Applications of the Random Walk (North-Holland Publishing Co., Amsterdam).\n[2] Haus J W and Kehr K W 1987, Diffusion in regular and disordered lattices, Phys. Rep. 150, 263–406.\n[3] F ¨urth R 1920, Die Brownsche Bewegung bei Ber ¨ucksichtigung einer Persistenz der Bewegungsrichtung. Mit\nAnwendungen auf die Bewegung lebender Infusorien, Zeit. f. Physik 2, 244.\n[4] Taylor G I 1922, Diffusion by continuous movements, Proc. London Math. Soc. 20, 196.\n[5] Kuhn W 1934, Kolloid Z. 68, 2; 1936 76, 258.\n[6] Manning J R 1959, Correlation effects in impurity diffusion, Phys. Rev. 116, 819.\n[7] Weiss G H 2002, Some applications of persistent random walks and the telegrapher’s equation, Physica A 311,\n381.\n[8] Montroll EW 1950, Markoff chains and excluded volume effect in polymer chains J. Chem. Phys. 18, 734.\n[9] Bender E A and Richmond L B, 1984 Correlated random walks Ann. Prob. 12, 274.\n[10] Renshaw E and Henderson R 1981, The correlated random walk J. Appl. Probab. 18, 403; 1994, The general\ncorrelated random walk, J. Appl. Probab. 31, 869.\n[11] Gilbert T and Sanders DP 2009, Persistence effects in deterministic diffusion. Preprint, arXiv:0908.0600v1.\n[12] Larralde H 1997, Transport properties of a two-dimensional “chiral” persistent random walk Phys. Rev. E 56,\n5004.\n[13] Fink T M and Mao Y 1999, Designing tie knots using random walks, Nature 398, 31."    },    {        "title": "1812.07244v2.Thermal_gradient_driven_domain_wall_dynamics.pdf",        "content": "arXiv:1812.07244v2  [cond-mat.mes-hall]  26 May 2019Thermal gradient driven domain wall dynamics\nM. T. Islam,1,2X. S. Wang,3,4and X. R. Wang1,5,∗\n1Physics Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong\n2Physics Discipline, Khulna University, Khulna, Banglades h\n3School of Electronic Science and Engineering and State Key L aboratory of Electronic Thin Film and Integrated Devices,\nUniversity of Electronic Science and Technology of China, C hengdu 610054, China\n4Center for Quantum Spintronics, Department of Physics,\nNorwegian University of Science and Technology, NO-7491 Tr ondheim, Norway\n5HKUST Shenzhen Research Institute, Shenzhen 518057, China\nThe issue of whether a thermal gradient acts like a magnetic fi eld or an electric current in the\ndomain wall (DW) dynamics is investigated. Broadly speakin g, magnetization control knobs can\nbe classified as energy-driving or angular-momentum drivin g forces. DW propagation driven by a\nstatic magnetic field is the best known example of the former i n which the DW speed is proportional\nto the energy dissipation rate, and the current-driven DW mo tion is an example of the latter. Here\nwe show that DW propagation speed driven by a thermal gradien t can be fully explained as the\nangular momentum transfer between thermally generated spi n current and DW. We found DW-\nplane rotation speed increases as DW width decreases. Both D W propagation speed along the wire\nand DW-plane rotation speed around the wire decrease with th e Gilbert damping. These facts\nare consistent with the angular momentum transfer mechanis m, but are distinct from the energy\ndissipation mechanism. We further show that magnonic spin- transfer torque (STT) generated by a\nthermal gradient has both damping-like and field-like compo nents. By analyzing DW propagation\nspeed and DW-plane rotational speed, the coefficient ( β) of the field-like STT arising from the non-\nadiabatic process, is obtained. It is found that βdoes not depend on the thermal gradient; increases\nwith uniaxial anisotropy K/bardbl(thinner DW); and decreases with the damping, in agreement w ith the\nphysical picture that a larger damping or a thicker DW leads t o a better alignment between the\nspin-current polarization and the local magnetization, or a better adiabaticity.\nI. INTRODUCTION\nManipulating domain walls (DW) in magnetic nanos-\ntructures has attracted much attention because of its po-\ntential applications in data storage technology [ 1] and\nlogic gates [ 2]. The traditional DW control knobs,\nnamely magnetic fields and spin-polarized currents, have\ncertain drawbacks in applications. In the magnetic-field-\ndriven DW motion, energy dissipation is the main cause\nofDWpropagationwhosespeedisproportionaltotheen-\nergy dissipation rate [ 3,4], and the magnetic field tends\ntodestroyunfavorabledomainsandDWs, insteadofdriv-\ning a series of DWs synchronously [ 5–7]. An electrical\ncurrent drives a DW to move mainly through the angu-\nlar momentum transfer so that it pushes multiple DWs\n[8–11] in the same direction. To achieve a useful DW\nspeed, it requires high electrical current densities that\nresult in a Joule heating problem [ 12–14]. To avoid these\nproblems, spin-wave spin current has been proposed as a\nmoreenergy-efficientcontrolparameter[ 15–18]. Thermal\ngradient, a way to generate spin-wave spin current, is an\nalternative control knob of the DW motion. The inves-\ntigation on thermal-gradient-driven domain wall motion\nis meaningful not only for conventional applications, but\nalso for the understanding of spin wave and domain wall\ndynamics [ 16,17,20–23], as well as for possible recycling\nof waste heat [ 19,24].\n∗[Corresponding author:]phxwan@ust.hkTo understand the mechanism behind thermal-\ngradient-drivenDWdynamics, therearemicroscopicthe-\nories [15–17,25,26] and macroscopic thermodynamic\ntheories [ 21,22]. Briefly speaking, the microscopic theo-\nries suggest that magnons populated in the hotter region\ndiffuses to the colder region to form a magnon spin cur-\nrent. The magnon spin currentpassesthrough a DWand\nexerts a torque on the DW by transferring spin angular\nmomentum to the DW. Thus, magnons drive the DW\npropagating toward the hotter region of the nanowire,\nopposite to the magnon current direction [ 15,16,18].\nThe thermodynamic theories anticipate that a thermal\ngradient generates an entropy force which always drives\nthe DW towards the hotter region in order to minimize\nthe system free energy. The macroscopic theories do not\nprovide any microscopic picture about DW dynamics al-\nthough a thermal gradient is often considered as an effec-\ntive magnetic field to estimate DW speed [ 21,22] from\nfield-driven DW theories. Thus, one interesting issue is\nwhether a thermal gradient in DW dynamics acts like a\nmagnetic field or an electric current. DW propagation\nspeed should be sensitive to both DW width and types\nof a DW (transverse DW) under an energy-driving force\nwhile the speed should be insensitive to the DW and DW\nstructure in the angular-momentum-driving force. This\nis the focus of the current work.\nIn this paper, we investigate DW motion along a uni-\naxial wire with the easy axis along the wire direction\nunder a thermal gradient. We found that the DW al-\nways propagates to the hotter region with an accom-\npanied DW-plane rotation. DW propagation speed and2\nz\nxy\nFIG. 1. Schematic diagram of a uniaxial magnetic nanowire\nwith a head-to-head DW at the center under a thermal gra-\ndient∇T. Black (white) color represents colder (hotter) end\nof the sample.\nDW-plane rotation speed increases as the magnetic easy-\naxis anisotropy and damping decreases. We show that\nDW motion can be attributed to the angular momen-\ntum transfer between magnonic spin current and the\nDW. Thus, we conclude that a thermal gradient in-\nteracts with DW through angular-momentum transfer\nrather than through energy dissipation. Similar to an\nelectric current [ 27], a thermal gradient can generate\nboth damping-like (or adiabatic) STT and field-like (or\nnon-adiabatic) STT. From the damping-dependence and\nanisotropy-dependence of the average DW velocity and\nDW-plane rotation angular velocity, we extract field-like\nSTT coefficient ( β). It is found that βis independent\nof thermal gradient; is bigger for a thinner DW; and de-\ncreases with the damping coefficient. We also show that\nin the presence of a weak hard-axis anisotropy perpen-\ndicular to the wire, the DW still undergoes a rotating\nmotion. The DW propagation speed increases slightly\nwhile the DW-plane rotation speed decreases with the\nstrength of the hard-axis anisotropy.\nII. MODEL AND METHOD\nWe consider a uniaxial nanowire of length Lxand\ncross-section Ly×Lzalong the x-axis (easy axis) with\na head-to-head DW at the center, as shown in Fig. 1.\nLy,Lzis much smaller than the DW width ∆, and ∆\nis much smaller than Lx. A thermal gradient is applied\nalong the wire. The highest temperature is far below\nthe Curie temperature Tc. The magnetization dynam-\nics is governed by the stochastic Landau-Lifshitz-Gilbert\n(LLG) equation [ 28,29],\ndm\ndt=−γm×(Heff+hth)+αm×∂m\n∂t,(1)\nwherem=M/MsandMsare respectively the magne-\ntization direction and the saturation magnetization. α\nis the Gilbert damping constant and γis the gyromag-netic ratio. Heff=2A\nµ0Ms/summationtext\nσ∂2m\n∂x2σ+2K/bardbl\nµ0Msmxˆx+hdipoleis\nthe effective field, where Ais the exchange constant, xσ\n(σ= 1,2,3) denote Cartesian coordinates x,y,z,K/bardblis\nthe easy-axis anisotropy, and hdipoleis the dipolar field.\nhthis the stochastic thermal field.\nThe stochastic LLG equation is solved numerically by\nMUMAX3 package [ 30] in which we use adaptive Heun\nsolver. To balance stability and efficiency, we choose the\ntime step 10−14s with the cell size (2 ×2×2) nm3. Mag-\nnetic charges at the two ends of the wire are removed to\navoid their attraction to the DW. The saturation mag-\nnetization Ms= 8×105A/m and exchange constant\nA= 13×10−12J/m are used to mimic permalloy in\nour simulations. The thermal field follows the Gaussian\nprocess characterized by following statistics [ 31]\n/angb∇acketlefthth,ip(t)/angb∇acket∇ight= 0,\n/angb∇acketlefthth,ip(t)hth,jq(t+∆t)/angb∇acket∇ight=2kBTiαi\nγµ0Msa3δijδpqδ(∆t),(2)\nwhereiandjdenote the micromagnetic cells, and p,q\nrepresent the Cartesian components of the thermal field.\nTiandαiare respectively temperature and the Gilbert\ndamping at cell i, andais the cell size. kBis the Boltz-\nmann constant [ 28]. The numerical results presented in\nthis study are averaged over 15 random configurations\n(for DW velocity) and 4000-5000 random configurations\n(for spin current).\nUnderthethermalgradient ∇xT,magnetizationatdif-\nferent positions deviate from their equilibrium directions\ndifferently and small transverse components myandmz\nare generated. The transverse components vary spatial-\ntemporally and depend on the local temperature. This\nvariation generates a magnonic spin current [ 16]. This\nmagnonic spin current can interact with spin textures\nsuch as DWs. In the absence of damping (the thermal\nfield also vanishes), the spin currentalong the xdirection\ncan be defined from the spin continuity equation derived\nfrom Eq. ( 1) as follows [ 15],\n∂m\n∂t=−1\n1+α2m׈xmxK/bardbl−∂J\n∂x,(3)\nwhere\nJ(x) =2γA\nµ0Msm×∂m\n∂x, (4)\nis the spin current density along x-direction due to the\nexchangeinteraction. J(x) can be numerically calculated\n[15,23]. In the presence of damping as well as the ther-\nmal field, the contribution of the damping term and the\nthermal term is proportional to α, which is relatively\nsmall. More importantly, according to the fluctuation-\ndissipation theorem [ 28], the damping term and the ther-\nmal term should cancel each other after average over a\nlong time. Since the time scale of DW dynamics is much\nlonger than the thermal fluctuation, the combined con-\ntribution of damping and thermal terms should be very\nsmall.3\n/s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s52/s56/s49/s50/s49/s54/s50/s48\n/s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s50/s52/s54/s56/s49/s48/s49/s50/s45/s56/s48/s48 /s45/s52/s48/s48 /s48 /s52/s48/s48 /s56/s48/s48/s45/s48/s46/s48/s50/s45/s48/s46/s48/s49/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50\n/s32/s32\n/s32/s118\n/s115/s105/s109/s117\n/s32/s118\n/s99/s117/s114/s114/s101/s110/s116/s118 /s32/s40/s109/s47/s115/s41\n/s120/s84 /s32/s40/s75/s47/s110/s109/s41/s32/s32/s32\n/s120/s84 /s32/s40/s75/s47/s110/s109/s41\n/s32/s32\n/s48/s46/s48/s55/s32 /s75/s47/s110/s109\n/s48/s46/s49 /s75/s47/s110/s109\n/s48/s46/s49/s53/s32 /s75/s47/s110/s109\n/s48/s46/s50/s32 /s75/s47/s110/s109\n/s48/s46/s50/s53/s32 /s75/s47/s110/s109\n/s48/s46/s51/s32 /s75/s47/s110/s109/s74\n/s116/s111/s116/s40/s120/s41/s40\n/s115/s41/s41\n/s120 /s32/s40/s110/s109/s41\n/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s49/s52/s49/s54/s49/s56/s50/s48/s50/s50/s40/s100/s41/s40/s98/s41/s40/s97/s41\n/s32\n/s75 /s32/s40/s49/s48/s52\n/s32/s74/s47/s109/s51\n/s41/s118 /s32/s40/s109/s47/s115/s41/s40/s99/s41/s100 /s47/s100/s116/s32/s40/s100/s101/s103/s47/s110/s115/s41\n/s100 /s47/s100/s116/s32/s40/s100/s101/s103/s47/s110/s115/s41\n/s51/s54/s51/s57/s52/s50\n/s32\nFIG. 2. (a) The spatial dependence of spin current densities\nJtot(x) for various ∇xT. The DW center is chosen as x=\n0. (b) Thermal gradient dependence of DW velocity vsimu\nfrom micromagnetic simulations (open squares) and vcurrent\ncomputedfrom total spin current(solid squares). (c)Therm al\ngradient dependence of DW-plane rotation angular velocity\n(squares). In (a)(b)(c) model parameters are Lx= 2048 nm,\nLy=Lz= 4 nm, α= 0.004 and K/bardbl= 5×105J/m3. (d)\nvsimu(solid squares) and dφ/dt(open squares) as a function\nofK⊥forLx= 1024 nm and ∇xT= 0.5 K/nm.\nIntegrating the x−component of Eq. ( 3) over a space\nenclosed the DW in the center and noticing the absence\nof the first term on the right, we have\nvcurrent=1\n2/integraldisplayLx/2\n−Lx/2∂mx\n∂tdx\n=−2γA\nµ0Ms/bracketleftbig1\n2(Jx|left−Jx|right)].(5)\nwhere we have assumed the fluctuations in the domains\nare small and the DW is not far from a symmetric one.\nJx|left,Jx|rightmean the x-components of the total spin\ncurrent on the left and right sides of the DW. The equa-\ntion clearly shows that the DW propagates opposite to\nthe spin current. This is the theoretical DW velocity un-\nder the assumption of angular momentum conservation,\nand it will be compared with the directly simulated DW\nvelocity below.\nIII. RESULTS\nA. Average spin current and DW velocity\nTosubstantiateourassertionthatDWpropagationun-\nder a thermal gradient is through angular-momentum ef-\nfect instead of energy effect, we would like to compare\nthe DW velocity obtained from micromagnetic simula-\ntions and that obtained from total spin current based on/s49 /s50 /s51 /s52 /s53 /s54 /s55/s56/s49/s50/s49/s54/s50/s48\n/s52/s56/s49/s50/s49/s54/s50/s48\n/s32/s118\n/s115/s105/s109/s117\n/s32/s118\n/s99/s117/s114/s114/s101/s110/s116\n/s100 /s47/s100/s116/s32/s40/s100/s101/s103/s47/s110/s115/s41/s32/s118 /s32/s40/s109/s47/s115/s41\n/s32/s76\n/s120/s61/s50/s48/s52/s56/s32/s110/s109\nFIG. 3. Damping αdependence of the DW dynamics: vsimu\n(Open squares); vcurrent(solid squares ); and dφ/dt(solid\ncircles). Model parameters are ∇xT= 0.2 K/nm, K/bardbl= 5×\n105J/m3,Lx= 2048 nm and Ly=Lz= 4 nm.\nEq. (5). Eq. (4) is used to calculate Jx(x). Fig.2(a) is\nspatial distribution of the ensemble averaged Jx(x) with\nDW atx= 0 for various thermal gradients. The sud-\nden sign change of Jx(x) at the DW center is a clear\nevidence of strong angular-momentum transfer from spin\ncurrent to the DW. Technically, magnetizationof the two\ndomains separated by the DW point to the opposite di-\nrections, thus the spin current polarization changes its\nsign. In calculating DW velocity vcurrentfrom Eq. ( 5),\nthe spin currents before entering DW and after passing\nDW are the averages of Jx(x) overx∈[−2∆,−∆] and\nx∈[∆,2∆], where ∆ is the DW width which is 16 nm\nin the current case. The thermal gradient dependence\nofvcurrentis shown in Fig. 2(b) (solid squares). vcurrent\ncompares well with the velocity vsimu(open squares) ob-\ntained directly from simulations by extracting the speed\nof the DW center along x-direction. The DW veloc-\nity is linearly proportional to the temperature gradient\nv=C∇xT, with the thermal mobility C= 6.66×10−8\nm2s−1K−1forvsimuorC= 6.59×10−8m2s−1K−1for\nvcurrent. It is noted that vcurrentalmost coincides with\nvsimuexcept a small discrepancy at very high thermal\ngradient when the nonlinear effects is strong. The small\ndiscrepancy may be attributed to the large fluctuations\nas well as the contribution from the damping, the dipo-\nlar and stochastic fields. These observations are consis-\ntent with magnonic STT [ 15,16,25,26]. It is observed\nthat the DW-plane rotates around the x-axis counter-\nclockwise for head-to-head DW and clockwise for tail-to-\ntail DW during DW propagation. DW rotation speed\ndφ/dt(squares) is shown in Fig. 2(c)) as a function of\n∇xT.4\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s52/s54/s56/s49/s48/s49/s50/s49/s52/s49/s54\n/s52/s54/s56/s49/s48/s49/s50/s49/s52/s32/s32\n/s75\n/s124/s124/s32/s40/s49/s48/s53\n/s32/s74/s47/s109/s51\n/s41/s32/s118\n/s115/s105/s109/s117\n/s32/s118\n/s99/s117/s114/s114/s101/s110/s116/s118 /s32/s40/s109/s47/s115/s41\n/s100 /s47/s100/s116/s40/s100/s101/s103/s47/s110/s115/s41/s32\nFIG. 4. Anisotropy K/bardbldependence of the DW dynamics:\nvsimu(open squares); vcurrent(solid squares); and dφ/dt(solid\ncircles). Model parameters are Lx= 2048 nm, Ly=Lz=4 nm,\nα= 0.004 and ∇xT= 0.2 K/nm.\nB. Damping and anisotropy dependence of DW\ndynamics\nAn energy-effect and angular-momentum-effect\nhave different damping-dependence and anisotropy-\ndependence of DW dynamics. To distinguish the roles of\nenergy and the angular-momentum in thermal-gradient\ndriven DW dynamics, it would be useful to probe how\nthe DW dynamics depends on αandK/bardbl. Damping have\ntwo effects on the spin currents: one is the decay of\nspin current during its propagation so that the amount\nof spin angular momentum deposited on a DW should\ndecrease with the increase of the damping coefficient.\nAs a result, the DW propagation speed and DW-plane\nrotation speed should also be smaller for a larger α.\nIndeed, this is what we observed in our simulations\nas shown in Fig. 3(a) for DW speed and DW-plane\nrotation speed (open squares for vsimu, solid circles for\nvcurrent, and stars for dφ/dt). The model parameters are\nLx= 2048, Ly=Lz= 4 nm, ∇xT= 0.2 K/nm and\nK/bardbl= 5×105J/m3. The second damping effect is that\nthe larger αhelps the spin current polarization to align\nwith the local spin. This second effect enhances the\nadiabatic process that is important for non-adiabatic\nSTT or field-like torque discussed in the next subsection.\nTherefore, α−dependence of DW dynamics supports\nthe origin of thermal driven DW dynamics to be the\nangular-momentum effect, not the energy effect that\nwould lead to a larger vsimuanddφ/dtfor a larger α\n[3,4,33–35] instead of a decrease observed here.\nHere we would like to see how the DW dynamics de-\npendsonuniaxialanisotropy K/bardbl. Fig.4showsboth vsimu\n(open squares), vcurrent(filled squares) and dφ/dt(cir-\ncles) for Lx= 2048 nm, α= 0.004 and ∇xT= 0.2. The\nDW propagation speed, vsimudecreases with K/bardblwhileDW-plane rotational speed increases with K/bardbl. These re-\nsults seem follow partially the behavior of magnetic-field\ninduced DW motion, in which DW propagation speed\nis proportional to DW width (∆ ∼/radicalBigg\nA\nK/bardbl) or decrease\nwithK/bardbl, and partially electric current driven DW mo-\ntion, in which DW-plane rotational speed increases with\nK/bardbl. Thus, one may tend to conclude that a thermal gra-\ndient behaves more like a magnetic field rather than an\nelectric current from the DW width dependence of DW\npropagation speed, opposite to our claim of the angular-\nmomentum effects of the thermal gradient. It turns out,\nthis is not true. The reason is that magnon spectrum,\nωk=2γ\nµ0Ms/parenleftbig\nAk2+K/bardbl/parenrightbig\n, has a gap in a system with mag-\nnetic anisotropy. The larger K/bardblis, the bigger the energy\ngap will be. Thus, it becomes harder to thermally excite\nmagnon. As a result, the spin current decreasesas K/bardblin-\ncreases. To see whether the thermal-gradient driven DW\nmotion is due to the angular-momentum transfer or not,\none should compare whether vsimuandvcurrentmaintain\na good agreement with each other as K/bardblvaries. Indeed,\na good agreement between vsimuandvcurrentis shown in\nFig.4. This conclusion is also consistent with existing\nmagnonic STT theories [ 33–35].\nC. Separation of adiabatic and non-adiabatic\ntorques\nWe have already demonstrated that a thermal gradi-\nent interacts with DW through magnonic STT rather\nthan through energy dissipation. It is then interesting to\nknow what kind of STTs a thermal gradient can gen-\nerate. Specifically, whether a magnonic spin current\ngenerates damping-like (adiabatic), or field-like (Non-\nadiabatic) torques, or both just like an electric current\n/s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53/s45/s48/s46/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52\n/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s73/s32/s40/s49/s48/s49/s48\n/s32/s65/s47/s109/s50\n/s41\n/s32\n/s120/s84 /s32/s40/s75/s47/s110/s109/s41\n/s32/s32\nFIG. 5. Model parameters are K/bardbl= 5×105J/m3,α=\n0.004,Lx= 1024 and Ly=Lz=4 nm. Effective electric current\ndensityI(open squares) and β(solid squares) are plotted as\nfunctions of ∇xT.5\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s48/s46/s48/s52/s48/s46/s48/s56/s48/s46/s49/s50\n/s48 /s49 /s50 /s51 /s52 /s53 /s54 /s55/s48/s46/s49/s52/s48/s46/s50/s49/s48/s46/s50/s56\n/s40/s49/s48/s45/s51\n/s41\n/s32/s32\n/s75\n/s124/s124/s32/s40/s49/s48/s53\n/s32/s74/s47/s109/s51\n/s41/s40/s97/s41\n/s40/s98/s41/s32/s32\nFIG. 6. Model parameters are ∇xT=0.5 K/nm, Lx= 1024\nnm and Ly=Lz=4 nm. (a) α-dependence of βforK/bardbl= 106\nand J/m3. (b)K/bardbl-dependence of βforα= 0.004.\n[27] does. To extract the STT generated from a thermal\ngradient, we approximate DW dynamics by the motion\nof its collective modes of DW center Xand the titled\nangleφof DW-plane. Subject to both damping-like and\nfield-like torques, using the travelling-wave ansatz [ 33–\n35], tan(θ/2) = exp[( x−X)/∆] where ∆ ∼/radicalbig\nA/K/bardbl, one\ncan derive the equations for X and φ,\nα\n∆dX\ndt+dφ\ndt=β\nαu,1\n∆dX\ndt−αdφ\ndt=u\nα.(6)\nFrom the above two equations, one can straightfor-\nwardly find DW propagating speed and DW-plane ro-\ntation speed,\nv=(1+αβ)\n(1+α2)u,˙φ=(β−α)\n(1+α2)u. (7)\nOne can extract βand equivalent electric current den-\nsityI= (2eMsu)/gµBPfromvanddφ/dtobtained in\nsimulations. For α= 0.004,K/bardbl= 106J/m3, theIandβ\nare obtained and plotted in Fig. 5as a function of ∇xT.\nIt is evident that Ilinearly increases with ∇xTandβ\nis independent of ∇xTas it should be. We then fixed\n∇xT= 0.5 K/nm, and repeat simulations and analysis\nmentioned above for various αandK/bardbl. Fig.6(a) and\n(b) shows βas a function of αandK/bardbl. From the figure,\nit is evident that βdecreases with α. This is because\nthe larger damping favors the alignment of spin current\npolarization with the local spin so that the non-adiabatic\neffect,β, becomes smaller. βincreases with K/bardblfor the\nsimilar reason: Larger K/bardblmeans a thinner DW so that\nit is much harder for the spin current polarization to re-\nverse its direction after passing through the thinner DW,\ni.e. a stronger non-adiabatic effect.\nIn some experiments, the temperature gradient is gen-\nerated by a laser spot[ 36]. The laser spot will induce a\nGaussian distribution of the temperature over the space\n[36,37]. In Fig. 7, weshowtheDWmotionin aGaussian\ntemperature profile T(x) =T0exp/parenleftBig\n−(x−xL)2\n2σ2/parenrightBig\nby plot-\nting the DW position against the time. Here we use the/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48\n/s32/s32/s68/s87/s32/s112/s111/s115/s105/s116/s105/s111/s110/s32/s40/s110/s109/s41\n/s116/s105/s109/s101/s32/s40/s110/s115/s41\nFIG. 7. Domain wall position versus time in a Gaussian tem-\nperature profile. The gray lines are raw data for different\nrandom seeds and the red line is the averaged result. The\ngreen dashed line is theoretical result using the thermal mo -\nbilityC= 6.66×10−8m2s−1K−1obtained from Fig. 2(b).\nsame parameters as those in Fig. 2(b), except a longer\nwireLx= 2048 nm, and T0= 400 K, σ= 200 nm,\nxL= 200 nm. Theoretically, if the instantaneous DW\nspeed under a Gaussian temperature is the same as that\nin the constant thermal-gradient case, we should expect\ndx\ndt=CdT\ndx, where the thermal mobility Cis the same as\nthat in Fig. 2(b). Using C= 6.66×10−8m2s−1K−1,\nthe above differential equation for x(t) can be numeri-\ncally solved with initial condition x(0) = 0. The result\nis plotted in Fig. 7in green dashed line. The simu-\nlated speed is smaller than this theoretical result. This\nis probably because, for the constant thermal-gradient,\nwe focus on the steady-state DW motion speed. In a\nGaussian temperature, the DW cannot immediately fol-\nlow the local temperature gradient. Before the DW can\nreach the steady-state speed corresponding to the local\ntemperature, it already moves to a position of smaller\ntemperature gradient. More details about DW motion in\nGaussian temperature profile may be an issue of future\nstudies.\nIV. DISCUSSION AND SUMMARY\nWe have studied the thermal gradient-driven DW dy-\nnamicsinanuniaxialnanowire. Inreality, thereisalways\ncertain hard anisotropy in a wire whose cross-section is\nnot a perfect ellipse. Thus, it is interesting to see how\nthe above results will change in a weak biaxial nanowire\nwith a small hard anisotropy K⊥= 1/2µ0M2\ns(Nz−Ny),\nsay along y-direction. Our simulations show that a DW\nstill propagatestowardsthe higher temperature region in\na similar way as that in a uniaxial wire. Interestingly, as\nshown in Fig. 2(d) for the K⊥-dependence of vsimu(solid\nsquares) and dφ/dt(open squares), DW speed increases6\nslightly with K⊥. This may be due to the increase of\ntorque along θ-direction [ 33] since Γ θis proportional to\n(Nz−Ny). This is also consistent with the early results\nforthe uniaxialwire that vsimu(which includes stochastic\nthermal field and demagnetisation fields) is always larger\nthanvcurrent(where the transverse fields are neglected).\nAt the meanwhile, dφ/dtdecreases with K⊥.\nThe main purpose of this paper is to study the\nmagnonic effects in thermal-gradient-driven domain wall\ndynamics. We consider the spin waves explicitly and\nall the material parameters (exchange constant A, crys-\ntalline anisotropy K, saturation magnetization Ms, and\nGilbert damping α) are assumed to be constant. Indeed,\nthe atomistic magnetic moments are independent of tem-\nperature. At the atomistic level, the exchange constant\nAoriginating from the Pauli exclusion principle and the\ncrystalline anisotropy Koriginating from the spin-orbit\ncoupling onlyweaklydepend on the temperature because\nof the vibration of atoms [ 39]. In micromagnetic models,\nbecause finite volumes that contains many magnetic mo-\nments are considered as unit cells, the parameters A,K,\nandMsdepend on the temperature. This is because the\nthermally excited spin waves with wavelengths shorter\nthan the length scale of the unit cells are included in\nthe effective A,K, andMsby doing an average [ 16,38].\nSince we use small mesh size 2 ×2×2 nm3, only spin\nwaves of very short wavelength affect the parameters A,\nK, andMsin our model. Those short-wavelength spin\nwavespossess high energyaswell as low density ofstates,\nso their contributions to the effective A,K, andMsare\nnot significant. The Gilbert damping αdepends on the\ntemperature non-monotonically [ 40–43]. The underlying\nmechanism is still under debate, but for many cases the\ndependence is not significant in a wide range of temper-\nature.\nIn summary, our results show that the uniform ther-\nmal gradient always drives a DW propagating towards\nthe hotter region and the DW-plane rotates around the\neasy axis. The DW velocity and DW-plane rotational\nspeed decrease with the damping coefficient. The DW\nvelocity obtained from simulation agrees with the veloc-\nity obtained from angular momentum conservation when\nthe magnon current density ( J(x)) from the simulation is\nusedtoestimatetheamountofangularmomentumtrans-\nferred from magnon current to the DW. All the above\nfindings lead to the conclusion that the thermal gradient\ninteracts with DW through angular-momentum transfer\nrather than through energy dissipation. Furthermore,\nwe demonstrated that the magnonic STT generated by\na thermal gradient has both damping-like and field-like\ncomponents. The field-like STT coefficient βis deter-\nmined from DW speed and DW-plane rotation speed. β\ndoes not depend on the thermal gradient as expected,\nbut increases with a decrease of DW width. This behav-\nior can be understood from the expected strongmisalign-\nment of magnon spin polarization and the local spin so\nthat non-adiabatic torque (also called field-like torque)\nis larger. For the same reason, a larger Gilbert dampingresults in a better alignment between spin current polar-\nization and the local spin, thus βshould decrease with\nα. The thermal gradientcan be a veryinteresting control\nknob for nano spintronics devices, especially those made\nfrom magnetic insulators.\nThis work was supported by the National Natural Sci-\nence Foundation of China (Grant No. 11774296) as well\nas Hong Kong RGC Grants Nos. 16300117, 16301518\nand 16301816. X.S.W acknowledges support from NSFC\n(GrantNo. 11804045),ChinaPostdoctoralScienceFoun-\ndation (Grant No. 2017M612932and 2018T110957),and\nthe Research Council of Norway through its Centres of\nExcellence funding scheme, Project No. 262633, “QuS-\npin.” M. T. I acknowledges the Hong Kong PhD fellow-\nship.7\n[1] Parkin S S P, Hayashi M and Thomas L 2008 Science\n320 190\n[2] Allwood D A, Xiong G, Faulkner C C, Atkinson D, Petit\nD and Cowburn R P 2005 Science309 1688\n[3] Wang X R, P Yan, Lu J and He C 2009 Ann. Phys. (N.\nY.)324 1815\n[4] Wang X R, Yan P and Lu J 2009 Europhys. Lett. 86\n67001\n[5] Atkinson D, Allwood D A, Xiong G, Cooke M D,\nFaulkner C C, and Cowburn R P 2003 Nat. Mater. 2\n85\n[6] Beach G S D, Nistor C, Knutson C, Tsoi M, and Erskine\nJ L 2005 Nat. Mater. 4 741\n[7] Hayashi M, Thomas L, Bazaliy Ya B , Rettner C, Moriya\nR, Jiang X, and Parkin S S P 2006 Phys. Rev. Lett. 96\n197207\n[8] Berger L 1996 Phys. Rev. B 54 9353\n[9] Slonczewski J 1996 J. Magn. Magn. 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B84 014412\n[43] Maier-Flaig H, Klingler S , Dubs C, Surzhenko O, Gross\nR, Weiler M, Huebl H, and Goennenwein S T B 2017\nPhys. Rev. B 95 214423"    },    {        "title": "2105.07376v1.Anatomy_of_inertial_magnons_in_ferromagnets.pdf",        "content": "arXiv:2105.07376v1  [cond-mat.mes-hall]  16 May 2021Anatomy of inertial magnons in ferromagnetic nanostructur es\nAlexey M. Lomonosov1,∗Vasily V. Temnov2,3,†and Jean-Eric Wegrowe3‡\n1Prokhorov General Physics Institute of the Russian Academy of Sciences, 119991, Moscow, Russia\n2Institut des Mol´ ecules et Mat´ eriaux du Mans, UMR CNRS 6283 ,\nLe Mans Universit´ e, 72085 Le Mans, France and\n3LSI, Ecole Polytechnique, CEA/DRF/IRAMIS, CNRS,\nInstitut Polytechnique de Paris, F-91128, Palaiseau, Fran ce\n(Dated: May 18, 2021)\nWe analyze dispersion relations of magnons in ferromagneti c nanostructures with uniaxial\nanisotropy taking into account inertial terms, i.e. magnet ic nutation. Inertial effects are\nparametrized by damping-independent parameter β, which allows for an unambiguous discrimi-\nnation of inertial effects from Gilbert damping parameter α. The analysis of magnon dispersion\nrelation shows its two branches are modified by the inertial e ffect, albeit in different ways. The up-\nper nutation branch starts at ω= 1/β, the lower branch coincides with FMR in the long-wavelength\nlimit and deviates from the zero-inertia parabolic depende nce≃ωFMR+Dk2of the exchange\nmagnon. Taking a realistic experimental geometry of magnet ic thin films, nanowires and nanodiscs,\nmagnon eigenfrequencies, eigenvectors and Q-factors are found to depend on the shape anisotropy.\nThe possibility of phase-matched magneto-elastic excitat ion of nutation magnons is discussed and\nthe condition was found to depend on β, exchange stiffness Dand the acoustic velocity.\nPACS numbers: Valid PACS appear here\nI. INTRODUCTION\nAfter the first description of the dynamics of the mag-\nnetization by Landau and Lifshitz [1], Gilbert proposed\nan equation that contains a correction due to the preces-\nsional damping [2, 3]. Since then, the so-called Landau-\nLifshitz-Gilbert (LLG) equation is known to give an ex-\ncellent description of the dynamics of the magnetization,\nincluding ferromagnetic resonance (FMR) and magneto-\nstatic waves [4, 5], as well as the magnetization reversal\n[6, 7]. Ferromagnetic resonance and time-resolved mag-\nnetization measurementsallow its spatially homogeneous\nprecession ( k= 0) but also non-uniform modes of the\nmagnetizationprecession( k∝ne}ationslash= 0, where kisthe wavevec-\ntor of spin waves) to be measured [8–10]. During the last\ndecades, these techniques have been advanced in the con-\ntext of ultrafast demagnetization dynamics [11, 12] that\npaved the way for the description of new physics at the\nsub-picosecond regime. High-frequency resonant modes\nof exchange magnons have been measured with ultrafast\ntime-resolved optical techniques [8, 10, 13]. Therefore,\nthe validity of the LLG equations has been confirmed\ndown to the picosecond time scale and below.\nHowever, limitations of LLG equations has been es-\ntablished in the stochastic derivation performed by W.\nF. Brown in a famous paper published in 1963 [14]. This\nlimit is due to the hypothesis that the typical time scales\nof magnetization dynamics are much longer than those\nof other degrees of freedom forming the dissipative envi-\nronment. In analogy to the common description of the\ndiffusion process of a Brownianparticle, the inertial (mo-\nmentum) degrees of freedom are supposed to relax much\n∗lom@kapella.gpi.ru\n†vasily.temnov@univ-lemans.fr\n‡jean-eric.wegrowe@polytechnique.edufaster than its spatial coordinate. This means that the\ndegrees of freedom related to the linear momentum (in\nthe case of the usual diffusion equation), or to the an-\ngular momentum (in the case of the magnetization) are\nincluded into the heat bath. As a consequence, the iner-\ntial terms do not explicitly appear in the equations, but\nare considered to be part of the damping term [15].\nThe possibility of measuring the contribution to iner-\ntial degrees of freedom led to a generalizationof the LLG\nequation with an additional term, incorporating the sec-\nond time-derivative of magnetization:\n˙m=−γm×Heff+αm×˙m+βmרm,(1)\nwherem=M/Msis the unit magnetization vector\nthatgivesthedirectionofthemagnetizationateachpoint\n(andMsis the modulus of the magnetization, which is\nconstant), γ=γ0µ0is the gyromagnetic ratio, αstands\nfor the Gilbert damping. Inertial effects are character-\nized by the parameter β, which is introduced in a phe-\nnomenological way, i.e. independent on αandγ[16].\nThis generalized LLG equation has been derived in the\nframework of different and independent theoretical con-\ntexts [15, 17–32]; its solutions have been studied in a\nseries of publications [33–36]. The main consequence of\ninertia for the uniform magnetization (magnon with the\nwavevector k= 0) is the existenceofnutation oscillations\nthat are superimposed to the precession. This leads to\nan appearance of the second resonance peak at a higher\nfrequency in FMR spectra. The direct measurement of\nnutation has been reported recently [37, 38].\nThe goal of the present report is to study the conse-\nquences of these inertial effects on the exchange magnons\n(i.e.k∝ne}ationslash= 0 modes), in the perspective of experimental\nstudies. Magnons are defined as linear magnetic exci-\ntations propagating in ferromagnets at the micromag-\nnetic limit. This work completes the first description2\nFMR \nmagnon ys(t) s(t) \nNutation \n   magnon (a)                                                                   (b) \nxΨz\nk\nFIG. 1. (a) Inertial magnons propagating in ferromagnetic\nnanostructures with wavevector kalong the zdirection under\nan external magnetic field Hresult in complex magnetization\ndynamics. (b) They can be decomposed in FMR magnon\nand nutation magnon precessing in opposite directions on el -\nliptical trajectories at different frequencies, giving ris e to a\ncharacteristic flower-shaped trajectory.\npublished in 2015, Section IV of the remarkable work of\nToru Kikuchi and Gen Tatara [22], and independently\nreconsidered by Makhfudz et al. in 2020 [39].\nThe paper is organized as follows. Section II presents\nthe derivation of the linear magnetic excitations deduced\nfrom (1). Section III describes the dispersion relation\nin a simple case of zero dipolar field (spherical symme-\ntry). The first consequence of the inertia is that the dis-\npersion relation splits in two branches: the lower one\ns1exp(ikz−iω1t) (FMR magnons ) and the upper one\nfors2exp(ikz−iω2t) (nutation magnons ). The second\nconsequence is that the quality factor Qincreases with\nthek-vector. SectionIVgeneralizesthedescriptiontothe\ncaseofauniaxialanisotropyquantifiedbythe dimension-\nless (shape) anisotropy parameter ξ. In the anisotropic\ncase the trajectories of both FMR magnons and nuta-\ntion magnons become elliptical and rotating in opposite\ndirections at each point in space. For a given k-vector\nthe magnetization vector corresponding to a superposi-\ntion of both magnons draws a typical trochoidal trajec-\ntory (see Fig. 1). Section V discusses the conditions\nfor phase-matched excitation the nutation magnons by\nco-propagatinglongitudinalacoustic phonons, illustrated\nby the material parameters for Gd-doped Permalloy thin\nfilms [13].\nII. EXCHANGE MAGNONS IN FERROMAGNETIC\nTHIN FILMS WITH MAGNETIC INERTIA\nWe start with the LLG equation for unit magnetiza-\ntion vector mwith aneffective field Heff, which includes\nexchange interactions with stiffness D, an external field\nH= (Hx,0,Hz) and a demagnetizing field induced by\nthe shape anisotropy Hd=−MS/hatwideNm. The demagne-\ntization tensor /hatwideNdepends on the specific shape of the\nferromagnetic sample. Hereafter we assume the diagonal\nformof/hatwideNwithdiagonalelements Nx,NyandNz. Damp-ing of the magnetization dynamics is described by the\nconventional Gilbert term with parameter α. In addition\nto the conventional LLG equation we take into account\nthe inertial effect characterized by the independent pa-\nrameterβ. Then the inertial LLG equation (ILLG) takes\nthe form of Eq.(1) with Heff=H+D∆m+Hd.\nThe coordinate system was chosen such that the ex-\nternal field lies in the y= 0 plane, as is shown in Fig. 1.\nThe material is assumed to be magnetically isotropic, so\nthat the unperturbed magnetization vector also lies in\nthey= 0 plane. We seek for time- and space-dependent\nsolutions in the form m=m0+s(z,t) with spin-wave\nsolutions\ns(z,t) = (sx,sy,sz)exp(ikz−iωt) (2)\npropagating as plane waves with a real wave vector k\nalong the z-axis, see Fig. 1. Substitution m(z,t) into\nequation (1) and its linearization with respect to small\nperturbations sx,sy,szresults in a homogeneous system\nof three linear equations:\n/hatwideA\nsx\nsy\nsz\n= 0 (3)\nwhere the matrix Ais given by\n\n−iω A 12(ω,k) 0\nA21(ω,k)−iω A 23(ω,k)\n0A32(ω,k)−iω\n (4)\nwith coefficients Aij(ω,k) defined as:\nA12=mz(γDk2+γMSξyz−iαω−βω2)+γHz\nA21=−mz(γDk2+γMSξxz−iαω−βω2)+γHz\nA23=mx(γDk2+γMSξzx−iαω−βω2)+γHx(5)\nA32=−mx(γDk2+γMSξyx−iαω−βω2)−γHx\nwhere coefficients ξij=Ni−Njcharacterize the shape\nanisotropy. The condition for the nontrivial solution of\nthehomogeneoussystem(3) toexist, i.e. det A= 0, gives\nrise to the secular equation\nω2+A12(ω,k)A21(ω,k)+A23(ω,k)A32(ω,k) = 0 (6)\nwhich is used to calculate the spin wave dispersion re-\nlationω(k) for different shapes/symmetries, i.e.charac-\nterized by different types of the /hatwideNtensor.\nIII. INERTIAL EXCHANGE MAGNONS IN SAMPLES\nWITH SPHERICAL SYMMETRY\nExamples of such symmetry are infinite homogeneous\nisotropic ferromagnetic media, or any spherical body. In3\nthesecasesthedemagnetizationtensor /hatwideNisdiagonalwith\nall nonzero elements equal 1 /3, so that its contribution\nto the magnetization dynamics (1) and correspondingly\nto the wave matrix components(5) vanishes. The secular\nequation (6) takes a concise form:\n/parenleftbig\nγH+γDk2−βω2−iαω+ω/parenrightbig\n×(7)\n×/parenleftbig\nγH+γDk2−βω2−iαω−ω/parenrightbig\n= 0.\nDue to the symmetry of /hatwideN, equation (7), and hence all\nits roots, remains independent on the direction of Hand\nthe equilibrium magnetization m0with respect to the\nwave propagation direction along the z-axis. For each\npositive wavenumber k, the determinant (7) is solved\nforω. Given that the presumed solution has a form\n∼exp(ikz−iωt), positive ωdesignates the waves trav-\nelling in the positive direction. The two positive roots\ncorresponding to the first parenthesis in (7) have the fol-\nlowing forms:\nω1=1\n2β/parenleftig\n−1−iα+/radicalbig\n4γβ(Dk2+H)+(1+iα)2/parenrightig\n(8)\nω2=1\n2β/parenleftbigg\n1−iα+/radicalig\n4γβ(Dk2+H)+(1−iα)2/parenrightbigg\n(9)\nThefirstrootisthelowermagnonbranchorprecession,\nslightly modified by the inertial term and the second one\nexhibits the inertial magnon branch or nutation. It is\nconvenient to split these roots into real and imaginary\nparts:ω1,2=ω′\n1,2+iω′′\n1,2. Taylor series approximation of\nthose roots assuming the smallness of γβDk2,γβH,α ≪\n1 results in the following expressions for their real parts:\nω′\n1≈γ[Dk2+H−2βγHDk2+...] (10)\nω′\n2≈1\nβ+ω′\n1 (11)\nIn this approximation the nutation magnon branch\nis simply shifted by 1 /βwith respect to FMR magnon\nbranch. The validity of this approximation is illustrated\nin Fig. 2, where the exact roots given by (8) and (9) are\ndepicted with solid lines, whereas dashed lines represent\nthe power series approximation of (10) and (11).\nLower branch emerges from the Larmor’s frequency\nγHand grows parabolically with k. Effect of inertia re-\nduces the coefficient at the term quadratic in k. Upper\nbranch is simply displaced by +1 /βand has the simi-\nlar shape. Imaginary parts of the roots ω′′\n1,2represent\nattenuation of the corresponding magnetization dynam-\nics in time as ∝exp/parenleftbig\nω′′\n1,2t/parenrightbig\n, and therefore they must be\nnegative. In the frequency domain they characterize the\nwidth ∆f=|ω′′|/π(FWHM) of the Lorentzian spectral\nline.0.3 0.5 0.8 1.0 1.3 1.5 0.25 0.50 0.75 1.00 1.25 \nβ= 0.17ps \nβ= 0.27ps β= 0.17ps Frequency, THz\nwavenumber  k, rad/nm β= 0.27ps \n0.25 0.50 0.75 1.00 1.25 1.50 10 20 30 \nβ= 0β= 0\nα= 0.023 α= 0.023 \nα= 0.01 \nα= 0.01 Line Width, GHz\nwavenumber  k, rad/nm β= 0.27ps \nFIG. 2. (a) Dispersion of the inertial magnon (red, magenta)\nand conventional magnon (blue,green). Dashed curves shows\nthe approximations by (10) and (11). (b) corresponding line\nwidths. In order to indicate the effect of inertia on the pre-\ncession, dashed curves show the line width without inertia\nβ= 0.\nω′′\n1≈ −αγ/bracketleftbig\nDk2+H−6βγ2HDk2+.../bracketrightbig\n(12)\nω′′\n2≈ −α\nβ−ω′′\n1 (13)\nNote that in the limiting case of (12), (13) field and\nexchange stiffness have opposite effects on the attenua-\ntion of the two magnon branches: they increase the at-\ntenuation in the lower branch ω1and decrease it for the\ninertial branch ω1. In the other limiting case of large\nfield and large kattenuation of both branches tends to\nexp/parenleftig\n−α\n2βt/parenrightig\n. The damping for both branches appears\nto be naturally proportional to the Gilbert damping pa-\nrameterα. A conspicuous decrease of nutation linewidth\nω′′\n2(k= 0) with growing αreported by Cherkasski et\nal. [36] roots back to the parametrization of the nu-\ntation phenomenon in terms of a product ατ, whereτ\ndenotes the characteristic nutation lifetime. Within this\nparametrization, a variation of α, while keeping τcon-\nstant, leads to the simultaneous decrease of the nutation4\n0.2 0.4 0.6 0.810203040Q\nk, nm-11/2H=0 2 T5 T\n0.2 0.4102030in a thin film\nFIG. 3. Q-factor dependencies on kfor different values of the\nexternal field. Dashed line shows the quadratic in kapprox-\nimation for H= 0. Inset shows Q-factors for the ordinary\nmagnon (blue curve) and inertial magnon (red curve) in a\nthin film, H= 0\nfrequency 1 /β= 1/(ατ) rendering the analysis of damp-\ning extremely difficult. An alternative notation in terms\nofαandβ, introduced in this paper, resolves this prob-\nlem and allows for an independent investigation of iner-\ntial and damping effects.\nAnother parameter, which characterizes the resonant\nspectral line centered at frequency f0, is its quality factor\ndefined as Q=f0/∆f=ω′/(2ω′′). As can be seen from\nequations (8) and (9), Q-factors for both branches coin-\ncide within the accuracy of ∼(ωα)2. Dependence of the\nQ-factor on the wavenumber klooks counterintuitive in\nthat it essentially grows with k. Assuming for simplicity\nH= 0, for small kthe Q-factor can be approximated by\nexpansion of ω′andω′′in the power series in k, which\nresults in:\nQ(k)∼1\n2α1+γβDk2\n1−γβDk2+...∼1\n2α/parenleftbig\n1+2γβDk2/parenrightbig\n(14)\nExact values for the Q-factor in comparison to the\nestimate of (14) are shown in Fig. 3 for the external\nfield ranging from 0 to 5 T.\nField effect for small kcan be approximated as Q∼\n1/(2α)(1+2γβH).\nIV. INERTIAL EXCHANGE MAGNONS IN SAMPLES\nWITH CYLINDRICAL SYMMETRY\nExamplesofsuchbodies aredisks, wires, infinite plates\nand films. Axial symmetry about the z-axis retains the\ndiagonal form of /hatwideNwith the diagonal elements satisfying\nthefollowingconditions: Nx=NyandNx+Ny+Nz= 1.\nAs a result, components of the matrix /hatwideAgiven by (5)acquire terms proportional to γMS. Lack of symmetry\nmakes the magnon propagation dependent on the orien-\ntation of vectors m0andHwith respect to the z-axis.\nWe consider two limiting cases: collinear arrangement\nwithm0parallel to the axis of symmetry (Θ = Ψ = 0\nin Fig. 1); and orthogonal arrangement with m0paral-\nlel to the x-axis and Θ = Ψ = π/2. In the collinear\ncase the demagnetizing field acts simply against the ex-\nternal field, hence the secularequation remains similarto\n(7), but with field Hsubstituted with the reduced field\nH′=H−ξMS:\n/parenleftbig\nγH′+γDk2−βω2−iαω+ω/parenrightbig\n×(15)\n×/parenleftbig\nγH′+γDk2−βω2−iαω−ω/parenrightbig\n= 0\nwhereξ=Nz−Nx=Nz−Nycharacterizes the shape\neffect on demagnetizing, so that in an infinite wire ξ=\n−1/2, in the spherical symmetric (or unbounded) body\nξ= 0 and in the infinite film ξ= 1. Correspondingly the\nroots to (15) are similar to ones given in (8) and (9) with\nmodified field:\nω1=1\n2β/parenleftbigg\n−1−iα+/radicalig\n4γβ(Dk2+H′)+(1+iα)2/parenrightbigg\n(16)\nω2=1\n2β/parenleftbigg\n1−iα+/radicalig\n4γβ(Dk2+H′)+(1−iα)2/parenrightbigg\n(17)\nAt the low- klimit the lower branch roughly tends\nto the Larmor’s frequency γ(H−ξMS) and the upper\nbranch limit is 1 /β+γ(H−ξMS), which is similar to the\ncase of spherical symmetry, but with modified field. In\nthe orthogonal configuration with m0andHperpendic-\nular to the axis of symmetry and to the magnon propa-\ngation direction, roots of the determinant (4) generally\ncannot be found in an analytical form. Therefore we\nfirstconsideranapproximatesolutions,andthendescribe\nbrieflythe numericalgorithmforobtainingthedispersion\ncurves. By neglecting the Gilbert attenuation ( α= 0),\nthe approximate solutions to (4) for the in-plane mag-\nnetization and field can be found in a concise analytical\nform:\nω1,2=1\nβ√\n2{2γβ(Dk2+H)+γβξM s+1\n∓/radicalbig\n4γβ(Dk2+H)+(γβξM s+1)2}1/2(18)\nHere indices 1 and 2 denote the frequencies of the\nconventional and inertial magnons respectively, sign ‘-\n‘ prior to the square root in (18) corresponds to the\nlower branch ω1; sign ‘+’ denotes the inertial branch\nω2. Numerical procedure for building the dispersion\nrelations of the magnonic modes for nonzero αor ar-\nbitrary orientation of the external field H starts with\ncalculation of the stationary equilibrium magnetization5\nm0= (mx,my,mz). This can be done by solving (1)\nin its stationary form, i.e.with all time derivatives set\nzero. In a thin film, for example, quantities Hiandmj\nare related by MSmxmz+Hxmz−Hzmx= 0. Thus\nobtained stationary magnetization components are then\nsubstituted into (5) and (4). At some fixed small kthe\ndeterminant (4) as a function of complex-valued ωpos-\nsesses two minima, which correspond to the FMR and\nnutational branches. Their exact locations can be evalu-\natedbyanumericalroutinewhichminimizestheabsolute\nvalue of the determinant (4) in the vicinity of the guess\nvalues for those branches, for example given by equa-\ntions (19) and (20). Then we give ka small increment\nand repeat the extremum search using the ωs obtained\nat the previous step as guess values, and so on. As a\nresult, calculated values for ω1andω2follow the disper-\nsion curves of both branches. Note that for nonzero α\nrootsω1,2possess imaginary parts, which determine the\nline width and Q-factor for each mode. Let us consider\nthe magnetization behavior in a thin film in more detail.\nFor this geometry /hatwideNpossesses the only nonzero compo-\nnentNz= 1, and correspondingly ξ= 1. In the small- k\nlimit, the lower branch approaches the Kittel’s frequency\nωFMR=γ/radicalbig\nH(H+MS) from below as β,k→0:\nω1≈ωFMR−1\n2γωFMR(2H+MS)β+...(19)\nEffect of the demagnetizing field on the inertial branch\nis exhibited by an upward shift by1\n2γMS; whereas effect\nof inertia is opposite:\nω2≈1\nβ+γH+1\n2γMS−/parenleftbigg\nω2\nFMR+1\n8γ2M2\nS/parenrightbigg\nβ+...(20)\nFor the orthogonal configuration, when both m0and\nHlie in the film plane, we can estimate the trajectories\nof the magnetization dynamics of both modes for small\nkand small α. For each root given by (18) we solve\nthe homogeneous equation for perturbations s(3). Nor-\nmalization of the solutions can be chosen in an arbitrary\nway,here for simplicity we define sz= 1. In orthogonal\ngeometry sxcomponent is obviously negligible or equals\nzero, so the system reduces to two equations in syand\nsz. Results shown as a power series expansion for small\ninertiaβω≪1 for the precession:\nsp=\n0\n−i/radicalig\n1+ξMS\nDk2+H/parenleftig\n1+βγξM S\n2/parenrightig\n1\nexp(−iω1t)\n(21)\nand nutation:\nsn=\n0\ni(1−βγξM S/2)\n1\nexp(−iω2t) (22)0.1 0.2 0.3 0.4 0.5 0.61.01.21.41.61.82.0\n1 T0.2 Tellipticity sy’/sx’\nk, nm-1sx’ sy’0.1 T\nFMR\nnutation\nFIG. 4. Ratio of the polarization axes for the external field\nof 0.1T, 0.2T, and 1T. Handm0are parallel to the inward\nnormal to the figure plane.\nwith the parameter of anisotropy ξ= 1 for a thin film\nnormal to the z−axis and ξ= 1/2 for a thin wire spread\nalong the z−axis. Both perturbations exhibit elliptical\npolarization within the ( x′,y′) plane. Precession trajec-\ntory is deformed by the demagnetizing effect so that the\ny-axis of the ellipse is stretched with the/radicalbig\n1+MS/H\nfactor due to demagnetizing effect, and in addition on\naccount of inertial effect. On the contrary, the nutation\nellipse is squeezed along the y-axis proportionally to the\ninertial parameter β. Ellipticity of the lower branch de-\npends on the external field (21), whereas that of the up-\nper branch in this approximation shows no dependence\non the field. Signs of sycomponents are opposite for\nnutation and precession, this indicates that they are ro-\ntating in the opposite directions. Exact polarizationscan\nbe found numerically for a reasonable set of material pa-\nrameters and fields, as is shown in Fig. 4.6\nV. EXCITATION MECHANISMS OF INERTIAL\nEXCHANGE MAGNONS\nThe only experimental evidence of inertial effects in\nferromagnets has been reported for k= 0 nutation man-\ngons in Py-thin films resonantly excited with a mag-\nnetic field of an intense quasi-monochromatic THz pulse\n[37]. In order to excite k∝ne}ationslash= 0 exchange magnon modes\none would need to have either spatially localized and\ninstantaneous stimuli [10] or any other source of effec-\ntive magnetic field characterized by spectral and spatial\noverlap with investigated magnon modes. The letter can\nbe provided through ultrashort large-amplitude acoustic\npulses [40, 41] producing effective magneto-elastic fields\nrapidly varying in time and space [42]. Acoustic pulses\npropagating through a thin ferromagnetic sample at an\nacoustic velocity vare quantified by a linearized disper-\nsion relation ωac=vk. Crossing between acoustic and\nmagnon brunches, i.e. satisfying the phonon-magnon\nphase-matchingcondition, usually facilitatesthe acoustic\nexcitation of magnetization dynamics [43, 44]. A ques-\ntion arises under which conditions the crossing between\ndispersion curves for longitudinal phonons and inertial\nmagnons can occur. Whereas for realistic magnetic fields\nthe acoustic dispersion always intersects the lower FMR-\nbranch at a frequency close to FMR frequency [42], the\ncrossing of the upper nutation brunch is less obvious.\nFIG. 5. The magneto-acoustic phase matching condition for\nnutation magnons can be tuned vie the reduction of exchange\nstiffness in Gd-doped Py samples. Gd concentration x varies\nfrom 0 to 13%. The dashed line displays the acoustic disper-\nsion relation ωac/(2π). Magneto-elastic coupling with inertial\nmangon is efficient when the dashed line lies within the pink\ntinted area. Material parameters are taken from Ref. [13] an d\nβ=0.276 ps.\nIt is possible to quantify the criterion for magneto-\nelastic crossing with nutation magnons analytically. To\ndo that we note that for larger wavenumbers ksatisfying\nDk2≫H,MStheexchangetermplaysthedominantrole\nand the asymptotic behaviourfor both branchesbecomeslinear in k:\nω1,2≈ ∓1\n2β+k/radicaligg\nγD\nβ. (23)\nIt follows from (23) that the condition for the nutation\nmagnon branch to intersect the acoustical dispersion re-\nlationωac(k), requires the asymptotic slope of ω2(k) to\nbe smaller than the acoustic velocity v:\n/radicaligg\nγD\nβ< v. (24)\nThis expression shows that for a given βthe magneto-\nelastic crossing is facilitated by small exchange stiffness\nDand small acousticvelocity. This approximateanalysis\nbreaks down for acoustic frequencies in above-THz spec-\ntral range, where the acoustic dispersion starts deviating\nfrom its linear approximation.\nFigure 5 highlights the remarkable role of exchange\nstiffness to achieve the dispersion crossing between nuta-\ntionmagnonsandlongitudinalacousticphonons. Doping\nPy thin films with Gadolinium has been shown to gradu-\nally reduce the exchange stiffness upon Gd-concentration\nfrom 300 to 100 [meV ·˚A2] [13]. For a fixed value of in-\nertial parameter β=0.276 ps, nutation magnons for pure\nPy samples do not display any crossing with acoustic\nphonons within the displayed range of k-vectors but the\nGd-doped Py with 13% Gd concentration does. The nu-\ntation magnon-phonon crossing point occurs at 0.75 THz\nfrequency and k= 0.85 nm−1(magnon wavelength of\napproximately 5 nm), i.e. magnon parameters readily\naccessible in ultrafast magneto-optical experiments [10].\nVI. CONCLUSIONS\nIn this paper we have theoretically studied exchange\ninertial magnons in ferromagnetic samples of different\nshapes under the action of an external magnetic field.\nThe parametrizationof magnetization dynamics in terms\nof two independent parameters, the Gilbert damping α\nand the inertial time β, allows for unambiguous discrim-\nination between the inertial and damping effects as well\nas their impact on both branches of magnon dispersion.\nInertial effects are found to strongly effect not only the\nfrequencies (magnon eigenvalues) of both branches but\nalso result in a monotonous increase of the Q-factor as\na function of the external magnetic field and magnon k-\nvector. The two magnon branchesare found to precessin\nopposite directions along the elliptical trajectories with\nperpendicularlyorientedlongaxisoftheellipses(magnon\neigenvectors). Their ellipticity is found to depend on the\ncomponents of the demagnetizing tensor. An analyti-\ncal criterion for the existence of phase-matched magneto-\nelastic excitation of nutation magnons has been derived\nandillustratedforGd-dopedpermalloysampleswithtun-\nable exchange stiffness.7\nACKNOWLEDGMENTS\nFinancial support by Russian Basic Research Founda-\ntion (Grant No. 19-02-00682)is gratefully acknowledged.\n[1] L. D. Landau and L. M. Lifshitz, Physik. Zeits. Sowjetu-\nnion8, 153 (1935).\n[2] T. L. Gilbert, Ph. D. Thesis (1956).\n[3] T. L. Gilbert, IEEE transactions on magnetics 40, 3443\n(2004).\n[4] R. W. Damon and J. R. Eshbach, Journal of Physics and\nChemistry of Solids 19, 308 (1961).\n[5] M. Farle, Reports on progress in physics 61, 755 (1998).\n[6] L. Thevenard, J.-Y. Duquesne, E. Peronne, H. J.\nVon Bardeleben, H. Jaffres, S. Ruttala, J.-M. George,\nA. Lemaitre, and C. Gourdon, Physical Review B 87,\n144402 (2013).\n[7] V. S. Vlasov, A. M. Lomonosov, A. V. Golov, L. N. Ko-\ntov, V. Besse, A. Alekhin, D. A. Kuzmin, I. V. Bychkov,\nand V. V. 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Olive, Journal of Physics: Con-\ndensed Matter 28, 106001 (2016).[24] M. Sayad, R. Rausch, and M. Potthoff, EPL (Euro-\nphysics Letters) 116, 17001 (2016).\n[25] D. Thonig, O. Eriksson, and M. Pereiro, Scientific re-\nports7, 1 (2017).\n[26] R. Mondal, M. Berritta, A. K. Nandy, and P. M. Oppe-\nneer, Physical Review B 96, 024425 (2017).\n[27] U. Bajpai and B. K. Nikoli´ c, Physical Review B 99,\n134409 (2019).\n[28] M. F ¨ahnle, Journal of Magnetism and Magnetic Materi-\nals469, 28 (2019).\n[29] R. Mondal and P. M. Oppeneer, Journal of Physics: Con-\ndensed Matter 32, 455802 (2020).\n[30] S. Giordano and P.-M. D´ ejardin, Physical Review B 102,\n214406 (2020).\n[31] R. Mondal, S. Großenbach, L. R´ ozsa, and U. Nowak,\nPhysical Review B 103, 104404 (2021).\n[32] S. Titov, W. Coffey, Y. P. Kalmykov, M. Zarifakis, and\nA. Titov, Physical Review B 103, 144433 (2021).\n[33] E. Olive, Y. Lansac, and J.-E. Wegrowe, AppliedPhysics\nLetters100, 192407 (2012).\n[34] E. Olive, Y. Lansac, M. Meyer, M. Hayoun, and J.-E.\nWegrowe, Journal of Applied Physics 117, 213904 (2015).\n[35] D. Bottcher and J. Henk, Physical Review B 86,\n020404(R) (2012).\n[36] M. Cherkasskii, M. Farle, and A. Semisalova, Physical\nReview B 102, 184432 (2020).\n[37] K. Neeraj, N. Awari, S. Kovalev, D. Polley, N. Z.\nHagstr¨om, S. S. P. K. Arekapudi, A. Semisalova, K. Lenz,\nB. Green, J.-C. Deinert, et al., Nature Physics 17, 245\n(2021).\n[38] Y. Li, A.-L. Barra, S. Auffret, U. Ebels, and W. E.\nBailey, Physical Review B 92, 140413(R) (2015).\n[39] I. Makhfudz, E. Olive, and S. Nicolis, Applied Physics\nLetters117, 132403 (2020).\n[40] V. V. Temnov, C. Klieber, K. A. Nelson,\nT. Thomay, V. Knittel, A. Leitenstorfer,\nD. Makarov, M. Albrecht, and R. Bratschitsch,\nNature Communications 4, 1468 (2013).\n[41] V. V. Temnov, I. Razdolski, T. Pezeril, D. Makarov,\nD. Seletskiy, A. Melnikov, and K. A. Nelson,\nJournal of Optics 18, 93002 (2016).\n[42] V. Besse, A. V. Golov, V. S. Vlasov, A. Alekhin,\nD. Kuzmin, I. V. Bychkov, L. N. Kotov, and V. V. Tem-\nnov, J. Magn. Magn. Mater. 502, 166320 (2020).\n[43] J. Januˇ sonis, C. L. Chang, T. Jansma, A. Gatilova, V. S.\nVlasov, A. M. Lomonosov, V. V. Temnov, and R. I.\nTobey, Phys. Rev. B 94, 024415 (2016).\n[44] C. L. Chang, A. M. Lomonosov, J. Janusonis, V. S.\nVlasov, V. V. Temnov, and R. I. Tobey, Phys. Rev. B\n95, 060409(R) (2017)."    },    {        "title": "1407.0635v1.Spin_Waves_in_Ferromagnetic_Insulators_Coupled_via_a_Normal_Metal.pdf",        "content": "Spin Waves in Ferromagnetic Insulators Coupled via a Normal Metal\nHans Skarsv\u0017 ag,\u0003Andr\u0013 e Kapelrud, and Arne Brataas\nDepartment of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway\n(Dated: May 27, 2022)\nHerein, we study the spin-wave dispersion and dissipation in a ferromagnetic insulator{normal\nmetal{ferromagnetic insulator system. Long-range dynamic coupling because of spin pumping and\nspin transfer lead to collective magnetic excitations in the two thin-\flm ferromagnets. In addition,\nthe dynamic dipolar \feld contributes to the interlayer coupling. By solving the Landau-Lifshitz-\nGilbert-Slonczewski equation for macrospin excitations and the exchange-dipole volume as well as\nsurface spin waves, we compute the e\u000bect of the dynamic coupling on the resonance frequencies and\nlinewidths of the various modes. The long-wavelength modes may couple acoustically or optically.\nIn the absence of spin-memory loss in the normal metal, the spin-pumping-induced Gilbert damp-\ning enhancement of the acoustic mode vanishes, whereas the optical mode acquires a signi\fcant\nGilbert damping enhancement, comparable to that of a system attached to a perfect spin sink. The\ndynamic coupling is reduced for short-wavelength spin waves, and there is no synchronization. For\nintermediate wavelengths, the coupling can be increased by the dipolar \feld such that the modes\nin the two ferromagnetic insulators can couple despite possible small frequency asymmetries. The\nsurface waves induced by an easy-axis surface anisotropy exhibit much greater Gilbert damping\nenhancement. These modes also may acoustically or optically couple, but they are una\u000bected by\nthickness asymmetries.\nPACS numbers: 76.50.+g,75.30.Ds,75.70.-i,75.76.+j\nI. INTRODUCTION\nThe dynamic magnetic properties of thin-\flm fer-\nromagnets have been extensively studied for several\ndecades.1,2Thin-\flm ferromagnets exhibit a rich vari-\nety of spin-wave modes because of the intricate inter-\nplay among the exchange and dipole interactions and the\nmaterial anisotropies. In ferromagnetic insulators (FIs),\nthese modes are especially visible; the absence of disturb-\ning electric currents leads to a clear separation of the\nmagnetic behavior. Furthermore, the dissipation rates\nin insulators are orders of magnitude lower than those\nin their metallic counterparts; these low dissipation rates\nenable superior control of travelling spin waves and facil-\nitate the design of magnonic devices.3\nIn spintronics, there has long been considerable in-\nterest in giant magnetoresistance, spin-transfer torques,\nand spin pumping in hybrid systems of normal met-\nals and metallic ferromagnets (MFs).4{7The experimen-\ntal demonstration that spin transfer and spin pumping\nare also active in normal metals in contact with insu-\nlating ferromagnets has generated a renewed interest in\nand refocused attention on insulating ferromagnets, of\nwhich yttrium iron garnet (YIG) continues to be the\nprime example.8{19In ferromagnetic insulators, current-\ninduced spin-transfer torques from a neighboring normal\nmetal (NM) that exhibits out-of-equilibrium spin accu-\nmulation may manipulate the magnetization of the insu-\nlator and excite spin waves.8,20,21The out-of-equilibrium\nspin accumulation of the normal metal may be induced\nvia the spin Hall e\u000bect or by currents passing through\nother adjacent conducting ferromagnets. Conversely, ex-\ncited spin waves pump spins into adjacent NMs, and this\nspin current may be measured in terms of the inverse spinHall voltages or by other conducting ferromagnets.8{14\nThe magnetic state may also be measured via the spin\nHall magnetoresistance.16{19,23,24Because of these devel-\nopments, magnetic information in ferromagnetic insula-\ntors may be electrically injected, manipulated, and de-\ntected. Importantly, an FI-based spintronic device may\ne\u000eciently transport electric information carried by spin\nwaves over long distances15without any excessive heat-\ning. The spin-wave decay length can be as long as cen-\ntimeters in YIG \flms.22These properties make FI{NM\nsystems ideal devices for the exploration of novel spin-\ntronic phenomena and possibly also important for future\nspintronic applications. Magnonic devices also o\u000ber ad-\nvantages such as rapid spin-wave propagation, frequen-\ncies ranging from GHz to THz, and the feasibility of cre-\nating spin-wave logic devices and magnonic crystals with\ntailored spin-wave dispersions.25\nTo utilize the desirable properties of FI{NM systems,\nsuch as the exceptionally low magnetization-damping\nrate of FIs, it is necessary to understand how the mag-\nnetization dynamics couple to spin transport in adjacent\nnormal metals. The e\u000bective damping of the uniform\nmagnetic mode of a thin-\flm FI is known to signi\f-\ncantly increase when the FI is placed in contact with\nan NM. This damping enhancement is caused by the loss\nof angular momentum through spin pumping.26{30Re-\ncent theoretical work has also predicted the manner in\nwhich the Gilbert damping for other spin-wave modes\nshould become renormalized.31For long-wavelength spin\nwaves, the Gilbert damping enhancement is twice as\nlarge for transverse volume waves as for the macrospin\nmode, and for surface modes, the enhancement can be ten\ntimes stronger or more. Spin pumping has been demon-\nstrated, both experimentally9and theoretically,31to be\nsuppressed for short-wavelength exchange spin waves.arXiv:1407.0635v1  [cond-mat.mes-hall]  2 Jul 20142\nA natural next step is to investigate the magnetization\ndynamics of more complicated FI{NM heterostructures.\nIn ferromagnetic metals, it is known that spin pumping\nand spin-transfer torques generate a long-range dynamic\ninteraction between magnetic \flms separated by normal\nmetal layers.32The e\u000bect of this long-range dynamic in-\nteraction on homogeneous macrospin excitations can be\nmeasured by ferromagnetic resonance. The combined ef-\nfects of spin pumping and spin-transfer torque lead to\nan appreciable increase in the resonant linewidth when\nthe resonance \felds of the two \flms are far apart and\nto a dramatic narrowing of the linewidth when the reso-\nnant \felds approach each other.32This behavior occurs\nbecause the excitations in the two \flms couple acous-\ntically (in phase) or optically (out of phase). We will\ndemonstrate that similar, though richer because of the\ncomplex magnetic modes, phenomena exist in magnetic\ninsulators.\nIn the present paper, we investigate the magnetization\ndynamics in a thin-\flm stack consisting of two FIs that\nare in contact via an NM. The macrospin dynamics in\na similar system with metallic ferromagnets have been\nstudied both theoretically and experimentally.32We ex-\npand on that work by focusing on inhomogeneous mag-\nnetization excitations in FIs.\nFor long-wavelength spin waves travelling in-plane in\na ferromagnetic thin \flm, the frequency as a function\nof the in-plane wave number Qstrongly depends on the\ndirection of the external magnetic \feld with respect to\nthe propagation direction. If the external \feld is in-\nplane and the spin waves are travelling parallel to this\ndirection, the waves have a negative group velocity. Be-\ncause the magnetization precession amplitudes are usu-\nally evenly distributed across the \flm in this geometry,\nthese modes are known as backward volume magneto-\nstatic spin waves (BVMSW). Similarly, spin waves that\ncorrespond to out-of-plane external \felds are known as\nforward volume magnetostatic spin waves (FVMSW),\ni.e., the group velocity is positive, and the precession\namplitudes are evenly distributed across the \flm. When\nthe external \feld is in-plane and perpendicular to the\npropagation direction, the precession amplitudes of the\nspin waves become inhomogeneous across the \flm, ex-\nperiencing localization to one of the interfaces. These\nspin waves are thus known as magnetostatic surface spin\nwaves (MSSW).33,34\nWhen two ferromagnetic \flms are coupled via a normal\nmetal, the spin waves in the two \flms become coupled\nthrough two di\u000berent mechanisms. First, the dynamic,\nnonlocal dipole-dipole interaction causes an interlayer\ncoupling to arise that is independent of the properties\nof the normal metal. This coupling is weaker for larger\nthicknesses of the normal metal. Second, spin pumping\nfrom one ferromagnetic insulator induces a spin accu-\nmulation in the normal metal, which in turn gives rise\nto a spin-transfer torque on the other ferromagnetic in-\nsulator, and vice versa. This dynamic coupling, is in\ncontrast to the static exchange coupling35rather long-ranged and is limited only by the spin-di\u000busion length.\nThis type of coupling is known to strongly couple the\nmacrospin modes. When two ferromagnetic \flms become\ncoupled, the characterization of the spin waves in terms\nof FVMSW, BVMSW, and MSSW still holds, but the\ndispersion relations are modi\fed. It is also clear that the\ndamping renormalization caused by spin pumping into\nthe NM may di\u000ber greatly from that in a simpler FI jN\nbilayer system. To understand this phenomenon, we per-\nform a detailed analytical and numerical analysis of a\ntrilayer system, with the hope that our \fndings may be\nused as a guide for experimentalists.\nThis paper is organized as follows. Section II intro-\nduces the model. The details of the dynamic dipolar\n\feld are discussed, and the boundary conditions associ-\nated with spin pumping and spin transfer at the FI jN\ninterfaces are calculated. Sec. III provides the analyti-\ncal solutions of these equations in the long-wavelength\nregime dominated by the dynamic coupling attributable\nto spin pumping and spin transfer. To create a more\ncomplete picture of the dynamic behavior of this system,\nwe perform a numerical analysis for the entire spin-wave\nspectrum of this system, which is presented in Sec. IV.\nWe conclude our work in Sec. V.\nII. EQUATIONS OF MOTION\nConsider a thin-\flm heterostructure composed of two\nferromagnetic insulators (FI1 and FI2) that are in elec-\ntrical contact via an NM layer. The ferromagnetic in-\nsulators FI1 and FI2 may have di\u000berent thicknesses and\nmaterial properties. We denote the thicknesses by L1,\ndN, andL2for the FI1, NM, and FI2 layers, respectively\n(see Fig. 1(a)). The in-plane coordinates are \u0010;\u0011, and the\ntransverse coordinate is \u0018(see Fig. 1(b)). We will \frst\ndiscuss the magnetization dynamics in isolated FIs and\nwill then incorporate the spin-memory losses and the cou-\npling between the FIs via spin currents passing through\nthe NM.\nA. Magnetization Dynamics in Isolated FIs\nThe magnetization dynamics in the ferromagnetic in-\nsulators can be described by using the Landau-Lifshitz-\nGilbert (LLG) equation,\n_Mi=\u0000\rMi\u0002He\u000b+\u000bMi\u0002_Mi; (1)\nwhere Miis the unit vector in the direction of the mag-\nnetization in layer i= 1;2,\ris the gyromagnetic ratio,\n\u000bis the dimensionless damping parameter, and He\u000bis\nthe space-time-dependent e\u000bective magnetic \feld. The\ne\u000bective magnetic \feld is\nHe\u000b=Hint+hex+hd+hsurface; (2)\nwhere Hintis the internal \feld attributable to an external\nmagnetic \feld and the static demagnetization \feld, hex=3\ndN2+L2\ndN2\n-dN2\n-dN2-L1NFI2\nFI1\nSUBx\n(a)\n (b)\nFIG. 1: (Color online) a) A cross section of the FI1 jNjFI2 het-\nerostructure. The ferromagnetic insulators FI1 and FI2 are\nin contact via the normal metal N. The transverse coordinate\n\u0018is indicated along with the thicknesses L1,dN, andL2of\nFI1, N, and FI2, respectively. b) The coordinate system of\nthe internal \feld (blue) with respect to the coordinate system\nof the FI1jNjFI2 structure (red). \u0012denotes the angle between\nthe \flm normal and the internal \feld, and \u001eis the angle be-\ntween the in-plane component of the magnetic \feld and the\nin-plane wave vector.\n2Ar2M=MSis the exchange \feld ( Ais the exchange\nconstant), hdis the dynamic demagnetization \feld, and\nhsurface =2KS\nM2\nS(Mi\u0001^n)\u000e(\u0018\u0000\u0018i)^n (3)\nis the surface anisotropy \feld located at the FI jN in-\nterfaces. In this work, hsurface is assumed to exist only\nat the FIjN interfaces and not at the interfaces between\nthe FIs and the substrate or vacuum. It is straightfor-\nward to generalize the discussion to include these surface\nanisotropies as well. We consider two scenarios: one with\nan easy-axis surface anisotropy ( KS>0) and one with no\nsurface anisotropy ( KS= 0). Note that a negative value\nofKS\u0018 \u0000 0:03 erg=cm2, which implies an easy-plane\nsurface anisotropy, has also been observed for sputtered\nYIGjAu bilayers.36In general, the e\u000bective \feld He\u000bmay\ndi\u000ber in the two FIs. We assume the two FIs consist of\nthe same material and consider external \felds that are\neither in-plane or out-of-plane. Furthermore, we consider\ndevices in which the internal magnetic \felds in the two\nFI layers are aligned and of equal magnitude.\nIn equilibrium, the magnetization inside the FIs is ori-\nented along the internal magnetic \feld, Mi=M0. In the\nlinear response regime, Mi=M0+mi, where the \frst-\norder correction miis small and perpendicular to M0.The magnetization vanishes outside of the FIs. Because\nthe system is translationally invariant in the \u0011and\u0010di-\nrections, we may, without loss of generality, assume that\nmconsists of plane waves travelling in the \u0010direction,\nmi(\u0010;\u0011;\u0018 ) =miQ(\u0018)ei(!t\u0000Q\u0010): (4)\nLinearizing Maxwell's equations in miimplies that the\ndynamic dipolar \feld must be of the same form,\nhd(\u0010;\u0011;\u0018 ) =hdQ(\u0018)ei(!t\u0000Q\u0010): (5)\nFurthermore, the total dipolar \feld (the sum of the static\nand the dynamic dipolar \felds) must satisfy Maxwell's\nequations, which, in the magnetostatic limit, are\nr\u0001(hd+ 4\u0019MSm) = 0; (6a)\nr\u0002hd= 0; (6b)\nwith the boundary equations\n(hd+ 4\u0019MSm)?;in= (hd)?;out; (7a)\n(hd)k;in= (hd)k;out; (7b)\nwhere the subscript in (out) denotes the value on the FI\n(NM, vacuum or substrate) side of the FI interface and ?\n(k) denotes the component(s) perpendicular (parallel) to\nthe FI{NM interfaces. Solving Maxwell's equations (6)\nwith the boundary conditions of Eq. (7) yields33\nhdQ(\u0018) =Z\nd\u00180^G(\u0018\u0000\u00180)mQ(\u00180); (8)\nwhere ^G(r\u0000r0) is a 3\u00023 matrix acting on min the (\u0011;\u0010;\u0018 )\nbasis,\n^G(\u0018) =0\n@GP(\u0018)\u0000\u000e(\u0018) 0\u0000iGQ(\u0018)\n0 0 0\n\u0000iGQ(\u0018) 0\u0000GP(\u0018)1\nA: (9)\nHere,GP(\u0018) =Qe\u0000Qj\u0018j=2, andGQ(\u0018) =\u0000sign(\u0018)GP.\nNote that the dynamic dipolar \feld of Eq. (8) accounts\nfor both the interlayer and intralayer dipole-dipole cou-\nplings because the magnetization varies across the two\nmagnetic insulator bilayers and vanishes outside these\nmaterials.\nIt is now convenient to perform a transformation from\nthe\u0010-\u0011-\u0018coordinate system de\fned by the sample geome-\ntry to thex-y-zcoordinate system de\fned by the internal\n\feld (see Fig. 1(b)). In the linear response regime, the\ndynamic magnetization milies in thex-yplane, and the\nlinearized equations of motion become33\n\u0014\ni!\u0012\n\u000b\u00001\n1\u000b\u0013\n+11\u0012\n!H+2A\nMS\u0014\nQ2\u0000d2\nd\u00182\u0015\u0013\u0015\nmiQxy(\u0018) =2X\ni=1Z\nd\u00180^Gxy(\u0018\u0000\u00180)miQxy(\u00180): (10)4\nN\nm1,QFI1m2,QFI2\nee\nFIG. 2: (Color online) Two coupled spin waves with ampli-\ntudem1Qin ferromagnet FI1 and amplitude m2Qin ferro-\nmagnet FI2. The spin-waves inject a spin current into the nor-\nmal metal (NM) via spin pumping. In the NM, the spins dif-\nfuse and partially relax, inducing a spin accumulation therein.\nIn turn, the spin accumulation causes spin-transfer torques to\narise on FI1 and FI2. The combined e\u000bect of spin transfer and\nspin pumping leads to a dynamic exchange coupling that, to-\ngether with the dynamic demagnetization \feld, couples the\nspin waves in the two FIs.\nHere, miQxy = (miQx;miQy) is the Fourier transform of\nthe dynamic component of the magnetization in the x-\nyplane and ^Gxy(\u0018) is the 2\u00022 matrix that results from\nrotating ^G(\u0018) into thex-y-zcoordinate system (see Ap-\npendix A), and considering only the xx,xy,yxandyy-\ncomponents.\nB. Boundary Conditions and Spin Accumulation\nThe linearized equations of motion (10) must be sup-\nplemented with boundary conditions for the dynamic\nmagnetization at the FI jN interfaces. A precessing mag-\nnetization at the FI jN boundaries injects a spin-polarized\ncurrent, jSP, into the NM, an e\u000bect known as spin\npumping .8,28{30The emitted spin currents at the lower\nand upper interfaces ( i= 1;2) are\njSP\ni=~\neg?Mi\u0002_Mi\f\f\f\f\n\u0018=\u0018i; (11)\nwhere\u0018i=\u0007dN=2 at the lower and upper interfaces,\nrespectively, and g?is the real part of the transverse spin-\nmixing conductance per unit area.37We disregard the\nimaginary part of the spin-mixing conductance because\nit has been found to be small at FI jN interfaces.38The\nreciprocal e\u000bect of spin pumping is spin transfer into the\nFIs because of a spin accumulation \u0016Sin the NM. In the\nnormal metal at the lower and upper interfaces ( i=1,2),the associated spin-accumulation-induced spin current is\njST\ni=\u00001\neg?Mi\u0002(Mi\u0002\u0016S)\f\f\f\f\n\u0018=\u0018i: (12)\nThe signs of the pumped and spin-accumulation-induced\nspin currents in Eqs. (11) and (12) were chosen such that\nthey are positive when there is a \row of spins from the\nNM toward the FIs.\nThe pumped and spin-accumulation-induced spin cur-\nrents of Eqs. (11) and (12) lead to magnetic torques act-\ning on the FI interfaces. The torques that correspond to\nthe spin pumping and spin transfer localized at the FI jN\ninterfaces are\n\u001cSP\ni=\r~2\n2e2g?\u000e(\u0018\u0000\u0018i)Mi\u0002_Mi; (13a)\n\u001cST\ni=\u0000\r~\n2e2g?Mi\u0002(Mi\u0002\u0016S)\u000e(\u0018\u0000\u0018i);(13b)\nrespectively. In the presence of spin currents to and from\nthe normal metal, the magnetization dynamics in the\nFIs is then governed by the modi\fed Landau-Lifshitz-\nGilbert-Slonczewski (LLGS) equation,\n_M=\u0000\rMi\u0002He\u000b+\u000bMi\u0002_Mi+X\ni=1;2\u001cSP\ni+\u001cST\ni:(14)\nBy integrating Eq. (14) over the FI jN interfaces and the\ninterfaces between the FI and vacuum/substrate, we \fnd5\nthatmimust satisfy the boundary conditions21,31\n\u0012\n\u0006Lidmi\nd\u0018+\u001fi\u0014\n_mi\u00001\n~M0\u0002\u0016\u0015\n+LiKS\nAcos (2\u0012)mi\u0013\nx\f\f\f\f\n\u0018=\u0007dN=2= 0;(15a)\n\u0012\n\u0006Lidmi\nd\u0018+\u001fi\u0014\n_mi\u00001\n~M0\u0002\u0016\u0015\n+LiKs\nAcos2(\u0012)mi\u0013\ny\f\f\f\f\f\n\u0018=\u0007dN=2= 0;(15b)\ndm1\nd\u0018\f\f\f\f\n\u0018=\u0000dN=2\u0000L1= 0;dm2\nd\u0018\f\f\f\f\n\u0018=dN=2+L2= 0:(15c)\nHere, we have introduced the timescale \u001fi=\nLi~2g?=4Ae2. The subscripts xandyin Eqs. (15a) and\n(15b) denote the xandycomponents, respectively. In\nour expressions for the boundary conditions (15), we have\nalso accounted for the possibility of a surface anisotropy\narising from the e\u000bective \feld described by Eq. (3),\nwhereKS>0 indicates an easy-axis surface anisotropy\n(EASA). The boundary conditions of Eq. (15), in combi-\nnation with the transport equations in the NM , which we\nwill discuss next, determine the spin accumulation in the\nNM and the subsequent torques caused by spin transfer.\nIn the normal metal, the spins di\u000buse, creating a spa-\ntially dependent spin-accumulation potential \u0016Q, and\nthey relax on the spin-di\u000busion length scale lsf. The\nspin accumulation for an FI jNjFI system has been cal-\nculated in the macrospin model.39The result of this\ncalculation can be directly generalized to the present\nsituation of spatially inhomogeneous spin waves by re-\nplacing the macrospin magnetization in each layer with\nthe interface magnetization and substituting the spin-\ndi\u000busion length with a wave-vector-dependent e\u000bective\nspin-di\u000busion length lsf!~lsf(Q) such that\n\u0016Q=\u0000~\n2M0\u0002[(_mQ(\u00181) +_mQ(\u00182))\u00001(\u0018)\n\u0000(_mQ(\u00181)\u0000_mQ(\u00182))\u00002(\u0018)]:(16)\nSee Appendix B for the details of the functions \u0000 1and\n\u00002. The e\u000bective spin-di\u000busion length is found by Fouriertransforming the spin-di\u000busion equation (see Appendix\nC), resulting in\n~lsf=lsf=p\n1 + (Qlsf)2: (17)\nWe thus have all the necessary equations to de-\nscribe the linear response dynamics of spin waves in the\nFI1jNjFI2 system. We now provide analytical solutions\nof the spin-wave modes in the long-wavelength limit and\nthen complement these solutions with an extensive nu-\nmerical analysis that is valid for any wavelength.\nIII. ANALYTIC SOLUTIONS FOR THE SPIN\nWAVE SPECTRUM\nThe e\u000bect that the exchange and dipolar \felds have\non the spin-wave spectrum depends on the in-plane wave\nnumberQ. WhenQLi\u001c1, the dipolar \feld dominates\nover the exchange \feld. In the opposite regime, when\nQLi\u001d1, the exchange \feld dominates over the dipo-\nlar \feld. The intermediate regime is the dipole-exchange\nregime. Another length scale is set by the spin-di\u000busion\nlength. When Qlsf\u001d1, the e\u000bective spin-relaxation\nlength ~lsfof Eq. (17) becomes small, and the NM acts\nas a perfect spin sink. In this case, only the relatively\nshort-ranged dipolar \feld couples the FIs. We therefore\nfocus our attention on the dipole-dominated regime, in\nwhich the interchange of spin information between the\ntwo FIs remains active.\nIn the limit QLi\u001c1, the magnetization is homoge-\nneous in the in-plane direction. We may then use the\nansatz that the deviation from equilibrium is a sum of\ntransverse travelling waves. Using the boundary condi-\ntions on the outer boundaries of the stack, Eq. (15c), we\n\fnd\nmiQxy(\u0018) =\u0012\nXi\nYi\u0013\ncos\u001a\nki\u0014\n\u0018\u0006(Li+dN\n2)\u0015\u001b\n;(18)\nwherei= 1 when\u0018is inside FI1 and i= 2 when\u0018is inside\nFI2.k1andk2are the out-of-plane wave vectors of the\nlower and upper \flms, respectively. The eigenfrequencies\nof Eq. (10) depend on ki. To \frst order in the damping\nparameter\u000b, we have\n!(ki) =!M\"\n\u0006s\u0012!H\n!M+A\n2\u0019M2\nSk2\ni\u0013\u0012!H\n!M+A\n2\u0019M2\nSk2\ni+ sin2\u0012\u0013\n+i\u000b\u0012!H\n!M+A\n2\u0019M2\nSk2\ni+1\n2sin2\u0012\u0013#\n: (19)\nWe can, without loss of generality, consider only those frequencies that have a positive real part. The eigen-6\nfrequency!is a characteristic feature of the entire sys-\ntem, so we must require !(k1) =!(k2), which implies\nthatk1=\u0006k2. We will discuss the cases of symmetric\n(L1=L2) and asymmetric ( L16=L2) geometries sepa-\nrately.\nA. Symmetric FI \flms without EASA\nConsider a symmetric system in which the FIs are of\nidentical thickness and material properties. We assume\nthat the e\u000bect of the EASA is negligible, which is the\ncase for thin \flms and/or weak surface anisotropy ener-\ngies such that KSL=A\u001c1, whereL=L1=L2. The\nother two boundary conditions, (15a) and (15b), cou-\nple the amplitude vectors\u0000X1Y1\u0001Tand\u0000X2Y2\u0001Tof\nEq. (18). A non-trivial solution implies that the deter-\nminant that contains the coe\u000ecients of the resulting 4 \u00024\nmatrix equation vanishes. Solving the secular equation,\nwe \fnd the following constraints on k,\ni\u001fA!A=kLtan(kL); (20a)\ni\u001fO!O=kLtan(kL); (20b)\nwhere\n\u001fA=\u001f \n1\u0000\u0014\n1 +2g?lsf\n\u001btanh(dN=2lsf)\u0015\u00001!\n;(21a)\n\u001fO=\u001f \n1\u0000\u0014\n1 +2g?lsf\n\u001bcoth(dN=2lsf)\u0015\u00001!\n;(21b)\nand\u001f=L~2g?=4Ae2. The two solutions correspond\nto a symmetric mode (acoustic) and an antisymmetric\nmode (optical). This result can be understood in terms\nof the eigenvectors that correspond to the eigenvalues of\nEqs. (20), which are m1= +m2andm1=\u0000m2for\nthe acoustic and optical modes, respectively. Typically,\nbecause spin pumping only weakly a\u000bects the magne-\ntization dynamics, the timescale \u001fthat is proportional\nto the mixing conductance g?is much smaller than the\nFMR precession period. In this limit, kLtan(kL)\u001c1.\nThis result allows us to expand the secular equations (20)\naroundkL=n\u0019, wherenis an integral number, which\nyields\ni\u001f\u0017!\u0017;n\u0019(kL+\u0019n)kL; (22)\nwhere\u0017= A;O. This result can be reinserted into the\nbulk dispersion relation of Eq. (19), from which we can\ndetermine the renormalization of the Gilbert damping\ncoe\u000ecient attributable to spin pumping, \u0001 \u000b. We de\fne\n\u0001\u000b=\u000b\u0010\nIm[!(SP)]\u0000Im[!(0)]\u0011\n=Im[!(0)] (23)\nas a measure of the spin-pumping-enhanced Gilbert\ndamping, where !(0)and!(SP)are the frequencies of\nthe same system without and with spin pumping, respec-\ntively.Similar to the case of a single-layer ferromagnetic\ninsulator,31we \fnd that all higher transverse volume\nmodes exhibit an enhanced magnetization dissipation\nthat is twice that of the macrospin mode. The enhance-\nment of the Gilbert damping for the macrospin mode\n(n= 0) is\n\u0001\u000b\u0017;macro =\r~2g?\n2LMSe2\u001f\u0017\n\u001f; (24)\nand for the other modes, we obtain\n\u0001\u000b\u0017;n6=0= 2\u0001\u000b\u0017;macro: (25)\nCompared with single-FI systems, the additional fea-\nture of systems with two FIs is that the spin-pumping-\nenhanced Gilbert damping di\u000bers signi\fcantly between\nthe acoustic and optical modes via the mode-dependent\nratio\u001f\u0017=\u001f. This phenomenon has been explored both\nexperimentally and theoretically in Ref. 32 for the\nmacrospin modes n= 0 when there is no loss of spin\ntransfer between the FIs, lsf!1 . Our results repre-\nsented by Eqs. (24) and (25) are generalizations of these\nresults for the case of other transverse volume modes and\naccount for spin-memory loss. Furthermore, in Sec. IV,\nwe present the numerical results for the various spin-wave\nmodes when the in-plane momentum Qis \fnite. When\nthe NM is a perfect spin sink, there is no transfer of spins\nbetween the two FIs, and we recover the result for a sin-\ngle FIjN system with vanishing back \row, \u001f\u0017!\u001f.31\nNaturally, in this case, the FI jNjFI system acts as two\nindependent FIjN systems with respect to magnetiza-\ntion dissipation. The dynamical interlayer dipole cou-\npling is negligible in the considered limit of this section\n(QL\u001c1).\nIn the opposite regime, when the NM \flm is much thin-\nner than the spin-di\u000busion length and the spin conductiv-\nity of the NM is su\u000eciently large such that g?dN=\u001b\u001c1,\nthen\u001fA!0 and\u001fO!\u001f. This result implies that for\nthe optical mode, the damping is the same as for a sin-\ngle FI in contact with a perfect spin sink, even though\nthe spin-di\u000busion length is very large. The reason for\nthis phenomenon is that when the optical mode is ex-\ncited, the magnetizations of the two \flms oscillate out\nof phase such that one layer acts as a perfect spin sink\nfor the other layer. By contrast, there is no enhance-\nment of the Gilbert damping coe\u000ecient for the acoustic\nmode; when the \flm is very thin and the magnetizations\nof the two layers are in phase, there is no net spin \row or\nloss in the NM \flm and no spin-transfer-induced losses\nin the ferromagnets. Finally, when the NM is a poor con-\nductor despite exhibiting low spin-memory loss such that\ng?dN=\u001b\u001d(lsf=dN)\u001d1, then\u001f\u0017!0 because there is no\nexchange of spin information. For the macrospin modes\nin the absence of spin-memory loss, these results are in\nexact agreement with Ref. 32. Beyond these results, we\n\fnd that regardless of how much spin memory is lost, it\nis also the case that in trilayer systems, all higher trans-\nverse modes experience a doubling of the spin-pumping-\ninduced damping. Furthermore, these modes can still7\nbe classi\fed as optical and acoustic modes with di\u000berent\ndamping coe\u000ecients.\nB. Symmetric Films with EASA\nMagnetic surface anisotropy is important when the\nspin-orbit interaction at the interfaces is strong. In this\ncase, the excited mode with the lowest energy becomes\ninhomogeneous in the transverse direction. For a \fnite\nKS, the equations for the xandycomponents of the\nmagnetization in the boundary condition (15) di\u000ber, re-\nsulting in di\u000berent transverse wave vectors for the two\ncomponents, kxandky, respectively. Taking this situa-\ntion into account, we construct the ansatz\nmiQxy(\u0018) =\u0012\nXicos (kx;i\u0018\u0006kx;i(L+dN=2))\nYicos (ky;i\u0018\u0006ky;i(L+dN=2))\u0013\n;(26)\nwhich, when inserted into the boundary conditions of\nEqs. (15a) and (15b), yields\ni\u001f\u0017!\u0017+LKS\nAcos (2\u0012) =kxdtan (kxd);(27a)\ni\u001f\u0017!\u0017+LKS\nAcos2(\u0012) =kydtan (kyd);(27b)\nwhere\u0017continues to denote an acoustic (A) or optical\n(O) mode, \u0017= A;O. Depending on the sign of KSand\nthe angle\u0012, the resulting solutions kxandkycan be-\ncome complex numbers, which implies that the modes\nare evanescent. Let us consider the case of KS>0 and\nan in-plane magnetization ( \u0012=\u0019=2). Although kyis\nunchanged by the EASA, with LKS=A> 1\u001d\u001f\u0017!\u0017,kx\nis almost purely imaginary, \u0014=ik=KS=A\u0000i!\u0017\u001f\u0017, so\nthat\nmiQx(\u0018) =Xcosh(\u0014\u0018\u0006\u0014(d+dN=2)): (28)\nThe magnetization along the xdirection is exponentially\nlocalized at the FI jN surfaces. Following the same proce-\ndure as in Sec. III A for the KS= 0 case, we insert this\nsolution into the dispersion relation (19) and extract the\nrenormalization of the e\u000bective Gilbert damping:\n\u0001\u000bEASA\n\u0017 =\r~2g?\n2LMSe2\u001f\u0017\n\u001f1 +!H\n!M\u0002\n1 +2LKS\nA\u0003\n\u0000K2\nS\n2\u0019M2\nSA\n1 + 2!H\n!M\u0000K2s\n2\u0019M2\nSA:\n(29)\nIn the presence of EASA, the damping coe\u000ecient is a ten-\nsor; thus, the e\u000bective damping of Eq. (29) is an average,\nas de\fned in Eq. (23). This Gilbert damping enhance-\nment may become orders of magnitude larger than the\n\u0001\u000bmacro of Eq. (24). For thick \flms, \u0001 \u000bmacro\u0018L\u00001,\nwhereas \u0001\u000bEASA\n\u0017 reaches a constant value that is in-\nversely proportional to the localization length at the FI jN\ninterface. Note that for large EASA, the equilibrium\nmagnetization is no longer oriented along the external\n\feld, and Eq. (29) for \u0001 \u000bEASA\n\u0017 becomes invalid.C. Asymmetric FI Films\nLet us now consider an asymmetric system in which\nL16=L2. In this con\fguration, we will \frst consider\nKS= 0, but we will also comment on the case of a \f-\nniteKSat the end of the section. Because the analytical\nexpressions for the eigenfrequencies and damping coe\u000e-\ncients are lengthy, we focus on the most interesting case:\nthat in which the spin-relaxation rate is slow.\nAs in the case of the symmetric \flms, the dispersion\nrelation of Eq. (10) dictates that the wave numbers in the\ntwo layers must be the same. To satisfy the boundary\nequations (15), we construct the ansatz\nmiQxy(\u0018) =\u0012\nXicos (k\u0018\u0006k(L+dN=2))\nYicos (k\u0018\u0006k(L+dN=2))\u0013\n: (30)\nThe di\u000berence between this ansatz and the one for the\nsymmetric case represented by Eq. (26) is that the mag-\nnitudes of the amplitudes, XiandYi, of the two layers,\ni= 1;2, that appear in Eq. (30) is no longer expected to\nbe equal.\nWhen the two ferromagnets FI( L1) and FI(L2) are\ncompletely disconnected, the transverse wave vectors\nmust be equivalent to standing waves, qn;1=\u0019n=L 1and\nqm;2=\u0019m=L 2in the two \flms, respectively, where nand\nmmay be any integral numbers. Because spin pumping\nis weak, the eigenfrequencies of the coupled system are\nclose to the eigenfrequencies of the isolated FIs. This\n\fnding implies that the wave vector kof the coupled sys-\ntem is close to either qn;1orqm;2. The solutions of the\nlinearized equations of motion are then\nk=kn;1=qn;1+\u000ekn;1or (31a)\nk=km;2=qm;2+\u000ekm;2; (31b)\nwhere\u000ekn;1and\u000ekm;2are small corrections attributable\nto spin pumping and spin transfer, respectively. Here,\nthe indices 1 and 2 represent the di\u000berent modes rather\nthan the layers. However, one should still expect that\nmode 1(2) is predominantly localized in \flm 1(2). In\nthis manner, we map the solutions of the wave vectors in\nthe coupled system to the solutions of the wave vectors\nin the isolated FIs. Next, we will present solutions that\ncorrespond to the qn;1of Eq. (31a). The other family of\nsolutions, corresponding to qm;2, is determined by inter-\nchangingL1$L2and making the replacement n!m.\nInserting Eq. (31a) into the boundary conditions of\nEq. (15) and linearizing the resulting expression in the\nweak spin-pumping-induced coupling, we \fnd, for the\nmacrospin modes,\ni!~\u001fA,O\n1;macro = (L1\u000ek0;1)2; (32)\nwhere\n~\u001fA\n1;macro\u00191\n2dN\nlsf\u001b\ng?lsfL1\nL1+L2\u001f1; (33a)\n~\u001fO\n1;macro\u00191\n2L1+L2\nL2\u001f1: (33b)8\nHere,\u001f1=L1~2g?=4Ae2. Inserting this parameter into\nthe dispersion relation of Eq. (19), we obtain the follow-\ning damping renormalizations:\n\u0001\u000bA\nmacro =\r~2g?\n2MSe21\n2dN\nlsf\u001b\ng?lsf1\nL1+L2;(34a)\n\u0001\u000bO\nmacro =\r~2g?\n2MSe21\n2\u00121\nL1+1\nL2\u0013\n: (34b)\nThese two solutions correspond to an acoustic mode\nand an optical mode, respectively. The corresponding\neigenvectors are m1=m2for the acoustic mode and\nL1m1=\u0000L2m2for the optical mode. As in the sym-\nmetric case, the damping enhancement of the acoustic\nmode vanishes in the thin-NM limit. In this limit, the\nbehavior of the acoustic mode resembles that of a single\nFI of thickness L1+L2. It is the total thickness that\ndetermines the leading-order contribution of the damp-\ning renormalization. The optical mode, however, experi-\nences substantial damping enhancement. For this mode,\nthe damping renormalization is the average of two sepa-\nrate FIs that are in contact with a perfect spin sink. The\ncause of this result is as follows. When there is no spin-\nmemory loss in the NM, half of the spins that are pumped\nout from one side return and rectify half of the angular-\nmomentum loss attributable to spin pumping. Because\nthe magnetization precessions of the two \flms are com-\npletely out of phase, the other half of the spin current\ncauses a dissipative torque on the opposite layer. In ef-\nfect, spin pumping leads to a loss of angular momentum,\nand the net sum of the spin pumping across the NM and\nthe back \row is zero. The total dissipation is not a\u000bected\nby spin transfer, and thus, the result resembles a system\nin which the NM is a perfect spin sink.\nFor the higher excited transverse modes, there are two\nscenarios, which we treat separately. I. The allowed wave\nnumber for one layer matches a wave number for the\nother layer. Then, for some integer n > 0,qn;1=qm;2\nfor some integer m. In this case, we expect a coupling\nof the two layers. II. The allowed wave number for one\nlayer does not match any of the wave numbers for the\nother layer, and thus, for some integer n > 0, we have\nqn;16=qm;2for all integers m. We then expect that the\ntwo layers will not couple.\nI. In this case, we \fnd two solutions that correspond\nto acoustic and optical modes. These modes behave very\nmuch like the macrospin modes; however, as in the sym-\nmetric case, the damping renormalization is greater by a\nfactor of 2:\n\u0001\u000bA,O\nn6=0= 2\u0001\u000bA,O\nmacro;Case I: (35)\nThe eigenvectors of these coupled modes have the same\nform as for the macrospin modes, such that m1=m2\nandL1m1=\u0000L2m2for the acoustic and optical modes,\nrespectively.\nII. In this case, the two layers are completely decou-pled. To the leading order in dN=lsf, we \fnd\n\u0001\u000bn6=0=\r~2g?\n2L1MSe2;Case II; (36)\nfor all modes that correspond to excitations in FI1.\nThe damping renormalization is thus half that of the\nFI(L1)jN(lsf= 0) system.31This result can be explained\nby the zero loss of spin memory in the NM. Although half\nof the spins are lost to the static FI2, half of the spins\nreturn and rectify half of the dissipation attributable\nto spin pumping. The amplitudes of these modes are\nstrongly suppressed in FI2 (or FI1, upon the interchange\nof FI1$FI2), such thatjm2j=jm1j\u0018!\u001f2.\nFinally, let us discuss the case in which EASA is\npresent. In the limit KSLi=A\u001d1, the excitation en-\nergies of the surface modes are independent of the FI\nthicknesses. However, the surface modes do not behave\nlike the macrospin modes for the asymmetric stack. The\nexcitation volume of these modes is determined by the\ndecay length A=KSin accordance with Eq. (28). This\n\fnding is in contrast to the result for the macrospin\nmodes, where the excitation volume spans the entire FI.\nThus, the surface modes couple in the same manner as in\nthe symmetric case. With a good experimental control\nof surface anisotropy, the coupling of the surface modes\nis thus robust to thickness variations. The higher ex-\ncited transverse modes, in the presence of EASA, have\nthickness-dependent frequencies, which means that these\nmodes behave similarly to the n>0 modes in the KS= 0\ncase.\nIV. NUMERICAL RESULTS\nWhen the spin-wave wavelength becomes comparable\nto the \flm thickness, the dipolar \feld becomes a compli-\ncated function of the wavelength. We study the proper-\nties of the system in this regime by numerically solving\nthe linearized equations of motion (10) with the bound-\nary conditions (15). We use the method presented in\nRef. 31, which solves the spin-wave excitation spectrum\nfor an FIjN system, and extend this approach to the\npresent trilayer system. The physical parameters used\nin the numerical calculations are listed in Table I. We\ninvestigate two geometries: I. the BWMSW geometry, in\nwhich the spin wave propagates parallel to the external\n\feld, and II. the MSSW geometry, in which the spin wave\npropagates perpendicular to the external \feld.\nTo calculate the renormalization of the Gilbert damp-\ning, we perform one computation without spin pumping\nand one computation with spin pumping, in which the\nintrinsic Gilbert damping is excluded. Numerically, the\nrenormalization can then be determined by calculating\n\u0001\u000b=\u000bIm[!(SP)]\u000b=0=Im[!(0)], where!(0)is the eigenfre-\nquency obtained for the computation without spin pump-\ning and!(SP)is the frequency obtained for the compu-\ntation with spin pumping.319\nTABLE I: Physical parameters used in the numerical calcu-\nlations\nConstant Value Units\ng?a3:4\u00011015cm\u00002e2=h\n\u001bb5:4\u00011017s\u00001\n4\u0019MSc1750 G\nAc3:7\u000110\u00007erg=cm\nHint 0:58\u00014\u0019MS\n\u000bc3\u000110\u00004\nKS 0;d0:05 erg=cm2\na) Ref. [47], b) Ref. [48], c) Ref. [34]\nd) Reported to be in the range of 0 :1\u00000:01 erg=cm2in\nRef. [21]\nA. BVMSW\nFIG. 3: (Color online) FI(100nm) jN(50nm)jFI(101nm): a)\nSpin-pumping-enhanced Gilbert damping \u0001 \u000bas a function\nofQL1of the uniform modes and the n= 1 modes. The inset\npresents the corresponding dispersion relation. b) Relative\nphase and c) amplitude between the out-of-plane magnetiza-\ntions along xat the edges of FI1 jN and FI2jN. The apparent\ndiscontinuity in the green line in c) appears because the phase\nis de\fned on the interval \u0000\u0019to\u0019.\nLet us \frst discuss the BVMSW geometry. The cou-\npling of the uniform modes in the two \flms is robust;it is not sensitive to possible thickness asymmetries. In\ncontrast, at Q= 0, the sensitivity to the ratio between\nthe thickness and the rather weak dynamic coupling at-\ntributable to spin pumping implies that the coupling of\nthe higher transverse modes in the two bilayers is fragile.\nSmall asymmetries in the thicknesses destroy the cou-\npling. This e\u000bect can best be observed through the renor-\nmalization of the damping. However, we will demon-\nstrate that a \fnite wave number Qcan compensate for\nthis e\u000bect such that the higher transverse modes also\nbecome coupled. To explicitly demonstrate this result,\nwe numerically compute the real and imaginary parts\nof the eigenfrequencies of a slightly asymmetric system,\nFI(100nm)jN(50nm)jFI(101nm) with lsf= 350 nm. The\nasymmetry between the thicknesses of the ferromagnetic\ninsulators is only 1%. The surface anisotropy is consid-\nered to be small compared with the ratio Li=A, and we\nsetKS= 0.\nIn Fig. 3, the numerical results for the e\u000bective Gilbert\ndamping, the dispersion of the modes, and the relative\nphase and amplitude between the magnetizations in the\ntwo FIs are presented. As observed in the relative phase\nresults depicted in Fig. 3(c), the two uniform modes in\nwidely separated FIs split into an acoustic mode and\nan optical mode when the bilayers are coupled via spin\npumping and spin transfer. Figure 3(a) also demon-\nstrates that the acoustic mode has a very low renor-\nmalization of the Gilbert damping compared with the\noptical mode. Furthermore, there is no phase di\u000berence\nbetween the two modes with a transverse node ( n= 1) in\nFig. 3(a), which indicates that the modes are decoupled.\nThesen= 1 modes are strongly localized in one of the\ntwo \flms; see Fig. 3(b). For small QL1, Fig. 3(a) demon-\nstrates that these modes have approximately the same\nrenormalization as the optical mode, which is in agree-\nment with the analytical results. Because the magnetiza-\ntion in the layer with the smallest amplitude is only a re-\nsponse to the spin current from the other layer, the phase\ndi\u000berence is \u0019=2 (Fig. 3(b)). When Qincreases, the dipo-\nlar and exchange interactions become more signi\fcant.\nThe interlayer coupling is then no longer attributable\nonly to spin pumping but is also caused by the long-range\ndipole-dipole interaction. This additional contribution to\nthe coupling is su\u000ecient to synchronize the n= 1 modes.\nThe relative amplitude between the two layers then be-\ncomes closer to 1 (see Fig. 3(b)). Again, we obtain an\nacoustic mode and an optical n= 1 mode, which can be\nobserved from the phase di\u000berence between the two lay-\ners in Fig. 3(c). The spin-pumping-induced coupling only\noccurs as long as the e\u000bective spin-di\u000busion length ~lsfis\nlarge or on the order of dN. Once this is no longer the\ncase, the modes rapidly decouple, and the system reduces\nto two separate FI jN systems with a relatively weak in-\nterlayer dipole coupling. In the limit of large QL1, the\nexchange interaction becomes dominant. The energy of\nthe wave is then predominantly attributable to the mo-\nmentum in the longitudinal direction, and the dynamic\npart of the magnetization goes to zero at the FI jN inter-10\nfaces, causing the renormalization attributable to spin\npumping to vanish.31\nWe also note that the dispersion relation depicted in\nthe inset of Fig. 3(a) reveals that the acoustic mode (blue\nline) exhibits a dip in energy at lower QL1than does the\noptical mode (red line). We suggest that this feature\ncan be understood as follows: The shift in the position\nof the energy dip can be interpreted as an increase in\nthe e\u000bective FI thickness for the acoustic mode with re-\nspect to that for the optical mode. When ~lsfis larger\nthan the NM thickness, the uniform mode behaves as\nif the NM were absent and the two \flms were joined.\nThis result indicates that the dispersion relation for the\nacoustic mode exhibits frequency behavior as a function\nofQ~L=2, where the e\u000bective total thickness of the \flm is\n~L=L1+L2. The optical mode, however, \\sees\" the NM\nand thus behaves as if ~L=L1. Consequently, the dip in\nthe dispersion occurs at lower QL1for the acoustic mode\nthan for the optical mode.\nB. MSSW\nFinally, let us study the dynamic coupling of mag-\nnetostatic surface spin waves (MSSWs). We now con-\nsider a perfectly symmetric system, FI(1000 nm) jN(200\nnm)jFI(1000 nm), with lsf= 350 nm. For such thick\n\flms, surface anisotropies may play an important role.\nWe therefore discuss a case in which we include a surface\nanisotropy of KS= 0:05 erg=cm2. According to the an-\nalytical result presented in Eq. (28), the lowest-energy\nmodes with QL1\u001c1 are exponentially localized at the\nFIjN surfaces, with a decay length of A=KS\u0018200 nm.\nWe now compute the eigenfrequencies, !, as a function\nof the wave vector in the range 10\u00004< QL 1<103. In\nFig. 4(a), we present the real part of the frequency for\nthe six lowest-energy modes with a positive real part, and\nin Fig. 4(b), we present the corresponding renormaliza-\ntions of the Gilbert damping for the four lowest-energy\nmodes. The dispersion relations indicate that the mode\npairs that are degenerate at QL1\u001c1 rapidly split in\nenergy when QL1approaches 10\u00002. Strong anticrossings\ncan be observed between the n= 1 andn= 2 modes.\nSuch anticrossings are also present between the surface\nmode and the n= 1 mode; they are almost too strong to\nbe recognized as anticrossings. The enhanced damping\nrenormalizations exhibit very di\u000berent behavior for the\ndi\u000berent modes. We recognize the large-\u0001 \u000bmode of one\npair as the surface optical mode and the low-\u0001 \u000bmode\nas the volume n= 1 acoustic mode. Without EASA,\nthe anticrossings in Fig. 4(a) would become crossings.\nThe lowest-energy modes at QL1\u001c1 would then cut\nstraight through the other modes. In the case considered\nhere, this behavior is now observed only as steep lines at\nQL1\u00180:05 and atQL1\u00180:5.\nWhenQis increased, the e\u000bective spin-di\u000busion\nlength decreases (see Eq. (17)), which reduces the spin-\npumping-induced coupling between the modes at largeQ. WhenQL1\u0018100, the coupling becomes so weak\nthat the two FIs decouple. This phenomenon can be ob-\nserved from the behavior of \u0001 \u000bin Fig. 4(b), where the\ndamping of the acoustic modes become the same as for\nthe optical modes.\nFIG. 4: (Color online) FI(1000nm) jN(200nm)jFI(1000nm)\nlsf= 350 nm, KS= 0:05 erg=cm2: a) The dispersion rela-\ntion as a function of QL1for the six lowest positive-real-part\nmodes. b) The renormalization of the damping attributable\nto spin pumping for the four lowest modes with frequencies\nwith positive real parts as a function of QL1. At largeQL1,\nthe computation becomes increasingly demanding, and the\npoint density of the plot becomes sparse. We have therefore\nindividually marked the plotted points in this region.\nIn the MSSW geometry, an isolated FI has magneto-\nstatic waves that are localized near one of the two sur-\nfaces, depending on the direction of propagation with\nrespect to the internal \feld.34Asymmetries in the exci-\ntation volume are therefore also expected for the trilayer\nin this geometry. In Fig. 5, we present the eigenvectors\nof the surface modes as functions of the transverse co-\nordinate\u0018for increasing values of the wave vector Q.\nAtQL1= 0:5, the modes have already begun to ex-\nhibit some asymmetry. Note that the renormalization\nof the damping observed in Fig. 4(b) is approximately\none order of magnitude larger than the intrinsic Gilbert\ndamping for the optical mode and that the damping of\nany one mode may vary by several orders of magnitude\nas a function of QL1.31Therefore, these e\u000bects should\nbe experimentally observable. The greatest damping oc-\ncurs when the two layers are completely decoupled; see\nFigs. Fig. 4(b) and 5. Because the damping of the opti-\ncal mode is equivalent to that of a system with a perfect\nspin sink, one might expect that the greatest damping11\nFIG. 5: (Color online)FI(1000nm) jN(200nm)jFI(1000nm),\nlsf= 350nm, KS= 0:05 erg=cm2: a) and b) present the\nreal parts of the xcomponents of the out-of-equilibrium mag-\nnetization vectors for the acoustic and optical surface modes,\nrespectively, for several values of QL1. For values of QL1&1,\nthe modes decouple and become localized in one of the two\nlayers. For large values of QL1\u0018100, the two modes are\nstrongly localized at one of the two FI jN interfaces, which\ncorrespond to the peaks in the damping that are apparent in\nFig. 4(b).\nshould occur for this mode. However, the large localiza-\ntion, which is achieved only at large QL1, in combination\nwith the vanishing of the e\u000bective spin-di\u000busion length\nleads to damping that is much greater than that of the\nsynchronized optical mode.\nV. CONCLUSIONS\nWe investigated the dynamic coupling of spin-wave ex-\ncitations, which are present in single FI thin \flms, pri-\nmarily through spin pumping and spin transfer but also\nthrough the dynamic demagnetization \feld created when\ntwo FI thin \flms are in contact via an NM layer. Because\nof this coupling, the modes are split into acoustical and\noptical excitations. When the NM is thin compared with\nlsf, the renormalization of the Gilbert damping vanishes\nfor the acoustic modes, whereas for the optical modes,\nthe renormalization is equally as large as for a single-\nFIjN system in which the NM is a perfect spin sink. A\nspin current pumped by a travelling magnetic wave has a\nwavelength of equal magnitude, which leads to traversal\npaths across the NM that are longer than the thickness\nof the NM. Consequently, the spin-memory loss is greater\nfor short-wavelength spin currents. This phenomenonleads to an e\u000bective spin-di\u000busion length in the NM that\ndecreases for increasing values of Q. As a result, the dy-\nnamic coupling strength is reduced for short-wavelength\nspin waves. At some critical value of Q, the coupling be-\ncomes so weak that the acoustic- and optical-mode con-\n\fgurations are lost in favor of modes that are localized\nin one of the two FIs. At these values of Q, the inter-\nlayer dipole coupling is also dominated by the intralayer\nexchange coupling. For these high-wave-number modes,\nthe system behaves similar to two separate FI jN(lsf= 0)\nsystems.\nWhen the two \flms are of di\u000berent thicknesses, the\nexchange energies of the higher-order transverse n > 1\nmodes di\u000ber between the two layers. Because of the rel-\natively small coupling attributable to spin pumping, the\nsynchronization of these modes at QL1\u001c1 requires that\nthe FI thicknesses be very similar. A small asymmetry\nbreaks the synchronization; however, for larger QL1\u00181,\nthe modes can again become coupled through interlayer\ndipole interaction. This coupling arises in addition to\nthe spin-pumping- induced coupling. For even larger Q,\nthe e\u000bective spin-di\u000busion length becomes small, and the\ncoupling attributable to spin pumping vanishes. The rel-\natively small dipole coupling alone is not su\u000ecient to\ncouple the modes when there is a \fnite di\u000berence in \flm\nthickness , and the synchronization breaks down.\nDepending on the quality of the interface between the\nFIs and the strength of the spin-orbit coupling in the\nNM , additional e\u000bective surface \felds may be present\nbecause of surface anisotropy energies. For the EASA\ncase, the lowest-energy modes are localized at the FI jN\nsurfaces. These modes couple in the same manner as the\nmacrospin modes. For \flms that are much thicker than\nthe decay length A=KS, the energies of the surface modes\ndo not depend on the \flm thickness. Consequently, the\ncoupling of these modes is independent of the thickness\nof the two FIs. Similar to the simpler FI jN system, the\ndamping enhancement may attain values as high as an or-\nder of magnitude larger than the intrinsic Gilbert damp-\ning. However, in the trilayer system, the presence of both\nacoustic and optical modes results in large variations in\nthe e\u000bective damping within the same physical sample.\nBecause of this wide range of e\u000bective damping, which\nspans a di\u000berence in \u0001 \u000bof several orders of magnitude\nas a function of Q, we suggest that trilayer modes should\nbe measurable in an experimental setting.\nWith more complicated FI structures in mind, we be-\nlieve that this work may serve as a guide for experimen-\ntalists. 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(9) can be rotated by the xyzcoordinate system\nwith the rotation matrix\nR=0\n@s\u0012\u0000c\u0012s\u0012\u0000c\u0012c\u001e\n0c\u001e\u0000s\u001e\nc\u0012s\u0012s\u001es\u0012c\u001e1\nA; (A1)where we have introduced the shorthand notation s\u0012\u0011\nsin\u0012,c\u0012\u0011cos\u0012and so on. We then get that\n(*\n^Gxyz=R^GRT\n=0\n@s2\n\u0012G\u0018\u0018\u0000c\u001es2\u0012G\u0018\u0010+c2\n\u0012c2\n\u001eG\u0010\u0010\u0000s\u001es\u0012G\u0018\u0010+s\u001ec\u001ec\u0012G\u0010\u0010s\u0012c\u0012G\u0018\u0018\u0000s\u0012c\u0012c2\n\u001eG\u0010\u0010+c\u001e(s2\n\u0012\u0000c2\n\u0012)G\u0018\u0010\n\u0000s\u001es\u0012G\u0018\u0010+s\u001ec\u001ec\u0012G\u0010\u0010 s2\n\u001eG\u0010\u0010 \u0000s\u001ec\u0012G\u0018\u0010+s\u001es\u0012c\u001eG\u0010\u0010\ns\u0012c\u0012G\u0018\u0018\u0000s\u0012c\u0012c2\n\u001eG\u0010\u0010+c\u001e(s2\n\u0012\u0000c2\n\u0012)G\u0018\u0010\u0000s\u001ec\u0012G\u0018\u0010+s\u001es\u0012c\u001eG\u0010\u0010c2\n\u0012G\u0018\u0018+s2\u0012c\u001eG\u0018\u0010+c2\n\u001es2\n\u0012G\u0010\u00101\nA:\n(A2)\nBecause we work in the linear respons regime the equilibrium magnetization should be orthogonal to the dynamic\ndeviation, mi\u0001^z= 0, it is therefor su\u000ecient to only keep the xypart of ^Gxyz. We then \fnd\n^Gxy=\u0012s2\n\u0012G\u0018\u0018\u0000c\u001es2\u0012G\u0018\u0010+c2\n\u0012c2\n\u001eG\u0010\u0010\u0000s\u001es\u0012G\u0018\u0010+s\u001ec\u001ec\u0012G\u0010\u0010\n\u0000s\u001es\u0012G\u0018\u0010+s\u001ec\u001ec\u0012G\u0010\u0010 s2\n\u001eG\u0010\u0010\u0013\n: (A3)\nAppendix B: Spin Accumulation\nThe functions \u0000 1(\u0018) and \u0000 2(\u0018) are taken directly from\nRef.39, and modi\fed to cover the more complicated mag-\nnetic texture model. We then have\n\u00001(\u0018)\u0011cosh\u0010\n\u0018=~lsf\u0011\ncosh\u0010\n\u0018=~lsf\u0011\n+\u001bsinh\u0010\n\u0018=~lsf\u0011\n=2g?~lsf;\n\u00002(\u0018)\u0011sinh\u0010\n\u0018=~lsf\u0011\nsinh\u0010\n\u0018=~lsf\u0011\n+\u001bcosh\u0010\n\u0018=~lsf\u0011\n=2g?~lsf:(B1)\nForQlsf\u001d1 the e\u000bective spin di\u000busion length becomes\nshort, \u0000 1!1 and \u0000 2!0 at the FIjN interfaces.\nAppendix C: E\u000bective spin di\u000busion length\nThe di\u000busion in the NM reads\n@t\u0016S=Dr2\u0016S\u00001\n\u001csf\u0016S; (C1)whereDis the di\u000busion constant and \u001csfis the spin \rip\nrelaxation time. We assume that the FMR frequency is\nmuch smaller than the electron traversal time, D=d2\nN, and\nthe spin-\rip relaxation rate, 1 =\u001csf.39This means the LHS\nof Eq. (C1) can be disregarded. In linear response the\nspin accumulation, which is a direct consequence of spin\npumping, must be proportional to the rate of change of\nmagnetization at the FI jN interfaces. We do the same\nFourier transform, as for the magnetization, so that \u0016\u0018\nexpfi(!t\u0000Q\u0010)g. The spin di\u000busion equation then takes\nthe form\n@2\n\u0018\u0016S=\u0012\nQ2+1\nD\u001csf\u0013\n\u0016S: (C2)\nThe spin di\u000busion length is then lsf=pD\u001csf, and\nby introducing the e\u000bective spin di\u000busion length ~lsf=\nlsf=q\n1 + (Qlsf)2one gets\n@2\n\u0018\u0016S=1\n~l2\nsf\u0016S: (C3)"    },    {        "title": "2312.13093v1.An_effective_field_theory_of_damped_ferromagnetic_systems.pdf",        "content": "Prepared for submission to JHEP\nAn effective field theory of damped ferromagnetic\nsystems\nJingping Li\nDepartment of Physics, Carnegie Mellon University, Pittsburgh, PA 15213\nE-mail: jingpinl@andrew.cmu.edu\nAbstract: Using the in-in formalism, we generalize the recently constructed magnetoelastic\nEFT [1] to describe the damping dynamics of ferromagnetic systems at long wavelengths. We find\nthat the standard Gilbert damping term naturally arises as the simplest leading-order symmetry-\nconsistentnon-conservativecontributionwithinthein-inframework. TheEFTiseasilygeneralized\nto scenarios with anisotropy and inhomogeneity. In particular, we find the classic Landau-Lifshitz\ndamping term emerges when isotropy is broken by a constant external background field. This\nprovides a first principle explanation for distinguishing the two types of damping dynamics that\nwere originally constructed phenomenologically. Furthermore, the EFT framework could also in-\ncorporate intrinsic anisotropy of the material in a straightforward way using the spurion method.\nFor systems with inhomogeneity such as nontrivial spin textures, we find that the leading order\nderivative correction yields the generalized Gilbert damping equations that were found in con-\ndensed matter literature. This shows that the EFT approach enables us to derive the form of\nhigher-derivative-order corrections in a systematic way. Lastly, using the phonon-magnon cou-\npling deduced in the magnetoelastic EFT, we are able to make a prediction for the generic form\nof the phononic contribution to the damping equation.arXiv:2312.13093v1  [hep-th]  20 Dec 2023Contents\n1 Introduction 1\n2 Review of the magnon EFT and Schwinger-Keldysh formalism 2\n2.1 Symmetry breaking in magnetoelastic systems 2\n2.2 The magnon EFT and the conservative equation of motion 4\n2.3 The Schwinger-Keldysh formalism 4\n3 Gilbert damping term from EFT 6\n3.1 Coupling at the leading order 6\n3.2 Gilbert damping 6\n4 More general materials 9\n4.1 Anisotropic materials 9\n4.2 Inhomogeneous materials 10\n5 Magnon damping term from phonons 11\n6 Conclusion and discussions 12\n1 Introduction\nIt has long been established that the methodology of coset construction serves as a powerful tool\nof relativistic effective field theories (EFTs) of Goldstone bosons (e.g. pions from spontaneously\nbroken approximate chiral symmetry) [2–5]. In recent years, many works have demonstrated\nthat its versatility is extendable to condensed matter systems where we are interested in the\nmacroscopic behavior which are usually the massless low energy excitations [6, 7]. Using this\napproach, a recent paper [1] constructed an EFT of magnetoelastic systems where the phonons\nand magnons are considered Goldstones associated with translations spontaneously broken by\nthe ground state location of the material (the lattice) and an SO(3)symmetry of the magnetic\nmoments by their ground state orientations. The EFT approach provides a systematic way to\nunderstand phonon-magnon interactions from first principles and predict the forms of higher-order\ncorrections which has not been done previously.\nWhile the paper was focused on conservative dynamics, there has also been extensive study\non the theoretical description of non-conservative dynamics of damped magnetic systems since\nthe seminal work of Landau, Lifshitz, and later Gilbert [8, 9]. However, to our knowledge, the\nprior works were mostly model-dependent phenomenological descriptions. It is therefore desirable\nto have a first principle derivation from a similar many-body EFT perspective.\nOn the other hand, the Schwinger-Keldysh formalism [10, 11] (and the related in-in formalism\n[12, 13]) has been known to describe the quantum field theory of open systems and hence fully\n– 1 –capable of describing dissipative effects. In recent years, its power has been successfully extended\nto the EFT framework of dissipative dynamics in astrophysics and black holes [14, 15, 17] that\nsystematically derives dissipative equations of motion. Therefore, it is natural to consider its\nutility in describing deriving damping equations in condensed matter EFTs.\nIn this paper, combining the power of the two techniques, we apply the Schwinger-Keldysh\nformalism to incorporate dissipative effects into the EFT of magnons to reproduce the known\nresults of magnetic damping. In section 2, we review the coset construction of the magnon EFT\nas well as the techniques in Schwinger-Keldysh formalism to be applied in this paper. Section 3\nderives the original Gilbert damping equation for homogeneous and isotropic materials. In section\n4, we move on to more general materials and recover the Landau-Lifshitz damping equation for\nanisotropic systems and generalized Gilbert damping for spatially inhomogeneous materials. In\nsection 5, we derive the damping terms originating from the magnon-phonon interaction.\nConventions: we use natural units where ℏ= 1. Unless specified, the uppercase Latin indices\nA, B, C . . . denote the full internal spin symmetry space which runs over 1,2,3while the lower\ncase ones a, b, c . . . in the begining alphabet index the broken subspace 1,2. Those in the middle\nalphabet i, j, k . . . run over the three spatial dimensions (to generalize to higher dimensions, the\ninternal spin symmetry will have to be modified accordingly).\n2 Review of the magnon EFT and Schwinger-Keldysh formalism\nIn this section, we first provide a self-contained review on the symmetries and the corresponding\ncoset construction of magnon-phonon EFT proposed by [1]. Furthermore, we summarize the\nSchwinger-Keldysh formalism which is the central tool for deriving the dissipative equations of\nmotion.\n2.1 Symmetry breaking in magnetoelastic systems\nWe follow the derivations in [1]. The symmetries under consideration are the spatial Galilean\ngroup (generated by translations Pi, rotations, Li, boost Ki) and internal symmetries (internal\ntranslations Ti, internal rotations Qi, spin rotations SA). The algebra of the generators is given\nby\n[Li, Kj] =iϵijkKk,[Li, Pj] =iϵijkPk, (2.1)\n[Ki, H] =−iPk,[Ki, Pj] =−iMδ ij, (2.2)\n[Qi, Tj] =iϵijkTk,[Qi, Qj] =iϵijkQk, (2.3)\n[SA, SB] =iϵijkKk,[Li, Lj] =iϵijkLk. (2.4)\nIn particular, the TiandQigenerators generate translations and rotations on the “comoving”\ncoordinates ϕI(x)(or the Lagrangian coordinates in continuum mechanics which simply gives an\ninitial labeling to the continuum)\nϕI(x)→ϕI(x) +aI, ϕI(x)→RI\nJϕJ(x), (2.5)\nandSAgenerates the internal rotation on the orientation of the spin\nNA→OA\nBNB, (2.6)\n– 2 –where O=eiχaSaand the Néel vector ⃗Nis the order parameter for the spin orientation.\nIn the ground state, the order parameters gain vacuum expectation values (VEVs) which we\nchoose to be D\n⃗ϕ(x)E\n=⃗ x,D\n⃗NE\n= ˆx3. (2.7)\nThe VEVs are not invariant under the transformations and hence spontaneously break some of\nthe symmetries\nUnbroken =\n\nH\nPi+Ti≡¯Pi\nLi+Qi\nS3\nM, Broken =\n\nKi\nTi\nQi\nS1, S2≡Sa. (2.8)\nThe parametrization of the ground state manifold which is simply the broken symmetry transfor-\nmations plus the unbroken translations is given by\nΩ =e−itHeixi¯PieiηiKieiπiTieiθiQieiχaSa, (2.9)\nwhere ηi,θi,χa, and πi=ϕi−xiare the corresponding Goldstone fields.\nFor any Goldstone fields ψicorresponding to the broken generators Xi, their covariant deriva-\ntives are the basic building blocks of the low energy EFT. They are systematically computed by\nthe coset construction using the Mauer-Cartan forms of the broken group\nΩ−1∂µΩ⊃\u0000\n∇µψi\u0001\nXi, (2.10)\nby computing the coefficients of the broken generator Xi. In addition, in the case that one broken\ngenerator X′appears in the commutation algebra of the other X’ with the unbroken translations\n\u0002¯P, X\u0003\n⊃X′, (2.11)\nit means that the two Goldstones are not independent and one of them can be eliminated. This\nis known as the inverse Higgs phenomenon.\nThe result of this exercise is that ηi,θiare eliminated and the only independent degrees of\nfreedom are the magnons χaand phonons πi, and at leading order in derivatives, they appear in\nthe following combinations:\n∇(iπj)= (D√\nDTDD−1)ij−δij, (2.12)\n∇tχa=1\n2ϵaBCn\nO−1h\n∂t−∂tπk(D−1)j\nk∂ji\nOo\nBC, (2.13)\n∇iχa=1\n2ϵaBC\u0000\nO−1∂iO\u0001\nBC, (2.14)\nwhere Dij=δij+∂iπj. [1] found the most general action for ferromagnetic material in the form\nL=c1\n2det (D)ϵabh\u0000\nO−1∂tO\u0001\nab−∂tπk\u0000\nD−1\u0001j\nk\u0000\nO−1∂jO\u0001\nabi\n(2.15)\n−1\n2Fij\n2(∇(iπj))∇iχa∇jχa−1\n2F3(∇(iπj))∇tχa∇tχa, (2.16)\nwhere the first term is similar to a Wess-Zumino-Witten (WZW) term that differs by a total\nderivative under the symmetry transformation.\n– 3 –2.2 The magnon EFT and the conservative equation of motion\nTo derive the equations of motion for magnons, it is more convenient to express the magnon fields\nin the nonlinear form\nˆn=R(χ)ˆx3= (sin θcosϕ,sinθsinϕ,cosθ), (2.17)\nwhere the two magnon fields are related to the angular fields by\nχ1=θsinϕ, χ 2=θcosϕ. (2.18)\nPhysically, this unit vector represents the direction of the magnetic moment. Under this repre-\nsentation, the pure magnon Lagrangian (in the absence of phonon excitations) becomes\nL →c2\n2ϵab\u0000\nO−1∂tO\u0001\nab+c6\n2(∂tˆn)2−c7\n2(∂iˆn)2, (2.19)\nwhere F3(0) = c6andFij\n2(0) = c7δij.\nThe dispersion relation for the quadratic Lagrangian has two solutions [1]\nω2\n+=\u0012c2\nc6\u00132\n+O(k2), ω2\n−=\u0012c7\nc2\u00132\nk4+O(k6). (2.20)\nFor ferromagnetic materials, where c2(c6c7)3/4, the first mode is gapped around the EFT cutoff\nscale, while the second has ω∼k2scaling and exits the EFT. In the long wavelength limit, we\nmay assign the scaling ∂1/2\nt∼∂ito the derivatives for ferromagnets.\nTo derive the equation of motion, we notice that an action of this form has a symmetry under\nthe infinitesimal spin rotation δˆn=⃗ ω׈n, where ⃗ ωis the constant infinitesimal parameter. It can\nbe shown that the Wess-Zumino-Witten term contributes to a total derivative ∂µ⃗Fµunder this\ntransformation. Using the equation for Noether current in Lagrangian mechanics\n⃗Jµ= ˆn×∂L\n∂∂µˆn−⃗Fµ, (2.21)\nwe find the conserved current\n⃗J0=−c2ˆn−c6∂tˆn׈n,⃗Ji=c7∂in׈n. (2.22)\nThe continuity equation ∂µ⃗Jµ= 0is then explicitly\nc2∂tˆn=−\u0000\nc6∂2\ntˆn−c7∇2ˆn\u0001\n׈n, (2.23)\nwhich is the equation of motion for Landau-Lifshitz model of magnetism.\n2.3 The Schwinger-Keldysh formalism\nThe appropriate formalism for non-conservative system is the so-called in-in or Schwinger-Keldysh\nformalism [10–13]. The basic idea is that there is an external sector Xthat the energy is dissipated\ninto since the total energy needs to be conserved. The external sector could evolve into any final\nstate which we do not observe, so all the dynamics are inclusive of the final states in the Hilbert\nspace of XX\nXout⟨Xin|. . .|Xout⟩⟨Xout|. . .|Xin⟩ ≡ ⟨. . .⟩in, (2.24)\n– 4 –and depends only on the initial state (hence the name in-in). We can generate an effective action\nfor the in-in observables via the Schwinger-Keldysh closed time path integral\nexp\u0014\niΓ[q; ˜q]\u0015\n=Z\ninitialDXD˜Xexp\u0014\niS[q, X]−iS[˜q,˜X]\u0015\n, (2.25)\nwhere we are integrating over an additional copy of variables ˜Xwhich corresponds to evolving\nback to the boundary conditions fixed at the initial time.\nThe equation of motion for the degrees of freedom in the observed sector qcan be derived\nfrom the action functional Γ[q; ˜q]by\nδ\nδqΓ[q; ˜q]\f\f\f\f\f\nq=˜q= 0. (2.26)\nAny external sector operator O(X)coupled to some operator in the observable sector F(q)by\nthe interaction termR\ndxO(X(x))F(q(x′))(where xis the corresponding spacetime coordinates)\nwould enter the equations of motion in terms of\n⟨O(X(x))⟩in=Z\ninitialDXD˜Xexp\u0014\niS[q, X]−iS[˜q,˜X]\u0015\nO(X), (2.27)\nwhere we have abbreviated ⟨O(X)⟩in≡ ⟨Xin|O(X)|Xin⟩. Just as in the perturbative quantum\nfield theory correlation functions calculated by Feynman propagators, this can be similarly calcu-\nlated using the Schwinger-Keldysh propagators\n⟨Oa(x)Ob(x′)⟩= \n⟨TO(x)O(x′)⟩ ⟨O (x′)O(x)⟩\n⟨O(x)O(x′)⟩ ⟨˜TO(x)O(x′)⟩!\n, (2.28)\nwhere Tand ˜Trepresent time and anti-time orderings. The sub-indices label the first and the\nsecond copy, which determines the relative time-ordering of the operators.\nExplicitly, the linear response gives\n⟨O(x)⟩=iZ\ndx′{⟨TO(x)O(x′)⟩ − ⟨O (x′)O(x)⟩}F\u0000\nq(x′)\u0001\n+O(F2). (2.29)\nor equivalently\n⟨O(x)⟩=Z\ndx′GR(x, x′)F\u0000\nx′\u0001\n, (2.30)\nwith the retarded Green’s function given by\nGR(x, x′) =iθ(t−t′)⟨[O(x),O(x′)]⟩\n=i(⟨TO(x)O(x′)⟩ − ⟨O (x′)O(x)⟩). (2.31)\nTherefore, the exact form of the damping term in the equation of motion would depend on the\ndetailed structure of these retarded response functions.\n– 5 –3 Gilbert damping term from EFT\n3.1 Coupling at the leading order\nThe composite operators Or(X)that encapsulate the external sector transform under arbitrary\nrepresentations (labeled by r), provided they form invariants of the unbroken SO(2)with the\nmagnon χaand derivatives. In order to achieve this, the operators have to be dressed with\nthe broken SO(3)/SO(2)subgroup parametrized by the Goldstones TR(χ)in the corresponding\nrepresentation\n˜Or(X)≡ Rr(χ)Or(X), (3.1)\nsuch that they transform covariantly under the unbroken subgroup [16].\nIn the long wavelength regime, the theory is organized by a spatial derivative expansion.\nIn fact, the simplest invariant operator at zeroth-order in the derivative expansion is the singlet\naligned along the ground state orientation ˆx3\nSint=Z\nd4x˜O3(X)≡Z\nd4xˆx3·˜⃗O(X), (3.2)\n(note that we are adopting manifestly relativistic notations for spacetime and energy-momentum\nfor convenience, albeit the system may or may not be relativistic). Equivalently, we may write\nSint=Z\nd4xˆn·⃗O(X) (3.3)\nwhere ˆn=O(χ)ˆx3as defined previously.\nAt the same order in this expansion, there could be more operators that can be added, such\nas when the operator is a two-index tensor operator O(2)and we may have combinations of the\nform ˆn· O(2)·ˆn. However, as will be explained in the next subsection, due to the constraint\nˆn2= 1, these will not lead to any new contributions, and from the perspective of the EFT, they\nare redundant operators. Hence, Eq. (3.3) is the only non-trivial operator at this order.\nFurthermore, we observe that the form of the operator restricts the possible external sector\nthat can couple in this way. For example, one can form such operators from fermions, where\n⃗O(ψ) =ψ†⃗ σψ+ψ⃗ σψ†, (3.4)\nwhere ⃗ σare the Pauli matrices acting as the intertwiner between the spinor and SO(3)spin space.\nOn the other hand, phonons cannot form operators in this form, and hence will not contribute in\nthis way.\n3.2 Gilbert damping\nThe undamped equation of motion is derived from the continuity equation\n∂µ⃗Jµ= 0 (3.5)\ncorresponding to the spin rotation transformation δˆn=⃗ ω׈n. To derive the non-conservative\nequation of motion, we need to find out how the additional term Eq. (3.3) affects the continuity\nequation.\n– 6 –The spin rotation transformation on ˆnalone is itself a valid symmetry for the pure magnon\naction Eq. (2.19), but the interaction term Eq. (3.3) is not invariant if we keep the external sector\nfixed. A standard trick of Noether theorem is that for an arbitrary symmetry transformation\nϕ(x)7→ϕ(x) +f(x)ϵ, if we promote the global symmetry variation parameter to be an arbitrary\nlocal variation ϵ(x), the total variation takes the form\nδS=−Z\nd4xJµ∂µϵ, (3.6)\nsuch that when ϵis a constant, invariance under the symmetry transformation is guaranteed\nδS= 0even off-shell (equation of motion is not satisfied). Integrate by parts, we find that\non-shell\nδS=Z\nd4x(∂µJµ)ϵ. (3.7)\nHowever, for an arbitrary ϵ(x), this is also the equation of motion, since\nδS=Z\nd4xδS\nδϕf(x)ϵ(x). (3.8)\nTherefore, the effect of an additional term in the action ∆Sis adding a term to the current\ndivergence\n∂µJµ= 0→∂µJµ+δ∆S\nδϕf(x) = 0 . (3.9)\nFor the spin rotations, the variation of the pure magnon EFT with local ω(x)is given by\nδS=Z\nd4x\u0010\n∂µ⃗Jµ\u0011\n·⃗ ω. (3.10)\nCorrespondingly, the addition of Eq. (3.3) leads to the modification\nδS=Z\nd4x\u0010\n∂µ⃗Jµ+ ˆn×⃗O(X)\u0011\n·⃗ ω. (3.11)\nThus, the (non-)continuity equation becomes\n∂µ⃗Jµ=−ˆn×⃗O(X). (3.12)\nWhen we focus on the measurements of the magnons, the effect of the external sector enters as\nan in-in expectation valueD\n⃗OE\nin. This may be evaluated via the in-in formalism, and the leading\norder contribution is given by\n⃗ ω·Z\nd4x∂µ⃗Jµ=−⃗ ω·Z\nd4xˆn×D\n⃗OE\nin=−⃗ ω·Z\nd4xˆn×Z\nd4x′\u0000\nGR·ˆn′\u0001\n, (3.13)\nwhere GR(t′, ⃗ x′;t, ⃗ x)is the retarded response function of the operator ⃗O.\nIn frequency space, we have GR(t, ⃗ x) =Rd3⃗kdω\n(2π)4e−iωt+⃗k·⃗ x˜GR(ω,⃗k)and furthermore, using the\nspectral representation (making the spin space indices explicit temporarily)\n˜GAB\nR(ω,⃗k) =Z∞\n−∞dω0\nπi\nω−ω0+iϵρAB(ω0,⃗k), (3.14)\n– 7 –we can separate the prefactor using the identity\ni\nω−ω0+iϵ=πδ(ω−ω0) +Pi\nω−ω0, (3.15)\ninto a δ-function and a principal part. The dissipative part is captured by the former\n˜GAB\nR,diss (ω,⃗k) =Z∞\n−∞dω0δ(ω���ω0)ρAB(ω0,⃗k) =ρAB(ω,⃗k). (3.16)\nTheindicesofthisspectralfunctionlivesinthespin SO(3)spaceand, forisotropicsystems, should\nbe built from invariant tensors δAB,ϵABC. However, the latter could not neither form a two-index\nobject nor respect parity invariance by itself, so the only symmetry-consistent possibility is\nρAB(ω,⃗k) =f(ω,|⃗k|2)δAB, (3.17)\nwith fassumed to be an analytic function of its arguments such that it has a smooth limit as\nωgoes to zero. Dissipative dynamics is antisymmetric under time reversal, so it should be odd\nunder the simultaneous transformation ω↔ −ωand(A, B)↔(B, A), meaning that the leading\norder contribution is given by\nρAB(ω,⃗k) =−iCωδAB(3.18)\nor in real spacetime\nGAB\nR,diss (t, ⃗ x) =C∂\n∂tδ(t)δ3(⃗ x)δAB. (3.19)\nCcould be understood as a Wilson coefficient in this non-conservative sector.\nFrom this, we arrive at the equation\n∂µ⃗Jµ=−Cˆn×∂\n∂tˆn. (3.20)\nCombining with the conservative part of the continuity equation Eq. (2.23), we find the Gilbert\ndamping equation\n∂\n∂t⃗ m=−γ ⃗ m×∂\n∂t⃗ m+. . . , (3.21)\nwhere γ=C/(c2ms),msˆn=⃗ m(for uniform materials), and the higher-order terms on the right-\nhand side of the original equation of motion Eq. (2.23) are contained in . . .which will be omitted\nin the following.\nWhen we have another singlet of the form ˆn·O(2)·ˆn, the effect of the extra ˆns is a replacement\nofthespectraldensity ρAB→ρACBDˆnCˆnD. ThetensorbasisstillconsistsofKroneckerdeltasince\nthe only structure that Levi-Civita tensors could contract to ˆnand would vanish automatically.\nTherefore, any additional structures will appear in the form of the inner product ˆn·ˆndue to\ncontractions with the Kronecker delta. Consequently, they do not lead to anything new due to\nthe normalization condition ˆn2= 1.\nWe see that using the in-in formalism, the form of the damping equation and the coeffi-\ncients are completely fixed by the principles of EFT: the symmetries, power counting, and Wilson\ncoefficients.\n– 8 –4 More general materials\nBy choosing the spectral function to depend only on invariant tensors and the frequency, we\nnaturally arrive at the Gilbert damping Eq. (3.21) which applies to isotropic and homogeneous\nsystems. However, in generic materials, we may be interested in situations with more general\nmaterials which for instance have non-trivial spin textures or highly anisotropic lattices and thus\ninhomogeneous or anisotropic. The advantage of the EFT framework is that these generalizations\ncan be systematically incorporated by including additional couplings. In this section, we explore\nseveral possibilities along these lines.\n4.1 Anisotropic materials\nFor homogeneous systems, one can still have anisotropy due to a background field. The retarded\nresponse function can then depend on the background. Given a homogeneous background vector\nfield⃗heff, the Levi-Civita tensor can now be incorporated into the response ρAB=DϵAB\nChC\neff.\nSince dissipative effects need to be antisymmetric under the simultaneous transformation A↔B,\nω↔ − ωandA↔Bantisymmetry is already included in the Levi-Civita tensor structure,\nthe response function has to be symmetric under ω↔ −ω. This means that the leading order\ncontribution is now independent of ω.1The corresponding response function is\nGA,B\nR,diss(t, ⃗ x) =DϵABChC\neffδ(t)δ3(⃗ x) (4.1)\nand gives rise to the conservation equation\n∂\n∂t⃗ m=−λ⃗ m×(⃗ m×⃗heff). (4.2)\nFor systems with conserved parity, the external field ⃗heffis an effective magnetic field in the sense\nof being parity odd to ensure an even-parity response function.\nThis damping equation involving an effective external background magnetic field is known\nas the Landau-Lifshitz damping equation [8]. We observe that from the EFT point of view, the\nLandau-Lifshitz and Gilbert damping terms are distinguished by symmetries.\nFor completeness, we note that in the literature there are generalizations to Gilbert damping\nby introducing anisotropic damping tensors. In field theoretic language, anisotropy corresponds\nto explicitly breaking of SO(3)and is thus straightforwardly realized in the EFT by a spurion\ncondensation. Instead of introducing a symmetry-breaking VEV to an explicit spurion operator\nin the action, one may assume the 2-point functions acquire a VEV and the most general resulting\ndissipative response function at LO is given by\nGA,B\nR,diss(ω) =SABω+AAB, (4.3)\nwhere SABandAABare general symmetric and antisymmetric tensors which are not SO(3)\ninvariant. Thegeneralizeddampingequationforananisotropicbuthomogeneousmagneticsystem\nthen become∂\n∂t⃗ m=−⃗ m×S·∂\n∂t⃗ m+⃗ m×A·⃗ m. (4.4)\n1This is analogous to the dissipative EFT of a spinning black hole in which case the role of this background is\nplayed by the direction of the spin vector [17].\n– 9 –The exact form of these anisotropy tensors can then be extracted from the microscopic details of\nthe given full theory.\n4.2 Inhomogeneous materials\nFor inhomogenous materials, e.g. configurations with background spin textures, we may have\ncontributions from higher orders in (spatial) derivative expansion. The simplest possible operator\nis given by the coupling\nSint=Z\nd4x∂iχa˜Oai(X). (4.5)\nIn terms of the orientation vector, this is equivalent to\nSint≈Z\nd4x(ˆn×∂iˆn)·⃗Oi(X), (4.6)\nat leading order in χ-field.\nAgain promoting the spin rotation transformation δˆn=⃗ ω׈nto a local parameter ⃗ ω(x), we\nfind an additional contribution to the current divergence\nδSint=Z\nd4x⃗ ω·(2∂iˆnˆn+ (ˆnˆn−δ)·∂i)·⃗Oi, (4.7)\nwhere we have used the fact that ˆn·∂iˆn=1\n2∂iˆn2= 0to simplify the expression. For the spectral\nfunction of the form\nρiA,jB(ω) =EωδijδAB, (4.8)\nthis leads to the damping term\nc2∂tˆn=E\u0010\n2∂iˆnˆn·(∂tˆn×∂iˆn) + (ˆnˆn−δ)·∂t\u0010\nˆn×⃗∇2ˆn\u0011\u0011\n=E\u0010\n2ˆn×(ˆn×∂iˆn) (ˆn×∂iˆn)·∂tˆn+ (ˆnˆn−δ)·∂t\u0010\nˆn×⃗∇2ˆn\u0011\u0011\n, (4.9)\nwhere we have used that ˆn×(ˆn×∂iˆn) =−∂iˆnon the first term. More compactly, this is\n∂\n∂t⃗ m=⃗ m×A·∂\n∂t⃗ m+E(ˆnˆn−δ)\nc2ms·∂t\u0010\n⃗ m×⃗∇2⃗ m\u0011\n, (4.10)\nwhere the first term contains the generalized damping tensor\nA=2E\nc2ms(⃗ m×∂i⃗ m) (⃗ m×∂i⃗ m). (4.11)\nThis corresponds to the generalized Gilbert damping in the presence of non-trivial spin textures\n(i.e. when ∇⃗ m̸= 0) [18].\nWe note that Eq. (4.6) is only the leading order derivative correction to the damping dynam-\nics. The EFT framework is capable of systematically generating higher derivative corrections. For\nexample, other types of inhomogeneity may be attributed to interactions in the lattice model [19]\nby\n⃗ mi×X\nijGij·⃗ mj, (4.12)\n– 10 –where the i, jindices label the lattice sites associated with the magnetic moments. In the contin-\nuum field theory, the lattice variables become ⃗ mi7→⃗ m(⃗ xi)and the tensorial structure becomes\nthe response function Gij7→GR,diss (t, ⃗ xi−⃗ xj), except that unlike the one in Eq. (3.19), it is\nnon-local (no longer proportional to δ(⃗ x)). In the simpler case that the long-range coupling falls\noff sufficiently quickly, these terms are traded for a series expansion\n⃗ m×X\nnAi1...in∂n\ni1...in∂t⃗ m, (4.13)\nfor some coefficient tensors Ai1...in. In terms of the action, this means the spectral functions are\nnow dependent on the wave vectors\nρAB(ω,⃗k) =X\nn˜Ai1...inki1. . . k inωδAB. (4.14)\n5 Magnon damping term from phonons\nFrom the magnetoelastic EFT, the generic magnon-phonon couplings are given by [1]\nLph=−1\n2Fij\n2(∇(iπj))∂iˆn·∂jˆn+1\n2ρ˜F3(∇(iπj))Dtˆn·Dtˆn, (5.1)\nwhere Dt≡∂t+vi∂iwith the velocity of the material given by vi=−∂tϕ(D−1)i\nj. The \"full\ntheory\" (technically the EFT at the next level of the hierarchy) action constrains the form of the\ncouplings between magnons the external sector to be\nLph=1\n2∂iˆn·∂jˆnOij\n2(π) +1\n2∂tˆn·∂tˆnO3(π) +1\n2∂tˆn·∂iˆnOi\n4(π), (5.2)\nwhere the last term arises from the linear-in- vcontribution in the expansion Dtˆn·Dtˆn.\nFor ferromagnets, the dispersion relation dictates the first term to be dominant. After inte-\ngrating out the external sector using the Schwinger-Keldysh methodR\nDπD˜π, we find its contri-\nbution to the in-in equation of motion is given by\nc2∂tˆn= ˆn×∂i\u0010\n∂jˆnD\nOij\n2E\u0011\n, (5.3)\nwhere the in-in expectation value is given by\nD\nOij\n2(x)E\n=Z\nd4x′Gij,kl\nR,2(x−x′)∂kˆn(x′)·∂lˆn(x′). (5.4)\nFor 4-index tensor structures under SO(3), there are two invariant tensors corresponding to the\nsymmetric-traceless and trace irreps. Therefore, one may write the leading-order dissipative con-\ntribution to the retarded response function as\nGij,kl\nR,2(x)≃\u00121\n2δijδklC2+δi(kδl)jD2\u0013\nδ3(⃗ x)∂tδ(t), (5.5)\nwhere C2andD2are the independent (Wilson) coefficients.\n– 11 –Substituting the results, we find the damping equation in a similar form\nc2∂tˆn=1\n2C2ˆn×∂i\u0000\n∂iˆn∂t\u0000\n∂jˆn·∂jˆn\u0001\u0001\n+D2ˆn×∂i\u0000\n∂jˆn∂t\u0000\n∂iˆn·∂jˆn\u0001\u0001\n=C2ˆn×∂i\u0000\n∂iˆn∂jˆn·∂j∂tˆn\u0001\n+D2ˆn×∂i\u0000\n∂jˆn∂iˆn·∂j∂tˆn\u0001\n+D2ˆn×∂j\u0000\n∂iˆn∂iˆn·∂j∂tˆn\u0001\n,(5.6)\nor more compactly\n∂\n∂t⃗ m=⃗ m×D·∂\n∂t⃗ m, (5.7)\nwhere the \"damping tensor\" Dis given by\nD=1\nc2msh\n∂i\u0000\nC2∂iˆn∂jˆn+D2∂jˆn∂iˆn\u0001\n+D2∂j\u0000\n∂iˆn∂iˆn\u0001\n+\u0000\nC2∂iˆn∂jˆn+D2∂jˆn∂iˆn\u0001\n∂i+D2∂iˆn∂iˆn∂ji\n∂j. (5.8)\nWe notice that the form of the couplings restricts the damping tensor to appear at higher-\norders in derivative expansions and hence they are expected to be small compared with contribu-\ntions from fermions (e.g. electrons) in the long wavelength limit. However, for insulating materials\nthat have electron-magnon coupling suppressed, we expect their effects to be more significant.\n6 Conclusion and discussions\nIn this paper, we used the in-in (Schwinger-Keldysh) formalism to generalize the recently con-\nstructed EFT of magnetoelasticity [1] to describe damped magnetic dynamics. We discover that\nthe Gilbert damping term naturally arises as the simplest symmetry consistent dissipative cor-\nrection within the in-in formalism. Systematic generalizations to anisotropic and inhomogeneous\nsetups also yield desired results such as the Landau-Lifshitz magnetic damping equation. More-\nover, we are able to predict the form of phononic contribution to the damping dynamics. Thus\nwe have shown that this is a useful framework to derive dissipative dynamics from first principles\nand to predict the forms of higher-order corrections in a systematic way.\nIt would be interesting to investigate the explicit full theory model of the “external sector”\nsuch as the fermionic fields in Eq. (3.4) and extract the relevant Wilson coefficients by matching\nthe response functions. In this way, we may gain better insights into what controls the damping\nparameter and give more predictive power to the EFT approach. It would also be interesting to\nmatch the relevant coefficients in Eq. (4.14) to obtain an EFT framework for a generalized class\nof models.\nFurthermore, various applications of the magnetoelastic EFT [20, 21] have appeared more\nrecently. It would be interesting to investigate the effects of adding dissipative terms into these\nproblem. There are also further developments in the technical aspects of such EFTs [22, 23]. It\nis natural to consider their implications on the non-conservative sector. We leave these problems\nfor future works.\nAcknowledgement\nThe author thanks Ira Rothstein for advising throughout the project and a careful reading of the\nmanuscript. The author also thanks Riccardo Penco for important discussions, Shashin Pavaskar\n– 12 –for other useful discussions, and Witold Skiba for comments on the draft. This work is partially\nsupported by the grants DE- FG02-04ER41338 and FG02- 06ER41449.\nReferences\n[1] S. Pavaskar, R. Penco, I. Z. Rothstein, An Effective Field Theory of Magneto-Elasticity , SciPost\nPhys.12.5.155 (2022), arXiv:2112.13873 [hep-th].\n[2] S. R. Coleman, J. Wess, and B. Zumino, Structure of phenomenological Lagrangians. 1. , Phys.Rev.\n177 2239 (1969).\n[3] J. Callan, Curtis G., S. R. Coleman, J. Wess, and B. Zumino, Structure of phenomenological\nLagrangians. 2. , Phys.Rev. 177 2247 (1969).\n[4] D. V. Volkov, Phenomenological Lagrangians , Fiz. Elem. Chast. Atom. Yadra 4 3 (1973).\n[5] V. I. Ogievetsky, Nonlinear Realizations of Internal and Space-time Symmetries , Proc. of. X-th\nWinter. School of Theoretical Physics in Karpacz, Vol. 1, Wroclaw 227 (1974) .\n[6] M. Baumgart et. al., Snowmass White Paper: Effective Field Theories for Condensed Matter\nSystems, in2021 Snowmass Summer Study. 10, 2022, arXiv:2210.03199[hep-ph].\n[7] T. Brauner et. al., Snowmass White Paper: Effective Field Theories for Condensed Matter Systems ,\nin2022 Snowmass Summer Study. 3, 2022„ arXiv:2203.10110[hep-th].\n[8] L. D. Landau and E. M. Lifshitz, Theory of the dispersion of magnetic permeability in ferromagnetic\nbodies, Phys. Z. Sowjetunion. 8, 153 (1935).\n[9] T. L. Gilbert, A phenomenological theory of damping in ferromagnetic materials , IEEE\nTransactions on Magnetics, vol. 40, no. 6, (2004).\n[10] J. Schwinger, Brownian Motion of a Quantum Oscillator , J. Math. Phys. 2 407 (1961).\n[11] L. V. Keldysh, Diagram technique for nonequilibrium processes , Zh. Eksp. Teor. Fiz. 47 1515 (1964).\n[12] C. R. Galley, The classical mechanics of non-conservative systems , Phys. Rev. Lett. 110, 174301\n(2013), arXiv:1210.2745 [gr-qc].\n[13] C. R. Galley, D. Tsang, and L. C. Stein, The principle of stationary nonconservative action for\nclassical mechanics and field theories (2014), arXiv:1412.3082 [math-ph].\n[14] S. Endlich, R. Penco, An effective field theory approach to tidal dynamics of spinning astrophysical\nsystems, Phys. Rev. D. 93.064021 (2016), arXiv:1510.08889 [gr-qc].\n[15] W. D. Goldberger and I. Z. Rothstein, Horizon radiation reaction forces , JHEP 10 026 (2020),\narXiv:2007.00731[hep-th].\n[16] L. V. Delacrétaz, S. Endlich, A. Monin, R. Penco, F. Riva, (Re-)Inventing the Relativistic Wheel:\nGravity, Cosets, and Spinning Objects , JHEP 11 (2014) 008, arXiv:1405.7384 [hep-th].\n[17] W. D. Goldberger, J. Li, and I. Z. Rothstein, Non-conservative effects on spinning black holes from\nworld-line effective field theory , JHEP 06 053 (2021) arXiv:2012.14869[hep-th].\n[18] S. Zhang, S. S.-L. Zhang, Generalization of the Landau-Lifshitz-Gilbert Equation for Conducting\nFerromagnets , Phys. Rev. Lett. 102, 086601 (2009).\n[19] S. Brinker, M. dos Santos Dias, S. Lounis, Generalization of the Landau-Lifshitz-Gilbert equation by\nmulti-body contributions to Gilbert damping for non-collinear magnets , J. Phys.: Condens. Matter\n34 285802 (2022), arXiv:2202.06154 [cond-mat.mtrl-sci].\n– 13 –[20] A. Esposito, S. Pavaskar, Optimal anti-ferromagnets for light dark matter detection (2022),\narXiv:2210.13516 [hep-ph].\n[21] S. Pavaskar, I. Z. Rothstein, The Dynamics of Line Defects and Their Sensitivity to the Lattice\nStructure (2022), arXiv:2212.10587 [hep-th].\n[22] A. Nicolis, I. Z. Rothstein, Apparent Fine Tunings for Field Theories with Broken Space-Time\nSymmetries (2022), arXiv:2212.08976 [hep-th].\n[23] C. O. Akyuz, G. Goon, R. Penco, The Schwinger-Keldysh Coset Construction (2023),\narXiv:2306.17232 [hep-th].\n– 14 –"    },    {        "title": "2307.00903v1.Magnetic_lump_motion_in_saturated_ferromagnetic_films.pdf",        "content": "Magnetic lump motion in saturated ferromagnetic films\nXin-Wei Jin,1, 2Shi-Jie Shen,2Zhan-Ying Yang,1, 3and Ji Lin2,∗\n1School of Physics, Northwest University, Xi’an 710127, China\n2Department of Physics, Zhejiang Normal University, Jinhua 321004, China\n3Peng Huanwu Center for Fundamental Theory, Xi’an 710127, China\n(Dated: July 4, 2023)\nIn this paper, we study in detail the nonlinear propagation of magnetic soliton in a ferromagnetic film. The\nsample is magnetized to saturation by an external field perpendicular to film plane. A new generalized (2+1)-\ndimensional short-wave asymptotic model is derived. The bilinear-like forms of this equation are constructed\nand exact magnetic line soliton solutions are exhibited. It is observed that a series of stable lumps can be\ngenerated by an unstable magnetic soliton under Gaussian disturbance. Such magnetic lumps are highly stable\nand can maintain their shapes and velocities during evolution or collision. The interaction between lump and\nmagnetic soliton, as well as interaction between two lumps, are numerically investigated. We further discuss\nthe nonlinear motion of lumps in ferrites with Gilbert-damping and inhomogeneous exchange effects. The\nresults show that the Gilbert-damping effects make the amplitude and velocity of the magnetic lump decay\nexponentially during propagation. And the shock waves are generated from a lump when quenching the strength\nof inhomogeneous exchange.\nI. INTRODUCTION\nThe propagation of electromagnetic wave in ordered\nmagnetic materials, especially in a ferromagnetic medium,\nplays a vital role in faster and higher density storage fields [1–\n3]. In particular, magnetic soliton(MS), which exists in both\nferro- and antiferro-magnets, is becoming a very promising\ninformation carrier because of its particle-like behavior and\nmaneuverability [4–9]. In the past few decades, a wide range\nof soliton-type propagation phenomena has been theoretically\npredicted [10–13], and some of them have been confirmed\nexperimentally [14, 15].\nIndeed, wave propagation in ferromagnetic media is well-\nknown as a highly nonlinear problem. A complete description\nof all types of nonlinear excitations is governed by the\nMaxwell equations coupled with Landau-Lifschitz equation.\nFor this moment, let us notice that a fully nonlinear theory has\nnot been developed. But the linear theory for sufficiently small\namplitudes was established and validated experimentally [16].\nIn order to obtain results valid in nonlinear regimes, or at\nleast weakly nonlinear, one has to resort to intermediate\nmodels (by introducing a small perturbative parameter related\nto the soliton wavelength) [17]. These models include long-\nwave model [18–20], modulational asymptotic model [21],\nand short-wave model [22–25]. Both long-wave model and\nmodulational asymptotic model are mainly used to explain\nand predict the behavior of large-scale phenomena owing to\ntheir long-wave-type approximate condition [26]. However,\nthis condition is not always applicable because the scale of\nmagnetic materials and devices are getting more refined and\nmore sophisticated. Moreover, the main practical interest of\nferrites is that they propagate microwaves [27, 28]. On the\ncontrary, from the viewpoint of applied physics, the short-\nwave-type approximation is much more relevant to available\nexperiments than the former one.\nSince Kraenkel et al. first proposed the short-wave model\n[29], quite a few related nonlinear evolution equations have\nbeen derived, which belong to the Kraenkel-Manna-Merle\n(KMM) system [22, 23, 30–32]. Some significant works\n∗Corresponding author: linji@zjnu.edu.cnhave been devoted to searching and explaining different\nexcitation patterns of ferromagnetic insulators. As for (1+1)-\ndimensional KMM system, the existence of multi-valued\nwaveguide channel solutions has been verified, and the\nnonlinear interaction properties were investigated between\nthe localized waves alongside the depiction of their energy\ndensities [22]. By applying the Hirota bilinear transformation\nmethod, the one- and two-soliton solutions were constructed\nwhile studying in details the solitons scattering properties\n[23]. This system is also solvable using the inverse scattering\nmethod [25]. It is noteworthy that this system possesses\nthe loop-soliton and spike-like soliton [33, 34], and the\nmagnetic loop-soliton dynamics have been extensively studied\n[35–37]. The propagation of electromagnetic waves in\nhigher-dimensional ideal ferromagnets has also been studied,\ncorresponding to the (2+1)-dimensional KMM system [26, 31,\n38, 39]. The analytical one-line-soliton solution as well as its\ntransverse stability have been reported [26]. It has been shown\nthat these structures were stable under certain conditions.\nOn the other hand, most previous studies have only focused\non the propagation of MS in ideal ferrites, which means\nsome important properties of the magnetic material were\nneglected. The main reason is that the nonlinear wave\nequation describing the propagation of electromagnetic waves\nin non-ideal ferromagnetic materials is no longer integrable.\nHowever, the Gilbert-damping and inhomogeneous exchange\neffects are essential features in a real ferromagnetic film, and\ntheir connection with MS motion is an important issue that has\nnot been explored so far. In this paper, we aim to investigate\ntheoretically and numerically the dynamics of the MS in a\nferromagnetic film including damping and the inhomogeneous\nexchange effect. The rest of this paper is organized as follows.\nIn Section 2, we review the physical background and derive\na new (2+1)-dimensional short-wave asymptotic model in\nferromagnetic media. In Section 3, the bilinear-like form\nof the reduced system is constructed and the analytical MS\nsolutions are acquired. In Section 4, the transmission stability\nof the magnetic soliton is numerically explored. The results\nshow that an unstable MS will split to some magnetic lumps\nby a small perturbation. The motions of these lumps under\nthe influence of damping and inhomogeneous exchange are\nanalysed in detail. We end this work in Section 5 with a brief\nconclusion and perspectives.arXiv:2307.00903v1  [nlin.PS]  3 Jul 20232\nII. PHYSICAL BACKGROUND\nA. Basic equations\nThe physical system under consideration is a saturated\nmagnetized ferrite film lying in the x−yplane, as shown in\nFig. 1. Different from Ref. [32], we consider the external\nfield H∞\n0perpendicular to the film, i.e., M0= (0,0,m). So the\ntransverse drift is avoided. The typical thickness of the film\nis about 0.5mm, and the width is approximately 10mm. We\nassume the propagation distance is large enough with regard\nto the wavelength, say more than 50cm. The evolution of the\nmagnetic field Hand the magnetization density Mis governed\nby the Maxwell equations coupled with Landau-Lifschitz-\nGilbert equation, which read as\n−∇(∇·H)+∆H=1\nc2∂2\n∂t2(H+M), (1a)\n∂\n∂tM=−γµ0M×Heff+σ\nMsM×∂\n∂tM, (1b)\nwhere c=1/p\nµ0˜εis the speed of light with the scalar\npermittivity ˜εof the medium, γis the gyromagnetic ratio,\nµ0being the magnetic permeability of the vacuum, σis the\ndamping constant, and Msis the saturation magnetization. The\neffective field Heffis given by [30]\nHeff=H−βn(n·M)+α∆M. (2)\nHere αandβare the constants of the inhomogeneous\nFigure 1. Ferrite film under consideration. The sample is magnetized\nto saturation by long strong magnetic field H∞\n0applied in the\nz-direction. The x-direction of the short wave propagation is\nperpendicular to the direction of static magnetization.\nexchange and the magnet anisotropy ( β>0 corresponds to\nthe easy-plane case), respectively. For a simple tractability, the\nunit vector nof the anisotropy axis is assumed to be along the\nzaxis (i.e., n≡ez). In order to transform the above systems to\ndimensionless equation, we rescale the quantities M,H, and t\nintoµ0γM/c,µ0γH/c, and ct. Thus, the constants µ0γ/cand\ncin Eqs.(2) and (3) are replaced by 1, Msbym=µ0γMs/c,\nandσby˜σ=σ/µ0γ[30].\nB. Linear analysis\nTo study the linear regime we look at a small perturbation of\na given solution. Equations (1) are linearized about the steady\nstate:\nM0= (0,0,m),H0=µM0. (3)where µis the strength of the internal magnetic field. Before\nproceeding further we assume that the ferromagnetic materials\nhave weak damping ¯σ∼ε˜σ. The exchange interaction\nparameter αand anisotropy parameter βare of order ε2and\nε3, respectively (i.e. ¯α=ε2α,¯β=ε3β). Let us seek for\nthe plane wave perturbation solution propagating along the x-\ndirection such as\nM=M0+εmexp[i(kx+ly−ωt)],\nH=H0+εhexp[i(kx+ly−ωt)],(4)\nwhere kandlare the wave numbers in the xandydirections, ω\nis the frequency. Vectors m= (mx,my,mz)andh= (hx,hy,hz)\nare arbitrary real scalar quantities.\nSubstituting Eq. (4) into (1) and (2) in the linear limit, it is\nreduced to\n\nω20 0 ω2−l2kl 0\n0 ω20 kl ω2−k20\n0 0 ω20 0 ω2−k2−l2\n−iωmµ 0 0 −m 0\n−mµ−iω 0 m 0 0\n0 0 −iω 0 0 0\n·\nmx\nmy\nmz\nhx\nhy\nhz\n=0\nThen we obtain the following dispersion relation\nm2(µ+1)\u0002\nµ(k2+l2−ω2)−ω2\u0003\n−ω2(k2+l2−ω2) =0\n(5)\nNote that we focus on studying the short-wave approximation\nk→∞[2]. It comes k0∼ε−1through a small parameter ε≪1\nlinked to the magnitude of the wavelength. Consequently, the\nfrequency expands accordingly as\nω=ω−1ε−1+ω1ε+ω3ε3+.... (6)\nThis assumption guarantees the phase velocity ω(k)/kand\nthe group velocity ∂ω/∂kare always bounded [3]. Now,\nreplacing Eq. (6) into the dispersion relation above, we obtain\na set of equations:\n•At order of ε−4:ω−1=±k0\n•At order of ε−2:ω1=\u0002\n(µ+1)m2+l2\u0003\n/2k0\n•higher order equations which determines ω3,ω5,...\nThe direction of the wave propagation is assumed to be\nclose to the xaxis, thus yvariable gives only account of a\nslow transverse deviation[40, 41]. Therefore lis assumed\nto be very small with respect to kand we write l=l0of\norder 0 with respect to ε. The phase up to order εis thus\n(x−t)/ε+l0y−εω1t,which motivates the introduction of\nnew variables:\nζ=1\nε(x−Vt),y=y,τ=εt. (7)\nThe variable ζdescribes the shape of the wave propagating at\nspeed V; it assumes a short wavelength about 1 /ε. The slow\ntime variable τaccounts for the propagation during very long\ntime on very large distances with regard to the wavelength.\nThe transverse variable yhas an intermediate scale, as in KP-\ntype expansions [26, 41]\nC. Multiple scale approach\nIn order to derive the nonlinear model, fields MandHare\nexpanded in power series of εas\nM=M0+εM1+ε2M2+ε3M3+...,\nH=H0+εH1+ε2H2+ε3H3+....(8)3\nwhere M0,H0,M1,H1,...are functions of (ζ,y,τ).\nWe consider the boundary conditions: lim\nζ→−∞M0=\n(0,0,m),lim\nζ→−∞Mj=lim\nζ→−∞Hj=0,(j̸=0). We derive\nthe following expressions by substituting Expansions (8) into\nequation (1):\n•At order ε−2:\nM0is a constant vector M0=(0,0,m),\n•At order ε−1:\nHx\n0=0,My\n1=0,Mz\n1=0,\n•At order ε0:\nMx\n1ζ=mHy\n0,\nMx\n2ζζ=−Hx\n2ζζ−Hy\n1ζτ\nMy\n2ζζ=−Hx\n1ζy+Hx\n0ζy\nMz\n2ζζ=Hz\n2ζτ+Hz\nyy\n•At order ε1:\nMx\n2ζ=−mHy\n1\nMy\n2ζ=m¯αMx\n1ζζ+¯σM1ζx−Mx\n1Hz\n0+mHx\n1\nMz\n2ζ=Mx\n1Hy\n0\nlet us introduce some independent variables XandTdefined\nasX=−mζ/2,Y=my,T=mτ.\nAfter eliminating H2andM2, we finally obtain the (2+1)-\ndimensional KMM equation:\nCXT=−BBX+CYY,\nBXT=BCX+BYY−sBX+ρBXX,(9)\nwhere observables B,Cand constants s,ρare defined by\nC=−X−ZX\n(Hz\n0/m)dX,B=Mx\n1/2m,\ns=−¯σ/2,ρ=¯αm2/4.(10)\nThis equation is new, which describes the evolution of\nmagnetization field Mand magnetic field Hwithin a ferrite\nfilm in presence of Gilbert-damping and inhomogeneous\nexchange. The quantities H0andM1refer to the zeroth and\nfirst-order expansion coefficients of the external magnetic field\nand the magnetization, respectively. For some simplicity,\nin the next, the independent variables X,YandTwill be\nrewritten as their lower cases x,yandt, respectively.\nIII. HIROTA’S BILINEARIZATION AND SOLITON\nSOLUTIONS OF THE (2+1)-DIMENSIONAL KMM\nEQUATION\nTo explore soliton solutions for the (2+1)-dimensional\nKMM equation (9), we consider a specific dependent variable\ntransformation\nB=G\nF,C=δx−2(lnF)t−2(lnF)y, (11)\nwhere δis an arbitrary constant. Consequently, the bilinear-\nlike forms of the (2+1)-dimensional KMM equation can be\nderived as follow\nF·(DxDt+sDx−D2\ny)G·F+G·(DxDy+D2\ny)F·F=δF2G\n(12a)\n∂x\u0014G2\n2F2−(DyDt+D2\nt)F·F\nF2\u0015\n+∂y\u0014(DyDt+D2\nt)F·F\nF2\u0015\n=0\n(12b)where G,Fare all differential functions of (x,y,t)to be\ndetermined. The symbols Dx,Dtrefer to the Hirota’s operators\nwith respect to the variable x,t, respectively. In order to\nconstruct the solitary wave solutions of Eq.(6), we expand\nGandFwith respect to a formal expansion parameter as\nG=εG1+ε3G3+ε5G5+...,F=1+ε2F2+ε4F4+ε6F6+...,\nin which εis a perturbation parameter and functions Gi,Fi,(i=\n1,2,3,...)are expansion coefficients of the above series. The\none-soliton solution could be constructed by truncating the\nperturbation expansion of GandFas follow\nG=eη1,F=1+k2A2\n16δ2e2η1. (13)\nSubstituting these expressions into Eq.(9) and solving the\nbilinear system recursively, in the absence of damping,\nthe analytical one-soliton solution of the (2+1)-dimensional\nKMM equation can be transformed into\nB=2δ\nksech(η1+η0),C=δx−2δ\nk[tanh(η1+η0)+1],\n(14)\nwhere η1=kx+ly+ [(l2−kl)/2k]t,η0=ln(k/4δ),kandl\nare arbitrary real constants. It should be noted that this soliton\nsolution exists only when the damping is neglected (s=0).\nSimilar to the procedure for constructing one-soliton solution,\nthe two-soliton solution can be given by treating the truncated\nperturbation expansions of GandFas\nG=A1eξ1+A2eξ2+C12eξ1+2ξ2+C21e2ξ1+ξ2, (15a)\nF=1+B11e2ξ1+B22e2ξ2+B12eξ1+ξ2+E12e2ξ1+2ξ2,(15b)\nwhere A1,A2,k1,k2are real constants, ξi=kix+liy+\u0002\n(l2\ni+δ)/ki\u0003\nt,(i=1,2), and the remaining parameters have\nthe following forms:\nBii=A2\nik2\ni\n16δ2,B12=A1A2\n2δ2k2\n1k2\n2\nk2+,k1l2=k2l1,\nCi j=AiA2\nj\n16δ2k2\njk2\n−\nk2+,E12=A2\n1A2\n2\n256δ4k2\n1k2\n2k4\n−\nk4+,(16)\nwhere k+=k1+k2,k−=k1−k2. Parameters Ai,Aj,ki,kj\nandli,(i=1,2,j=3−i)are arbitrary real constants.\nIV . NUMERICAL INVESTIGATION OF LINE-SOLITON\nAND MAGNETIC LUMPS\nA. Unstable MS splits into lumps\nWe now turn to the stability and interactions between MSs\nin a ferromagnetic film. The initial data is a MS perturbed\nby some position-dependent Gaussian wave packets with the\nfollowing expression:\nf=bexp\"\n−\u0012x−x0\nxr\u00132\n−\u0012y\nyr\u00132#\n, (17)\nwhere b,xrandyrcorrespond to the shape of the wave packet\nandx0is related to the perturbation position.\nThe time evolution results clearly show the instability of\nthe MS. For small bi, the MS will break up and eventually4\n(a)\n (b)\n(c)\n (d)\nFigure 2. Propagation of MS perturbed by a Gaussian disturbance.\n(a) Component Hz, (b) Component Hy, (c) and (d) are enlarged views\nof the indicated areas circled in red and black, respectively. The\nparameters are chosen as A1=A2=1,δ=−1,l1=l2=0,k1=\n1,k2=2,x0=−29,b=0.1,xr=1.5,yr=2.5 in (16) and (17).\nevolve into some stable two-dimensionally localized lumps , as\ndisplayed in Figs. 2(a) and 2(b). We observe that most of the\nenergy is always propagated as a lump, even if its speed may\ndiffer from the input. Such a magnetic lump is a solitary wave\npacket that maintains its shape and speed during propagation\nor collision.\nA complete single lump of magnetic field component Hz\n(component Hy) is circled in red (black) in Fig.2. The enlarged\nviews (see Figs.2(c) and 2(d)) provide a clear picture of the\nshape and contour map of the lump. It can be found that\ncomponent Hzis a dipole-mode lump, whereas component\nHyis a standard KP-lump. We also show the vector field\nof the magnetic lump in Fig.3(a). Note that magnetic field\ncomponent Hxis zero, the magnetic field is always in the y−z\nplane, hence the lump can be regarded as a 360◦domain wall\nlocalized in xandydirections. Fig.3(b) presents the magnetic\nfield along y=0. The blue and red arrows correspond to the\nmagnetic field intensity of component Hz,Hy, respectively.\nThe rest of this work is concerned with the propagation and\ninteraction behavior of these lumps in ferrite medium.\n(a)\n (b)\nFigure 3. (a) The vector field of the magnetic lump. (b) The magnetic\nlump along y=0. The blue and red arrows correspond to the\nmagnetic field intensity of components Hz,Hy, respectively.B. Lump motion in ferromagnets with damping or\ninhomogeneous exchange effects\nFigure 4. Three dimensional projections of lump at t=0,HandW\nrepresent the definitions of lump height and width, respectively.\nThe evolution behavior of the magnetic lump in the ideal\nferrite is quite simple and imaginable. Each lump maintains\nits shape while it travels at a constant speed. However, in most\nof real ferromagnetic materials, we have to take the Gilbert-\ndamping into account . For instance, the dimensionless\ndamping constant sranges from 0.048 to about 0.385 in\ngarnet ferrite films. Here we are going to study the dynamics\nof magnetic lump in a damped ferrite film. The typical\nferromagnetic film under consideration is a garnet ferrite film\nwith the dimensionless damping constant s=0.1. For a\nclearer view of the change in shape of the lump, we define\nHandWas the height and width of the lump, which\nare the vertical distance between the highest point and the\nlowest point and the horizontal distance along the propagation\ndirection, respectively. All of these are summarized in Fig.4.\nThe propagation of a lump on the garnet ferrite film\nis presented in Fig.5. As shown in Fig.5(a), the lump\ntravels forward a visible distance in the damped ferrite.\nBeyond that, comparing the profiles of lump between t=0\nand t=10, we evidently observe that the lump becomes\nsmaller and narrower. Fig.5(b) shows the lump height\nand width exhibit a tendency of exponential decay. The\nsolid blue line is the exponential fitting curve to H(t),\nwith the function expression being H(t) = A0e−st. We\nconfirm the above-mentioned amplitude attenuation law is\nuniversal by simulating the motion of lump in ferrites with\nvirous damping factors. Moreover, a definite relationship\nbetween the amplitude and the localization region of solitons\nis important for the soliton excitations. We analyze different\nsizes of numerical lumps and mark the width and height of\nlumps in the phase diagram (see Fig. 5(c)). The results show\nthat for a magnetic lump excitation, its width and height meet\na linear relationship within the error range ( W/H∼0.305).\nSo the lump excitation, upon decay, retains a soliton form.\nTherefore, in this system, the Gilbert-damping plays a role of\ndissipating energy during the motion of magnetic lumps and\nit is characterized by decreasing the amplitude and width of\nlump.\nThe inhomogeneities otherwise referred to as deformities is\ninevitable in real magnetic materials, and it can be caused by\neither external fields or the presence of defects, voids and gaps\nin the material. It has already been reported that the MS may\nbe deformed by the presence of inhomogeneities, in particular5\n(a)\n (b)\n (c)\nFigure 5. Evolution of a magnetic lump in a damped ferrite film with dimensionless damping constant s=0.1. (a) Comparison picture of lump\nwave at t=0 and t=10. (b) The variation of lump height H, lump width Wand velocity V . (c) Numerical relationship between the width and\nheight of magnetic lump.\nits structure and speed [35, 42]. In this present system,\nthe inhomogeneous exchange process is unignorable when\nthe wavelength of lump is comparable to the characteristic\nexchange length.\n(a)\n (b)\n(c)\n (d)\nFigure 6. Propagation of lump with and without the inhomogeneous\ninteraction, respectively.\nWe now move to study the lump motion in the presence of\ninhomogeneous exchange effect. The initial data is the stable\nmagnetic lump shown in Fig.5. As can be observed from Fig.\n6(a) and 6(b), in ferrite without exchange interaction, the lump\nsolution propagates at a constant speed and along the previous\npath. We then consider the non-equilibrium dynamics of lump\nby performing a sudden interaction quench. The pictures\nof component Hyat dimensionless times t=2 and t=4.5\nare shown in Fig. 6(c) and 6(d). As we see, for a quench\nfrom the non-interacting to strong inhomogeneous exchange\nferrite film, the lump oscillates rapidly and diffracts alongthe propagation direction. A two-dimensional shock wave\nis generated and propagates forward. The shock wave front\ncontinues to propagate in the negative direction along x-axis.\nFinally, the energy of lump will be dissipated into numberless\ntiny waves. Accordingly, considering that the lump would be\ndestroyed by the inhomogeneous exchange process, one has to\nconsider keeping its wavelength away from the characteristic\nexchange length in the lump-based microwave applications.\nC. Some examples of excitations and interactions\nThe evolution pattern given in Fig.2 reveals that the lump\nmoves at a larger velocity than the broken MS in the\npropagation. The reason is that the velocity of soliton solution\nis proportional to the soliton amplitude. During the formation\nof the lump, the original MS will be destroyed, and most of the\nenergy is concentrated in some certain centers, which causes\nthe amplitude (and velocity) of the lump to be greater than that\nof MS. These lumps with various speeds enable us to explore\nthe interaction between lump and soliton, as well as between\ntwo lumps.\nA typical example of lump-MS collision is shown in\nFig.7(a). The MS begins to break up around at t=4.\nSubsequently, the splitting lump is going to catch up and\ncollide with the front-MS. After the collision, the front-\nMS is destroyed and broken into several lumps with various\nsizes. It is remarkable that the lump keep its localized\nform before and after the collision almost unchanged. This\nphenomenon implies such two-component lumps are natural\nresults from this nonlinear propagation equations. Further\nsimulation shows these lump structures could be generated\nby a MS with random disturbance. Fig.7(b) depicts a\ncharacteristic inelastic collision between two lumps. We\ninitially generate two adjoining lumps. They are emitted by\nMS at dimensionless time t=6.5. The merging process can\nbe performed as follows. From t=7.5 tot=9.5, two lumps\nmerge simultaneously together and give birth to a new lump\nwhose amplitude is significantly greater than the amplitude of\nprevious lumps. Obviously there is a weak attraction between\ntwo lumps which results in their fusion. In addition to the\nfusion of the two lumps, we also observed an extraordinary\npeak at a specific moment (about t=9.5), which looks like a6\n(a)\n(b)\nFigure 7. (a) Collision between lump and MS. (b) Mergence of two lumps and the formation of a second-order rogue wave-like structure.\nsecond-order rogue wave. It appears to be the result of the\ninteraction between the ripples surrounding the two lumps.\nAfter the fusion, the rouge wave-like structure disappears and\nthe dynamics of the output is determined mainly by a single\nhigh-amplitude lump.\nV . CONCLUSION\nAs a conclusion, the nonlinear propagation of MS in a\nsaturation magnetized ferromagnetic thick film is studied in\ndetail. In the starting point, we derive the (2+1)-dimensional\nKMM system that governs the evolution of short MS waves\nin a saturated ferromagnetic film. The bilinear form of the\nKMM system is constructed and the MS solutions are obtained\nanalytically.\nAfter that, numerical simulations are performed to analyse\nthe evolution behaviours of MS. A significant observation\nis that the unstable MS can be destroyed by Gaussian\nperturbation and broken into some stable magnetic lumps.\nThese lumps exhibit high stability during the propagation.\nFurthermore, some examples are given to analyse the collision\nbehaviours between lump and MS, and the interaction between\ntwo lumps. It is found the lump keeps its shape and speed in\nthe collision with MS. The results confirm that the lump is astable propagation mode in this system and, more to the point,\nthe velocity of lump can be adjusted by its amplitude. Their\nrobustness and controllability provide the possibility for future\ninformation memory and logic devices. We also study the\npropagation of such a lump in ferrites subjected to influence\nof damping and inhomogeneous exchange effects. When the\nGilbert-damping of ferrite is considered, the lumps undergo\nthe following changes: the amplitude and the speed of lump\nare decreased, and the width of lump along the propagation\ndirection is getting narrow. It would cause a strong diffraction\nof the lump if we quench the interaction strength.\nWe hope our work will invoke follow-up experimental\nstudies of lump-based microwave applications. Addition-\nally, since only one- and two-line-soliton are obtained,\nthe integrability of the (2+1)-dimensional system Kraenkel-\nManna-Merle (KMM) remains an open issue. The existence\nof the higher-dimensional evolution system as well as the\nbulk polariton solution is an intriguing avenue for future\nexploration.\nACKNOWLEDGMENT\nThis work was supported by the National Natural Science\nFoundation of China under Great Nos. 11835011; 11675146;\n11875220;.\n[1] M Daniel, V Veerakumar, and R Amuda. 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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": "1902.07563v1.CoFeB_MgO_CoFeB_structures_with_orthogonal_easy_axes__perpendicular_anisotropy_and_damping.pdf",        "content": "CoFeB/MgO/CoFeB structures with orthogonal easy\naxes: perpendicular anisotropy and damping\nH. G lowi\u0013 nskia, A. _Zywczakb, J. Wronac, A. Kryszto\fka, I. Go\u0013 scia\u0013 nskaa,\nT. Stobieckid,e, J. Dubowika,\u0003\naInstitute of Molecular Physics, Polish Academy of Sciences, ul. Smoluchowskiego 17,\n60-179 Poznan, Poland\nbAGH University of Science and Technology, Academic Centre of Materials and\nNanotechnology, Al. Mickiewicza 30, 30-059 Krakow, Poland\ncSingulus Technologies AG, Hanauer Landstrasse 103, 63796 Kahl am Main, Germany\ndAGH University of Science and Technology, Department of Electronics, Al. Mickiewicza\n30, 30-059 Krakow, Poland\neAGH University of Science and Technology, Faculty of Physics and Applied Computer\nScience, Al. Mickiewicza 30, 30-059, Krakw Poland\nAbstract\nWe report on the Gilbert damping parameter \u000b, the e\u000bective magnetization\n4\u0019Meff, and the asymmetry of the g-factor in bottom-CoFeB(0.93 nm)/MgO(0.90{\n1.25 nm)/CoFeB(1.31 nm)-top as-deposited systems. Magnetization of CoFeB\nlayers exhibits a speci\fc noncollinear con\fguration with orthogonal easy axes\nand with 4\u0019Meffvalues of +2 :2 kG and\u00002:3 kG for the bottom and top\nlayers, respectively. We show that 4 \u0019Meffdepends on the asymmetry g?\u0000gk\nof theg-factor measured in the perpendicular and the in-plane directions re-\nvealing a highly nonlinear relationship. In contrast, the Gilbert damping is\npractically the same for both layers. Annealing of the \flms results in collinear\neasy axes perpendicular to the plane for both layers. However, the linewidth\n\u0003Corresponding author\nEmail address: dubowik@ifmpan.poznan.pl (J. Dubowik)\nPreprint submitted to Journal of Physics: Condensed Matter February 21, 2019arXiv:1902.07563v1  [cond-mat.mes-hall]  20 Feb 2019is strongly increased due to enhanced inhomogeneous broadening.\nKeywords: ferromagnetic resonance, perpendicular magnetic anisotropy,\nmagnetization precession damping\nPACS: 75.30.Gw, 75.70.Tj, 75.78.-n, 76.50.+g\n1. Introduction\nCoFeB/MgO/CoFeB systems are extensively employed in magnetic tun-\nnel junctions (MTJs), which are important for modern spintronic devices\nsuch as read-heads and magnetic random-access memory [1]. In these ap-\nplications the two key features are the perpendicular magnetic anisotropy\n(PMA) with PMA constant K?and magnetization damping with inhomoge-\nneous (extrinsic) and Gilbert (intrinsic) contributions to the ferromagnetic\nresonance (FMR) linewidth.\nThe FMR linewidth is usually enhanced in Ta/CoFeB/MgO stacks for\nwhich the values of PMA and the Gilbert damping parameter \u000bare scattered\n[2, 3, 4]. Recent experimental results [4, 5] indicate that there is no correlation\nbetweenK?and\u000bin these systems. Speci\fcally, \u000bis approximately constant\nwhile the PMA tends to improve on annealing. However, systems with a high\nPMA have often an increased linewidth due to an inhomogeneous broadening\n[6, 7] so that an extrinsic contribution to the linewidth may be as high as\n400{500 Oe [8] despite \u000bis of 0.01 { 0.02 in these systems. An increase in\nlinewidth is attributed to an angular dispersion of the easy PMA axis, which\nresults in a high inhomogeneous broadening attributed to the zero-frequency\nlinewidth \u0001 H0[6].\nIt has been shown that PMA in CoFe/Ni multilayers is linearly propor-\n2tional to the orbital-moment asymmetry [7, 9] in accordance with the Bruno's\nmodel [see Ref. [7] for discussion]. On the other hand, substantial PMA in\nTa/CoFeB/MgO systems [2] has been considered as related to an inhomoge-\nneous concentration of the anisotropy at the interface [10] so that the Bruno's\nmodel may be not valid in this case. Based on our experimental results, we\naim to shed some light on possible correlation between asymmetry of the\ng-factor and the e\u000bective magnetization 4 \u0019Meff, which are the magnetic\nparameters measured directly in a broadband FMR experiment. According\nto well known Kittel's formula, a departure from the free electron g-factor\nis proportional to \u0016L=\u0016S[11] so that we can discuss the asymmetry of the\ng-factor as well as on the asymmetry of the orbital moment on equal footing.\nHere, we prefer to use asymmetry in g-factor for evaluating the relationship\nbetween orbital moment and PMA.\nAs far as we know, FMR has not yet been thoroughly investigated in\n\"full\" Ta/CoFeB/MgO/CoFeB/Ta MTJ structures. In particular, a depen-\ndence of PMA on the asymmetry in the g-factor has not yet been proved\nin CoFeB/MgO/CoFeB systems. In this paper, we aim to independently\ncharacterize each CoFeB layer separated by a MgO tunnel barrier in terms\nof the\u000bparameter and 4 \u0019Meff. By analyzing FMR measurements in the\nin-plane and out-of-plane con\fgurations, we \fnd that PMA correlates with\ntheg-factor asymmetry in a highly nonlinear relationship.\n2. Experimental methods\nThe samples were sputtered in an Ar atmosphere using a Singulus Timaris\nPVD Cluster Tool. The CoFeB magnetic \flms were deposited by dc-sputtering\n3from a single Co 40Fe40B20target, whereas the MgO barriers were deposited\nby rf-sputtering directly from a sintered MgO target. The samples were de-\nposited on an oxidized silicon wafer with 5 Ta/ 20 Ru /Ta 3 bu\u000ber layers\nand capped with 5 Ta/ 5 Ru (numbers indicate the nominal thickness in\nnanometres). The studied structures consist of two ferromagnetic CoFeB\n(0.93 nm { bottom and 1.31 nm { top) \flms separated by a MgO barrier of\ndi\u000berent thicknesses (0.90, 1.1, and 1.25 nm). It is important to note that we\ninvestigated the as-deposited samples so that the CoFeB layers were amor-\nphous [3, 12]. The e\u000bect of annealing treatment (330oC for 1 hr) on magnetic\nproperties of the system will be discussed at the end of the paper.\nHysteresis loops of the samples were measured by vibrating sample mag-\nnetometer (VSM) with the perpendicular and in-plane magnetic \felds. The\nsaturation magnetization Msof 1200 G in the as-deposited state was deter-\nmined from magnetic moment per unit area vs. CoFeB thickness dependen-\ncies [13]. To investigate anisotropy and damping in studied samples, vector\nnetwork analyzer ferromagnetic resonance (VNA-FMR) spectra of the S21\nparameter were analyzed [14]. VNA-FMR was performed at a constant fre-\nquency (up to 40 GHz) by sweeping an external magnetic \feld, which was\napplied either in-plane or perpendicular to the sample plane. These two con-\n\fgurations will be referred to as the in-plane and out-of-plane con\fgurations.\nExperimental data were \ftted using the Kittel formula\n!\n\rk=q\n(Hr+Ha) (Hr+Ha+ 4\u0019Meff) (1)\nfor the in-plane con\fguration and\n!\n\r?= (Hr\u00004\u0019Meff) (2)\n4for the out-of-plane con\fguration, where != 2\u0019fis the angular microwave\nfrequency,Hrthe resonance \feld, \rk;?=gk;?\u0016B=~the gyromagnetic ratio,\ngkandg?are the spectroscopic g-factors for the in-plane and out-of-plane\ncon\fgurations, respectively, ~the reduced Planck constant, \u0016Bthe Bohr\nmagneton, and Hathe in-plane uniaxial anisotropy \feld. 4 \u0019Meff= 4\u0019Ms\u0000\nH?is the e\u000bective magnetization , where Msis the saturation magnetization,\nandH?= 2K?=Msis the perpendicular anisotropy \feld and K?is the\nperpendicular anisotropy constant. For the in-plane easy axis 4 \u0019Meff>0\nwhereas for the perpendicular to the plane easy axis 4 \u0019Meff<0. According\nto Eqs. (1) and (2), 4 \u0019Meff=\u00002Keff=Ms, whereKeffis the e\u000bective\nanisotropy constant de\fned as K?\u00002\u0019M2\ns[15].\n3. Results and discussion\nFigure 1 (e) presents hysteresis loops of the sample with a 1.25 nm thick\nMgO barrier measured in the out-of-plane (red line) and in-plane con\fgu-\nration (black line). The shape of the loops in both directions is nearly the\nsame for each con\fguration as the saturation \felds (of Hs\u00192 kOe) for both\nlayers have nearly the same magnitude with the opposite signs in 4 \u0019Meff.\nEach hysteresis loop is a sum of the loops typical for the easy and hard axis\nand, as explained below, we can infer from magnetization reversals which\nlayer possesses PMA.\nLet us assume that the bottom CoFeB layer (B) has an in-plane easy axis\nand the top layer (T) has a perpendicular to the plane easy axis so that their\nmagnetization directions are orthogonal at remanence. Three con\fgurations\nof a magnetic \feld Happlied for the magnetization measurements are shown\n5-10 -5 0 5 10-101\n  Normalized moment\nH (kOe)HTT\nBH\ne.a.e.a.\nB\nHe.a.e.a. T\nBe.a.\ne.a.a) b) c)\nB\nT\nB+T\nd)\n 10 51\n-1\n-10 -5 0e)\nH (kOe)Figure 1: (a)-(c) Con\fgurations used for the magnetic measurements with a magnetic\n\feld applied perpendicular or parallel to the \flm plane. (d) Example of schematic pictures\nof the magnetization reversals of a CoFeB/MgO/CoFeB structure for con\fguration (a).\n(e) Hysteresis loops of a CoFeB/MgO/CoFeB structure measured in con\fgurations (a)\n- black line and (b) - red line. The inset shows schematically the model reversals for\ncon\fgurations (a)-black and (b)-red\n.\nin Figs. 1 (a) - (c). These con\fgurations enable magnetization reversals to be\nobserved with Horiented parallel- (a) (perpendicular- (b)) to the easy axis of\nB (T) layer, respectively, or perpendicular to both easy axes (c). Further, we\nwill refer to these con\fgurations as (a), (b), and (c) con\fgurations. As it is\nschematically shown in Fig. 1 (d), an apparent magnetization reversal of B+T\nfor the con\fguration (a) is a sum of independent magnetization reversals of\nB and T. For the perfectly asymmetric structure with 4 \u0019MB\neff=\u00004\u0019MT\neff\nwith the same thickness (i.e. with the same magnetic moments MSVT;B) the\n6apparent magnetization reversals taken in con\fgurations (a) and (b) would\noverlay. However, as it is seen in Fig. 1 (e) they do not completely overlay\nso that the curve taken in the con\fguration (b) lies a bit higher than that\ntaken in (a). As it is shown in the inset of (e), a simple model explains that\nthe T layer (i.e. the with nominal thickness tof 1.3 nm) possesses an easy\naxis perpendicular to the plane, while the B layer with t= 0:93 nm has an\nin-plane easy axis.\nIn the model, the magnetization reversals in each layer can be approxi-\nmated with a normalized relation [16] M(H;S) = arctan[H=H s\u0002tan(\u0019S=2)]=\narctan[H=H max\u0002tan(\u0019S=2)], where Hsof 2 kOe is a saturation \feld for\nthe hard direction and Sis de\fned as a ratio of remanence to the satura-\ntion moment. For Hkparallel to the easy axis, S= 1 (B layer in Fig. 1\n(d)) and for H?perpendicular to the easy axis (T layer in Fig. 1 (d)),\nS= 0:66 as well as Hmax= 10 kOe are arbitrary chosen for the sake of\nsimplicity. The apparent magnetization curve for con\fguration (a) is a sum\n[tB\u0002M(H;S = 1) +tT\u0002M(H;S = 0:66)]=(tB+tT). For the con\fguration\n(b),tTandtBare reversed in the sum. In order to satisfy the experimental\ndata shown in (e), a ratio tB=tT= 0:79. It is easily seen that if the B layer\nhad an in-plane easy axis and the T layer had an easy axis perpendicular to\nthe plane, a curve taken in con\fguration (b) would lie lower than that taken\nin con\fguration (a). Hence, the thin B layer is that with the in-plane easy\naxis.\nFigures 2 (a) and (b) show typical VNA-FMR spectra of the CoFeB/MgO(1.25\nnm)/CoFeB system measured (see Figs. 1) in con\fguration (a) and (b) , re-\nspectively. Two FMR peaks associated with the bottom and top CoFeB lay-\n76 8ImS21(a.u.)\nH(kOe)topbottom(a) 20GHz\nin-planeconfiguration\n4 6 8 10(b)ImS21(a.u.)\nH(kOe)top\nbottom20GHz\nout-of-planeconfiguration\n( )\n()Figure 2: Typical VNA-FMR spectrum of the as-deposited CoFeB/MgO(1.25 nm)/CoFeB\nstructure with resonance peaks from bottom (B) and top (T) layers measured in the in-\nplane (a) and out-of-plane (b) con\fgurations. Solid red lines represent the Lorentzian\n\fts to the experimental data. (c) Dependence of the FMR \feld on the polar angle \u0002\nof applied \feld in X band (9.1 GHz). The easy axis of magnetization of the B is in the\nin-plane orientation. For the T layer, the out-of-plane direction becomes the easy axis.\ners are clearly visible. To determine the resonance \feld Hrand the linewidth\n\u0001Hat constant frequency with a high precision, the spectra were \ftted with\nLorentzians (marked by solid lines in Fig. 2 (a) and (b)). Figure 2 (c) shows\ndependencies of the X-band (9.1 GHz) resonance \felds of the B and T layers\non the polar angle between the \flm normal and the direction of an applied\n\feld. It is clearly seen that the T layer has 4 \u0019Meff<0 (i.e., a perpendicular\neasy axis) and the B layer with 4 \u0019Meff>0 has an in-plane easy axis. From\nFigs. 2 (a) and (b), we can clearly see that the intensity (area under the FMR\n8peak) of the T layer is higher than that of the B layer. This additionally\ncon\frms that the bottom layer has the lower magnetic moment than that of\nthe top layer.\nA typicalHrvs.fdependence, observed for the CoFeB/MgO(1.25 nm)/CoFeB\nsystem is shown in Fig. 3 (a) and (b) for the in-plane (a) and out-of-plane\n(b) con\fguration, respectively. The observed data points are \ftted using\nEqs. (1) and (2). The values of 4 \u0019Meff, obtained from the \ftting are found\nto be of +2 :2 kG and\u00002:3 kG for the bottom and top layers, respectively.\nThefversusHrdata for the B layer were \ftted assuming Haof 30 Oe as\ncon\frmed by VSM measurements (not shown) in the con\fguration presented\nin Fig. 1(c). The values of gkof the top and bottom layers are equal to 2.04\nand 2.08, respectively, in contrast, the values of g?for these layers are 2.06\nand 2.22. One can notice the di\u000berences in values of g?resulting from clear\ndi\u000berences in the slopes of the f(Hr) dependencies (see, Fig. 3 (b)) for the\nbottom (\r?= 2:88 MHz/Oe) and top ( \r?= 3:11 MHz/Oe) layer, respec-\ntively.\nTo sum up, VSM and FMR measurements con\frmed the presence of or-\nthogonal easy axes in our CoFeB/MgO/CoFeB systems and showed that the\nthickness ratio tB=tT= 0:79 is slightly higher than the ratio of nominal thick-\nness (tB\nnom=tT\nnom= 0:71). The thinner B layer has an in-plane easy axis while\nthe T layer has a perpendicular easy axis. However, keeping in mind our for-\nmer studies of a dead magnetic layer (DML) in the Ta/CoFeB/MgO (B) and\nMgO/CoFeB/Ta (T) structures [13] deposited in the same Timaris system,\nwe estimated DMLB'0:23 nm and DMLT'0:4. With such asymmetric\nDMLs the e\u000bective thickness tB\neff'0:7 nm andtT\neff'0:9 nm which satis\fes\n90 5 1005101520\n0 5 10010203040\nf (GHz)\nHr(kOe)\n(b)(a)\nin-plane configuration\nout-of-plane configuration\nf (GHz)\nHr(kOe)1.25 nm MgO\nbottom\ntop0 2 46 8 10048121620\ntop 1.25 nm MgO \n 1.25 nm MgO \n 0.96 nm MgO\n 0.96 nm MgO\n 0.85 nm MgO\n 0.85 nm MgOf (GHz)\nH (kOe)bottomFigure 3: FMR dispersion relations of the as-deposited CoFeB/MgO(1.25 nm)/CoFeB\nstructure measured in the in-plane con\fguration (a) and out-of-plane con\fguration (b).\nThe solid lines show the \fts given in accordance with Eqs. (1) and (2). Inset in (a) shows\nthat the \ftting parameter practically do not depend on the MgO thickness.\ntB=tT= 0:78. VNA-FMR measurements, which o\u000ber a greater precision than\nVSM measurements, give 4 \u0019Meff=\u00002:3 kG (K?= 10:4\u0002106erg/cm3) and\n4\u0019Meff= +2:2 kG (K?= 7:7\u0002106erg/cm3) for the T and B layers, respec-\ntively. All \ftting parameters for a CoFeB/MgO(1.25 nm)/CoFeB structure\nare juxtaposed in Table 1. As it is shown in the inset of Fig. 3 (a), the thick-\nness of MgO spacer within a range of 0.9 { 1.25 nm had almost no in\ruence\non the \ftting parameters, therefore, the values of \ftting parameters 4 \u0019Meff,\ng,\u000b, and \u0001H0are typical for all samples with various MgO thickness.\n10Table 1: Parameters determined from VNA-FMR spectra for the as-deposited\nCoFeB(0.93 nm)/MgO (1.25 nm)/CoFeB(1.31 nm) for the in-plane and out-of-plane con-\n\fgurations: the in-plane anisotropy \feld ( Ha), the e\u000bective magnetization (4 \u0019M eff), spec-\ntroscopicg-factors for in-plane and out-of-plane con\fguration, Gilbert damping ( \u000b), the\nfrequency-independent FMR linewidth (\u0001 H0). The values of the \ftting parameters do\nnot depend on the MgO thickness. The values of g?are marked by asterisks.\nIn-plane con\fguration\nHa(Oe) 4\u0019Meff(kG)gk,g? \u000b \u0001H0(Oe)\ntop 0 -2.29\u00060.05 2.04\u00060.02 0.018\u00060.002 102\u000622\nbottom 30 2.22\u00060.15 2.08\u00060.03 0.017\u00060.002 69\u000623\nOut-of-plane con\fguration\ntop { -2.3\u00060.01 2.22\u00060.01?0.018\u00060.001 95\u000613\nbottom { 2.19\u00060.04 2.06\u00060.02?0.017\u00060.003 160\u000630\nAlthough it is counter-intuitive that the thinner B layer possesses an in-\nplane easy axis, the same feature has been reported for other Ta/CoFeB(1\nnm)/MgO systems deposited in the same Timaris equipment [17]. Similar ef-\nfect has been recently observed in a substrate/MgO/CoFeB/Ta/CoFeB/MgO\nstructure, where the thicker CoFeB layer exhibits a strong PMA in con-\ntrast to the relatively weak PMA in the thinner CoFeB layer [18, 19]. It is\npossible that the growth mode of the MgO layer in contact with an amor-\nphous CoFeB layer might be responsible. The perpendicular anisotropy in\nthese systems originates from the CoFe/MgO interface [20]. The structure\nof the unannealed CoFeB layers is amorphous regardless of underlying lay-\ners, whereas the MgO barrier deposited on the amorphous CoFeB has an\namorphous structure of up to four monolayers (that is about 0.9 nm) [21].\n11Hence, there are subtle di\u000berences between the CoFeB/MgO (bottom) and\nMgO/CoFeB (top) interfaces; the interface of the bottom CoFeB layer is\nmainly amorphous whereas the interface of the top layer is crystalline, be-\ncause the barrier thickness of the investigated samples is above the transition\nfrom amorphous to crystalline phase. Therefore, di\u000berent structures for the\nCoFeB/MgO interfaces may result in di\u000berent values of anisotropy constant.\nAnother explanation is that the measured dependence Keff\u0002teffvs.teff\nin \flms with PMA is often strongly nonlinear due to either intermixing at\ninterfaces [22] or magnetoelastic e\u000bects [15], with Keff\u0002teffexhibiting a\nmaximum as a function of decreasing teffand with the PMA eventually\nbeing lost for small teffof, for example, 0.7 nm.\nThe values of gfactor yield the ratio of the orbital \u0016Land spin\u0016Smag-\nnetic moments in accordance with equation [9, 11]\n\u0016L\n\u0016S=g\u00002\n2; (3)\nwhere\u0016S=\u0016B. Hence, the di\u000berence between orbital moments \u0001 \u0016Lalong\nthe easy and hard direction in the in-plane [Fig. 1 (a)] and out-of-plane [Fig. 1\n(b)] con\fgurations is proportional to ( g?\u0000gk) and reads \u0001 \u0016L=\u0016B(g?\u0000\ngk)=2. \u0001\u0016Lis of 0.09\u0016Band\u00000:01\u0016Bfor the T and B layer, respectively.\nIn CoFe/Ni multilayers [7], the PMA has been shown to be proportional\nto the orbital moment anisotropy in accordance to Bruno model [23]. How-\never, in the case of the CoFeB/MgO systems this direct relationship between\nthe orbital moment asymmetry and the perpendicular anisotropy is not ful-\n\flled. As can be seen in Table 1, ( g?\u0000gk)\u00190 for the B layer corresponds to\n4\u0019Meff= 2:2 kG. Hence, while ( g?\u0000gk) is negligible, a decrease in 4 \u0019Meff\ndue to PMA from 4 \u0019MS= 15 kG to 2.2 kG is substantial. In contrast,\n12(g?\u0000gk)\u00190:18 is exceptionally large for the T layer, while 4 \u0019Meffmerely\ndecreases to - 2.3 kG. In accordance with the earlier report [24], this con\frms\nthat any relationship between the orbital moment asymmetry and the per-\npendicular anisotropy in CoFeB/MgO systems is highly nonlinear. Of course,\nother factors controlled by annealing such as disorder at interfaces and over-\nor underoxidized interfaces would also play a signi\fcant role in PMA [20].\nFuture work con\frming such a nonlinear relationship for a broad range of\ntCoFeB might resolve this issue.\nAt present, there is no doubt that PMA in MgO/CoFeB structures is\nan interface e\u000bect and it is correlated with the presence of oxygen atoms\nat the interface despite the weak spin-orbit coupling [20, 25]. The origin\nof PMA is attributed to hybridization of the O-p with Co(Fe)-d orbitals at\nthe interface [20] and/or to a signi\fcant contribution of thickness dependent\nmagnetoelastic coupling [15]. A deviation of the g-factor from the 2.0 value\nis expressed by g'2\u00004\u0015=\u0001 , where\u0015 < 0 is the spin-orbit constant for\nFe(Co) and \u0001 is the energy levels splitting in the ligand \feld [11]. While\nthe deviation of the g-factor is inversely proportional to \u0001, PMA (and hence\n4\u0019Meff) is proportional to the enhanced spin-orbit-induced splitting around\nthe Fermi level [20]. This may result in a complex relationship between PMA\nandg-factor anisotropy.\nThe Gilbert damping parameter \u000bis evaluated from the dependence of\nthe linewidth \u0001 Hon the resonance frequency as shown in Fig. 4 for the\nin-plane (a) and the out-of-plane (b) con\fgurations. The lines are linear \fts\nto\n\u0001H=\u000b4\u0019f\n\rk;?+ \u0001H0; (4)\n13/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s50/s48/s48/s52/s48/s48\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48\n/s40/s98/s41/s40/s97/s41\n/s111/s117/s116/s45/s111/s102/s45/s112/s108/s97/s110/s101/s32/s99/s111/s110/s102/s105/s103/s117/s114/s97/s116/s105/s111/s110/s32/s98/s111/s116/s116/s111/s109/s32/s67/s111/s70/s101/s66/s32/s32 /s61/s48/s46/s48/s49/s55/s32\n/s32/s116/s111/s112/s32/s67/s111/s70/s101/s66/s32/s32 /s61/s48/s46/s48/s49/s56/s32/s32/s72/s32/s40/s79/s101/s41\n/s102/s32/s40/s71/s72/s122/s41/s105/s110/s45/s112/s108/s97/s110/s101/s32/s99/s111/s110/s102/s105/s103/s117/s114/s97/s116/s105/s111/s110\n/s32/s98/s111/s116/s116/s111/s109/s32/s67/s111/s70/s101/s66/s32/s32 /s61/s48/s46/s48/s49/s55/s32\n/s32/s116/s111/s112/s32/s67/s111/s70/s101/s66/s32/s32 /s61/s48/s46/s48/s49/s56\n/s32/s32/s72/s32/s40/s79/s101/s41\n/s102/s32/s40/s71/s72/s122/s41Figure 4: Linewidth as a function of frequency measured in the in-plane con\fguration (a)\nand out-of-plane con\fguration (b). The \u000bdamping parameter is obtained using Eq. (4).\nThe thickness of MgO was 1.25 nm.\nwhere \u0001H0is the inhomogeneous broadening related to CoFeB layer quality.\nThe values of \u000band \u0001H0are shown in Table 1. The top and the bottom layers\nshow almost the same \u000bof 0.017 - 0.018. This suggests that the damping has\nno relation to PMA. While \u0001 H0for the top layer is almost the same for both\ncon\fgurations, \u0001 H0for the bottom layer at the (b) con\fguration is nearly\ntwice as large as that for the (a) con\fguration. Such a behavior suggests\nthat the layer B is rather inhomogeneous with a large angular dispersion of\nmagnetization across the layer [26, 27].\nSpin pumping to Ta layers (which are a part of the bu\u000ber and cap-\n14ping layers, as shown in Fig. 1 (e)) may also in\ruence the damping in\nCoFeB/MgO/CoFeB systems since magnetization precession induces a spin\ncurrent to the adjacent nonmagnetic Ta layers that result in an enhanced\ndamping [8]. This is an interface e\u000bect and hence scales inversely propor-\ntional to the CoFeB layer thickness. Because the bottom layer with an in-\nplane easy axis is thinner than the top layer with a perpendicular easy axis,\nthe spin pumping e\u000bect a\u000bects it more. To estimate spin pumping e\u000bect the\nstandard equation [28] without back\row is used\n\u0001\u000b=g\u0016Bg#\"\n4\u0019Msteff; (5)\nwhereteffis the e\u000bective thickness of CoFeB and g#\"is the mixing con-\nductance. The measured damping of both layers is of 0.017 - 0.018, while\ndamping of a bulk CoFeB is around 0.004 [12]. Therefore, an increase of \u0001 \u000b\ndue to spin pumping is of 0.014 which gives the mixing conductance g#\"= 0:8\nand 1\u00021015cm\u00002for the e\u000bective thickness 0.7 nm and 0.9 nm of B and\nT layer, respectively. The value of mixing conductance g#\"for Ta/CoFeB\ninterface found in the literature lies in a broad range from 1 :67\u00021014to\n1:4\u00021015cm\u00002[29, 30, 31, 32]. Taking into account our simpli\fcation (the\nlack of back\row), this estimation gives the maximal values of mixing conduc-\ntance. Hence, we can conclude that spin pumping substantially in\ruences\nthe damping in our structures. It is worth mentioning that the measured \u000b\nof 0.017 - 0.018 for CoFeB/MgO/CoFeB systems agrees with \u000b= 0:015 for\nthe Ta/CoFeB(1)/MgO structure reported in [3].\nFinally, we would like to make a further comment on postdeposition an-\nnealing of our CoFeB/MgO/CoFeB systems. We found that annealing at\n330oC for 1 hr, beside increasing Msto 1500 G, enhances also PMA so that\n15both layers possess easy axes perpendicular to the plane. 4 \u0019Meffattains\n-1 kG and -4 kG for the B and T layers, respectively. We found that an\nincrease in K?of 7:7\u0002106erg/cm3equally contributes to both layers and,\nfor example, K?= 17\u0002106erg/cm3for the T layer. On the other hand, the\nlinewidth \u0001 Hstrongly broadens to \u0018400 Oe and\u0018700 Oe for the B layer\nand the T layer, respectively. These values are in agreement with recently\nreported values for a similar systems [17]. Moreover, as it is shown in Fig. 5,\n\u0001Hdoes not follow the linear dependence described by Eq. (4). Therefore,\nit is impossible to determine \u000bprecisely for the annealed systems. Such a\nbehavior of \u0001 Hand the decreased remanence with respect to the saturation\nmagnetization (see, [17]) both con\frm a strong angular dispersion of the easy\nPMA axis in both layers. It has been observed that with increasing PMA\nthe dispersion of anisotropy also increases [6, 7, 27]. As a result, dispersion\nin PMA leads to a large two magnon scattering contribution to the linewidth\nfor in-plane magnetization and to an enhanced Gilbert damping [6]. While\nthe magnetic parameters practically do not depend on the MgO thickness in\nas-deposited structures, the annealed structures show a substantial spread in\n4\u0019Meffas it is shown in Fig. 6, which may imply some di\u000berent CoFeB/MgO\ninterfaces due to, for example, boron di\u000busion [30, 33].\n4. Conclusion\nWe investigated the CoFeB/MgO/CoFeB as-deposited systems with the\nin-plane and out-of-plane orthogonal easy axes due to the substantial dif-\nference in PMA for the bottom (B) and the top (T) CoFeB layers, respec-\ntively. The T and the B layer had comparable Gilbert damping \u000bsuggesting\n16/s53 /s49/s48 /s49/s53 /s50/s48/s48/s50/s48/s48/s52/s48/s48/s54/s48/s48/s56/s48/s48\n/s32/s32\n/s32/s116/s111/s112/s32/s67/s111/s70/s101/s66\n/s32/s98/s111/s116/s116/s111/s109/s32/s67/s111/s70/s101/s66/s72/s32/s40/s79/s101/s41\n/s102/s32/s40/s71/s72/s122/s41/s105/s110/s45/s112/s108/s97/s110/s101/s32/s99/s111/s110/s102/s105/s103/s117/s114/s97/s116/s105/s111/s110Figure 5: Linewidth as a function of frequency measured in the in-plane con\fguration for\nthe annealed structure. The thickness of MgO was 1.25 nm.\nthat there is no correlation between the Gilbert damping and PMA. We\nalso showed that 4 \u0019Meffcorrelates with the asymmetry in the g-factor (and\nhence with \u0001 \u0016L) and this correlation is highly nonlinear. Annealing enhances\nPMA in both layers but it has detrimental e\u000bect on the linewidth, however.\nTherefore, despite the Gilbert parameter shows no correlation with PMA, it\nseems that there is some correlation between the linewidth (see Eq. 4) and\nPMA in the annealed systems through a combined e\u000bect between dispersion\nof local anisotropy easy axes in crystallites with a high PMA.\n17/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s52/s56/s49/s50/s49/s54/s50/s48\n/s32/s32\n/s32/s49/s46/s50/s53/s32/s110/s109/s32/s77/s103/s79/s32\n/s32/s49/s46/s49/s48/s32/s110/s109/s32/s77/s103/s79/s32\n/s32/s49/s46/s48/s48/s32/s110/s109/s32/s77/s103/s79\n/s32/s48/s46/s57/s48/s32/s110/s109/s32/s77/s103/s79\n/s32/s49/s46/s50/s53/s32/s110/s109/s32/s77/s103/s79/s32\n/s32/s49/s46/s49/s48/s32/s110/s109/s32/s77/s103/s79/s32\n/s32/s49/s46/s48/s48/s32/s110/s109/s32/s77/s103/s79\n/s32/s48/s46/s57/s48/s32/s110/s109/s32/s77/s103/s79/s102/s32/s40/s71/s72/s122/s41\n/s72/s32/s40/s107/s79/s101/s41/s105/s110/s45/s112/s108/s97/s110/s101/s32/s99/s111/s110/s102/s105/s103/s117/s114/s97/s116/s105/s111/s110Figure 6: FMR dispersion relations of CoFeB/MgO(0.9 { 1.25 nm)/CoFeB annealed struc-\nture measured in the in-plane con\fguration.\nAcknowledgments\nWe acknowledge support from the the project \\Marie Sk lodowska-Curie\nResearch and Innovation Sta\u000b Exchange (RISE)\" Contract No. 644348 with\nthe European Commission, as part of the Horizon2020 Programme, and\npartially by the project NANOSPIN PSPB-045/2010 under a grant from\nSwitzerland through the Swiss Contribution to the enlarged European Union.\n18References\n[1] B. 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Kurinec,\n\\Study of boron di\u000busion in MgO in CoFeB/MgO \flm stacks using par-\nallel electron energy loss spectroscopy,\" Applied Physics Letters , vol. 94,\nno. 8, p. 082110, 2009.\n24"    },    {        "title": "0805.3495v1.Intrinsic_and_non_local_Gilbert_damping_in_polycrystalline_nickel_studied_by_Ti_Sapphire_laser_fs_spectroscopy.pdf",        "content": "Intrinsic and non-local Gilbert damping in\npolycrystalline nickel studied by Ti:Sapphire laser fs\nspectroscopy\nJ Walowski1, M Djordjevic Kaufmann1, B Lenk1, C Hamann2\nand J McCord2, M M unzenberg1\n1Universit at G ottingen, Friedirch-Hund-Platz 1, 37077 G ottingen, Germany\n2IFW Dresden, Helmholtzstra\u0019e 20, 01069 Dresden\nE-mail: walowski@ph4.physik.uni-goettingen.de\nAbstract. The use of femtosecond laser pulses generated by a Ti:Sapphire laser\nsystem allows us to gain an insight into the magnetization dynamics on time scales from\nsub-picosecond up to 1 ns directly in the time domain. This experimental technique is\nused to excite a polycrystalline nickel (Ni) \flm optically and probe the dynamics\nafterwards. Di\u000berent spin wave modes (the Kittel mode, perpendicular standing\nspin-wave modes (PSSW) and dipolar spin-wave modes (Damon-Eshbach modes)) are\nidenti\fed as the Ni thickness is increased. The Kittel mode allows determination of the\nGilbert damping parameter \u000bextracted from the magnetization relaxation time \u001c\u000b.\nThe non-local damping by spin currents emitted into a non-magnetic metallic layer\nof vanadium (V), palladium (Pd) and the rare earth dysprosium (Dy) are studied\nfor wedge-shaped Ni \flms 1 nm \u000030 nm. The damping parameter increases from\n\u000b= 0:045 intrinsic for nickel to \u000b > 0:10 for the heavy materials, such as Pd and\nDy, for the thinnest Ni \flms below 10 nm thickness. Also, for the thinnest reference Ni\n\flm thickness, an increased magnetic damping below 4 nm is observed. The origin of\nthis increase is discussed within the framework of line broadening by locally di\u000berent\nprecessional frequencies within the laser spot region.arXiv:0805.3495v1  [cond-mat.other]  22 May 2008Gilbert damping in Nickel thin \flms 2\n1. Introduction\nThe understanding of picosecond-pulsed excitation of spin packets, spin wave modes\nand spin currents is of importance in developing a controlled magnetic switching concept\nbeyond the hundred picosecond timescale and to test the speed of magnetic data storage\nmedia heading to the physical limits. Over the last years profound progress has been\nmade within that \feld by using femtosecond laser spectroscopy. The recent discoveries\nin ultrafast magnetization dynamics are heading to a new understanding [1{5] and\nnew all-optical switching concepts have been discovered [6]. In addition, the all-optical\nmethod has developed into a valuable tool to study the magnetization dynamics of\nthe magnetic precession and thereby access magnetocrystalline anisotropies and the\nmagnetic damping [7{11] or the dynamics of magnetic modes in nanometer sized arrays\nof magnetic structures [12, 13] and single magnetic nanostructures [14, 15]. Naturally,\none \fnds similarities and di\u000berences as compared to magnetic resonance techniques\nin frequency space (FMR) [16], optical techniques such as Brillouin light scattering\n(BLS) [17,18] and time-resolved techniques, for example pulsed inductive magnetometry\n(PIMM) [19]. Advantages and disadvantages of the di\u000berent techniques have already\nbeen compared in previous work [20{22]. The same concepts can be applied to the\nfemtosecond-laser-based all-optical spectroscopy techniques. Here we discuss their\nabilities, highlighting some aspects and peculiarities [11,23{27]:\ni. After excitation within the intense laser pulse, the nature of the magnetic relaxation\nmechanisms determine the magnetic modes observed on the larger time scale [5].\nFor a Ni wedge di\u000berent modes are found as the thickness is increased: coherent\nprecession (Kittel mode), standing spin waves (already found in [28]) and dipolar\nsurface spin waves (Damon-Eshbach modes) appear and can be identi\fed.\nii. Magnetic damping has been extracted by the use of fs spectroscopy experiments\nalready in various materials, epitaxial \flms, as a function of the applied \feld\nstrength, \feld orientation and laser excitation power [7{11]. Using the Kittel\nmode, we study the energy dissipation process caused by non-local damping by spin\ncurrents [29] in Ni by attaching a transition metal \flm (vanadium (V), palladium\n(Pd) and a rare earth \flm (dysprosium (Dy)) as a spin sink material and compare\nthem to a Ni reference sample. The present advantages and disadvantages of the\nmethod are discussed.\niii. A modi\fcation of the magnetic damping is found for the thinnest magnetic layers\nbelow 4 nm. The understanding of this e\u000bect is of high interest because of the\nincrease in methods used to study magnetic damping processes in the low \feld\nregion in the current literature. We present a simple model of line broadening\nknown from FMR [30{32] and adapted to the all-optical geometry that pictures\nthe e\u000bect of the increased intrinsic apparent damping observed. Therein a spread\nlocal magnetic property within the probe spot region is used to mimic the increased\napparent damping for the low \feld region.Gilbert damping in Nickel thin \flms 3\na)\nb)Side view:\nFigure 1. a) Schematics of the pump probe experiment to determine the change in\nKerr rotation as a function of the delay time \u001c. b) Experimental data on short and\nlong time scales. On top a schematic on the processes involved is given.\n2. Experimental Technique\nThe all-optical approach to measuring magnetization dynamics uses femtosecond laser\npulses in a pump-probe geometry. In our experimental setup a Ti:Sapphire oscillator\ngenerates the fs laser pulses which are then ampli\fed by a regenerative ampli\fer (RegA\n9050). This systems laser pulse characteristics are 815 nm central wave length, a\nrepetition rate of 250 kHz, a temporal length of 50 \u000080 fs and an energy of \u00181\u0016J\nper pulse. The beam is split into a strong pump beam (95% of the incoming power),\nwhich triggers the magnetization dynamics by depositing energy within the spot region,Gilbert damping in Nickel thin \flms 4\nand a weaker probe pulse (5% of the incoming power) to probe the magnetization\ndynamics via the magneto-optical Kerr e\u000bect delayed by the time \u001c, in the following\nabbreviated as time-resolved magneto-optic Kerr e\u000bect (TRMOKE). The schematic\nsetup and sample geometry is given in \fgure 1a). The spot diameters of the pump\nand probe beam are 60 \u0016m and 30\u0016m respectively. A double-modulation technique is\napplied to detect the measured signal adapted from [33]: the probe beam is modulated\nwith a photo-elastic modulator (PEM) at a frequency f1= 250 kHz and the pump\nbeam by a mechanic chopper at a frequency f2= 800 Hz. The sample is situated in\na variable magnetic \feld (0 \u0000150 mT), which can be rotated from 0\u000e(in-plane) to 90\u000e\n(out of plane) direction. The degree of demagnetization can be varied by the pump\n\ruence (10 mJ =cm2\u000060 mJ=cm2) to up to 20% for layer thicknesses around 30 nm and\nup to over 80% for layers thinner than 5 nm. The samples studied were all grown on\nSi(100) substrates by e-beam evaporation in a UHV chamber at a base pressure of\n\u00185\u000210\u000010mbar. For a variation of the thickness, the layers are grown as wedges with\na constant gradient on a total wedge length of 15 mm.\n3. Results and discussion\n3.1. Kittel mode, standing spin waves and Damon-Eshbach surface modes\nTo give an introduction to the TRMOKE signals \u0001 \u0012Kerr(\u001c) measured on the timescale\nfrom picoseconds to nanoseconds \frst, the ultrafast demagnetization on a characteristic\ntime scale\u001cMand the magnetic precessional motion damped on a time scale \u001c\u000bis shown\nfor a Ni \flm in \fgure 1b); the schematics of the processes involved on the di\u000berent time\nscales are given on the top. The change in Kerr rotation \u0001 \u0012Kerr(\u001c) shows a sudden drop\nat\u001c= 0 ps. This mirrors the demagnetization within a timescale of \u0018200 fs [34{36]. For\nthe short time scale the dynamics are dominated by electronic relaxation processes, as\ndescribed phenomenologically in the three temperature model [34] or by connecting the\nelectron-spin scattering channel with Elliot-Yafet processes, as done by Koopmans [36]\nand Chantrell [4] later. At that time scale the collective precessional motion lasting up\nto the nanosecond scale is initiated [28, 37]: the energy deposited by the pump pulse\nleads to a change in the magnetic anisotropy and magnetization, and thus the total\ne\u000bective \feld. Within \u001810 ps the total e\u000bective \feld has recovered to the old value\nand direction again. However, the magnetization, which followed the e\u000bective \feld,\nis still out of equilibrium and starts to relax by precessing around the e\u000bective \feld.\nThis mechanism can be imagined as a magnetic \feld pulse a few picoseconds long, and\nis therefore sometimes called an anisotropy \feld pulse. The resulting anisotropy \feld\npulse is signi\fcantly shorter than standard \feld pulses [38]. This makes the TRMOKE\nexperiment di\u000berent to other magnetization dynamics experiments.\nThe fact that the situation is not fully described by the model can be seen in the\nfollowing. Already van Kampen et al. [28] not only observed the coherent precessional\nmode, they also identi\fed another mode at a higher frequency than the coherentGilbert damping in Nickel thin \flms 5\nprecession mode, shifted by !k;n\u00182Ak2= 2An\u0019=t Ni2, the standing spin wave (PSSW)\nmode. It originates from the con\fnement of the \fnite layer thickness, where Ais the\nexchange coupling constant and nis a given order. Here we also present the \fnding of\ndipolar propagating spin waves. For all three, the frequency dependence as a function\nof the applied magnetic \feld will be discussed, a necessity for identifying them in the\nexperiments later on.\nFor the coherent precession the frequency dependence is described by the Kittel\nequation. It is derived by expressing the e\u000bective \feld in the Landau-Lifshitz-Gilbert\n(LLG equation) as a partial derivative of the free magnetic energy [39, 40]. Assuming\nnegligible in-plane anisotropy in case of the polycrystalline nickel (Ni) \flm and small\ntilting angles of the magnetization out of the sample plane (\feld is applied 35\u000eout of\nplane \fgure 1a)), it is solved as derived in [41]:\n!=\r\n\u00160s\n\u00160Hx\u0012\n\u00160Hx+\u00160Ms\u00002Kz\nMs\u0013\n; (1)\nFor the standing spin waves (PSSW) a similar equation is given. For the geometry\nwith the \feld applied 35\u000eout of plane (\fgure 1a) the frequencies !and!k;ndo not\nsimply add as in the \feld applied in plane geometry [41]:\n!=\r\n\u00160s\n(\u00160Hx+2Ak2\nMs)\u0012\n\u00160Hx+\u00160Ms\u00002Kz\nMs+2Ak2\nMs\u0013\n; (2)\nWhile the exchange energy dominates in the limit of small length scales, the\nmagnetic dipolar interaction becomes important at larger length scales. Damon and\nEshbach [42] derived by taking into account the dipolar interactions in the limit of\nnegligible exchange energy, the solution of the Damon-Eshbach (DE) surface waves\npropagating with a wave vector qalong the surface, decaying within the magnetic layer.\nThe wavelengths are found to be above the >\u0016m range for Ni [27].\n!=\r\n\u00160s\n\u00160Hx\u0012\n\u00160Hx+\u00160Ms\u00002Kz\nMs+M2\nS\n4[1\u0000exp(\u00002qtNi)]\u0013\n; (3)\nThe depth of the demagnetization by the femtosecond laser pulse is given by the\noptical penetration length \u0015opt\u001915 nm (\u0015= 800 nm). From the nature of the excitation\nprocess in the TRMOKE experiment one can derive that for di\u000berent thicknesses tNiit\nwill change from an excitation of the full \flm for a \u001810 nm \flm to a thin excitation\nlayer only for a few 100 nm thick \flm; thus the excitation will be highly asymmetric.\nThe model of the magnetic anisotropy \feld pulse fails to explain these e\u000bects since it is\nbased on a macrospin picture.\nAnother way to look at the excitation mechanism has been discussed by Djordjevic\net al. [5]. When the magnetic system is excited, on a length scale of the optical\npenetration depth short wavelength (high kvector) spin-wave excitations appear. As\ntime evolves, two processes appear: the modes with high frequency owning a fastGilbert damping in Nickel thin \flms 6\noscillation in space are damped very fast by giving part of the deposited energy to\nthe lattice. In addition, through multiple magnon interaction lower k-vector states are\npopulated, resulting in the highest occupation of the lowest energy modes at the end\n(e.g. the PSSW and DE modes here). As the Ni thickness is increased, the excitation\npro\fle becomes increasingly asymmetric, favoring inhomogeneous magnetic excitations,\nas the PSSW mode. The DE modes, due to their nature based on a dipolar interaction,\nare expected to be found only for higher thicknesses.\nFigure 2. Change in Kerr rotation after excitation on the long time scale for Cu 2nm =\nNi tNinm=Si(100) with tNi= 20 and b) their Fourier transform for di\u000berent applied\n\felds 0\u0000150 mT, (35\u000eout of plane (blue)). In c) the Fourier power spectra as color\nmaps for three Ni thicknesses tNi= 20, 40 and 220 nm are given. The data overlaid is\ndetermined form the peak positions. The straight lines are the analysis of the di\u000berent\nmodes and are identi\fed in the graph (Kittel model), perpendicular standing spin wave\n(PSSW) and dipolar surface spin wave (Damon Eshbach mode).Gilbert damping in Nickel thin \flms 7\nThe identi\fcation of the mode is important in determining a value for the magnetic\ndamping\u000b. Figure 2 pictures the identi\fcation of the di\u000berent modes and their\nappearance for di\u000berent Ni thicknesses. The data are handled as follows: for a\ntNi= 20 nm \flm on Si(100), covered with a 2 nm Cu protection layer, in a) the original\ndata after background subtraction and in b) its corresponding Fourier transform, shown\nfor increasing applied magnetic \feld. The evolution of the mode frequency and its\namplitude increase can be followed. An exponentially decaying incoherent background\nis subtracted from the data. This has to be done very carefully, to avoid a step-\nlike background which will be evident after Fourier transform as a sum of odd higher\nharmonics. The frequency resolution is limited by the scan range of 1 ns corresponding\nto \u0001!=2\u0019= 1 GHz. However, since the oscillation is damped within the scan range,\nthe datasets have been extended before Fourier transform to increase their grid points.\nA color map of the power spectrum is shown in \fgure 2c), where the peak positions are\nmarked by the data points overlaid. For the 20 nm thick \flm with tNi= 20 nm\u0018\u0015opt\nonly a single mode is observed. The mode is analyzed by 1 indicating the Kittel mode\nbeing present (data points and line in \fgure 2c), top) using Kz= 3:03\u0001104J=m3. With\nincreasing nickel thickness tNi= 40 nm> \u0015 opt, the perpendicular standing spin waves\n(PSSW) of \frst order are additionally excited and start to appear in the spectra (\fgure\n2c), middle). An exchange constant A= 9:5\u00011012J=m is extracted. In the limit of\ntNi= 220 nm\u001d\u0015opt(\fgure 1c)) the excitation involves the surface only. Hence, modes\nwith comparable amplitude pro\fle, e.g. with their amplitude decaying into the Ni layer,\nare preferred. Consequently DE surface waves are identi\fed as described by 3 and\ndominate the spectra up to critical \felds as high as \u00160Hcrit= 100 mT. For tNi= 220 nm\nthe wave factor is k= 2\u0016m (data points and line in \fgure 2c), bottom). For larger\n\felds than 100 mT the DE mode frequency branch merges into the Kittel mode [27].\nTo resume the previous \fndings for the \frst subsection, we have shown that in\nfact the DE modes, though they are propagating spin-wave modes, can be identi\fed\nin the spectra and play a very important role for Ni thicknesses above tNi= 80 nm.\nThey appear for thicknesses much thinner than the wavelength of the propagating\nmode. Perpendicular standing spin waves (PSSW) give an important contribution to\nthe spectra for Ni thicknesses above tNi= 20 nm. For thicknesses below tNi= 20 nm we\nobserve the homogeneously precessing Kittel mode only. This thickness range should\nbe used to determine the magnetic damping in TRMOKE experiments.\n3.2. Data analysis: determination of the magnetic damping\nFor the experiments carried out in the following with tNi<25 nm the observed dynamics\ncan be ascribed to the coherent precession of the magnetization (Kittel mode). The\nanalysis procedure is illustrated in the following using the data given in \fgure 3a). A\nPd layer is attached to a Ni \flm with the thicknesses (Ni 10 nm =Pd 5 nm=Si(100)) to\nstudy the non-local damping by spin currents absorbed by the Pd. The di\u000berent spectra\nwith varying the magnetic \feld strength from 0 mT \u0000150 mT are plotted from bottomGilbert damping in Nickel thin \flms 8\nto top (with the magnetic \feld tilted 35\u000eout of the sample plane).\n0 .0 4 5 0 .0 5 0 0 .0 5 5  \nα\n0 2 5 5 0 7 5 1 0 0 1 2 5 1 5 0 02468\nν [GHz]\nµ0 He x t  [m T ]0 2 5 0 5 0 0 7 5 0 1 0 0 0 - 8 - 4 048\n  \n∆θk  [a.u.]\nτ  [p s ]µ0He x t =\n1 5 0 m T \n1 4 0 m T \n1 3 0 m T \n1 2 0 m T \n1 1 0 m T \n1 0 0 m T \n 9 0 m T \n 8 0 m T \n 7 0 m T \n 6 0 m T \n 5 0 m T \n 4 0 m T \n 3 0 m T \n  0 m T a) b)5 nm Pd/ 10 n m Ni\n5 nm Pd/ 10 n m Ni\nFigure 3. a) Kerr rotation spectra for a Cu 2nm =Ni 10 nm=Pd 5 nm=Si(100) layer,\nmeasured for \felds applied from 0 \u0000150 mT (35\u000eout of plane, (blue)) and the \ftted\nfunctions (white, dashed). b) The magnetic damping \u000band precession frequencies\nextracted from the \fts to the measured spectra. The line is given by the Kittel mode\n(gray).\nThe data can be analyzed using the harmonic function with an exponential decay\nwithin\u001c\u000b:\n\u0001\u0012k\u0018exp\u0012\n\u0000\u001c\n\u001c\u000b\u0013\n\u0001sin(2\u0019(\u001c\u0000\u001c0)\u0017) +B(\u001c); (4)\nThe precession frequency \u0017=!=2\u0019and the exponential decay time \u001c\u000bof the\nprecession amplitude is extracted, where the function B(\u001c) stands for the background\narising from the uncorrelated magnetic and phonon excitations. To determine the\nGilbert damping parameter \u000bas given in the ansatz by Gilbert, the exponential decay\ntime\u001c\u000bhas to be related with \u000b. The LLG equation is solved under the same\npreconditions as for equation 1 using an exponential decay of the harmonic precession\nwithin\u001c\u000bfrom 4. Then the damping parameter \u000band can be expressed by the followingGilbert damping in Nickel thin \flms 9\nequation [41]:\n\u000b=1\n\u001c\u000b\r\u0010\nHx\u0000Kz\n\u00160Ms+Ms\n2\u0011: (5)\nIt is evident from 5 that in order to determine the Gilbert damping \u000bfrom the decay\nof the Kittel mode \u001c\u000b, the variables \r,MsandKzhave to be inserted, and therefore Kz\nhas to be determined beforehand.\nIn \fgure 3a), the background B(\u001c) is already subtracted. The \fts using 4 are\nplotted with the dashed lines on top of the measured spectra. The results are presented\nin b). The frequencies range from 3 GHz for 30 mT to 7 :5 GHz for the 150 mT applied\nmagnetic \feld. They increase linearly with the strength of the applied magnetic \feld for\nhigh \feld values. The extrapolated intersection with the ordinate is related to the square\nroot of the dipolar and anisotropy \feld. Using the Kittel equation (1), one determines\nthe out-of-plane anisotropy constant KzofKz= 6:8\u0001104J=m3. The calculated magnetic\ndamping\u000bas a function of the applied \feld is given in the graph below: this is mostly\nconstant but increases below 60 mT. Within the ansatz given by Gilbert, the damping\nconstant\u000bis assumed to be \feld-independent. We \fnd that this is ful\flled for most\nof the values: the average value of \u000b= 0:0453(4), consistent with earlier \fndings by\nBhagat and Lubitz from FMR experiments [43], is indicated by the line in the plot. The\n\u000bgiven in the following will always be averaged over a \feld region where the damping\nis Gilbert-like. A deviation from this value occurs for the small external \feld strengths.\nIt originates for two reasons: \ftting 5 with a few periods only does not determine a\nreliable value of the exponential precession decay time \u001c\u000band leads to a larger error.\nSecond, magnetic inhomogeneities mapping a spread in anisotropy energies within the\nprobe spot region can also be a source, and this becomes generally more important for\neven thicker \flms below 4 nm [31]. This will be discussed in more detail in the last\nsection of the manuscript.\n3.3. Intrinsic damping: nickel wedge\nFor our experiments Ni was chosen instead of Fe or Py as a ferromagnetic layer. The\nlatter would be preferable because of their lower intrinsic damping \u000bint, which make\nthe \flms more sensitive for detecting the non-local contribution to the damping. The\nreason for using Ni for our experiments is the larger signal excited in the TRMOKE\nexperiments. The magnetic damping \u000bintis used as a reference later on. The di\u000berent\nspectra with varying the Ni thickness tNiNixnm=Si(100) from 2 nm \u0014x\u001422 nm are\nplotted from bottom to top (with the constant magnetic \feld 150 mT and tilted 30\u000e\nout of plane) in \fgure 4a). The measurements were performed immediately after the\nsample preparation, in order to prevent oxidation on the nickel surface caused by the\nlack of a protection layer (omitted on purpose). The spectra show similar precession\nfrequency and initial excitation amplitude. However, the layers with tNi<10 nm show a\nfrequency shift visually recognized in the TRMOKE data. Furthermore, the precessionGilbert damping in Nickel thin \flms 10\namplitude decreases faster for the thinner layers. Figure 5 shows the frequencies and\nthe damping parameter extracted from the measured data in the intrinsic case for the\nnickel wedge sample (black squares). While the precession frequency given for 150 mT\nis almost constant above 8 nm Ni thickness, it starts to drop by about 25% for the\nthinnest layer. The magnetic damping \u000b(black squares) is found to increase to up to\n\u000b= 0:1, an indication that in addition to the intrinsic there are also extrinsic processes\ncontributing. It has to be noted that the change in \u000bis not correlated with the decrease\nof the precession frequency. The magnetic damping \u000bis found to increase below a\nthickness of 4 nm, while the frequency decrease is observed below a thickness of 10 nm.\nA priori\r,MsandKzcan be involved in the observed frequency shift, but they can\nnot be disentangled within a \ft of our \feld-dependent experiments. However, from our\nmagnetic characterization no evidence of a change of \randMsis found. A saturation\nmagnetization \u00160Ms= 0:659 T and g-factor of 2.21 for Ni are used throughout the\nmanuscriptzandKzis determined as a function of the Ni thickness, which shows a\n1=tNibehavior, as expected for a magnetic interface anisotropy term [44].\nThe knowledge of the intrinsic Gilbert damping \u000bintof the Ni \flm of a constant\nvalue for up to 3 nm thickness allows us to make a comparative study of the non-local\ndamping\u000b0, introduced by an adjacent layer of vanadium (V) and palladium (Pd) as\nrepresentatives for transition metals, and dysprosium (Dy) as a representative of the\nrare earths. Both damping contributions due to intrinsic \u000bintand non-local spin current\ndamping\u000b0are superimposed by:\n\u000b=\u000bint+\u000b0: (6)\nThey have to be disentangled by a study of the thickness dependence and compared\nto the theory of spin-current pumping, plus a careful comparison to the intrinsic value\n\u000binthas to be made.\n3.4. Non-local spin current damping: theory\nDynamic spin currents excited by a precessing moment in an adjacent nonmagnetic\nlayer (NM) are the consequence of the fact that static spin polarization at the interface\nfollows a dynamic movement of a collective magnetic excitation. The e\u000bect has already\nbeen proposed in the seventies [45,46] and later calculated within a spin reservoir model\nwith the spins pumped through the interfaces of the material by Tserkovnyak [29, 47].\nFor each precession, pumping of the spin current results in a corresponding loss in\nmagnetization, and thus in a loss of angular momentum. The spin information is lost\nand the backward di\u000busion damps the precession of the magnetic moment. In addition to\nthe \frst experiments using ferromagnetic resonance (FMR) [48{54] it has been observed\nin time- resolved experiments using magnetic \feld pulses for excitation [55,56]. In fact\nzAn altered g-factor by interface intermixing can not decrease its value below \u00182. Also, there is no\nevidence for a reduced Msfor lower thicknesses found in the Kerr rotation versus Ni thickness data.\nMore expected is a change in the magnetic anisotropy Kz. For the calculation of \u000blater on, the in\nboth cases (assuming a variation of Kzor an altered \r) the di\u000berences are negligible.Gilbert damping in Nickel thin \flms 11\na) b)\n0 250 500 750 10 00- 10 - 8 - 6 - 4 - 2 0\n2 2  n m 1 8  n m 1 0  n m 1 4  n m 8  n m 7  n m 6  n m 5  n m 4  n m 3  n m 2  n m \n \n∆θk  [a.u.]\nτ  [ p s ] 0 2 5 0 5 0 0 7 5 0 1 0 0 0 - 1 0 -8-6-4-20\n1 7  n m \n2 3  n m 1 4  n m 1 0  n m 8  n m 7  n m 6  n m 5  n m 4  n m 2  n m \n3  n m \n  \n∆θk [a.u.]\nτ  [p s]Ni r efer enc e x Ni/ 5 nm DydNi = dNi =\nFigure 4. a) Kerr rotation spectra for nickel layers from tNi= 2 nm\u000022 nm, measured\non the nickel wedge tNi= nm Ni=Si(100) and opposed in b) by a nickel wedge Al 2nm =\nDy 5nm= tNi= nm Ni=Si(100) with a 5 nm Dy spin-sink layer.\nthe non-local spin current damping is very closely related to the damping by spin-\rip\nscattering described within the s-d current model [57, 58] that uses the approximation\nof strongly localized d-states and delocalized s-states [59].\nA review describes the underlying circuit theory and dynamics of the spin currents\nat interfaces in detail [60]. The outcome of the theoretical understanding is that the\nadditional Gilbert damping is proportional to the angular momentum Ar;ltransmitted\nthrough the interface. Since each interface owns a characteristic re\rection and\ntransmission, the size of Ar;ldepends on the matching of the Fermi surfaces. The\nabsolute value is given by the total balance between transmitted angular momentum\nand the back \row. For the non-local damping \u000b0one \fnds:\n\u000b0=\r~G\"#\n4\u0019MstFM1\n1 +q\n\u001csf\n\u001celtanh\u0010\ntNM\n\u0015sd\u0011\u00001: (7)\nThe tanh function stems thereby from the di\u000busion pro\fle of the spin currents\ndetermined by the spin di\u000busion length \u0015sdwithin the non-magnetic material withGilbert damping in Nickel thin \flms 12\nthicknesstNM. Also, one \fnds from the analysis the ratio of the electron scattering\nrate\u001celversus the spin \rip rate \u001csf. The total amount of spin current through the\ninterfaces is determined by the interface spin mixing conductance G\"#. It is related\nto the magnetic volume. It is therefore that scales with the thickness of the magnetic\nlayertFM. The e\u000bective gyromagnetic ratio altered by the spin-current implies that in\naddition to an increased damping a small frequency shift will be observed. The non-local\nGilbert damping becomes important when it exceeds the intrinsic damping \u000bint.\n3.5. Non-local damping: vanadium, palladium and dysprosium\nDi\u000bering from other techniques, TRMOKE experiments require optical access for\nexcitation and detection, setting some restrictions to the layer stack assembly that can\nbe investigated with this method: a thick metallic layer on top of the magnetic layer is\nnot practical. Placing the damping layer below the magnetic layer is also unfavorable:\nby increasing the spin sink thickness the roughness of the metal \flm will increase with\nthe metals layer thickness and introduce a di\u000berent defects density, altering \u000bint. In the\nfollowing the nickel thickness will be varied and the spin sink thickness will be kept \fxed\nat 5 nm. To warrant that the nickel \flms magnetic properties are always comparable\nto the reference experiment ( Kz,\u000bint), they are always grown \frst on the Si(100). For\nthe Pd case the damping layer is below the Ni layer. Here the excitation mechanism\ndid not work and the oscillations were too weak in amplitude to analyze the damping\n\u000b, probably due to the high re\rectivity of Pd.\nThe results are presented in \fgure 4b) for the nickel wedge sample Ni xnm=Si(100)\nwith a 5 nm dysprosium (Dy) as a spin sink layer, covered by an aluminum protection\nlayer, as opposed to the nickel wedge sample data without this in a). The nickel layer\nthickness is varied from 2 nm \u0014x\u001422 nm. All spectra were measured in an external\nmagnetic \feld set to 150 mT and tilted 30\u000eout of plane. For the thinnest Ni thickness,\nthe amplitude of the precession is found to be smaller due to the absorption of the Dy\nlayer on the top. While the precession is equally damped for the Ni thicknesses ranging\nfrom 7 to 23 nm, an increased damping is found for smaller thicknesses below this. The\ndi\u000berence in damping of the oscillations is most evident for tNi= 4 and 5 nm.\nThe result of the analysis as described before is summarized in \fgure 5. In this\ngraph the data are shown for the samples with the 5 nm V, Pd, Dy spin-sink layer and\nthe Ni reference. While for the Ni reference, and Ni with adjacent V and Dy layer,\nthe frequency dependence is almost equal, indicating similar magnetic properties for\nthe di\u000berent wedge-like shaped samples, the frequency for Pd is found to be somewhat\nhigher and starts to drop faster than for the others. The most probable explanation is\nthat this di\u000berence is due to a slightly di\u000berent anisotropy for the Ni grown on top of\nPd in this case. Nevertheless, the magnetic damping found for larger thicknesses tNiis\ncomparable with the Ni reference. In the upper graph of \fgure 5 the Gilbert damping\nas a function of the Ni layer thickness is shown. While for the Pd and Dy as a spin\nsink material a additional increase below 10 nm contributing to the damping can beGilbert damping in Nickel thin \flms 13\nidenti\fed, for V no additional damping contribution is found.\n0 . 05 0 . 10 0 . 15 \n0 5 1 0 1 5 20 685 1 0 1 5 0 . 0 0 0 0 . 0 2 5 0 . 0 5 0 \ndN i  [ nm]  α\n   \nν [GHz]\n x  N i \nw i th : \n 5 n m  V  \n 5 n m  P d \n 5 n m  D y α−αint\ndN i  [n m ]\nFigure 5. Gilbert damping parameters \u000band frequency \u0017as a function of the nickel\nlayer thickness for the intrinsic case and for di\u000berent damping materials of 5 nm V,\nPd, and Dy adjacent to the ferromagnet. \u000bis extracted from experiments over a large\n\feld region. The \fts are made using equation 5 and equation 7. In the inset the data\nis shown on a reciprocal scale. Below, the frequency is given (150 mT). The lines are\nguides for the eye.\nFor the adjacent V layer, since it is a transition metal with a low spin orbit-\nscattering (light material with low atomic number Z), with a low spin-\rip scattering rate\nand thus a spin di\u000busion length larger than the thickness tNM(d\u001c\u0015sd), no additional\ndamping will occur. For Pd and Dy the situation is di\u000berent: whereas the heavier Pd\nbelongs to the transition metals with a strong orbit-scattering (heavy material with\nhigh atomic number Z), Dy belongs to the rare earth materials. It owns a localized 4fGilbert damping in Nickel thin \flms 14\nmagnetic moment: therefore, both own a high spin-\rip scattering rate and we expect\nthe latter two to be in the region where ( t\u001d\u0015sd). In their cases the thickness of 5 nm\nof the spin-sink layer is chosen to be larger than the spin di\u000busion length ( tNM\u001d\u0015sd).\nIn this limit the spin current emitted from the magnetic layer through the interface is\ntotally absorbed within the non-magnetic layer. One can simplify 6 to:\n\u000b0(1) =\r~G\"#\n4\u0019Mst\u00001\nFM: (8)\nThis is called the limit of a perfect spin sink. The additional non-local spin current\ndamping is expected to behave inversely proportional with the nickel layer thickness\n\u0018t\u00001\nFM. The inset gives the analysis and the data point on a reciprocal scale. The slope\nshows a linear increase for thinner nickel layers, as expected for an inverse proportionality\nfor both the Pd and the Dy. Since the value for the intrinsic damping of the nickel \flm\nincreases below 4 nm this contribution has to be subtracted to reveal the spin-current\ncontribution. The value for \u000b0is then found to be 0 :07 for the 2 nm Ni =5 nm Pd \flm,\nwhich is in the order found by Mizukami by FMR for sputtered Permalloy \flms with\na Pd spin sink ( \u000b0= 0:04 for 2 nm Py =5 nm Pd) [49, 50]. A further analysis of the\nthickness dependence of \u000byields values for the prefactor in 7 for Pd (0 :33(3) nm) and Dy\n(0:32(3) nm) with the \ft given in the graph. From that value the real part of the interface\nspin mixing conductance in 7 can be calculated. It is found to be G\"#= 4:5(5)\u00011015\ncm1\nfor the Ni/Pd and Ni/Dy interface. The increase of the intrinsic damping \u000binthas\nbeen analyzed using an inverse thickness dependence (prefactor of 0 :1 nm). While it\ndescribes the data in the lower thickness range, it can be seen that it does not describe\nthe thickness dependence for the thicker range and thus, probably the increase does not\noriginate from an interface e\u000bect.\n3.6. Increased damping caused by anisotropy \ructuations: consequences for the\nall-optical approach\nIn this last part we want to focus on the deviation from the intrinsic damping \u000bintfor\nthe thin nickel layers itself ( tNi<4 nm). In the low \feld range (10 \u000050 mT) small\nmagnetization inhomogeneities can build up even when the magnetization appears to\nbe still saturated from the hysteresis curve (the saturation \felds are a few mT). For\nthese thin layers the magnetization does not align parallel in an externally applied \feld\nany more, but forms ripples. The in\ruence of the ripples on the damping is discussed\nin reference [32]. In the following we adopt this ansatz to the experimental situation\nof the TRMOKE experiment. We deduce a length scale on which the magnetization\nreversal appears for two di\u000berent Ni thicknesses and relate it to the diameter of our\nprobe spot. Lateral magnetic inhomogeneities were studied using Kerr microscopy at\ndi\u000berent applied magnetic \felds [44]. Magnetization reversal takes place at low \felds\nof a -0.5 to 2 mT. The resolution of the Kerr microscopy for this thin layer thickness\ndoes not allow us to see the extent of the ripple e\u000bect in the external \feld where the\nincrease of \u000band its strong \feld dependence is observed. However, the domains in theGilbert damping in Nickel thin \flms 15\ndemagnetized state also mirror local inhomogeneities. For our Ni xnm=Si(100) sample\nthis is shown in \fgure 6a) and b). The domains imaged using Kerr microscopy are\nshown for a 3 nm and a 15 nm nickel layer in the demagnetized state. The domains of\nthe 15 nm layer are larger than the probe spot diameter of 30 \u0016m, whereas the domains\nof the 3 nm layer are much smaller.\nd =15nmNi\nd =3nmNi\ndemagnetized\ndemagnetizeda)\nb)c)\nd)20µm\n20µm\nFigure 6. a) and b) Kerr microscopy images for the demagnetized state for 15 nm\nand 3 nm. c) and d) corresponding model representing the areas with slightly varying\nanisotropy\nFrom that observation, the model of local anisotropy \ructuations known from\nFMR [30, 31] is schematically depicted in \fgure 6c) and d). A similar idea was also\ngiven by McMichael [61] and studied using micromagnetic simulations. While for the\nthick \flm the laser spot probes a region of almost homogeneous magnetization state, for\nthe thin layer case the spot averages over many di\u000berent regions with slightly di\u000berent\nmagnetic properties and their magnetization slightly tilted from the main direction\naveraging over it. The TRMOKE signal determined mirrors an average over the probed\nregion. It shows an increased apparent damping \u000band a smaller \u001c\u000bresulting from the\nline broadening and di\u000berent phase in frequency space. While for the thick layer the\ntypical scale of the magnetic inhomogeneity is as large as the probe laser spot given\nand only 1-2 regions are averaged, for the thinner \flm of dNi= 3 nm many regions\nare averaged within a laser spot, as can be seen in 6b) and d). Because the magnetic\ninhomogeneity mapping local varying anisotropies becomes more important for smaller\n\felds, it also explains the strong \feld dependence of \u000bobserved within that region.\nFigure 7 shows data calculated based on the model, in which the upper curve (i)Gilbert damping in Nickel thin \flms 16\nis calculated from the values extracted from the experimental data for the 10 nm nickel\nlayer, curve (ii) is calculated by a superposition of spectra with up to 5% deviation\nfrom the central frequency at maximum and curve (iii) is calculated by a superposition\nof spectra of 7% deviation from the central frequency at maximum to mimic the line\nbroadening. The corresponding amplitudes of the superposed spectra related to di\u000berent\nKzvalues is plotted in the inset of the graph to the given frequencies. The apparent\ndamping is increased by 0.01 (for 5%) and reaches the value given in \fgure 3b) for\nthe 10 nm \flm determined for the lowest \feld values of 30 mT. These e\u000bects generally\nbecome more important for thinner \flms, since the anisotropy \ructuations arising from\nthickness variations are larger, as shown by the Kerr images varying on a smaller length\nscale. These \ructuations can be vice versa determined by the analysis.\n/s48 /s50/s53/s48 /s53/s48/s48 /s55/s53/s48 /s49/s48/s48/s48/s48/s50/s52\n/s55/s46/s48 /s55/s46/s53 /s56/s46/s48\n/s32/s32/s77/s32/s91/s97/s46/s117/s46/s93\n/s32/s91/s112/s115/s93/s40/s105/s105/s105/s41/s40/s105/s41\n/s40/s105/s105/s41/s40/s105/s105/s105/s41/s40/s105/s41/s65/s109/s112/s108/s105/s116/s117/s100/s101/s32/s65\n/s32/s91/s71/s72/s122/s93/s40/s105/s105/s41\nFigure 7. a) Datasets generated by superposing the spectra with the frequency spread\naccording to the inset: (i) is calculated from the values extracted from the experimental\ndata for the 10 nm Ni layer, (ii) by a superposition of spectra with up to 5% and (iii) is\ncalculated by a superposition of spectra owing 7% variation from the central frequency\nat maximum. The average precession amplitude declines faster if a higher spread of\nfrequencies (i.e. di\u000berent anisotropies) are involved.Gilbert damping in Nickel thin \flms 17\n4. Conclusion\nTo conclude, we have shown that all-optical pump-probe experiments are a powerful\ntool to explore magnetization dynamics. Although the optical access to the magnetic\nlayer allows an access to the surface only, magnetization dynamics can be explored\ndirectly in the time domain, resolving di\u000berent types of spin-wave modes (Kittel mode,\nperpendicular standing spin waves and Damon-Eshbach dipolar surface waves). This is\nin contrast to FMR experiments, where the measured data is a response of the whole\nsample. The obtained data can be similar to the \feld-pulsed magnetic excitations and\nthe Gilbert damping parameter \u000b, needed for the analysis of magnetization dynamics\nand the understanding of microscopic energy dissipation, can be determined from these\nexperiments. We have evaluated the contributions of non-local spin current damping\nfor V, Pd and Dy. Yet there are limits, as shown for the nickel layer thicknesses\nbelow 4 nm, where the magnetic damping increases and may overlay other contributions.\nThis increase is attributed to an inhomogeneous line broadening arising from a strong\nsensitivity to local anisotropy variations. Further experiments will show whether the\nreduction of the probe spot diameter will improve the results by sensing smaller areas\nand thus reducing the line broadening.\nAcknowledgments\nThe \fnancial support by the Deutsche Forschungsgemeinschaft within the SPP 1133\nprogramme is greatfully acknowledged.Gilbert damping in Nickel thin \flms 18\nReferences\n[1] B. Koopmans, J. J. M. Ruigrok, F. Dalla Longa, and W. J. M. de Jonge. Phys. Rev. Lett. ,\n95(26):267207, 2005.\n[2] C. Stamm, T. Kachel, N. Pontius, R. Mitzner, T. Quast, K. Holldack, S. Khan, C. Lupulescu,\nE. F. Aziz, M. Wietstruk, H. A. Durr, and W. Eberhardt. Nat. Mater. , 6:740{743, 2007.\n[3] U. Atxitia, O. Chubykalo-Fesenko, N. Kazantseva, D. 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D., 2007."    },    {        "title": "1903.02812v1.Current_induced_motion_of_twisted_skyrmions.pdf",        "content": "1 \n Current -induced  motion of twisted skyrmions  \nChendong Jin1, Chunlei Zhang1, Chengkun Song1, Jinshuai Wang1, Haiyan Xia1, Yunxu \nMa1, Jianing Wang1, Yurui Wei1, Jianbo Wang1,2 and Q ingfang  Liu1,* \n \n1Key Laboratory for Magnetism and Magnetic Materials of the Ministry of Education, Lanzhou University, Lanzhou 730000, \nPeople’ s Republic of China . \n2Key Laboratory for Special Function Materials and Structural Design of the Ministry of the Education , Lanzhou University, \nLanzhou 730000, People ’s Republic of China.  \n \nAbstract  \nTwisted skyrmions, whose helicity angles are different from that of Bloch skyrmion s and Né el skyrmion s, have \nalready been demonstrated  in experiments recently. In this work, we first contrast  the magnetic structure and origin of the \ntwisted skyrmion with other three types of skyrmion  including Bloch skyrmion , Né el skyrmion  and antiskyrmion.  \nFollowing, we investigate the dynamics of twisted skyrmions driven by the spin transfer toque (STT) and the spin Hall \neffect (SHE) by using micromagnetic simulations. It is found that the spin Hall angle of the twisted skyrmion is related to \nthe dissipative force tensor and the Gilbert damping both for the motions induced by the  STT and the SHE, especially for \nthe SHE induced motion, the skyrmion Hall angle depends s ubstantially  on the skyrmion helicity. At last, we \ndemonstrate  that the trajectory  of the twisted skyrmion can be controlled in a t wo dimensional plane with  a Gilbert  \ndamping gradient . Our results provide the understanding of current -induced motion of twisted skyrmions, which may \ncontribute to the applications of skyrmion -based racetrack memories.  \n \nKeywords: Twisted skyrmion, spin transfer torque, spin Hall effect  \n \n \n \n \n \n_____________________________  \n*Corresponding author:  Qingfang Liu , liuqf @lzu.edu.cn  2 \n Introduction  \nIt has been recognized that the spin -polarized current -induced the motion and reversal of magnetic structures arises \nas a result of the spin transfer torque (STT) effect [1-4], which has attracted large interests due to the fundamental physics \nand potential applications in spintronic devices, such as magnetic random access memorie s (MRAM s)[5, 6] , racetrack \nmemori es[7, 8] , nano -oscillator s[9-11] and logic device s[12-14]. Recently, it has been reported that the spin Hall effect \n(SHE )[15, 16] , generated by the pure spin currents flowed from the heavy metal  substrate due to t he strong spin -orbit \ncoupling at the interface of ferromagnet/heavy -metal , is an  alternative  efficient method to manipulate the magnetization \ndynamics in magnetic materials [17-20]. Compared with the STT, the SHE  does not require currents flow through the \nmagnetic layer, and then reducing the Joule h eat and electromigration, i.e., avoiding the restricted effect of large current \ndensity in traditional STT devices [21].  \nMagne tic skyrmions are chiral spin magnetization structures with topological properties  and can be divided into the \nfollowing types according  to different types of Dzyaloshinskii -Moriya interaction (DMI) [22-26]: (i) Bloch skyrmmions  \nare first discovered in bulk non -centrosymmetric B20 -type lattice structures such as MnSi [27], FeCoS i[28-30], and \nFeGe [31, 32]  due to the presence of bulk DMI; (ii) Né el skyrmions are observed in multilayered ultrathin films lacking \ninversion symmetry with str ong spin -orbit coupling like Ir (111)/F e[33], Ta/CoFeB [34]  and Pt/Co [35] due to the presence \nof interfacial DMI ; (iii)  Antiskyrmions are reported in Heu sler compound s such as MnPtSn [36] due to the presence of \nanisotropic  DMI [37, 38] . Recently, at the interface of chiral bulk  Cu2OSeO 3 below a certain  thickness, the so -called \ntwisted skyrmions are demonstrated  directly  by the circularly polarized resonant e lastic x -ray scattering, due to the \nbreaking of translational symmetry at the surface of bulk ferromagnet [39, 40] . Up to now, the dynamics of twisted \nskyrmions driven by current have not been reported. Therefore, in this paper,  on the basis of comparing the magnetic \nstructure , origin and topological properties of the above four types of skyrmion,  we focus on the dynamics of twisted \nskyrmions driven by the STT and the SHE and also analysis the simulation results by using Thiele ’s equations [41].   3 \n Micromagnetic simulation details  \nOur magnetic simulation results are performed by using t he Object Oriented MicroMagnetic Framework (OOMMF) \npublic code [42], which includ es the additional modules for bulk DMI, interfacial DMI, anisotropic  DMI  and twisted DMI. \nThe magnetization  dynamics is described  by numerical ly solving the Landau -Lifshitz -Gilbert (LLG) equation  containing \nterms  of the STT and the SHE [17, 20] , as follow : \neff STT SHEd+ + ,m dmm H mdt dt      \n                                                    (1) \nwhere \nm  is the unit vector of the local magnet ization, is the gyromagnetic ratio,   is the Gilbert damping,  \neffH\n  is \nthe effective field including the exchange field, anisotropy field, demagnetization field and DMI  effective field . The STT \nterm is expressed  as \nSTT s s ( ) ( ),mmv m m v mxx    \n                                                        (2) \nwhere is the non -adiabatic factor , and vs is the velocity of the conduction electrons with the form \ns\n0s2PvJeM\n\n , \nwhere J is the current density, e is the electron char ge, P is the spin polarizatio n, \n is the reduced Planck c onstant , 0 is \nthe permeability of  free space , and Ms is the saturation magnetization . The electrons flowing toward + x direction when  vs > \n0. The SHE term is given by  \nSHE SH HM\n0s( ),2m m z jeM L   \n                                                     (3) \nwhere L is the thickness of the magnetic  layer with the value of 1 nm,  SH is the  spin-Hall angle  of Pt substrate with the \nvalue of 0.07 , \nz\n is the unit vectors o f the surface normal direction,  and \nHMj\n  is the current  density  injected into the \nheavy metal . \nIn order to eliminate the influence of the  boundary  effect on the size and dynamics of skyrmions, t he 2D plane  is \nassumed to 500 × 500 ×  1 nm3 (length × width × thickness) with the mesh size  of 1 ×  1 ×  1 nm3, and the initial position \nof the skyrmion is set in the center of the 2D plane. The material parameters  are chosen  similar to  Ref. [8]: saturation \nmagnetization Ms = 580  × 103 A/m, exchange constant A = 1.5  × 10-11 J/m, perpendicular magnetic anisotropy constant 4 \n Ku = 8 × 105 J/m3, and DMI strength DDMI = 2.5 ~  3.5 × 10-3 J/m2.  \nFour types of skyrmions  \nAccording  to the different helicity of skyrmions, there are four types of skyrmions: Bloch skyrmion, Né el skyrmion , \nantiskyrmion and twisted skyrmion as shown in Figs. 1(a)–(d), respectively. Figure s 1(e)–(h) display the corresponding \nspatial profiles of the  local magnetization  across the skyrmions. It can  be seen that the mz of the four types of skyrmions \nare consistent, while the mx and my of the four types of skyrmions are different, which again proves the different \ndistribution of the in-plane  magnetic moments of the four types of skyrmions. We emphasize that the distribut ion of the \nin-plane magnetic moments in the skyrmion structure is determined by the direction of the DMI vector , that is to say, the \nexistence  of the twisted skyrmion in this work is achieved by changing the DMI vector, which is much different as the \nreaso n that observed in the experiments.  Figure s 1(i)–(l) show the four types of DMIs: bulk DMI, interfacial DMI, \nanisotropic  DMI and twisted DMI that promise the existence of the Bloch skyrmion, Né el skyrmion , antiskyrmion and \ntwisted skyrmion, respectively. The four types of DMI considered in C4 symmetry can be written  as: \nˆ ˆ ˆ ˆ BulkDMI\nˆ ˆ ˆ ˆ InterDMI\nˆ ˆ ˆ ˆ AnisoDMI\nTwisDMIˆ ˆ ˆ ˆ ( ),2\nˆ ˆ ˆ ˆ ( ),2\nˆ ˆ ˆ ˆ ( ),2\n(2i i x i x i y i y\ni\ni i x i x i y i y\ni\ni i x i x i y i y\ni\niiDE S S x S x S y S y\nDE S S y S y S x S x\nDE S S y S y S x S x\nDE S S   \n   \n   \n        \n         \n        \n\n\n\nˆ ˆ ˆ ˆ ˆˆ ˆˆ ˆˆ ˆˆ ( ) ( ) ( ) ( ),x i x i y i y\nix y S x y S x y S x y              \n              (4) \nwhere D is the DMI constant representing the DMI strength , \niS\n is the  atomic moment unit vector, \nˆx  and \nˆy  are \nthe unit vectors in the model.  \nTopological properties  of four types of skyrmions  \nA. Helicity, winding number and topological number  \nIn order to better understand the helicity  and winding number of skyrmions , we use the  two-dimensional polar \ncoordinates to describe a general magnetic skyrmion structure, as shown in Fig . 2 which display s a Bloch skyrmion, as 5 \n example, in the polar coordinates with azimuthal angle ( ) and radial coordinate ( ). Therefore , the unit vector of  the \nlocal magnetization  mx, my and mz in the C artesian coordinates  can be written as [26, 43, 44] : \nsin ( )cos ( ),\nsin ( )sin ( ),\ncos ( ),x\ny\nzm\nm\nm   \n   \n\n\n\n                                                                     (5) \nwhere () is the radial profile of the perpendicular component of the magnetization, and fro m the center  to the boundary, \nits value chan ges from 0.5 to 0.5; () is the angle between the magnetic moment and the radial coordinate . The \nvorticity of skyrmions is obtained by calculat ing the full turns of the transverse magnetic moments on the perimeter and \nis defined by the winding num ber[45] \n2\n01d ( )2W\n . Therefore, the winding number W = 1 for twisted \nskyrmion , Bloch skyrmion  and Né el skyrmion  as shown in  Fig. 3(a), and W = 1 for antiskyrmion  as shown in  Fig. 3(b). \nThe helicity of a skyrmion is given by \n( ) ( 0) W           with the value ranging  form  to, that is, for \nthe Bloch skyrmion, 0.5; for Né el skyrmion , 0 or ; for twisted skyrmion , 0.5, 0 and , and the helicity \n of the twi sted skyrm ion shown in  Fig. 1 (d) equals to 0.25; for antiskyrmion  as shown in Fig.  1(c), . The \ntopological  number Q relates to the winding number and counts how many times the unit vector  along the magnetic \nmoment  wraps the unit sphere with the form [26] \n1,,4mmQ qdxdy q mxy     \n                                                      (6) \nwhere q is the topological density. Figure s 3(c) and (d) show the topological densities corresponding to the magnet ic \nskyrmions shown in  Figs.  3(a) and (b), respectivel y. It can be seen that Q = 1 in Fig.  3(c) and Q = 1 in Fig.  3(d), i.e., Q \n= W when the spins point down in the central region and  point up in the boundary region.  \nB. Skyrmion size  and d issipative force tensor  \nThe diameter of twisted skyrmion size ( d) is usually defined as the distance from in -plane to in -plane magnetization, \ni.e., the distance between the region mz = 0, as shown in the inset of Fig.  4. The dissipative force tensor  D is used to \ndescribe the effect of the dissipative forces on the moving skyrmion [46-48]. For a single twisted skyrmion, D is given by  6 \n \n0 14 , ,0 4mmdxdyxx    \nDDDD                                                    (7) \nwher e D is the diagonal element of the dissipative tensor  and also called dissipative  parameter. The dissipative  parameter \nD is determined by t he diameter  and domain wall width of the twisted skyrmion. Therefore, both d and D are affected by  \nDMI strength as shown in Fig.  4. With  the increase in DDMI from 2.5  to 3.5 mJ/m2, d increases from 7.9 to 34.8 nm and D \nincreases from 1.0577 to 1.961, respectively, for the twisted skyrmion.  \nDynamics of twisted skyrmion driven by the STT  \nTo understand the STT-induced motion of the twisted skyrmion s, we first use the Thiele equation [41] to describe the \ndynamics  of the four kinds of skyrmions  mentioned above  by casting  the L LG Eqs. (1) and (2)  to the following \nequation [46, 47] : \ns d s d( - ) ( ) 0,v v v v      D G\n                                                             (8) \nwhere G is the gyrovector with the form G = (0 0 G) = (0 0 4Q), and vd is the drift velocity of the skyrmion. When the \nvelocity of the conduction electrons vs applied along the x direction, vd = (vx, vy) is derived from Eq.  (8) as  \n2\nxs 2 2 2\nys 2 2 2()+,()\n().QvvQ\nv Q vQ  \n  \n\n     D\nD\nD\n                                                              (9) \nIt can be seen that the direction of the skyrmion deviates from the direction of the conduction elect rons when, and \nthis phenomenon  is called the skyrmion Hall effect and can be further defined  by the skyrmion Hall angle  \n \nx\nSky y22\nxy= sign( ) arccos( ),vv\nvv \n                                                          (10) \nwhich defines the angle in the range from 180o to 180o. For the situation of STT -induced skyrmion motion, the sign of \nthe vx is always  the same with vs, i.e., the skyrmion Hall angle is in the range of (-90o, 90o), and therefore the Eq. (10) can \nbe reduced to \nSky 22()= arctan( )Q\nQ\nD\nD .  \nThe trajectories  of the f our types of skyrmion driven by the in -plane STT with  vs = 100 m/s,  = 0.4,  = 0.2 and 7 \n DDMI = 3 mJ/m2 is shown in Fig.  5. The positions of the skyrmions are obtain ed by solving  the guiding center ( Rx, Ry) \nwith the form [49, 50]  \nxy ,xqdxdy yqdxdy\nR = , R =\nqdxdy qdxdy \n \n                                                         (11) \nwhere q is the topological density. One can see that the antiskyrmion deflects to the y direction, while for Bloc h \nskyrmion, Né el skyrmion  and twisted skyrmion  deflect to the y direction, i.e., θSky of the skyrmions with Q =1 \n(antiskyrmion) and Q =1 (Bloch,  Né el and twisted skyrmion ) equal to 12.89o and 12.89o, respectively. Following we \nfocus  on the STT -induced motion of twisted skyrmion with differen t conditions, as shown in Fig.  6. Figure s 6 (a) and (b) \nshow the vx and vy as a function of vs for different  with  = 0.2 and DDMI = 3 mJ/m2, respectively.  It can be seen that vx \nand vy both increase linearly with the increase in  vs for different α, it should be also note that vy is a negative value for  < \n, a positive value  for  > , and zero for  = . Then we chose the situation of vs = 100 m/s to investigate the skyrmion \nHall angle of the twisted skyrmion as a function of vs, as shown in Fig.  6(c), the skyrmion Hall angel Sky remains almost \nunchanged with the increase in vs. Figure 6(d) shows the simulation and calculation of Sky as a function of  with  = 0.2 , \nthe skyrmion Hall angle θSky decreases from 13.7 o to 12.89 o with the  increasing from 0. 01 to 0.4. According to Eq s. \n(9) and (10), both the velocity and the skyrmion Hall angle Sky are affected by the dissipative  parameter D, and the \ndissipative parameter D is determined by the DMI strength DDMI. Therefore, it is necessary to investigate the dynamics of \nthe twisted skyrmion under different DDMI, as shown in Figs.  6 (e) and (f) wit h vs = 100 m/s,  = 0.4 and  = 0.2 . vx \nincreases at first and then decreases w ith DDMI increasing from 2.5 to 3.5 mJ/m2, while vy keeps decreasing (the a bsolute \nvalue  of vy is continuously increasing ), and both simulation and calculation results support that the corresponding \nskyrmion Hall angle Sky decreases from  11.4 o to 16.8 o ( the absolute value of Sky is proportional to the DDMI). \nWe have known that the STT -induced twisted skyrmion motion is affected by the damping in the previous paragraph . \nFollowing, we investigate the dynamics of twisted skyrmion induced by the STT under a damp ing gradient, as shown in \nFig. 7. Figure 7(a) shows t he position along the y axis of the twisted skyrmion as a functio n of distance along  the x axis 8 \n with vs = 100 m/s,  = 0.4 and DDMI = 3 mJ/m2. The damping  decreases from 0.5 to 0.25 linearly from 0 to 50 nm along \nthe x axis, as indicated by the color code.  Figure 7 (b) sho ws the skyrmion Hall angle Sky of the twisted skyrmion as a \nfunction of its position along the x axis. In the region  > , the twisted skyrmion moves along the x axis direction  from 0 \nnm and deflects in the –y direction  until moving to the x axis of  20 nm , where  =  = 0.4 ; from the  region of 20 to 50 \nnm along x axis, the twisted skyrmion begins to deflect in the + y direction because of  < . Therefore , the trajectory of \ntwisted skyrmion induced by the ST T can be controlled  under a damping gradient.  \nDynamics of twisted skyrmion driven by the SHE  \nSHE -induced motion of antiskyrmion has already been studied  in Ref. [38], which demonstrates that the \nantiskyrmion Hall angle depends  on the direction of the current strongly. In thi s section, we focus on  the SHE -induced \nmotion s of the skyrmio ns whose winding number W = 1(Bloch, Né el and twisted skyrmion ). The LLG Eqs. (1) and (3)  \ncan be cast into the following form : \nd d HM 4 ( ) 0 v v B J      D GR\n                                                         (12) \nwhere G = (0 0 4) due to Q = 1, B is linked to  the SHE , and the sign of B is determined by th e SHE angle;  R(χ) is the \nin-plane rotation matrix with the form \ncos sin()sin cosR [49, 51] . When the current  JHM injected into the \nheavy metal  along the x direction, vd = (vx, vy) is derived from Eq.  (12) as  \nx HM 22\ny HM 22cos sin,1\nsin cos.1v B J\nv B J  \n\n  \n    D\nD\nD\nD\n                                                            (13) \nThe skyrmion Hall angle Sky can be obtained by the Eq.  (10), which is in the range of  180o to 180o.  \n The Eq. (13) suggests that the direction of motion of the skyrmions depends on their helicities. Therefore, we first \ninvestigate the trajectories of skyrmions driven by the SHE with  JHM = 10 ×  1010 A/m2,  = 0.2 and DDMI = 3 mJ/m2 for \ndifferent  helicities of skyrmions, as shown in Fig.  8. These  skyrmions with different helicities are achieved by changing \nthe direction of DMI vector. The simulation results in Fig.  8(a) show that the skyrmion Hall angles Sky are 150.4o, 9 \n 165.4o, 121.4o, 75.6o, 29.6o, 14.6o, 58.6o and 104.4o for the helicities χ = 0.75, 0.5, 0.25, 0, 0.25, 0.5, 0.75 \nand , respective ly. Figure 8(b) shows the skyrmion Hall angle as a function of the helicity both supported by simulations \nand calculations . Following we take the case of χ = 5 (the twisted skyrmion  shown in Fig. 1(d) ) and investigate the \nmotion induced by the SHE , as shown in Fig.  9. Figure 9(a) shows the simulation results of vx and vy of the twisted \nskyrmion as a function of JHM with  = 0.2 and DDMI = 3 mJ/m2. It can be seen that vx and vy both increase linearly with \nthe increase in  JHM, and the corresponding skyrmion Hall angle Sky is shown in Fig.  9(b). The skyrmion Hall angle  Sky \nalmost remains at 29.6o when JHM is no more than 200 ×  1010 A/m2, while for the case JHM =500 ×  1010 A/m2, the \nskyrmion Hall angle  Sky decreases to 28.9o. This is because  the size of the twisted skyrmion, i.e., the dissipative  \nparameter D, increases slightl y with JHM increasing to 500 ×  1010 A/m2, the skyrmion Hall angle  Eq. (10) can be reduced \nto  \nSky1= arctan( ).1+D\nD\n                                                                     (14) \nFor χ = 5, which indicates that the skyrmion Hall angle  Sky decreases with the increase in D. Figure 9(c) shows the \nsimulation results of vx and vy of the twisted skyrmion as a function of  with JHM =100 ×  1010 A/m2 and DDMI = 3 mJ/m2, \nvx first increases and then decreases w ith  increasing from 0.01 to 1, while vy keeps decreasing (the a bsolute value  of vy \ndecreases at first and then increases), and therefore the corresponding  skyrmion Hall angle  Sky decreases from  44.3o to \n8.2o (the trend of Sky is consistent with  vy), which also supported by  calculation, as shown in Fig.  9(d). Figure 9(e) \nshows that vx and vy both increases with DDMI increasing  from 2.5 to 3.5 mJ/m2 when JHM = 100 ×  1010 A/m2 and  = 0.2.  \nFigure 9(f) shows that the corresponding skyrmion Hall angle Sky decreases with the increase in DDMI, which is similar \nto the res ults by calculating the Eq.  (14) with the increase in D.  \nIn contrast to  the STT -induced twisted skyrmion motion under a damping gradient , we investigate th e dynamics of \ntwisted skyrmion driven  by the S HE under a damping gradient , as shown in Fig.  10. Figure 10 (a) shows the trajectory of \nthe twisted skyrmion as a functio n of its position  along  the x axis with JHM = 100 ×  1010 A/m2 and DDMI = 3 mJ/m2. The \ndamping  increases from 0.2 to 1.2 linearly from 0 to 20 0 nm along the x axis, as indicated by the color code. Figure 10 (b) 10 \n shows the corresponding skyrmion Hall angle Sky as a function of its position along the x axis. The Eq. (14)  implies that:  \nin the region 1D > 0, the twisted skyrmion moves along the x axis direction  from 0 nm and deflects in the y direction  \nuntil moving to the x axis of  114 nm where 1D = 0; in the region 1D < 0, i.e., from 114 to 200 nm along x axis, the \ntwisted skyrmion deflects in the y direction . Therefore , the trajectory of the SHE -induced motion of twisted skyrmion \ncan also be controlled  by a damping gradient.  \nConclusions  \nIn summary,  we first introduce the magnetic structure and the corresponding DMI of the twisted skyrmion in contr ast \nto that of Bloch skyrmion , Né el skyrmion  and antiskyrmion. Furthermore, we discuss and calculate the helicity, winding \nnumber, topological number, size and dissipative force tensor  of the twisted skyrmion, which pave the way for the \nfollowing study of the dynamics  of twisted skyrmion driven by the STT and the SHE . For the STT -induced motion of \ntwisted skyrmion , it is found that the skyrmion Hall angle is determined by the topological number, the dissipative force \ntensor  and the difference  between  the Gilbert damping  and the non -adiabatic factor . For the SHE -induced motion of \ntwisted skyrmion, apart from the dissipative force tensor and the Gilbert damping , the skyrmion angle depends  on the \nhelicity significantly.  At last, we demonstrate  that the trajectories of both the STT -induced and the SHE -induced motion \nof twisted skyrmion can be controlled by a Gilbert  damping gradient . These results may present guidance  for the design \nof twisted skyrmion -based  racetrack memories . \nAcknowledgments  \nThis work is supported  by National Science Fund of China  (11574121 and 51771086 ). C. J.  acknowledges the \nfunding by the China  Scholarship Council.  \nReferences  \n[1] J.C. Slonczewski, Current -driven excitation of magnetic multilayers, J. Magn. Magn. Mater.  159, L1 (1996) . \n[2] L. Berger, Emission of spin waves by a magnetic multilayer traversed by a current, Phys. Rev. B  54, 9353  (1996) . \n[3] S. Zhang, Z. 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We only intercept the central region of the 2D plane with the size of 50 nm × 50 nm. The red, white \nand blue represent where the z comp onent of the magnetization is positive, zero and negative, respectively. The black arrows  denote  \nthe distribution of the in -plane magnetization. (e)–(h) are the spatial profiles of the  local magnetization  corresponding to the yellow \ndotted line which marked in the Fig.  1(a). (i)–(l) are the configuration s of bulk DMI, interfacial DMI ，anisotropic DMI and twisted \nDMI , respectively. The orange arrows denote  the directions of the DMI vector.   \n \n \n14 \n  \nFIG. 2. Schematic of a general  skyrmion in two-dimensional polar co ordinates ., ,  and  () indica te the radial coordinate, \nazimuthal angle, skyrmion helicity and the angle between the magnetic moment and the radial coordinate, respectively.  \n \n \n \n \n \n \n \n \n15 \n  \nFIG. 3. (a) display s the magnetization distribution s of twisted skyrmion , Bloch skyrmion and Né el skyrmion  with W = 1. (b)  display s \nthe magnetization distribution  of anti skyrmion  with W = 1. (c) and (d) show the distribution s of topological density  corresponding to \nthe magnetizations shown in (a) and (b) with Q = 1 and Q = 1, respectively.  \n \n \n \n \n \n \n \n \n \n16 \n  \nFIG. 4. Skyrmion  diameter  (d) and the diagonal element of the dissipative tensor  (D) as a function of DMI stre ngth. The inset is the \nspatial profile of mz across the twisted skyrmion. It should be note that the twisted skyrmion exist s stably  in region of 250 nm × 250 \nnm, the diagram only show the central part of 50 nm × 50 nm . \n \n \n \n \n \n \n \n \n \n17 \n  \nFIG. 5. The trajectories of four types of skyrmion driven by the STT. The initial  position of the skyrmions is at  the center  of the 2D \nmagnetic film, the size of 2D plane is 250 nm × 250 nm, vs = 100 m/s  in x direction ,  = 0.4,  = 0.2 and DDMI = 3 mJ/m2. The big \nyellow  solid arrow and white dotted arrow s represent the direction of conduction electrons and the trajectories of skyrmions, \nrespectively. It should be note that the four types of skyrmions are enlarged with the purpose to see their helicities clearly . The actual \nsizes of the four skyrmions are almost the same as the skyrmion at the center position.  \n \n \n \n \n \n \n \n \n \n18 \n  \nFIG. 6. The STT-induced motion of the twisted skyrm ion (0.25). (a) and (b) display  the vx and vy as a function of vs for  = 0.01, \n0.1, 0.2, 0.3 and 0.4 with  = 0.2 and DDMI = 3 mJ/m2, respectively. (c) T he skyrmion Hall an gle Sky as a function of vs corresponding \nto the situation of  = 0.4 shown in Figs.  (a) and (b). (d) The skyrmion Hall a ngle Sky as a function of α corresponding to the situation \nof vs = 100 m/s shown in Figs.  6 (a) and (b). (e) and (f) display  the skyrmon velocity and the skyrmion Hall angle as a function of DDMI \nwith vs = 100 m/s,  = 0.4 and  = 0.2 , respectively.  \n \n \n \n19 \n  \nFIG. 7. The STT -induced motion of the twisted skyrmion under a damping gradient. (a) and (b) show the transverse distance  (y axis) of \nthe skyrmion  and the corresponding skyrmion Hall angle Sky as a function of radial distance ( x axis), respectively.  The initial  position \nof the skyrmions is  defined as 0 nm both in x and y axis, vs = 100 m/s,  = 0.4  and DDMI = 3 mJ/m2. The c olor code  represents that the \ndamping  decreases from 0.5 to 0.25 linearly in the region from 0 to 50 nm along the x direction. The red dotted line represents the \nposition w here  = .  \n20 \n  \nFIG. 8. The SHE -induced motion of skyrm ions with different  (a) The trajectories of eight types of  skyrmions with χ = 0.75, 0.5, \n0.25, 0, 0.25, 0.5, 0.75 and  driven by the SHE. The initial  position of the  eight  skyrmions is in the center of the 2D magnetic \nfilm whose size is 250 nm × 250 nm,  = 0.2 and DDMI = 3 mJ/m2. The big yellow  solid arrow denote s the direction of current JHM = 10 \n×  1010 A/m2. The wh ite dotted arrow s represent the trajectories of skyrmions. It also should be note here that the eight  types of \nskyrmi ons are enlarged to see their helicities clearly . The actual sizes of the eight skyrmions are almost the same as them at the center \nposition. (b) The skyrmion Hall angle Sky as a function of  the helicitiy . The black solid squares c orrespond to the e ight types of \nskyrmion in Fig.  8(a), and the black  hollow squares are calculated by the equation.  \n \n21 \n  \nFIG. 9. The SHE -induced motion of the twisted skyrm ion (0.25). (a) and (b) display  the skyrmion velocity and skyrmion Hall \nangle Sky as a function of JHM with  = 0.2 and DDMI = 3 mJ/m2, respectively. (c) and (d) denote the skyrmion velocity and skyrmion \nHall angle Sky as a function of  with JHM = 100 ×  1010 A/m2 and DDMI = 3 mJ/m2, respectively. (e ) and (f) represent the skyrmion \nvelocity and skyrmion Hall angle Sky as a function of DDMI with JHM = 100 ×  1010 A/m2 and  = 0.2, respectively.  \n \n \n \n \n22 \n  \nFIG. 10. The SHE -induced motion of the twisted skyrmion under a damping gradient. (a) and (b) show the transverse distance  (y axis) \nof the skyrmion  and the corresponding skyrmion Hall angle Sky as a function of radial distance ( x axis), respectively.  The initial  \nposition of the skyrmions is  defined as 0 nm both in x and y axis, JHM = 100 ×  1010 A/m2 and DDMI = 3 mJ/m2. The c olor code  \nrepresents that the damping  increases from 0.2 to 1.2 linearly in the region from 0 to 200 nm along the x direction. The red dotted \nline represents the position w here 1D = 0, i.e., the skyrmion Hall angle Sky = 0o. \n"    },    {        "title": "2106.13702v1.Perturbed_primal_dual_dynamics_with_damping_and_time_scaling_coefficients_for_affine_constrained_convex_optimization_problems.pdf",        "content": "arXiv:2106.13702v1  [math.OC]  25 Jun 2021Perturbed inertial primal-dual dynamics with damping and s caling terms for\nlinearly constrained convex optimization problems⋆\nXin Hea, Rong Hub, Ya-Ping Fanga,∗\naDepartment of Mathematics, Sichuan University, Chengdu, S ichuan, P.R. China\nbDepartment of Applied Mathematics, Chengdu University of I nformation Technology, Chengdu, Sichuan, P.R. China\nAbstract\nWeproposeaperturbedinertialprimal-dualdynamicwithdampingan dscalingcoefficients, whichinvolvesinertial\nterms both for primal and dual variables, for a linearly constrained convex optimization problem in a Hilbert\nsetting. With different choices of damping and scaling coefficients, by a Lyapunov analysis approach we discuss\nthe asymptotic properties of the dynamic and prove its fast conve rgence properties. Our results can be viewed\nextensions of the existing ones on inertial dynamical systems for t he unconstrained convex optimization problem\nto the linearly constrained convex optimization problem.\nKeywords: Perturbed inertial primal-dual dynamic, linearly constrained conve x optimization problem, damping\nand scaling, Lyapunov analysis approach, convergence rate\n1. Introduction\n1.1. Problem statement\nLetH1andH2be two real Hilbert spaces with inner /angb∇acketleft·,·/angb∇acket∇ightand norm /ba∇dbl·/ba∇dbl. Letf:H1→Rbe a differentiable\nconvex function and A:H1→ H2be a continuous linear operator with its adjoint operator AT. Consider the\nperturbed inertial primal-dual dynamical system\n\n\n¨x(t)+α(t)˙x(t) =−β(t)(∇f(x(t))+AT(λ(t)+δ(t)˙λ(t))+σAT(Ax(t)−b))+ǫ(t),\n¨λ(t)+α(t)˙λ(t) =β(t)(A(x(t) +δ(t)˙x(t))−b)(1)\nwheret∈[t0,+∞) witht0≥0,σ≥0,α: [t0,+∞)→(0,+∞) is a viscous damping coefficient, β: [t0,+∞)→\n(0,+∞) is a scaling coefficient, δ: [t0,+∞)→(0,+∞) is an extrapolation coefficient, and ǫ: [t0,+∞)→ H1\nis an integrable source term that can be interpreted as a small exte rnal perturbation. In terms of the dynamic\n(1), in this paper, we shall develop a fast primal-dual dynamic appro ach to solve the linearly constrained convex\noptimization problem\nmin\nxf(x), s.t. Ax =b. (2)\n⋆This work was supported by the National Science Foundation o f China (11471230) and the Scientific Research Foundation of the\nEducation Department of Sichuan Province (16ZA0213).\n∗Corresponding author\nEmail addresses: hexinuser@163.com (Xin He), ronghumath@aliyun.com (Rong Hu), ypfang@aliyun.com (Ya-Ping Fang)\nPreprint submitted to Elsevier June 28, 2021The primal-dualdynamic(1) involvesthree important parameters: the dampingcoefficient α(t), the extrapolation\ncoefficient δ(t), and the scalingcoefficient β(t), which play crucialrolesin derivingthe fast convergencepropert ies.\nThe importance of the damping coefficient and the scaling coefficient h as been widely recognized in inertial\ndynamical approaches[4, 19, 40] as well as fast algorithms [35, 40, 9, 39, 44] for unstrained optimization problems.\nRecently, the damping technique and the scaling technique were also used to develop inertial primal-dual dynamic\napproachesand inertial primal-dual algorithms for linearly constra ined optimization problems, see [47, 27, 28, 26].\nExtrapolation coefficients were also considered in [47, 27].\nLetL(x,λ) andLσ(x,λ) be the Lagrangian function and the augmented Lagrangian funct ion of the problem\n(2) respectively, i.e.,\nL(x,λ) =f(x)+/angb∇acketleftλ,Ax−b/angb∇acket∇ight\nand\nLσ(x,λ) =L(x,λ)+σ\n2/ba∇dblAx−b/ba∇dbl2=f(x)+/angb∇acketleftλ,Ax−b/angb∇acket∇ight+σ\n2/ba∇dblAx−b/ba∇dbl2, (3)\nwhereσ≥0 is the penalty parameter and λis the Lagrangian multiplier. Let Ω ⊂ H1×H2be the saddle point\nset ofL(Lσ). It is known that ( x∗,λ∗)∈Ω if and only if\n\n\n−ATλ∗=∇f(x∗),\nAx∗−b= 0.(4)\nThroughout this paper, we always assume that fis a convex continuously differentiable function and Ω /negationslash=∅.\nWe will investigate the asymptotical behavior of the dynamic (1) with the damping coefficient α(t) =α\ntrand the\nextrapolation coefficient δ(t) =δts, whereα >0,δ >0, and 0 ≤r≤s≤1.\n1.2. Related works\n1.2.1. Inertial dynamical systems with damping coefficients\nLet’s recall some important inertial dynamical systems with damping coefficients for the unstrained optimiza-\ntion problem\nminΦ(x), (5)\nwhere Φ( x) is a smooth convex function. The following inertial gradient system :\n(IGSα) ¨x(t)+α(t)˙x(t)+∇Φ(x(t)) = 0,\nand its perturbed version\n(IGSα,ǫ) ¨x(t)+α(t)˙x(t)+∇Φ(x(t)) =ǫ(t),\nhave been intensively studied in the literature. When damping coefficie ntα(t) =αwithα >0: (IGS α) becomes\nthe heavy ball with friction system, which was introduced by Polyak [ 36], and the asymptotic behavior has\nbeen investigated in [1, 15]; under the assumption/integraltext+∞\nt0/ba∇dblǫ(t)/ba∇dbldt <+∞, Haraux and Jendoubi [25] studied the\n2asymptotic behavior of solutions of (IGS α,ǫ). When α(t) =α\ntrwithα >0, r∈(0,1): Cabot and Frankel [20] and\nMay [33] investigated the asymptotic behavior of (IGS α) astgoes to infinity; Jendoubi and May [29] generalized\nthe results of [20] to (IGS α,ǫ) with/integraltext+∞\nt0/ba∇dblǫ(t)/ba∇dbldt <+∞and/integraltext+∞\nt0t/ba∇dblǫ(t)/ba∇dbldt <+∞respectively; Balti and May\n[13] obtained the O(1/t2r) convergence rate with/integraltext+∞\nt0tr/ba∇dblǫ(t)/ba∇dbldt <+∞and theo(1/t1+r) convergence rate with\n/integraltext+∞\nt0t(1+r)/2/ba∇dblǫ(t)/ba∇dbldt <+∞for (IGS α,ǫ); Sebbouh et al. [37] investigated the convergence rate of the va lues along\nthe trajectory of (IGS α,ǫ) under some additional geometrical conditions on Φ( x). When α(t) =α\nt: Su et al. [40]\npointed out that (IGS α) withα= 3 can be viewed as a continuous version of the Nesterov’s accelera ted gradient\nalgorithm ([14, 34]), and obtained the convergence rate Φ( x(t))−minΦ = O(1/t2) asα≥3; Attouch et al. [6]\ninvestigated the asymptotic behavior of (IGS α,ǫ) asα≥3 under the assumption/integraltext+∞\nt0t/ba∇dblǫ(t)/ba∇dbldt <+∞; May [32]\nproved an improved convergence rate Φ( x(t))−minΦ = o(1/t2) withα >3; in the case α≤3 of (IGS α) and\n(IGSα,ǫ), theO(1/t2α/3) rate of convergence can be found in [7, 43]; the optimal converge nce rates under some\nadditional geometrical conditions was studied by [11] for (IGS α) withα >0. For general damping coefficient\nα(t), it has been investigated by [4, 8, 19].\n1.2.2. Inertial dynamical systems with scaling coefficients\nBalhag el al. [12] considered following inertial gradient system with t ime scaling and constant damping\ncoefficient:\n¨x(t)+α˙x(t)+β(t)∇Φ(x(t)) = 0, (6)\nfor solving problem (5), under the assumption β(t) =eβtwithβ≤α, they can obtain the linear convergence\nwithout strong convexity of Φ. From the calculus of variations, Wibis ono et al. [44] proposed the following\ndynamic\n¨x(t)+α\nt˙x(t)+C(α−1)2tα−3∇Φ(x(t)) = 0, (7)\nwith time scaling β(t) =C(α−1)2tα−3for problem (5) where α >1 andC >0, and obtained the O(1/tα−1) rate\nof convergence. Fazlyab et al. [23] extended the dynamic (7) to fo llowing dual dynamic for solving problem (2) :\n¨λ(t)+α\nt˙λ(t)+C(α−1)2tα−3∇G(λ(t)) = 0,\nwhereG(λ) = min xL(x,λ),α >1C >0, the convergence rate G(λ∗)−G(λ(t)) =O(1/tα−1) also obtained. In\n[9], they consider following dynamic:\n¨x(t)+α\nt˙x(t)+β(t)∇Φ(x(t)) = 0\nfor problem (5), and showed O(1/t2β(t)) rate of convergence under assumption t˙β(t)≤(α−3)β(t). The general\ndamped inertial gradient system with time scaling can be found in [3, 10 , 17].\n1.2.3. Inertial primal-dual dynamics\nFortheaffineconstrainedconvexoptimizationproblem(2), themos tpopularnumericalmethodsanddynamics\nare based on the primal-dual framework. In recent years, many fi rst-order dynamical systems were proposed for\n3a better understanding of iterative schemes of the numerical algo rithms, (see [5, 16, 31, 38]). How to extend the\ndynamics (IGS α) and (IGS α,ǫ) to second-order primal-dual dynamics for solving problem (2) is a p roblem worth\nstudying. Recently, Zeng et al. [47] proposed the following damped p rimal-dual dynamical system for solving the\nproblem (2):\n\n¨x(t)+α\nt˙x(t) =−∇f(x(t))−AT(λ(t)+δt˙λ(t))−σAT(Ax(t)−b),\n¨λ(t)+α\nt˙λ(t) =A(x(t)+δt˙x(t))−b,(8)\nIn this dynamic, the damping coefficients α(t) =α\nt,δ(t) =δt. When α >3 andδ=1\n2, they showed that the\ntrajectory satisfies the following asymptotic convergence rate\nL(x(t),λ∗)−L(x∗,λ∗) =O(1/t2),/ba∇dblAx(t)−b/ba∇dbl=O(1/t), (9)\nthey also obtained L(x(t),λ∗)− L(x∗,λ∗) =O(1/t2α/3) withα≤3, δ=3\n2α. He et al. [27] and Attouch et\nal. [3] extended dynamic (8) to solve separable convex optimization p roblems with general conditions. The\n“second-order” + “first-order” primal-dual dynamics with time sc aling was investigated by [26, 28].\nIn the next, by the substitution of variables in dynamic (8), let’s illust rate the role of time scaling β(t) in\ndynamic (1). Suppose that α >3 andδ=1\n2in (8), (x∗,λ∗)∈Ω. Let’s make the change of time variable t=υ(p),\nwhereυ:R→Rand lim p→+∞υ(p) = +∞. Set ¯x(p) =x(υ(p)) and¯λ(p) =λ(υ(p)). By the chain rule, we have\n˙¯x(p) = ˙x(υ(p))˙υ(p),¨¯x(p) = ˙x(υ(p))¨υ(p)+ ¨x(υ(p))˙υ(p)2\nand\n˙¯λ(p) =˙λ(υ(p))˙υ(p),¨¯λ(p) =˙λ(υ(p))¨υ(p)+¨λ(˙υ(p))υ(p)2.\nThen rewritten (8) in terms of ¯ x(·),¯λ(·) and its derivatives, we obtain\n\n\n¨¯x(p)+/parenleftBig\nα˙υ(p)\nυ(p)−¨υ(p)\n˙υ(p)/parenrightBig\n˙¯x(p) =−˙υ(p)2(∇f(¯x(p))+AT(¯λ(p)+υ(p)\n2˙υ(p)˙¯λ(p))+σAT(A¯x(p)−b),\n¨¯λ(p)+/parenleftBig\nα˙υ(p)\nυ(p)−¨υ(p)\n˙υ(p)/parenrightBig˙¯λ(p) = ˙υ(p)2(A(¯x(p)+υ(p)\n2˙υ(p)˙¯x(p))−b).(10)\nThis leads to the time scaling coefficient β(p) = ˙υ(p)2and the damping coefficients α(p) =α˙υ(p)\nυ(p)−¨υ(p)\n˙υ(p), δ(p) =\nυ(p)\n2˙υ(p).The convergence rate (9) becomes\nL(¯x(p),λ∗)−L(x∗,λ∗) =O(1\nυ(p)2),/ba∇dblA¯x(p)−b/ba∇dbl=O(1\nυ(p)).\nIn the next, we investigate two model examples. First, taking υ(p) =ep, then (10) reads\n\n\n¨¯x(p)+(α−1)˙¯x(p) =−e2p(∇f(¯x(p))+AT(¯λ(p)+1\n2˙¯λ(p))+σAT(A¯x(p)−b)),\n¨¯λ(p)+(α−1)˙¯λ(p) =e2p(A(¯x(p)+1\n2˙¯x(p))−b).(11)\nInthiscase, thedampingcoefficients α(p) =α−1,δ(p) =1\n2areconstants,thetimescalingcoefficientis β(p) =e2p,\nand the convergence rate becomes\nL(¯x(p),λ∗)−L(x∗,λ∗) =O(1\ne2p),/ba∇dblA¯x(p)−b/ba∇dbl=O(1\nep).\n4Takingυ(p) =pκwithκ >0, then (10) reads\n\n\n¨¯x(p)+1+(α−1)κ\np˙¯x(p) =−κ2p2(κ−1)(∇f(¯x(p))+AT(¯λ(p)+p\n2κ˙¯λ(p))+σAT(A¯x(p)−b)),\n¨¯λ(p)+1+(α−1)κ\ns˙¯λ(p) =κ2p2(κ−1)(A(¯x(p)+p\n2κ˙¯x(p))−b).(12)\nthe convergence rate becomes\nL(¯x(p),λ∗)−L(x∗,λ∗) =O(1\np2κ),/ba∇dblA¯x(p)−b/ba∇dbl=O(1\ntκ),\nthe damping coefficient α(p) =1+(α−1)κ\np. Forκ≥1, we have 1+( α−1)κ≥α, so damping coefficient similar to\n(8), where α(t) =α\nt.\n1.3. Organisation\nIn Section 2, we present the rate of convergence in the different c hoice of damping coefficient and extrapola-\ntion coefficient under the suitable assumptions on time scaling coefficie nt and external perturbation. Section 3\nconcludes the paper. Some technical proofs and lemmas are postp oned to Appendix .\n2. Main results\nIn this paper, we will investigate the dynamic (1) with damping coefficie ntα(t) =α\ntrand extrapolation\ncoefficient δ(t) =δts, whereα >0,δ >0, 0≤r≤s≤1. The the dynamic (1) becomes:\n\n\n¨x(t)+α\ntr˙x(t) =−β(t)(∇f(x(t)) +AT(λ(t)+δts˙λ(t))+σAT(Ax(t)−b))+ǫ(t),\n¨λ(t)+α\ntr˙λ(t) =β(t)(A(x(t) +δts˙x(t))−b).(13)\nBeforeinvestigatingtherateofconvergence,wefirstdiscussth eexistenceanduniquenessofsolutionsfordynamical\nsystem (13).\nWhen∇f(x) is Lipschitz continuous on H1, from [3, Theorem 4.2], for any ( x0,λ0,u0,v0), the dynamic (13)\nhas a unique strong global solution ( x(t),λ(t)), in which (i): x(t)∈ C2([t0,+∞),H1),λ(t)∈ C2([t0,∞),H2); (2):\n(x(t),λ(t)) and (˙x(t),˙λ(t)) are locally absolutely continuous; (3): for almost every t∈[0,+∞), (13) holds, and\n(x(t0),λ(t0)) = (x0,λ0) and (˙x(t0),˙λ(t0)) = (u0,v0).\nWhen∇f(x) is locally Lipschitz continuous on H1, following from the Picard-Lindelof Theorem (see [42,\nTheorem 2.2]), we can establish the local existence and uniqueness s olution of dynamic (13) as follows:\nProposition 2.1. Letfbe continuously differentiable function such that ∇fis locally Lipschitz continuous,\nβ: [t0,+∞)→(0,+∞)be a continuous function, ǫ: [t0,+∞)→ H1be locally integrable. Then for any\n(x0,λ0,u0,v0), there exists a unique solution (x(t),λ(t))withx(t)∈ C2([t0,T),H1),λ(t)∈ C2([t0,T),H2)of the\ndynamic (13)satisfying (x(t0),λ(t0)) = (x0,λ0)and(˙x(t0),˙λ(t0)) = (u0,v0)on a maximal interval [t0,T)⊆\n[t0,+∞).\n5So under the assumptions in Proposition 2.1, we obtain that there ex ists a unique solution ( x(t),λ(t)) defined\non maximal interval [ t0,T)⊆[t0,+∞). If we can prove that the derivative of trajectory (˙ x(t),˙λ(t)) is bounded on\n[t0,T), it follows from assumptions that (¨ x(t),¨λ(t)) is also bounded on [ t0,T). This implies that ( x(t),λ(t)) and\nits derivative (˙ x(t),˙λ(t)) have a limit at t=T, and therefore can be continued, a contradiction. Thus T= +∞,\nwe obtain the existence and uniqueness of global solution of dynamic (13). To simplify the proof process, we\nassume that the global solution of dynamic (1) exists. We will discuss the existence and uniqueness of global\nsolution of dynamics (13) in the case r= 0,s∈[0,1] later, and it can be proved similarly for other cases.\nIn order to investigate the convergence rates of dynamic (13) un der different choices of r,s. We construct the\ndifferent energy functions, fixed ( x∗,λ∗)∈Ω, for any λ∈ H2, define the energy function Eλ,ρ\nǫ: [t0,+∞)→Ras\nEλ,ρ\nǫ(t) =Eλ,ρ(t)−/integraldisplayt\nt0/angb∇acketleftθ(w)(x(w)−x∗)+wρ˙x(w),wρǫ(w)/angb∇acket∇ightdw, (14)\nwhere\nEλ,ρ(t) =E0(t)+E1(t)+E2(t), (15)\nwith\n\nE0(t) =t2ρβ(t)(Lσ(x(t),λ)−Lσ(x∗,λ)),\nE1(t) =1\n2/ba∇dblθ(t)(x(t)−x∗)+tρ˙x(t)/ba∇dbl2+η(t)\n2/ba∇dblx(t)−x∗/ba∇dbl2,\nE2(t) =1\n2/ba∇dblθ(t)(λ(t)−λ)+tρ˙λ(t)/ba∇dbl2+η(t)\n2/ba∇dblλ(t)−λ/ba∇dbl2,\nθ,η: [t0,+∞)→Rare two smooth functions, and ρ≥0.\nThe key point of our proof is to find the appropriate θ(t),η(t) to ensure that the energy function Eλ,ρ\nǫ(t) is\ndecreasing. To avoid repeated calculations, we list the main calculatio n procedures in Appendix A.1.\n2.1. Case r= 0, s∈[0,1]\nLet us first consider the case when r= 0,s∈[0,1], i.e., the dynamic (13):\n\n\n¨x(t)+α˙x(t) =−β(t)(∇f(x(t))+AT(λ(t)+δts˙λ(t))+σAT(Ax(t)−b))+ǫ(t),\n¨λ(t)+α˙λ(t) =β(t)(A(x(t) +δts˙x(t))−b),(16)\nwithα >0, δ >0, σ≥0, t≥t0>0.\nTheorem 2.1. Assume that β: [t0,+∞)→(0,+∞)is continuous differentiable function with\nts˙β(t)≤(1\nδ−sts−1)β(t), (17)\nandǫ: [t0,+∞)→ H1is a integrable function with\n/integraldisplay+∞\nt0ts\n2/ba∇dblǫ(t)/ba∇dbldt <+∞.\nSuppose αδ >1whens= 0;δ≤1whens= 1, andσ >0. Let(x(t),λ(t))be a global solution of the dynamic\n(16)and(x∗,λ∗)∈Ω. Then(x(t),λ(t))is bounded, and the following conclusions hold:\n6(i)/integraltext+∞\nt0((1\nδ−sts−1)β(t)−ts˙β(t))(Lσ(x(t),λ∗)−Lσ(x∗,λ∗))dt <+∞.\n(ii)/integraltext+∞\nt0ts(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2)dt <+∞,/integraltext+∞\nt0β(t)/ba∇dblAx(t)−b/ba∇dbl2dt <+∞.\n(iii)/ba∇dbl˙x(t)/ba∇dbl+/ba∇dbl˙λ(t)/ba∇dbl=O(1\nts/2).\n(iv) When limt→+∞tsβ(t) = +∞:\nL(x(t),λ∗)−L(x∗,λ∗) =O(1\ntsβ(t)),/ba∇dblAx(t)−b/ba∇dbl=O(1\nts/2/radicalbig\nβ(t)).\nProof.Givenλ∈ H2, define energy functions Eλ,ρ(t) andEλ,ρ\nǫ(t) same as (15), (14) with r= 0,s∈[0,1],ρ=s\n2,\nand\nθ(t) =1\nδt−s/2, η(t) =1\nδ(α−1\nδt−s). (18)\nBy computations, we can verify that (A.2) and (A.4) hold.\nCases= 0:θ(t) =1\nδandη(t) =αδ−1\n2δ2. Sinceαδ >1, we obtain that (A.1), (A.3) hold, and then (A.5) holds,\nθ(t)+ρtρ−1−αtρ−r=1\nδ−α <0. (19)\nIt follows from (17) that\ntρ˙β(t)+(2ρtρ−1−θ(t))β(t) =˙β(t)−1\nδβ(t)≤0 (20)\nfor allt≥t0. Taking λ=λ∗, thenLσ(x(t),λ∗)−Lσ(x∗,λ∗)≥0, it follows from (19), (20) and (A.5) that\n˙Eλ∗,ρ\nǫ(t)≤(1\nδ−α)(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2)−σβ(t)\n2δ/ba∇dblAx(t)−b)/ba∇dbl2\n+(˙β(t)−1\nδβ(t))(Lσ(x(t),λ∗)−Lσ(x∗,λ∗)) (21)\n≤0.\nSoEλ∗,ρ\nǫ(·) is nonincreasing on [ t0,+∞), and then\nEλ∗,ρ\nǫ(t)≤ Eλ∗,ρ\nǫ(t0),∀t≥t0. (22)\nBy the definition of Eλ∗,ρ(·) andEλ∗,ρ\nǫ(·), we have\n1\n2/ba∇dbl1\nδ(x(t)−x∗)+ ˙x(t)/ba∇dbl2≤ Eλ∗,ρ\nǫ(t0)+/integraldisplayt\nt0/angb∇acketleft1\nδ(x(w)−x∗)+ ˙x(w),ǫ(w)/angb∇acket∇ightdw.\nBy Cauchy-Schwarz inequality, we get\n1\n2/ba∇dbl1\nδ(x(t)−x∗)+ ˙x(t)/ba∇dbl2≤ |Eλ∗,ρ\nǫ(t0)|+/integraldisplayt\nt0/ba∇dbl1\nδ(x(w)−x∗)+ ˙x(w)/ba∇dbl/ba∇dblǫ(w)/ba∇dbldw,\nthen applying Lemma Appendix A.1 with µ(t) =/ba∇dbl1\nδ(x(t)−x∗)+ ˙x(t)/ba∇dbl, we obtain\nsup\nt≥t0/ba∇dbl1\nδ(x(t)−x∗)+ ˙x(t)/ba∇dbl ≤/radicalBig\n2|Eλ∗,ρ\nǫ(t0)|+/integraldisplay+∞\nt0/ba∇dblǫ(t)/ba∇dbldt <+∞. (23)\n7It is easy to verify Eλ∗,ρ(t)≥0 for all t≥t0, then we have\ninf\nt≥t0Eλ∗,ρ\nǫ(t)≥ −sup\nt≥t0/ba∇dbl1\nδ(x(t)−x∗)+ ˙x(t)/ba∇dbl×/integraldisplay+∞\nt0/ba∇dblǫ(s)/ba∇dblds >−∞\nand\nsup\nt≥t0Eλ∗,ρ(t)≤ Eλ∗,ρ\nǫ(t0)+ sup\nt≥t0/ba∇dbl1\nδ(x(t)−x∗)+ ˙x(t)/ba∇dbl×/integraldisplay+∞\nt0/ba∇dblǫ(s)/ba∇dblds <+∞.\nThistogetherwith(22)andthedefinitionof Eλ∗,ρ(·)yieldstheboundednessof Eλ∗,ρ(·)andEλ∗,ρ\nǫ(·). Byintegrating\ninequality (21) on [ t0.+∞), it follows the boundedness of Eλ∗,ρ\nǫ(·) that\n(α−1\nδ)/integraldisplay+∞\nt0/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2dt+/integraldisplay+∞\nt0(1\nδβ(t)−˙β(t))(Lσ(x(t),λ∗)−Lσ(x∗,λ∗))dt\n+σ\n2δ/integraldisplay+∞\nt0β(t)/ba∇dblAx(t)−b)/ba∇dbl2\n<+∞.\nThis together with1\nδ< αyields (i)−(ii).\nSinceη(t) =αδ−1\n2δ2>0, By the boundedness of Eλ∗,ρ(·), we obtain that /ba∇dblx(t)−x∗/ba∇dbl2,/ba∇dblλ(t)−λ∗/ba∇dbl2,/ba∇dbl1\nδ(x(t)−\nx∗)+ ˙x(t)/ba∇dbland/ba∇dbl1\nδ(λ(t)−λ∗)+˙λ(t)/ba∇dblare bounded, and then the trajectory ( x(t),λ(t)) is bounded,\nsup\nt0∈[t0,+∞)/ba∇dbl˙x(t)/ba∇dbl ≤1\nδsup\nt∈[t0,+∞)/ba∇dblx(t)−x∗/ba∇dbl+ sup\nt∈[t0,+∞)/ba∇dbl1\nδ(x(t)−x∗)+ ˙x(t)/ba∇dbl<+∞,\nsimilarly, supt0∈[t0,+∞)/ba∇dbl˙λ(t)/ba∇dbl<+∞, this is ( iii). When lim t→+∞β(t) = +∞, following from the boundedness\nofEλ∗,ρ(·), we get\nLσ(x(t),λ∗)−Lσ(x∗,λ∗) =O(1\nβ(t)).\nSinceLσ(x(t),λ∗)−Lσ(x∗,λ∗) =L(x(t),λ∗)−L(x∗,λ∗)+σ\n2/ba∇dblAx(t)−b/ba∇dbl2, then we obtain ( iv).\nCases∈(0,1]:There exists t1≥t0such that\n1\nδt−s+s\n2t−1≤α\n2,∀t≥t1, (24)\nthis together with (18) yields\nη(t)≥α\n2δ>0,∀t≥t1. (25)\nWe can compute that\nθ(t)˙θ(t)+˙η(t)\n2= 0.\nThen (A.1)-(A.4) are satisfied for any t≥t1. It follows from (24) that\nθ(t)+ρtρ−1−αtρ−r=ts/2(1\nδt−s+s\n2t−1−α)≤ −α\n2ts/2,∀t≥t1. (26)\nBy computation,\ntρ(tρ˙β(t)+(2ρtρ−1−θ(t))β(t)) =ts˙β(t)−(1\nδ−sts−1)β(t)≤0,∀t≥t0. (27)\n8Letλ=λ∗,Lσ(x(t),λ∗)−Lσ(x∗,λ∗)≥0. Combining (26), (27) and (A.5), we get\n˙Eλ∗,ρ\nǫ(t)≤ −α\n2ts(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2)−σβ(t)\n2δ/ba∇dblAx(t)−b)/ba∇dbl2\n+(ts˙β(t)−(1\nδ−sts−1)β(t))(Lσ(x(t),λ∗)−Lσ(x∗,λ∗)) (28)\n≤0\nfor allt≥t1.Eλ∗,ρ\nǫ(·) is nonincreasing on [ t1,+∞),\nEλ∗,ρ\nǫ(t)≤ Eλ∗,ρ\nǫ(t1),∀t≥t1.\nBy the definition of Eλ∗,ρ(·) andEλ∗,ρ\nǫ(·), for all t≥t1we have\n1\n2/ba∇dbl1\nδt−s/2(x(t)−x∗)+ts/2˙x(t)/ba∇dbl2≤ Eλ∗,ρ\nǫ(t1)+/integraldisplayt\nt1/angb∇acketleft1\nδw−s/2(x(w)−x∗)+ws/2˙x(w),ws/2ǫ(w)/angb∇acket∇ightdw.\nBy similar arguments in Case s=0 , we obtain the boundedness of Eλ∗,ρ(·) andEλ∗,ρ\nǫ(·). Integrating inequality\n(28) on [ t1,+∞), we get the results ( i)−(ii).\nSinceEλ∗,ρ(·) is bounded, following from the definition of Eλ∗,ρ(·), we obtain ( iv),\nsup\nt≥t0η(t)/ba∇dblx(t)−x∗/ba∇dbl2<+∞\nand\nsup\nt≥t0/ba∇dbl1\nδt−s/2(x(t)−x∗)+ts/2˙x(t)/ba∇dbl<+∞.\nThis together with (25) and s∈(0,1] implies\nsup\nt≥t0/ba∇dblx(t)−x∗/ba∇dbl<+∞\nand\nsup\nt≥t0ts/2/ba∇dbl˙x(t)/ba∇dbl ≤1\nδsup\nt≥t0t−s/2/ba∇dblx(t)−x∗/ba∇dbl+ sup\nt≥t0/ba∇dbl1\nδt−s/2(x(t)−x∗)+ts/2˙x(t)/ba∇dbl\n≤1\nδts/2\n0sup\nt≥t0/ba∇dblx(t)−x∗/ba∇dbl+ sup\nt≥t0/ba∇dbl1\nδt−s/2(x(t)−x∗)+ts/2˙x(t)/ba∇dbl\n<+∞.\nSimilarly we have supt≥t0/ba∇dblλ(t)−λ∗/ba∇dbl<+∞, supt≥t0ts/2/ba∇dbl˙λ(t)/ba∇dbl<+∞. Then we obtain the boundedness of\n(x(t),λ(t)) and (iii).\nRemark 2.1. From Proposition 2.1, there exists a unique local solution (x(t), λ(t))of the dynamic (16)defined\non a maximal interval [t0,T)withT≤+∞. If we pick a appropriate t0>0, following from the proof process\nin Theorem 2.1 and (iii), we can obtain supt∈[t0,T)/ba∇dbl˙x(t)/ba∇dbl+/ba∇dbl˙λ(t)/ba∇dbl<+∞, and then T= +∞, the existence and\nuniqueness of global solution of the dynamic (16)is established.\n9Remark 2.2. From Theorem 2.1, we can see that for same damping α(t) =α, choosing another damping\nδ(t) =δ\ntsdifferent, the different rates of convergence can be obtained . Taking A= 0,b= 0, we can obtain the\nO(1/tsβ(t))convergence rate for dynamic (6)under the assumption ts˙β(t)≤(1\nδ−sts−1)β(t)withδ >0, so\nTheorem 2.1 complements the results in [12]. The assumption/integraltext+∞\nt0/ba∇dblǫ(t)/ba∇dbldt <+∞for perturbation ǫ(t)has been\nused in [25] for asymptotic analysis of heavy ball dynamic.\nRemark 2.3. Whens= 0, choosing β(t)≡1, then(17)is automatically satisfied. Then from (i), we have\n/integraltext+∞\nt0Lσ(x(t),λ∗)−Lσ(x∗,λ∗)dt <+∞. SinceLσ(·,λ∗)is a convex function with respect to first variable, taking\n¯x(t) =/integraltextt\nt0x(s)ds\nt−t0, we have\nLσ(¯x(t),λ∗)−Lσ(x∗,λ∗)≤1\nt−t0/integraldisplayt\nt0Lσ(x(s),λ∗)−Lσ(x∗,λ∗)ds\n≤1\nt−t0/integraldisplay+∞\nt0Lσ(x(s),λ∗)−Lσ(x∗,λ∗)ds.\nFollowing from the definition of Lσ(¯x(t),λ∗), we obtain L(¯x(t),λ∗)− L(x∗,λ∗) =O(1/t)and/ba∇dblA¯x(t)−b/ba∇dbl=\nO(1/√\nt), theO(1/t)ergodic convergence rate corresponds to the convergence ra te of the discrete heavy ball\nalgorithm in [24]; for general β(t)withs= 0, the similar convergence rate results can be found in [28]. W hen\ns= 1, choosing β(t)≡1andδ≤1, theO(1/t)rate of convergence also was investigated in [27, Theorem 4. 4] with\nr= 0for problem (2), and it is consistent with results of heavy ball dynamic and a lgorithm in [41] for problem\n(5).\nIn Theorem 2.1, when lim t→+∞tsβ(t) = +∞, we show the O(1/tsβ(t)) convergence rate of Lagrangian\nfunction and O(1/ts/2/radicalbig\nβ(t)) convergence rate of constraint, then\n|f(x(t))−f(x∗)| ≤ L(x(t),λ∗)−Lσ(x∗,λ∗)+/ba∇dblλ∗/ba∇dbl/ba∇dblAx(t)−b/ba∇dbl=O/parenleftBigg\n1\nts/2/radicalbig\nβ(t)/parenrightBigg\n.\nWe only can obtain the O(1/ts/2/radicalbig\nβ(t)) convergence rate of objection function.\nIn the next, we will investigate the best convergence rates of obj ection function and constrain for suitable\nβ(t). When s= 0, let ˙β(t) =1\nδβ(t). then˙β(t)\nβ(t)=1\nδ, integrating it on [ t0,t], we have\nβ(t) =β(t0)\net0/δet\nδ.\nIn this case, from Theorem 2.1, we can obtain the O(1\net\n2δ) convergence rate of objective function and constraint.\nLetβ(t) =µet/δwithµ >0, we list the following improved convergence rate results, which also can be found in\n[3, Proposition 6.2] with ǫ(t) = 0.\nTheorem 2.2. Letβ(t) =µet/δwithµ >0,αδ >1,s= 0,σ≥0. Assume/integraltext+∞\nt0/ba∇dblǫ(t)/ba∇dbldt <+∞. Let(x(t),λ(t))\nbe a solution of dynamic (16)and(x∗,λ∗)∈Ω. Then:\n|f(x(t))−f(x∗)|=O(1\net/δ),/ba∇dblAx(t)−b/ba∇dbl=O(1\net/δ).\n10Proof.Givenλ∈ H2, recall the energy functions Eλ,ρ(t) andEλ,ρ\nǫ(t) from Theorem 2.1 with β(t) =µet/δ, s=\nρ= 0. Then\ntρ˙β(t)+(2ρtρ−1−θ(t))β(t) = 0,\nthis together with (19) and (A.5) yields\n˙Eλ,ρ\nǫ(t)≤(1\nδ−α)(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2)−σβ(t)\n2δ/ba∇dblAx(t)−b)/ba∇dbl2≤0,∀t≥t0,λ∈ H2. (29)\nSo for any λ∈ H2,Eλ,ρ\nǫ(·) is nonincreasing on [ t0,+∞) such that,\nEλ,ρ\nǫ(t)≤ Eλ,ρ\nǫ(t0),∀t≥t0.\nBy the definition of Eλ,ρ\nǫ(·) andσ≥0, we have\nf(x(t))−f(x∗)+/angb∇acketleftλ,Ax(t)−b/angb∇acket∇ight ≤1\nµet/δ/parenleftbigg\nEλ,ǫ(t0)+ sup\nt≥t0/ba∇dbl1\nδ(x(t)−x∗)+ ˙x(t)/ba∇dbl/integraldisplay+∞\nt0/ba∇dblǫ(t)/ba∇dbldt/parenrightbigg\nfor anyλ∈ H1andt≥t0. Taking ̺ >/ba∇dblλ∗/ba∇dbl, it follows from Lemma Appendix A.2 that\nf(x(t))−f(x∗)+̺/ba∇dblAx(t)−b/ba∇dbl ≤1\nµet/δ/parenleftBigg\nsup\n/bardblλ/bardbl≤̺Eλ,ǫ(t0)+ sup\nt≥t0/ba∇dbl1\nδ(x(t)−x∗)+ ˙x(t)/ba∇dbl/integraldisplay+∞\nt0/ba∇dblǫ(t)/ba∇dbldt/parenrightBigg\n.(30)\nDenoteC= sup/bardblλ/bardbl≤̺Eλ,ǫ(t0)+supt≥t0/ba∇dbl1\nδ(x(t)−x∗)+ ˙x(t)/ba∇dbl/integraltext+∞\nt0/ba∇dblǫ(t)/ba∇dbldt. Since̺ >/ba∇dblλ∗/ba∇dbl, sup/bardblλ/bardbl≤̺Eλ,ǫ(t0)≥\nEλ∗,ǫ(t0)≥0, this together with (23) yields 0 ≤C <+∞. Following from (4), we have\nf(x(t))−f(x∗)≥ −/ba∇dblλ∗/ba∇dbl/ba∇dblAx(t)−b/ba∇dbl,\nthis together with (30) implies\n/ba∇dblAx(t)−b/ba∇dbl ≤C\nµ(̺−λ∗)et/δ\nand then\n−/ba∇dblλ∗/ba∇dblC\nµ(̺−λ∗)et/δ≤f(x(t))−f(x∗)≤C\nµet/δ.\nWe obtain results from above inequalities.\nRemark 2.4. Whens= 0andβ(t) =µet/δ, Theorem 2.1 obtains O(1\net/2δ)convergence rate of objective function\nand constraint, it is consistent with convergence rates of d ynamic(11), which is derived from dynamic (8).\nTheorem 2.2 shows that the rate of convergence is actually O(1\net/δ). Then we can obtain the linear convergence\nrate of dynamic (16)merely under the convexity assumption of f, and in this case we also allow the penalty\nparameter σof augmented Lagrangian function to be zero, which is differe nt in Theorem 2.1.\nWhens∈(0,1), letts˙β(t) = (1\nδ−sts−1)β(t). It leads\nβ(t) =ts\n0β(t0)\ne1\nδ(1−s)t1−s\n0e1\nδ(1−s)t1−s\nts.\nTakeβ(t) =µe1\nδ(1−s)t1−s\ntswithµ >0. We investigate the following optimal results.\n11Theorem 2.3. Letβ(t) =µe1\nδ(1−s)t1−s\ntswithµ >0, s∈(0,1), σ≥0. Suppose/integraltext+∞\nt0ts/2/ba∇dblǫ(t)/ba∇dbldt <+∞. Let\n(x(t),λ(t))be a solution of dynamic (16)and(x∗,λ∗)∈Ω. Then\n|f(x(t))−f(x∗)|=O/parenleftbigg1\ne1\nδ(1−s)t1−s/parenrightbigg\n,/ba∇dblAx(t)−b/ba∇dbl=O/parenleftbigg1\ne1\nδ(1−s)t1−s/parenrightbigg\n.\nProof.Givenλ∈ H2,recalltheenergyfunctions Eλ,ρ(t)andEλ,ρ\nǫ(t)fromTheorem2.1with β(t) =µe1\nδ(1−s)t1−s\nts, s∈\n(0,1), ρ=s\n2. Then\ntρ˙β(t)+(2ρtρ−1−θ(t))β(t) = 0,\nthis together with (19) and (A.5) yields\n˙Eλ,ρ\nǫ(t)≤ −α\n2ts(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2)−σβ(t)\n2δ/ba∇dblAx(t)−b)/ba∇dbl2≤0,∀t≥t1,λ∈ H2, (31)\nfor some t1≥t0. By similar arguments in Theorem 2.1, we obtain the results.\nWhens= 1, lett˙β(t) = (1\nδ−1)β(t). It leads\nβ(t) =β(t0)\nt1\nδ−1\n0t1\nδ−1.\nTakingβ(t) =µt1\nδ−1withµ >0. By similar arguments in Theorem 2.2 and Theorem 2.3, we obtain the f ollowing\nresults.\nTheorem 2.4. Letβ(t) =µt1\nδ−1withµ >0,δ≤1,s= 1,σ≥0. Suppose/integraltext+∞\nt0t1/2/ba∇dblǫ(t)/ba∇dbldt <+∞. Let\n(x(t),λ(t))be a solution of dynamic (16)and(x∗,λ∗)∈Ω. We have\n|f(x(t))−f(x∗)|=O(1\nt1/δ),/ba∇dblAx(t)−b/ba∇dbl=O(1\nt1/δ).\nRemark 2.5. Whens= 1, takingδ= 1andβ(t)≡1, from Theorem 2.4, we obtain O(1\nt)convergence rates of\nobjective function and constraint, which improves results in Theorem 2.1 with time scaling β(t)≡1.\nRemark 2.6. For damping α(t) =α,δ=δ\ntswiths∈[0,1], theO(1/tsβ(t))convergence rate in Theorem 2.1\nshows that convergence results is better as slarger in [0,1]. Conversely, following from Theorem 2.2-Theorem\n2.4, when sis smaller in [0,1], we can obtain better optimal convergence rates with suitab leβ(t).\n2.2. Case r∈(0,1), s∈[r,1]\nIn the case r∈(0,1),s∈[r,1], the dynamic (1) reads:\n\n\n¨x(t)+α\ntr˙x(t) =−β(t)(∇f(x(t)) +AT(λ(t)+δts˙λ(t))+σAT(Ax(t)−b))+ǫ(t),\n¨λ(t)+α\ntr˙λ(t) =β(t)(A(x(t) +δts˙x(t))−b).(32)\nwithα >0, δ >0, σ≥0, t≥t0>0. We will investigate the convergence properties of dynamic (32).\n12Theorem 2.5. Assume that β: [t0,+∞)→(0,+∞)is continuous differentiable function with\nts˙β(t)≤(1\nδ−τts−1)β(t) (33)\nandǫ: [t0,+∞)→ H1satisfies/integraldisplay+∞\nt0tτ/2/ba∇dblǫ(t)/ba∇dbldt <+∞,\nwhereτ∈(0,r+s). Assume αδ >1whens=r;τδ≤1whens= 1. Let(x(t),λ(t))be a global solution of the\ndynamic (32)and(x∗,λ∗)∈Ω. The following results hold:\n(i)/integraltext+∞\nt0((1\nδtτ−s−τtτ−1)β(t)−tτ˙β(t))(Lσ(x(t),λ∗)−Lσ(x∗,λ∗))dt <+∞.\n(ii)/integraltext+∞\nt0tτ−sβ(t)/ba∇dblAx(t)−b/ba∇dbl2dt <+∞,/integraltext+∞\nt0tτ−r(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2)dt <+∞.\n(iii)/ba∇dbl˙x(t)/ba∇dbl+/ba∇dbl˙λ(t)/ba∇dbl=O(1\ntτ/2).\n(iv) When limt→+∞tτβ(t) = +∞,\nL(x(t),λ∗)−L(x∗,λ∗) =O(1\ntτβ(t)),/ba∇dblAx(t)−b/ba∇dbl=O(1\ntτ/2/radicalbig\nβ(t)).\nProof.Givenλ∈ H2, recall energy functions Eλ,ρ(t) andEλ,ρ\nǫ(t) from (15), (14) with r∈(0,1),s∈[r,1],ρ=τ\n2\nand\nθ(t) =1\nδtτ/2−s, η(t) =−1\nδtτ−s−r(1\nδtr−s+(τ−s)tr−1−α). (34)\nThen the equations (A.2) and (A.4) are automatically satisfied.\nWe claim that there exists C1<0 andt1≥t0such that\n1\nδtr−s+τ\n2tr−1−α≤C1,∀t≥t1. (35)\nIndeed, when s=r, sinceαδ >1 andr∈(0,1), there exists t1≥t0such that1\nδtr−s+τ\n2tr−1−α=1\nδ−α+τ\n2tr−1≤\n1\n2(1\nδ−α)<0; whens∈(r,1], since r∈(0,1), there exists t1≥t0such that1\nδtr−s+τ\n2tr−1−α≤ −α\n2<0. Since\nτ\n2<r+s\n2≤s, it follows from (35) that\n1\nδtr−s+(τ−s)tr−1−α≤C1,∀t≥t1,\nand it yields\nη(t)≥−C1\nδtτ−s−r≥0,∀t≥t1. (36)\nSinceτ∈(0,s+r), then there exist t2≥t1such that\nα(τ−s−r)−(τ−s−1)(τ−s)tr−1<0,∀t≥t2,\nso we can compute\nθ(t)˙θ(t)+˙η(t)\n2=1\n2δtτ−s−r−1(α(τ−s−r)−(τ−s−1)(τ−s)tr−1)<0\n13for allt≥t2. Then (A.1) and (A.3) hold for any t≥t2.\nIt follows from (35) that\nθ(t)+ρtτ\n2−1−αtτ\n2−r=tτ\n2−r(1\nδtr−s+τ\n2tr−1−α)≤C1tτ\n2−r<0,∀t≥t1. (37)\nBy computation, and from (33), we have\ntρ˙β(t)+(2ρtρ−1−θ(t))β(t) =tτ\n2−s(ts˙β(t)−(1\nδ−τts−1)β(t))≤0.\nfor allt≥t0. Letλ=λ∗, thenLσ(x(t),λ∗)−Lσ(x∗,λ∗)≥0, this together with (34), (37) and (A.5) yields\n˙Eλ∗,ρ\nǫ(t)≤C1tτ−r(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2)−σtτ−sβ(t)\n2δ/ba∇dblAx(t)−b)/ba∇dbl2(38)\n+(tτ˙β(t)−(1\nδtτ−s−τtτ−1)β(t))(Lσ(x(t),λ∗)−Lσ(x∗,λ∗))\n≤0\nfor allt≥t2. ThenEλ∗,ρ\nǫ(·) is nonincreasing on [ t2,+∞),\nEλ∗,ρ\nǫ(t)≤ Eλ∗,ρ\nǫ(t2),∀t≥t2.\nSince/integraltext+∞\nt0tτ/2/ba∇dblǫ(t)/ba∇dbldt <+∞, by similar arguments in proof of Theorem 2.1 and using the fact C1<0, we\nobtain that Eλ∗,ρ(·) andEλ∗,ρ\nǫ(·) are bounded on [ t0,+∞), and then ( i),(ii),(iv) hold. It follows from (34), (36)\nand the definition of Eλ∗,ρ(·) that\nsup\nt≥t0t(τ−s−r)/2/ba∇dblx(t)−x∗/ba∇dbl<+∞,sup\nt≥t0/ba∇dbl1\nδtτ/2−s(x(t)−x∗)+tτ/2˙x(t)/ba∇dbl<+∞.\nSinces∈[r,1], then\nsup\nt≥t0tτ/2/ba∇dbl˙x(t)/ba∇dbl ≤1\nδsup\nt≥t0tτ/2−s/ba∇dblx(t)−x∗/ba∇dbl+ sup\nt≥t0/ba∇dbl1\nδtτ/2−s(x(t)−x∗)+tτ/2˙x(t)/ba∇dbl\n≤1\nδsup\nt≥t0t(τ−s−r)/2/ba∇dblx(t)−x∗/ba∇dbl+ sup\nt≥t0/ba∇dbl1\nδtτ/2−s(x(t)−x∗)+tτ/2˙x(t)/ba∇dbl\n<+∞.\nSimilarly, supt≥t0tτ/2/ba∇dbl˙λ(t)/ba∇dbl<+∞, the result ( iii) holds.\nIf we take β(t) satisfying\nts˙β(t)≤(1\nδ−(r+s)ts−1)β(t),\nthen for any τ∈(0,r+s), (33) is satisfied, and then we obtain the following results from The orem 2.5.\nCorollary 2.1. Assume that\nts˙β(t)≤(1\nδ−(r+s)ts−1)β(t),/integraldisplay+∞\nt0t(r+s)/2/ba∇dblǫ(t)/ba∇dbldt <+∞. (39)\nSuppose αδ >1whens=r;δ(r+s)≥1whens= 1. Let(x(t),λ(t))be a global solution of the dynamic (32).\nThen for any (x∗,λ∗)∈Ωandτ∈(0,r+s):\n14(i)/ba∇dbl˙x(t)/ba∇dbl+/ba∇dbl˙λ(t)/ba∇dbl=O(1\ntτ/2).\n(ii) When limt→+∞tτβ(t) = +∞,\nL(x(t),λ∗)−L(x∗,λ∗) =O(1\ntτβ(t)),/ba∇dblAx(t)−b/ba∇dbl=O(1\ntτ/2/radicalbig\nβ(t)).\nRemark 2.7. In proof process of Theorem 2.5, we can note that the boundedn ess of trajectory (x(t),λ(t))is not\nguaranteed. If (39)holds, we can obtain\nsup\nt≥t0t(τ−s−r)/2(/ba∇dblx(t)−x∗/ba∇dbl+/ba∇dblλ(t)−λ∗/ba∇dbl)<+∞\nis satisfied for any (x∗,λ∗)∈Ωandτ∈(0,r+s), then we get that tp(/ba∇dblx(t)−x∗/ba∇dbl+/ba∇dblλ(t)−λ∗/ba∇dbl)is bounded for\nanyp <0. When objective function fsatisfying the following coercive condition:\nlim\n/bardblx/bardbl→+∞f(x) = +∞, (40)\nwe also can obtain the boundedness of x(t)of dynamic (32)from(iv)of Theorem 2.5.\nRemark 2.8. Takingβ(t)≡1,s= 1, and letting/integraltext+∞\nt0t(r+1)/2/ba∇dblǫ(t)/ba∇dbldt <+∞andδ(r+1)≥1. We obtain the\nO(1/tτ)rate of convergence for any τ∈(0,r+1), sincer∈(0,1),r+ 1>2r, so the results in Corollary 2.1\nimprove the corresponding results in [27, Theorem 3.4] whic h only obtain the O(1/t2r)convergence rate. In the\ncaseα(t) =α\ntrwithα >0, r∈(0,1), theo(1/tr+1)convergence of (IGSα)and(IGSα,ǫ)for problem (5)have\nbeen obtained in [4, Corollary 4.5] and [13, Theorem 1.2] res pectively, which have subtle differences of dynamic\n(32)for problem (2). The assumption/integraltext+∞\nt0t(r+1)/2/ba∇dblǫ(t)/ba∇dbldt <+∞also can find in [13].\nBy similar discussions in Section 2.1, we obtain the following optimal conv ergence rates of Theorem 2.5, and\nthe proof is similar to Theorem 2.2, so we omit it.\nTheorem 2.6. Letβ(t) =µe1\nδ(1−s)t1−s\ntτwithµ >0, τ∈(0,r+s), r∈(0,1), s∈[r,1), σ≥0. Assume αδ >1\nwhens=r. Suppose/integraltext+∞\nt0tτ/2/ba∇dblǫ(t)/ba∇dbldt <+∞. Let(x(t),λ(t))be a solution of dynamic (16)and(x∗,λ∗)∈Ω.\nThen\n|f(x(t))−f(x∗)|=O/parenleftbigg1\ne1\nδ(1−s)t1−s/parenrightbigg\n,/ba∇dblAx(t)−b/ba∇dbl=O/parenleftbigg1\ne1\nδ(1−s)t1−s/parenrightbigg\n.\nTheorem 2.7. Letβ(t) =µt1\nδ−τwithµ >0,τ∈(0,r+ 1),r∈(0,1), s= 1, δτ≤1, σ≥0. Suppose\n/integraltext+∞\nt0tτ/2/ba∇dblǫ(t)/ba∇dbldt <+∞. Let(x(t),λ(t))be a solution of dynamic (16)and(x∗,λ∗)∈Ω. Then\n|f(x(t))−f(x∗)|=O(1\nt1/δ),/ba∇dblAx(t)−b/ba∇dbl=O(1\nt1/δ).\nRemark 2.9. Whens= 1, taking τ=1\nδ, thenβ(t) =µ >0is a positive constant time scaling. For any\n1\nδ< r+1, we can obtain the O(1\nt1/δ)convergence rates of objective function and constraint.\n152.3. Case r= 1, s= 1\nConsider the case when r= 1,s= 1, i.e., the dynamic (1) becomes:\n\n\n¨x(t)+α\nt˙x(t) =−β(t)(∇f(x(t))+AT(λ(t)+δt˙λ(t))+σAT(Ax(t)−b))+ǫ(t),\n¨λ(t)+α\nt˙λ(t) =β(t)(A(x(t) +δt˙x(t))−b).(41)\nWe will discuss dynamic (41) with α≤3 andα >3 respectively.\nTheorem 2.8. Assume that β: [t0,+∞)→(0,+∞)is continuous differentiable function with\nt˙β(t)≤τβ(t), (42)\nandǫ: [t0,+∞)→ H1satisfies/integraldisplay+∞\nt0tα−τ\n3/ba∇dblǫ(t)/ba∇dbldt <+∞.\nLet0≤τ≤α≤3,δ=3\n2α+τand(x(t),λ(t))be a global solution of the dynamic (41). Then for any (x∗,λ∗)∈Ω,\nthe following conclusions hold:\n(i)/integraltext+∞\nt0t2(α−τ)\n3−1β(t)/ba∇dblAx(t)−b/ba∇dbl2dt <+∞.\n(ii) When τ∈[0,α): for any ρ∈[0,α−τ\n3),\n/integraldisplay+∞\nt0t2ρ−1β(t)(Lσ(x(t),λ∗)−Lσ(x∗,λ∗))dt <+∞,/integraldisplay+∞\nt0t2ρ−1/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2dt <+∞.\n(iii) When limt→+∞t2(α−τ)\n3β(t) = +∞:\nL(x(t),λ∗)−L(x∗,λ∗) =O(1\nt2(α−τ)\n3β(t)),/ba∇dblAx(t)−b/ba∇dbl=O(1\ntα−τ\n3/radicalbig\nβ(t)).\n(iv) When τ= 0andα= 3:\n/ba∇dbl˙x(t)/ba∇dbl+/ba∇dbl˙λ(t)/ba∇dbl=O(1\ntρ),∀ρ∈(0,1).\nOtherwise:\n/ba∇dbl˙x(t)/ba∇dbl+/ba∇dbl˙λ(t)/ba∇dbl=O(1\ntα−τ\n3).\nProof.Givenλ∈ H2, defineEλ,ρ(t) andEλ,ρ\nǫ(t) as (15), (14) with r=s= 1,ρ∈[0,α−τ\n3] and\nθ(t) =2α+τ\n3tρ−1, η(t) =2α+τ\n3(1+α−τ\n3−2ρ)t2ρ−2. (43)\nBy computation, we have\nη(t)≥2α+τ\n3(1−α−τ\n3)t2ρ−2≥0, (44)\nand (A.2), (A.4) are satisfied. Since 0 ≤τ≤α≤3 andρ∈[0,α−τ\n3], we also can verify that\nθ(t)˙θ(t)+˙η(t)\n2=2α+τ\n3(α+1−2ρ)(ρ−1)t2ρ−3≤0.\n16Then (A.1)-(A.4) hold for any t≥t0.\nIt is easy to verify that\nθ(t)+ρtρ−1−αtρ−r= (ρ−α−τ\n3)tρ−1≤0 (45)\nand\ntρ˙β(t)+(2ρtρ−1−θ(t))β(t) =tρ−1(t˙β(t)−τβ(t)+(τ+2ρ−2α+τ\n3)β(t))\n≤2(ρ−α−τ\n3)tρ−1β(t) (46)\n≤0\nfor allt≥t0. This together with (A.5) in case λ=λ∗implies\n˙Eλ∗,ρ\nǫ(t)≤(ρ−α−τ\n3)t2ρ−1(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2)+2(ρ−α−τ\n3)t2ρ−1β(t)(Lσ(x(t),λ∗)−Lσ(x∗,λ∗))\n−σt2ρ−1β(t)\n2δ/ba∇dblAx(t)−b)/ba∇dbl2(47)\n≤0.\nThenEλ∗,ρ\nǫ(·) is nonincreasing on [ t0,+∞),\nEλ∗,ρ\nǫ(t)≤ Eλ∗,ρ\nǫ(t0),∀t≥t0.\nSince/integraltext+∞\nt0tα−τ\n3/ba∇dblǫ(t)/ba∇dbldt <+∞, for any ρ∈[0,α−τ\n3], we have\n/integraldisplay+∞\nt0tρ/ba∇dblǫ(t)/ba∇dbldt <+∞.\nBy similar arguments in proof of Theorem 2.1, we obtain the boundedn ess ofEλ∗,ρ(·) andEλ∗,ρ\nǫ(·). Since (47)\nholds for any ρ∈[0,α−τ\n3], integrating it on [ t0,+∞), and following from the boundedness of Eλ∗,ρ\nǫ(·), we get the\nresults (i)−(ii).\nSinceEλ∗,ρ(·) is bounded for any ρ∈[0,α−τ\n3], by the definition of Eλ∗,ρ(·) and (3), (43), we obtain ( iii),\nsup\nt≥t0/radicalbigg\n1+α−τ\n3−2ρ×tρ−1/ba∇dblx(t)−x∗/ba∇dbl<+∞ (48)\nand\nsup\nt≥t0/ba∇dbl2α+τ\n3tρ−1(x(t)−x∗)+tρ˙x(t)/ba∇dbl<+∞ (49)\nfor anyρ∈[0,α−τ\n3].\nWhenτ= 0 and α= 3: for any ρ∈(0,1), 1+α−τ\n3−2ρ= 2(1−ρ)>0, it follows from (48) and (49) that\nsup\nt≥t0tρ−1/ba∇dblx(t)−x∗/ba∇dbl<+∞\nand then\nsup\nt≥t0tρ/ba∇dbl˙x(t)/ba∇dbl ≤2sup\nt≥t0tρ−1/ba∇dblx(t)−x∗/ba∇dbl+ sup\nt≥t0/ba∇dbl2tρ−1(x(t)−x∗)+tρ˙x(t)/ba∇dbl)<+∞,\n17for anyρ∈(0,1). Similarly supt≥t0tρ/ba∇dbl˙λ(t)/ba∇dbl<+∞.\nOtherwise α−τ <3, taking ρ=α−τ\n3, then 1+α−τ\n3−2ρ= 1−α−τ\n3>0, by similar discussions in above, we\nget (iv).\nRemark 2.10. Following from above proof process, when τ= 0andα= 3:\nsup\nt≥t0tρ−1(/ba∇dblx(t)−x∗/ba∇dbl+/ba∇dblλ(t)−λ∗/ba∇dbl)<+∞,∀ρ∈(0,1),\nOtherwise:\nsup\nt≥t0tα−τ\n3−1(/ba∇dblx(t)−x∗/ba∇dbl+/ba∇dblλ(t)−λ∗/ba∇dbl)<+∞.\nWhen the coercive condition (40)satisfied , we also can obtain the boundedness of x(t)of dynamic (42)with\nα≤3.\nRemark 2.11. Theorem 2.8 extends the results in [27, Corollary 2.9] and [4 7, Theorem 3.2] to general case.\nTakingA= 0,b= 0, the dynamic (41)reduces to\n¨x(t)+α\nt˙x(t)+β(t)∇f(x(t)) =ǫ(t),\nwithα≤3for solving unconstrained optimization problem, then Theo rem 2.8 also complements the results in [9,\nTheorem A.1], which considered the case α≥3.\nTakingt˙β(t) =τβ(t), in which β(t) =µtτwithµ >0, we investigate the improved rate of convergence.\nTheorem 2.9. Letβ(t) =µtτwithµ >0,0≤τ≤α≤3,δ=3\n2α+τ, σ≥0. Suppose/integraltext+∞\nt0t(α−τ)/3/ba∇dblǫ(t)/ba∇dbldt <\n+∞. Let(x(t),λ(t))be a solution of dynamic (41). For any (x∗,λ∗)∈Ω:\n|f(x(t))−f(x∗)|=O(1\nt(2α+τ)/3),/ba∇dblAx(t)−b/ba∇dbl=O(1\nt(2α+τ)/3).\nProof.Recall the energy functions Eλ,ρ(t) andEλ,ρ\nǫ(t) from Theorem 2.8 with β(t) =µtτ,ρ=α−τ\n3. Then\ntρ˙β(t)+(2ρtρ−1−θ(t))β(t) = 0,\nthis together with (45) and (A.5) yields\n˙Eλ,ρ\nǫ(t)≤ −σβ(t)\n2δ/ba∇dblAx(t)−b)/ba∇dbl2≤0,∀t≥t0,λ∈ H2. (50)\nBy similar arguments in Theorem 2.2, we obtain the results.\nFrom Theorem 2.9, we obtain the following results in the case τ= 0 and τ=α, respectively.\nCorollary 2.2. Letβ(t) =β >0,α≤3,δ=3\n2α, σ≥0. Suppose/integraltext+∞\nt0tα/3/ba∇dblǫ(t)/ba∇dbldt <+∞. Let(x(t),λ(t))be a\nsolution of dynamic (41). For any (x∗,λ∗)∈Ω:\n|f(x(t))−f(x∗)|=O(1\nt2α/3),/ba∇dblAx(t)−b/ba∇dbl=O(1\nt2α/3).\n18Corollary 2.3. Letβ(t) =µtαwithµ >0,α≤3,δ=1\nα, σ≥0. Suppose/integraltext+∞\nt0/ba∇dblǫ(t)/ba∇dbldt <+∞. Let(x(t),λ(t))\nbe a solution of dynamic (41). For any (x∗,λ∗)∈Ω:\n|f(x(t))−f(x∗)|=O(1\ntα),/ba∇dblAx(t)−b/ba∇dbl=O(1\ntα).\nRemark 2.12. Takingβ= 1, the dynamic (41)has been investigate in [27] and [47] for α≤3. Corollary\n2.2 improves the convergence rates of [27, Corollary 2.9] an d [47, Theorem 3.2], which only obtain O(1\ntα/3)\nconvergence rate of |f(x(t))−f(x∗)|and/ba∇dblAx(t)−b/ba∇dbl, and it also can be viewed as analogs of the results in [7, 43],\nwhere the convergence rate analysis of (IGSα,ǫ)withα(t) =α\nt, α≤3for unconstrained optimization problem\n(5). Corollary 2.3 shows the optimal convergence rate we can exp ect of dynamic (41)withα≤3.\nNext, we investigate the convergence rate of dynamic (41) with α >3. The similar results can be found in\n[26].\nTheorem 2.10. Assume that β: [t0,+∞)→(0,+∞)is continuous differentiable function with\nt˙β(t)≤(1\nδ−2)β(t),\nand2≤1\nδ< α−1. Letǫ: [t0,+∞)→ H1with\n/integraldisplay+∞\nt0t/ba∇dblǫ(t)/ba∇dbldt <+∞.\nLet(x(t),λ(t))be a global solution of the dynamic (41)and y(x∗,λ∗)∈Ω. Then(x(t),λ(t))is bounded and the\nfollowing conclusions hold:\n(i)/integraltext+∞\nt0t((1\nδ−2)β(t)−t˙β(t))(Lσ(x(t),λ∗)−Lσ(x∗,λ∗))dt <+∞.\n(ii)/integraltext+∞\nt0tβ(t)/ba∇dblAx(t)−b/ba∇dbl2dt <+∞,/integraltext+∞\nt0t/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2dt <+∞.\n(iii)/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2=O(1\nt).\n(iv) When limt→+∞t2β(t) = +∞:\nL(x(t),λ∗)−L(x∗,λ∗) =O(1\nt2β(t)),/ba∇dblAx(t)−b/ba∇dbl=O(1\nt/radicalbig\nβ(t)).\nProof.Givenλ∈ H2, defineEλ,ρ(t) andEλ,ρ\nǫ(t) as (15), (14) with r=s=ρ= 1 and\nθ(t) =1\nδ, η(t) =αδ−δ−1\nδ2.\nSinceα−1>1\nδ≥2, by simple computations we can verify (A.1)-(A.4). It follows from a ssumptions that\ntρ˙β(t)+(2ρtρ−1−θ(t))β(t) =t˙β(t)+(2−1\nδ)β(t)≤0.\nTakingλ=λ∗, this together with (A.5) implies\n˙Eλ∗,ρ\nǫ(t)≤(1\nδ+1−α)t(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2)+t(t˙β(t)+(2−1\nδ)β(t))(Lσ(x(t),λ∗)−Lσ(x∗,λ∗))\n19−σtβ(t)\n2δ/ba∇dblAx(t)−b/ba∇dbl2\n≤0.\nBy similarly arguments in proof of Theorem 2.1, we obtain the bounded ness ofEλ∗,ρ(·) andEλ∗,ρ\nǫ(·). This yields\n(i),(ii),(iv). Sinceη(t) =αδ−δ−1\nδ2>0, we get that ( x(t),λ(t)) is bounded and\nsup\nt≥t0t(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dblλ(t)/ba∇dbl2)<+∞.\nThis implies ( iii).\nRemark 2.13. Theorem 2.10 extends the results in [9, Theorem A.1] and [10, Section 3.2] from (IGSα,ǫ)with\nα(t) =α\nt, α >3for problem (5)to primal-dual dynamic for problem (2). Taking β(t)≡1, we recover the\nconvergence rate of [27, Corollary 2.9] and [47, Theorem 3.1 ], moreover when A= 0, b= 0, we get the classical\nresults for (IGSα)and(IGSα,ǫ)withα(t) =α\ntwithα >3, which can be seen as a continuous version of the\nNesterov method, see [6, 11, 33, 40].\nLett˙β(t) = (1\nδ−2)β(t). We have β(t) =µt1\nδ−2withµ >0. By similar proof of Theorem 2.2, we obtained\nfollowing results, and the corresponding results of unperturbed c ase can be found in [3, Proposition 6.3].\nTheorem 2.11. Letβ(t) =µt1/δ−2withµ >0,2≤1\nδ< α−1,σ≥0. Suppose/integraltext+∞\nt0t/ba∇dblǫ(t)/ba∇dbldt <+∞. Let\n(x(t),λ(t))be a solution of dynamic (41). For any (x∗,λ∗)∈Ω:\n|f(x(t))−f(x∗)|=O(1\nt1/δ),/ba∇dblAx(t)−b/ba∇dbl=O(1\nt1/δ).\nFrom Theorem (2.11), we have following result.\nCorollary 2.4. Letβ(t) =β >0,δ=1\n2,α >3,σ≥0. Suppose/integraltext+∞\nt0t/ba∇dblǫ(t)/ba∇dbldt <+∞. Let(x(t),λ(t))be a\nsolution of dynamic (41). For any (x∗,λ∗)∈Ω:\n|f(x(t))−f(x∗)|=O(1\nt2),/ba∇dblAx(t)−b/ba∇dbl=O(1\nt2).\nRemark 2.14. Theorem 2.11 shows the optimal convergence rates of dynamic (41)in the case α >3. The\nO(1/tp+2)convergence rate results associated with the time scaling β(t) =µtpfor unconstrained optimization\nproblem (5)can be found in [9, 44], it also can be found in [23] with Euclid ean setting of Bregman distance for\nproblem (2). Corollary 2.4 showst the convergence rate of objective fun ction and constraint of dynamical system\n(8)isO(1\nt2)instead of O(1\nt).\n2.4. Summary of results\nIn the subsection, we complete the tables giving a synthetic view of c onvergence results in before.\nFor dynamic (13) with different rands, chose suitable parameters α, δ. Table 1 lists the convergences rates\nforL(x(t),λ∗)− L(x∗,λ∗) of dynamic (1) under different assumptions of β(t) andǫ(t). Table 2 summarizes\n20the properties of trajectory ( x(t),λ(t)) and its derivates (˙ x(t),˙λ(t)). (See Theorem 2.1, Corollary 2.1, Theorem\n2.8, Theorem 2.10, Remark 2.7, Remark 2.10). The results extend th e inertial dynamic with time scaling in\n[9, 10, 12, 44] for problem (5) to primal-dual dynamic (1) for proble m (2). Taking A= 0,b= 0, our results also\ncan complement the existing results the inertial dynamic with time sca ling.\nTable 1: Convergence rates for L(x(t),λ∗)−L(x∗,λ∗) of dynamic (1)\nr,s β(t) ǫ(t) L(x(t),λ∗)−L(x∗,λ∗)\nr= 0,s∈[0,1] ts˙β(t)≤(1\nδ−sts−1)β(t)/integraltext+∞\nt0ts\n2/ba∇dblǫ(t)/ba∇dbldt <+∞ O(1\ntsβ(t))\nr∈(0,1),s∈[r,1]ts˙β(t)≤(1\nδ−(r+s)ts−1)β(t)/integraltext+∞\nt0tr+s\n2/ba∇dblǫ(t)/ba∇dbldt <+∞O(1\ntρβ(t)),∀ρ∈(0,r+s)\nr=s= 1α≤3t˙β(t)≤τβ(t),τ∈[0,α]/integraltext+∞\nt0tα−τ\n3/ba∇dblǫ(t)/ba∇dbldt <+∞ O/parenleftBig\n1\nt2(α−τ)/3β(t)/parenrightBig\nα >3 t˙β(t)≤(1\nδ−2)β(t)/integraltext+∞\nt0t/ba∇dblǫ(t)/ba∇dbldt <+∞ O(1\nt2β(t))\nTable 2: Summary of trajectory properties\nr,s /ba∇dbl˙x(t)/ba∇dbl+/ba∇dbl˙λ(t)/ba∇dbl I=/ba∇dblx(t)−x∗/ba∇dbl+/ba∇dblλ(t)−λ∗/ba∇dbl\nr=s= 0 bounded Ibounded\nr= 0, s∈(0,1] O(1\nts/2) Ibounded\nr∈(0,1), s∈[r,1] O(1\ntρ),∀ρ∈(0,r+s\n2) tρIbounded, ∀ρ∈(−r+s\n2,0)\nr=s= 1α= 3,τ= 0 O(1\ntρ),∀ρ∈(0,1) tρIbounded, ∀ρ∈(−1,0)\nα≤3,0≤α−τ <3 O(1\nt(α−τ)/3) tα+τ\n3−1Ibounded\nα >3 O(1\nt) Ibounded\nSelect a specific time scaling β(t) with suitable parameters α, δ. Table 3 shows optimal convergence rates\nwe can expect for different choices of coefficients. (See Theorem 2 .2, Theorem 2.3, Theorem 2.4, Theorem 2.6,\nTheorem 2.7, Corollary 2.3, Theorem 2.11)\nTaking time scaling β(t)≡1, Table 4 lists the corresponding convergence rates (See Remark 2.3, Theorem\n2.1, Theorem 2.4, Theorem 2.5, Theorem 2.7, Corollary 2.2, Corollary 2 .4), it extends the convergence rates of\n(IGSα) and (IGSα,ǫ) in [6, 7, 13, 40, 41, 43] for unconstrained optimization problems to primal-dual dynamic\n(1) for linear equality constrained optimization problems. It also ext end and complements the existing results of\ninertial primal-dual dynamic in [5, 26, 27, 28, 47].\n21Table 3: Optimal convergence rates of |f(x(t)−f(x∗))|and/bardblAx(t)−b/bardbl\nr,s β(t) |f(x(t)−f(x∗))|and/ba∇dblAx(t)−b/ba∇dbl\nr= 0,s∈[0,1) µe1\nδ(1−s)t1−s\nts O/parenleftbigg\n1\ne1\nδ(1−s)t1−s/parenrightbigg\nr= 0,s= 1 µt1\nδ−1O(1\nt1/δ)\nr∈(0,1),s∈[r,1) µe1\nδ(1−s)t1−s\ntτ,∀τ∈(0,r+s) O/parenleftbigg\n1\ne1\nδ(1−s)t1−s/parenrightbigg\nr∈(0,1),s= 1 µt1\nδ−τ,∀τ∈(0,r+1) O(1\nt1/δ)\nr=s= 1α≤3 µtαO(1\ntα)\nα >3 µt1/δ−2O(1\nt1/δ)\nTable 4: Convergence rates of dynamic (1) with β(t)≡1\nr,s |f(x(t))−f(x∗)|and/ba∇dblAx(t)−b/ba∇dbl L(x(t),λ∗)−L(x∗,λ∗)\nr= 0,s= 0 O(1√\nt) ergodic sence O(1\nt) ergodic sence\nr= 0,s∈(0,1) O(1\nts/2) O(1\nts)\nr= 0,s= 1 O(1\nt)\nr∈(0,1),s∈[r,1) O(1\ntτ/2),∀τ∈(0,r+s) O(1\ntτ),∀τ∈(0,r+s)\nr∈(0,1),s= 1 O(1\ntτ),∀τ∈(0,r+1)\nr=s= 1α≤3 O(1\nt2α/3)\nα >3 O(1\nt2)\n3. Conclusion\nIn this paper, we propose a family of damped inertial primal-dual dyn amical systems with time scaling for\nsolving problem (2) in Hilbert space. We extend the inertial dynamic in [6 , 7, 9, 12, 13, 40, 41, 43, 44] for\nsolving unconstrained optimization problems to primal-dual dynamic ( 1) for solving linear equality constrained\nconvex optimization problems. Our results also extend and compleme nt the existing results of inertial primal-\ndual dynamics in [5, 26, 27, 28, 47]. Taking A= 0,b= 0, our results also complement the convergence rate\nresults of existing inertial dynamic for solving unconstrained conve x optimization problems. By discretization\nof primal-dual dynamic (13), it may lead to new primal-dual algorithms for solving problem (2), how to chose\nsuitable discretization scheme of (13) to get rate-matchingalgorit hms is an interesting direction of research. From\n22references [26, 28], it seems achievable, and we will consider it in the f uture works.\nAppendix A. Some auxiliary results\nAppendix A.1. Differentiating the energy function\nIn this part, we list the main calculation procedures for differentiatin g the energy function Eλ,ρ\nǫ(t).\nMultiplying the first equation of (13) by tρ, we have\ntρ¨x(t) =−αtρ−r˙x(t)−tρβ(t)(∇f(x(t)) +AT(λ(t)+δts˙λ(t))+σAT(Ax(t)−b))+tρǫ(t).\nThis yields\n˙E1(t) =/angb∇acketleftθ(t)(x(t)−x∗)+tρ˙x(t),˙θ(t)(x(t)−x∗)+θ(t)˙x(t)+ρtρ−1˙x(t)+tρ¨x(t)/angb∇acket∇ight\n+˙η(t)\n2/ba∇dblx(t)−x∗/ba∇dbl2+η(t)/angb∇acketleftx(t)−x∗,˙x(t)/angb∇acket∇ight\n=/angb∇acketleftθ(t)(x(t)−x∗)+tρ˙x(t),˙θ(t)(x(t)−x∗)+(θ(t)+ρtρ−1−αtρ−r)˙x(t)\n−tρβ(t)(∇f(x(t)) +AT(λ(t)+δts˙λ(t))+σAT(Ax(t)−b))+tρǫ(t)/angb∇acket∇ight\n+˙η(t)\n2/ba∇dblx(t)−x∗/ba∇dbl2+η(t)/angb∇acketleftx(t)−x∗,˙x(t)/angb∇acket∇ight\n= (θ(t)˙θ(t)+˙η(t)\n2)/ba∇dblx(t)−x∗/ba∇dbl2+tρ(θ(t)+ρtρ−1−αtρ−r)/ba∇dbl˙x(t)/ba∇dbl2\n+(θ(t)(θ(t)+ρtρ−1−αtρ−r)+tρ˙θ(t)+η(t))/angb∇acketleftx(t)−x∗,˙x(t)/angb∇acket∇ight\n−δθ(t)tρ+sβ(t)/angb∇acketleftx(t)−x∗,AT˙λ(t)/angb∇acket∇ight−δt2ρ+sβ(t)/angb∇acketleftA˙x(t),˙λ(t)/angb∇acket∇ight\n−θ(t)tρβ(t)(/angb∇acketleftx(t)−x∗,∇f(x(t))+ATλ(t)+σAT(Ax(t)−b)/angb∇acket∇ight\n−t2ρβ(t)/angb∇acketleft˙x(t),∇f(x(t))+ATλ(t)+σAT(Ax(t)−b)/angb∇acket∇ight\n+/angb∇acketleftθ(t)(x(t)−x∗)+tρ˙x(t),tρǫ(t)/angb∇acket∇ight.\nSimilarly, we have\n˙E2(t) = (θ(t)˙θ(t)+˙η(t)\n2)/ba∇dblλ(t)−λ/ba∇dbl2+tρ(θ(t)+ρtρ−1−αtρ−r)/ba∇dbl˙λ(t)/ba∇dbl2\n+(θ(t)(θ(t) +ρtρ−1−αtρ−r)+tρ˙θ(t)+η(t))/angb∇acketleftλ(t)−λ,˙λ(t)/angb∇acket∇ight\n+θ(t)tρβ(t)/angb∇acketleftλ(t)−λ,Ax(t)−b/angb∇acket∇ight+δθ(t)tρ+sβ(t)/angb∇acketleftλ(t)−λ,A˙x(t)/angb∇acket∇ight\n+t2ρβ(t)/angb∇acketleft˙λ(t),Ax(t)−b/angb∇acket∇ight+δt2ρ+sβ(t)/angb∇acketleft˙λ(t),A˙x(t)/angb∇acket∇ight.\nDifferentiating of E0(t) to get\n˙E0(t) =t2ρβ(t)/angb∇acketleft∇f(x(t)) +ATλ+σAT(Ax(t)−b),˙x(t)/angb∇acket∇ight\n+(2ρt2ρ−1β(t)+t2ρ˙β(t))(Lσ(x(t),λ)−Lσ(x∗,λ)).\nLetθ(t) satisfy t2ρβ(t) =δθ(t)tρ+sβ(t). Adding ˙E0(t),˙E1(t),˙E2(t) together, using Ax∗=band rearranging the\nterms, we get\n˙Eλ,ρ(t) =˙E0(t)+˙E1(t)+˙E2(t) =5/summationdisplay\ni=1Vi(t),\n23where\nV1(t) =/parenleftbigg\nθ(t)˙θ(t)+˙η(t)\n2/parenrightbigg\n(/ba∇dblx(t)−x∗/ba∇dbl2+/ba∇dblλ(t)−λ/ba∇dbl2),\nV2(t) = (θ(t)(θ(t) +ρtρ−1−αtρ−r)+tρ˙θ(t)+η(t))(/angb∇acketleftx(t)−x∗,˙x(t)/angb∇acket∇ight+/angb∇acketleftλ(t)−λ,˙λ(t)/angb∇acket∇ight),\nV3(t) =tρ(θ(t)+ρtρ−1−αtρ−r)(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2),\nV4(t) =tρ(tρ˙β(t)+(2ρtρ−1−θ(t))β(t))(Lσ(x(t),λ)−Lσ(x∗,λ))\n+θ(t)tρβ(t)(f(x(t))−f(x∗)−/angb∇acketleftx(t)−x∗,∇f(x(t))/angb∇acket∇ight)−σθ(t)tρβ(t)\n2/ba∇dblAx(t)−b/ba∇dbl2,\nV5(t) =/angb∇acketleftθ(t)(x(t)−x∗)+tρ˙x(t),tρǫ(t)/angb∇acket∇ight.\nTo investigate the rates of convergence of dynamical system (13 ), we need to find the appropriate θ(t) andη(t)\nto satisfy the following conditions:\nθ(t)≥0, η(t)≥0, (A.1)\nt2ρβ(t)−δθ(t)tρ+sβ(t) = 0, (A.2)\nθ(t)˙θ(t)+˙η(t)\n2≤0, (A.3)\nθ(t)(θ(t) +ρtρ−1−αtρ−r)+tρ˙θ(t)+η(t) = 0, (A.4)\nThenV1≤0,V2= 0, this together with the convexity of fyields\n˙Eλ,ρ\nǫ(t) =˙Eλ,ρ(t)−/angb∇acketleftθ(t)(x(t)−x∗)+tρ˙x(t),tρǫ(t)/angb∇acket∇ight\n≤tρ(θ(t)+ρtρ−1−αtρ−r)(/ba∇dbl˙x(t)/ba∇dbl2+/ba∇dbl˙λ(t)/ba∇dbl2)−σθ(t)tρβ(t)\n2/ba∇dblAx(t)−b)/ba∇dbl2\n+tρ(tρ˙β(t)+(2ρtρ−1−θ(t))β(t))(Lσ(x(t),λ)−Lσ(x∗,λ)) (A.5)\nfor anyλ∈ H2.\nAppendix A.2. Technical lemmas:\nIn convergence analysis for the dynamical system, we shall recall the following lemmas.\nLemma Appendix A.1. [18, Lemma A.5] Let ν: [t0,T]→[0,+∞)be integrable, and M≥0. Suppose\nµ: [t0,T]→Ris continuous and\n1\n2µ(t)2≤1\n2M2+/integraldisplayt\nt0ν(s)µ(s)ds\nfor allt∈[t0,T]. Then|µ(t)| ≤M+/integraltextt\nt0ν(s)dsfor allt∈[t0,T].\nLemma Appendix A.2. [46, Lemma 2.1.] For problem (2), letx∗be a solution. Given a function φand a fix\npointx, if for any λit holds that\nf(x)−f(x∗)+/angb∇acketleftλ,Ax−b/angb∇acket∇ight ≤φ(λ),\nthen for any ̺ >0, we have\nf(x)−f(x∗)+̺/ba∇dblAx−b/ba∇dbl ≤sup\n/bardblλ/bardbl≤̺φ(λ),\n24References\n[1] Alvarez F. On the minimizing property of a second order dissipative system in Hilbert spaces. SIAM Journal\non Control and Optimization. 2000;38(4):1102-1119.\n[2] Attouch H. Fast inertial proximal ADMM algorithms for convex st ructured optimization with linear con-\nstraint. 2020;hal-02501604.\n[3] Attouch H, Balhag A, Chbani Z, Riahi H. 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Dynamicalprimal-dualacceleratedmeth od with applicationsto networkoptimization.\n2019;arXiv:1912.03690.\n28"    },    {        "title": "1910.10977v2.Topological_damping_Rashba_spin_orbit_torque_in_ballistic_magnetic_domain_walls.pdf",        "content": "arXiv:1910.10977v2  [cond-mat.mes-hall]  11 Feb 2020Topological damping Rashba spin orbit torque in ballistic\nmagnetic domain walls\nD. Wang1,∗and Yan Zhou2,†\n1College of Engineering Physics, Shenzhen Technology\nUniversity, Guangdong 518118, P. R. China\n2School of Science and Engineering,\nThe Chinese University of Hong Kong,\nShenzhen, Guangdong 518172, P. R. China\n(Dated: February 12, 2020)\nAbstract\nRashba spin orbit torque derived from the broken inversion s ymmetry at ferromagnet/heavy\nmetal interfaces has potential application in spintronic d evices. In conventional description of the\nprecessional and damping components of the Rashba spin orbi t torque in magnetization textures,\nthe decomposition coefficients are assumed to be independent of the topology of the underlying\nstructure. Contrary to this common wisdom, for Schr¨ odinge r electrons trespassing ballistically\nacross a magnetic domain wall, we found that the decompositi on coefficient of the damping\ncomponent is determined by the topology of the domain wall. T he resultant damping Rashba\nspin orbit torque is protected by the topology of the underly ing magnetic domain wall and robust\nagainst small deviations from the ideal domain wall profile. Our identification of a topological\ndamping Rashba spin orbit torque component in magnetic doma in walls will help to understand\nexperiments on current driven domain wall motion in ferroma gnet/heavy metal systems with\nbroken inversion symmetry and to facilitate its utilizatio n in innovative device designs.\n1One main theme in the field of nanomagnetism is to search for new appr oaches to\nrealize fast and energy efficient manipulation of magnetic state, rat her than using the\nconventional magnetic field. In the past three decades, several promising candidates, such\nas electric field1, laser pulses2and spin current through the spin transfer torque (STT)3–6,\nwere proposed. A recent development along this line is the emergenc e of the Rashba spin\norbittorque(RSOT)inmagneticsystemswithoutinversionsymmetr y. Inasimplepicture7,\nthe electric field along the symmetry breaking direction is equivalent t o a magnetic field,\ndubbed the Rashba field, in the rest reference frame of an electro n in motion. Due to the\ns-dexchange between the local and itinerant spin degrees of freedom , the Rashba field is\ntransformed into the RSOT acting on the local magnetization.\nWhen it was first proposed, only the precessional component8–11of the RSOT, corre-\nsponding to the torque caused by an effective Rashba field acting on the local magnetiza-\ntion, was considered. Upon considering the impurity and spin-flip sca ttering, an additional\ndamping torque in accordance with the effective Rashba field can aris e12. Subsequent the-\noretical investigations were devoted to exposition of the physics o f the RSOT, adopting\ndifferent approachesandconsideringsamplegeometrieswithfinitee xtension13–20. However,\nmost of the previous theoretical investigations focus on the case of uniform magnetization\ndistribution or slowly varying magnetization textures, the moreimpo rtant case of magnetic\ndomain walls (DWs), which will be the focus of the current work, is almo st not touched\nupon.\nThetopologicaldescriptionofelectrontransportinperiodicpoten tialsappearsnaturally\nby considering the geometric Berry phase21of itinerant electrons. In the simplest case of\none dimensional (1D) motion of electrons, it leads to the Zak phase22, and the Thouless-\nKohmoto-Nightingale-den Nijs (TKNN) invariant23for two dimensional (2D) motion. The\nBerry phase is generically caused by the existence of a gauge field24, which is given by the\nspatial variationof the periodic modulationwave function inthe case of Bloch electrons. In\nthe presence of spin orbit interaction and a background magnetic fi eld, which is generated\nbyamagneticDW, theitinerant electronswill alsoexperience aspatia llyvarying, emergent\ngauge field. By analogy with the TKNN invariant and the Zak phase, we speculate that\ntopological phase factors should arise for electrons traversing c ross a DW. Actually, the\neffect of spatially varying magnetization on the motion of electrons w as already discussed\ntheoretically by Bruno et al.25. Whether a similar topological effect will emerge in RSOT\nremains a question.\nFor a simple demonstration of the physics, we will use the following minim al model\nHamiltonian to study the magnetization dynamics of itinerant electro ns confined to the\n2interface between a ferromagnet and a heavy metal8–11,\nH=p2\n2me+µBσ·M+αR\n¯hσ·(p׈z). (1)\np=−i¯h∇is the momentum operator, meis the electron mass, ¯ his the Planck constant\ndivided by 2 π, andµBis the Bohr magneton. αRis the Rashba constant, which measures\nthe degree of the inversion symmetry breaking26. We consider only the motion of the\nelectrons in the interface, which is a 2D xyplane in our coordinate system, since previous\ndensity functional theory investigation found that the RSOT is prim arily an interface\neffect13. The third term in the Hamiltonian (1) is the Rashba spin orbit interact ion term,\nshowing that the main effect of the broken inversion symmetry is to in troduce an effective\nin-plane magnetic field, which is everywhere tangential to the in-plan e linear momentum\np.σ= ˆxσx+ ˆyσy+ ˆzσzis a vector in the spinor space where σx,σyandσzare the Pauli\nmatrices, and ˆ x, ˆyand ˆzare unit vectors along the x,yandzdirections, respectively.\nThe Hamiltonian (1) gives the energy of conduction electrons intera cting through the\ns-dexchange interaction with the local magnetization M. In our model treatment, we\nconsider only the itinerant Hamiltonian as given in Eq. (1), while the loca l magnetic\nmoments are assumed to be static, as described by M. The variation of the vector M\ninsidemagnetizationtexturesisusedtoprovideaneffective’exchan ge’ fieldfortheitinerant\nmagnetization.\nThe Walker DW profile27considered for the study of the RSOT is characterized by an\nangleθthrough the expression M=M(ˆxsinθ+ ˆzcosθ) with cosθ=−qtanh(x/λ) and\nsinθ=χsech(x/λ), whereλ=/radicalBig\nA/KistheDWwidth. Aistheexchange constant and K\ntheanisotropyconstantoftheferromagnet. Forageneraldes cription, weconsider explicitly\nthe charge qand chirality χof a DW28. Using the time dependent Pauli-Schr¨ odinger\nequationi¯h∂ψ/∂t=Hψfor the spinor wave function ψ, the equation of motion for the\nspin density s=ψ†σψof conduction electrons is given by\n2me\n¯h∂s\n∂t=∇·Q+2k2\nBˆM×s+τ, (2)\nwhere the spin current density is defined as\nQ=i(ψ†∇σψ−∇ψ†σψ)+kαǫij3ˆiˆjψ†ψ. (3)\nǫijkis the antisymmetric Levi-Civita symbol and a summation over repeat ed indices is\nimplied in the expression for Q. A substitution of x,yandzby numbers 1, 2 and 3 is\nmade to compactify the expression. The parameter kBis related to the Zeeman energy\nsplitting ¯h2k2\nB/2me=µBM, and the constant kα= 2meαR/¯h2is an effective wave number\n3characterizing the strength of the Rashba interaction. ˆMis a unit direction vector for\nthe local magnetization, ˆM=M/M. The precessional term τfollows directly from the\nRashba term in Eq. (1), and is given by\nτ(k,ρ) = 2kαℑ(ˆzψ†σ·∇ψ−ψ†σz∇ψ). (4)\nFor later convenience, the momentum and position dependence of τis explicitly written\nout in Eq. (4). Our equation of motion for the spin density is identical in form to a pre-\nvious result29, if the angular momentum operator is replaced by the Rashba field op erator\nconsidered here. However, this connection is superficial, as the dy namics for the angular\nmomentum are not considered here.\nWith Eq. (2), it is obvious that the itinerant magnetization dynamics is governed by\nthree torques. The first term on the right hand side of Eq. (2) cor responds to the spin\ncurrent torque acting on the itinerant magnetization, which is just the divergence of the\nspin current density. In the ground state, the spin current torq ue reduces to the exchange\ntorqueformagnetizationtextures, whichisproportionalto ˆ m×∇2ˆm, with ˆmanunit vector\nfor the itinerant magnetization. The second term describes the to rque originating from the\nstatic local magnetization, whose net effect can be viewed as an effe ctives-dexchange field\nacting on the itinerant magnetization. The Rashba term in the Hamilto nianHgives rise\nto the last torque on the right hand side of Eq. (2). In equilibrium, th is Rashba torque\nhas a form identical to the Dzyaloshinskii-Moriya torque30–34. If a steady state electronic\ncurrent is allowed to flow, the spin current torque and the Rashba t orque transform into\nthe conventional STT and RSOT, respectively. In the current car rying steady state, there\nis no time variation of the itinerant magnetization. Hence the various torques on the right\nhand side must sum to zero. Due to this torque balance, the torque induced by the spin\naccumulation, which corresponds to the second term on the right h and side of Eq. (2),\ncontains both the STT and RSOT contributions.\nEq. (4) gives only the RSOT for a single Bloch state in the momentum sp ace. Using the\nrelaxation time approximation35, the physical RSOT induced in the presence of an electric\nfieldEalong thexdirection can be obtained through an integration in the momentum\nspace as\nτ(ρ) =−eEτ0\n(2π)2¯h/contintegraldisplay\ndϕkxτ(k,ρ), (5)\nwhereτ0is the relaxation time constant, ethe electron charge, and ϕthe angle of the\nwave vector relative to the x-axis. As the temperature is assumed to be absolute zero, the\nintegration is confined to the 2D Fermi surface, which is a circle.\n4-20 -10 0 10 20-2-1012Spin orbit torque (a. u.)x\ny\nz\n(a)\n-700 -350 0 350 700-4-202Spin orbit torque (a. u.)x\ny\nz\n(b)\nFIG. 1. RSOT with q= 1 and χ= 1. The DW widths correspond to λkF= 2 (a) and λkF= 70\n(b). For the small DW width λkF= 2 (a), both the precessional ( xandz) and the damping\n(y) components are comparable in magnitude. As the DW width inc reases to λkF= 70 (b), the\ndamping component decreases in comparison to the precessio nal one. For the long DW width\nλkF= 70, although the damping component is negligibly small at t he DW center, its magnitude\nis sizable far away from the DW center.\nWe adopt a scattering matrix method36,37to numerically solve the eigenvalue problem\nHψ=ǫkψ (6)\nfor the Pauli-Schr¨ odinger equation with energy ǫk. The idea behind this scattering matrix\nmethod is intuitively simple. In order to construct the eigenfunction s of Eq. (6), we first\nsolve it at infinity to obtain the asymptotic wave functions with specifi c momentum and\nspin. Then we evolve the obtained asymptotic wave functions towar ds the DW center,\naccording to Eq. (6). Generally, the evolved wave functions are no t continuous at the\nDW center, and are thus not eigenfunctions in the whole space. This problem can be\novercome by forming linear combinations of the evolved wave functio ns with the same\nenergy but different momenta and spin projections along the zdirection, requiring that\nthe continuity condition is satisfied at the DW center. The resultant wave functions are\neigenfunctions over the whole space. Previously, the same method was successfully applied\nto the discussion of STT in DWs38. In the actual numerical implementation, we can\nemploy a particle-hole or charge-parity-time-reversal symmetry of the Hamiltonian (1),\nH=σxPTHTPσx, to reduce the number of the wave functions to be computed. Wav e\nfunctions related to each other by the particle-hole symmetry, ψandσxPψ, are conjugate\npairs with opposite momenta but identical spin projections along the zdirection, injecting\nfromoppositeendsoftheDW.Itisinteresting tonotethatasimilarp article-holesymmetry\nwas found formagnons inside DWs39. Further numerical details of thecalculation are given\n5-20 -10 0 10 201.21.62.02.4 &  (a. u.)\nFIG. 2. Precessional ( α) and damping ( β) RSOT coefficients with q= 1 and χ= 1. The DW\nwidth is λkF= 2. The quantum confinement induced oscillation around the D W center ( xkF=\n0) and far away from the DW center ( xkF=±20), which is obvious for the displayed DW width,\nis smoothed out as the DW width is increased to λkF= 70, as shown in Fig. 3. Unspecified\nparameters are the same as those used to generate Fig. 1.\nin Ref. 40.\nWith the numerical wave functions thus obtained, the RSOT can be c omputed using\nEqs. (4) and (5). The resultant RSOT for the DW width λkF= 2 andλkF= 70 with\nkB/kF= 0.4 andkα/kF= 0.1 is shown in Fig. 1, where we have measured the DW width\nin terms of the inverse Fermi wave vector k−1\nFfor the free electron system that is described\nby only the kinetic energy term in the Hamiltonian (1). For the shorte r DW width λkF\n= 2, the RSOT has both sizable precessional and damping component s. The precessional\ncomponent is caused by the effective Rashba field, which has the for m ˆm׈y, while the\ncorresponding damping component is ˆ m×(ˆm׈y)9. The total RSOT is a sum of both\ncomponents,\nτ=αˆm׈y+βˆm×(ˆm׈y). (7)\nThe corresponding decomposition coefficients αandβare displayed in Fig. 2. Due to\nthe confinement of electrons caused by such short DWs, quantum interference of wave\nfunctions shows up as the observable spatial variation of the RSOT and decomposition\ncoefficients far away from the DW center. This spatial variation dec ays out as the DW\nwidth is increased (cf. Figs. 1 (a) and (b), 2 and 3).\nAstheDWwidthincreases, themagnitudeoftheprecessionalcomp onentincreaseswhile\nthe magnitude of the damping component decreases, as can be exp ected from a previous\ninvestigation on STT38. However, the scaling of the non-adibaticity for the RSOT, which\nis defined as β/α, is algebraic instead of exponential40. At the DW center, the residue\n6-700 -350 0 350 7002.63.03.4 (a. u.)(a)\n-700 -350 0 350 700-0.8-0.400.40.8 (a. u.)(b)\nFIG. 3. Topological behaviour of the precessional ( α) and damping ( β) RSOT coefficients for all\nfour combinations of qandχ. The DW width is λkF= 70. Other parameters are the same as\nthose used to generate Fig. 1.\ndamping component is negligible, but it is sizable far away from the DW ce nter, as evident\nfrom Fig. 3 (b) for the longer DW width λkF= 70. This finite residue damping component\nof the RSOT will demonstrate itself in the current driven magnetizat ion dynamics of mag-\nnetization textures, and warrants further attention in consider ing its effects in spintronic\ndevices. Furthermore, our numerical result shows that the coeffi cientβdepends on the\ntopology of the underlying DW. As shown in Fig. 3, for the four possib le combinations of\nthe DW charge and chirality, we have only two traces for β, reversed to each other, for the\nlonger DW width λkF= 70: the product of the DW charge and chirality, qχ, determines\nthe sign of β.\nThe physical origin of the damping RSOT can be determined through a perturbation\nanalysis of the same Pauli-Schr¨ odinger equation (6) which is used fo r our numerical sim-\nulation. Using the first order wave function, the damping RSOT comp onent atx=±∞\ncan be calculated. It has the form as given in Eq. (7) with the coefficie nt40\nβ���qχk2\nα/parenleftbigg\nc+a\nλ2+be−γλ/parenrightbigg\n(8)\nto the lowest order in kα, wherea,b,candγare all constants. The constant cis of the\norder of unity, hence as the DW width increases to a very large value ,λ≫λc, the damping\nRSOT will approach to a constant value cat±∞. The critical length λc=kF/k2\nB, which\nisλckF= 6.25 using our parameters, determines the DW width where transition from non-\nadiabatic to adiabatic behaviour occurs for STT in DWs without spin or bit interaction38.\nTheappearanceofthefactor qχintheexpression of βindicates thatthedampingRSOT\nis a topological quantity. The factor k2\nαsignifies that the damping RSOT is a higher order\neffect, askαis proportional to αR. In the perturbation calculation, the adiabatic or zeroth\n7order wave functions give rise to only the precessional RSOT. Due t o this origin from\nthe zeroth order wave functions, the adiabatic coefficient αis almost independent of the\ntopological features of the underlying DW, whether in the adiabatic limit or not: For α,\nthe dominant contribution does not sense the topology of the DW, a nd the topological\ncontribution only enters as a higher order correction (cf. Fig. 3 (a )). Inclusion of the\nfirst order wave functions brings about the damping RSOT. The firs t order wave functions\nat infinity are determined by the scattering of the incident, zeroth order waves under the\ninfluence of the perturbation potential. To the first order of kα, the explicit form of the\nperturbation potential in momentum space V(kf,ki) for incident and scattered momenta\nkiandkfis give by\nV(kf,ki) =pcschp\n4πλ−χkα\n4ky\nk2\nBπ2+4p2\n2π2λsechp+qχkα\n4sechp\n−qχks\n4/parenleftBigg\nsechp−2χkαky\nπk2\nBpcschp/parenrightBigg\nσy+χλkα\n2kyσzsechp, (9)\nwithp=Qλπ/2 andks=kf+ki.Q=kf−kiis the momentum transfer. In comparison to\nthe original potential in (1), the potential (9) corresponds to a m agnetic field with only y\nandzcomponents and a scalar electric potential, while the Rashba interac tion is absorbed\ninto the magnetic field and electric potential. When the momentum tra nsfer is zero, the\nscaling ofV(ki,ki) with respect to the DW width λis algebraic. For finite momentum\ntransferQ=kf−ki,V(kf,ki) brings about theexponential decay ofthe physical quantities\non the DW width through the hyperbolic secant and cosecant funct ions41.\nNot all of the topological terms in potential (9) contribute to the e xpression for the\ndamping RSOT. In the case of zero momentum transfer, Q= 0, theycomponent of the\neffective magnetic field in(9), which is the coefficient of σy, does not contribute at all; while\nthezeffective field, which is the coefficient of σz, andthe scalar potential contribute partly:\nThe product of the first and second terms in the scalar potential g ives rise to the term\nproportional to λ−2in (8), while the product of the zcomponent of the effective magnetic\nfield with the first term of the scalar potential contributes the con stant term in β. Both\nthose two contributions areproportionalto thechirality χ. The dependence of qinthe final\nexpression for βis derived from its dependence on the zcomponent of the magnetization,\nmz. Hence the topological feature of βis characterized by the relation β∝χmz, far away\nfrom the DW center. This behaviour is similar to that of Bloch wave fun ctions in periodic\npotentials, as demonstrated by the Zak phase22. Themzis a dynamical contribution, and\nχis a manifestation of the existence of a topological phase with the va lue of 0 or π. The\ntopological dependence of βobtained using the potential (9), Eq. (8), is actually borne\nout by the numerical results, as shown in Fig. 3.\n8The topological nature of βexplains mathematically why the damping RSOT remains\nfinite even when the DW width is large, λ≫λc. Due to the different topologies of the\nDW and a uniformly magnetized state, a continuous transition betwe en the two states\nis prohibited. Thus βcannot be reduced to zero, which is the value for βin a uniform\nstate. Physically, the topological protection of the damping RSOT c an be traced back\nto the nonlocal character of quantum particles, which means that the wave functions are\nnot determined locally by the potential. In particular, the damping RS OT atx=±∞\nis determined by V(ki,ki) in the adiabatic limit ( λ≫λc), which is an integration of the\nperturbation potential over the whole space and gives rise to the t opological characteristics\nof the damping RSOT. Therefore, the damping RSOT at x=±∞is finite due to the\npure existence of the DW, even though the magnetization variation there is infinitesimal,\napproaching to the value for a uniform magnetization distribution.\nToseehowthenewlyidentifieddampingRSOTinfluencesthecurrentd rivenDWmotion\n(CDWM), we consider the expression for the normalized DW velocity v\nnαGv=qhzcosθh−nξu+qχβ∆θ0, (10)\nobtained using a simple 1D model description of CDWM42. A detailed derivation of the\nvelocity (10) is given in the Supplemental Material [43], and referenc es [44, 45] therein.\nαGis the Gilbert damping constant, and n,θhand ∆θ0are constants related to the equi-\nlibrium DW configuration. ξis the non-adiabaticity and uis an equivalent speed for the\nSTT.hzis a perpendicularly applied magnetic field, normalized to the anisotrop y field.\nIn obtaining Eq. (10), we have assumed that the current density is small, and the RSOT\nonly causes infinitesimal deviation from the equilibrium DW configuratio n. Even with this\nrather simple assumption, Eq. (10) shows that the CDWM can exhibit very complicated\nbehaviour: For short DWs, βis not completely determined by the product qχ, then the\nRSOT contribution to the DW velocity has both qχdependent and independent compo-\nnents. In the adiabatic limit, the qχdependent component of βfades out, and the RSOT\ncontribution to the DW velocity is qχindependent, resembling the behaviour of the STT\ncontribution.\nIn systems with a sizable Rashba interaction, the sign of the produc tqχfor a stable\nN´ eel wall is determined by the sign of the Dzyaloshinskii-Moriya inter action constant D, as\nqχD<0 gives a lower energy. Additional control over the DW chirality can b e realized by\napplying an in-plane magnetic field hx46, with the DW charge fixed. With this freedom in\nmanipulating the DW chirality, the velocity of CDWM can be tuned by the application of\nan in-plane magnetic field, for DW width in the non-adiabatic limit. Furth er complication\n9can arise from the chirality dependence of the Gilbert damping const ant and gyromagnetic\nfactor47–49, as well as the STT non-adiabaticity50. Before turning to our conclusion, it is\nappropriate to mention that our above discussion is based on a simple 1D treatment of the\nCDWM, which is a very rough approximation based on the assumption t hat the ground\nstate of the DW is a N´ eel configuration. The applicability of this assu mption is dubious in\nthe presence of an electric current, since the current induced eff ective Rashba field tends to\nstabilize a Bloch wall. Our discussion is only to illustrate the complication o f the CDWM\nin the presence of the RSOT. Further detailed investigation is neede d for a thorough\nunderstanding of the CDWM in systems with sizable Rashba spin orbit in teraction.\nIn conclusion, we have studied the RSOT in magnetic DWs, which is deriv ed from the\nbroken inversion symmetry at ferromagnet/heavy metal interfa ces. By numerically solving\nthe Pauli-Schr¨ odinger equation for 2D electrons moving inside a N´ e el DW, a topological\ndamping RSOT component is identified. Even in the adiabatic limit, the ma gnitude of\nthe topological damping component is sizable, in stark contrast to t he negligible non-\nadiabatic STT in the same limit. This finite damping RSOT is a manifestation of the\nnontrivial topology of the underlying DW. The identification of a topo logical damping\nRSOT component in magnetization textures will promote the applicat ion of RSOT in\nspintronic devices and facilitate a thorough understanding of the e xperimental data in\ncurrent driven motion of magnetic DWs in ferromagnet/heavy meta l bilayer systems.\nACKNOWLEDGEMENTS\nWewouldliketoexpressourgratitudetoProf. JiangXiaoforhisvalua blecommentsand\ndiscussions, especially for bringing us to the topic of RSOT in magnetic DWs and sharing\nhis code on STT simulation. Y. 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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. 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Lett. 88, 056404\n(2002)."    },    {        "title": "2103.03407v3.Multilevel_quasi_Monte_Carlo_for_random_elliptic_eigenvalue_problems_II__Efficient_algorithms_and_numerical_results.pdf",        "content": "arXiv:2103.03407v3  [math.NA]  6 Oct 2022Multilevel quasi-Monte Carlo for random elliptic eigenval ue\nproblems II: Efficient algorithms and numerical results\nAlexander D. Gilbert1Robert Scheichl2\nOctober 7, 2022\nAbstract\nStochastic PDE eigenvalue problems often arise in the field of uncert ainty quan-\ntification, whereby one seeks to quantify the uncertainty in an eige nvalue, or its eigen-\nfunction. Inthis paperwepresentanefficient multilevelquasi-Mont eCarlo(MLQMC)\nalgorithm for computing the expectation of the smallest eigenvalue o f an elliptic eigen-\nvalue problem with stochastic coefficients. Each sample evaluation re quires the solu-\ntion of a PDE eigenvalue problem, and so tackling this problem in practic e is noto-\nriously computationally difficult. We speed up the approximation of this expectation\nin four ways: we use a multilevel variance reduction scheme to sprea d the work over a\nhierarchyofFEmeshes andtruncationdimensions; weuse QMCmeth ods to efficiently\ncompute the expectations on each level; we exploit the smoothness in parameter space\nand reuse the eigenvector from a nearby QMC point to reduce the n umber of itera-\ntions of the eigensolver; and we utilise a two-grid discretisation sche me to obtain the\neigenvalue on the fine mesh with a single linear solve. The full error ana lysis of a basic\nMLQMC algorithm is given in the companion paper [Gilbert and Scheichl, 20 22], and\nso in this paper we focus on how to further improve the efficiency and provide theo-\nretical justification for using nearby QMC points and two-grid meth ods. Numerical\nresults are presented that show the efficiency of our algorithm, an d also show that the\nfour strategies we employ are complementary.\n1 Introduction\nIn this paper we develop efficient methods for computing the ex pectation of an eigenvalue\nof the stochastic eigenvalue problem (EVP)\n−∇·/parenleftbig\na(x,y)∇u(x,y)/parenrightbig\n+b(x,y)u(x,y) =λ(y)c(x,y)u(x,y),forx∈D,\nu(x,y) = 0 for x∈∂D,(1.1)\nwhere the differential operator ∇is with respect to x, which belongs to the physical\ndomainD⊂Rdford= 1,2,3. Randomness is incorporated into the PDE (1.1) through\nthe dependence of the coefficients a,bon the stochastic parameter y= (yj)j∈N, which\nis a countably infinite-dimensional vector with i.i.d. unif ormly distributed entries: yj∼\nU[−1\n2,1\n2] forj∈N. The whole stochastic parameter domain is denoted by Ω := [−1\n2,1\n2]N.\n1School of Mathematics and Statistics, University of New Sou th Wales, Sydney NSW 2052, Australia.\nalexander.gilbert@unsw.edu.au\n2Institute for Applied Mathematics & Interdisciplinary Cen tre for Scientific Computing, Universit¨ at\nHeidelberg, 69120 Heidelberg, Germany and Department of Ma thematical Sciences, University of Bath,\nBath BA2 7AY UK.\nr.scheichl@uni-heidelberg.de\n1The study of stochastic PDE problems is motivated by applica tions in uncertainty\nquantification—where one is interested in quantifying how u ncertain input data affect\nmodeloutputs. Inthecaseof (1.1)theuncertaininputdataa rethecoefficients aandb,and\nthe outputs of interest are the eigenvalue λ(y) and its correspondingeigenfunction u(x,y),\nwhich are now also stochastic objects. As such, to quantify u ncertainty we would like to\ncompute statistics of the eigenvalue (or eigenfunction), a nd in particular, in this paper\nwe compute the expectation of the smallest eigenvalue λwith respect to the countable\nproduct of uniform densities. This is formulated as the infin ite-dimensional integral\nEy[λ] =/integraldisplay\nΩλ(y) dy:= lim\ns→∞/integraldisplay\n[−1\n2,1\n2]sλ(y1,y2,...,ys,0,0...) dy1dy2···dys.\nEVPs corresponding to differential operators appear in many a pplications from engi-\nneering and the physical sciences, e.g., structural vibrat ion analysis [43], nuclear reactor\ncriticality [14, 27] or photonic crystal structures [13, 29 , 34]. In addition, stochastic EVPs,\nsuch as (1.1), have recently garnered more interest due to th e desire to quantify the un-\ncertainty present in such applications [41, 2, 44, 3, 38]. Th us, significant development has\nrecently also gone into efficient numerical methods for tackl ing such stochastic EVPs in\npractice, the most common being Monte Carlo [41], stochasti c collocation [1] and stochas-\ntic Galerkin/polynomial chaos methods [16, 44]. The latter two classes perform poorly for\nhigh-dimensional problems, so in order to handle the high-d imensionality of the param-\neter space, sparse and low-rank versions of those methods ha ve been developed, see e.g.,\n[1, 22, 15]. Furthermore, to improve upon classical Monte Ca rlo, while still performing\nwell in high dimensions, the present authors with their coll eagues have analysed the use\nof quasi-Monte Carlo methods [17, 18].\nIn practice, for each parameter value y∈Ω the elliptic EVP (1.1) must be solved\nnumerically, which we do here by the finite element (FE) metho d, see, e.g., [4]. First,\nthe spatial domain is discretised by a family of triangulati ons{Th}h>0indexed by the\nmeshsize h >0, and then (1.1) is solved on the finite-dimensional FE space corresponding\ntoTh. This leads to a large, sparse, symmetric matrix EVP, which i s typically solved by\nan iterative method (such as Rayleigh quotient iteration or the Lanczos algorithm, see,\ne.g., [37, 40]), requiring several solves of a linear system . To speed up the solution of each\nEVP we use the accelerated two-grid method developed independently in [25, 26] and [47].\nIn particular, to obtain an eigenvalue approximation corre sponding to a “fine” mesh Th,\none first solves the FE EVP on a “coarse” mesh TH, withH≫h, to obtain a coarse\neigenpair ( λH,uH). An eigenvalue approximation λhon the fine grid This then obtained\nby performing a single step of shifted inverse iteration wit h shiftλHand start vector uH.\nTypically, the fast convergence rates of FE methods and of sh ifted inverse iteration for\neigenvalue problems allow for a very large difference between the coarse and fine meshsize,\ne.g., for piecewise polynomial FE spaces it is sufficient to ta keH/equalorsimilarh1/4, so that the cost\nof the two-grid method essentially reduces to the cost of a si ngle linear solve on the fine\nmesh. Multilevel sampling schemes also exploit a hierarchy of FE meshes, and so the\ntwo-grid method very naturally fits into this framework.\nThe multilevel Monte Carlo (MLMC) method [20, 23] is a varian ce reduction scheme\nthat has achieved great success when applied to problems inc luding path integration [23],\nstochastic differential equations [20] and also stochastic P DEs [6, 8]. For stochastic PDE\nproblems, it is based on a hierarchy of L+1 increasingly fine FE meshes {Thℓ}L\nℓ=0(i,.e.,\nthe meshwidths are decreasing h0> h1>···> hL>0), and an increasing sequence\nof truncation dimensions s0< s1<···< sL<∞of the infinite-dimensional parameter\ndomain Ω. For the eigenvalue problem (1.1), letting the trun cation-FE approximation on\nlevelℓbe denoted by λℓ:=λhℓ,sℓ, the key idea is to write the expectation on the desired\n2finest level Las a telescoping sum of differences:\nEy[λL] =Ey[λ0]+L/summationdisplay\nℓ=1Ey[λℓ−λℓ−1], (1.2)\nand then compute each expectation Ey[λℓ−λℓ−1] by an independent MC approximation.\nAsℓ→∞, provided hℓ→0 andsℓ→∞, we have λℓ→λand hence also λℓ−λℓ−1→0.\nThus the variance on each level decreases, and so less sample s will be needed on the\nfiner levels. In this way, the MLMC method achieves a significa nt cost reduction by\nspreading the work across the hierarchy of levels, instead o f performing all evaluations on\nthe finest level L. For any linear functional G, we can write a similar telescoping sum for\nEy[G(uL)], where we define uℓ:=uhℓ,sℓ. The smallest eigenvalue is simple, therefore we\ncan ensure that the corresponding eigenfunction is unique b y normalising it and choosing\nthe sign consistently. Similarly, we normalise each approx imationuℓand choose the sign\nto match the eigenfunction, which ensures they are also well defined. Our method can\nalso be applied to approximate the expectation of other simp le eigenvalues higher up\nthe spectrum, without any essential modifications. If the ei genvalue in question is well-\nseparated from the rest of the spectrum (uniformly in y) then our analysis can also be\nextended in a straightforward way. However, for simplicity and clarity of presentation in\nthis paper we focus on the smallest eigenvalue.\nQuasi-Monte Carlo methods are equal-weight quadrature rul es that are tailored to\nefficiently approximate high-dimensional integrals, see, e .g., [11, 12]. In particular, by\ndeterministically choosing well-distributed quadrature points, giving preference to more\nimportant dimensions, QMC rules can be constructed such tha t the error converges faster\nthan for MC methods, whilst still being independent of dimen sion. Using a QMC rule\nto approximate the expectation on each level in (1.2) instea d of Monte Carlo gives a\nMultilevel quasi-Monte Carlo (MLQMC) method. MLQMC method s were first developed\nin [21] for option pricing, and since then have also had great success for UQ in stochastic\nPDE problems [19, 32, 31]. The gains are complementary, so th at for several problems\nMLQMCmethodscanbeshowntoresultinfasterconvergenceth aneitherMLMCmethods\nor single level QMC approximations.\nIn this paper, we present an efficient MLQMC method for computi ng the expectation\nof the smallest eigenvalue of stochastic eigenvalue proble ms of the form (1.1). We employ\nfour complementary strategies: 1) we use the ML strategy to r educe the variance and\nspread the work across a hierarchy of FE meshes and truncatio n dimensions; 2) we use\nQMC methods to compute the expectation on each level more effic iently; 3) we use the\ntwo-grid method for eigenvalue problems [25, 26, 47] to comp ute the eigenpair on finer\ngrids using an eigensolve on a very coarse grid followed by a s ingle linear solve; and 4) we\nreuse the eigenvector corresponding to a nearby QMC point as the starting vector for the\nRayleigh quotient algorithm to solve each eigenvalue probl em.\nThefocusofthispaperisondevelopingtheabovepracticals trategies togiveanefficient\nMLQMC method. A rigorous analysis of MLQMC methods for the st ochastic EVP (1.1)\nis the focus of a separate paper [19]. However, we do give a the oretical justification of the\nenhancement strategies 3) and 4). First, we extend the two-g rid method and its analysis\nto stochastic EVPs, allowing also for a reduced truncation d imension on the “coarse grid”.\nSecond, we analyse the benefit of using an eigenvector corres ponding to a nearby QMC\npoint as the starting vector for the iterative eigensolve.\nThe structure of the paper is as follows. In Section 2 we prese nt the necessary back-\nground material. Thenin Section 3we extend the two-grid met hodfor deterministic EVPs\nto stochastic EVPs and analyse the error. In Section 4 we desc ribe a basic MLQMC algo-\nrithm, and thenoutline how onecan reducethecost by usingtw o-grid methods andnearby\n3QMC points. Finally, in Section 5 we present numerical resul ts for two test problems.\n2 Mathematical background\nIn this section we briefly summarise the relevant material on variational EVPs, two-grid\nFE methods and QMC methods. For further details we refer the r eader to the references\nindicated throughout or to [17].\nWe make the following assumptions on the physical domain and on the boundedness\nof the coefficients (from above and below). These ensure the we ll-posedness of (1.1) and\nadmit a fast convergence rate of our MLQMC algorithm.\nAssumption A1.\n1.D⊂Rd, ford= 1,2,3, is bounded and convex.\n2.aandbare of the form\na(x,y) =a0(x)+∞/summationdisplay\nj=1yjaj(x)andb(x,y) =b0(x)+∞/summationdisplay\nj=1yjbj(x),(2.1)\nwhereaj, bj∈L∞(D), for allj≥0, andc∈L∞(D)depend on xbut noty.\n3. There exists amin>0such that a(x,y)≥amin,b(x,y)≥0andc(x)≥amin, for all\nx∈D,y∈Ω.\n4. There exists p∈(0,1)andq∈(0,1)such that\n∞/summationdisplay\nj=1max/parenleftbig\n/ba∇dblaj/ba∇dblL∞,/ba∇dblbj/ba∇dblL∞/parenrightbigp<∞and∞/summationdisplay\nj=1/ba∇dbl∇aj/ba∇dblq\nL∞<∞.\nFor convenience, we then let amax<∞be such that\nmax/braceleftbig\n/ba∇dbla(y)/ba∇dblL∞,/ba∇dbl∇a(y)/ba∇dblL∞,/ba∇dblb(y)/ba∇dblL∞,/ba∇dblc/ba∇dblL∞/bracerightbig\n≤amax,for ally∈Ω.(2.2)\n2.1 Variational eigenvalue problems\nFor the variational form of the EVP (1.1), we introduce the us ual function space setting\nfor second-order elliptic PDEs: the first-order Sobolev spa ce of functions with zero trace\nis denoted by V:=H1\n0(D) and equipped with the norm /ba∇dblv/ba∇dblV:=/ba∇dbl∇v/ba∇dblL2. Its dual space\nisV∗:=H−1(D). We will also use the Lebesgue space L2(D), equipped with the usual\ninner product/a\\}b∇acketle{t·,·/a\\}b∇acket∇i}htL2, and the induced norm /ba∇dbl·/ba∇dblL2.\nNext, for each y∈Ω define the bilinear form A(y) :V×V→Rby\nA(y;w,v):=/integraldisplay\nDa(x,y)∇w(x)·∇v(x)dx+/integraldisplay\nDb(x,y)w(x)v(x)dx,\nwhich is also an inner product on Vand admits the induced norm /ba∇dblv/ba∇dblA(y):=/radicalbig\nA(y;v,v).\nWe define also the inner product M:V×V→Rby\nM(w,v):=/integraldisplay\nDc(x)w(x)v(x)dx,\nagain with induced norm given by /ba∇dblv/ba∇dblM:=/radicalbig\nM(v,v). Further, letM(·,·) also denote\nthe duality paring on V×V∗.\n4The variational form of the EVP (1.1) is: Find λ(y)∈R,u(y)∈Vsuch that\nA(y;u(y),v) =λ(y)M(u(y),v) for all v∈V , (2.3)\n/ba∇dblu(y)/ba∇dblM= 1.\nThevariational EVP(2.3)is symmetricandsoit is well-know n that(2.3)admits countably\nmany, strictly positive eigenvalues, see, e.g., [4]. The ei genvalues – labelled in ascending\norder, counting multiplicities – and the corresponding eig enfunctions are denoted by\n0< λ1(y)≤λ2(y)≤···,andu1(y), u2(y), ... .\nFory∈Ω, we define the solution operator T=T(y) :V∗→Vby\nA(y;Tf,v) =M(f,v) for all v∈V.\nClearly, if λ(y) is an eigenvalue of (2.3) then µ(y) = 1/λ(y) is an eigenvalue of Tand the\ncorresponding eigenspaces are the same.\nThe Krein–Rutmann Theorem ensures the smallest eigenvalue is simple, and then in\n[17, Prop. 2.4] it was shown that the spectral gap can be bound ed away from 0 indepen-\ndently of y. That is, there exists ρ >0, independent of y, such that\nλ2(y)−λ1(y)≥ρfor ally∈Ω. (2.4)\nThe eigenfunctions {u(y)}k∈Ncan be chosen to form a basis for Vthat is orthonormal\nwith respect toM(·,·), and hence, by (2.3), also orthogonal with respect to A(y;·,·).\nFory∈Ω, let the eigenspace E(λk(y)) be the subspace spanned by all eigenfunctions\ncorresponding to λk(y), and let/hatwideE(λk(y)):={v:v∈E(λk(y)),/ba∇dblv/ba∇dblV= 1}.\nSince the coefficients are uniformly bounded away from 0 and fr om above, theA(y)-\nandM-norms are equivalent to the V- andL2-norms, respectively, with\ncA/ba∇dblv/ba∇dblV≤/ba∇dblv/ba∇dblA(y)≤CA/ba∇dblv/ba∇dblV, (2.5)\ncM/ba∇dblv/ba∇dblL2≤/ba∇dblv/ba∇dblM≤CM/ba∇dblv/ba∇dblL2, (2.6)\nwherethe constants are independentof y, see [19, eqs. (2.7), (2.8)] for their explicit values.\nByCPoin>0 we denote the Poincar´ e constant, which is independent of yand such that\n/ba∇dblv/ba∇dblL2(D)≤CPoin/ba∇dblv/ba∇dblV,for allv∈V. (2.7)\nFor the remainder of the paperwe denote the smallest eigenva lue and its corresponding\neigenfunction by λ=λ1andu=u1, respectively.\n2.2 Stochastic dimension truncation\nIn order to evaluate the stochastic coefficients a(y) andb(y) in practice, we must first\ntruncate the infinite-dimensional stochastic domain Ω. Thi s is done by choosing a finite\ntruncation dimension s∈Nand by setting yj= 0 for all j > s. We define the following\nnotation: ys= (y1,y2,...,ys),\nas(x,y):=a0(x)+s/summationdisplay\nj=1yjaj(x) and bs(x,y):=b0(x)+s/summationdisplay\nj=1yjbj(x).\nIn this way, the truncated coefficients as(y) andbs(y) can be evaluated in practice, since\nthey only depend on finitely many terms.\n5Similarly, the truncated approximations of the eigenvalue and eigenfunction are de-\nnoted by λs(y),us(y), respectively. Defining the bilinear form As(y) :V×V→R\ncorresponding to the truncated coefficients by\nAs(y;w,v):=/integraldisplay\nDas(x,y)∇w(x)·∇v(x)dx+/integraldisplay\nDbs(x,y)w(x)v(x)dx,(2.8)\nwe have that λs(y),us(y) satisfy\nAs(y;us(y),v) =λs(y)M(us(y),v),for allv∈V . (2.9)\n2.3 Finite element methods for eigenvalue problems\nThe eigenvalue problem (2.3) will be discretised in the spat ial domain using piecewise\nlinear finiteelements (FE). First, wepartitionthespatial domainDusingafamilyofshape\nregular triangulations {Th}h>0, indexed by the meshwidth h= max{diam(τ) :τ∈Th}.\nThen, for h >0 letVhbe the conforming FE space of continuous functions that are\npiecewise linear on the elements of the triangulation Th, and let Mh:= dim(Vh)<∞\ndenote the dimension of this space. Additionally, we assume that each mesh This such\nthat the dimension of the corresponding FE space Vhis\nMh/equalorsimilarh−d, (2.10)\nwhich will be satisfied by quasi-uniform meshes, but also all ows for locally refined meshes.\nFor each y∈Ω, the FE eigenvalue problem is: Find λh(y)∈R,uh(y)∈Vhsuch that\nA(y;uh(y),vh) =λh(y)M(uh(y),vh) for all vh∈Vh, (2.11)\n/ba∇dbluh(y)/ba∇dblM= 1.\nThe FE eigenvalue problem (2.11) admits Mheigenvalues and corresponding eigenvectors\n0< λ1,h(y)≤λ2,h(y)≤···≤ λMh,h(y),andu1,h(y), u2,h(y), ..., u Mh,h(y),\nwhich converge from above to the first Mheigenvalues and eigenfunctions of (2.3) as\nh→0, see, e.g., [4] or [17] for the stochastic case.\nAsbefore,let E(λk,h(y))betheeigenspacecorrespondingto λk,h(y)anddefine/hatwideE(λk,h(y)):=\n{v∈E(λk,h(y)) :/ba∇dblv/ba∇dblV= 1}. If the exact eigenvalue λk(y) has multiplicity m(and we\nassume without loss of generality that λk(y) =λk+1(y) =···=λk+m−1(y)), then there\nexistmFE eigenvalues, λk,h(y), λk+1,h(y),...,λ k+m−1,h(y), that converge to λk(y), but\nare not necessarily equal. As such, we also define Eh(λk(y)) to be the direct sum of all the\neigenspaces E(λℓ,h(y)) such that λℓ,h(y)→λk(y). Finally, we define /hatwideEh(λk(y):={v∈\nEh(λk(y)) :/ba∇dblv/ba∇dblV= 1}.\nIn Assumption A1 we have only assumed that the physical domai nDis convex and\nthata∈W1,∞(D). Hence, piecewise linear FEs are sufficient to achieve the op timal rates\nof convergence with respect to hin general. In particular, in [17, Thm. 2.6] it was shown\nthat the FE error for the minimal eigenpair can be bounded ind ependently of ywith the\nusual rates in terms of h. Explicitly, if h >0 is sufficiently small, then for all y∈Ω\n/ba∇dblu(y)−uh(y)/ba∇dblV≤Cuh,|λ(y)−λh(y)|≤Cλh2, (2.12)\nand forG∈H−1+t(D) witht∈[0,1]\n/vextendsingle/vextendsingleG(u(y))−G(uh(y))/vextendsingle/vextendsingle≤CGh1+t, (2.13)\n6where 0< Cλ, Cu, CG<∞are independent of yandh.\nIn the companion paper [19], it is shown that for hsufficiently small1the spectral gap\nof the FE eigenvalue problem (2.11) satisfies the uniform low er bound\nλ2,h(y)−λ1,h(y)≥ρ\n2>0, (2.14)\nand that the eigenvalues and eigenfunctions of both (2.3) an d (2.11) satisfy the bounds\nλk≤λk(y)≤λk,h(y)≤λk,and (2.15)\nmax/braceleftbig\n/ba∇dbluk(y)/ba∇dblV,/ba∇dbluk,h(y)/ba∇dblV/bracerightbig\n≤uk, (2.16)\nwhereλk,λk,ukare also independent of both yandh.\nNote that the use of piecewise linear FEs is not a restriction on our MLQMC methods.\nThe algorithms presented in Section 4 are very general, and w ill work with higher order\nFE methods as well, without any modification of the overall al gorithm structure.\n2.4 Iterative solvers for eigenvalue problems\nThe discrete EVP (2.11) from the previous section leads to a g eneralised matrix EVP of\nthe form Ahuh=λhBhuh, where, in general, the matrices AhandBhare large, sparse\nand symmetric positive definite.\nSince we are only interested in computing a single eigenpair , we will use Rayleigh\nquotient (RQ) iteration to compute it. It is well-known that for symmetric matrices RQ\niteration converges cubically for almost all starting vect ors, see, e.g., [37].\n2.5 Quasi-Monte Carlo integration\nA quasi-Monte Carlo (QMC) method is an equal weight quadratu re rule\nQs,Nf=1\nNN−1/summationdisplay\nk=0f(tk) (2.17)\nwithN∈Ndeterministically-chosen quadrature points {tk}N−1\nk=0, as opposed to random\nquadrature points as in Monte Carlo. The key feature of QMC me thods is that the points\nare cleverly constructed to be well-distributed within hig h-dimensional domains, which\nallows for efficient approximation of high-dimensional inte grals such as\nIsf:=/integraldisplay\n[−1\n2,1\n2]sf(y)dy.\nThere are many different types of QMC point sets, and for furthe r details we refer the\nreader to, e.g., [11].\nIn this paper, we use a simple to construct, yet powerful, cla ss of QMC methods called\nrandomly shifted rank-1 lattice rules . A randomly shifted lattice rule approximation to\nIsfusingNpoints is given by\nQs,N(∆)f:=1\nNN−1/summationdisplay\nk=0f({tk+∆}−1\n2) (2.18)\nwherez∈Nsis thegenerating vector and the points tkare given by\ntk=/braceleftbiggkz\nN/bracerightbigg\nfork= 0,1,...,N−1,\n1The explicit condition is that h≤hwithh:=/radicalbig\nρ/(2Cλ).\n7∆∈[0,1)sis a uniformly distributed random shift , and{·}denotes the fractional part of\neach component of a vector. Note that we have subtracted 1 /2 in each dimension to shift\nthe quadrature points from [0 ,1]sto [−1\n2,1\n2]s.\nGoodgeneratingvectorscanbeconstructedinpracticeusin gthecomponent-by-component\n(CBC) algorithm, or the more efficient Fast CBC construction [35, 36]. In fact, it can\nbe shown that for functions in certain first-order weighted S obolev spaces such as those\nintroduced in [42], the root-mean-square (RMS) error of a ra ndomly shifted lattice rule\nusing a CBC-constructed generating vector achieves almost the optimal rate of O(N−1).\nTo state the CBC error bound, we briefly introduce the followi ng specific class of\nweighted Sobolev spaces, which are useful for the analysis o f lattice rules. Given a col-\nlection of weightsγ:={γu>0 :u⊆{1,2,...,s}}, which represent the importance of\ndifferent subsetsof variables, let Ws,γbethes-dimensionalweighted (unanchored)Sobolev\nspace of functions with square-integrable mixed first deriv atives, equipped with the norm\n/ba∇dblf/ba∇dbl2\nWs,γ=/summationdisplay\nu⊆{1:s}1\nγu/integraldisplay\n[−1\n2,1\n2]|u|/parenleftbigg/integraldisplay\n[−1\n2,1\n2]s−|u|∂|u|\n∂yuf(y) dy−u/parenrightbigg2\ndyu,(2.19)\nwhere we use the notation {1 :s}={1,2,...,s},yu= (yj)j∈uandy−u= (yj)j∈{1:s}\\u.\nThen, for f∈Ws,γandNa power of 2, the RMS error of a CBC-constructed randomly\nshifted lattice rule approximation satisfies\n/radicalBig\nE∆/bracketleftbig\n|Isf−Qs,Nf|2/bracketrightbig\n/lessorsimilarN−1+δ/ba∇dblf/ba∇dblWs,γ, δ > 0, (2.20)\nwhere under certain conditions on the decay of the weights γthe constant is independent\nof the dimension. Note that similar results also hold for gen eralN, but with Non the\nRHS of (2.20) replaced by the Euler Totient function, which c ounts the number of integers\nless than and coprime to N, see, e.g., [11, Theorem 5.10]. For more details on the gener al\ntheory of lattice rules see [11], and for a theoretical analy sis of randomly shifted lattice\nrules for MLQMC applied to (1.1) see [19].\nThe generating vectors given by the CBC algorithm are extens ible in dimension, how-\never, they are constructed for a fixed value of N. By modifying the error criterion that is\nminimised in each step of the CBC algorithm, one can construc t a generating vector that\nworks well for a range of values of N, where now Nis given as some power of a prime base,\ne.g.,Nis a power of 2. The resulting quadrature ruleis called an embedded lattice rule and\nwas developed in [9]. Not only do embedded lattice rules work well for a range of values\nofN, but the resulting point sets are nested. Hence, one can impr ove the accuracy of a\npreviously computed embedded lattice rule approximation b y simply adding the function\nevaluations corresponding to the new points to the sum from t he previous approximation.\nAs will be clear later, the extensibilty in both sandNof embedded lattice rules makes\nthem extremely convenient for use in MLQMC methods in practi ce.\nCurrently there is not any theory for the error of embedded la ttice rules, however, a\nseries of comprehensive numerical tests conducted in [9] sh ow empirically that the optimal\nrateofN−1isstillobserved, andthattheworst-caseerrorforanembed dedlattice increases\nat most by a factor of 1.6 as compared to the normal CBC algorit hm with Nfixed.\nFinally, instead of using a single random shift, in practice it is better to average over\nseveral randomlyshiftedapproximationsthatcorrespondt oasmallnumberofindependent\nrandom shifts. The practical benefits are (i) that averaging gives more consistent results,\nby reducing the chance of using a single “bad” shift, and (ii) that the sample variance of\nthe shifted approximations provides a practical error esti mate. Let ∆(1),∆(2),...,∆(R)\nbeRindependent uniform random shifts, then the average of the Q MC approximations\n8corresponding to the random shifts is\n/hatwideQs,N,Rf:=1\nRR/summationdisplay\nr=1Qs,N(∆(r))f,\nand the mean-square error of /hatwideQs,N,Rfcan be estimated by the sample variance\n/hatwideV[/hatwideQs,N,R]:=1\nR(R−1)R/summationdisplay\nr=1/bracketleftbig/hatwideQs,N,Rf−Qs,N(∆(r))f/bracketrightbig2. (2.21)\n2.6 Discrepancy theory\nMuch of the modern theory for QMC rules is based on weighted fu nction spaces as dis-\ncussed in Section 2.5, however, the traditional analysis of QMC rules is based on the\ndiscrepancy of the quadrature points. Loosely speaking, for a given poin t set the discrep-\nancy measures the difference between the number of points that actually lie within some\nsubset of the unit cube and the number of points that are expec ted to lie in that subset\nif the point set were perfectly uniformly distributed. This more geometric notion of the\nquality of a QMC point set will be useful later when we analyse the use of an eigenvector\ncorresponding to a nearby QMC point as the starting vector fo r the RQ iteration.\nWe now recall some basic notation and definitions from the fiel d of discrepancy theory\nfor a point setPN={t0,t1,...,tN−1}⊂[0,1]son the unit cube. Note that by a simple\ntranslation the results from this section are also applicab le on [−1\n2,1\n2]s. The axis-parallel\nbox with corners a,b∈[0,1]swithaj< bjis denoted by [ a,b):= [a1,b1)×[a2,b2)×···×\n[as,bs). The number of points from PNthat lie in [ a,b) is denoted by|{PN∩[a,b)}|and\nthe Lebesgue measure on [0 ,1]sbyLs.\nDefinition 2.1. Thestar discrepancy of a point setPNis defined by\nD∗\nN(PN):= sup\nb∈[0,1]s/vextendsingle/vextendsingle/vextendsingle/vextendsingle|{PN∩[0,b)}|\nN−Ls/parenleftbig\n[0,b)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (2.22)\nPNis called a low discrepancy point set if there exists CPN<∞, independent of s, such\nthat\nD∗\nN(PN)≤CPNlog(N)s−1\nN. (2.23)\nThere exist several well-known points sets that have low-di screpancy, such as Ham-\nmersley point sets, see [12] for more details.\nThe connection between star discrepancy and quadrature is g iven by the Koksma–\nHlawka inequality, which for a function fwith bounded Hardy–Krause variation states\nthat the quadrature error of a QMC approximation (2.17) sati sfies the bound\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle/integraldisplay\n[0,1]sf(y) dy−Qs,Nf/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤/parenleftBigg/summationdisplay\n∅/\\e}atio\\slash=u⊆{1:s}/integraldisplay\n[0,1]|u|/vextendsingle/vextendsingle/vextendsingle/vextendsingle∂|u|\n∂yuf(yu;1)/vextendsingle/vextendsingle/vextendsingle/vextendsingledyu/parenrightBigg\nD∗\nN(PN),(2.24)\nsee, e.g., [12]. Here, ( yu;1) denotes the anchored point with jth component yjifj∈uand\n1 otherwise. Hence, low-discrepancy point sets lead to QMC a pproximations for which\nthe error converges like O(log(N)s−1/N).\nLattice rules can also be constructed such that their discre pancy is log( N)s/N(see\n[12, Corollary 3.52]). By the Koksma–Hlawka inequality (2. 24), they then admit error\nbounds similar to (2.20), but with an extra log( N)sfactor. By instead considering the\n9weighted discrepancy, one can construct lattice rules that have a weighted discrepancy\n(and similarly error bounds) without this log factor, see [2 8].\nFinally, we also define the extreme discrepancy of a point set , which removes the\nrestriction that the boxes are anchored to the origin.\nDefinition 2.2. Theextreme discrepancy of a point setPNis defined by\n/hatwideDN(PN):= sup\n[a,b)⊂[0,1]s/vextendsingle/vextendsingle/vextendsingle/vextendsingle|{PN∩[a,b)}|\nN−Ls/parenleftbig\n[a,b)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (2.25)\n3 Two-grid-truncation methods for stochastic EVPs\nTwo-grid FEdiscretisation methodsforEVPswerefirstintro ducedin[46]andlater refined\nindependentlyin[25,26]and[47]. Theideabehindthemissi mple: tocombineFEmethods\nwith iterative solvers for matrix EVPs. Letting H > h > 0 be the meshwidths of a coarse\nand a fine FE mesh, THandTh, respectively, one first solves the EVP (2.11) on the\ncoarse FE space VHto giveλH,uH. This coarse eigenpair ( λH,uH) is then used as the\nstarting guess for an iterative eigensolver for the EVP on th e fine FE space Vh. Since FE\nmethods for PDE EVPs and iterative methods for matrix EVPs bo th converge very fast,\nHandhcan be chosen such that a single linear solve is all that is req uired to obtain the\nsame order of accuracy as can be expected from solving the ori ginal FE EVP on the fine\nmesh. This strategy can be adapted to a full multigrid method for EVPs as in, e.g., [45].\nHowever, it was shown in [25, 26, 47] that the maximal ratio H/hbetween the coarse and\nfine meshwidth in a two grid method is so large that, in general , two grids are sufficient.\nHere, we present a new algorithm that extends the two-grid me thod to stochastic (or\nparametric) EVPs, by also using a reduced (i.e., cheaper and less accurate) truncation of\nthe parameter space when solving the parametric EVP on the in itial coarse mesh. Since\nour new algorithm combines this truncation and FE approxima tions, we first introduce\nsome notation. For some h >0 ands∈N, the FE EVP that approximates the truncated\nproblem (2.9) is: Find λh,s(y)∈Randuh,s(y)∈Vhsuch that\nAs(y;uh,s(y),vh) =λh,s(y)M(uh,s(y),vh) for all vh∈Vh, (3.1)\n/ba∇dbluh,s(y)/ba∇dblM= 1.\nWe also define the solution operator Th,s=Th,s(y) :V∗→Vhfor (3.1), which for f∈V∗\nsatisfies\nAs(y;Th,sf,vh) =M(f,vh) for all vh∈Vh, (3.2)\nand theAs(y)-orthogonal projection operator Ph,s=Ph,s(y) :V→Vh, which for u∈V\nsatisfies\nAs(y;u−Ph,su,vh) = 0 for all vh∈Vh. (3.3)\nAlthough both operators depend on ywe will not specify this dependence.\nIn Algorithm 1 below we detail our new two-grid and truncatio n method for paramet-\nric EVPs. The algorithm is based on the accelerated version o f the two-grid algorithm\n(see [25, 26] and also [47]), which uses the shifted-inverse power method for the update\nstep. In addition, we add a normalisation step so that /ba∇dbluh(y)/ba∇dblM= 1, which simplifies\nthe RQ update but does not affect the theoretical results. As in the papers above, we\nwill consistently use the notation that two-grid approxima tions use superscripts whereas\nordinary approximations (i.e., eigenpairs of truncated/F E problems) use subscripts. To\nperform step 2 in practice, one must interpolate the uH,S(y) at the nodes of the fine mesh\nThto obtain the corresponding start vector. Since we use piece wise linear FE methods,\nlinear interpolation is sufficient.\n10Algorithm 1 Two-grid-truncation method for parametric EVPs\nGivenH > h > 0, 0< S < s andy∈Ω:\n1:FindλH,S(y)∈RanduH,S(y)∈VHsuch that\nAS(y;uH,S(y),vH) =λH,S(y)M(uH,S(y),vH) for all vH∈VH,\n/ba∇dbluH,S(y)/ba∇dblM= 1.\n2:Finduh,s∈Vhsuch that\nAs(y;uh,s(y),vh)−λH,S(y)M(uh,s(y),vh) =M(uH,S(y),vh) for all vh∈Vh.(3.4)\n3:uh,s(y)←uh,s(y)//ba∇dbluh,s(y)/ba∇dblM ⊲normalise the eigenfunction approximation\n4:\nλh,s(y) =As(y;uh,s(y),uh,s(y)). (3.5)\nThe following lemmas will help us to extend the error analysi s of two-grid methods to\ninclude a component that corresponds to truncating the para meter dimension.\nLemma 3.1. LetB,/tildewideB:V×V→R, be two bounded, coercive, symmetric bilinear forms.\nSuppose that (λ,u)is an eigenpair of\nB(u,v) =λM(u,v)for allv∈V ,\nand letw∈V. Then\n/tildewideB(w,w)\nM(w,w)−λ=/ba∇dblu−w/ba∇dbl2\n/tildewideB\n/ba∇dblw/ba∇dbl2\nM−λ/ba∇dblu−w/ba∇dbl2\nM\n/ba∇dblw/ba∇dbl2\nM+1\n/ba∇dblw/ba∇dbl2\nM/parenleftbig\nB(u,u−2w)−/tildewideB(u,u−2w)/parenrightbig\n.(3.6)\nProof.Expanding, then using the fact that ( λ,u) is an eigenpair gives\n/ba∇dblu−w/ba∇dbl2\n/tildewideB−λ/ba∇dblu−w/ba∇dbl2\nM\n=/tildewideB(u,u)+/tildewideB(w,w)−2/tildewideB(u,w)−λM(u,u)−λM(w,w)+2λM(u,w)\n=/tildewideB(u,u)+/tildewideB(w,w)−2/tildewideB(u,w)−B(u,u)−λM(w,w) +2B(u,w)\n=/tildewideB(w,w)−λM(w,w)−B(u,u−2w)+/tildewideB(u,u−2w).\nDividing by/ba∇dblw/ba∇dbl2\nMand rearranging leads to the desired result.\nLemma 3.2. Let Assumption A1 hold, then\n/ba∇dblT−Th,s/ba∇dbl≤CT(s−1/p+1+h), (3.7)\nwhereCTis independent of y,sandh.\nProof.The differential operator A(y)v=−∇·(a(y)∇v)+b(y)vfrom the EVP (1.1) fits\ninto the general framework of [10]. Defining Ts:=T(ys) to be the solution operator for\nthe truncated EVP (2.9), it follows from the triangle inequa lity that\n/ba∇dblT−Th,s/ba∇dbl≤/ba∇dblT−Ts/ba∇dbl+/ba∇dblTs−Th,s/ba∇dbl≤C1s−1/p+1+C2h,\nwhere we have used [10, Theorem 2.6 and eq. (2.17)] in the last step.\n11The error of the outputs of Algorithm 1 are given in the theore m below. The proof\nfollows a similar proof technique as used in [47], and also re lies on an abstract approxi-\nmation result for operators from that paper. Note that the FE component of the error is\nthe same as the results in [25, 26, 47], but here we have extra t erms corresponding to the\ntruncation error. The proof is deferred to the appendix.\nTheorem 3.1. Suppose that Assumption A1 holds, let S∈Nbe sufficiently large and let\nH >0be sufficiently small. Then, for s > Sand0< h < H ,\n/ba∇dblu(y)−uh,s(y)/ba∇dblV/lessorsimilarH4+h+S−2(1/p−1)+s−(1/p−1)+H2S−(1/p−1),and(3.8)\n|λ(y)−λh,s(y)|/lessorsimilarH8+h2+S−4(1/p−1)+s−(1/p−1)+H4S−2(1/p−1), (3.9)\nwhere both constants are independent of s,S,h,H andy.\nIt follows that in our two-grid-truncation method, to maint ain the optimal order h\nconvergence for the eigenfunction we should take H/equalorsimilarh1/4,s/equalorsimilarh−p/(1−p)andS/equalorsimilars1/2,\nwhereas for the eigenvalue error we should take a higher trun cation dimension, namely\ns/equalorsimilarh−2p/(1−p)andS/equalorsimilars1/4. The difference in conditions comes from the fact that for\nEVPs, the truncation error for the eigenvalue and eigenfunc tion are of the same order,\nwhereas the FE error for the eigenvalue is double the order of the eigenfunction FE error.\nItis similar to howahigher precision numerical quadrature ruleshouldbeusedtocompute\nthe elements of the stiffness matrix for eigenvalue approxima tion, see, e.g., [5].\n4 MLQMC algorithms for random eigenvalue problems\nIn this section, we present two MLQMC algorithms for approxi mating the expectation of a\nrandom eigenvalue. First, we briefly give a straightforward MLQMC algorithm, for which\na rigorous theoretical analysis of the error was presented i n [19]. After analysing the cost\nof this algorithm we then present a second, more efficient MLQM C algorithm, where we\nfocus on reducing the overall cost by reducing the cost of eva luating each sample.\n4.1 A basic MLQMC algorithm for eigenvalue problems\nThe starting point of our basic MLQMC algorithm is the telesc oping sum (1.2), along with\na collection of L+1 FE meshes corresponding to meshwidths, h0> h1>···> hL>0,\nandL+ 1 truncation dimensions, 0 < s0≤s1≤···≤ sL<∞. Recall that we denote\nthe eigenvalue approximation on level ℓbyλℓ:=λhℓ,sℓwithλ−1≡0. The expectation on\neach level ℓin the sum (1.2) can be approximated by a QMC rule using Nℓpoints, which\nwe denote by Qℓ:=Qsℓ,Nℓas in (2.18), so that our MLQMC approximation of E[λ] is\nQML\nL(∆)λ:=L/summationdisplay\nℓ=0Qℓ(∆ℓ)/parenleftbig\nλℓ−λℓ−1/parenrightbig\n. (4.1)\nHere, each ∆ℓ∈[0,1)sℓis an independent random shift, so that the QMC approximatio ns\non different levels are independent. To simplify the notation , we also concatenate the L+1\nshifts into a single random shift ∆= (∆0;∆1;...;∆L) (where “;” denotes concatenation\nof column vectors). For a linear functional G ∈V∗, the MLQMC approximation to\nEy[G(u)] can be defined analogously.\nAs described in Section 2.5, in practice it is beneficial to us eRindependent random\nshifts∆(1),∆(2),...,∆(R). Then the shift-averaged MLQMC approximation is\n/hatwideQML\nL,Rλ:=L/summationdisplay\nℓ=01\nRR/summationdisplay\nr=1Qℓ(∆(r)\nℓ)/parenleftbig\nλℓ−λℓ−1/parenrightbig\n. (4.2)\n12In this case, the variance on each level can be estimated by th e sample variance as given\nin (2.21) and denoted by Vℓ. Due to the independence of the QMC approximations across\nthe levels, the total variance of the MLQMC estimator is\n/hatwideV/bracketleftbig/hatwideQML\nL,Rλ/bracketrightbig\n=L/summationdisplay\nℓ=0Vℓ. (4.3)\n4.2 Cost & error analysis\nThe cost of the MLQMC estimator (4.2) for the expected value o fλis given by\ncost(/hatwideQML\nL,Rλ) =RL/summationdisplay\nℓ=0Nℓcost(λℓ−λℓ−1),\nwhere cost( λℓ−λℓ−1) denotes the cost of evaluating the difference at a single para meter\nvalue. Since cost( λℓ−λℓ−1)≤2cost(λℓ), the cost of evaluating λℓat a single parameter\nvalue,\ncost(/hatwideQML\nL,Rλ)/lessorsimilarRL/summationdisplay\nℓ=0Nℓcost(λℓ).\nThe cost of evaluating the eigenvalue approximation λℓconsists of two parts:\ncost(λℓ) =Csetup\nℓ+Csolve\nℓ,\nwhereCsetup\nℓdenotes the setup cost of constructing the stiffness and mass m atrices, and\nCsolvedenotes the cost of solving the eigenvalue problem. Since th e coefficient cis inde-\npendent of yso too is the mass matrix, and as such we only compute it once pe r level.\nThus,Csetup\nℓis dominated by constructing the stiffness matrix for each qua drature point.\nConstructing the stiffness matrix at each parameter value inv olves evaluating the co-\nefficients, which are sℓ-dimensional sums, at the quadrature points for each elemen t in the\nmesh. Under the assumption (2.10) on the number of FE degrees of freedom, the number\nof elements in the mesh is also O(h−d), which implies the setup cost is\nCsetup\nℓ/lessorsimilarsℓh−d\nℓ.\nAt each each step of an iterative eigensolver a linear system must be solved, and this\nforms the dominant component of the cost for that step. Essen tially, the cost of each\neigenproblem solve is of the order of a source problem solve ( on the mesh Thℓ) multiplied\nby the number of iterations. As in the case of the source probl em (see e.g., [32, 31]),\nwe assume that the linear systems occurring in each iteratio n of the eigensolver can be\nsolved inO(h−γ) operations, with d < γ < d +1. Assuming that the number of iterations\nrequired is independent of y, the cost of each eigensolve is then\nCsolve/lessorsimilarh−γ\nℓ. (4.4)\nWe discuss how to bound the number of iterations of the eigens olver in Section 4.4.\nIt then follows that the cost of evaluating λℓat a single parameter value satisfies\ncost(λℓ)/lessorsimilarsℓh−d\nℓ+h−γ\nℓ, and hence the total cost of the MLQMC estimator (4.2) satisfi es\ncost(/hatwideQML\nL,Rλ)/lessorsimilarRL/summationdisplay\nℓ=0Nℓ(sℓh−d\nℓ+h−γ\nℓ). (4.5)\n13Since the eigenfunction approximation uℓis computed at the same time as λℓ, and we\nassume that the cost of applying a linear functional Gis constant, the cost of the MLQMC\nestimator/hatwideQML\nL,RG(u) is of the same order as the cost of the eigenvalue estimator i n (4.5).\nThe error of the approximation (4.1) is analysed rigourousl y in [19], and so here we\nonly give a brief summary of one of the key results. First, sup pose that Assumption A1\nholds with 0 < p < q < 1, and let each Qℓuse a generating vector given by the CBC\nconstruction. Next, choose hℓ/equalorsimilar2−ℓandsℓ=sL/equalorsimilarh2p/(2−p)\nL. Then, it was shown in [19,\nCorollary 3.1] that for 0 ≤ε <exp(−1), we can choose Land the number of points on\neach level, Nℓ, such that the mean-square error of the estimator (4.1) is bo unded by\nE∆/bracketleftbig\n|Ey[λ]−QML\nL(∆)λ|2/bracketrightbig\n≤ε2,\nand forδ >0 the cost is bounded by\ncost/parenleftbig\nQML\nL(λ)/parenrightbig\n/lessorsimilar\n\nε−2/η−p/(2−p)ifη <4/d,\nε−2/η−p/(2−p)log2(ε−1)1+1/ηifη= 4/d,\nε−d/2−p/(2−p)ifη >4/d,(4.6)\nwhereηis the convergence rate of the variance of Vℓwith respect to Nℓ, which by [19,\nTheorem 5.3] is given by\nη=\n\n2−δifq∈(0,2\n3]\n2\nq−1 ifq∈(2\n3,1).\nIn practice, we typically set hℓ/equalorsimilar2−ℓ,sℓ/equalorsimilar2ℓand use the adaptive algorithm from [21]\nto choose{Nℓ}andL. Although this is a greedy algorithm, it was shown in [31, Sec tion\n3.3] that the resulting choice of Nℓleads to the same asymptotic order for the overall cost\nas the choice of Nℓin the theoretical complexity estimate from [19, Corollary 3.1].\n4.3 An efficient MLQMC method with reduced cost per sample\nTo reduce the cost of computing each sample in the MLQMC appro ximation (4.1) in\npractice, we employ the following two strategies for each ev aluation of the difference λℓ−\nλℓ−1on a given level: 1) we use the two-grid-truncation method (c f. Algorithm 1) to\nevaluate the eigenpairs in the difference; and 2) we use the eig envector from a nearby\nquadrature point as the starting vector for the eigensolve o n the coarse mesh.\nTwo-grid-truncation methods Our strategy for how to use the two-grid-truncation\nmethod from Section 3 for a given sample yis as follows. First, we solve the EVP (3.4)\ncorresponding to a coarse discretisation, with meshwidth a nd truncation dimension given\nby\nHℓ= min/parenleftbig\nh1/4\nℓ,h0/parenrightbig\nandSℓ= max/parenleftbig/ceilingleftbig\ns1/2\nℓ/ceilingrightbig\n,s0/parenrightbig\n,\nto get the coarseeigenpair ( λHℓ,Sℓ(y),uHℓ,Sℓ(y)). Then, we let uℓ(y):=uhℓ,sℓ(y)∈Vℓbe\nthe solution to the following source problem\nAsℓ(y;uℓ(y),v)−λHℓ,sℓ(y)M(uℓ(y),v) =M(uHℓ,sℓ(y),v) for all v∈Vℓ,(4.7)\nand define the eigenvalue approximations for ℓ= 1,2,...,Lby the Rayleigh quotient\nλℓ(y):=λhℓsℓ(y):=Asℓ(y;uℓ(y),uℓ(y))\nM(uℓ(y),uℓ(y)). (4.8)\n14The eigenpair on level ℓ−1 for the same sample is computed in the same way and we\nsetλ−1(y) = 0 and λ0(y) =λh0,s0(y).\nIn this way, the MLQMC approximation with two-grid update, a ndRrandom shifts,\nis given by\n/hatwideQTG\nL,Rλ:=1\nRR/summationdisplay\nr=1L/summationdisplay\nℓ=0Qℓ(∆(r)\nℓ)/parenleftbig\nλℓ−λℓ−1/parenrightbig\n. (4.9)\nNote that for a given sampleon level ℓwe use(λHℓ,Sℓ(y),uHℓ,Sℓ(y)) to compute λℓ−1as\nwell. Technically, this violates the telescoping property , sinceλℓ−1from the previous level\n(ℓ−1) will use ( λHℓ−1,Sℓ−1(y),uHℓ−1,Sℓ−1(y)), but in practice this difference is negligible\nand does not justify an extra coarse solve. Furthermore, sin ce the two-grid method allows\nfor such a large difference in parameters of the coarse grid and the fine grid ( H/equalorsimilarh1/4and\nS/equalorsimilars1/2), often we will have the case where Hℓ−1=Hℓ=h0andSℓ−1=Sℓ=s0. So that\nreusing ( λHℓ,Sℓ(y),uHℓ,Sℓ(y)) to compute λℓ−1in the difference on level ℓdoes not violate\nthe telescoping property. As an example, if we take h0= 1/8, then we can use Hℓ=h0\nas the coarse meshwidth for all levels ℓup tohℓ≤2−12= 1/4096. In all of our numerical\nresults, 1 /4096 was below the finest grid size hLrequired.\nUsing the two-grid method still involves solving a source pr oblem on the fine mesh,\nso that the cost of a two-grid solve is of the same order as Csolvebut with an improved\nconstant. The reduction in cost is proportional to the numbe r of RQ iterations that are\nrequired to solve the eigenproblem on the fine mesh without tw o-grid acceleration, so that\nthe highest gains will be achieved for problems where the RQ i teration converges slowly.\nReusing samples from nearby QMC points Now, foreach samplewemuststillsolve\nthe EVP (4.7) corresponding to a coarse mesh and a reduced tru ncation dimension, which\nwe do using the RQ algorithm (see, e.g., [37]). To reduce the n umber of RQ iterations\nto compute this coarse eigenpair at some QMC point tk, we use the eigenvector from a\nnearby QMC point (say t′) as the starting vector: v0=uHℓ,Sℓ(t′). For the initial shift in\nthe RQ algorithm we use the Rayleigh quotient of this nearby v ector with respect to the\nbilinear form at the currentQMC point: σ0=ASℓ(tk;v0,v0). In practice, we have found\nthat a good choice of the nearby QMC point is simply the previo us point tk−1.\nExplicit details on how these two strategies areimplemente d to construct theestimator\n(4.9) in practice are given in Algorithm 2. First we introduc e some notation to simplify\nthe presentation. Denote the kth randomly shifted rank-1 lattice point on level ℓby\ntℓ,k:=/braceleftbiggkzℓ\nNℓ+∆ℓ/bracerightbigg\n, (4.10)\nwherezℓis ansℓ-dimensional generating vector and ∆ℓ∼U[0,1)sℓ.\nFinally, by relaxing the restriction that approximations o n different levels are inde-\npendent from one another we can use the same set of random shif ts for all levels. In this\ncase, the variance decomposition (4.3) becomes an inequali ty with a factor Lin front of\nthe sum. Following the arguments in [7, Section 3.1 and Remar k 2] it can be shown that\nthis does not significantly change the overall complexity, a t worst the cost increases by a\nfactor of|log(ǫ)|.\nThen, if we also use nested QMC rules we can reuse approximati ons from lower levels\non the higher levels. In particular, for ℓ≥1 we will have tℓ,k=t0,kand can set Hℓ=h0so\nthat we can omit the coarse eigenvalue solves (steps 6 and 8) i n Algorithm 2. Furthermore,\nthereis noneed tocalculate λℓ−1,uℓ−1again either, sosteps 12 and14 can also beskipped.\nIn this case, because the optimal choice for the parameters i n the two-grid-truncation\nmethods are H/equalorsimilarh1/4andS/equalorsimilars1/2, the range of possible meshwidths and truncation\n15Algorithm 2 Two-grid MLQMC for eigenvalue problems\nGivenv0,L,R,{sℓ}L\nℓ=0,{hℓ}L\nℓ=0and{Nℓ}L\nℓ=0:\n1:forℓ= 0,1,2,...,Ldo\n2:Hℓ←min(h1/4\nℓ,h0) andSℓ←max(s1/2\nℓ,s0)\n3:forr= 1,2,...,Rdo\n4: generate ∆ℓ∼U[0,1)sℓ\n5: fork= 0,1,...,N ℓdo\n6: generate tℓ,kusing the shift ∆ℓas in (4.10) ⊲shifted QMC point\n7: compute ( λHℓ,Sℓ(tℓ,k),uHℓ,Sℓ(tℓ,k)) usingv0as start value\n8: v0←uHℓ,Sℓ(tℓ,k) ⊲update starting value\n9: ifℓ >0then\n10: solve the source problem (4.7) for uℓ(tℓ,k)∈Vℓ\n11: setλHℓ−1,Sℓ−1←λHℓ,SℓanduHℓ−1,Sℓ−1←uHℓ,Sℓ\n12: solve the source problem (4.7) for uℓ−1(tℓ,k)∈Vℓ−1\n13: λℓ(tℓ,k)←Asℓ(tℓ,k;uℓ(tℓ,k),uℓ(tℓ,k))\nM(uℓ(tℓ,k),uℓ(tℓ,k))⊲two-grid updates\n14: λℓ−1(tℓ,k)←Asℓ−1(tℓ,k;uℓ−1(tℓ,k),uℓ−1(tℓ,k))\nM(uℓ−1(tℓ,k),uℓ−1(tℓ,k))\n15: end if\n16: Q(r)\nℓλ←Q(r)\nℓλ+(λℓ(tℓ,k)−λℓ−1(tℓ−1,k)) ⊲update QMC sum\n17: end for\n18: Q(r)\nℓλ←1\nNℓQ(r)\nℓλ\n19:/hatwideQℓ,Rλ←/hatwideQℓ,R+Q(r)\nℓλ\n20:end for\n21:/hatwideQℓ,Rλ←1\nR/hatwideQℓ,Rλ ⊲average over shifts\n22:/hatwideQML\nR,Lλ←/hatwideQML\nL,Rλ+/hatwideQℓ,Rλ ⊲ update ML estimator\n23:end for\ndimensions are restricted. With this in mind, we let meshwid th of the finest triangulation\nbe denoted by hand let the coarsest possible triangulation have h0≤h1/4, then we define\nthe maximum number of levels Land the meshwidth on each level so that\nh=hL< hL−1<···< h1< h0≤h1/4.\nFor example, if h= 2−8, thenh0= 2−2and we could take L= 6 with hℓ= 2−ℓ−2.\nAgain, this does not affect the asymptotic complexity bounds p roved in [19]. To overcome\nthis restriction on the coarse and fine meshwidths, one could instead use a full multigrid\nmethod as in [39]. Such an extension would be an interesting t opic for future work.\nNote that for a given problem, it may be possible that a meshwi dth ofh1/4is not\nsufficiently fine to resolve the coefficients. For this reason, w e only demand that h0≤h1/4\nand not equality. Note also that, asymptotically, the coars est meshwidth h0must decrease\nwith the finest meshwidth hL, but only at the rate h0/equalorsimilarh1/4\nL. Similarly, defining sLto be\nthe highest truncation dimension, the lowest truncation di mension increases like s0/equalorsimilars1/2\nL.\n4.4 Analysis of using nearby QMC samples\nThe argument for why starting from the eigenvector of a nearb y QMC point reduces the\nnumber of RQ iterations is very intuitive: As the number of po intsNin a QMC rule\nincreases the points necessarily become closer, and since t he eigenvectors are Lipschitz in\n16the parameter (see [17, Proposition 2.3]) this implies that the eigenvectors corresponding\nto nearby QMC samples become closer as Nincreases. Hence the starting guess for the\nRQ algorithm becomes closer to the eigenvector that is to be f ound, and so for a fixed\ntolerance the number of RQ iterations also decreases.\nIn this section we provide some basic analysis to justify our intuition above. Through-\nout it will be convenient to use the more geometric notions fr om the classical discrepancy\ntheory of QMC point sets on the unit cube [0 ,1]s, which were discussed in Section 2.6.\nFrom the definition of the star discrepancy (see Definition 2. 1) follows a simple upper\nbound on how close nearby points are in a low-discrepancy poi nt set. The result is given\nin terms of dist(·,·), the distance function with respect to the ℓ∞norm.\nProposition 4.1. LetPNbe a low-discrepancy point set for N >1, then\nmax\nt∈PNdist(t,PN\\{t})≤3C1/s\nPNlog(N)1−1/sN−1/s, (4.11)\nwhereCPNis the constant from the discrepancy bound (2.23)onPN.\nProof.Lett∈PN. Clearly dist( t,PN\\{t})≤1 holds trivially because supx,y∈[0,1]s/ba∇dblx−\ny/ba∇dblℓ∞≤1. Hence, we can assume, without loss of generality, that the upper bound in\n(4.11) satisfies\n3C1/s\nPNlog(N)1−1/sN−1/s<1, (4.12)\nwhich will be satisfied for Nsufficiently large.\nFor any box [ a,b)⊂[0,1]s, it follows from the definition of the extreme discrepancy\n/hatwideDNin Definition 2.2 that\n/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle|{PN∩[a,b)}|\nN−Ls/parenleftbig\n[a,b)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle≤sup\na≤b∈[0,1]s/vextendsingle/vextendsingle/vextendsingle/vextendsingle|{PN∩[a,b)}|\nN−Ls/parenleftbig\n[a,b)/parenrightbig/vextendsingle/vextendsingle/vextendsingle/vextendsingle=:/hatwideDN(PN).\nBy the reverse triangle inequality it then follows that\n|{PN∩[a,b)}|≥N/parenleftbig\nLs/parenleftbig\n[a,b)/parenrightbig\n−/hatwideDN(PN)/parenrightbig\n. (4.13)\nNow, define τ=/parenleftbig\n2/N+/hatwideDN(PN)/parenrightbig1/sand consider the box [ a,b) given by\n[aj,bj) =/braceleftBigg\n[tj,tj+τ) iftj+τ <1,\n[1−τ,1) otherwise.(4.14)\nFrom [12, Proposition 3.14], the extreme discrepancy can be bounded by the star dis-\ncrepancy:/hatwideDN(PN)≤2sD∗\nN(PN), and then due to (2.23) and (4.12) we have the upper\nbound\nτ≤/parenleftbigg2\nN+2sCPNlog(N)s−1\nN/parenrightbigg1/s\n≤3C1/s\nPNlog(N)1−1/sN−1/s<1, (4.15)\nwhere we have also used the fact that CPNlog(N)s−1>1 forNsufficiently large and\n2+2s≤3s. As such, we have [ a,b)⊂[0,1]s, witht∈[a,b), andLs([a,b)) =τs<1.\nApplying the lower bound (4.13) to the box [ a,b) defined in (4.14) gives\n|{PN∩[a,b)}|≥N/parenleftbig\nτs−/hatwideDN(PN)/parenrightbig\n=N/parenleftbig\n2/N+/hatwideDN(PN)−/hatwideDN(PN)/parenrightbig\n= 2,\nwhich implies that there are at least 2 points in the box [ a,b). By the construction of the\nbox [a,b) it then follows from (4.15) that there exists a t′∈PNsuch that t′/\\e}atio\\slash=tand\n/ba∇dblt−t′/ba∇dblℓ∞≤τ≤3C1/s\nPNlog(N)1−1/sN−1/s.\n17Since the eigenvalue and eigenfunction are analytic and thu s Lipschitz in y, we can\nnow bound how close eigenpairs corresponding to nearby QMC p oints are, explicit in N.\nProposition 4.2. LetPNbe a low-discrepancy point set, let s∈N, leth >0be sufficiently\nsmall and suppose that Assumption A1holds. Then for any t∈PNthere exists t/\\e}atio\\slash=t′∈PN\nsuch that the eigenvalue and eigenfunction satisfy\n|λh,s(t)−λh,s(t′)|/lessorsimilarlog(N)1−1/sN−1/s,and (4.16)\n/ba∇dbluh,s(t)−uh,s(t′)/ba∇dblV/lessorsimilarlog(N)1−1/sN−1/s, (4.17)\nwhere the constants are independent of t,t′,sandh.\nProof.We only prove the result for the eigenfunction. The eigenval ue result follows the\nsame argument. For hsufficiently small, the eigenfunction uhis analytic. In particular,\nuhadmits a Taylor series that converges in Vfor ally∈Ω. Hence, for any y,y′∈Ω the\nzeroth order Taylor expansion of uh(y) abouty′(see [24]) gives\nuh(y) =uh(y′)+∞/summationdisplay\nj=1(yj−y′\nj)/integraldisplay1\n0∂j\nyuh(τy)dτ.\nRearranging and taking the V-norm, this can be bounded by\n/ba∇dbluh(y)−uh(y′)/ba∇dblV≤/ba∇dbly−y′/ba∇dblℓ∞∞/summationdisplay\nj=1sup\nτ∈[0,1]/ba∇dbl∂j\nyuh(τy)/ba∇dblV\n≤/ba∇dbly−y′/ba∇dblℓ∞∞/summationdisplay\nj=1uCβmax/parenleftbig\n/ba∇dblaj/ba∇dblL∞,/ba∇dblbj/ba∇dblL∞/parenrightbig\n,\nwhere in the last inequality we have used the upper bound [19, eq. (4.4)] on the stochastic\nderivatives of uh, andCβis independent of handy. From Assumption A1.4 the sum is\nfinite, and hence uhis globally Lipschitz in ywith a constant that is independent of h.\nSince this bound holds for all y, it also holds for all ywithyj= 0 forj > s, and thus\nclearlyuh,sis also Lipschitz with a constant that is independent of sandh.\nThe Lipschitz continuity of uh,stogether with Proposition 4.1 then imply (4.17). Since\nC1/s\nPN≤max(1,CPN), the result holds with a constant independent of s.\nSuppose now that for t∈PNwe wish to compute the eigenpair ( λh,s(t),uh,s(t)) using\nthe RQ algorithm with the initial vector v0=uh,s(t′) and initial shift σ0=As(t;v0,v0),\nwheret′∈PNis the nearby QMC point from Proposition 4.2. Then, there exi sts anN\nsufficiently large, such that these starting values satisfy\n/ba∇dbluh,s(t)−v0/ba∇dblV<1,and|λ2,h,s(t)−σ0|>ρ\n2,\ni.e., distance between the initial vector and the eigenvect or to be found is less than one,\nand the initial shift is closer to λh,s(t) than to λ2,h,s(t). In particular, for any t∈PNwe\ncan choose the starting values such that this holds.\nSince the RQ algorithm converges cubically (see, e.g., [37] ) for all sufficiently close\nstarting vectors, for a fixed tolerance ε >0 it follows that the number of iterations will be\nbounded independently of the current QMC point t, if the starting vector is sufficiently\nclose. For Nsufficiently large, Proposition 4.1 implies that for each QMC point there\nis a starting vector (taken to be the eigenvector correspond ing to a nearby QMC point)\nthat is sufficiently close to the target eigenvector, with a un iform upper bound on the\n18distance (4.11). This uniform upper bound implies that for a ll QMC points the target\neigenvector and the starting vector will be sufficiently clos e, and hence that the number of\nRQ iterations is bounded independently of the QMC point. Fur thermore, as Nincreases\nthe starting vector becomes closer to the eigenvector to be f ound due to (4.17), and so the\nnumber of iterations decreases with increasing N.\n5 Numerical results\nIn this section we present numerical results for two different test problems, which demon-\nstrate the efficiency of MLQMC and also show the computational gains achieved by our\nefficient MLQMC algorithm using two-grid methods and nearby Q MC points as described\nin Section 4.3. The superiority of MLQMC for the two test prob lems is also clearly\ndemonstrated by a comparison with single level Monte Carlo ( MC), multilevel Monte\nCarlo (MLMC) and single level QMC. All tests were performed o n a single node of the\ncomputational cluster Katana at UNSW Sydney. Note also that we use “e” notation for\npowers of 10, e.g., 5e −3 = 5×10−3.\nThenumberof quadraturepoints for all methods ( NorNℓ), includingthe MC/MLMC\ntests, are chosen to be powers of 2, and for the QMC methods we u se a randomly shifted\nembedded lattice rule [9] in base 2 given by the generating ve ctorlattice-39102-1024-\n1048576.3600 from [30] with R= 8 random shifts. For base-2 embedded lattice rules,\nthe points are enumerated in blocks of powers of 2, where each subsequent block fills in\nthe gaps between the previous points and retains a lattice st ructure, see [9] for further\ndetails. The FE triangulations are uniform, with geometric ally decreasing meshwidths\ngiven by hℓ= 2−(ℓ+3),ℓ≥0. For the two-grid method, we take as the coarse meshwidth\nHℓ=h0= 2−3= 1/8, which satisfies Hℓ≤h1/4\nℓfor allℓ≤10. Note that none of our tests\nrequired a FE mesh as ��ne as h=h10= 1/4096. To choose NℓandL, we use the adaptive\nMLQMC algorithm from [21], with error tolerances ranging fr omε= 0.625,...,6.1e−5.\nFor the eigensolver we use the RQ algorithm with an absolute e rror tolerance of 5e −8,\nwhich is below the smallest error tolerance εgiven as input to our MLQMC algorithm.\nNumerical tests in [17] for almost the same EVPs, show that th e error corresponding\nto dimension truncation with s= 64 is less than 1e −5. The smallest error tolerance we\nuse below is bigger than 5e −5. Thus, for simplicity we take a single truncation dimensio n\nsℓ=s= 64 for all ℓbelow. Consequently, the “coarse” truncation dimension fo r the\ntwo-grid method is then Sℓ=S=s1/2= 8 for all ℓ.\n5.1 Problem 1\nFirst let D= (0,1)2and consider the eigenvalue problem (1.1) with b≡0,c≡1 andaas\nin (2.1) with a0= 1 orπ/√\n2 and\naj(x) =1\nj/tildewidepsin(jπx1)sin((j+1)πx2), (5.1)\nfor several different values of the decay parameter /tildewidep >1.\nTaking the L∞(D) norm of the basis functions we get /ba∇dblaj/ba∇dblL∞=j−/tildewidep, so that for all /tildewidep\nthe bounds on the coefficient are given by\namin=a0−ζ(/tildewidep)\n2andamax=a0+ζ(/tildewidep)\n2,\nwhereζis the Riemann Zeta function and thus, for /tildewidep <2, a choice of a0=π/√\n2 ensures\n1910-410-310-210-1\neps10-1100101102103104105time (s)1MC\nSLQMC\nMLMC\nMLQMC\nTG-MLQMC\n10-410-310-210-1\neps10-1100101102103104105time (s)\n1MC\nSLQMC\nMLMC\nMLQMC\nTG-MLQMC\nFigure 1: Problem 1: Complexity (measured as time in seconds ) of MC, QMC, MLMC,\nplain-vanilla MLQMC and the enhanced MLQMC method using the two-grid method and\nnearby QMC points for /tildewidep= 4/3 (left) and/tildewidep= 2 (right).\namin>0. For/tildewidep≥2 we choose a0= 1. Furthermore\n∇aj(x) =\njπ\nj/tildewidepcos(jπx1)sin((j+1)πx2)\n(j+1)π\nj/tildewidepsin(jπx1)cos((j+1)πx2)\n, (5.2)\nso that/ba∇dblaj/ba∇dblW1,∞= (j+1)π/j/tildewidep≤2πj−(/tildewidep−1). Thus, Assumption A1 holds for p >1//tildewidepand\nq >1/(/tildewidep−1).\nIn Figure 1, we plot the cost, measured as computational time in seconds, against the\ntolerance εfor our two MLQMC algorithms, benchmarked against single le vel MC and\nQMC, and against MLMC. To ensure an identical bias error for t he single- and multilevel\nmethods, the FE meshwidth for the single-level methods is ta ken to be equal to hL, the\nmeshwidth on the finest level of the multilevel methods. The d ecreasing sequence of\ntolerances εcorresponds to a reduction in the finest meshwidth hLby a factor 2 at each\nstep. The axes are in log-log scale, and as a guide the black tr iangle in the bottom left\ncorner of each plot indicates a slope of −1. As expected, the MLQMC algorithms are\nclearly superior in all cases, and for /tildewidep= 2, the MLMC and single level QMC methods\nseem to converge at the same rate of approximately −2. Also, as we expect the cost of\nthe two MLQMC algorithms grow at the same rate of roughly −1, but the enhancements\nintroduced in Algorithm 2 yields a reduction in cost by a (rou ghly) constant factor of\nabout 2. Note that for this problem the RQ algorithm requires only 3 iterations for\nalmost all cases tested, and so at best we can expect a speedup factor of 3. In almost all\nof our numerical tests using the eigenvector of a nearby QMC p oint as the starting vector\nreduced the number of RQ iterations to 2. A similar speedup by a factor of 2 was also\nobserved in [39], which recycled samples from the multigrid hierarchy within a MLQMC\nalgorithm for the elliptic source problem.\nFrom [19, Corollary 3.1], for our MLQMC algorithms we expect a rate of−1 (with a\nlog factor) when q≤2/3, or equivalently /tildewidep≥5/2. However, we observe for our MLQMC\nalgorithms are close to −1, regardless of the decay /tildewidep. A possible explanation of this is\nthat we use an off-the-shelf lattice rule that hasn’t been tail ored to this problem, and so\nwe observe nearly the optimal rate but the constant may still depend on the dimension\n(which is fixed for these experiments). For the other methods we observe the expected\nrates, with the exception of QMC, which appears to not yet be i n the asymptotic regime.\nResults for/tildewidep= 3 are very similar to those for /tildewidep= 2, and so have been omitted.\n205.2 Problem 2: Domain with interior islands\nConsider again the domain D= (0,1)2, and the subdomainconsisting of four islandsgiven\nbyDf:= [1\n8,3\n8]2∪[5\n8,7\n8]2∪[1\n8,3\n8]×[5\n8,7\n8]∪[5\n8,7\n8]×[1\n8,3\n8],see Figure 2 for a depiction. Since\nwe use uniform triangular FE meshes with hℓ= 2−ℓ+3the FE triangulation aligns with\nthe boundaries of the components Dfon all levels ℓ= 0,1,2,....\n1\n1 01\n81\n8\n3\n83\n8\n7\n87\n8\n5\n85\n8\nFigure 2: Domain Dwith four islands forming Df(in grey).\nThe coefficients are now given by\na0(x) =/braceleftBigg\nσdiff:= 0.01 ifx∈Df,\nσ′\ndiff:= 0.011 ifx∈D\\Df,aj(x) =/braceleftBigg\nσdiffw(j+1)/2(/tildewidepa;x) forjodd,\nσ′\ndiffw′\nj/2(/tildewidep′\na;x) for jeven,\nb0(x) =/braceleftBigg\nσabs:= 2 if x∈Df,\nσ′\nabs:= 0.3 ifx∈D\\Df,bj(x) =/braceleftBigg\nσabsw(j+1)/2(/tildewidepb;x) forjodd,\nσ′\nabsw′\nj/2(/tildewidep′\nb;x) for jeven,\nwhere\nwk(q;x) =/braceleftBigg\n1\nkqsin/parenleftbig\n8kπx1/parenrightbig\nsin/parenleftbig\n8(k+1)πx2/parenrightbig\nforx∈Df,\n0 for x∈D\\Df,and\nw′\nk(q;x) =/braceleftBigg\n0 for x∈Df,\n1\nkqsin/parenleftbig\n8kπx1/parenrightbig\nsin/parenleftbig\n8(k+1)πx2/parenrightbig\nforx∈D\\Df,\nand where the parameters /tildewidepa,/tildewidep′\na,/tildewidepb,/tildewidep′\nb≥4/3 give the different decays of the coefficients\non the different areas of the domain. As for Problem 1, if any of /tildewidepa,/tildewidep′\na,/tildewidepb,/tildewidep′\nbare less than\n2, then we scale the corresponding zeroth term in the coefficie nt byπ/√\n2.\nThe complexity of MLMC, MLQMC and the enhanced MLQMC using tw o-gird meth-\nods and nearby QMC points for this problem is given in Figure 3 . As expected for both\nMLQMC algorithms we observe a convergence rate of −1, and the MLMC results ap-\nproach the expected convergence rate of −2. Also, since this problem is more difficult for\neigensolvers to handle, we now observe that the two-grid MLQ MC gives a speedup by a\nfactor of more than 3. Other tests using different values of /tildewidepa,/tildewidep′\na,/tildewidepb,/tildewidep′\nbyielded similar\nresults. Note also that numerical results for single level Q MC methods applied to this\nproblem (with slightly different aj,bj) were given previously in [17].\n6 Conclusion\nWe have developed an efficient MLQMC algorithm for random elli ptic EVPs, which uses\ntwo-grid methods and nearby QMC points to obtain a speedup co mpared to an ordinary\nMLQMC implementation. We provided theoretical justificati on for the use of both strate-\ngies. Finally, we presented numerical results for two test p roblems, which validate the\n2110-510-410-3\neps100101102103104time (s)\n1MLMC\nMLQMC\nTG-MLQMC\n10-510-410-3\neps10-1100101102103104time (s)\n1MLMC\nMLQMC\nTG-MLQMC\nFigure 3: Problem 2: Complexity (measured as time in seconds ) of MLMC, plain-vanilla\nMLQMC and the enhanced MLQMC method using the two-grid metho d and nearby QMC\npoints for/tildewidepa=/tildewidepb= 4/3,/tildewidep′\na=/tildewidep′\nb= 2 (left), and /tildewidepa=/tildewidep′\na=/tildewidepb=/tildewidep′\nb= 2 (right).\ntheoretical results from the accompanying paper [19] and al so demonstrate the speedup\nobtained by our new MLQMC algorithm.\nAcknowledgements. This work is supported by the Deutsche Forschungsgemeinsch aft\n(GermanResearchFoundation)underGermany’sExcellenceS trategyEXC2181/1-390900948\n(theHeidelbergSTRUCTURESExcellenceCluster). 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Anal. , 49:1602–1624, 2011.\nA Proof of Theorem 3.1\nProof.First of all, we can use the triangle inequality to split the eigenfunction error into\n/ba∇dblu(y)−uh,s(y)/ba∇dblV≤/ba∇dblu(y)−us(y)/ba∇dblV+/ba∇dblus(y)−uh,s(y)/ba∇dblV+/ba∇dbluh,s(y)−uh,s(y)/ba∇dblV\n/lessorsimilarh+s−(1/p+1)+/ba∇dbluh,s(y)−uh,s(y)/ba∇dblV, (A.1)\nwhere we have used [17, Theorem 4.1] and (2.12), and the constant is independent of h,sandy.\nAll that remains for the eigenfunction result is to bound the third te rm above.\nTo this end, we can rewrite Step 2 of Algorithm 1 using (3.2) as\nAs(y;uh,s(y)−λH,S(y)Th,suh,s(y),vh) =As(y;Th,suH,S(y),vh) for all vh∈V ,\nwhich is equivalent to the operator equation: Find uh,s(y)∈Vhsuch that\n/parenleftbigg1\nλH,S(y)−Th,s/parenrightbigg\nuh,s(y) =1\nλH,S(y)Th,suH,S(y).\n24This is in turn equivalent (up to a constant scaling factor) to the pro blem: find/tildewideu∈Vhsuch that\n/parenleftbigg1\nλH,S(y)−Th,s/parenrightbigg\n/tildewideu=λH,S(y)Th,suH,S(y)\n/ba∇dblλH,S(y)Th,suH,S(y)/ba∇dblV=:u0. (A.2)\nExplicitly,\n/tildewideu=λH,S(y)uh,s(y)\n/ba∇dblTh,suH,S(y)/ba∇dblV,\nbut after normalisation (Step 3) uh,s(y) =/tildewideu//ba∇dbl/tildewideu/ba∇dblM.\nWe now apply Theorem 3.2 from [47] to (A.2), using the space X=Vand with\nµ0=1\nλH,S(y)andu0=λH,S(y)Th,suH,S(y)\n/ba∇dblλH,S(y)Th,suH,S(y)/ba∇dblV.\nTo do so, we must first verify that µ0andu0satisfy the required assumptions of [47, Theorem 3.2],\nnamely,/ba∇dblu0/ba∇dblV= 1,µ0is not an eigenvalue of Th,s, and for all y∈Ω\ndist(u0,Eh(λs(y))≤1\n2and (A.3)\n|µ0−µ2,h,s(y)|≥µ1,s(y)−µ2,s(y)\n2=:/tildewideρs(y)\n2. (A.4)\nRecall that µk(y) = 1/λk(y) is an eigenvalue of T, and similarly, subscripts handsdenote their\nFE and dimension-truncated counterparts, respectively. Clearly , the first two assumptions hold,\nand so it remains to verify (A.3) and (A.4).\nTo show (A.3), since λs(y) is simple we have\ndist(u0,Eh(λs(y))\n= inf\nα∈R/ba∇dblu0−αuh,s(y)/ba∇dblV\n=1\nλH,S(y)/ba∇dblTh,suH,S(y)/ba∇dblVinf\nα∈R/ba∇dblλH,S(y)Th,suH,S(y)−αuh,s(y)/ba∇dblV.(A.5)\nToshowthatthefirstfactorcanbeboundedbyaconstant,weus ethereversetriangleinequality\nalong with the lower bound (2.15), which since H >0 was assumed to be sufficiently small gives\nλH,S(y)/ba∇dblTh,suH,S(y)/ba∇dblV≥λ/vextendsingle/vextendsingle/ba∇dblTuH,S(y)/ba∇dblV−/ba∇dbl(T−Th,s)uH,S(y)/ba∇dblV/vextendsingle/vextendsingle. (A.6)\nNow, by the equivalence of norms (2.5) we have\n/ba∇dblTuH,S(y)/ba∇dblV≥1\nCA/radicalBig\nA(y;TuH,S(y),TuH,S(y)).\nThen using the definition of T, along with the facts that uH,S(y) is an eigenfunction and A(y) is\nsymmetric, we can simplify this as\nA(y;TuH,S(y),TuH,S(y)) =M(uH,S(y),TuH,S(y))\n=1\nλH,S(y)A(y;uH,S(y),TuH,S(y))\n=1\nλH,S(y)M(uH,S(y),uH,S(y))≥1\nλ,\nwhere for the last inequality we have used (2.15) and /ba∇dbluH,S(y)/ba∇dblM= 1. Hence, we have the\nconstant lower bound\n/ba∇dblTuH,S(y)/ba∇dblV≥C−1\nAλ−1/2, (A.7)\nwhich is independent of y,SandH.\nFor the second term in (A.6), by (3.7) we have the upper bound\n/ba∇dbl(T−Th,s)uH,S(y)/ba∇dbl=/ba∇dblT−Th,s/ba∇dbl/ba∇dbluH,S(y)/ba∇dblV\n≤CT(s−1/p+1+h)/radicalBig\nλH,S(y)/ba∇dbluH,S(y)/ba∇dblM\n≤λ1/2CT(s−1/p+1+h), (A.8)\n25where we have used that uH,S(y) is an eigenfunction, normalised in M, and also (2.15).\nReturning to (A.6), since /ba∇dblTuH,S(y)/ba∇dblVis bounded from below by a constant, by (A.8) there\nexistsS0∈Nsufficiently large and H0>0 sufficiently small such that for all s≥S0andh≤H0\nwe have/ba∇dblTuH,S(y)/ba∇dblV>/ba∇dbl(T−Th,s)uH,S(y)/ba∇dblV. Thus, substituting (A.7) and (A.8) into (A.6), we\nhave the lower bound\nλH,S(y)/ba∇dblTh,suH,S(y)/ba∇dblV≥C−1\nAλλ−1/2−λλ1/2CT(s−1/p+1+h)\n≥C−1\nAλλ−1/2−λλ1/2CT(S−1/p+1\n0+H0) =:1\nCu0>0,\nwhere 0< Cu0<∞is independent of s,S,h,H andy. It follows that\ndist/parenleftbig\nu0,Eh(λs(y))/parenrightbig\n≤Cu0/ba∇dbluh,s(y)−λH,S(y)Th,suH,S(y)/ba∇dblV.\nFor the second factor in (A.5), using (3.2), for all vh∈Vhwe have the identity\nAs(y;uh,s(y)−λH,STh,suH,S(y),vh) =/bracketleftbig\nλh,s(y)−λH,S(y)/bracketrightbig\nM(uH,S(y),vh)\n+λh,s(y)M(uh,s(y)−uH,S(y),vh).\nLettingvh=uh,s(y)−λH,S(y)Th,suH,S(y), then using that Asis coercive, as well as applying the\ntriangle and Cauchy–Schwarz inequalities, we have\n/ba∇dbluh,s(y)−λH,STh,suH,S(y)/ba∇dbl2\nV\n/lessorsimilar/vextendsingle/vextendsingleλh,s(y)−λH,S(y)/vextendsingle/vextendsingle/ba∇dbluH,S/ba∇dblM/ba∇dbluh,s(y)−λH,S(y)Th,suH,S(y)/ba∇dblM\n+λh,s(y)/ba∇dbluh,s(y)���uH,S(y)/ba∇dblM/ba∇dbluh,s(y)−λH,S(y)Th,suH,S(y)/ba∇dblM.\nDividing through by /ba∇dbluh,s(y)−λH,S(y)Th,suH,S(y)/ba∇dblVand applying the Poincar´ e inequality (2.7)\ngives\n/ba∇dbluh,s(y)−λH,STh,suH,S(y)/ba∇dblV/lessorsimilar/vextendsingle/vextendsingleλh,s(y)−λH,S(y)/vextendsingle/vextendsingle+λ/ba∇dbluh,s(y)−uH,S(y)/ba∇dblM,\nwhere we have also used that /ba∇dbluH,S(y)/ba∇dblM= 1 and (2.15). We can incorporate λinto the constant,\nand then split the right hand side again using the triangle inequality to g ive\n/ba∇dbluh,s(y)−λH,STh,suH,S(y)/ba∇dblV/lessorsimilar/vextendsingle/vextendsingleλ(y)−λs(y)|+|λs(y)−λh,s(y)|\n+|λ(y)−λS(y)|+|λS(y)−λH,S(y)/vextendsingle/vextendsingle+/ba∇dblu(y)−us(y)/ba∇dblV\n+/ba∇dblus(y)−uh,s(y)/ba∇dblM+/ba∇dblu(y)−uS(y)/ba∇dblV+/ba∇dbluS(y)−uH,S(y)/ba∇dblM,\nwhere we have also applied the Poincar´ e inequality (2.7) again to switc h to the V-norms for the\neigenfunction truncation errors.\nNow, each of the terms in (A.5) can be bounded by [17, Theorems 2.6 & 4.1] to give\ndist(u0,Eh(λs(y))/lessorsimilar/ba∇dbluh,s(y)−λH,STh,suH,S(y)/ba∇dblV\n/lessorsimilars−1/p+1+S−1/p+1+h2+H2, (A.9)\nwhere to bound the FE error in the M-norm we have used [17, eqn. (2.35)] with the functional\nG=M(·,us(y)−uh,s(y))//ba∇dblus(y)−uh,s(y)/ba∇dblM∈L2(D) (and similarly for uS(y)−uH,S(y)). It\nfollowsfrom(A.9)thatthereexists Ssufficientlylargeand Hsufficientlysmall—bothindependent\nofy— such that (A.3) holds.\nNext, to verify (A.4), since µ0= 1/λH,S(y) =:µH,S(y) and since the FE eigenvalues converge\nfrom above and thus µ2,h,s(y)≤µ2,s(y),\n|µ0−µ2,h,s(y)|=µH,S(y)−µ2,h,s(y)≥µH,S(y)−µ2,s(y)\n=/tildewideρs(y)−/parenleftbig\nµs(y)−µH,S(y)/parenrightbig\n. (A.10)\nNow, suppose that/parenleftbig\nµs(y)−µH,S(y)/parenrightbig\n≤0, then (A.10) simplifies to\n|µ0−µ2,h,s(y)|≥/tildewideρs(y)≥/tildewideρs(y)\n2,\n26as required. Alternatively, if/parenleftbig\nµs(y)−µH,S(y)/parenrightbig\n>0 then (A.10) becomes\n|µ0−µ2,h,s(y)|≥/tildewideρs(y)−/vextendsingle/vextendsingleµs(y)−µH,S(y)/vextendsingle/vextendsingle.\nBy the triangle inequality we can bound the second term on the right, again using the bounds\nfrom [17, Theorems 2.6 & 4.1], as well as (2.15), to give\n/vextendsingle/vextendsingleµs(y)−µH,S(y)/vextendsingle/vextendsingle≤|λ(y)−λs(y)|\nλ(y)λs(y)+|λ(y)−λS(y)|\nλ(y)λS(y)+|λS(y)−λH,S(y)|\nλS(y)λH,S(y)\n≤C\nλ2(s−1/p+1+S−1/p+1+H2).\nThe upper bound is independent of y, thus we can take Ssufficiently large and Hsufficiently\nsmall, such that, using the bound on the spectral gap in (2.4) toget her with (2.15),\n/vextendsingle/vextendsingleµs(y)−µH,S(y)/vextendsingle/vextendsingle≤1\n2ρ\nλ1λ2≤1\n2λ2,s(y)−λs(y)\nλs(y)λ2,s(y)=/tildewideρs(y)\n2. (A.11)\nThen, to show (A.4) we can substitute the bound above into (A.10).\nHence, we have verified the assumptions for [47, Theorem 3.2] for ally. Sinceλs(y),λh,s(y)\nare simple, dist( uh,s(y),/hatwideEh(λs(y)) =/ba∇dbluh,s(y)−uh,s(y)/ba∇dblVand hence, it now follows from [47,\nTheorem 3.2] that\n/ba∇dbluh,s(y)−uh,s(y)/ba∇dblV≤16\n/tildewideρs(y)|λh,s(y)−λH,S(y)|\nλh,s(y)λH,S(y)/ba∇dbluh,s(y)−λH,S(y)Th,suH,S(y)/ba∇dblV.(A.12)\nWe handle the three factors in turn. For the first factor, by the a rgument used in (A.11) we\nhave 1//tildewideρs(y)≤λ1λ2/ρ, independently of y. For the second factor, we can use the uniform lower\nbound (2.15), and then the triangle inequality to give the upper boun d\n|λh,s(y)−λH,S(y)|\nλh,s(y)λH,S(y)≤1\nλ2/parenleftBig\n|λ(y)−λs(y)|+|λs(y)−λh,s(y)|\n+|λ(y)−λS(y)|+|λS(y)−λH,S(y)|/parenrightBig\n.\nEach term above can be bounded by using one of Theorems 2.6 or 4.1 f rom [17] to give\n|λh,s(y)−λH,S(y)|\nλh,s(y)λH,S(y)/lessorsimilars−(1/p−1)+h2+S−(1/p−1)+H2, (A.13)\nwhere the constant is again independent of s,S,h,H andy.\nFinally, the third factor in (A.12) can be bounded using (A.9). Hence, substituting (A.13) and\n(A.9) into (A.12) we obtain the upper bound\n/ba∇dbluh,s(y)−uh,s(y)/ba∇dblV/lessorsimilars−2(1/p−1)+h4+S−2(1/p−1)+H4\n+2/parenleftbig\ns−(1/p−1)h2+s−(1/p−1)S−(1/p−1)+s−(1/p−1)H2+h2S−(1/p−1)+h2H2+S−(1/p−1)H2/parenrightbig\n/lessorsimilarH4+S−2(1/p−1)+H2S−(1/p−1), (A.14)\nwhere we have used the fact that s≥Sandh≤Hto obtain the last inequality. Then to give the\nerror bound (3.8), we simply substitute (A.14) into (A.1).\nThe second result (3.9) follows from Lemma 3.1, by choosing B=A(y;·,·),/tildewideB=As(y;·,·) =\nA(ys;·,·),u=u(y) andw=uh,s(y). Noting that/ba∇dbluh,s/ba∇dblM= 1 and using the definition of λh,s(y)\nin (3.5), this gives\nλh,s(y)−λ(y) =/ba∇dblu(y)−uh,s(y)/ba∇dbl2\nA(ys)−λ(y)/ba∇dblu(y)−uh,s(y)/ba∇dbl2\nA(y)\n+A/parenleftbig\ny−ys;u(y),u(y)−2uh,s(y)/parenrightbig\n/lessorsimilar/ba∇dblu(y)−uh,s(y)/ba∇dbl2\nV+A/parenleftbig\ny−ys;u(y),u(y)−2uh,s(y)/parenrightbig\n, (A.15)\n27where we simplified using the linearity of A(y) inyand used the equivalence of norms in (2.5)\nand (2.15), which both hold for all y.\nThe last term from (A.15) is bounded as follows\nA(y−ys;u(y),u(y)−2uh,s(y))\n=/integraldisplay\nD/summationdisplay\nj>s/parenleftbig\nyjaj(x)∇u(y)·∇[u(y)−2uh,s(y)]\n+yjbj(x)u(y)[u(y)−2uh,s(y)]/parenrightbig\ndx\n≤1\n2/summationdisplay\nj>s/bracketleftbig\n/ba∇dblaj/ba∇dblL∞/ba∇dblu(y)/ba∇dblV(/ba∇dblu(y)/ba∇dblV+2/ba∇dbluh,s(y)/ba∇dblV)\n+/ba∇dblbj/ba∇dblL∞/ba∇dblu(y)/ba∇dblL2(/ba∇dblu(y)/ba∇dblL2+2/ba∇dbluh,s(y)/ba∇dblL2)/bracketrightbig\n/lessorsimilar/summationdisplay\nj>smax/parenleftbig\n/ba∇dblaj/ba∇dblL∞,/ba∇dblbj/ba∇dblL∞/parenrightbig\n/lessorsimilars−(1/p−1), (A.16)\nwhere in the second last inequality we have bounded the V-norms using (2.16) and the L2-norms\nusing the equivalence to the M-norm (2.6), and then used that u(y) anduh,s(y) are normalised.\nThe tail sum in the last inequality is bounded using [33, Theorem 5.1].\nFinally, the result (3.9) is obtained by substituting (3.8) and (A.16) int o (A.15).\n28"    },    {        "title": "1803.10064v2.Dynamics_of_a_Magnetic_Needle_Magnetometer__Sensitivity_to_Landau_Lifshitz_Gilbert_Damping.pdf",        "content": "arXiv:1803.10064v2  [physics.gen-ph]  19 Oct 2018Dynamics of a Magnetic Needle Magnetometer: Sensitivity to\nLandau–Lifshitz–Gilbert Damping\nY. B. Band1,2, Y. Avishai2,3,4, Alexander Shnirman3,5,6\n1Department of Chemistry, Department of Physics,\nDepartment of Electro-Optics, and the Ilse Katz Center for N ano-Science,\nBen-Gurion University, Beer-Sheva 84105, Israel\n2New York University and the NYU-ECNU Institute of Physics at NYU Shanghai,\n3663 Zhongshan Road North, Shanghai, 200062, China\n3Department of Physics, and the Ilse Katz Center for Nano-Sci ence,\nBen-Gurion University, Beer-Sheva 84105, Israel\n4Yukawa Institute for Theoretical Physics, Kyoto, Japan\n5Institut f¨ ur Theorie der Kondensierten Materie,\nKarlsruhe Institute of Technology, D-76128 Karlsruhe, Ger many\n6Institute of Nanotechnology, Karlsruhe Institute of Techn ology, D-76344 Eggenstein-Leopoldshafen, Germany\nAn analysis of a single-domain magnetic needle (MN) in the pr esence of an external magnetic\nfieldBis carried out with the aim of achieving a high precision magn etometer. We determine the\nuncertainty ∆ Bof such a device due to Gilbert dissipation and the associate d internal magnetic\nfield fluctuations that give rise to diffusion of the MN axis dir ectionnand the needle orbital angular\nmomentum. The levitation of the MN in a magnetic trap and its s tability are also analyzed.\nA rigid single-domain magnet with large total spin,\ne.g.,S≃1012/planckover2pi1, can be used as a magnetic needle magne-\ntometer (MNM). Recently Kimball, Sushkov and Budker\n[1] predicted that the sensitivity of a precessing MNM\ncan surpass that of present state-of-the-art magnetome-\nters by orders of magnitude. This prediction motivates\nour present study of MNM dynamics in the presence of\nan external magnetic field B. Such analysis requires in-\nclusion of dissipation of spin components perpendicular\nto the easy magnetization axis (Gilbert damping). It is\ndue to interactions of the spin with internal degrees of\nfreedom such as lattice vibrations (phonons), spin waves\n(magnons), thermal electric currents, etc. [2, 3]. Once\nthere is dissipation, fluctuations are also present [6], and\nresult in a source of uncertainty that can affect the ac-\ncuracy of the magnetometer. Here we determine the un-\ncertainty in the measurement of the magnetic field by a\nMNM. We also analyze a related problem concerning the\ndynamics of the needle’s levitation in an inhomogeneous\nmagnetic field, e.g., a Ioffe-Pritchard trap [8].\nThe Hamiltonian for a MN, treated asa symmetric top\nwith body-fixed moments of inertia IX=IY≡ I ∝negationslash=IZ,\nsubject to a uniform magnetic field Bis,\nH=1\n2IˆL2+(1\n2IZ−1\n2I)ˆL2\nZ\n/bracehtipupleft /bracehtipdownright/bracehtipdownleft /bracehtipupright\nHR−(ω0//planckover2pi1)(ˆS·ˆn)2\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nHA−ˆµ·B/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\nHB,\n(1)\nwhere a hat denotes quantum operator. In the rotational\nHamiltonain HR,ˆLis the orbital angular momentum op-\nerator and ˆLZ=ˆL·ˆZis its component along the body-\nfixed symmetry axis. ˆSis the needle spin angular mo-\nmentum operator, and ˆnis the operator for nthat is the\nunit vector in the direction of the easy magnetization\naxis. The frequency appearing in the anisotropy Hamil-\ntonianHA[4] isω0= 2γ2KS/V, whereKis the strengthofthe anisotropy, Vis the needle volume, and γ=gµB//planckover2pi1\nis the gyromagnetic ratio, in which µBis the Bohr mag-\nnetron, and gis theg-factor (taken to be a scalar for\nsimplicity). In the expression for the Zeeman Hamilto-\nnianHB,ˆµ=gµBˆSis the magnetic moment operator.\nThe Heisenberg equations of motion are\n˙ˆS=−gµBB׈S+2ω0\n/planckover2pi1(ˆS׈n)(ˆS·ˆn),(2)\n˙ˆL=-2ω0\n/planckover2pi1(ˆS׈n)(S·ˆn), (3)\n˙ˆJ=−gµBB׈S, (4)\n˙ˆn=I−1\n/planckover2pi1[ˆL׈n+i/planckover2pi1ˆn], (5)\nwhereˆJ=ˆL+ˆSis the total angularmomentum operator\nandIis the moment of inertia tensor.\nThe dynamics of a MN can be treated semiclassically\nbecause Sis very large. A mean–field approximation\n[9–11] is obtained by taking quantum expectation values\nof the operator equations and assuming that for a given\noperator ˆA, the inequality/radicalBig\n∝angbracketleftˆA2∝angbracketright−∝angbracketleftˆA∝angbracketright2≪ |∝angbracketleftˆA∝angbracketright|holds,\n(an assumption warranted for large S). Hence, the ex-\npectation values of a product of operators on the RHS\nof Eqs. (2)-(5) can be replaced by a product of expecta-\ntion values. The semiclassical equations are equivalent\nto those obtained in a classical Lagrangian formulation.\nDissipation is accounted for by adding the Gilbert term\n[2, 4]−αS×(˙S//planckover2pi1−Ω×S//planckover2pi1) to the RHS of the expecta-\ntion value of Eq. (2) and subtracting it from the RHS of\nEq. (3). Here αis the dimensionless friction parameter,\nand the term Ω×Stransforms from body fixed to space\nfixed frames. Note that Gilbert damping is due to inter-\nnalforces, hence Jis not affected and Eq. (4) remains\nintact.2\nIt is useful to recast the semiclassical dynamical equa-\ntions of motion in reduced units by defining dimension-\nless vectors: the unit spin m≡S/S, the orbital angu-\nlar momentum ℓ≡L/S, the total angular momentum,\nj=m+ℓand the unit vector in the direction of the\nmagnetic field b=B/B:\n˙m=ωBm×b+ω0(m×n)(m·n)−αm×(˙m−Ω×m),(6)\n˙ℓ=−ω0(m×n)(m·n)+αm×(˙m−Ω×m),(7)\n˙n=Ω×n, (8)\n˙j=ωBm×b, (9)\nwhere the angular velocity vector Ωis given by\nΩ= (ω3−ω1)(ℓ·n)n+ω1ℓ\n= (ω3−ω1)[(j−m)·n]n+ω1(j−m).(10)\nHereωB=γ|B|is the Larmor frequency, ω1=S/IX,\nandω3=S/IZ. Similar equations were obtained in\nRef. [5], albeit assuming that the deviations of n(t) and\nm(t) frombare small. We show below that the dynam-\nics can be more complicated than simply precession of\nthe needle about the magnetic field, particularly at high\nmagnetic fields where nutation can be significant.\nFor the numerical solutions presented below we are\nguided by Ref. 1, which uses parameters for bulk cobalt,\nand take ω1= 100 s−1,ω3= 7000 s−1, anisotropy fre-\nquencyω0= 108s−1, Gilbert constant α= 0.01, tem-\nperature T= 300 K, and N=S//planckover2pi1= 1012. First, we elu-\ncidate the effects of Gilbert dissipation, and consider the\nshorttimebehaviorin aweakmagneticfield, ωB= 1s−1.\nThe initial spin direction is intentionally chosen notto be\nalong the easy magnetic axis; n(0) = (1 /2,1/√\n2,1/2),\nm(0) = (1 /√\n2,1/√\n2,0),ℓ(0) = (0 ,0,0). Figure 1(a)\nshows the fast spin dissipation as it aligns with the easy\naxis of the needle, i.e., m(t)→n(t) after a short time,\nand Fig. 1(b) shows relaxation of the oscillations in ℓ(t),\nwhileℓx(t) andℓy(t) approach finite values. Figure 1(c)\nshowsthe innerproduct m·n, which clearlytendstounity\nonthe timescaleofthe figure. Increasing αleadsto faster\ndissipation of m(t), but the short-time saturation values\nof bothm(t) andℓ(t) are almost independent of α.\nWe consider now the long time dynamics (still in\nthe weak field regime) and take the initial value of the\nspin to coincide with the easy magnetization axis, e.g.,\nm(0) =n(0) = (1 /√\n2,1/√\n2,0), with all other param-\neters unchanged. The spin versus time is plotted in\nFig. 2(a). The unit vectors m(t) andn(t) are almost\nidentical, andsincetheir z-componentisnearlyzero,they\nmove together in the x-yplane. In this weak field case,\nthe nutation is small, and the fast small-oscillations due\nto nutation are barely visible. The orbital angular mo-\nmentum dynamics is plotted in Fig. 2(b) [note the differ-\nenttimescalein (a)and(b)] andshowsthat ℓ(t) oscillates\nwith a frequency equal to that of the fast tiny-oscillation\u0001\u0001\u0001\u0002\u0001\u0003\u0001\n\u0001\u0002\u0001\u0003\u0001\u0002\u0004\u0001\u0001\u0002\u0004\u0003\u0001\u0002\u0005\u0001\u0001\u0002\u0005\u0003\u0001\u0002\u0006\u0001\u0004\u0001\u0001\u0002\n-\u0001\u0002\u0003\u0001\u0002\u0003\u0004\u0002\u0001{\u0001\u0001\u0007\u0001\u0002\u0007\u0001\u0003\u0007\u0001}\n\u0001\u0001\u0001\u0002\u0001\u0003\n\u0001\u0002\u0003\u0004\u0002\u0001\u0004\u0002\u0003\u0005\u0002\u0001\u0005\u0002\u0003\u0006\u0002\u0001\u0002\n-\u0001\u0002\u0001\u0005\u0001-\u0001\u0002\u0001\u0004\u0003-\u0001\u0002\u0001\u0004\u0001-\u0001\u0002\u0001\u0001\u0003\u0001\u0002\u0001\u0001\u0003\u0001\u0002\u0001\u0004\u0001{\u0001\u0001\u0007\u0001\u0002\u0007\u0001\u0003}\n\u0001\u0002\u0001\u0003\u0001\u0002\u0004\u0001\u0001\u0002\u0004\u0003\u0001\u0002\u0005\u0001\u0001\u0002\u0005\u0003\u0001\u0002\u0006\u0001\u0004\u0001\u0001\u0001\u0001\u0002\u0007\u0001\u0001\u0002\u0007\u0003\u0004\u0002\u0001\u0001{\u0002\u0002\u0003}\nFIG. 1: (color online) (a) The normalized spin vector mver-\nsus time for the low-field case at short times (5 orders of\nmagnitude shorter than in Fig. 2) when the initial spin is\nnot along the fast axis. (b) The reduced orbital angular mo-\nmentum vector ℓ(t). (c) The inner product m(t)·n(t) (the\nprojection of the spin on the fast magnetic axis of the needle .\nofm(t) [the oscillation amplitude is 0 .02|m(t)|]. Fig-\nure 2(c) shows a parametric plot of m(t) versus time.\nThe nutation is clearly very small; the dynamics of m(t)\nconsists almost entirely of precession at frequency ωB.\nFigure 3 shows the dynamics at high magnetic field\n(ωB= 105s−1) with all the other parametersunchanged.\nFigure 3(a) shows mversus time, and now the nutation\nis clearly significant. For the high magnetic field case,\nm(t) is also almost numerically equal to n(t).ℓ(t) is\nplotted in Fig. 3(b). Its amplitude is very large, ℓ(t)≈\n40m(t). However, its oscillation frequency is comparable\nwith that of m(t). In contrast with the results in Fig. 2,\nhere, in addition to precession of the needle, significant\nnutation is present, as shown clearly in the parametric\nplot of the needle spin vector m(t) in Fig. 3(c).\nWe now determine the uncertainty of the MNM due to\ninternal magnetic field fluctuations related to the Gilbert\ndamping. A stochastic force ξ(t), whose strength is de-\ntermined by the fluctuation–dissipation theorem [6], is3\n\u0001\u0001\u0001\u0002\u0001\u0003\u0001\n\u0001\u0002\u0003\u0004\u0005\u0006\u0002\n-\u0005\u0007\u0006-\u0006\u0007\b\u0006\u0007\b\u0005\u0007\u0006{\u0001\u0001\t\u0001\u0002\t\u0001\u0003\t\u0001}\n\u0001\u0001\u0001\u0002\u0001\u0003\n\u0001\u0002\u0003\u0004\u0002\u0001\u0004\u0002\u0003\u0005\u0002\u0001\u0005\u0002\u0003\u0006\u0002\u0001\u0002\n-\u0001\u0002\u0001\u0005\u0001-\u0001\u0002\u0001\u0004\u0003-\u0001\u0002\u0001\u0004\u0001-\u0001\u0002\u0001\u0001\u0003\u0001\u0002\u0001\u0001\u0003\u0001\u0002\u0001\u0004\u0001{\u0001\u0001\u0007\u0001\u0002\u0007\u0001\u0003}\nFIG. 2: (color online) Dynamics for thelow-field case ( ωB= 1\ns−1), over relatively long timescales relative to those in Fig. 1.\n(a)mversus time in units of seconds (note that nis indistin-\nguishable from mon the scale of the figure). (b) ℓ(t) (note\nthat it stays small compared to S). (c) Parametric plot of the\nneedle spin vector m(t) showing that nutation is almost im-\nperceptible for small fields [contrast this with the large fie ld\nresult in Fig. 3(c)]; only precession is important.\nadded to Eq. (6), in direct analogy with the treatment of\nBrownianmotionwherebothdissipationandastochastic\nforce are included [12]:\n˙m=m×(ωBb+ξ)+ω0(m×n)(m·n)\n−αm×(˙m−Ω×m). (11)\nξ(t) is internal to the needle and therefore it does not\naffect the total angular momentum jdirectly, i.e., ξ(t)\ndoes not appear in Eq. (9) [since the term −m×ξis also\nadded to the RHS of (7)]. However, as shown below, ξ(t)\naffectsℓas well as m, causing them to wobble stochas-\ntically. This, in turn, makes jstochastic as well via the\nZeeman torque [see Eq. (9)].\nThe fluctuation-dissipation theorem [6] implies\n∝angbracketleftξαξβ∝angbracketrightω≡/integraldisplay\ndt∝angbracketleftξα(t)ξβ(0)∝angbracketrighteiωt\n=δαβαωcoth(/planckover2pi1ω/2kBT)\nN≈δαβ2αkBT\n/planckover2pi1N,(12)\u0001\u0001\u0001\u0002\u0001\u0003\u0001\u0002\u0001\u0003\u0001\u0002\u0001\u0004\u0001\u0002\u0001\u0005\u0001\u0002\u0001\u0006\u0001\u0002\u0007\u0001\u0002\n-\u0007\u0002\u0001-\u0001\u0002\b\u0001\u0002\b\u0007\u0002\u0001{\u0001\u0001\t\u0001\u0002\t\u0001\u0003}\n\u0001\u0001\u0001\u0002\u0001\u0003\n\u0001\u0002\u0001\u0003\u0001\u0002\u0001\u0004\u0001\u0002\u0001\u0005\u0001\u0002\u0001\u0006\u0001\u0002\u0007\u0001\u0002\n-\u0004\u0001-\u0003\u0001\u0003\u0001\u0004\u0001{\u0001\u0001\n\b\u0001\u0002\n\u0000\u0001\u0003}\nFIG. 3: (color online) High-field case ( ωB= 105s−1). (a)\nm(t) [which is almost numerically equal to n(t)]. (b)ℓ(t)\n(note the ordinate axis scale is [ −40,40]). (c) Parametric plot\nof the needle spin vector m(t) showing that strong nutation\noccurs for large fields in addition to precession.\nwhereN=S//planckover2pi1, and the last approximation is ob-\ntained under the assumption that /planckover2pi1ω≪kBT. Note that\nEq. (11) should be solved together with Eqs. (8) and (9).\nThe presence of the anisotropy term in Eq. (11) makes\nnumerical solution difficult for large ω0. Hence, we con-\nsider a perturbative expansion in powers of λ≡ω1/ω0:\nm(t) =n0(t)+λδm(t)+...,n(t) =n0(t)+λδn(t)+...,\nj(t) =j0(t) +λδj(t) +.... Sinceω0is the largest fre-\nquency in the problem, the inequalities αω0≫ωB,ω1,ω3\nhold. Moreover, the Gilbert constant αis large enough\nto effectively pin m(t) ton(t) [hencej(t) =ℓ(t)+m(t)≈\nℓ(t)+n(t)]. Therefore, anadiabaticapproximationtothe\nset of dynamical stochastic equations can be obtained.\nThe zero order term in λreads:\n˙j0=ωBn0×b,˙n0=ω1j0��n0,(13)4\nwhereΩwas approximated by Ω0= (ω3−ω1)(j0·n0−\n1)n0+ω1(j0−n0) in Eqs. (8) and (10) in obtaining (13)\n[7]. The solution to Eqs. (13) [for times beyond which\nGilbert dissipation is significant so m(t)≈n(t)] is very\nclose to that obtained from Eqs. (6)-(8).\nExpanding Eq. (11) in powers of λand keeping only\nthe first order terms (the zeroth order term on the LHS\nvanishes since m0=n0), we get: ω1(δm−δn)×n0=\n˙n0−ωBn0×b+αn0×(˙n0−Ω0×n0)−n0×ξ. Taking\nEq. (13) into account and introducing the notation δη≡\nδm−δn, we obtain\nδη×n0=j0×n0−(ωB/ω1)n0×b−(1/ω1)n0×ξ,(14)\nand from Eqs. (8) and (9) we find\nd\ndtδj=ωB(δn+δη)×b, (15)\nd\ndtδn=ω1(j0−n0)×δn+ω1(δj−δn−δη)×n0\n=ω1j0×δn+ω1(δj−δη)×n0. (16)\nTo first order in λ,δn⊥n0(sincenmust be a unit\nvector), and δm⊥n0, henceδη⊥n0. Therefore, δη×\nb= [j0−(j0·n0)n0]×b+(ωB/ω1)[b−(b·n0)n0]×b+\nω−1\n1[ξ−(ξ·n0)n0]×bon the RHS of Eq. (15) and\nd\ndtδj=ωBδn×b+ωB[j0−(j0·n0)n0]×b\n−ω2\nB\nω1(b·n0)n0×b+ωB\nω1[ξ−(ξ·n0)n0]×b.(17)\nEquations (13), (16) and (17) form a closed system of\nstochastic differential equations [upon using Eq. (14) to\nsubstitute for δη×n0on the RHS of Eq. (16)]. With\nthe largest frequency ω0eliminated, a stable numerical\nsolution is obtained. Moreover, for small magnetic field\n(whereωBis the smallest frequency in the system), an\nanalytic solution of these equations is achievable. To ob-\ntain an analytic solution to Eqs. (13), let us transform\nto the frame rotating around Bwith frequency ωBto\nget equations of the formd\ndτv=d\ndtv+ωBb×v(which\ndefinesτ):\nd\ndτn0=−ω1n0×/parenleftbigg\nn0−j0+ωB\nω1b/parenrightbigg\n,(18)\nd\ndτj0=ωBb×/parenleftbigg\nn0−j0+ωB\nω1b/parenrightbigg\n.(19)\nIf the initial condition is n0(0)−j0(0)+(ωB/ω1)b= 0,\nthen, in the rotating frame j0(τ) andn0(τ) are constant\nvectors. Note that this initial condition is only slightly\ndifferent from the “ordinary” initial condition n0(0) =\nj0(0)since( ωB/ω1)≪1forsmallmagneticfields. Hence,\nin the rotating frame,\nd\ndτδn=ω1n0×(δn−δj+δη),(20)d\ndτδj=−ωBb×(δn−δj+δη).(21)\nWith the special initial conditionbeing satisfied, Eq. (14)\nbecomes δη×n0=−(1/ω1)n0×ξ, and Eqs. (20)-(21)\nbecome a set of first order differential equations with\ntime-independent coefficients. Their solution for initial\nconditions, δn(t= 0) = 0, δj(t= 0) = 0 is,\n/parenleftbiggδn(t)\nδj(t)/parenrightbigg\n=t/integraldisplay\n0dt1exp[C(t−t1)]C/parenleftbiggδη(t1)\n0/parenrightbigg\n,(22)\nwhere the constant matrix C=/parenleftbiggA−A\n−B B/parenrightbigg\nhas di-\nmension 6 ×6 and the 3 ×3 matrices AandBare given by\nAij=−ω1ǫijknk\n0,Bij=−ωBǫijkbk. Without loss of gen-\neralitywecanchoose n0=ˆzandb=ωB(cosθˆz+sinθˆx),\nwhereθis the angle between the easy magnetization\naxis and the magnetic field. In this basis, ∝angbracketleftδηxδηx∝angbracketrightω=\n∝angbracketleftδηyδηy∝angbracketrightω≈ω−2\n0∝angbracketleftξxξx∝angbracketrightω=ω−2\n0∝angbracketleftξyξy∝angbracketrightω=Sa(ω), and\n∝angbracketleftδηzδηz∝angbracketrightω= 0. Here ∝angbracketleftxx∝angbracketrightω≡/integraltext\ndteiωt∝angbracketleftx(t)x(0)∝angbracketrightand [see\nEq. (12)] Sa(ω) =αωcoth(/planckover2pi1ω/2kBT)\nω2\n0N≈2αkBT\nN/planckover2pi1ω2\n0.\nWe are particularly interested in the quantities\n∝angbracketleftδn2\ny(t)∝angbracketright ≡ ∝angbracketleftδny(t)δny(t)∝angbracketrightand∝angbracketleftδj2\ny(t)∝angbracketright ≡ ∝angbracketleftδjy(t)δjy(t)∝angbracketright\nbecause, in the basis chosen above, the y-axis is the di-\nrection of precession of n0aroundb. Using Eq. (22) we\nobtain∝angbracketleftδn2\ny(t)∝angbracketright ≈tω2\n1Sa(ω∼ω1). Assuming the pre-\ncession of nis measured, [or equivalently, the precession\nofm, since they differ only for short timescales of or-\nder (αω0)−1], the uncertainty in the precession angle is\n∝angbracketleft(∆ϕ)2∝angbracketright ≈tω2\n1Sa(ω∼ω1). We thus arrive at our central\nresult: the precision with which the precession frequency\ncan be measured is, ∆ ωB=√\n/angbracketleft(∆ϕ)2/angbracketright\nt≈ω1\nω0/radicalBig\n2αkBT\n/planckover2pi1N1√\nt.\nEquivalently, the magnetic field precision is,\n∆B=∆ωB\nγ≈/planckover2pi1\ngµBω1\nω0/radicalbigg\n2αkBT\n/planckover2pi1N1√\nt.(23)\nFor the parameters used in this paper we find ∆ B≈\n5×10−18√\nt[s]Tesla (independent of ωB). This result should\nbe compared with the scaling ∆ B∝t−3/2obtained in\nRef. 1. Therein, the initial uncertainty of the spin di-\nrection relative to the needle axis was estimated from\nthe fluctuation-dissipation relation and the deterministic\nprecession resulted in the t−3/2scaling of the precession\nangle uncertainty (in addition this angle was assumed to\nbe small). In contrast, we consider the uncertainty ac-\nquired due to Gilbert dissipation duringthe precession,\nallowing the precession angle to be large. Thus, the stan-\ndard1/√\ntdiffusion scalingis obtained and dominates for\ntimes that are even much longer than those considered\nin Ref. 1.\nIntheSupplementalMaterial[13]wediscussthreerele-\nvant related issues. (a) The time at which diffusion stops\nbecause equipartition is reached (we estimate the time5\nwhen the energy stored in stochastic orbital motion be-\ncomes of order kBT). (b) The uncertainty of the mag-\nnetic field for experiments in which the fast precession of\nnaroundjis averaged out in the measurement, and the\ndiffusion of jdetermines ∆ B. (c) We consider the related\nproblem of the dynamics and stability of a rotating MN\nin an inhomogeneous field (e.g., levitron dynamics in a\nIoffe-Pritchard trap [14, 15]).\nIn conclusion, we show that ∆ Bdue to Gilbert damp-\ning is very small; external noise sources, as discussed in\nRef. [1], will dominate over the Gilbert noise for weak\nmagnetic fields. A closed system of stochastic differen-\ntial equations, (13), (16) and (17), can be used to model\nthe dynamics and estimate ∆ Bfor large magnetic fields.\nA rotating MN in a magnetic trap can experience levi-\ntation, although the motion does not converge to a fixed\npoint or a limit cycle; an adiabatic–invariant stability\nanalysis confirms stability [13].\nThis work was supported in part by grants from the\nDFG through the DIP program (FO703/2-1). Useful\ndiscussions with Professor Dmitry Budker are gratefully\nacknowledged. A. S. was supported by DFG Research\nGrant No. SH 81/3-1.\n[1] D. F. J. Kimball, A. O. Sushkov, and D. Budker, Phys.\nRev. Lett. 116, 190801 (2016).\n[2] T. L. Gilbert, IEEE Transactions on Magnetics 40, 3443\n(2004)\n[3] L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjet. 8153\n(1935). In L. D. Landau, Collected Papers. Ed. by D. ter\nHaar, (Gordon and Breach, New York, 1967), p. 101.\n[4] W. F. Brown Jr., Phys. Rev. 130, 1677 (1963).\n[5] H. Keshtgar, et al., Phys. Rev. B 95, 134447 (2017).\n[6] H. B. Callen and T. A. Welton, Phys. Rev. 83, 34 (1951).\n[7] We note in passing that Eqs. (13) are equivalent to the\nequations of motion of a symmetric top in a gravita-\ntional field when the top is anchored at a point a=an\non its axis a distance afrom the center of mass. The\nequations of motion are: dL/dt=T, where Land\nT=an×(−mgz) are taken with respect to the fixedpoint, and dn/dt=Ω×n. The angular velocity is given\nbyΩ=I−1\n1[L−(L·n)n] +I−1\n3(L·n)n, where the mo-\nments of inertia ( I1,I1,I3) are calculated relative to the\nfixed point. Introducing a characteristic scale L0so that\nL=L0j(jis not a unit vector and its length is not\nconserved) we obtain Eqs. (13) with ωB=mga/L 0and\nω1=L0/I1. Here, the analog of the magnetic field is the\ngravitational field and the analog of bisz.\n[8] S. Gov, S. Shtrikman, and H. Thomas, J. Appl. Phys. 87,\n3989 (2000), and references therein.\n[9] O. Zobay and B. M. Garraway, Phys. Rev. A 61, 033603\n(2000); J. Liu, L. Fu, B.-Y. Ou, S.-G. Chen, D.-I. Choi,\nB. Wu, and Q. Niu, Phys. Rev. A 66, 023404 (2002).\n[10] Y. B. Band, I. Tikhonenkov, E. Pazyy, M. Fleischhauer,\nand A. Vardi, J. of Modern Optics 54, 697-706 (2007).\n[11] Y. B. Band, Phys. Rev. E 88, 022127 (2013); Y. B. Band\nand Y. Ben-Shimol, Phys. Rev. E 88, 042149 (2013).\n[12] H. P. Breuer and F. Petruccione, The Theory of\nOpen Quantum Systems (Oxford University, Cambridge,\n2002); M. Schlosshauer, Decoherence and the Quantum-\nto-Classical Transition (Springer, Berlin, 2007).\n[13] See Supplemental Material at\nhttp://link.aps.org/supplemental/10.1103/PhysRevLet t.121.160801\nwhich contains a discussion of the three issues enumer-\nated in the text, and which includes Refs. 16-20.\n[14] M. V. Berry, Proc. R. Soc. A 452, 1207 (1996).\n[15] A movie showing the dynamics of a Levitron can be seen\nathttps://www.youtube.com/watch?v=wyTAPW_dMfo .\n[16] Y. B. Band, Y. Avishai, A. Shnirman, “Dynamics of a\nMagnetic Needle Magnetometer: Sensitivity to Landau–\nLifshitz–Gilbert Damping”, Phys. Rev. Lett. (to be pub-\nlished).\n[17] S. Gov, S. Shtrikman, and H. Thomas, J. Appl. Phys. 87,\n3989 (2000), and references therein; D. E. Pritchard,\nPhys. Rev. Lett. 51, 15 (1983).\n[18] C. C.Rusconi, V.P¨ ochhacker, K.Kustura, J.I.Ciracan d\nO. Romero-Isart, Phys. Rev. Lett. 119, 167202 (2017);\nC. C. Rusconi, V. P¨ ochhacker, K. Kustura, J. I. Cirac\nand O. Romero-Isart, Phys. Rev B 96, 134419 (2017); C.\nC. Rusconi and O. Romero-Isart, Phys. Rev B 93, 054427\n(2016).\n[19] S. Earnshaw, Trans. Camb. Phil. Soc. 7, 97-112 (1842).\n[20] D. R. Merkin, Introduction to the Theory of Stability ,\n(Springer–Verlag, New York, 1997); F. Verhulst, Non-\nlinear Differential Equations and Dynamical Systems ,\n(Springer–Verlag, Berlin, 1990).arXiv:1803.10064v2  [physics.gen-ph]  19 Oct 2018Supplemental Material for “Dynamics of a Magnetic Needle Ma gnetometer:\nSensitivity to Landau–Lifshitz–Gilbert Damping”\nY. B. Band1,2, Y. Avishai2,3,4, Alexander Shnirman3,5,6\n1Department of Chemistry, Department of Physics,\nDepartment of Electro-Optics, and the Ilse Katz Center for N ano-Science,\nBen-Gurion University, Beer-Sheva 84105, Israel\n2New York University and the NYU-ECNU Institute of Physics at NYU Shanghai,\n3663 Zhongshan Road North, Shanghai, 200062, China\n3Department of Physics, and the Ilse Katz Center for Nano-Sci ence,\nBen-Gurion University, Beer-Sheva 84105, Israel\n4Yukawa Institute for Theoretical Physics, Kyoto, Japan\n5Institut f¨ ur Theorie der Kondensierten Materie,\nKarlsruhe Institute of Technology, D-76128 Karlsruhe, Ger many\n6Institute of Nanotechnology, Karlsruhe Institute of Techn ology, D-76344 Eggenstein-Leopoldshafen, Germany\nIn this supplemental material we expand the discussion of the main t ext [1] and address the following three issues.\n(a) The time τeat which the diffusion of the magnetic needle axis direction nand the magnetic needle orbital\nangular momentum ℓstops because equipartition is reached, i.e., we estimate the time req uired for the energy stored\nin stochastic orbital motion to become of order kBT. (b) The uncertainty ∆ Bof the magnetic field for experiments\nin which the fast precession of naroundjis averaged out in the measurement process and the uncertainty ∆ Bis\ndetermined by the diffusion of j. (c) The dynamics of a magnetic needle in an inhomogeneous field, e.g., levitron\ndynamics of a rotating magnetic needle in a Ioffe-Pritchard trap [2], s ee Refs. [3–5].\n(a):τecan be estimated by noting that the diffusion determined in [1] stops once equipartition is reached. The\nenergy ∆ Estored in stochastic orbital motion is given by\n∆E∼/planckover2pi1ω1N/angbracketleftδℓ2/angbracketright, (1)\nwhere where N=S//planckover2pi1(note that δj−δn=δℓ). By requiring ∆ E∼kBTwe can estimate that the diffusion given\nby Eqs. (20-21) of [1] stops when τe∼ω2\n0/(αω3\n1) (this result can also be obtained by expanding Eq. (11) further in\npowers of λ≡ω1/ω0). For the parameters used in [1] this is an extremely long time ( τe∼1012s∼5 years). Hence,\nwe conclude that the diffusion of Eqs. (20-21) and the error estima tes given for ∆ Bin Ref. [1] are relevant for all\nreasonable times.\n(b): In [1] we calculate ∆ Bassuming the experimental measurement follows the temporal dyn amics of nandj.\nAn alternative assumption is that the precession of naroundjis averaged out by the measurement process and one\nmeasures the diffusion of j. For the latter we obtain the leading term\n/angbracketleftδj2\ny(t)/angbracketright ≈tω2\nBcos2θSa(ω∼ω1), (2)\nwhereSa(ω) is given in Eq. (23) of [1]. At θ=π/2 the leading contribution obtained in Eq. (2) vanishes and the\nremaining sub-leading term is\n/angbracketleftδn2\ny(t)/angbracketright ≈t2ω4\nB\nω2\n1Sa(ω∼ω1), (3)\nhence for θ/negationslash=π/2 we obtain\n∆B=∆ωB\nγ≈/planckover2pi1\ngµBωB\nω0cosθ/radicalbigg\n2αkBT\n/planckover2pi1N1√\nt, (4)\nwhereas at θ=π/2,\n∆B=∆ωB\nγ≈/planckover2pi1\ngµBω2\nB\nω0ω1/radicalbigg\n4αkBT\n/planckover2pi1N1√\nt. (5)\nTakingωB= 1s−1we obtain ∆ B≈cosθ×5×10−23√\nt[s]Tesla for θ/negationslash=π/2, and ∆ B≈7×10−25√\nt[s]Tesla for θ=π/2.2\n(c): A rotating magnet can be levitated in an inhomogeneous magnet ic field [3–5]. This is possible despite Earn-\nshaw’s theorem [6] from which one can conclude that levitation of a non-rotating ferromagnetin a static magnetic field\nis not possible. Two important factors regarding magnetic levitation are the forces on the magnet and its stability\n(ensuring that it does not spontaneously slide or flip into a configura tion without lift). The dynamics of a magnetic\nneedle in an inhomogeneous magnetic field can be modelled using Eqs. (6 ), (7) and (8) of [1] augmented by the\nequations of motion for the center of mass (CM) degrees of freed om of the needle,\n˙p=∇(µ·B(r)), (6)\n˙r=p/m , (7)\nwhererandpare the needle CM position and momentum vectors. Our numerical re sults show levitation of the\nmagnetic needle when the initial rotational angular momentum vecto r of the needle is sufficiently large and points\nin the direction of magnetic field at the center of the trap. We shall s ee that the dynamical variables do not evolve\nto a fixed point or a simple cyclic orbit. Moreover, a linear stability analy sis yields a 15 ×15 Jacobian matrix with\neigenvalues having a positive real part, so the system is unstable. However, a stability analysis of the system using\nthe adiabatic invariant |µ||B|[3] does yield a stable fixed point (contrary to the full numerical re sults which show a\nmore complicated levitation dynamics).\nFigure 1 shows the dynamics of the system over time in the trap. We u se the same magnetic needle parameters\nused in Fig. 2 of [1] and a Ioffe-Prichard magnetic field [2]\nB(r) =ex/parenleftbigg\nB′x−B′′\n2xz/parenrightbigg\n+ey/parenleftbigg\nB′y−B′′\n2zy/parenrightbigg\n+ez/parenleftbigg\nB0+B′′\n2(z2−x2+y2\n2)/parenrightbigg\n, (8)\nwith field bias B0, gradient B′, and curvature B′′parameters chosen so that the Zeeman energy and its variation ov er\nthe trajectory of the needle in the trap are substantial (as is clea r from the results shown in the figure). We start\nthe dynamics with initial conditions: r(0) = (0,0,0),p(0) = (0,0,0),m(0) = (0,0.0011/2,−(1−0.001)1/2) (almost\nalong the −zdirection), n(0) =m(0),ℓ(0) = (0 ,0,0.001) [this is large orbital angular momentum since ℓis the\norbital angular momentum divided by S]. Figure 1(a) shows the needle CM position r(t) versus time. Fast and slow\noscillations are seen in the xandymotion, whereas z(t) remains very close to zero. Figure 1(b) shows oscillations of\nthe CM momentum p(t) with time. px(t) andpy(t) oscillate with time, and pz(t) remains zero. Figure 1(c) plots the\nspinm(t) versus time. Initially, m(0) points almost in the −zdirection, and the tip of the needle n(t) =m(t) carries\nout nearly circular motion in the nx-nyplane. Figure 1(d) plots the orbital angular momentum ℓ(t). The components\nℓx(t) andℓy(t) undergo a complicated oscillatory motion in the ℓx(t)-ℓy(t) plane but ℓz(t)≈ℓz(0). Figure 1(e) is a\nparametric plot of m(t); the motion consists of almost concentric rings that are slightly dis placed one from the other.\nThe full dynamics show levitation but they do not converge to a fixed point or a limit cycle.\nQuite generally, for a system of dynamical equations, ˙ yi(t) =fi(y1,...,y n),i= 1,...n, a linear stability analysis\nrequires calculating the eigenvalues of the Jacobian matrix evaluate d at the equilibrium point y∗wheref(y∗) =0,\nJij=/parenleftBig\n∂fi\n∂yj/parenrightBig\ny∗[7]. The system is unstable against fluctuations if any of the eigenvalu es ofJijhave a positive real\npart. Equations (6), (7) and (8) of [1] together with Eqs. (6) and (7) above have a Jacobian matrix with eigenvalues\nwhose real part are positive, so the linear stability test fails. Howev er, if the Zeeman force −∇HZin Eq. (6) is\nreplaced by the gradient of the adiabatic invariant, µ·∇|B(r)|, none of the eigenvalues of the Jacobian matrix have\na positive real part and the system is linearly stable, i.e., the stability a nalysis using the adiabatic-invariant predicts\nstability. Note that substituting the adiabatic invariant for the Zee man energy in the full equations of motion yields\nr(t) andp(t) vectors that are constant with time and n(t),m(t) andℓ(t) are similar to the results obtained with\nthe full equations of motion (but the parametric plot of m(t) is a perfectly circular orbit). Thus, adiabatic–invariant\nstability analysis of a rotating magnetic needle in a magnetic trap confi rms stability of its levitation as obtained in\nthe numerical solution of the dynamical equations.3\n\u0001\u0002\u0001\u0003\u0001\u0001\u0003\u0002\u0001\u0004\u0001\u0001-\u0001\u0005\u0001\u0001\u0004-\u0001\u0005\u0001\u0001\u0003\u0001\u0005\u0001\u0001\u0001\u0001\u0005\u0001\u0001\u0003\u0001\u0005\u0001\u0001\u0004\n\u0001{\u0001\u0001\u0002\u0001\u0003}\u0002\u0003\u0004\n\u0001\u0002\u0001\u0003\u0001\u0001\u0003\u0002\u0001\u0004\u0001\u0001-\u0001\u0005\u0003\u0001-\u0001\u0005\u0001\u0002\u0001\u0005\u0001\u0001\u0001\u0005\u0001\u0002\u0001\u0005\u0003\u0001\n\u0001{\u0001\u0001\u0001\u0001\u0002\u0001\u0001\u0003}\u0002\u0001\u0002\u0002\u0002\u0003\n\u0001\u0002\u0001\u0003\u0001\u0001\u0003\u0002\u0001\u0004\u0001\u0001-\u0003\u0005\u0001-\u0001\u0005\u0006-\u0001\u0005\u0007-\u0001\u0005\b-\u0001\u0005\u0004\u0001\u0005\u0001\n\u0001{\u0001\u0001\u0001\u0001\u0002\u0001\u0001\u0003}\u0002\u0001\u0002\u0002\u0002\u0003\n\u0001\u0002\u0001\u0003\u0001\u0001\u0003\u0002\u0001\u0004\u0001\u0001-\u0001\u0005\u0001\u0001\u0001\u0006-\u0001\u0005\u0001\u0001\u0001\u0004\u0001\u0005\u0001\u0001\u0001\u0001\u0001\u0005\u0001\u0001\u0001\u0004\u0001\u0005\u0001\u0001\u0001\u0006\u0001\u0005\u0001\u0001\u0001\u0007\u0001\u0005\u0001\u0001\u0001\b\u0001\u0005\u0001\u0001\u0003\u0001\n\u0001{\u0001\u0001\u0001\u0001\u0002\u0001\u0001\u0003}\u0002\u0001\u0002\u0002\u0002\u0003\nFIG. 1: (color online) Dynamics of a needle in a Ioffe-Pritcha rd magnetic field. (a) rversus time, (b) pversus time, (c) m\nversus time (note that n(t) is indistinguishable from m(t) on the scale of the figure). (d) ℓversus time (note that |ℓ(t)|is small\ncompared to Sbut rotational angular momentum L(t) =Sℓ(t) is large since S= 1012). (e) Parametric plot of the needle spin\nvectorm(t) (nutation is very small for this case of small magnetic field ).4\n[1] Y. B. Band, Y. Avishai, A. Shnirman, “Dynamics of a Magnet ic Needle Magnetometer: Sensitivity to Landau–Lifshitz–\nGilbert Damping”, Phys. Rev. Lett. (to be published).\n[2] S. Gov, S. Shtrikman, and H. Thomas, J. Appl. Phys. 87, 3989 (2000), and references therein; D. E. Pritchard,\nPhys. Rev. Lett. 51, 15 (1983).\n[3] M. V. Berry, Proc. R. Soc. A 452, 1207 (1996).\n[4] A movie a a Levitron can be seen at https://www.youtube.com/watch?v=wyTAPW_dMfo .\n[5] C. C. Rusconi, V. P¨ ochhacker, K. Kustura, J. I. Cirac and O. Romero-Isart, Phys. Rev. Lett. 119, 167202 (2017); C. C.\nRusconi, V. P¨ ochhacker, K. Kustura, J. I. Cirac and O. Romer o-Isart, Phys. Rev B 96, 134419 (2017); C. C. Rusconi and\nO. Romero-Isart, Phys. Rev B 93, 054427 (2016).\n[6] S. Earnshaw, Trans. Camb. Phil. Soc. 7, 97-112 (1842).\n[7] D. R. Merkin, Introduction to the Theory of Stability , (Springer–Verlag, New York, 1997); F. Verhulst, Nonlinear Differential\nEquations and Dynamical Systems , (Springer–Verlag, Berlin, 1990)."    },    {        "title": "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": "2303.07025v2.Experimental_investigation_of_the_effect_of_topological_insulator_on_the_magnetization_dynamics_of_ferromagnetic_metal___BiSbTe__1_5_Se__1_5___and__Ni__80_Fe__20___heterostructure.pdf",        "content": "Experimental investigation of the effect of topological insulator on the magnetization\ndynamics of ferromagnetic metal: BiSbTe 1.5Se1.5andNi80Fe20heterostructure\nSayani Pal, Soumik Aon, Subhadip Manna, Sambhu G Nath, Kanav Sharma & Chiranjib Mitra∗\nIndian Institute of Science Education and Research Kolkata,\nMohanpur 741246, West Bengal, India\n(Dated: November 27, 2023)\nWe have studied the spin-pumping phenomenon in ferromagnetic metal( Ni80Fe20)/topological\ninsulator( BiSbTe 1.5Se1.5) bilayer system to understand magnetization dynamics of ferromagnetic\nmetal (FM) in contact with a topological insulator (TI). TIs embody a spin-momentum-locked\nsurface state that spans the bulk band gap. Due to this special spin texture of the topological surface\nstate, the spin-charge interconversion efficiency of TI is even higher than that of heavy metals. We\nevaluated the parameters like effective damping coefficient ( αeff), spin-mixing conductance ( g↑↓\neff)\nand spin current density ( j0\nS) to demonstrate an efficient spin transfer in Ni80Fe20/BiSbTe 1.5Se1.5\nheterostructure. To probe the effect of the topological surface state, a systematic low-temperature\nstudy is crucial as the surface state of TI dominates at lower temperatures. The exponential increase\nof ∆Hfor all different thickness combinations of FM/TI bilayers and the enhancement of effective\ndamping coefficient ( αeff) with lowering temperature confirms that the spin chemical potential bias\ngenerated from spin-pumping induces spin current into the TI surface state. Furthermore, low-\ntemperature measurements of effective magnetization (4 πMeff) and magnetic anisotropy field ( Hk)\nshowed anomaly around the same temperature region where the resistivity of TI starts showing\nmetallic behavior due to the dominance of conducting TI surface state. The anomaly in Hkcan\nresult from the emerging exchange coupling between the TI surface state and the local moments of\nthe FM layer at the interface without any long-range ferromagnetic order in TI at the interface.\nINTRODUCTION\nSpintronics is one of the emerging fields that has\nwitnessed remarkable progress on both fundamen-\ntal and technological fronts over the past couple of\ndecades. Phenomena like spin-orbit torque [1], spin\nHall effect[2], giant magnetoresistance [3], tunnelling\nmagnetoresistance [4], domain wall motion [5] provide\nbasics for applications in memory devices[6], storage\ntechnology[7], logic gates [8] and magnetic sensors [9].\nThese devices utilize the spin degrees of freedom of\nelectrons and their interaction with orbital moments\nthrough spin-orbit coupling. Complete knowledge of the\nprocess of generation, manipulation, and detection of\nspin degrees of freedom or the spin current is essential for\nwidespread applications in this field. If one focuses on\nthe currently available spin current generation processes,\nspin-pumping [10, 11] is one of the most efficient methods\nwhere the precessing magnetization in the ferromagnet\n(FM) injects spin current into the adjacent layer by\ntransferring spin angular momentum. This raises a\nneed to study the effect of spin pumping with special\nemphasis on exploring new materials which can give\nrise to significant spin-charge interconversion efficiency.\nTopological insulators (TI) are a new class of materials\nthat have an interesting spin texture of the surface\nstate, owing to spin-momentum locking[14–17]. The\nmomentum direction of the electron in the surface state\nof TI is perpendicularly locked to its spin polarization\n∗Corresponding author:chiranjib@iiserkol.ac.indirection. Thus the spin-charge interconversion for TI\nis even higher than the heavy metals which makes TIs\nsuitable for spintronics application[12]. As surface states\nare robust against deposition of FM layers on top of TI\n[18], the topological surface states remain intact and\ngapless [19]. TI/FM bilayers have been successfully\nused for the spin current generation in spin-pumping\nexperiments[20, 34–37]. The effect of spin pumping can\nbe witnessed in the enhanced damping coefficient ( αeff)\nvalue of the ferromagnet upon excitations of ferromag-\nnetic resonance (FMR) because, in the spin pumping\nprocess, the net transfer of spin angular momentum into\nTI layer brings about an additional damping torque on\nthe precessing magnetization in the FM. It is difficult\nto fabricate a perfect TI thin film where the bulk state\nof TI is completely insulating. Thus for a complete\nunderstanding of the effect of TI surface state on FM\nmagnetization dynamics, the low-temperature study is\nnecessary where the surface states of TI dominate.\nIn this paper, we present the study of the spin-pumping\nphenomenon in ferromagnetic metal (FM)/ topological\ninsulator (TI) bilayer system. We chose Ni80Fe20\nas the FM layer and BiSbTe 1.5Se1.5as the TI layer.\nCurrently, BiSbTe 1.5Se1.5is one of the best 3D TI\nmaterials in which bulk conduction in thin films is\nnegligible even at room temperature and the dominance\nof surface state is very prominent at lower temperatures\n[21–23]. In our low-temperature measurements, we have\nwitnessed exponential enhancement of FMR linewidth\n(∆H) and effective damping coefficient ( αeff) at lower\ntemperatures. It supports the proposal of the spin\nchemical potential bias induced spin current injectionarXiv:2303.07025v2  [cond-mat.mes-hall]  24 Nov 20232\ninto the surface state of TI given by Abdulahad et al. [50].\nFor further investigation of the effect of the TI surface\nstate on the FM magnetization, we have also studied\nlow-temperature variations of effective magnetization\nand anisotropy field. We calculated the interfacial\nmagnetic anisotropy of the bilayer to be in-plane of the\ninterface. At low temperatures, this magnetic anisotropy\nfield shows a hump-like feature concomitant with the\nresistivity behavior of BiSbTe 1.5Se1.5with temperature.\nIt predicts the existence of exchange coupling between\nthe surface states of TI and the local moments of the FM\nlayer which acts perpendicular to the TI/FM interface.\nWe have also evaluated the values of spin-transport\nparameters like spin-mixing conductance, g↑↓\neffand spin\ncurrent density, j0\nsat room temperature to ensure a suc-\ncessful spin injection into the TI layer from the FM layer.\nSAMPLE PREPARATION AND\nCHARACTERIZATION\nFor this particular work, we have prepared dif-\nferent thickness combinations of topological insu-\nlator(TI)/ferromagnet(FM) bilayer heterostructure.\nBiSbTe 1.5Se1.5(BSTS ) has been taken as the TI\nmaterial and Permalloy( Ni80Fe20) has been used as\nthe ferromagnetic material. BSTS thin films were\ngrown on silicon (Si 111) substrate using pulsed laser\ndeposition(PLD) technique [24, 25]. The target material\nwas prepared using 99 .999% pure Bi, Sb, Te, and Se in a\n1:1:1.5:1.5 stoichiometric ratio. The films were deposited\nthrough ablation of the target by a KrF excimer laser\n(248 nm, 25 ns pulse width) at a low repetition rate of\n1Hz and 1 .2Jcm−2laser fluence keeping the substrate\ntemperature fixed at 2500Cand the chamber partial\npressure at 0.5 mbar (base pressure 2 ×10−5mbar)\nwith a continuous flow of Ar gas. After deposition,\nTI films were immediately transferred into the thermal\nevaporation chamber for the deposition of the FM\nlayer. Commercially available 99 .995% pure permalloy\n(Ni80Fe20) pallets were used for deposition. The Py\nfilm was deposited [26] on top of TI film at a rate of\n1.2˚A(crystal monitor: Inficon SQM 160) keeping the\nchamber pressure fixed at 1 ×10−6torr (base pressure\n1×10−7torr). For the characterization of the films\nX-ray diffraction analysis (XRD), field emission scanning\nelectron microscope (FE-SEM) imaging, and atomic\nforce microscopy (AFM) facilities have been used. X-ray\nreflectometry technique has been used for thickness\nmeasurements here. For convenience we are defining the\nBSTS of different thicknesses as follows: 10nm BSTS as\nBSTS1, 21nm BSTS as BSTS2, 28nm BSTS as BSTS3,\nand 37nm BSTS as BSTS4.RESULTS AND DISCUSSION\nFor a systematic study of the FM/TI bilayer system,\nwe have done in-plane FMR measurements in reflection\nmode geometry using a short-circuited CPW as shown\nin fig.1a. We obtained typical FMR signal at different\nmicrowave frequencies for Py(15nm)/BSTS2 sample in\nfig.1b. From the Lorentz formula fitting [53] of the FMR\nsignal we extracted the frequency dependence of the field\nlinewidth (∆ Hvs.f) and the resonance frequency vs.\nresonance field ( fvsH) data as shown in fig.2a and fig.2b\nrespectively. These give us valuable information about\nthe magnetization dynamics in ferromagnet which can\nbe described within the framework proposed by Landau,\nLifshitz, and Gilbert [30],\nd⃗M\ndt=−γ⃗M×⃗Heff+αeff\nMS⃗M×d⃗M\ndt(1)\nwhere, γis the gyromagnetic ratio, ⃗Mis the magneti-\nzation vector, MSis the saturation magnetization, Heff\nis the effective magnetic field which includes the exter-\nnal field, demagnetization and crystalline anisotropy field\nandαeffis the effective damping coefficient of the sys-\ntem.\nFor a given magnetic material at ferromagnetic res-\nonance, the resonance field and frequency follow Kittel\nequation[27] given by,\nf=γ\n2πq\n(H+Hk)(H+Hk+ 4πMeff) (2)\nwhere H,Hk, and 4 πMeffare the externally ap-\nplied field, magnetic anisotropy field, and effective mag-\nnetization respectively. We have obtained Hkand\n4πMefffor different FM/TI bilayer systems by fitting\nthe Kittel equation to the fvs. Hcurve as shown\nin fig.2b. The obtained 4 πMeffvalue contains satura-\ntion magnetization(4 πMs) and other anisotropic contri-\nbutions. We can evaluate 4 πMsvalue by analyzing the\nthickness dependent measurement of 4 πMeffof the FM\nlayer. In the lower thickness region of the ferromagnetic\nthin films, 4 πMeffis inversely proportional to the film\nthickness and follows the equation[28],\n4πMeff= 4πMs−2Ks\nMsd(3)\nwhere Ksis the surface anisotropy constant and dis the\nthickness of the FM film. The slope of the linear fit\ngives the anisotropy field contribution to 4 πMeffand\nthe intercept gives the 4 πMsvalue as shown in fig.2c.\nThe 4 πMeffdoes not depend on the thickness varia-\ntion of BSTS at room temperature but 4 πMefffor Py(t)\nmonolayer samples and for Py(t)/BSTS2 bilayer sam-\nples vary linearly with the inverse Py thickness as shown\nin Fig.2c. From the linear fitting (Eq.3) of 4 πMeff3\n(a)\n (b)\nFIG. 1. (a) In the left diagram, a schematic illustration of the experimental set-up has shown where the FM/TI bilayer is\nplaced upside down on top of a CPW, and in the right diagram, net injected spin current ( Ipump\nS ) due to spin-pumping into\nthe TI layer (BSTS) from the FM layer (Py) has shown, it results faster magnetization relaxation in FM; (b) Ferromagnetic\nResonance spectra of absorption at different frequencies for Py/BSTS bilayer system at room temperature after background\nsubtraction.\n(a)\n (b)\n (c)\nFIG. 2. (a) Field linewidth (∆ H) variation with resonance frequencies ( f) at 300K for Py/BSTS bilayer samples with different\nPy thicknesses. Eq.4 has been used for fitting the curve and to determine the damping coefficient ;(b) Resonance field ( H) vs.\nresonance frequency ( f) for Py(20nm)/BSTS2 system at different temperatures . Eq.2 has been used for fitting the curve and\nto determine the effective magnetization; (c) Effective magnetization (4 πMeff) variation with thickness of Py(t), Py(t)/BSTS2\nand Py(15nm/BSTS(t) at room temperature. Eq3 has been used for fitting the curve and to evaluate saturation magnetization\n(4πMS) and magnetic anisotropy field( Hk).\n(a)\n (b)\nFIG. 3. (a) αeffvariation with Py thickness for Py(t)/BSTS2 heterostructure at room temperature which fits in Eq.5; (b)\nαeffas a function of BSTS thickness for Py(15nm)/BSTS(t) heterostructure at room temperature.4\n(a)\nFIG. 4. Temperature dependence of resistivity of the BSTS\nsample of thickness 21nm deposited on Si(111) substrate.\nvs. 1 /tPydata for the Py(t) and Py(t)/BSTS2 sam-\nples, we evaluated the saturation magnetization, Msof\nthe Py/BSTS bilayer that has been decreased from that\nof the bare Py sample by an amount of 183 emu/cc3.\nIt is a result of the loss of ferromagnetic order in the\nPermalloy layer. Due to the intermixing of the Py and\nBSTS at the interface, a magnetic dead layer could have\nformed at the interface which resulted in the reduction\nof saturation magnetization value as suggested by some\nprevious studies [42–44] also. The Ksvalue has de-\ncreased from 0 .092±0.008erg/cm2in bare Py film to\n0.091±0.015erg/cm2in Py/BSTS2 bilayer. So interfa-\ncial anisotropy constant, Ki(=KPy/TI\ns −KPy\ns) for the\nPy/BSTS2 sample is −0.001erg/cm2. From the nega-\ntive value of Ki, we can ensure an in-plane magnetic\nanisotropy in the Py/BSTS interface at room temper-\nature. A detailed discussion of magnetic anisotropy has\nbeen provided in the last section where the temperature\nvariation of Hkis discussed.\nαeffcan be determined by analysing ∆ Hat different\nfrequencies. ∆ Hcontains both the intrinsic and extrin-\nsic contributions to the damping. Linewidth due to in-\ntrinsic damping is directly proportional to the resonance\nfrequency( f) and follows the equation[29],\n∆H= ∆H0+ (2παeff\nγ)f (4)\nwhere ∆ H0describes inhomogeneous linewidth broad-\nening [38, 39] due to different extrinsic contributions\nlike magnetic inhomogeneities [40, 41], surface roughness,\nand defects in the sample. We have evaluated the αeff\nvalues by fitting the ∆ Hvsfcurve for FM/TI bilayers\nas shown in fig.2a. This αeffconsists of Gilbert damp-\ning in the bulk ferromagnet( αFM) and the enhanced\ndamping( αSP) resulting from spin pumping into the ad-\njacent TI layer [31–33], αeff=αFM+αSP. The αFM\nvalue for bare Py film of thickness 15nm was calculated\nto be 0.0074 and for the FM/TI bilayer system there has\nbeen significant enhancement in the αeffvalue over thebare Py value due to spin pumping, αSP. In this het-\nerostructure, αeffincreases gradually as the thickness of\nPy decreases both for Py(t) and Py(t)/BSTS2 samples as\nshown in fig.3a. From the linear fit of αeffvs. 1/tPydata\nwe have obtained the spin-mixing coefficient, g↑↓\nefffor the\nBSTS/Py interface to be 5 .26×1018±0.71×1018m−2\nby using the equation[34, 37],\nαeff−αFM=gµB\n4πMstFMg↑↓\neff(5)\nwhere, gandµBare the g-factor and Bohr magneton\nrespectively. We have also calculated the spin current\ndensity( j0\ns) for the FM/TI heterostructure using the g↑↓\neff\nvalue in the following equation[20, 36],\nj0\ns=g↑↓\neffγ2h2\nmℏ[4πMsγ+p\n(4πMs)2γ2+ 4ω2]\n8πα2[(4πMs)2γ2+ 4γ2](6)\nwhere γ,hm,ℏ,ω, and αare the gyromagnetic ratio,\nmicrowave magnetic field, Planck’s constant, Larmour\nprecession frequency, and effective damping parameter\nrespectively. Using Eq.6 the j0\nsvalue for Py/BSTS2\nwas obtained to be 0 .901×10−10±0.122×10−10Jm−2\nin our experiment. The g↑↓\neffandj0\nsvalues obtained\nfrom Py thickness-dependent study of αeffare in a\ncomparable range of the previously reported values\nfor other combinations of ferromagnet and TI bilayer\nstructures [34, 35, 37]. This gives evidence of successful\nspin injection into the BSTS layer from the Py layer\ndue to spin pumping [31–33, 50]. We also report the TI\nthickness-dependent study of αeffas shown in fig.3b.\nFor bilayer structures of Py(15nm)/BSTS2(t) there is\na sudden jump in the αeffvalue from that of the bare\nFM film ( αFM = 0.0074) because of spin pumping.\nThen with the thickness variation of TI layer in the\nrange of 10nm to 37nm, αeffincreases slowly from\n0.015 to 0.02. The TI thickness dependence of αeff\nfor Py(15nm)/BSTS(t) bilayer is almost linear which\ncertainly can not be described by the conventional\nspin diffusion theory [48] for FM/NM proposed by\nTserkovnyak et al. [47]. For Py/BSTS heterostructure,\nαeffvs. tBSTS study suggests an efficient spin-sink\nnature of the TI bulk with increasing thickness at\nroom temperature [49]. From the room temperature\nstudy we certainly can not distinguish the TI surface\nstate contribution from the TI bulk state contribution\nbecause growing a BSTS thin film with a perfectly\ninsulating bulk state is still very challenging. Thus it\nwas imperative to study the effect of topological surface\nstate at low-temperature where bulk states of TI get\nsuppressed and surface states of TI starts to dominate.\nIn this section, we have focused on low-temperature\nmeasurements specifically to understand the effect of\ntopological surface states (TSS) on the magnetization\nrelaxation of FM. At higher temperatures, a significant\namount of bulk carriers are available to participate\nin the transport but with the reduction of phonon5\n(a)\n (b)\nFIG. 5. (a)Temperature dependence of the field linewidth (∆ H) for different thickness combinations of Py/BSTS bilayer\nsystems and for a bare Py thin film. The solid lines are the fits in the expression exp(−T/T 0); (b)Temperature dependence of\neffective damping coefficient, αeffof Py(20nm)/BSTS2 and bare Py(20nm) film.\n(a)\n (b)\nFIG. 6. (a)Temperature dependence of effective magnetization of Py(20nm/BSTS2); (b)Temperature dependence of the\nanisotropy field of Py(20nm)/BSTS2.\nscattering, surface carriers dominate at a lower tem-\nperature. From the resistivity vs. temperature data of\nBSTS2 in fig.4, we can see an insulating behavior of\nresistivity due to the enhanced insulating nature of the\nbulk state of TI at higher temperatures and a metallic\nbehavior of resistivity below a certain temperature\nwhere the topological surface states dominate. We\nmeasured temperature variation of FMR linewidth\n(∆H), enhanced damping coefficient ( αeff), anisotropy\nfield ( Hk) and effective magnetization (4 πMeff). For\ndifferent thickness combinations of Py/BSTS bilayer,\nwe obtained the ∆ Hvariation with temperature. It\nincreases exponentially with decreasing temperature\nthat fits the expression, exp(−T/T 0) as shown in fig.5a.\nFor bare Py(15nm) film, we can note that there is no\nsignificant variation in ∆ Hat low temperatures as can\nbe seen from the curve at the bottom of fig.5a. To\ngain further understanding, the temperature variation\nofαeffhas also been studied for Py(20nm)/BSTS2\nas shown in fig.5b and compared with αefffor barePy film. From the enhancement of αeffvalue for\nPy(20nm)/ BSTS2 at room temperature we can ensure a\nsuccessful spin injection due to the spin pumping effect.\nBut the exponential increase of αeffwith decreasing\ntemperature for the bilayer implies a huge increment in\nthe amount of spin angular momentum transfer into the\nTI layer at lower temperatures. We attribute the origin\nof the exponential increase of αeffand ∆ Hat lower\ntemperatures to the spin chemical potential bias induced\nspin current into the surface state of TI as proposed by\nAbdulahad et al. [50]. The induced spin current into\nthe TI surface state at lower temperatures corresponds\nto the rapid relaxation of magnetization precession of\nFM which is reflected in the exponential increase of ∆ H\nandαeffof the ferromagnet.\nTo further investigate the effect of TI surface state on\nthe magnetization of FM, we studied the temperature\nvariation of 4 πMeffandHkfor Py(20nm)/BSTS2. In\nour previous study [26] with bare Py thin films, we have6\nseen that 4 πMeffincreases monotonically as saturation\nmagnetization increases with lowering the temperature.\nBut from fig.6a, we can see that the low-temperature\ndependence of 4 πMefffor Py/BSTS2 bilayer deviates\nfrom the single layer Py film [Supplementary fig.S11(a)].\nThis anomaly in 4 πMeffis related to the change of mag-\nnetic anisotropy energy of the system as well as the other\neffects like spin chemical potential induced current and\nexchange coupling between TSS and FM as mentioned\nby Abdulahad et al. [50]. In a previous section, we\nevaluated the interfacial magnetic anisotropy coefficient\n(Ki=−0.001erg/cm2) to be in-plane of the interface of\nthe Py/BSTS2 bilayer. The anisotropy field associated\nwith the system anisotropy energy shows an interesting\nnature as we lower the temperature. We can see from\nfig.6b that Hkincreases initially with decreasing tem-\nperature until a certain value is reached and then the\nanisotropy field weakens against a further decrease in\ntemperature. Thus we get a hump-like feature of HK\nfor the same temperature region where 4 πMeffshows\nthe anomaly and it is concomitant with the resistivity vs\ntemperature behavior of the BSTS2 sample. The low-\ntemperature behavior of Hkand 4 πMeffcan be justified\nby the argument proposed by Abdulahad et al. [50]. In\ntheir phenomenological model, they propose an existence\nof exchange interaction between the surface states of TI\nand local moments of the ferromagnetic layer. Several\ntheoretical as well as experimental predictions confirm\nthe existence of gapless topological surface states even\nafter transition metal deposition on TI [51, 52]. These\nsurface states can couple with the local moments of the\nFM through exchange interaction without any long-range\nferromagnetic order. This exchange coupling acts per-\npendicular to the TI surface and weakens the in-plane\nanisotropy at lower temperatures where the surface states\nof TI dominate.\nCONCLUSIONS\nIn summary, we have carried out spin-pumping ex-\nperiment in BiSbTe 1.5Se1.5(TI)/ Ni80Fe20(FM) bilayer\nsystem. From the thickness-dependent measurements of\nFM/TI bilayers, we obtained the spin-transport param-\neters like damping coefficient due to spin-pumping, spin\nmixing conductance, and spin current density at room\ntemperature. These results demonstrate a successful spin\ntransfer from the FM layer to the TI layer due to spin-\npumping. We have performed low-temperature measure-\nments to specifically understand the surface state con-\ntribution of TI on the FM magnetization because the\nsurface states of TI are more pronounced at lower tem-\nperatures. We have confirmed the suppression of the\ninsulating bulk state of TI at lower temperatures from\nthe resistivity vs. temperature data of TI. In our low-\ntemperature measurements of FMR linewidth and ef-fective damping coefficient, we have witnessed an expo-\nnential increase in both parameters with the decrease in\ntemperature. It suggests a spin chemical potential bias-\ninduced spin current injection into the surface states of\nTI that gets enhanced at low temperatures [50]. We have\nalso studied temperature variations of the effective mag-\nnetization of the system. It showed a deviation from\nthe bare Py film [26] in the temperature regime where\nTI surface states dominate. This deviation of effective\nmagnetization results from the change in the anisotropy\nenergy of the system. At room temperature, we eval-\nuated the magnetic anisotropy energy coefficient which\nis found to be in-plane of the interface. This in-plane\nanisotropy weakens when conducting surface state of TI\nstarts to dominate. It reflects from the hump-like feature\nin the magnetic anisotropy field vs. temperature data of\nthe bilayer system. The decrease in in-plane magnetic\nanisotropy below a certain temperature can result from\nthe exchange coupling between the surface states of TI\nand the local moments of the FM layer which act per-\npendicular to the interface [50]. Combining the results of\nour low-temperature measurements we can conclude that\nthere exists an exchange coupling between the TI surface\nstate and FM which does not create any long-range ferro-\nmagnetic order in the TI and is unable to alter the overall\nspin texture of the TI surface state at the interface[18].\nHowever, it affects the magnetization dynamics of the\nferromagnetic metal quite significantly. These added fea-\ntures of enhancing the damping coefficients enables an-\nother fast control of magnetization dynamics in the FM\nlayer.\nACKNOWLEDGEMENTS\nThe authors sincerely acknowledge the Ministry\nof Education, Government of India and Science\nand Engineering Research Board (SERB) (grant no:\nEMR/2016/007950), and Department of Science and\nTechnology (grant no. DST/ICPS/Quest/2019/22) for\nfinancial support. S.P. acknowledges the Department\nof Science and Technology(DST)-INSPIRE fellowship In-\ndia, S. A. acknowledges the Ministry of Education of the\nGovernment of India, S.M. acknowledges the Council Of\nScientific and Industrial Research(CSIR), India, S.G.N\nand K.S acknowledges the University Grant Commis-\nsion, India for research fellowship. The authors would\nlike to thank Dr. Partha Mitra of the Department of\nPhysics, Indian Institute of Science Education and Re-\nsearch Kolkata, for providing the lab facilities for sample\ndeposition. 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Journal of Magnetism and Mag-\nnetic Materials, 166(1-2), pp.6-26."    },    {        "title": "1703.09444v2.Temperature_dependent_magnetic_damping_of_yttrium_iron_garnet_spheres.pdf",        "content": "Temperature dependent magnetic damping of yttrium iron garnet spheres\nH. Maier-Flaig,1, 2,\u0003S. Klingler,1, 2C. Dubs,3O. Surzhenko,3R.\nGross,1, 2, 4M. Weiler,1, 2H. Huebl,1, 2, 4and S. T. B. Goennenwein1, 2, 4, 5, 6\n1Walther-Mei\u0019ner-Institut, Bayerische Akademie der Wissenschaften, 85748 Garching, Germany\n2Physik-Department, Technische Universit at M unchen, 85748 Garching, Germany\n3INNOVENT e.V. Technologieentwicklung, 07745 Jena, Germany\n4Nanosystems Initiative Munich, 80799 M unchen, Germany\n5Institut f ur Festk orperphysik, Technische Universit at Dresden, 01062 Dresden, Germany\n6Center for Transport and Devices of Emergent Materials,\nTechnische Universit at Dresden, 01062 Dresden, Germany\n(Dated: June 5, 2017)\nWe investigate the temperature dependent microwave absorption spectrum of an yttrium iron\ngarnet sphere as a function of temperature (5 K to 300 K) and frequency (3 GHz to 43 :5 GHz). At\ntemperatures above 100 K, the magnetic resonance linewidth increases linearly with temperature\nand shows a Gilbert-like linear frequency dependence. At lower temperatures, the temperature\ndependence of the resonance linewidth at constant external magnetic \felds exhibits a characteristic\npeak which coincides with a non-Gilbert-like frequency dependence. The complete temperature and\nfrequency evolution of the linewidth can be modeled by the phenomenology of slowly relaxing rare-\nearth impurities and either the Kasuya-LeCraw mechanism or the scattering with optical magnons.\nFurthermore, we extract the temperature dependence of the saturation magnetization, the magnetic\nanisotropy and the g-factor.\nI. INTRODUCTION\nThe magnetization dynamics of the ferrimagnetic insu-\nlator yttrium iron garnet (YIG) recently gained renewed\ninterest as YIG is considered an ideal candidate for spin-\ntronic applications as well as spin-based quantum infor-\nmation storage and processing1{4due to the exception-\nally low damping of magnetic excitations as well as its\nmagneto-optical properties5{7. In particular, consider-\nable progress has been made in implementing schemes\nsuch as coupling the magnetic moments of multiple YIG\nspheres2,8or interfacing superconducting quantum bits\nwith the magnetic moment of a YIG sphere3,9.\nMagnetization dynamics in YIG have been investi-\ngated in a large number of studies in the 1960s.10{12\nHowever, a detailed broadband study of the magnetiza-\ntion dynamics in particular for low temperatures is still\nmissing for bulk YIG. Nevertheless, these parameters are\nessential for the design and optimization of spintronic\nand quantum devices. Two recent studies13,14consider\nthe temperature dependent damping of YIG thin \flms.\nHaidar et al.13report a large Gilbert-like damping of\nunknown origin, while the low damping thin \flms inves-\ntigated by Jermain et al.14show a similar behavior as\nreported here.\nOur systematic experiments thus provide an impor-\ntant link between the more recent broadband studies on\nYIG thin \flms and the mostly single-frequency studies\nfrom the 1960s: We investigate the magnetostatic spin\nwave modes measured in a YIG sphere using broadband\nmagnetic resonance up to 43.5 GHz in the temperature\nrange from 5 to 300K. We extract the temperature de-\npendent magnetization, the g-factor and the magnetic\nanisotropy of YIG. Additionally, we focus our analysis on\nthe temperature dependent damping properties of YIGand identify the phenomenology of slowly relaxing rare-\nearth impurities and either the Kasuya-LeCraw mecha-\nnism or the scattering with optical magnons as the mi-\ncroscopic damping mechanism.\nThe paper is organized as follows. We \frst give a short\nintroduction into the experimental techniques followed by\na brief review of the magnetization damping mechanisms\nreported for YIG. Finally, we present the measured data\nand compare the evolution of the linewidth with temper-\nature and frequency with the discussed damping models.\nThe complete set of raw data and the evaluation routines\nare publicly available.15\nII. EXPERIMENTAL DETAILS AND\nFERROMAGNETIC RESONANCE THEORY\nThe experimental setup for the investigation of the\ntemperature dependent broadband ferromagnetic reso-\nnance (bbFMR) is shown schematically in Fig. 1. It\nconsists of a coplanar wave guide (CPW) onto which\na 300 µm diameter YIG sphere is mounted above the\n300µm wide center conductor. The [111] direction of\nthe single crystalline sphere is aligned along the CPW\nsurface normal as con\frmed using Laue di\u000braction (not\nshown). We mount a pressed diphenylpicrylhydrazyl\n(DPPH) powder sample in a distance of approximately\n5 mm from the sphere. The identical sample with the\nsame alignment has been used in Ref. 16. This assembly\nis mounted on a dip stick in order to place the YIG sphere\nin the center of a superconducting magnet (Helmholtz\ncon\fguration) in a Helium gas-\row cryostat. End-launch\nconnectors are attached to the CPW and connected to\nthe two ports of a vector network analyzer (VNA) mea-\nsuring the phase sensitive transmission of the setup uparXiv:1703.09444v2  [cond-mat.mtrl-sci]  2 Jun 20172\nliquid Helium\nside view\nCPWP1\nP2H0hMW hMWVNAP1P2\nside viewside view5mm\nDPPH\nYIG\ncom ±1‰ ±1‰\nFIG. 1. The coplanar waveguide (CPW), on which the YIG\nsphere and a DPPH marker are mounted (right), is inserted\ninto a magnet cryostat (left). The microwave transmission\nthrough the setup is measured phase sensitively using a vec-\ntor network analyzer (VNA). As the Oersted \feld hMW(red)\naround the center conductor of the CPW extends into the\nYIG sphere and the DPPH, we can measure the microwave\nresponse spectra of both samples. A superconducting magnet\nprovides the static external magnetic \feld H0(orange) at the\nlocation of the sample. Also shown are the lines correspond-\ning to the speci\fed 1 ‰homogeneity of the \feld for a on-axis\ndeviation from the center of magnet (com).\nto 43:5 GHz.\nThe sphere is placed within the microwave Oersted\n\feldhMWof the CPW's center conductor which is excited\nwith a continuous wave microwave of variable frequency.\nWe apply a static external magnetic \feld H0perpen-\ndicular to the CPW surface and thus hMWis oriented\nprimarily perpendicular to H0. The microwave Oersted\n\feld can therefore excite magnetization precession at fre-\nquencies that allow a resonant drive. The magnetization\nprecession is detected by electromagnetic induction via\nthe same center conductor.17This induction voltage in\ncombination with the purely transmitted microwave sig-\nnal is measured phase sensitively as the complex scatter-\ning parameter Sraw\n21(!) at port 2 of the VNA.\nThe frequency-dependent background is eliminated as\nfollows: A static external magnetic \feld su\u000eciently large\nthat no resonances are expected in the given microwave\nfrequency range is applied and the transmission at this\n\feld is recorded as the background reference SBG\n21. Then,\nthe external \feld is set to the value at which we ex-\npect resonances of YIG in the given frequency range and\nrecord the transmission Sraw\n21. We \fnally divide Sraw\n21by\nSBG\n21. This corrects for the frequency dependent attenu-\nation and the electrical length of the setup. We choose\nthis background removal method over a microwave cal-\nibration because it additionally eliminates the \feld andtemperature dependence of S21that arises from the ther-\nmal contraction and movement of the setup and magnetic\nmaterials in the microwave connectors. In the following\nwe always display S21=Sraw\n21=SBG\n21.\nFor the evaluation of the magnetization dynamics, we\n\ft the transmission data to S21=\u0000ifZ\u001f +A1+A2f\nfor eachH0. Here,A1;2describe a complex-valued back-\nground and\n\u001f(f;H 0) =\u00160Ms\r\n2\u0019\u0000\r\n2\u0019\u00160H0\u0000i\u0001f\u0001\nf2res\u0000f2\u0000if\u0001f(1)\nis the ferromagnetic high-frequency susceptibility.17,18\nThe free parameters of the \ft are the resonance frequency\nfres, the full width at half maximum (FWHM) linewidth\n\u0001fas well as the complex scaling parameter Z, which\nis proportional to the strength of the inductive coupling\nbetween the speci\fc magnetic resonance mode and the\nCPW. For a given \fxed magnetic \feld the \ft parame-\nters\r\n2\u0019(gyromagnetic ratio) and Ms(saturation magne-\ntization) are completely correlated with Zand are thus\n\fxed. They are later determined from \ftting the disper-\nsion curves.19\nIn spheres various so-called magnetostatic modes\n(MSM) arise due to the electromagnetic boundary\nconditions.20These modes can be derived from the\nLandau-Lifshitz equation in the magnetostatic limit ( ~r\u0002\n~H= 0) for insulators.21The lineshape of all modes is\ngiven by Eq. 1. Due to the di\u000berent spatial mode pro\fles\nand the inhomogeneous microwave \feld, the inductive\ncoupling and thus Zis mode dependent.22A detailed re-\nview of possible modes, their distribution and dispersion\nis given in R oschmann and D otsch20. We will only dis-\ncuss the modes (110) and (440) in detail in the following\nas all the relevant characteristics of all other modes can\nbe related to these two modes. Their linear dispersions\nare given by20\nf110\nres=\r\n2\u0019\u00160(H0+Hani) (2)\nf440\nres=\r\n2\u0019\u00160\u0012\nH0+Hani+Ms\n9\u0013\n(3)\nwhereHaniis the magnetic anisotropy \feld and\r\n2\u0019is\nthe gyromagnetic ratio which relates to the g-factor by\n\r\n2\u0019=\u0016B\nhg. It is thus generally assumed that gis the\nsame for all modes. We note that the apparent g-factor\nmay still vary in between modes if the modes experience\na di\u000berent anisotropy.23,24Such an anisotropy contribu-\ntion can be caused by surface pit scattering as it a\u000bects\nmodes that are localized at the surface stronger than bulk\nlike modes25. In our experiment, no such variation in g\ncoinciding with a change in anisotropy was observed and\nwe use a mode number independent gin the following.\nKnowledge of the dispersion relations of the two modes\nallows to determine the saturation magnetization from\n\u00160Ms(T) = 92\u0019\n\r\u0001fM= 92\u0019\n\r\u0000\nf440\nres\u0000f110\nres\u0001\n:(4)3\nThe anisotropy \feld is extracted by extrapolating the\ndispersion relations in Eqs. (2) and (3) to H0= 0.\nThe temperature dependent linewidth \u0001 fof the modes\nis the central result of this work. For a short review of\nthe relevant relaxation processes we refer to the dedicated\nSec. III.\nIn this work, we investigate the T-dependence of Ms,\nHani,gand \u0001f. Accurate determination of the g-factor\nand the anisotropy Hanirequires accurate knowledge of\nH0. In order to control the temperature of the YIG\nsphere and CPW, they are placed in a gas-\row cryo-\nstat as displayed schematically in Fig. 1. The challenge\nin this type of setup is the exact and independent deter-\nmination of the static magnetic \feld and its spatial in-\nhomogeneity. Lacking an independent measure of H0,26\nwe only report the relative change of gandHanifrom\ntheir respective room temperature values which were de-\ntermined separately using the same YIG sphere.16Note\nthat we determine the resonance frequencies directly in\nfrequency space. Our results on linewidth and magneti-\nzation are hence independent of a potential uncertainty\nin the absolute magnitude of H0and its inhomogeneity.\nIII. RELAXATION THEORY\nWhen relaxation properties of ferromagnets are dis-\ncussed today, the most widely applied model is the so-\ncalled Gilbert type damping. This purely phenomeno-\nlogical model is expressed in a damping term of the form\n\u000bM\u0002dM\ndtin the Landau-Lifshitz equation. It describes\na viscous damping, i.e. a resonance linewidth that de-\npends linearly on the frequency. A linear frequency de-\npendence is often found in experiments and the Gilbert\ndamping parameter \u000bserves as a \fgure of merit of the\nferromagnetic damping that allows to compare samples\nand materials. It contains, however, no insight into the\nunderlying physical mechanisms.\nIn order to understand the underlying microscopic re-\nlaxation processes of YIG, extensive work has been car-\nried out. Improvements on both the experimental side\n(low temperatures27, temperature dependence10,28,29,\nseparate measurements of MzandMxy11) and on the\nsample preparation (varying the surface pit size25, pu-\nrifying Yttrium10, doping YIG with silicon30and rare-\nearth elements30{33) led to a better understanding of\nthese mechanisms.\nHowever, despite these e\u000borts the microscopic origin\nof the dominant relaxation mechanism for bulk YIG at\nroom temperature is still under debate. It has been\ndescribed by a two-magnon process by Kasuya and\nLeCraw28(1961). In this process, a uniformly-precessing\nmagnon (k= 0) relaxes under absorption of a phonon\nto ak6= 0 magnon. If the thermal energy kBTis\nmuch larger than the energy of the involved magnons and\nphonons (T > 100 K) and low enough that no higher-\norder processes such as four-magnon scattering play a\nrole (T < 350 K), the Kasuya-LeCraw process yields alinewidth that is linear in frequency and temperature:\n\u0001fKL/T;f.28,30This microscopic process is therefore\nconsidered to be the physical process that explains the\nphenomenological Gilbert damping for low-damping bulk\nYIG. More recently, Cherepanov et al.34pointed out\nthat the calculations by Kasuya and LeCraw28assume\na quadratic magnon dispersion in k-space which is only\ncorrect for very small wave numbers k. Taking into ac-\ncount a more realistic magnon dispersion (quadratic at\nlowk, linear to higher k), the Kasuya-LeCraw mechanism\ngives a value for the relaxation rate that is not in line\nwith the experimental results. Cherepanov therefore de-\nveloped an alternative model that traces back the linear\nfrequency and temperature dependence at high tempera-\ntures (150 K to 300 K) to the interaction of the uniform-\nprecession mode with optical magnons of high frequency.\nRecently, atomistic calculations by Barker and Bauer35\ncon\frmed the assumptions on the magnon spectrum that\nare necessary for the quantitative agreement of the latter\ntheory with experiment.\nBoth theories, the Kasuya-LeCraw theory and the\nCherepanov theory, aim to describe the microscopic ori-\ngin of the intrinsic damping. They deviate in their\nprediction only in the low-temperature ( T < 100 K)\nbehavior.30At these temperatures, however, impurities\ntypically dominate the relaxation and mask the contri-\nbution of the intrinsic damping process. Therefore, the\ndominant microscopic origin of the YIG damping at tem-\nperatures above 150K has not been unambiguously de-\ntermined to date.\nIf rare-earth impurities with large orbital momentum\nexist in the crystal lattice, their exchange coupling with\nthe iron ions introduces an additional relaxation chan-\nnel for the uniform precession mode of YIG. Depend-\ning on the relaxation rate of the rare-earth impurities\nwith respect to the magneto{dynamics of YIG, they are\nclassi\fed into slowly and fast relaxing rare-earth impu-\nrities. This is an important distinction as the e\u000eciency\nof the relaxation of the fundamental mode of YIG via\nthe rare-earth ion to the lattice at a given frequency de-\npends on the relaxation rate of the rare-earth ion and the\nstrength of the exchange coupling. In both the slow and\nthe fast relaxor case, a characteristic peak-like maximum\nis observed in the linewidth vs. temperature dependence\nat a characteristic, frequency-dependent temperature12.\nThe frequency dependence of this peak temperature al-\nlows to distinguish fast and slowly relaxing rare-earth\nions: The model of a fast relaxing impurity predicts that\nthe peak temperature is constant, while in the case of\nslowly relaxing rare-earth ions the peak temperature is\nexperted to increase with increasing magnetic \feld (or\nfrequency). The relaxation rate of rare-earths \u001cREis typ-\nically modeled by a direct magnon to phonon relaxation,\nan Orbach processes36,37that involves two phonons, or\na combination of both. The inverse relaxation rate of\nan Orbach process is described by1\n\u001cOrbach =B\ne\u0001=(kBT)\u00001\nwith the crystal \feld splitting \u0001 and a proportionality\nfactorB. A direct process leads to an inverse relax-4\nation rate of1\n\u001cdirect =1\n\u001c0coth\u000e\n2kBTwith\u001c0, the relaxation\ntime atT= 0 K. It has been found experimentally that\nmost rare-earth impurities are to be classi\fed as slow\nrelaxors.30The sample investigated here is not intention-\nally doped with a certain rare-earth element and the peak\nfrequency and temperature dependence indicates a slow\nrelaxor. We therefore focus on the slow relaxing rare-\nearth impurity model in the following.\nDeriving the theory of the slowly relaxing impurities\nhas been performed comprehensively elsewhere.30The\nlinewidth contribution caused by a slowly relaxing rare-\nearth impurity is given by31:\n\u0001fSR=C\n2\u0019f\u001cRE\n1 + (f\u001cRE)2(5)\nwithC/1\nkBTsech\u0010\n\u000ea\n2kBT\u0011\n. Therein, \u000eais the splitting\nof the rare-earth Kramers doublet which is given by the\ntemperature independent exchange interaction between\nthe iron ions and the rare-earth ions.\nAlso Fe2+impurities in YIG give rise to a process\nthat leads to a linewidth peak at a certain temperature.\nThe physical origin of this so-called valence exchange or\ncharge-transfer linewidth broadening is electron hopping\nbetween the iron ions.30Simpli\fed, it can be viewed as\na two level system just like a rare-earth ion and thus re-\nsults in the same characteristic linewidth maximum as\na slowly relaxing rare-earth ion. For valence exchange,\nthe energy barrier \u0001 hopthat needs to be overcome for\nhopping determines the time scale of the process. The\ntwo processes, valence exchange and rare-earth impurity\nrelaxation, can therefore typically not be told apart from\nFMR measurements only. In the following, we use the\nslow relaxor mechanism exclusively. This model consis-\ntently describes our measurement data and the resulting\nmodel parameters are in good agreement with literature.\nWe would like to emphasize, however, that the valence\nexchange mechanism as the relevant microscopic process\nresulting for magnetization damping can not be ruled out\nfrom our measurements.\nIV. EXPERIMENTAL RESULTS AND\nDISCUSSION\nTwo exemplary S21broadband spectra recorded at two\ndistinct temperatures are shown in Fig. 2. The color-\ncoded magnitude jS21jis a measure for the absorbed mi-\ncrowave power. High absorption (bright color) indicates\nthe resonant excitation of a MSM in the YIG sphere or\nthe excitation of the electron paramagnetic resonance of\nthe DPPH. In the color plot the color scale is truncated\nin order to improve visibility of small amplitude reso-\nnances. In addition, the frequency axis is shifted relative\nto the resonance frequency of a linear dispersion with\ng= 2:0054 (fg=2:0054\nres =g\u0016B\nh\u00160H) for each \feld. In this\nway, modes with g= 2:0054 appear as vertical lines. A\n0.20.40.60.81.01.21.4¹0H0 (T)290 K\nminmax\nAbsorption\n−1.0 −0.5 0.0 0.5 1.0\nf¡f(g=2:0054)\nres  (GHz)0.20.40.60.81.01.21.4¹0H0 (T)20 KΔfAΔfM\nf (GHz)10.05 10.06 10.07 10.08\nIm(S21)\n2\n046810\n0\n-4-224Re(S21)\nȴffres(a)\n(b)FIG. 2. Eigenmode spectra of the YIG sphere at (a) 290 K\nand (b) 20 K. The (110) and (440) MSM are marked with red\ndashed lines. The change in their slope gives the change of the\ng-factor of YIG. Their splitting (\u0001 fM, red arrow) depends lin-\nearly on the YIG magnetization. The increase in Msto lower\ntemperatures is already apparent from the increased splitting\n\u0001fM. Marked in orange is the o\u000bset of the resonance fre-\nquency \u0001fAextrapolated to H0= 0 resulting from anisotropy\n\feldsHanipresent in the sphere. The green marker denotes\nthe position of the DPPH resonance line which increases in\namplitude considerably to lower temperatures. Inset: S21pa-\nrameter (data points) and \ft (lines) at \u00160H= 321 mT and\nT= 20 K.\ndeviatingg-factor is therefore easily visible as a di\u000ber-\nent slope. Comparing the spectra at 290 K [Fig. 2 (a)]\nto the spectra at 20 K [Fig. 2 (b)], an increase of the g-\nfactor is observed for all resonance modes upon reducing\nthe temperature. The rich mode spectrum makes it nec-\nessary to carefully identify the modes and assign mode\nnumbers. Note that the occurrence of a particular mode\nin the spectrum depends on the position of the sphere\nwith respect to the CPW due to its inhomogeneous exci-\ntation \feld. We employ the same method of identifying\nthe modes as used in Ref. 16 and \fnd consistent mode\nspectra. As mentioned before, we do not use the DPPH\nresonance (green arrow in Fig. 2) but the (110) YIG\nmode as \feld reference. For this \feld reference, we take\ng(290 K) = 2 :0054 and\r\n2\u0019\u00160Hani(290 K) = 68 :5 MHz de-\ntermined for the same YIG sphere at room temperature\nin an electromagnet with more accurate knowledge of the\napplied external magnetic \feld.16The discrepancy of the\nDPPHg-value from the literature values of g= 2:0036 is5\nattributed to the non-optimal location of the DPPH spec-\nimen in the homogeneous region of the superconducting\nmagnet coils.\nIn Fig. 2, the \ftted dispersion of the (110) and (440)\nmodes are shown as dashed red lines. As noted previ-\nously, we only analyze these two modes in detail as all\nparameters can be extracted from just two modes. The\n(110) and (440) mode can be easily and unambiguously\nidenti\fed by simply comparing the spectra with the ones\nfound in Ref 16. Furthermore, at high \felds, both modes\nare clearly separated from other modes. This is necessary\nas modes can start hybridizing when their (unperturbed)\nresonance frequencies are very similar (cf. low-\feld re-\ngion of Fig. 2 (b)) which makes a reliable determination of\nthe linewidth and resonance frequency impossible. These\nattributes make the (110) and the (440) mode the ideal\nchoice for the analysis.\nAs described in Sec. II, we simultaneously \ft the (110)\nand the (440) dispersions with the same g-factor in or-\nder to extract Ms,Haniandg. In the \ft, we only take\nthe high-\feld dispersion of the modes into account where\nno other modes intersect the dispersion of the (110) and\n(440) modes. The results are shown in Fig. 3. Note that\nthe statistical uncertainty from the \ft is not visible on the\nscale of any of the parameters Ms,Haniandg. Following\nthe work of Solt38, we model the resulting temperature\ndependence of the magnetization (Fig. 3 (a)) with the\nBloch-law taking only the \frst order correction into ac-\ncount:\nMs=M0\u0010\n1\u0000aT3\n2\u0000bT5\n2\u0011\n: (6)\nThe best \ft is obtained for \u00160M0 =\n249:5(5) mT, a = (23\u00063)\u000210\u00006K\u00003=2and\nb= (1:08\u00060:11)\u000210\u00007K\u00005=2. The obtained \ft\nparameters depend strongly on the temperature window\nin which the data is \ftted. Hence, the underlying\nphysics determining the constants aandbcannot be\nresolved.39Nevertheless, the temperature dependence\nofMsis in good agreement with the results determined\nusing a vibrating sample magnetometer.40\nIn particular, also the room temperature saturation\nmagnetization of \u00160Ms(300 K) = (180 :0\u00060:8) mT is in\nperfect agreement with values reported in literature.41,42\nNote that the splitting of the modes is purely in frequency\nspace and thus errors in the \feld do not add to the uncer-\ntainty. We detect a small non-linearity of the (110) and\n(440) mode dispersions that is most likely due to devia-\ntions from an ideal spherical shape or strain due to the\nYIG mounting. This results in a systematic, temperature\nindependent residual of the linear \fts to these disper-\nsions. This resulting systematic error of the magnetiza-\ntion is incorporated in the uncertainty given above. How-\never, a deviation from the ideal spherical shape, strain in\nthe holder or a misalignment of the static magnetic \feld\ncan also modify the splitting of the modes and hence re-\nsult in a di\u000berent Ms.43This fact may explain the small\ndiscrepancy of the value determined here and the valuedetermined for the same sphere in a di\u000berent setup at\nroom temperature.16\nFrom the same \ft that we use to determine the mag-\nnetization, we can deduce the temperature dependence\nof the anisotropy \feld \u00160Hani[Fig. 3 (b)]. Most notably,\nHanichanges sign at 200 K which has not been observed\nin literature before and can be an indication that the\nsample is slightly strained in the holder. The resonance\nfrequency of DPPH extrapolated to \u00160H0= 0 (\u0001fani, red\nsquares) con\frms that the error in the determined value\nHaniis indeed temperature independent and very close\nto zero. Thus, the extracted value for the anisotropy is\nnot merely given by an o\u000bset in the static magnetic \feld.\nThe evolution of the g-factor with temperature is\nshown in Fig. 3 (c). It changes from 2 :005 at room tem-\nperature to 2 :010 at 10 K where it seems to approach a\nconstant value. As mentioned before, the modes' disper-\nsion is slightly non-linear giving rise to a systematic, tem-\nperature independent uncertainty in the determination of\ngof\u00060:0008. Theg-factor of YIG has been determined\nusing the MSMs of a sphere for a few selected tempera-\ntures before.12Comparing our data to these results, one\n\fnds that the trend of the temperature dependence of g\nagrees. However, the absolute value of gand the mag-\nnitude of the variation di\u000ber. At the same time, we \fnd\na change of the g-factor of DPPH that is on the scale\nof 0:0012. This may be attributed to a movement of\nthe sample slightly away from the center of magnet with\nchanging temperature due to thermal contraction of the\ndip stick. In this case, the YIG g-factor has to be cor-\nrected by this change. The magnitude of this e\u000bect on\nthe YIGg-factor can not be estimated reliably from the\nchange of the DPPH g-factor alone. Furthermore, the\ntemperature dependence of the DPPH g-factor has not\nbeen investigated with the required accuracy in literature\nto date to allow excluding a temperature dependence of\ntheg-factor of DPPH. We therefore do not present the\ncorrected data but conclude that we observe a change in\nthe YIGg-factor from room temperature to 10 K of at\nleast 0.2 %.\nNext, we turn to the analysis of the damping properties\nof YIG. We will almost exclusively discuss the damping of\nthe (110) mode in the following but the results also hold\nquantitatively and qualitatively for the other modes.16\nVarying the applied microwave excitation power P(not\nshown) con\frms that no nonlinear e\u000bects such as a power\nbroadening of the modes are observed with P= 0:1 mW.\nNote that due to the microwave attenuation in the mi-\ncrowave cabling, the microwave \feld at the sample loca-\ntion decreases with increasing frequency for the constant\nexcitation power.\nFirst, we evaluate the frequency dependent linewidth\nfor several selected temperatures [Fig. 4 (a)]. At temper-\natures above 100 K, a linear dependence of the linewidth\nwith the resonance frequency is observed. This depen-\ndence is the usual so-called Gilbert-like damping and\nthe slope is described by the Gilbert damping param-\neter\u000b. A linear frequency dependence of the damping6\n180200220240260¹0M (mT)\n(a)model\n−20246¹0Hani (mT)\n(b)YIG\nDPPH\n0 50 100 150 200 250 300\nTemperature (K)2.0042.0082.0122.0162.020g-factor (unitless)\n0.08%\n0.26%\n(c)\nFIG. 3. (a)YIG magnetization as function of tempera-\nture extracted from the (110) and (440) mode dispersions us-\ning Eq. 4. The purple line shows the \ft to a Bloch model\n(cf. parameters in the main text). (b)YIG anisotropy \feld\n\u00160Hani(T) =2\u0019\n\r\u0001fani. Red squares: Same procedure applied\nto the DPPH dispersion as reference. (c)YIGg-factor (blue\ncircles). For reference, the extracted DPPH g-factor is also\nshown (red squares). The gray numbers indicate the rela-\ntive change of the g-factors from the lowest to the highest\nmeasured temperature (gray horizontal lines). As we use the\nYIG (110) mode as the magnetic \feld reference, the extracted\nvalue ofgandHaniat 300 K are \fxed to the values determined\nin the room temperature setup.16\nin bulk YIG has been described by the theory developed\nby Kasuya and LeCraw28and the theory developed by\nCherepanov et al.34(cf. Sec. III). We extract \u000bfrom\na global \ft of a linear model to the (110) and (440)\nlinewidth with separate parameters for the inhomoge-\nneous linewidths \u0001 f110\n0and \u0001f440\n0and a shared Gilbert\ndamping parameter \u000bfor all modes:16\n\u0001f= 2\u000bf+ \u0001f110;440\n0 (7)\nThe \ft is shown exemplarily for the 290 K (red) data in\nFig. 4 (a).\nThe Gilbert damping parameter \u000bextracted using this\n\ftting routine for each temperature is shown in Fig. 5 (a).\nConsistently with both theories, \u000bincreases with increas-\ning temperature. The error bars in the \fgure correspond\nto the maximal deviation of \u000bextracted from separate\n\fts for each mode. They therefore give a measure of\nhow\u000bscatters in between modes. The statistical error\nof the \ft (typically \u00060:00001) is not visible on this scale.\nThe Gilbert damping parameter \u000blinearly extrapolatedto zero temperature vanishes. Note that this is consis-\ntent with the magnon-phonon process described by Ka-\nsuya and LeCraw28but not with the theory developed\nby Cherepanov et al.34. For room temperature, we ex-\ntract a Gilbert damping of 4 \u000210\u00005which is in excel-\nlent agreement with the literature value.16,44From the\n\ft, we also extract the inhomogeneous linewidth \u0001 f0,\nwhich we primarily associate with surface pit scattering\n(Sec. III, Ref 16). In the data, a slight increase of \u0001 f0\ntowards lower temperatures is present [Fig. 5 (b)]. Such a\nchange in the inhomogeneous linewidth can be caused by\na change in the surface pit scattering contribution when\nthe spin-wave manifold changes with Ms.16,25\nNote that according to Fig. 5 (b) \u0001 f0is higher for the\n(440) mode than for the (110) mode. This is in agreement\nwith the theoretical expectation that surface pit scatter-\ning has a higher impact on \u0001 f0for modes that are more\nlocalized at the surface of the sphere like the (440) mode\ncompared to the more bulk-like modes such as the (110)\nmode25.45\nTurning back to Fig. 4 (a), for low temperatures (20 K,\nblue data points), a Gilbert-like damping model is obvi-\nously not appropriate as the linewidth increases consider-\nably towards lower frequencies instead of increasing lin-\nearly with increasing frequency. Typically, one assumes\nthat the damping at low frequencies is dominated by so-\ncalled low \feld losses that may arise due to domain for-\nmation. The usual approach is then to \ft a linear trend\nto the high-frequency behavior only. Note, however, that\neven though the frequency range we use is already larger\nthan usually reported13,14,46, this approach yields an un-\nphysical, negative damping. We conclude that the model\nof a Gilbert-like damping is only valid for temperatures\nexceeding 100 K (Fig. 5) for the employed \feld and fre-\nquency range.\nThe linewidth data available in literature are typi-\ncally taken at a \fxed frequency and the linewidth is dis-\nplayed as a function of temperature12,29,30. We can ap-\nproximately reproduce these results by plotting the mea-\nsured linewidth at \fxed H0as a function of temperature\n[Fig. 4 (b)].47A peak-like maximum of the linewidth be-\nlow 100 K is clearly visible. For increasing magnetic \feld\n(frequency), the peak position shifts to higher tempera-\ntures. This is the signature of a slowly relaxing rare-earth\nimpurity (Sec. III). A fast relaxing impurity is expected\nto result in a \feld-independent linewidth vs. temperature\npeak and can thus be ruled out. At the peak position,\nthe linewidth shows an increase by 2 :5 MHz which trans-\nlates with the gyromagnetic ratio to a \feld linewidth in-\ncrease of 0:08 mT. For 0.1 at. % Terbium doped YIG, a\nlinewidth increase of 80 mT has been observed48. Con-\nsidering that the linewidth broadening is proportional to\nthe impurity concentration and taking the speci\fed pu-\nrity of the source material of 99.9999% used to grow the\nYIG sphere investigated here, we estimate an increase of\nthe linewidth of 0 :08 mT, in excellent agreement with the\nobserved value.\nModeling the linewidth data is more challenging: The7\n0 5 10 15 20 25 30 35 40 45\nf110\nres (GHz)0¢f023456¢f110 (MHz)(a) 20 K\n290 K\n0 50 100 150 200\nTemperature (K)0.51.01.52.02.53.03.5¢f110 (MHz)\n(b) 341 mT, 9.6 GHz\n1007 mT, 28.3 GHzTmax\n0:3TTmax\n1:0T\nFIG. 4. (a)Full width at half maximum (FWHM) linewidth \u0001 f110\nresof the (110) mode as a function of frequency for di\u000berent\ntemperatures. A linear Gilbert-like interpretation is justi\fed in the high- Tcase (T > 100 K) only. Below 100 K, the slope of\n\u0001f110(f110\nres) is not linear so that a Gilbert type interpretation is no longer applicable. (b)FWHM linewidth as a function of\ntemperature for two di\u000berent \fxed external magnetic \felds. The linewidth peaks at a magnetic \feld dependent temperature\nthat can be modeled using the phenomenology of rare-earth impurities resulting in Tmax(vertical dotted lines).\n01234® (unitless)£10−5\n(a)\n0 50 100 150 200 250 300\nTemperature (K)0123¢f0 (MHz) (b)110\n440\nFIG. 5. (a)Gilbert damping parameter \u000bdetermined from the slope of a linear \ft to the \u0001 f(f;T) data for frequencies\nabove 20 GHz. The red line shows the linear dependence of the linewidth with temperature expected from the Kasuya-LeCraw\nprocess. (b)Inhomogeneous linewidth \u0001 f0(intersect of the aforementioned \ft with f110\nres= 0) as a function of temperature.\nThe inhomogeneous linewidth shows a slight increase with decreasing temperature down to 100 K. In the region where the\nslow relaxor dominates the linewidth (gray shaded area, cf. Fig. 4), the linear \ft is not applicable and unphysical damping\nparameters and inhomogeneous linewidths are extracted.\nmodel of a slowly relaxing rare-earth ion contains the\nexchange coupling of the rare-earth ion and the iron\nsublattice, and its temperature dependent relaxation fre-\nquency as parameters. As noted before, typically a di-\nrect and an Orbach process model the relaxation rate,\nand both of these processes have two free parameters.\nUnless these parameters are known from other experi-\nments for the speci\fc impurity and its concentration in\nthe sample, \ftting the model to the temperature behav-\nior of the linewidth at just one \fxed frequency gives am-\nbiguous parameters. In principle, frequency resolved ex-\nperiments as presented in this work make the determina-\ntion of the parameters more robust as the mechanism re-\nsponsible for the rare-earth relaxation is expected not to\nvary as a function of frequency. The complete frequency\nand \feld dependence of the linewidth is shown in Fig. 6.At temperatures above approx 100 K, the linewidth in-\ncreases monotonically with \feld, in agreement with a\ndominantly Gilbert-like damping mechanisms, which be-\ncomes stronger for higher temperatures. On the same\nlinear color scale, the linewidth peak below 100 K and its\nfrequency evolution is apparent. Fig. 4 (b) corresponds\nto horizontal cuts of the data in Fig. 6 at \u00160H0= 341\nand 1007 mT.\nFor typical YIG spheres, that are not speci\fcally en-\nriched with only one rare-earth element, the composition\nof the impurities is unknown. Di\u000berent rare-earth ions\ncontribute almost additively to the linewidth and have\ntheir own characteristic temperature dependent relax-\nation frequency respectively peak position. This is most\nprobably the case for the YIG sphere of this study. The\nconstant magnitude of the peak above 0 :3 T and the con-8\nstant peak width indicates that fast relaxing rare-earth\nions play a minor role. The evolution of the linewidth\nwithH0andfcan therefore not be \ftted to one set\nof parameters. We thus take a di\u000berent approach and\nmodel just the shift of the peak position in frequency\nand temperature as originating from a single slowly re-\nlaxing rare-earth impurity. For this, we use a value\nfor the exchange coupling energy between the rare-earth\nions and the iron sublattice in a range compatible with\nliterature30of\u000ea= 2:50 meV. To model the rare-earth\nrelaxation rate as a function of temperature, we use the\nvalues determined by Clarke et al.32for Neodymium\ndoped YIG: \u001c0= 2:5\u000210\u000011s for the direct process and\n\u0001 = 10:54 meV and B= 9\u00021011s\u00001for the Orbach pro-\ncess. The model result, i.e. the peak position, is shown as\ndashed line in Fig. 6 and shows good agreement with the\ndata. This indicates that, even though valence exchange\nand other types of impurities cannot be rigourously ex-\ncluded, rare-earth ions are indeed the dominant source\nfor the linewidth peak at low temperatures.\nV. CONCLUSIONS\nWe determined the ferromagnetic dispersion and\nlinewidth of the (110) magnetostatic mode of a polished\nYIG sphere as a function of temperature and frequency.\nFrom this data, we extract the Gilbert damping param-\neter for temperatures above 100 K and \fnd that it varies\nlinearly with temperature as expected according to the\ntwo competing theories of Kasuya and LeCraw28andCherepanov et al.34. At low temperatures, the temper-\nature dependence of the linewidth measured at constant\nmagnetic \feld shows a peak that shifts to higher tempera-\ntures with increasing frequency. This indicates slowly re-\nlaxing impurities as the dominant relaxation mechanism\nfor the magnetostatic modes below 100 K. We model\nthe shift of the peak position with temperature and fre-\nquency with values reported for Neodymium impurities32\nin combination with a typical value for the impurity-ion\nto iron-ion exchange coupling. We \fnd that these param-\neters can be used to describe the position of the linewidth\npeak. We thus directly show the implications of (rare\nearth) impurities as typically present in YIG samples\non the dynamic magnetic properties of the ferrimagnetic\ngarnet material. Furthermore, we extract the temper-\nature dependence of the saturation magnetization, the\nanisotropy \feld and the g-factor.\nACKNOWLEDGEMENTS\nThe authors thank M. S. Brandt for helping out with\nthe microwave equipment. C.D. and S.O. would like to\nacknowledge R. Meyer, M. Reich, and B. Wenzel (IN-\nNOVENT e.V.) for technical assistance in the YIG crys-\ntal growth and sphere preparation. We gratefully ac-\nknowledge funding via the priority program Spin Caloric\nTransport (spinCAT), (Projects GO 944/4 and GR\n1132/18), the priority program SPP 1601 (HU 1896/2-\n1) and the collaborative research center SFB 631 of the\nDeutsche Forschungsgemeinschaft.\n\u0003hannes.maier-\raig@wmi.badw.de\n1X. Zhang, C.-l. Zou, L. Jiang, and H. X. Tang, Physical\nReview Letters 113, 156401 (2014).\n2X. Zhang, C.-l. Zou, N. Zhu, F. Marquardt, L. Jiang, and\nH. X. Tang, Nature Communications 6, 8914 (2015).\n3Y. Tabuchi, S. Ishino, T. Ishikawa, R. Yamazaki, K. Usami,\nand Y. Nakamura, Physical Review Letters 113, 083603\n(2014).\n4L. Bai, M. Harder, Y. P. Chen, X. Fan, J. Q. Xiao, and\nC.-M. Hu, Physical Review Letters 114, 227201 (2015).\n5S. Klingler, H. Maier-Flaig, R. Gross, C.-M. Hu, H. Huebl,\nS. T. B. Goennenwein, and M. Weiler, Applied Physics\nLetters 109, 072402 (2016).\n6S. Viola Kusminskiy, H. X. Tang, and F. Marquardt, Phys-\nical Review A 94, 033821 (2016).\n7R. Hisatomi, A. Osada, Y. Tabuchi, T. Ishikawa,\nA. Noguchi, R. Yamazaki, K. Usami, and Y. Nakamura,\nPhysical Review B 93, 174427 (2016).\n8N. J. Lambert, J. A. Haigh, S. Langenfeld, A. C. Doherty,\nand A. J. Ferguson, Physical Review A 93, 021803 (2016).\n9Y. Tabuchi, S. Ishino, A. Noguchi, T. Ishikawa, R. Ya-\nmazaki, K. Usami, and Y. Nakamura, Science 349, 405\n(2015).\n10E. G. Spencer, R. C. Lecraw, and A. M. Clogston, Physical\nReview Letters 3, 32 (1959).11M. Sparks and C. Kittel, Physical Review Letters 4, 232\n(1960).\n12K. P. Belov, L. A. Malevskaya, and V. I. Sokoldv, Soviet\nPhysics JETP 12, 1074 (1961).\n13M. Haidar, M. Ranjbar, M. Balinsky, R. K. Dumas,\nS. Khartsev, and J. \u0017Akerman, Journal of Applied Physics\n117, 17D119 (2015).\n14C. L. Jermain, S. V. Aradhya, J. T. Brangham, M. R. Page,\nN. D. Reynolds, P. C. Hammel, R. A. Buhrman, F. Y.\nYang, and D. C. Ralph, arXiv preprint arXiv:1612.01954\n(2016).\n15H. Maier-Flaig, \\Temperature dependent damp-\ning of yttrium iron garnet spheres { Measure-\nment data and analysis programs,\" (2017),\nhttps://dx.doi.org/10.17605/OSF.IO/7URPT.\n16S. Klingler, H. Maier-Flaig, C. Dubs, O. Surzhenko,\nR. Gross, H. Huebl, S. T. B. Goennenwein, and M. Weiler,\nApplied Physics Letters 110, 092409 (2017).\n17S. S. Kalarickal, P. Krivosik, M. Wu, C. E. Patton, M. L.\nSchneider, P. Kabos, T. J. Silva, and J. P. Nibarger, Jour-\nnal of Applied Physics 99, 093909 (2006).\n18M. L. Schneider, J. M. Shaw, A. B. Kos, T. Gerrits, T. J.\nSilva, and R. D. McMichael, Journal of Applied Physics\n102, 103909 (2007).9\n50 100 150 200 250\nTemperature (K)0.20.40.60.81.01.21.4¹0H0 (T)\n123456\n¢f110 (MHz)\n5152535\nf110\nres (GHz)\nFIG. 6. Full map of the FWHM linewidth of the (110) mode as function of temperature and \feld resp. resonance frequency\nf110\nres. At low temperatures, only the slow relaxor peak is visible while at high temperatures the Gilbert-like damping becomes\ndominant. The position of the peak in the linewidth modeled by a slow relaxor is shown as dashed orange line. The model\nparameters are taken from Clarke31and taking \u000ea= 2:50 meV. The dotted lines indicate the deviation of the model for 0 :5\u000ea\n(lowerTmax) and 2\u000ea(higherTmax).\n19H. T. Nembach, T. J. Silva, J. M. Shaw, M. L. Schneider,\nM. J. Carey, S. Maat, and J. R. Childress, Physical Review\nB84, 054424 (2011).\n20P. R oschmann and H. D otsch, physica status solidi (b) 82,\n11 (1977).\n21L. R. Walker, Physical Review 105, 390 (1957).\n22Due to the comparable dimensions of center conductor\nwidth and sphere diameter, we expect that the sphere ex-\nperiences an inhomogeneous microwave magnetic \feld with\nits main component parallel to the surface of the CPW and\nperpendicular to its center conductor. As the microwave\nmagnetic \feld is su\u000eciently small to not cause any non-\nlinear e\u000bects, a mode dependent excitation e\u000eciency is the\nonly e\u000bect of the microwave magnetic \feld inhomogeneity.\n23C. Kittel, Physical Review 76, 743 (1949).\n24P. Bruno, Physical Review B 39, 865 (1989).\n25J. Nemarich, Physical Review 136, A1657 (1964).\n26The resonance frequency of the DPPH sample that has\nbeen measured simultaneously was intended as a \feld cal-\nibration but can not be utilized due to the magnetic \feld\ninhomogeneity. In particular, since the homogeneity of our\nsuperconducting magnet system is speci\fed to 1 ‰for an\no\u000b-axis deviation of 2 :5 mm, the spatial separation of 5 mm\nof the DPPH and the YIG sphere already falsi\fes DPPH\nas an independent magnetic \feld standard. Placing DPPH\nand YIG in closer proximity is problematic as the stray\n\feld of the YIG sphere will a\u000bect the resonance frequency\nof the DPPH. Note further that we are not aware of any re-\nports showing the temperature independence of the DPPH\ng-factor with the required accuracy.\n27J. F. Dillon, Physical Review 111, 1476 (1958).\n28T. Kasuya and R. C. LeCraw, Physical Review Letters 67,\n223 (1961).\n29E. G. Spencer, R. C. Lecraw, and J. Linares, Physical\nReview 123, 1937 (1961).\n30M. Sparks, Ferromagnetic-Relaxation Theory , edited by\nW. A. Nierenberg (McGraw-Hill, 1964).\n31B. H. Clarke, Physical Review 139, A1944 (1965).\n32B. H. Clarke, K. Tweedale, and R. W. Teale, Physical\nReview 139, A1933 (1965).33M. Sparks, Journal of Applied Physics 38, 1031 (1967).\n34V. Cherepanov, I. Kolokolov, and V. L'vov, Physics Re-\nports 229, 81 (1993).\n35J. Barker and G. E. W. Bauer, Physical Review Letters\n117, 217201 (2016).\n36R. Orbach, Proceedings of the Physical Society 77, 821\n(1961).\n37B. H. Clarke, Journal of Applied Physics 36, 1211 (1965).\n38I. H. Solt, Journal of Applied Physics 33, 1189 (1962).\n39Note that we failed to reproduce the \ft of Ref. 38 using\nthe data provided in this paper and that the reasonable\nagreement with the there-reported \ft parameters might\nbe coincidence.\n40E. E. Anderson, Physical Review 134, A1581 (1964).\n41P. Hansen, Journal of Applied Physics 45, 3638 (1974).\n42G. Winkler, Magnetic Garnets , Tracts in pure and applied\nphysics; Vol. 5 (Vieweg, 1981).\n43R. L. White, Journal of Applied Physics 31, S86 (1960).\n44P. R oschmann and W. Tolksdorf, Materials Research Bul-\nletin18, 449 (1983).\n45In comparison to Klingler et al.16, here, we do not see\nan increased inhomogeneous linewidth of the (110) mode\nand no secondary mode that is almost degenerate with the\n(110) mode. The di\u000berence can be explained by the ori-\nentation of the sphere which is very di\u000ecult to reproduce\nvery accurately ( <1°) between the experimental setups:\nThe change in orientation either separates the mode that\nis almost degenerate to the (110) mode or makes the degen-\neracy more perfect in our setup. The di\u000berent placement of\nthe sphere on the CPW can also lead to a situation where\nthe degenerate mode is not excited and therefore does not\ninterfere with the \ft.\n46Y. Sun, Y.-Y. Song, H. Chang, M. Kabatek, M. Jantz,\nW. Schneider, M. Wu, H. Schultheiss, and A. Ho\u000bmann,\nApplied Physics Letters 101, 152405 (2012).\n47Naturally, the resonance frequency varies slightly\n(\u00060:9 GHz) between the data points because the magneti-\nzation and the anisotropy changes with temperature.\n48J. F. Dillon and J. W. Nielsen, Physical Review Letters 3,\n30 (1959)."    },    {        "title": "2108.02090v1.Nonlinear_fluid_damping_of_elastically_mounted_pitching_wings_in_quiescent_water.pdf",        "content": "This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics 1\nNonlinear fluid damping of elastically mounted\npitching wings in quiescent water\nYuanhang Zhu1y, Varghese Mathai2and Kenneth Breuer1\n1Center for Fluid Mechanics, School of Engineering, Brown University, Providence, RI 02912, USA\n2Department of Physics, University of Massachusetts, Amherst, MA 01003, USA\n(Received xx; revised xx; accepted xx)\nWe experimentally study the nonlinear fluid damping of a rigid but elastically mounted\npitching wing in the absence of a freestream flow. The dynamics of the elastic mount\nare simulated using a cyber-physical system. We perturb the wing and measure the fluid\ndamping coefficient from damped oscillations over a large range of pitching frequencies,\npitching amplitudes, pivot locations and sweep angles. A universal fluid damping scaling\nis proposed to incorporate all these parameters. Flow fields obtained using particle image\nvelocimetry are analyzed to explain the nonlinear behaviors of the fluid damping.\n1. Introduction\nThe interaction between elastically mounted pitching wings and unsteady flows is central\nto many applications. With a free-stream flow, this interaction can lead to self-sustained,\nflow-induced oscillations, which have been studied for understanding classic aeroelastic\nbehaviour (Dowell et al.1989; Dugundji 2008), as well as in developing oscillating foil\nenergy harvesting devices (Xiao & Zhu 2014; Young et al.2014). Without a free stream,\nbut with prescribed heaving or flapping (i.e. hovering), the passive flow-induced pitching\nmotionsareusedinmodellingthethrustgenerationandmaneuveringinanimalflight(Wang\n2005; Bergou et al.2007; Shinde & Arakeri 2013; Kang & Shyy 2014; Beatus & Cohen\n2015).\nOne of the critical parameters that govern the flow-structure interactions of passively\npitching wings is the fluid damping. According to the semi-empirical Morison equation\n(Morison etal.1950),thetotalfluidforceexertedonawingsubmergedinunsteadyviscous\nfluid can be divided into two parts – the force associated with fluid inertia (i.e. the added\nmass force), which is in phase with acceleration (Brennen 1982; Corkery et al.2019), and\nthe force induced by vortices in the flow (i.e. the fluid damping force), which is in phase\nwith velocity (Shih & Buchanan 1971; Kang & Shyy 2014; Su & Breuer 2019). While the\nstructuraldampingforceistypicallyproportionaltovelocitybecauseoftheconstantstructural\ndampingcoefficient,thefluiddampingforceisexpectedtoscalequadraticallywithvelocity\n(Morison et al.1950; Keulegan & Carpenter 1958), and due to this nonlinearity, the fluid\ndamping coefficient is usually obtained empirically as a function of the reduced frequency,\nthe Reynolds number, the oscillation amplitude, etc (Shih & Buchanan 1971). For pitching\nflexible wings (Alben 2008) and heaving membrane wings (Tzezana & Breuer 2019), the\nfluid damping coefficient is found to scale inversely with the oscillation frequency.\nyEmail address for correspondence: yuanhang_zhu@brown.eduarXiv:2108.02090v1  [physics.flu-dyn]  4 Aug 20212 Y. Zhu, V. Mathai and K. Breuer\nFor elastically mounted pitching wings with a free stream, the interplay between the fluid\ndamping and the structural damping governs the flow-induced oscillation. By mapping out\nthe cycle-averaged energy transfer between the elastic system and the ambient fluid using\nprescribedkinematics,Menon&Mittal(2019)andZhu etal.(2020)showedthattheenergy\ninjectedbythenegativefluiddampingmustbeequaltotheenergydissipatedbythepositive\nstructural damping in order for the flow-induced oscillations to sustain. In other words,\nthe total damping of the system must be zero (Dugundji 2008). The negative fluid damping\narisesprimarilyfromtheformationandsheddingofdynamicstallvortices(McCroskey1982;\nCorke&Thomas2015).Intheabsenceofafreestream,however,thefluiddampingbecomes\npositive and counteracts the pitching motion because of the drag effect. With both the fluid\ndamping and the structural damping being positive, any perturbations to the system will\nbe damped out. However, little is known about how the fluid damping shapes the damped\noscillations, and understanding this is of critical importance for understanding the fluid-\nstructure interactions of elastically mounted pitching wings under external perturbations\nsuch as gusts.\nIn the present study, we use laboratory experiments to characterise the fluid damping\nof elastically mounted pitching wings in quiescent water, with the elastic mount simulated\nusingacyber-physicalsystem(§2).Weperform‘ringdown’experimentstoextractthefluid\ndamping(§3.1).Theeffectsofmanyparametersareinvestigated,includingtheeffectsofthe\npitching frequency, the pitching amplitude, the pivot location and the sweep angle (§3.2).\nWe propose a universal fluid damping scaling to incorporate these parameters (§3.3), and\ncorrelate the nonlinear behaviour of the fluid damping with the dynamics of the vortical\nstructures measured using particle image velocimetry (§3.4). Finally, the key findings are\nsummarised in §4.\n2. Experimental set-up\nFigure 1(a) shows a schematic of the experimental set-up. We conduct all the experiments\nin the Brown University free-surface water tunnel (test section width\u0002depth\u0002length=\n08 m\u000206 m\u000240 m), with the flow speed kept at zero ( 𝑈1=0m/s). A NACA 0012\nwing, made of clear acrylic, is mounted vertically in the tunnel, with an endplate on the\ntop to skim surface waves and eliminate wingtip vortices at the root. The wing is connected\nto a six-axis force/torque transducer (ATI 9105-TIF-Delta-IP65), which measures the fluid\ntorque𝜏𝑓exerted on the wing. This 𝜏𝑓is then fed into the cyber-physical system (CPS).\nDepending on the input virtual structural parameters, specifically the torsional stiffness 𝑘𝑣,\ndamping𝑏𝑣andinertia𝐼𝑣,theCPScalculatesthepitchingpositionofthewingandoutputs\nthe signal to the servo motor (Parker SM233AE). An optical encoder (US Digital E3-2500)\nwhich is independent of the CPS is used to measure the pitching position 𝜃. The CPS is\noperated at 4000 Hz to minimise any phase delay between the input 𝜏𝑓and the output 𝜃. A\ndetailed explanation of the CPS can be found in Zhu et al.(2020).\nWeusetwo-dimensionalparticleimagevelocimetry(PIV)tomeasuretheflowfieldaround\nthewing.Theflowisseededusing50 𝜇mdiameterhollowceramicspheresandilluminated\nbyalasersheetatthemid-spanplane.Thelasersheetisgeneratedbyadouble-pulseNd:YAG\nlaser(532nm,QuantelEverGreen)withLaVisionsheetoptics.Thetransparentwingenables\nflow field measurements on both sides of the wing. Due to the limitation of space beneath\nthe tunnel, a 45\u000emirror is used to reflect the images into two co-planar sCMOS cameras\n(LaVision).WeusetheDaVissoftware(LaVision)tocalculate(twopassesat 64\u000264pixels,\ntwo passes at 32\u000232pixels, both with 50% overlap) and stitch the velocity fields from the\ntwo cameras to form a field of view of 32𝑐\u000232𝑐, where𝑐is the chord length of the wing.\nFigure 1(b) sketches the two types of wings we use in the present study. For the unsweptNonlinear fluid damping of pitching wings 3\nServo motor\n& gearbox\nForce \ntransducer\nLaser sheet\n@ mid-spanEndplate\n2×sCMOS\nwith 35mm lensMirrorOptical encoder\nU   = 0 m/sNACA 0012 \n(transparent)CPS\nUnswept wing\nΛx/c0 10.5\nSwept wing(a) (b)\ncs\ncsLE \nLE TE \nTE k\nb\nIv\nv\nv\n∞\nF/i.pc/g.pc/u.pc/r.pc/e.pc 1. (a) A schematic of the experimental set-up. The structural dynamics of the wing is simulated by\na cyber-physical system (CPS). ( b) Sketches of unswept and swept wings. The leading edge (LE) and the\ntrailing edge (TE) are parallel. Dashed lines represent the pivot axis.\nwing, a wing holder mechanism (not shown) enables the pivot axis to be adjusted between\n𝑥𝑐=0and 1 with a step size of 0.125. For the swept wings, the sweep angle Λis defined\nastheanglebetweentheleadingedgeandtheverticalaxis.Foursweptwingswith Λ=10\u000e,\n15\u000e,20\u000eand25\u000eareused.Asshowninthefigure,thepivotaxisofsweptwingsisavertical\nline passing through the mid-chord point ( 𝑥𝑐=05) of the mid-span plane. All the wings\nhaveaspanof 𝑠=03mandachordlengthof 𝑐=01m,whichresultsinanaspectratioof\n𝐴𝑅=3.\nThe governing equation of the system is\n𝐼¥𝜃¸𝑏¤𝜃¸𝑘𝜃=𝜏𝑓 (2.1)\nwhere𝜃,¤𝜃,and¥𝜃aretheangularposition,velocityandacceleration,respectively. 𝐼,𝑏and𝑘\naretheeffectiveinertia,dampingandstiffnessofthesystem.Theeffectiveinertia 𝐼isthesum\nof the virtual inertia 𝐼𝑣, which we prescribe with the CPS, and the physical inertia 𝐼𝑝of the\nwing (i.e.𝐼=𝐼𝑣¸𝐼𝑝). The effective damping 𝑏equals the virtual damping 𝑏𝑣(i.e.𝑏=𝑏𝑣)\nbecause the friction in the system is negligible. The effective stiffness 𝑘equals the virtual\nstiffness(i.e. 𝑘=𝑘𝑣).𝜏𝑓isthenonlinearfluidtorqueexperiencedbythewing,whichcanbe\ndivided into the added mass torque, 𝜏𝑎=\u0000𝐼𝑎¥𝜃, where𝐼𝑎is the added fluid inertia, and the\nfluid damping torque, for simplicity 𝜏𝑏=\u0000𝑏𝑓¤𝜃, where𝑏𝑓is the fluid damping coefficient\n(see §1). Note that 𝑏𝑓is expected to be a function of ¤𝜃(Mathaiet al.2019). Equation 2.1\ncan thus be rearranged as\n¹𝐼¸𝐼𝑎º¥𝜃¸¹𝑏¸𝑏𝑓º¤𝜃¸𝑘𝜃=0 (2.2)\nAfter a perturbation of amplitude 𝐴0is applied at time 𝑡0, the damped oscillations of the\nsystem can be described as\n𝜃=𝐴0𝑒\u0000𝛾¹𝑡\u0000𝑡0ºcos»2𝜋𝑓𝑝¹𝑡\u0000𝑡0º¼ (2.3)\nwhere\n𝛾=𝑏¸𝑏𝑓\n2¹𝐼¸𝐼𝑎ºand𝑓𝑝=1\n2𝜋√︂\n𝑘\n𝐼¸𝐼𝑎\u0000𝛾2¹24abº4 Y. Zhu, V. Mathai and K. Breuer\n0 50 100 150-2-1012\n0 0.5 1 1.5 2 2.5024610-3\n51 5501\n0 1 20.380.400.42(a) (b)\nn+1 n\nF/i.pc/g.pc/u.pc/r.pc/e.pc 2. (a) System response and amplitude decay in a typical ‘ring down’ test, where an elastically\nmounted unswept wing ( Λ=0\u000e) pivots around the mid-chord ( 𝑥𝑐=05) at a frequency of 𝑓𝑝=040Hz.\nThe inset shows the measurements of the pitching amplitude 𝐴𝑛and the pitching frequency 𝑓𝑝of the𝑛-th\npeak. The fluid damping 𝑏𝑓at𝐴𝑛is extracted by fitting an exponential curve (i.e. the red solid line) to the\nadjacent three peaks. ( b) Extracted𝑏𝑓in air and in water. The zero value is indicated by the black dashed\nline. The inset compares the measured pitching frequency 𝑓𝑝in air and in water.\n3. Results and Discussion\n3.1.Extracting the fluid damping from ‘ring down’ experiments\nWeconduct‘ringdown’experimentstomeasurethefluiddampingexperiencedbyelastically\nmountedpitchingwings.Inthe‘ringdown’experiment,ashort-timeconstant-torqueimpulse\nisappliedtotheCPSastheperturbation,afterwhichthesystemresponseandtheamplitude\ndecay of the wing is recorded and analysed. Figure 2( a) shows the results from a typical\n‘ringdown’experiment.Inthisspecificcase,weuseanunsweptwing( Λ=0\u000e)whichpivots\naround the mid-chord ( 𝑥𝑐=05) at a frequency of 𝑓𝑝=040Hz. We conduct the ‘ring\ndown’experimenttwice–onceinairandonceinwater.Thepitchingamplitudeofthewing\ndecays faster in water than in air, indicating a higher total damping in water.\nToquantifythisamplitudedecay,thepositivepeaksofthesystemresponseareidentified.\nAsshownintheinset,theamplitudeofthe 𝑛-thpeakisdenotedby 𝐴𝑛,andthecorresponding\npitchingfrequencyismeasuredas 𝑓𝑝=2¹𝑡𝑛¸1\u0000𝑡𝑛\u00001º.Tomeasurethetotaldamping 𝑏¸𝑏𝑓\nat amplitude 𝐴𝑛, we fit an exponential, 𝑦=𝛼𝑒\u0000𝛾𝑡, to the three adjacent peaks, 𝑛\u00001,𝑛and\n𝑛¸1,andextractthecorresponding 𝛾(seeequation2.3).Nowtheonlyunknowninequation\n2.4bistheaddedmass, 𝐼𝑎.Afterobtaining 𝐼𝑎,thefluiddamping, 𝑏𝑓,isthencalculatedusing\nequation 2.4 a(Rao 1995). Since 𝑓𝑝and𝛾are both measured, 𝐼𝑎and𝑏𝑓are alsomeasured\nquantities. Moreover, both 𝐼𝑎and𝑏𝑓arecycle-averaged , meaning they cannot reflect the\ninstantaneousvariationofthefluidinertiaanddamping.Themeasuredfluiddamping, 𝑏𝑓,in\nbothairandwaterarecomparedinfigure2( b).Since𝜏𝑓inequation2.1isnegligibleinairas\ncomparedtootherforcesintheequation, 𝑏𝑓staysnearzero,whichisindicatedbythegood\nagreementbetweentheredcirclesandtheblackdashedline.Asshownbythegreensquares,\n𝑏𝑓in water is significant because of the existence of the fluid damping torque, 𝜏𝑏. It is also\nobserved that 𝑏𝑓in water increases non-monotonically with 𝐴. This nonlinear behaviour\nwillberevisitedlaterin§3.4.Theinsetoffigure2( b)showsthemeasuredpitchingfrequency,\n𝑓𝑝, in both air and water. Due to the combined effect of the fluid inertia and damping, we\nsee that𝑓𝑝is slightly lower in water than in air.Nonlinear fluid damping of pitching wings 5\n0 0.5 1 1.5 2 2.5051015\n0 0.5 1 1.5 2 2.504812(a) (b) 10-310-3\nF/i.pc/g.pc/u.pc/r.pc/e.pc 3. (a) Extracted fluid damping 𝑏𝑓at different pitching frequencies for 𝑥𝑐=05. (b) A frequency\nscaling for the fluid damping which collapses 𝑏𝑓at different𝑓𝑝into one curve. Note that ( a) and (b) share\nthe same legend.\n3.2.Frequency scaling of the fluid damping\nWe repeat the ‘ring down’ experiment for the unswept wing ( Λ = 0\u000e) pivoting at the mid-\nchord,𝑥𝑐=05, and change the pitching frequency by tuning the virtual inertia, 𝐼𝑣, and\nthe virtual stiffness, 𝑘𝑣, while keeping the virtual damping 𝑏𝑣constant (Onoue & Breuer\n2016, 2017). The extracted fluid damping, 𝑏𝑓, are shown in figure 3( a). Note that figure\n3(a) and (b) share the same legend. We observe that 𝑏𝑓increases monotonically with the\npitching frequency, 𝑓𝑝, and that the trend of 𝑏𝑓remains consistent for all frequencies.\nThis observation agrees with those observed in heaving rigid plates (Keulegan & Carpenter\n1958; Shih & Buchanan 1971), where the fluid damping coefficient scales inversely with\nthe oscillation period. As we discussed earlier, 𝑏𝑓derives from the fluid damping torque\n𝜏𝑏, which depends strongly on the vortex-induced forces on the wing (Kang & Shyy 2014).\nOnoue & Breuer (2016, 2017) have shown that the circulation of LEVs scales with the\nstrength of the feeding shear-layer velocity. In our case without a free-stream flow, the\nfeedingshear-layer velocityequals theleading-/trailing-edgevelocity, whichisproportional\nto𝑓𝑝. Based on this, we divide 𝑏𝑓by𝑓𝑝(figure 3b). It is seen that with this scaling, all of\nthe fluid damping curves collapse nicely.\nWe extend this frequency scaling to unswept wings with different pivot axes (figure 4 a)\nand to swept wings with different sweep angles (figure 4 b). For comparison, we include the\nprevious results (figure 3 b) using purple circles in both figure 4( a) and (b). Note that each\nsymbol shape in figure 4 contains fivedifferent pitching frequencies, 𝑓𝑝=020, 0.28, 0.40,\n0.56 and 0.78 Hz.\nFor the unswept wing ( Λ = 0\u000e), we change the pivot axis from 𝑥𝑐=0to 1 with a step\nsizeof0.125(seetheinsetoffigure4 a).Weobservethat 𝑏𝑓𝑓𝑝increasesasthepivotaxisis\nmoved away from the mid-chord, 𝑥𝑐=05. For pivot axes that are symmetric with respect\nto the mid-chord (i.e. 𝑥𝑐=0375& 0.625, 0.25 & 0.75, 0.125 & 0.875 and 0 & 1), 𝑏𝑓𝑓𝑝\nroughlyoverlap.Theslightinconsistencybetween 𝑏𝑓𝑓𝑝for𝑥𝑐¡05and𝑥𝑐05comes\nfromtheasymmetryoftheNACA0012winggeometrywithrespecttothemid-chord;wesee\nthat the scaled damping, 𝑏𝑓𝑓𝑝, is always slightly higher for 𝑥𝑐 05. In these cases, the\ndamping at the trailing edge dominates due to the higher velocity and longer moment arm,\nand is stronger than the cases when 𝑥𝑐 ¡05, where the leading-edge damping dominates.\nWe will show in §3.4 that this is due to differences in the vortex structures generated by the\nsharp and rounded geometries.\nThisfrequencyscaling, 𝑏𝑓𝑓𝑝,alsoholdsforthree-dimensional(3D)sweptwings(figure6 Y. Zhu, V. Mathai and K. Breuer\n0 0.5 1 1.5 2 2.500.030.060.09\n0 0.5 1 1.5 2 2.500.050.100.15\n0 0.5 1(a) ( b)\nNACA 0012 \nΛPivot axis\nF/i.pc/g.pc/u.pc/r.pc/e.pc 4. (a)𝑏𝑓𝑓𝑝for an unswept wing ( Λ=0\u000e) pivoting at 𝑥𝑐=0to 1 with a step size of 0.125. The\npivotlocationforeachdatasetisshownbytheinset.( b)𝑏𝑓𝑓𝑝forsweptwingswith Λ=0\u000e,10\u000e,15\u000e,20\u000e\nand25\u000e.Theinsetshowssideviewsofthefivesweptwingsandthedashedlineindicatesthepivotaxis.The\ncolours of the wings correspond to the colours of 𝑏𝑓𝑓𝑝curves in the figure. The purple circles in ( a) and\n(b) are replotted from figure 3( b). Note that each dataset in ( a) and (b) includes fivedifferent𝑓𝑝.\n4b). Again, each curve includes data from five pitching frequencies. Here, the pivot axes\nof swept wings are kept as a vertical line passing through the mid-chord of the mid-span\nplane (see the inset of figure 4 b). AsΛincreases, the average pivot axes of the top and the\nbottom portion of the swept wing move away from the mid-chord, leading to the increase\nof the scaled damping, 𝑏𝑓𝑓𝑝, in a manner similar to that observed for unswept wings with\ndifferent pivot locations (figure 4 a). This argument will be revisited in the next section.\n3.3.Universal fluid damping scaling for unswept and swept wings\nFigure 4( a) indicates that the pivot axis plays an important role in determining the fluid\ndamping of unswept wings. We extend the frequency scaling of 𝑏𝑓to take into account\nthis effect. First, we divide the wing into two parts, the fore part from LE to the pivot axis\nwith a chord length of 𝑐𝐿𝐸, and the aft part from the pivot axis to TE with a chord length\nof𝑐𝑇𝐸(see the inset of figure 5 for an example when the wing pivots at 𝑥𝑐=05). The\nMorisonequation(Morison etal.1950)indicatesthatthefluiddampingforce 𝐹scaleswith\n05𝜌𝑈2𝑠𝑐, where𝜌is the fluid density, 𝑈\u0018¤𝜃𝑐is the characteristic velocity and 𝑠𝑐is the\nwingarea.Wecanexpressthetotalfluiddampingtorqueasthesumofthetorqueexertedon\nthe fore and aft portions of the wing,\n𝜏𝑏\u0018𝐾𝐿𝐸𝐹𝐿𝐸𝑐𝐿𝐸¸𝐾𝑇𝐸𝐹𝑇𝐸𝑐𝑇𝐸 (3.1)\nwhere the subscripts 𝐿𝐸and𝑇𝐸refer to the leading- and trailing-edge contributions, and\n𝐾𝐿𝐸and𝐾𝑇𝐸are empirical factors that account for the subtle differences in the damping\nassociated with the specific geometries of the leading and trailing edges (figure 4 a). Since\nthe differences are small, 𝐾𝐿𝐸and𝐾𝑇𝐸should be close to one, and for consistency, their\naverage value must equal one ( ¹𝐾𝐿𝐸¸𝐾𝑇𝐸º2=1).\nSince the damped oscillations are observed to be near-sinusoidal (figure 2 a), the average\nangular velocity is given by 4𝑓𝑝𝐴. Simplifying, we arrive at an expression for the fluid\ndamping:\n𝑏𝑓\u00182𝜌𝑓𝑝𝐴𝑠¹𝐾𝐿𝐸𝑐4\n𝐿𝐸¸𝐾𝑇𝐸𝑐4\n𝑇𝐸º (3.2)Nonlinear fluid damping of pitching wings 7\n0 0.5 1 1.5 2 2.50123\ncLE cTE s\ncLE,botcTE,bots/2 s/2 ΛcLE,topcTE,topFLE FTE \nF/i.pc/g.pc/u.pc/r.pc/e.pc 5. Non-dimensional fluid damping coefficient 𝐵\u0003\n𝑓versus pitching amplitude 𝐴for unswept wings\npivotingat𝑥𝑐=0to1andsweptwingswithsweepangles Λ=0\u000eto25\u000e.Theinsetshowsthedefinitionof\nthe leading-edge chord 𝑐𝐿𝐸and the trailing-edge chord 𝑐𝑇𝐸, with black dashed lines indicating the pivot\naxes. The black dotted line indicates the small amplitude prediction for a drag coefficient of 𝐶𝐷=28.\nor, in non-dimensional form,\n𝐵\u0003\n𝑓\u0011𝑏𝑓\n2𝜌𝑓𝑝𝑠¹𝐾𝐿𝐸𝑐4\n𝐿𝐸¸𝐾𝑇𝐸𝑐4\n𝑇𝐸º/𝐴 (3.3)\nForsweptwings,becausethepivotaxispassesthrough 𝑥𝑐=05atthemid-span,thetop\nhalf of the wing has an average pivot axis 𝑥𝑐 ¡05, while the bottom half has an average\npivot axis𝑥𝑐 05. Ignoring three-dimensional effects, we approximate the swept wing\nby two ‘equivalent’ unswept wing segments. We choose not to divide the wing into a large\nnumberofnarrow‘bladeelements’(Glauert1983),becausethepivotaxisofsomeelements\nnear the wing root/tip for large sweep angles may lie outside the range 𝑥𝑐=»01¼, where\nour scaling has not been tested. The inset of figure 5 shows how these two unswept wing\nsegmentsareconfigured(rectangleswithreddottedlines).Basedonthewinggeometry,we\nsee that\n𝑐𝐿𝐸𝑡𝑜𝑝=𝑐𝑇𝐸𝑏𝑜𝑡=𝑐\n2¸𝑠\n4tanΛ\n𝑐𝑇𝐸𝑡𝑜𝑝=𝑐𝐿𝐸𝑏𝑜𝑡=𝑐\n2\u0000𝑠\n4tanΛ(3.4)\nFollowing the same analysis as for the unswept wing, and adding the fluid damping of the\ntopandthebottomwingsegmentstogether,wefindthatthefluiddampingforthefullswept\nwing is given by\n𝑏𝑓\u0018𝜌𝑓𝑝𝐴𝑠¹𝐾𝐿𝐸𝑐4\n𝐿𝐸𝑡𝑜𝑝¸𝐾𝑇𝐸𝑐4\n𝑇𝐸𝑡𝑜𝑝¸𝐾𝐿𝐸𝑐4\n𝐿𝐸𝑏𝑜𝑡¸𝐾𝑇𝐸𝑐4\n𝑇𝐸𝑏𝑜𝑡º(3.5)\nIf we define an effective leading-edge chord 𝑐𝐿𝐸=𝑐𝐿𝐸𝑡𝑜𝑝=𝑐𝑇𝐸𝑏𝑜𝑡and an effective\ntrailing-edgechord 𝑐𝑇𝐸=𝑐𝑇𝐸𝑡𝑜𝑝=𝑐𝐿𝐸𝑏𝑜𝑡,thisscalingreducestoequation3.2with 𝐾𝐿𝐸\nand𝐾𝑇𝐸cancelled out. This cancellation results because the effective pivot axes of the top\nand the bottom segments are symmetric about 𝑥𝑐=05at the mid-span, which averages\nout the slight differences in fluid damping experienced by the top and the bottom segments.\nFor the same reason, 𝐾𝐿𝐸and𝐾𝑇𝐸also cancel out in equation 3.3 for swept wings.8 Y. Zhu, V. Mathai and K. Breuer\n(a)\n-1 0 1 (b) (c) (d)\n(e)\n-1 0 1 -1 0 1 (f)\n-1 0 1 (g)\n-1 0 1 (h)\n-1 0 1 \n-50-25025 50 \nLEVTEV\nLEVTEVLEVTEVLEVTEV\nLEVTEV\nLEV TEV LEVTEVLEV\nTEV= 0.52 = 1.05 = 1.57 = 2.09\nF/i.pc/g.pc/u.pc/r.pc/e.pc 6. PIV flow field measurements for an unswept wing undergoing prescribed sinusoidal pitching\nmotions in quiescent water. ( a–d) Pivot axis (shown by green dots) 𝑥𝑐=05, pitching frequency 𝑓𝑝=05\nHz, pitching amplitude 𝐴=052¹30\u000eº105¹60\u000eº157¹90\u000eºand209¹120\u000eº. (e–h) Same as ( a–d),\nexcept that the pivot axis is at 𝑥𝑐=025. All the velocity fields are phase-averaged over 20 cycles. Only\neveryfifthvelocityvectorisshown.Spanwisevorticity 𝜔:positive(red),counterclockwise;negative(blue),\nclockwise. See supplementary materials for the full video.\nFigure 5 shows the non-dimensional fluid damping, 𝐵\u0003\n𝑓, as a function of the pitching\namplitude,𝐴,forunsweptandsweptwings.Here,wehaveused 𝐾𝐿𝐸=095and𝐾𝑇𝐸=105.\nWe see that all of our measurements collapse remarkably well under the proposed scaling,\nespecially for 𝐴  157¹90\u000eº, despite the wide range of pitching frequencies ( 𝑓𝑝=020\nto 0.78 Hz), pivot axes ( 𝑥𝑐=0to 1) and sweep angles ( Λ = 0\u000eto25\u000e) tested in the\nexperiments.Inthesmall-amplitudelimit( 𝐴05),𝐵\u0003\n𝑓scaleslinearlywith 𝐴,withaslope\nthat corresponds to the drag coefficient, 𝐶𝐷. We note that 𝐶𝐷\u001928, which is comparable\nto that of an accelerated normal flat plate (Ringuette et al.2007). At higher pitching angles\n(𝐴¡05),however,thelinearapproximationnolongerholdsandweseeadecreasingslope\nof𝐵\u0003\n𝑓as a function of 𝐴. This is presumably because the shed vortices no longer follow\nthe rotating wing and the fluid force becomes non-perpendicular to the wing surface as 𝐴\nincreases.For 𝐴¡157¹90\u000eº,thescalingworksreasonablywellexceptforthecase Λ=0\u000e,\n𝑥𝑐=05,whereadecreasing 𝐵\u0003\n𝑓isobserved.Inthenextsection,wewilluseinsightsfrom\nthe velocity fields to explain this non-monotonic behaviour.\n3.4.Insights obtained from velocity fields\nTo gain more insight regarding the nonlinear behaviour of 𝐵\u0003\n𝑓, we conduct 2D PIV\nexperiments to measure the surrounding flow fields of an unswept wing ( Λ = 0\u000e) with\na prescribed pitching motion: 𝜃=𝐴sin¹2𝜋𝑓𝑝𝑡º. The results are shown in figure 6. The\npitching frequency is kept at 𝑓𝑝=05Hz for all the cases and the pitching amplitude is\nvariedfrom𝐴=052¹30\u000eºto209¹120\u000eºwithastepsizeof 052¹30\u000eº.Twopivotaxesare\ntested,𝑥𝑐=05(figure6a–d)and𝑥𝑐=025(figure6e–h).Notethattheflowfieldsshown\nin figure 6 are notsequential. Instead, all the snapshots are taken right before 𝑡𝑇=025\nfor different pitching amplitudes, where 𝑇is the pitching period. This specific time instant\nis chosen because it best reflects the difference in dynamics associated with the different\npitching amplitudes and pivot axes.\nFor both pivot locations (figure 6 a–d:𝑥𝑐=05ande–h:𝑥𝑐=025), the spanwise\nvorticity of the pitch-generated leading-edge vortex (LEV) and trailing-edge vortex (TEV)Nonlinear fluid damping of pitching wings 9\nincreases with the pitching amplitude, 𝐴. This can be explained by the increase in the\nfeeding shear-layer velocities associated with the higher pitching amplitudes (Onoue &\nBreuer 2016). The boundary vortices near the wing surface, which are related to the added\nmass effect (Corkery et al.2019), also become more prominent due to the increase of the\nangular acceleration. When the wing pivots at 𝑥𝑐=05(figure 6a–d), the leading-edge\nvelocityequalsthetrailing-edgevelocity.Asaresult,theLEVandTEVarefairlysymmetric\nabout the pivot axis, with some subtle differences caused by the rounded and sharp edges,\nrespectively. This confirms the arguments given earlier for the differences between 𝑏𝑓𝑓𝑝\nfor𝑥𝑐 ¡05and𝑥𝑐 05(figure 4a). For𝑥𝑐=025(figure 6e–h), however, the TEV is\nmuchmoreprominentthantheLEVbecauseofthehighertrailing-edgevelocity.Duetothe\nlow leading-edge velocity and the pitch-induced rotational flow, the sign of the LEV even\nreverses and becomes negative for 𝐴=105to209(figure 6f–h).\nForbothpivotlocations,duetotheabsenceofaconvectivefreestream,andtheexistence\nof the pitch-induced rotational flow, the LEV and TEV (only the TEV for 𝑥𝑐=025) are\nentrainedclosertothewingsurfaceas 𝐴increases.For 𝑥𝑐=05,asshowninfigure6( c–d),\nthe LEV moves towards the aft portion of the wing and the TEV moves towards the fore\nportionofthewingwhen 𝐴>157¹90\u000eº.Thetorquegeneratedbythesetwovortices,which\ncounteracts the wing rotation for small 𝐴, now assists the rotation as the wing pitches up\ntowardshigherangularpositions.Thisassistreducesthefluiddragexperiencedbythewing\nandthuslowersthefluiddamping.Thiseffectcanaccountforthenon-monotonicbehaviour\nof𝐵\u0003\n𝑓for𝑥𝑐=05(figure 5). For 𝑥𝑐=025(figure 6g–h), a similar scenario is observed,\nin which the TEV moves towards the fore portion of the wing and gets closer to the wing\nsurfaceas𝐴increases.However,becauseoftheexistenceofacounter-rotatingLEV,theTEV\nisnotabletoapproachthewingsurfaceascloselyasinthe 𝑥𝑐=05case.Thisexplainswhy\na flattening behaviour, rather than a non-monotonic trend of 𝐵\u0003\n𝑓, is observed for 𝑥𝑐=025\nand presumably for other pivot locations at high pitching amplitudes.\n4. Conclusions\nBy utilising a cyber-physical control system to create an elastically mounted pitching wing,\nwehaveexperimentallymeasuredthenonlinearfluiddampingassociatedwithvorticesshed\nfrom a bluff body. A theoretical scaling has been proposed and validated, based on the\nMorison equation, which incorporates the frequency, amplitude, pivot location and sweep\nangle. The nonlinear behaviour of the scaled fluid damping has been correlated with the\nvelocity fields measured using particle image velocimetry.\nOne should note that our scaling may not be applicable for instantaneous fluid damping,\nbecausethedampingcharacterisedinthepresentstudyiscycle-averagedovernear-sinusoidal\noscillations.Inaddition,wehavenotconsideredthree-dimensionaleffects,whicharepresent\ndue to the wing tip flows. Incorporating these may further improve the collapse of the fluid\ndamping coefficient, 𝐵\u0003\n𝑓(figure 5).Lastly, in §3.4, onlyqualitative analysis ofthe flow field\nhasbeenperformedthusfar.Inordertogetmoreaccuratecorrespondencebetweenthefluid\ndampingandtheflowdynamics,quantitativeanalysisofthevortextrajectoryandcirculation\nis needed, which will be the focus of future study.\nDespitetheselimitations,theproposedscalinghasbeenshowntocollapsethedataovera\nwiderangeofoperatingconditions( 𝑓𝑝=020to0.78Hzand 𝐴=0to2.5)forbothunswept\n(𝑥𝑐=0to1)andsweptwings( Λ=0\u000eto25\u000e).Itcanbeusedtopredictdampingassociated\nwithshedvortices,andthusbenefitthefuturemodellingofawidevarietyofflows,including\nunswept and swept wings in unsteady flows as well as other bluff body geometries. The\nuniversality of this scaling reinforces the underlying connection between swept wings and10 Y. Zhu, V. Mathai and K. Breuer\nunswept wings with different pivot locations. 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Fluid Mech. 899, A35."    },    {        "title": "1509.01807v1.Study_of_spin_dynamics_and_damping_on_the_magnetic_nanowire_arrays_with_various_nanowire_widths.pdf",        "content": "1 \n Study of spin dynamics  and damping  on the magnetic nanowire  arrays   \nwith various nanowire widths  \n \nJaehun Cho a, Yuya  Fujii b, Katsunori  Konioshi b, Jungbum Yoon c, Nam -Hui Kim a, Jinyong Jung a, \nShinji Miwa b, Myung -Hwa Jung d, Yoshishige Suzuki b, and Chun -Yeol You  a,* \n \na Department of Physics, Inha University , Inch eon, 402-751, South Korea  \nb Graduate School of Engineering Science,  \nOsaka University,  Toyonaka, Osaka 560 -8531,  Japan  \nc Department of Electrical and Computer  Engineering ,  \nNational University of Singapore , Singapore  117576  \nd Department of Physics, Sogang University, Seoul, 121 -742, South Korea  \n \nAbstract  \nWe investigate the spin dynamics including Gilbert damping in the ferromagnetic  nanowire \narray s. We have measured  the ferromagnetic resonance of ferromagnetic nanowire arrays \nusing vector -network analyzer ferromagnetic resonance  (VNA -FMR)  and analyzed the results \nwith the micromagnetic si mulations . We find excellent agree ment  between the experimental \nVNA -FMR spectra and micromagnetic simulations result  for various applied magnetic fields . \nWe find that  the demagnetization factor for longitudinal conditions, Nz (Ny) increases  \n(decreas es) as decreasing the nanowire width  in the micromagnetic simulation s. For the \ntransverse magnetic field , Nz (Ny) increas es (decreas es) as increasing the nanowire width . We \nalso find that t he Gilbert damping constant increases  from 0.018 to 0.051 as the incr easing \nnanowire width  for the transverse case , while it is almost constant as 0.021  for the \nlongitudinal case . \n 2 \n * Corresponding author.  FAX: +82 32 872 7562.  \nE-mail address:  cyyou@inha.ac.kr  \nKeywords : Nanowires , Ferromagnetic Resonance , Micromagnetic  Simulations , Gilbert \ndamping  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 3 \n  \nFerromagnetic nanostructures have recently attracted much interest for the wide potential \napplications in high density spintronic information storage , logic devices and various spin \norbit torque phenomena .1,2,3,4,5 It is well known that the detail spin dynamics of nanostructure \nis far from the one of the bulk’s because of many reasons, different boundary conditions, \nchanges of the magnetic properties including the saturat ion magnetization, anisotropy energy, \nand exchange stiffness constant, etc. Since the magnetic properties are usually sensitive \nfunctions of the sample fabrication conditions, it has been widely accepted that the detail \nsample fabrications are also importa nt in the study of spin dynamics. However, the relatively \nless caution has been made for the boundary conditions of the spin dynamics in the \nnanostructure.  \nIn the spin transfer torque magnetic random access memory  (STT -MRAM), the magnetic \ndamping constant is important because the switching current density  is proportional to the \ndamping constant .6 In the nanowire, damping constant also plays crucial role in the spin \ndynamics including domain wall motion with magnetic field7 and spin transfer torque .8 \nFurthermore, it is the most important material parameter in spin wave (SW) dynamics .9 \nDespite of the importance of the damping constant, many studies about spin dynamics in \nferromagnetic nanowires  have not taken into account the damping constant  properly .10,11,12 \nOnly a few studies paid attention to the magnetic  damping in the nanowires  spin 4 \n dynamics .13,14 \nIn this study, arrays of CoFeB nanowire s are prepared by e -beam lithography , and they are  \ncovered coplanar  wave -guide  for the ferromagnetic resonance  (FMR) measurement  as shown \nin Fig. 1 . We measured FMR signal with longitudinal (wire direction) and transverse \nmagnetic field s in order to investigate the spin dynamics with different boundary conditions. \nAlso w e extract Gilbert damping constant  using micromagnetic  simulations  with the different \napplied magnetic field directions in various nanowire arrays . We find the damping constant \ndecreas es with increasing the nanowire width for the transverse magnetic field  with constant \ninput damping consta nt in micromagnetic simulations, while we obtain almost constant \ndamping constant for the longitudinal field.  \nThe film s were  prepared using DC magnetron sputtering. The stack s consist of Ta (5  \nnm)/Co 16Fe64B20 (30 nm)/Ta (5  nm) on single crystal MgO (001) substrate s. The film s are \npatterned as 100 -nm-width wire array s with 200 -nm-space each wires using e -beam \nlithography and an Ar ion milling technique  as shown in Fig. 1. The width is determined with \na scanning electron microscope (SEM).  These nanowire arrays are covered by coplanar wave \nguide in order to characterized with the Vector Network Analyzer ( VNA )-FMR technique \ndescribed elsewhere .15 We prepare nanowire arrays  as shown in Fig. 1 , and external DC \nmagnetic field direction  for FMR measurement  is also depicted.  \nWe use  VNA -FMR spectra to measure  imaginary parts of the susceptibility of the samples.16 5 \n The measured imaginary parts of the susceptibility raw data are calibrated with the careful \ncalibration procedures .16 The calibrated  imaginary parts of the susceptibility are shown in Fig \n2(a) and (b)  for an applied magnetic field at 0.194  T for the nanowire arrays . The un -\npatterned thin film is also examined for the reference. We find two resonance frequencies, \n17.2 and 26 .4 GHz, as shown in Fig. 2(a) for the nanowire array, while  there is only one peak \nat 16.8 GHz  for the  un-patterned  thin film  as shown in Fig 2(b). We believe that the smaller \npeak  (17.2 GHz) in Fig. 2(a) is originated  from the un-patterned part of the nanowire array, \nbecause the frequency is closed to the un -patterned thin film’s peak (16.8 GHz). Probably, the \nun-patterned part of the nanowire is formed due to poor e -beam lithography processes. On the \nother hand, t he resonance f requency  near 26.0 GHz is calculated from micromagnetic \nsimulation  at an applied  magnetic  field at 0.200 T , as shown in Fig. 2 (c).  We clarif y the \nsource of the main  peak (26.4 GHz) is nanowire  arrays  by using micromagnetic simulation.  \nThese two peaks name d as the uniform FMR  mode (smaller peak position) and nanowire \nmode (higher peak position).  \nIn order to determine the saturation magnetization, the resonance frequencies are measured \nas a function of the applied magnetic field, and the results are fitted with the Kittel ’s \nequation .17 This equation employs the corresponding demagnetization factors of Nx = 0, Ny = \n0 and Nz = 1 for un -patterned film, when applied magnetic field H is x- direction  with \nfollowing equations , 6 \n  \n    2y x s z x s f H N N M H N N M\n    \n.   (1) \n \n Here,  is the gyromagnetic ratio, H is the applied magnetic field, Ms is saturated \nmagnetization, Nx, Ny, and Nz are the demagnetization factor s applying the cyclic permutation \nfor the  applied magnetic field direction .  \nThe micromagnetic simulations are performed by using the Objective -Oriented -\nMicroMagnetic Framework (OOMMF)18 with 2-dimensional  periodic boundary  condition \n(PBC ).19 We select a square slat of 100 nm × 100 nm × 30 nm  nanowire separated 200 nm in \ny- direction with a cell size of 5 nm × 5 nm ×  30 nm. The material parameters of CoFeB used \nin our simulation are summarized as follows: Ms = 15.79 × 105 A/m, the exchange stiffness \n1.5 × 1011 J/m, the gyromagnetic ratio 2.32  × 1011 m/(A ∙s) and we ignore the magneto -\ncrystalline anisotropy. In this simulation, the Gilbert damping constant  of 0.0 27 is fixed. The \nsaturation magnetization and Gilbert damping constant are determined by using VNA -FMR  \nmeasurement for un -patterned thin film . For t he exchange  stiffness constant, experimentally \ndetermined values are range of 0.98 to 2.84 × 1011 J/m which value  has dependence on the \nfabrication processes20 and composition of ferromagnetic materials ,21 while we have picked \n1.5 × 1011 J/m as the exchange  stiffness  constant . The determination  method of Gilbert \ndamping constant will be described later.  7 \n In order to mimic FMR experiments in the micromagnetic simulations , a “sinc”  function\n0 0 0 ( ) sin 2 / 2y H HH t H f t t f t t    \n, with H0 = 10 mT, and field frequency fH = 45 \nGHz, is applied the whole nanowire  area.22 We obtain the FMR spectra in the corresponding \nfrequency range from 0 to 45 GHz . The FMR spectra due to the RF -magnetic field are \nobtained by the fast Fourier  transform  (FFT) of stored My(x) (x, y, t ) in longitudinal  (transverse)  \nH0 field. More details can be described elsewhere .23 \nThe closed blue circles  in Fig. 3 is the calculated values with the fitting parameter using Eq. \n(1) which are fitted with the experimental data of un -patterned thin film. The obt ained Ms is \n15.79 × 105 A/m while gyromagnetic ratio is fixed as 2.32 × 1011 m/(A∙s) . The obtained Ms \nvalue is similar  with vibrating sample magnetometer  method24 which CoFeB structure has Ta \nbuffer layer. The resonance frequencies of uniform FMR mode in nanowire arrays are plotted \nas open red circles in Fig. 3. The resonance frequencies of uniform FMR mode is measured \nby VNA -FMR are agreed well with resonance freq uency of un-patterned thin film measured \nby VNA -FMR. In Fig. 3, the applied field dependences of  the resonance frequencies \nMeasured by VNA -FAM for the nanowire are plotted as open black  rectangular , along with \nthe result of micromagnetics calculated with E q. (1)  as depicted  closed black rectangular . It is \nalso well agreed with the experimental result in nanowire mode and micromagnetic \nsimulation result  in the nanowire arrays.  \nIn order to reveal the effect s of spin dynamics properties with various  nanowire width s, we 8 \n perform micromagnetic simulat ions. The nanowire width s are varied from 50 to 150 nm in \n25-nm step for fixed 200 -nm-space with PBC , it causes changes of the demagnetization \nfactor  of the nanowire . In Fig. 4 (a) shows  the nanowire width  dependences of the resonance \nfrequencies for the longitudinal magnetic field (open symbols) along with the resonance \nfrequencies calculated with Eq. (1)  (solid lines) . The demagnetization factors can be \ndetermined by fitting Eq. (1) while Nx is fixed as 0  to represent infinitely long wire . The \nagreements between the results of micromagnetic simulations  (open circles)  and Eq. (1)  \n(solid lines)  are excellent.  \nFor the transverse  magnetic  field, the direction of applied magnetic field is y - axis, Eq. (1) \ncan be rewritten as follows:  \n \n    2x y s z y s f H N N M H N N M\n    \n.    (2) \n \nIn this equation, we use the relation of demagnetization factors , \n1x y zN N N   , in \norder to remove uncertainty in the fitting procedure . In the transverse field, the \ndemagnetization factors are determined by Eq. (2). The resonance frequencies  for transverse \nmagnetic field  which are obtained by micromagnetic simulation  (open circles)  and \ncalculated  by Eq. (2)  (solid lines)  as a function of the appl ied magnetic field with various \nnanowire width  are displayed in Fig. 4(b). The longitudinal case, when the field direction is 9 \n easy axis, they are saturated with small field. However, the transverse case, when the field \ndirection is hard axis, certain amoun t of field is necessary to saturate along the transverse \ndirections. The narrower nanowire, the larger field is required as shown in Fig. 4 (b).  \nFig. 5(a) and (b) show the changes of demagnetization factors in longitudinal and transverse \nmagnetic  fields as a functi on of  the nanowire width , respectively.  The demagnetization \nfactors play important role in the domain wall dynamics, for example the Walker breakdown \nis determined by the demagnetization factors .25 Furthermore, they are essential physical \nquantities to analyze the details of the spin dynamics. It is clear ly shown  that the Nz (Ny) \nincrease s (decrease s) with increasing the nanowire width  in longitudinal magnetic  field. For \nthe transverse magnetic  field, Nz (Ny) increase s (decrease s) with increasing the nanowire \nwidth , during Nx is almost zero value.  The demagnetization factors both longitudinal and \ntransverse have similar tendency with the effective demagnetization factors of dynamic \norigin26 and the static demagnetization factors for the prism geometry.27  \nNow, let us discuss about the Gilbert damping constant . The relation of the full width and \nhalf maxim a (f) of a resonance peak s as a function of applied field are shown in Fig. 6 for \nlongitudinal (a) and transverse (b). The f is given by15: \n \n,\n,2\n22xy\ns z ex yxN\nf H M N N f\n         \n. (3)  \n 10 \n where, fex is the extrinsic line width  contributions , when  the applied magnetic field  is x-(y-\n)axis for longitudinal (transverse) case . The symbol s are the result s of the micromagnetic \nsimulations and the solid lines are the fitting result of Eq. (3) . We use pre -determined \ndemagnetization factors (Fig. 5) during fitting procedures, and the agree ments are excellent.  \nWe have plotted the Gilbert damping constant as a function of the wavevector in nanowire \nwidth  (\n/ qa , a is the nanowire width ) in Fig. 7. The black open rectangles are data \nextracted from the transverse field and the red open circles are longitudinal field data. We \nfind that the Gilbert damping constant varied from 0.051 to 0.018 by changing  the \nwavevector  in nanowire width  in transverse field. On the other hand, lon gitudinal field case \nthe damping constant is almost constant as 0.021. Let us discuss about the un -expected \nbehavior of the damping constant of transverse case. T he wire width acts as a kind of cut -off \nwavelength of the SW excitations in the confined geome try. SWs whose wavelength are \nlarger than 2 a are not allowed in the nanowire. Therefore, only limited SW can be excited for \nthe narrower wire, while more SW can be existed in the wider wire. For example, we show \ntransverse standing SW as profiled  in the inset of Fig. 6 for 150 -nm width nanowire in our \nmicromagnetic simulations. More possible SW excitations imply more energy dissipation \npaths, it causes larger damping constant. For narrower nanowire (50 -nm), only limited SWs \ncan be excited, so that the damping constant is smaller. However, for the limit case of infinite \na case, it is the same with un -patterned thin films, there is no boundar y so that only uniform 11 \n mode can be excited, the obtained damping constant must be the input  value.  \nIn summary, the VNA -FMR experiments is employed  to investigate the magnetic properties \nof CoFeB nanowire  arrays  and the micromagnetic simulations is proposed to understand the \nmagnetic properties including Gilbert damping constant of various CoFeB nanowire  arrays  \nwidth. We f ind that the demagnetization factors are similar with the dynamic origin and static \nfor the prism geometry. The wire width or SW wavevector dependent damping constants can \nbe explained with number of SW excitation modes.  \n \nACKNOWLEDGMENTS  \nThis work  was supported by the National Research Foundation of Korea  (NRF) Grants  (Nos. \n616-2011 -C00017  and 2013R1A12011936 ). \nReferences  \n                                            \n1 S. S. P. Parkin,  M. Hayashi, and L. Thomas, Science 320, 190 (2008) . \n2 D. A. Allwood,  G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, and R. P. Cowburn,  \nScience 309, 1688 (2005) . \n3 I. M. Miron,  G. Gaudin, S. Aufftet, B. Rodmacq, A. Schuhl, S. Pizzini, J. V ogel and P. \nGambardella, Nature Materials 9, 230 (2010) . \n4 H-R Lee , K. Lee, J. Cho, Y . -H. Choi, C. -Y . You, M. -H. Jung, F. Bonell, Y. Shiota, S. Miwa \nand Y . Suzuki, Sci. Rep.  4, 6548 (2014).  \n5 J. Cho, et al.  Nat. Commun. 6, 7635 (2015).  \n6 S. Ikeda , K. 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Let t. 102, 127202 (2009).  \n23 J. Yoon, C. -Y . You, Y . Jo, S. -Y. Park, M. H. Jung , J. Korean Phys. Soc. 57, 1594 (2010) .  \n24 Y . Shiota, F. Bonell, S. Miwa, N. Muzuochi, T. Shinjo and Y . Suzuki , Appl. Phys. Le tt. 103. \n082410 (2013),  \n25 I. Purnama, I. S. Kerk, G. J. Lim  and W. S. Lew , Sci. Rep. 5, 8754 (2015).  \n26 J. Ding, M. Kostylev, and A. O. Adeyeye, Phys. Rev. B. 84, 054425 (2011) . \n27 A. Aharoni, J. Appl. Phys. 83, 3432 (2011) . 13 \n Figure  Captions  \n \nFig. 1. Measurement geometry with SEM image s of the 100 -nm-width nanowires with a gap \nof 200 nm between nanowires . The longitudinal nanowire arrays are shown. After the \nnanowire patterns have been  defined by e -beam lithography, they are covered by co -planar \nwave guides.  \n \nFig. 2. (a) The measured FMR  spectrum of the CoFeB nanowire with H =0.194 T. The red  (lower \npeak)  and blue  (higher peak)  arrows indicate t he resonance frequencies of the  uniform FMR  mode and \nthe nanowire mode, respectively. (b) The measured FMR spectrum of the CoFeB thin  film with H \n=0.194 T. (c) Simulated FMR spectrum of the CoFeB nanowire with H= 0.200 T.  \n \nFig. 3. Measured and calculated FMR frequencies with the applied magnetic field for 100 -\nnm-width nanowire. The open black rectangles  are nanowire mode and open red circles are \nthe uniform FMR mode for CoFeB thin film. The closed black rectangles  are calculated by \nOOMMF and the closed blue circles  are theoretical ly calculated by Eq. (1)  using fitted \nparameters form un -patterned film . \n \nFig. 4. Variation of resonanc e frequencies with the applied magnetic field for the different \nPBC wire width for (a) longitudinal field and (b) transverse field.  Inset: The geometry of 2 -\ndimensional PBC micromagnetic simulation with nanowire width a and a gap of 200 nm \nbetween nanowire s. The black open rectangles, red open circles, green open upper triangles, \nblue open down triangles, cyan open diamonds represent as nanowire width as 50 nm, 75nm, \n100 nm, 125nm, and 150 nm, respectively.  \n \nFig. 5. Demagnetization factor with PBC wire widt h for (a) longitudinal and (b) transverse \nfield.  The black open circles, red open rectangles, blue open upper triangles represent as \ndemagnetization factors, Ny, Nz, and Nx, respectively.  \n \nFig. 6. Full width and half maxim a with the applied magnetic field for (a) longitudinal and (b) \ntransverse field.  The black open rectangles, red open circles, green open upper triangles, blue \nopen down triangles, cyan open diamonds represent as nanowire width as 50 nm, 75nm, 100 \nnm, 125nm, and 150 nm, respectively.  \n \nFig. 7. Damping constants with wavevector for transverse ( the black open rectangles ) and \nlongitudinal ( the red open circles) field with errors. The black line is the input value which is \ndetermined from un -patterned film. Inset presents the profile of the trans verse spin density as SWs.  \n Fig. 1 \n \n \n    \n \n      Fig. 2. \n \n \n \nFig. 3. \n \n \n \n  \nFig. 4 \n \n` \n \n \n \nFig. 5 \n \n \n \nFig. 6 \n \n \n \nFig. 7 \n \n \n"    },    {        "title": "1805.01230v1.Exact_Intrinsic_Localized_Excitation_of_an_Anisotropic_Ferromagnetic_Spin_Chain_in_External_Magnetic_Field_with_Gilbert_Damping__Spin_Current_and_PT_Symmetry.pdf",        "content": "Exact Intrinsic Localized Excitation of an Anisotropic Ferromagnetic Spin Chain in External\nMagnetic Field with Gilbert Damping, Spin Current and PT-Symmetry\nM. Lakshmanan1,a)and Avadh Saxena2,b)\n1)Centre for Nonlinear Dynamics, School of Physics, Bharathidasan University,\nTiruchirappalli - 620 024, India\n2)Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory,\nLos Alamos, NM 87545, USA\nWe obtain the exact one-spin intrinsic localized excitation in an anisotropic Heisenberg\nferromagnetic spin chain in a constant/variable external magnetic field with Gilbert damping\nincluded. We also point out how an appropriate magnitude spin current term in a spin\ntransfer nano-oscillator (STNO) can stabilize the tendency towards damping. Further, we\nshow how this excitation can be sustained in a recently suggested PT-symmetric magnetic\nnanostructure. We also briefly consider more general spin excitations.\na)Electronic mail: lakshman.cnld@gmail.com\nb)Electronic mail: avadh@lanl.gov\n1arXiv:1805.01230v1  [cond-mat.mes-hall]  3 May 2018I. INTRODUCTION\nThe study of dynamics of classical Heisenberg ferromagnetic spin chain with anisotropic inter-\naction is of considerable importance in applied magnetism1,2and from application point of view3,4.\nWhile several continuum versions are known to be completely integrable soliton systems5–7, such\nas the isotropic case, no discrete integrable case is known in the literature, except for a modified\nversion, namely the Ishimori lattice8. On the other hand, the present authors9have shown the exis-\ntence of several classes of exact solutions in terms of Jacobian elliptic functions which exist for the\ncase of the discrete lattice including onsite anisotropy and external magnetic field. Identifying such\ninteresting classes of solutions and their relevance in the context of appropriate physical situations\nconstitute one of the important areas of investigation in spin dynamics9,10.\nFrom another point of view, occurrence of intrinsic localized breathers/oscillations in suitable\nanisotropic ferromagnetic spin chains is of practical relevance11,12and is being explored for the past\nseveral years. Apart from many numerical studies, in recent times the present authors and Subash13\nhave obtained explicit analytical solutions for the Heisenberg anisotropic spin chain with additional\nonsite anisotropy and constant external magnetic field corresponding to excitations of one, two and\nthree spins and also investigated their stability. Additionally, relevant situations were pointed out\nwhere such excitations can be physically identified.\nInrecenttimes,onehasalsoseenthatatnanoscalelevelspintransfernano-oscillator(STNO)14,15,\nwhich essentially consists of a trilayer structure of two nanoscale ferromagnetic films separated by a\nnon-ferromagnetic but conducting layer, can lead to switching of spin angular momentum directions\nand allow for the generation of microwave oscillations16,17. The ferromagnetic film even when it is\nhomogeneous is dominated by anisotropic interactions besides the presence of external magnetic\nfields (both dc and ac) and spin current terms. The equation of motion defining the evolution of\nthe spins is the Landau-Lifshitz-Gilbert-Slonczewski (LLGS) equation18where the spin current term\nis given by the Slonczewski form. One notices that the LLGS equation is a simple generalization of\nthe Landau-Lifshitz-Gilbert (LLG) equation which describes the nonlinear magnetization dynamics\n2in bulk materials as in the case of ferromagnetic lattices. Then it becomes important to ask what is\nthe influence of spin current term on the spin excitations, particularly intrinsic localized oscillations\n(ILOs) and identify the conditions under which damping effect can be off-set by the spin current\nterm.\nFrom yet another point of view, one may consider the possibility of designing a PT-symmetric\nferromagnetic nanoscale device by considering two nano-film structures interspersed by a nonmag-\nnetic but conducting thinner layer (i.e. a sandwich structure) as suggested by Lee, Kottos and\nShapiro19very recently. These authors have proposed a class of synthetic magnetic nanostructures\nwhich utilize natural dissipation (loss) mechanisms along with suitable chosen gain mechanism so\nas to control the magnetization dynamics. We will also explore how the spin ILOs can be identified\nin these structures.\nIn this paper we deduce an explicit one-spin excitation in an anisotropic ferromagnetic lattice\n(without onsite anisotropy, to start with) in the presence of external magnetic field and explore the\neffect of spin current term to maintain the oscillatory nature of the spin excitation. We then point\nout how this can be generalized to more general spin excitations and in PT-symmetric nanostruc-\ntures.\nThe organization of the paper is as follows. In Sec. 2 we deduce the dynamical equation for an\nanisotropic ferromagnetic spin in the presence of external magnetic field and set up the appropriate\nequation for a one-spin excitation in the presence of Gilbert damping. In Sec. 3, we deduce the\nexplicit one-spin excitation including the damping effect and analyze how the spin excitation gets\naffected by the damping. In Sec. 4, we incorporate the spin current term and point out how an\nappropriate strength of spin current can off-set the effect of damping so as to control the spin\noscillations. In Sec. 5, we point out how the above analysis can be extended to a PT-symmetric\nnanostructure. We briefly indicate how this study can be extended to consider more general spin\nexcitations in Sec. 6. Finally in Sec. 7, we present our conclusions.\n3II. DYNAMICS OF THE ANISOTROPIC SPIN CHAIN AND ONE-SPIN\nEXCITATION\nConsidering the evolution of spins of a one-dimensional anisotropic Heisenberg ferromagnetic\nspin chain modeled by the Hamiltonian12\nH=−N/summationdisplay\n{n}(ASx\nnSx\nn+1+BSy\nnSy\nn+1+CSz\nnSz\nn+1)−D/summationdisplay\nn(Sz\nn)2−/vectorH·/summationdisplay\nn/vectorSn, (1)\nwhere the spin components /vectorSn= (Sx\nn,Sy\nn,Sz\nn)are classical unit vectors satisfying the constant length\ncondition\n(Sx\nn)2+ (Sy\nn)2+ (Sz\nn)2= 1, n = 1,2,...,N. (2)\nHereA,B, andCare the exchange anisotropy parameters, Dis the onsite anisotropy parameter and\nthe external magnetic field /vectorH= (H,0,0)is chosen along the x-axis for convenience. By introducing\nthe appropriate spin-Poisson brackets and deducing the LLG spin evolution equation one can obtain\nthe equation for the spin lattice (1) as\nd/vectorSn\ndt=/vectorSn×/vectorHeff+α/vectorSn×(/vectorSn×/vectorHeff), (3)\nwhere\n/vectorHeff=A(Sx\nn+1+Sx\nn−1)ˆi+B(Sy\nn+1+Sy\nn−1)ˆj+C(Sz\nn+1+Sz\nn−1)ˆk+ 2DSz\nnˆk+/vectorH,(4)\nandαis the Gilbert damping parameter. In component form Eq. (3) with Eq. (4) reads as\ndSx\nn\ndt=CSy\nn(Sz\nn+1+Sz\nn−1)−BSz\nn(Sy\nn+1+Sy\nn−1)−2DSy\nnSz\nn+α/bracketleftbigg\nBSy\nnSz\nn(Sy\nn+1+Sy\nn−1)\n−A(Sx\nn+1+Sx\nn−1)((Sy\nn)2+ (Sz\nn)2) +CSz\nnSx\nn(Sz\nn+1+Sz\nn−1)−2DSx\nn(Sz\nn)2−H((Sx\nn)2+ (Sz\nn)2)/bracketrightbigg\n,(5)\ndSy\nn\ndt=ASz\nn(Sx\nn+1+Sx\nn−1)−CSx\nn(Sz\nn+1+Sz\nn−1) + 2DSx\nnSz\nn+HSz\nn+α/bracketleftbigg\nCSz\nnSy\nn(Sz\nn+1+Sz\nn−1)\n−B((Sx\nn)2+ (Sz\nn)2)(Sy\nn+1+Sy\nn−1) +ASx\nnSy\nn(Sx\nn+1+Sx\nn−1)−2DSy\nn(Sz\nn)2+HSx\nnSy\nn/bracketrightbigg\n, (6)\n4dSz\nn\ndt=BSx\nn(Sy\nn+1+Sy\nn−1)−ASy\nn(Sx\nn+1+Sx\nn−1)−HSy\nn+α/bracketleftbigg\nASx\nnSy\nn(Sx\nn+1+Sx\nn−1)\n−C((Sx\nn)2+ (Sy\nn)2)(Sz\nn+1+Sz\nn−1) +BSy\nnSz\nn(Sy\nn+1+Sy\nn−1) + 2DSz\nn((Sx\nn)2+ (Sy\nn)2) +HSx\nnSz\nn/bracketrightbigg\n.(7)\nNow looking for the one spin excitation for (1) as\n/vectorSn=...,(1,0,0),(1,0,0),(Sx\ni(t),Sy\ni(t),Sz\ni(t)),(1,0,0),(1,0,0),..., (8)\nwhere we have used nto denote a general spin in the lattice and used ito specify the localized spin\nexcitation, and redesignating (Sx\ni(t),Sy\ni(t),Sz\ni(t))as(Sx\n0(t),Sy\n0(t),Sz\n0(t)), the equation of motion\n(LLG equation) for the excited spin can be given as\ndSx\n0\ndt=−2DSy\n0Sz\n0−α/bracketleftbigg\n(2A+H)((Sy\n0)2+ (Sz\n0)2) + 2DSx\n0(Sz\n0)2/bracketrightbigg\n, (9)\ndSy\n0\ndt= (2A+H)Sz\n0+ 2DSx\n0Sz\n0+α/bracketleftbigg\n(2A+H)Sx\n0Sy\n0−2DSy\n0(Sz\n0)2/bracketrightbigg\n, (10)\ndSz\n0\ndt=−(2A+H)Sy\n0+α/bracketleftbigg\n(2A+H)Sx\n0Sz\n0+ 2D((Sx\n0)2+ (Sy\n0)2)Sz\n0/bracketrightbigg\n. (11)\nNote that from Eqs. (9) - (11), one can check that\nSx\n0dSx\n0\ndt+Sy\n0dSy\n0\ndt+Sz\n0dSz\n0\ndt= 0, (12)\nso that/vectorS2= (Sx\no)2+ (Sy\n0)2+ (Sz\n0)2=Constant = 1is conserved.\nNext, further confining to the case where the onsite anisotropy vanishes, D= 0, we have the\nLLG equation for the one-spin excitation,\ndSx\n0\ndt=−α(2A+H)(1−(Sx\n0)2), (13)\ndSy\n0\ndt= (2A+H)Sz\n0+α(2A+H)Sx\n0Sy\n0, (14)\ndSz\n0\ndt=−(2A+H)Sy\n0+α(2A+H)Sx\n0Sz\n0, (15)\nwith the constraint /vectorS2= (Sx\no)2+ (Sy\n0)2+ (Sz\n0)2= 1. The system (13) - (15) can be exactly solved\nas shown below.\n5III. EXPLICIT ONE-SPIN EXCITATION\nNow the above system of nonlinear differential equations can be straightforwardly solved. Inte-\ngrating (14) we obtain\nSx\n0(t) =c2e−2α(2A+H)t−1\nc2e−2α(2A+H)t+ 1, (16)\nwherecis an arbitrary constant. We also note that when α= 0, that is no damping, Sx\n0(t) =\n(c2−1)/(c2+ 1) =const =√\n1−a2as noted in ref. [13], Eq. (11). Also we note that Sx\n0(0) =\n(c2−1)/(c2+ 1)andSx\n0(∞) =−1, indicating a switching from a given initial value to the other\nground state, Sx\n0=−1.\nTo findSy\n0andSz\n0, we proceed as follows. Considering Eq. (14) and differentiating once with\nrespect toton both sides to obtain (d2Sy\n0/dt2), after making use of the forms of (dSx\n0/dt)and\n(dSz\n0/dt)from (13) and (15), respectively, we have\nd2Sy\n0\ndt2=−(2A+H)2(1 +α2)Sy\n0+ 2α(2A+H)2Sx\n0Sz\n0+ 2α2(2A+H)2(Sx\n0)2Sy\n0,(17)\nso that\nSz\n0(t) =1\n2α(2A+H)2Sx\n0/bracketleftBiggd2Sy\n0\ndt2+ (2A+H)2(1 +α2)Sy\n0−2α2(2A+H)2(Sx\n0)2Sy\n0/bracketrightBigg\n.(18)\nAlso from (14) we can write\nSz\n0(t) =1\n2A+H/bracketleftBiggdSy\n0\ndt−αSx\n0Sy\n0/bracketrightBigg\n. (19)\nEquating the right hand sides of (18) and (19), we obtain\nd2Sy\n0\ndt2=−2α(2A+H)Sx\n0dSy\n0\ndt+ (2A+H)2(1 +α2)Sy\n0= 0. (20)\nAfterastandardtransformationandtwointegrations(asindicatedinAppendixA),wecanexplicitly\nwrite the solution for Sy\n0as\nSy\n0=cexp(−α(2A+H)t)\nc2exp(−2α(2A+H)t) + 1ˆacos(Ωt+δ),Ω = 2A+H, (21)\n6where ˆais an arbitrary constant. Also from (14) we have\nSz\n0(t) =1\n(2A+H)/bracketleftBiggdSy\n0\ndt−αSx\n0Sy\n0/bracketrightBigg\n=−ˆasin(Ωt+δ)ce−α(2A+H)t\nc2e(−2α(2A+H)t+ 1. (22)\nNowinordertofixtheconstant ˆawedemandthatthespinlengthconstraint (Sx\n0)2+(Sy\n0)2+(Sz\n0)2= 1\nbe valid. This leads to\nˆa2= 4orˆa= 2, (23)\nso that we have now the complete solution of the excited spin as\nSx\n0(t) =c2e−2α(2A+H)t−1\nc2e−2α(2A+H)t+ 1, (24)\nSy\n0(t) =2ce−α(2A+H)t\nc2e−2α(2A+H)t+ 1cos[(2A+H)t+δ], (25)\nSz\n0(t) =−2ce−2α(2A+H)t\nc2e−2α(2A+H)t+ 1sin[(2A+H)t+δ]. (26)\nNote that the arbitrary constant corresponding to the undamped case ( α= 0) is\nˆa=c2−1\nc2+ 1. (27)\nIt is obvious from the above that for α= 0,Sx\n0=constant =c2−1\nc2+1, whileSy\n0andSz\n0are periodic\nfunctions of t. In this case Eqs. (14)-(15) are linear in Sy\n0andSz\n0so that the perturbation around\nthe origin in the ( Sy\n0−Sz\n0) plane admits pure imaginary eigenvalues. When α/negationslash= 0, they get damped\nas shown in Fig. 1 corresponding to the explicit forms (24)-(26). Note that in the above we have\nassumed the external magnetic field to be a constant in time. However, even in the case where the\nfield is a variable function of time, say\nH(t) =h0+h1cosωt, (28)\nwhereh0,h1andωare constants, we observe from the equations of motion of the spin components\n(13) - (15), that Hoccurs always as a linear combination 2A+H(t) = (2A+h0+h1cosωt).\nTherefore by redefining the time (2A+H)tas\nτ= (2A+h0)t−h1ωsinωt, (29)\n7all the previous analysis goes through. The final spin excitations are of the same form as (24) - (26)\nbut with the transformed time variable given by Eq. (29).\ntSx\n0(t)(i)\n2000 1500 1000 500 01\n0.5\n0\n-0.5\n-1\ntSy\n0(t)(ii)\n2000 1500 1000 500 01\n0.5\n0\n-0.5\n-1\ntSz\n0(t)(iii)\n2000 1500 1000 500 01\n0.5\n0\n-0.5\n-1\nFIG. 1. Damped spin excitation: One-spin excitation (Eqs. (24)-(26)) showing the three spin components\nfor the damped cases ( α= 0.005).\nIV. EFFECT OF SLONCZEWSKI SPIN CURRENT\nNext we consider the influence of spin current term in a trilayer structured STNO (see Fig. 2),\nwhere we consider the excitation of a single spin of magnetization in the outer uniformly magnetized\nlayer under anisotropic interaction and external magnetic field in the presence of spin current. The\ncorresponding spin excitation is given by the Landau-Lifshitz-Gilbert-Slonczewski equation for the\nspin as\nFIG. 2. A schematic representation of STNO.\nd/vectorSn\ndt=/vectorSn×/vectorHeff+α/vectorSn×(/vectorSn×/vectorHeff) +j/vectorSn×(/vectorSn×/vectorSp), (30)\n8where/vectorHeffis the effective field given by Eq. (4) and jis the magnitude of the spin current and\nthe polarization vector /vectorSpis\n/vectorSp= (1,0,0), (31)\ncorresponding to the flow of electrons in the x-direction. Consequently,\n/vectorSn×(/vectorSn×/vectorSp) =/vectorSn(/vectorSn·/vectorSp)−/vectorSp=−((Sy\nn)2+ (Sz\nn)2)ˆi+Sx\nnSy\nnˆj+Sx\nnSz\nnˆk, (32)\nwhere (ˆi,ˆj,ˆk)form the unit orthonormal trihedral. As a result, the equations for the one-spin\nexcitations get modified from (13) - (15) as\ndSx\n0\ndt=−[α(2A+H)−j](1−(Sz\n0)2), (33)\ndSy\n0\ndt= (2A+H)Sz\n0+ [α(2A+H)−j]Sx\n0Sy\n0, (34)\ndSz\n0\ndt=−(2A+H)Sy\n0+ [α(2A+H)−j]Sx\n0Sz\n0. (35)\nNow choosing the spin current as\nj=α(2A+H), (36)\none can check that\ndSx\n0\ndt= 0, (37)\ndSy\n0\ndt= (2A+H)Sz\n0, (38)\ndSz\n0\ndt=−(2A+H)Sz\n0. (39)\nConsequently, the spin vector evolves as\n/vectorS0=/parenleftBig√\n1−ˆa2,ˆacos(Ωt+δ),−ˆasin(Ωt+δ/parenrightBig\n, (40)\nwhere ˆa=constant and Ω = (2A+H), and the effect of damping is exactly offset by the spin\ncurrent term. Thus the spin current acts effectively as an “external magnetic field plus anisotropy\"\nand the system can generate microwave oscillations. When j <α (2A+H), damping will overtake\nasymptotically and the spin will switch its direction.\n9V.PT-SYMMETRIC MAGNETIC DEVICE\nRecentlyaclassofsyntheticmagneticnanostructuresthatmakesuseofthenatureofloss/dissipation\nmechanism together with appropriate amplification (gain) process has been suggested by Lee, Kot-\ntos and Shapiro19to control magnetization dynamics. The suggested arrangement consists of two\ncoupled nano-ferromagnetic films, n= 1,2(when separated by a spacer) in the presence of an\nexternal magnetic field along the x-axis, for example out-of-plane geometry (so that the z-axis is\nperpendicular to the films) as shown in Fig. 3.\nFIG.3. APT-symmetrictrilayerstructurecomprisingtwomagneticthinfilmsandaspacerlayersuggested\nby Lee, Kottos and Shapiro19.\nConsidering the effective instantaneous local fields as /vectorH1effand/vectorH2efffor the two layers 1 and 2,\nrespectively, associated with the homogeneous magnetization vectors /vectorS1= (/vectorM1/|/vectorM1|)and/vectorS2=\n(/vectorM2/|/vectorM2|), we have the associated dynamical equations\nd/vectorS1\ndt=/vectorS1×/vectorH1eff+k/vectorS1×/vectorS2+α/vectorS1×d/vectorS1\ndt, (41)\nd/vectorS2\ndt=/vectorS2×/vectorH2eff+k/vectorS2×/vectorS1−α/vectorS2×d/vectorS2\ndt, (42)\nwherekistheferromagneticcouplingand αisthedamping/gaincoefficient. Notethatthecombined\nsystems (41)-(42) are invariant under the simultaneous changes of the variables /vectorS1,2→ −/vectorS2,1,\n/vectorH1,2eff→−/vectorH2,1effandt→−t, which may be treated as equivalent to combined PT-symmetry\n10operation19. Now we choose the two layers such that\n/vectorS2=/vectorS1×/vectorSp,/vectorS1=−/vectorS2×/vectorSp, (43)\nwhere/vectorSp= (1,0,0)is a fixed polarization vector. Equation (43) implies that /vectorSpis perpendicular\nto the plane of /vectorS1and/vectorS2. Then, similar to the analysis in Sec. 4, we can choose the ferromagnetic\ncouplingksuch that for simple anisotropy as in Eqs. (13) - (15) and external magnetic field H, we\ncan choose\nk=α(2A+H), (44)\nso that the gain/loss terms are exactly cancelled by the ferromagnetic coupling, leaving out\nd/vectorS1\ndt=/vectorS1×/vectorH1eff, (45)\nd/vectorS2\ndt=/vectorS2×/vectorH2eff, (46)\nleading to spin oscillations and thereby to an effective control of magnetization oscillations.\nVI. MORE GENERAL SPIN EXCITATIONS\nOne can consider more general localized spin excitations like two, three, etc. spin excitations.\nFor example, in the case of localized two-spin excitations,\n/vectorSn=...,(1,0,0),(1,0,0),(Sx\ni,Sy\ni,Sz\ni),(Sx\ni+1,Sy\ni+1,Sz\ni+1),(1,0,0),(1,0,0),...\n=...,(1,0,0),(1,0,0),(Sx\n0,Sy\n0,Sz\n0),(Sx\n1,Sy\n1,Sz\n1),(1,0,0),(1,0,0),..., (47)\n11we obtain the dynamical equations from (5) - (7) as\ndSx\n0\ndt=CSy\n0Sz\n1−BSz\n0Sy\n1−2DSy\n0Sz\n0\n+α[BSx\n0Sy\n0Sy\n1−(A(Sx\n1+ 1) +H)((Sy\n0)2+ (Sz\n0)2) +CSx\n0Sz\n0Sz\n1−2DSx\n0(Sz\n0)2], (48)\ndSy\n0\ndt=ASz\n0(Sx\n1+ 1)−CSx\n0Sz\n1+ 2DSx\n0Sz\n0+HSz\n0\n+α[(A(Sx\n1+ 1) +H)Sx\n0Sy\n0−BSy\n1((Sx\n0)2+ (Sz\n0)2) +CSy\n0Sz\n0Sz\n1−2DSy\n0(Sz\n0)2], (49)\ndSz\n0\ndt=BSx\n0Sy\n1−ASy\n0(Sx\n1+ 1)−HSy\n0\n+α[(A(Sx\n1+ 1) +H)Sx\n0Sz\n0+BSy\n0Sz\n0Sy\n1−CSz\n1((Sx\n0)2+ (Sy\n0)2) + 2DSz\n0((Sx\n0)2+ (Sy\n0)2)],(50)\ndSx\n1\ndt=CSz\n0Sy\n1−BSy\n0Sz\n1−2DSy\n1Sz\n1\n+α[BSy\n0Sx\n1Sy\n1−(A(Sx\n0+ 1) +H)((Sy\n1)2+ (Sz\n1)2) +CSz\n0Sx\n1Sz\n1−2DSx\n1(Sz\n1)2], (51)\ndSy\n1\ndt=ASz\n1(Sx\n0+ 1)−CSx\n1Sz\n0+ 2DSx\n1Sz\n1+HSz\n1\n+α[(A(Sx\n0+ 1) +H)Sx\n1Sy\n1−BSy\n0((Sx\n1)2+ (Sz\n1)2) +CSz\n0Sy\n1Sz\n1−2DSy\n1(Sz\n1)2], (52)\ndSz\n1\ndt=BSx\n1Sy\n0−ASy\n1(Sx\n0+ 1)−HSy\n1\n+α[(A(Sx\n0+ 1) +H)Sx\n1Sz\n1+BSy\n0Sy\n1Sz\n1−CSz\n0((Sx\n1)2+ (Sy\n1)2) + 2DSz\n1((Sx\n1)2+ (Sy\n1)2)].(53)\nNote that the terms proportional to αare generalizations for the present two-spin excitation case\ncompared to those given in Eqs. (9) - (11). As such the system (48) - (53) does not seem to be\nanalytically solvable. In Fig. 4, we numerically integrate the system for both the undamped case\n(α= 0) and the damped case ( α/negationslash= 0) for nonzero Dand present the solutions in the undamped\nand damped cases to show the existence of more general internal localized excitations. The analysis\ncan be extended to even more general situations, which will be presented elsewhere.\nVII. CONCLUSION\nBy looking at the simplest internal localized excitations in an anisotropic Heisenberg ferromag-\nnetic spin chain in external magnetic field with additional Gilbert damping, we deduced the explicit\nsolutions which characteristically show the effect of damping. Then applying a spin current in an\n12tSx\n0(t)(i)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSy\n0(t)(ii)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSz\n0(t)(iii)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSx\n1(t)(iv)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSy\n1(t)(v)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSz\n1(t)(vi)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1Fig. 4 (a): Undamped two-spin excitations\ntSx\n0(t)(i)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSy\n0(t)(ii)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSz\n0(t)(iii)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSx\n1(t)(iv)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSy\n1(t)(v)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\ntSz\n1(t)(vi)\n200 150 100 50 01\n0.5\n0\n-0.5\n-1\nFig. 4 (b): Damped two-spin excitations\nFIG. 4. Solution of Eqs. (48)-(53) for two-spin excitations S0andS1for the (a) undamped ( α= 0) and\n(b) damped cases ( α= 0.005), with the anisotropy parameters A= 0.1,B= 0.23,C= 1.0andD= 0.3\nand the magnetic field H= 113Oe.\nSTNO of appropriate magnitude, we pointed out how the tendency toward damping can be offset\nexactly and thereby sustaining the magnetic oscillations. Our prediction about a PT-symmetric\n13STNO could be tested in magnetic multilayer structures with carefully balanced gain and loss. We\nhave also pointed out how such controlled oscillations can be effected in a recently suggested nano-\nmagnetic trilayer device. It will be insightful to observe these oscillations in appropriate magnetic\nsystems experimentally. Finally, in a related context we note that nonreciprocal optical modes can\nexist at an interface between two PT-symmetric magnetic domains near a frequency corresponding\nto almost zero effective permeability20.\nVIII. ACKNOWLEDGMENTS\nThe authors wish to thank Dr. D. Aravinthan for his help in the numerical analysis. The\nresearch work of ML was supported by a NASI Senior Scientist Platinum Jubilee Fellowship\n(NAS 69/5/2016-17) and a DST-SERB Distinguished Fellowship (No.: SERB/F/6717/2017-18).\nML was also supported by a Council of Scientific and Industrial Research, India research project\n(No.: 03/1331/15/EMR-II) and a National Board for Higher Mathematics research project (No.:\n2/48(5)/2015/NBHM(R.P.)/R&D II/14127). ML also wishes to thank the Center for Nonlinear\nStudies, Los Alamos National Laboratory, USA for its warm hospitality during his visit in the\nsummer of 2017. This work was supported in part by the U.S. Department of Energy.\nAPPENDIX A\nHere we briefly point out how to solve Eq. (20). Introducing the transformation\nSy\n0(t) =eα(2A+H)/integraltext\nSx\n0dt·ˆSy\n0(t) (A. 1)\ninto Eq. (20), we obtain\nd2ˆSy\n0(t)\ndt2+ (2A+H)2ˆSy\n0(t) = 0. (A. 2)\nConsequently, we have\nˆSy\n0(t) = ˆacos(Ωt+δ),Ω = 2A+H, (A. 3)\n14where ˆaandδare arbitrary constants. Then, the prefactor on the right hand side of (21) can be\ndeduced as follows. Since\nI=/integraldisplay\nSx\n0dt=/integraldisplayc2exp(−2α(2A+H)t)−1\nc2exp(−2α(2A+H)t) + 1dt=−1\n2α(2A+H)log(c2exp(−2α(2A+H)t) + 1)2\nc2exp(−2α(2A+H)t,\n(A. 4)\nthe prefactor becomes\nexp/bracketleftbigg\nα(2A+H)/integraldisplay\nSx\n0dt/bracketrightbigg\n=cexp(−α(2A+H)t)\nc2exp(−2α(2A+H)t) + 1. (A. 5)\nCorrespondingly\nSy\n0=cexp(−α(2A+H)t)\nc2exp(−2α(2A+H)t) + 1ˆacos(Ωt+δ),Ω = 2A+H, (A. 6)\nwhich is Eq. (21).\nREFERENCES\n1B. Hillerbrands and K. Ounadjela, Spin Dynamics in Confined Magnetic Structures , Vols. I & II\n(Springer, Berlin) 2002.\n2M. Lakshmanan, Philos. Trans. R. Soc. A 369(2011) 1280.\n3B. Georges, V. Cros and A. Fert, Phys. Rev. B 73(2006) 0604R.\n4Z. Yang, S. Zhang and Y. C. Li, Phys. Rev. Lett. 99(2007) 134101.\n5M. Lakshmanan, Phys. Lett. A 61(1977) 53.\n6K. Nakamura and T. Sasada, J. Phys. C 15(1982) L915.\n7E. K. Sklyanin, LOMI preprint E-3-79, Leningrad (1979).\n8Y. Ishimori, Prog. Theor. Phys. 72(1984) 33.\n9M. Lakshmanan and A. Saxena, Physica D 237(2008) 885.\n10H. Zabel and M. Farle (Eds.), Magnetic Nanostructures: Spin Dynamics and Spin Transport\n(Springer, Berlin) 2013.\n11A. Sievers and S. Takeno, Phys. Rev. Lett. 61(1988) 970.\n1512Y. Zolotaryuk, S. Flach and V. Fleurov, Phys. Rev. B 63(2003) 214422.\n13M. Lakshmanan, B. Subash and A. Saxena, Phys. Lett. A 378(2014) 1119.\n14G. Bertotti, I. Mayergoyz and C. Serpico, Nonlinear Magnetization Dynamics in Nanosystems\n(Elsevier, Amsterdam) 2009.\n15B. Georges, J. Grollier, V. Cros and A. Fert, Appl. Phys. Lett. 92(2008) 232504.\n16B. Subash, V. K. Chandrasekar and M. Lakshmanan, Europhys. Lett. 102(2013) 17010; 109\n(2015) 17009.\n17J. Turtle, K. Beauvais, R. Shaffer, A. Palacios, V. In, T. Emery and P. Langhini, J. Appl. Phys.\n113(2013) 114901.\n18J. C. Slonczewski, J. Magn. & Magn. Mater. 159(1996) L261.\n19J. M. Lee, T. Kottos and B. Shapiro, Phys. Rev. B 91(2015) 094416.\n20J. Wang, H. Y. Dong, C. W. Ling, C. T. Chan and K. H. Fung, Phys. Rev. B 91(2015) 235410.\n16"    },    {        "title": "2212.12016v1.Novel_Bottomonium_Results.pdf",        "content": "Novel Bottomonium Results\nBen Page,𝑎\u0003Chris Allton𝑎and Seyong Kim𝑏\n𝑎Department of Physics, Swansea University, Swansea SA2 8PP, United Kingdom\n𝑏Department of Physics, Sejong University, Seoul 143-747, Korea\nE-mail: {b.page.9/zero.alt3/zero.alt3727,c.r.allton}@swansea.ac.uk, skim@sejong.ac.kr\nWe present the latest results from the use of the Backus-Gilbert method for reconstructing the\nspectra of NRQCD bottomonium mesons using anisotropic FASTSUM ensembles at non-zero\ntemperature. We focus in particular on results from the 𝜂𝑏,Υ,𝜒𝑏1andℎ𝑏generated from\nTikhonov-regularized Backus-Gilbert coefficient sets. We extend previous work on the Laplace\nshifting theorem as a means of resolution improvement and present new results from its use.\nWe conclude with a discussion of the limitations of the improvement routine and elucidate a\nconnection with Parisi-Lepage statistical scaling.\nThe 39th International Symposium on Lattice Field Theory,\n8th-13th August, 2022,\nRheinische Friedrich-Wilhelms-Universität Bonn, Bonn, Germany\n\u0003Speaker\n©Copyright owned by the author(s) under the terms of the Creative Commons\nAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/arXiv:2212.12016v1  [hep-lat]  22 Dec 2022Novel Bottomonium Results Ben Page\n1. Introduction\nBottomoniumhaspreviouslybeenstudiedasameansofestimatingthepropertiesofthequark-\ngluonplasmaasproducedinrelativisticheavy-ioncollisions. [1–3]. Theproblemofreconstructing\nthespectrumofbottomoniumstatesatnon-zerotemperatureisanill-posedoneduetothepresenceof\nfiniteuncertaintiesindatacollectedfromlatticeQCDsimulations. Thisworkrepresentsoneofthe\nlatestinacollectionofstudiesperformedbythe F/a.pc/s.pc/t.pc/s.pc/u.pc/m.pc Collaborationfocussingonbottomonium\nstates simulated using anisotropic lattices at nonzero temperature. In the following, we build upon\nthe results of previous work [4] and present a discussion on the use of the Backus-Gilbert method\nfor reconstructing bottomonium spectra, showing how the Laplacian nature of the reconstruction\nformula my be exploited to give improved resolution.\n2. Lattice details\nNon-relativisticQCD(NRQCD)isaneffectivefieldtheoryforheavyquarkonia,whichapprox-\nimates fully relativistic QCD by expanding the Lagrangian in powers of the heavy quark velocity\n[5]. One of the principal benefits of the NRQCD formulation is that the calculation of the time\nevolutionoftheheavyquarksreducestoaninitial-valueproblem,astheheavyquarksandantiquarks\ndecouple in the non-relativistic regime. This decoupling effect turns the spectral representation of\ntheEuclideancorrelator 𝐺¹𝜏ºintoaLaplacetransformationofthespectraldensityfunction 𝜌¹𝜔º:\n𝐺¹𝜏;𝑇º=∫𝜔max\n𝜔min𝑑𝜔\n2𝜋𝐾¹𝜏𝜔º𝜌¹𝜔;𝑇º (1)\nwhere𝐾¹𝜏𝜔º=𝑒\u0000𝜔𝜏is the temperature independent kernel function and 𝑇=¹𝑎𝜏𝑁𝜏º\u00001is the\nlattice temperature as a function of the temporal extent 𝑁𝜏. In order to relate NRQCD energies to\nphysicalenergies,thereconstructionwindow 𝜔2»𝜔min𝜔max¼mustbeadditivelyrenormalisedby\nthe NRQCD energy shift, Δ𝐸=746GeV.\nWe make use of F/a.pc/s.pc/t.pc/s.pc/u.pc/m.pc’s Generation 2L ensembles, generated using anisotropic lattices\n(𝜉=𝑎𝑠𝑎𝜏\u001835)with2+1flavour,clover-improvedWilsonfermionsusingaphysical 𝑠quarkand\nlighter, degenerate 𝑢and𝑑quarks[6]. The spatial extent of the lattice 𝑁𝑠=32and the temporal\nextent along with the corresponding lattice temperatures are detailed in Table 1.\n𝑁𝜏 128645648403632282420\n𝑇=1¹𝑎𝜏𝑁𝜏º[MeV] 4795109127152169190217253304\n𝑁cfg 1024104110421123110211191090103110161030\nTable 1: Temporal extent, corresponding lattice temperature in MeV and number of configurations for the\nF/a.pc/s.pc/t.pc/s.pc/u.pc/m.pc Generation 2L ensembles. The double vertical line mid-table represents our value of 𝑇pc[6].\n3. The Backus-Gilbert Method\nThe Backus-Gilbert method [7] is a reconstruction technique which extracts regularised solu-\ntions from the ill-posed inverse problem by imposing constraints on the stability of its predictions\n2Novel Bottomonium Results Ben Page\nunder a change of input. Since 𝐺¹𝜏ºis only known to at most O¹100ºbut𝜌¹𝜔ºis continuous\n(O¹1000¸ºpoints), there are theoretically an infinite number of possible spectra 𝜌which produce\nthe correct𝐺¹𝜏ºwithin numerical errors. Backus-Gilbert attempts to estimate a solution of Eq. 1,\ndenoted ˆ𝜌, on a point-by-point basis by constructing averaging functions 𝐴¹𝜔𝜔 0ºcentred about\nsome point𝜔0generated using the data kernel 𝐾¹𝜏𝜔º:\nˆ𝜌¹𝜔0º=∫𝜔max\n𝜔min𝐴¹𝜔𝜔 0º𝜌¹𝜔º𝑑𝜔 (2)\nwith𝐴¹𝜔𝜔 0º=Í\n𝜏𝑐𝜏¹𝜔0º𝐾¹𝜏𝜔º. In the limit 𝐴¹𝜔𝜔 0º\u0000!𝛿¹𝜔\u0000𝜔0º, we obtain a perfect\nreconstrucion of the target spectrum 𝜌. It is easily seen that plugging Eq. 1 into\nˆ𝜌¹𝜔0º=∑︁\n𝜏𝑐𝜏¹𝜔0º𝐺¹𝜏º (3)\ngives Eq. 2. The coefficients 𝑐𝜏¹𝜔0ºcontrol the shape of 𝐴¹𝜔𝜔 0ºand are found by minimising\nthecostfunction 𝐽¹𝜔0ºrepresentingtheleast-squaresdistancebetweentheaveragingfunctionand\nthe delta function at 𝜔0[8]:\n𝐽¹𝜔0º=∫𝜔max\n𝜔min»𝐴¹𝜔𝜔 0º\u0000𝛿¹𝜔\u0000𝜔0º¼2𝑑𝜔 (4)\nSetting𝜕𝑐𝜏𝐽¹𝜔0º=0reduces the problem to an inversion of a matrix-vector product:\nK𝜏𝜏0\u0001𝑐𝜏0¹𝜔0º=𝐾¹𝜔0𝜏ºwhereK𝜏𝜏0=∫𝜔max\n𝜔min𝐾¹𝜏𝜔º𝐾¹𝜏0𝜔º𝑑𝜔 (5)\nThekernelwidthmatrix Kisnear-singular(worsenedbytheexponentialnatureofthekernel)\nandsomustbetreatedbyaconditioningroutinebeforeinversion. Onesuchconditioningroutineis\nthe addition of a small constant term to the diagonal entries in a Tikhonov-like fashion[9]:\nK¹𝛼º=K¸𝛼𝐼 (6)\nwhere𝛼isaparameterwhichcontrolsthestrengthoftheregularisation. Thebenefitofsuchscalar\nconditioningoverothermethodsisthatthecoefficients 𝑐𝜏areconstructedwithoutpriorknowledge\nof𝐺¹𝜏º,enablingtheiruseinthereconstructionofanyspectraobeyingEq.1,regardlessofchoices\nof quantum numbers, see [4].\n4. Improvement via the Laplace Shift Transform\nIt has been previously shown [10] that, for the case of the maximum entropy method (MEM),\nthere exists a relationship between the choice of 𝜔minand the resolving power of the method. This\neffect also occurs in the Backus-Gilbert method which shares a similar basis-function mechanism\nof reconstruction as the MEM. The application of a Laplace shift transform:\n𝐺0¹𝜏º=𝑒Δ\u0001𝜏𝐺¹𝜏ºL=)𝜌0¹𝜔º=𝜌¹𝜔¸Δº (7)\nwhereΔ¡0shifts the spectral features closer to 𝜔minwhere the resolving power of the method\nis improved [4] offering improved predictions for mass and width estimates. Building upon this\nfeature, we have opted to combine 𝜔minand the Laplace shift transform parameter Δinto a single\nparameter eΔ=𝜔min¸Δ.\n3Novel Bottomonium Results Ben Page\n5. Systematic analysis: Removing eΔand𝛼dependence\nThere is a systematic dependence of the ground state mass 𝑀and widthΓon the value of eΔ\nand𝛼which must be removed during analysis. This is illustrated in Fig. 1 where the mass (Left)\nand width (Right) are shown as a function of eΔfor various𝛼values.\nAscanbeseen,forfixed 𝛼,themassincreasesandwidthdecreasesmonotonicallywith eΔdue\nto the resolution improvement. We also note that in the mass case, this slope approaches zero as\n𝛼!0,indicatingthatthe eΔdependencefallsaway. Finally,wenotethatthemaximumtheoretical\nshift occurs when eΔequals the mass. This is indicated by the boundary of the hatched region in\nFig.1(Left)). Wethereforeobtainourmassestimatebyfirstlyperformingalinearextrapolationof\nthe mass, at fixed 𝛼, to the boundary of the hatched region, obtaining 𝑀¹𝛼º. The𝛼dependence is\nthen removed by noting empirically that 𝑀¹𝛼ºbecomes independent of 𝛼, for small𝛼, and so can\nbe fit with a constant. A bootstrap analysis produces the error estimate.\nThegroundstatewidthisdeterminedbyasimilarprocedure. Firstalinearfitin eΔisperformed,\natfixed𝛼,andextrapolatedtotheboundaryregion. The 𝛼dependenceisthenremovedinthesame\nmanner as in the mass case. However we note that, due to the finite resolving power of the Backus\nGilbert method, the widths obtained should be considered upper bounds on the physical width of\nthe state.\n6 7 8 9 10\n (GeV)\n8910Mass (GeV) (s-s)        T=47 MeV\nPDG estimate\nBG estimate\n109\n107\n105\n103\n101\nlog10()\n6 7 8 9\n (GeV)\n050010001500 (GeV)\n (s-s)        T=47 MeV\n109\n107\n105\n103\n101\nlog10()\nFigure 1: Left:Plot of the ground state mass versus the shift parameter eΔfor the (smeared) Υmeson over\na range of𝛼values. The hatched region represents the maximum possible shift, beyond which the ground\nstate feature falls outside of the sampling window. The red dashed line is the PDG estimate [11] and the\nblack dashed line represents our best estimate after extrapolating the eΔand𝛼hyper-parameters. Right: Plot\nofourestimateoftheupperboundforthegroundstateFWHMwidthversus eΔfortheΥmesonoverarange\nof𝛼values. The black dashed line is our best estimate of the upper bound and the red band denotes its\nuncertainty. The temperature used in these plots is 𝑇=47MeV.\nThe value of 𝜔minis held fixed in this analysis (at a value of 𝜔min=\u000001𝑎\u00001\n𝜏\u001968GeV). We\nhave chosen to include all possible Euclidean times in our analysis, i.e. 0\u0014𝜏  𝑁 𝜏, because the\nresolving power of Backus Gilbert method is greatest for the largest time windows.\n4Novel Bottomonium Results Ben Page\n6. Results for the bottomonium sector: 𝜂𝑏,Υ,𝜒𝑏1andℎ𝑏\n0 100 200 300\nT (MeV)9.29.39.49.59.6Mass (GeV)\n (s-s) (2021)\n (s-s)\n0 100 200 300\nT (MeV)0250500750100012501500 (MeV)\n (s-s) (2021)\n (s-s)\nFigure 2: Comparisonbetweennewresultsfromthisworkandthosefrom[4],labelled(2021),forthemass\n(left)andupperboundonthewidth( right). ThemagentalineontheleftpaneisthePDGestimateforthe Υ\nmass[11]. Theredshadedbandontherightpaneindicatesourestimateofthemaximumresolvingpowerof\nthe new method.\nWe have improved our previous results [4]) by including the Laplace shifting (see §4) and an\nimproved fitting analysis (see §5). In Fig. 2, we display our results for the Υmass and width as\na function of temperature, including the results from our earlier analysis. We have also extended\nour previous work by including the 𝜂𝑏,𝜒𝑏1andℎ𝑏states, with results shown in Figs. 3 and 4. For\nthis analysis, we restricted ourselves to data generated using smeared quark sources which have\nimproved overlap with the ground state over local sources. We also conducted a more rigorous\nstudyofthedependenceofthemassandwidthontheparameters 𝛼andeΔinanattempttomeasure\ntheir contribution to the systematic error.\nIn particular, in the case of the Υmass, we wish to highlight in Fig.2 the comparison between\nthe results of this work and the estimates presented in [4] which appear to be contaminated by\nsystematics of the method, as the dependence of the the mass and width on 𝛼was not fully\naccounted for. We also note that below the pseudocritical temperature ( 𝑇pc=162MeV for our\nGen-2L ensembles[6, 12]), the experimental widths for the Υand𝜂𝑏are5402\u0006125keV and\n10¸4\n\u00005MeV respectively[11], over an order of magnitude smaller than even the minimum resolvable\nwidthforourmethod(seeFig.4). Weonceagainreiteratethatourpresentedvaluesfortheground\nstatewidthrepresentanupperboundonthetruevalue,andweleaveaninvestigationintotheeffect\nof changing the Euclidean time extent and the lower bound 𝜔minto a future study.\n5Novel Bottomonium Results Ben Page\n0.0 0.2 0.4 0.6 0.8 1.0\nTemperature (MeV)0.00.20.40.60.81.0Mass (GeV)0 100 200 3009.09.29.49.6\nb (s-s)\n0 100 200 3009.09.29.49.6\n (s-s)\n0 100 200 3009.509.7510.0010.25\nhb (s-s)\n0 100 200 3009.509.7510.0010.25\nb1 (s-s)\nFigure 3: Plots showing the estimate of the mass versus lattice temperature for select bottomonium states.\nThe horizontal dashed line represents the PDG estimate for the given state [11].\n0.0 0.2 0.4 0.6 0.8 1.0\nTemperature (MeV)0.00.20.40.60.81.0 (MeV)\n0 100 200 300050010001500\nb (s-s)\n0 100 200 300050010001500\n  (s-s)\n0 100 200 300050010001500\n hb (s-s)\n0 100 200 300050010001500\nb1 (s-s)\nFigure 4: Plots showing the estimate of the upper bound on the width versus lattice temperature for select\nbottomonium states. The shaded region represents our best estimate of the resolving power of the method.\n6Novel Bottomonium Results Ben Page\n0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14\na/2\n9.39.49.59.69.79.89.910.0Predicted Mass (GeV)T=47 MeV\nPredicted Masses\nb2\nhb\nb1\nb0\nb\nPDG Masses\nb2\nhb\nb1\nb0\nb\nFigure 5: Value of the energy shift Δsing(i.e. the predicted mass) which gives the most singular shifted\ncovariance matrix (see Eq. 9) for a variety of bottomonium channels as a function of 1𝜏2. The covariance\nmatricesaredefinedoverthetimeinterval 0\u0014𝜏 𝜏 2,andthereforethebestresultsareobtainedas 𝜏2!1.\nThelatticetemperatureis47MeV.Thepredictedmasseseachmesontendstowardtheexperimentalestimate\nfor the pseudoscalar mass. Experimental values for the meson masses are shown as horizontal lines [11].\n7. Connection with Parisi-Lepage Statistical Scaling\nToestimatetheerrorintheresultingreconstruction,theuncertaintyintheEuclideancorrelator\nΔ𝐺¹𝜏ºmust be combined with the Backus Gilbert coefficients 𝑐𝜏. The uncertainty corresponding\nto Eq. 3 is simply\nΔˆ𝜌2=∑︁\n𝜏𝜏0𝑐𝜏Cov»𝐺¼𝜏𝜏0𝑐𝜏0 (8)\nwhere Cov»𝐺¼is the covariance in 𝐺¹𝜏º. Under the Laplace transformation outlined in Eq. 7, the\ncovariance matrix transforms as\nCov»𝐺;Δ¼𝜏𝜏0=𝑒Δ\u0001𝜏Cov»𝐺¼𝜏𝜏0𝑒Δ\u0001𝜏0(9)\nwhichinturninfluencesthespectralerror Δˆ𝜌. Thiseffectcanbeprobedbymeasuringthecondition\nnumber of the resulting matrix, defined by\n𝜅¹Cov»𝐺;Δ¼º=𝜎max\n𝜎min(10)\nwhere𝜎are the singular values of the matrix.\nIt is interesting to study 𝜅as a function of Δand determine the value, Δsingwhen Cov»𝐺;Δ¼\nbecomes singular. One may imagine that Δsingis the ground state mass of the 𝐺¹𝜏ºchannel.\nHowever,aspointedoutbyParisi[13]andelucidatedfurtherbyLepage[14],thecovariancematrix\nhasaspecialphysicalsignificance. Itcanbeexpressedasacorrelationfunctionofthe squareofthe\n7Novel Bottomonium Results Ben Page\ninterpolatingoperatorsoftheoriginalcorrelationfunction, 𝐺¹𝜏º. Analysingthisfurther,onefinds\nthatthelighteststatewhichcontributestothecovariancematrixisthepseudoscalar,nomatterwhat\nstate was being probed by 𝐺¹𝜏º. This therefore implies that Cov »𝐺;Δ¼becomes singular when\nΔ=Δsingis the pseudoscalar mass (i.e. the 𝜂𝑏mass in our case) independent of the channel.\nWe illustrate this in Fig. 5 where Δsingis plotted for a variety of channels. The covariance\nmatrix was defined over the time interval 0\u0014𝜏  𝜏 2meaning that the large time limit (where the\ngroundstatedominates)isobtainedas 𝜏2!1. Ascanbeseen,inthislimitwerecoverthe 𝜂𝑏(i.e.\npseudoscalar) mass, thereby confirming the prediction of [13, 14].\n8. Summary\nWehavepresentedresultsforthegroundstatemassandanestimatefortheupperboundonthe\nwidth for several bottomonium states using smeared quark sources, showing improved resolution\ncompared to our previous results. We have demonstrated the ability of the Laplace shift to naively\nincrease the resolving power of the method, but show that is still insufficient to resolve the ground\nstate widths of the system. The effect of the Laplace shift transform on the covariance matrix of\ntheEuclideancorrelationfunctionwasalsostudied,wheretheconditionnumberofthematrixwas\nfound to confirm Parisi-Lepage statistical scaling in the long-time limit.\nAcknowledgments\nThisworkissupportedbySTFCgrantST/T000813/1. SKissupportedbytheNationalResearch\nFoundationofKoreaundergrantNRF-2021R1A2C1092701andgrantNRF-2021K1A3A1A16096820,\nfunded by the Korean government (MEST). BP has been supported by a Swansea University Re-\nsearch Excellence Scholarship (SURES). This work used the DiRAC Extreme Scaling service at\ntheUniversityofEdinburgh,operatedbytheEdinburghParallelComputingCentreandtheDiRAC\nData Intensive service operated by the University of Leicester IT Services on behalf of the STFC\nDiRAC HPC Facility (www.dirac.ac.uk). This equipment was funded by BEIS capital funding via\nSTFC capital grants ST/R00238X/1, ST/K000373/1 and ST/R002363/1 and STFC DiRAC Opera-\ntionsgrantsST/R001006/1andST/R001014/1. DiRACispartoftheUKNationale-Infrastructure.\nThis work was performed using PRACE resources at Cineca (Italy), CEA (France) and Stuttgart\n(Germany) via grants 2015133079, 2018194714, 2019214714 and 2020214714. We acknowledge\nthesupportoftheSwanseaAcademyforAdvancedComputing,theSupercomputingWalesproject,\nwhichispart-fundedbytheEuropeanRegionalDevelopmentFund(ERDF)viaWelshGovernment,\nand the University of Southern Denmark and ICHEC, Ireland for use of computing facilities. We\nare grateful to the Hadron Spectrum Collaboration for the use of their zero temperature ensemble.\nReferences\n[1] G. Aarts, S. Kim, M. P. Lombardo, M. B. Oktay, S. M. Ryan, D. K. Sinclair et al.,\narXiv:1/zero.alt31/zero.alt3.3725 [hep-lat, physics:hep-ph, physics:nucl-th] .\n[2] G. Aarts, C. Allton, S. Kim, M. P. Lombardo, M. B. Oktay, S. M. Ryan et al.,\narXiv:11/zero.alt39.4496 [hep-lat, physics:hep-ph, physics:nucl-th] .\n8Novel Bottomonium Results Ben Page\n[3] G. Aarts, C. Allton, T. Harris, S. Kim, M. P. Lombardo, S. M. Ryan et al.,\narXiv:14/zero.alt32.621/zero.alt3 [hep-lat, physics:hep-ph] .\n[4] B. Page, G. Aarts, C. Allton, B. Jäger, S. Kim, M. P. Lombardo et al., arXiv:2112./zero.alt32/zero.alt375\n[hep-lat] .\n[5] G. Lepage, in Nuclear Physics B - Proceedings Supplements , vol. 26, pp. 45–56. DOI.\n[6] G. Aarts, C. Allton, J. Glesaaen, S. Hands, B. Jäger, S. Kim et al., arXiv:2/zero.alt3/zero.alt37./zero.alt34188\n[hep-lat, physics:hep-ph, physics:nucl-th] .\n[7] G. Backus and F. Gilbert, The Resolving Power of Gross Earth Data , vol. 16.\n10.1111/j.1365-246X.1968.tb00216.x.\n[8] D. W. Oldenburg, An introduction to linear inverse theory , vol. GE-22.\n10.1109/TGRS.1984.6499187.\n[9] A. N. Tikhonov, in On the Stability of Inverse Problems , vol. 39, pp. 195–198.\n[10] A. Rothkopf, arXiv:12/zero.alt38.5162 [hep-lat, physics:nucl-th, physics:physics] .\n[11] Particle Data Group, R. L. Workman, V. D. Burkert, V. Crede, E. Klempt, U. Thoma et al., .\n[12] G. Aarts, C. Allton, R. Bignell, T. J. Burns, S. C. García-Mascaraque, S. Hands et al.,\narXiv:22/zero.alt39.14681 [hep-lat, physics:nucl-th] .\n[13] G. Parisi, The strategy for computing the hadronic mass spectrum , vol. 103.\n10.1016/0370-1573(84)90081-4.\n[14] G. P. Lepage, in The Analysis of Algorithms for Lattice Field Theory , pp. 97–120.\n9"    },    {        "title": "2010.05614v2.Decays_rates_for_Kelvin_Voigt_damped_wave_equations_II__the_geometric_control_condition.pdf",        "content": "DECAY RATES FOR KELVIN-VOIGT DAMPED WAVE EQUATIONS II: THE\nGEOMETRIC CONTROL CONDITION\nNICOLAS BURQ AND CHENMIN SUN\nAbstract. We study in this article decay rates for Kelvin-Voigt damped wave equations under a geometric\ncontrol condition. When the damping coe\u000ecient is su\u000eciently smooth ( C1vanishing nicely, see (1.3)) we show\nthat exponential decay follows from geometric control conditions (see [5, 12] for similar results under stronger\nassumptions on the damping function).\n1.Introduction\nIn this paper we investigate decay rates for Kelvin-Voigt damped wave equations under geometric control\nconditions. We work in a smooth bounded domain \n \u001aRdand consider the following equation\n(1.1)8\n><\n>:(@2\nt\u0000\u0001)u\u0000div(a(x)rx@tu) = 0\nujt=0=u02H1\n0(\n); @tujt=0=u12L2(\n)\nuj@\n= 0\nwith a non negative damping term a(x). The solution can be written as\n(1.2) U(t) =\u0012u\n@tu\u0013\n=eAt\u0012u0\nu1\u0013\n;\nwhere the generator Aof the semi-group is given by\nA=\u0012\n0 1\n\u0001 divar\u0013\u0012\nu0\nu1\u0013\n;\nwith domain\nD(A) =f(u0;u1)2H1\n0\u0002L2; \u0001u0+ divaru12L2;u12H1\n0g:\nThe energy of solutions\nE(u)(t) =Z\n\n(jrxuj2+j@tuj2)dx\nsatis\fes\nE((u0;u1))(t)\u0000E((u0;u1))(0) =\u0000Zt\n0Z\n\na(x)jrx@tuj2(s;x)ds:\nOur purpose here is to show that if the damping ais su\u000eciently smooth, the exponential decay rate holds,\ndropping some unnecessary assumptions on the behaviour of the damping term where it becomes positive in\nprevious works [5]. Namely we shall assume a(x)>0 isC1(\n) and satisfy the regularity hypothesis\njraj6Ca1\n2: (1.3)\nOur main result is\nTheorem 1. Assume that \nis a compact Riemannian manifold with smooth boundary. Let a2C1(\n)be a\nnonnegative function satisfying (1.3) , such that the following geometric control condition is satis\fed:\n\u000fThere exists \u000e > 0such that all rays of geometric optics (straight lines) re\recting on the boundary\naccording to the laws of geometric optics eventually reach the set !\u000e=fx2\n :a(x)>\u000egin \fnite time.\n1arXiv:2010.05614v2  [math.AP]  19 Mar 20212 N. BURQ AND C-M. SUN\nThen there exists \u000b>0, such that for all t>0and every (u0;u1)2H1\n0(\n)\u0002L2(\n), the energy of solution u(t)\nof(1.1) with initial data (u0;u1)satis\fes\nE[u](t)6e\u0000\u000btE[u](0):\nTo prove this result, we \frst reduce it very classicaly in Section 2 to resolvent estimates. Since the low\nfrequency estimates are true, we are reduced to the high frequency regime. The proof relies on resolvent\nestimates which are proved through a contradiction argument that we establish in Section 2. In Section 3 we\nprove a priori estimates for our sequences. The main task then is to prove a propagation invariance for these\nmeasures. A main di\u000eculty to overcome is that it is not possible to put the damping term in the r.h.s. of the\nequation (1.1) and treat it as a perturbation . Instead we have to keep it on the left hand side and revisit the\nproof of the propagation property from [7]. In Section 4, we introduce the geometric tools necessary to tackle the\nboundary value problem and de\fne semi-classical measures associated to our sequences. In Section 5 we prove\nthe interior propagation result for our measures. Finally, in Section 6, we \fnish the proof of the contradiction\nargument by establishing the invariance of the semi-classical measures we de\fned up to the boundary. Here the\nproof uses crucially the main result in [7, Th\u0013 eor\u0012 eme 1].\nRemark 1.1. Throughout this note, we shall prove that some operators of the type P\u0000\u0015Id,\u00152R(resp.\n\u00152iR) are invertible with estimates on the inverse. All these operators share the feature that they have\ncompact resolvent, i.e. 9z02C; (P\u0000z0)\u00001exists and is compact (or it will be possible to reduce the question\nto this situation). As a consequence, since\n(P\u0000\u0015) = (P\u0000z0)\u00001(Id + (z0\u0000\u0015))\u00001);\nand (Id + (z0\u0000\u0015)\u00001) is Fredholm with index 0, to show that ( P\u0000\u0015) is invertible with inverse bounded in norm\nbyA, it is enough to bound the solutions of ( P\u0000\u0015)u=fand prove\n(P\u0000\u0015)u=f)kukL26AkfkL2:\nRemark 1.2. Assume that ais the restriction to \n of a nonnegative C2(Rd) function. Then the hypothesis (1.3)\nis satis\fed.\nProof. It is enough to prove (1.3) for \n = Rd,a2C2(\n). Letx02Rdand denote by z0=ra(x0) From\nTaylor's formula, we have for any s2R, there exists \u00122(0;1), such that\na(x0+sz0) =a(x0) +sjz0j2+s2\n2a00(x0+\u0012sz0)(z0;z0)>0\nSince this polynomial in sis non negative, we deduce tat its discriminant is non positive\njz0j4\u00002ka00k1jz0j2a(z0)60)jrxa(x0)jj262ka00k1a(z0):\nNotice that in the above lemma, the condition cannot be relaxed to a2C2(\n);a>0. Indeed, consider the\nfollowing example: \n = B(0;1) anda(x) = 1\u0000jxj2forjxj61. Then obviously a2C2(\n),a>0 , but on the\nboundary,rxa6= 0, whilea= 0. \u0003\nAcknowledgment. The \frst author is supported by Institut Universitaire de France and ANR grant ISDEEC,\nANR-16-CE40-0013. The second author is supported by the postdoc programe: \\Initiative d'Excellence Paris\nSeine\" of CY Cergy-Paris Universit\u0013 e and ANR grant ODA (ANR-18-CE40- 0020-01).\n2.Contradiction argument\nIt is well known that decay estimates for the evolution semi-group follow from resolvent estimates [1, 2, 4].\nHere we shall need the classical (see e.g. [6, Proposition A.1])\nTheorem 2. The exponential decay of the Kelvin Voigt semi-group is equivalent to the following resolvent\nestimate: There exists Csuch that for all \u00152R, the operator (A\u0000i\u0015)is invertible from D(A)toHand its\ninverse satis\fes\n(2.1) k(A\u0000i\u0015)\u00001kL(H)6CKELVIN-VOIGT DAMPING 3\nLet us \frst recall that\n(2.2) ( A\u0000i\u0015)\u0012u\nv\u0013\n=\u0012f\ng\u0013\n,\u001a\u0000i\u0015u+v=f\n\u0001u+ divarxv\u0000i\u0015v=g\nFrom [8, Section 2], we have the following low frequencies estimates of the resolvent of the operator A:\nProposition 2.1. Assume that a2L1is non negative a>0and non trivialR\n\na(x)dx > 0. Then for any\nM > 0, there exists C > 0such that for all \u00152R;j\u0015j6M, the operatorA\u0000i\u0015is invertible from D(A)toH\nwith estimate\n(2.3) k(A\u0000i\u0015)\u00001kL(H)6C:\nAs a consequence, to prove Theorem 1 it is enough to study the high frequency regime \u0015!+1and prove\nProposition 2.2. Assume that a2C1(\n)is a nonnegative function satisfying (1.3) . Then under the geometric\ncontrol condition, there exists \u00030>0such that for any j\u0015j>\u00030we have\nk(A\u0000i\u0015)\u00001kL(H)6C:\nBy standard argument, we can reduce the proof of Proposition 2.2 to a semi-classical estimate. We denote\nby 0<h=j\u0015j\u00001\u001c1 and\nPh=\u0000h2\u0001\u00001\u0000ihdiva(x)r:\nProposition 2.3. There exists C > 0, such that for all 0<h\u001c1,\nkP\u00001\nhkL(L2)6Ch\u00001: (2.4)\nFor the proof of Proposition 2.3, we argue by contradiction. Assume that there exist sequences ( un)\u001a\nH2\\H1\n0;(fn)\u001aL2andhn!0, such that Phnun=fn,kunkL2= 1 andkfnkL2=o(hn). We will use a\nsemi-classical notation and denote by ( uh;fh) the sequences with the properties\nkuhkL2= 1;kfhkL2=o(h); Phuh=fh: (2.5)\nSometimes we even omit the subindex for uh;fh. In the following subsections, we will prove propagation\nestimates for such sequences.\n3.A priori estimates\nIn this section we establish a series of a priori estimates for the sequence de\fned in (2.5).\nLemma 3.1. Assume that a2L1(\n)is a non-negative function, then\n(1)Z\n\n\u0000\njuhj2\u0000jhruhj2\u0001\n= ReZ\n\nfhuh=o(h); (3.1)\n(2)Z\n\na(x)jhruhj2=hImZ\n\nfhuh=o(h2):; (3.2)\n(3)h2kuhkH2=O(1): (3.3)\nProof. We get (1) and (2) by multiplying the equation Phu=fbyu, integrating by part and taking the real\nand imaginary parts repectively. For (3), from the equation, we have\nh2\u0001u+u+ihra\u0001ru+iha\u0001u=\u0000f;i.e.h2\u0001u=\u0000u+f+ira\u0001hru\n1 +ih\u00001a(x):\nFrom the global estimate of the Poisson equation\nkwkH26Ck\u0001wkL2;8w2H2\\H1\n0;\nwe obtain thatkh2ukH2=O(1). \u0003\nCorollary 3.2. Assume that a2C1(\n)is a non-negative function satisfying (1.3) , then\nka1\n2uhkL2+ka1\n2hruhk+ka1\n2h2\u0001uhkL2=o(h): (3.4)4 N. BURQ AND C-M. SUN\nProof. We only need to estimateR\n\na(x)juj2, sinceR\n\na(x)jhruj2=o(h2) is just (3.2). Multiplying Phu=fby\nauand taking the real part, we haveZ\n\na(x)juj2= ReZ\n\nhru\u0001hr(au)\u0000ImhZ\n\na(x)ru\u0001r(a(x)u) + ImZ\n\na(x)fu:\nSincejraj6Ca1\n2, the \frst term on the r.h.s. can be bounded by\nka1\n2hruk2\nL2+hkrahrukL2kukL2=o(h2):\nThe third term of r.h.s is bounded by o(h)ka1\n2ukL2, and the second term can be bounded by\nh\f\f\fZ\n\nara\u0001uru\f\f\f6hka1\n2ukL2ka1\n2rarukL26Cka1\n2hrukka1\n2ukL2=o(h)ka1\n2ukL2:\nFor the second derivative, we observe that\na(x)1\n2h2\u0001u=\u0000a1\n2u+a1\n2f+iha1\n2ra\u0001hru\n1 +ih\u00001a(x);\nthuska1\n2h2\u0001ukL2=o(h). This completes the proof of Corollary 3.2. \u0003\nLet\u0017be the out-normal vector \feld on @\n. We denote by L2(@) =L2(@\n). The following hidden regularity\nholds:\nLemma 3.3. Assume that a2C1(\n)is a nonnegative function satisfying (1.3) , then\nkh@\u0017ukL2(@)=O(1);ka1\n2h@\u0017ukL2(@)=O(h1\n2):\nProof. We use the standard multiplier method. Let L=bj(x)@jbe anC2extension of the out-normal vector\n\feld\u0017, wherebj's are supported in a neighborhood of @\n. WritePh=Ph;0+iMh, where\nPh;0=\u0000h2\u0001\u00001; Mh=\u0000hdiva(x)r\nare self-adjoint operators. Consider the commutator [ Ph;L] = [Ph;0;L] +i[Mh;L]. Note that [ Ph;0;L] =\n1\nh[Ph;0;hL] belongs to h2Op (S2)\u0003, we deduce that\u0000\n[Ph;0;L]u;u\u0001\nL2=O(1). By direct computation, we have\n\u0000\u0000\n[Mh;L]u;u\u0001\nL2=\u0000\nh@k[a@k;bj@j]u;u\u0001\nL2+h\u0000\n[@k;bj@j]a@ku;u\u0001\nL2\n=\u0000\nh@k(a(@kbj)@ju\u0000bj(@ja)@ku);u\u0001\nL2+\u0000\n(@kbj)h@j(a@ku);u\u0001\nL2\n=\u0000\u0000\n(a@kbj)@ju\u0000bj(@ja)@ku;h@ku\u0001\nL2\u0000\u0000\na@ku;h@j((@kbj)u):\u0001\nL2\nFrom Corollary 3.2, the absolute value of the r.h.s. can be bounded by constant times\nkarukL2+krarukL2=o(1):\nTherefore,\u0000\n[Ph;L]u;u\u0001\nL2=O(1). On the other hand, by developing the commutator and exploiting the\nequation, we have\u0000\n[Ph;L]u;u\u0001\nL2=\u0000\nPhLu;u\u0001\nL2\u0000\u0000\nLf;u\u0001\nL2\n=\u0000\nLu;P\u0003\nhu\u0001\nL2\u0000\u0000\nf;L\u0003u\u0001\nL2+h2k@\u0017uk2\nL2(@)+ihka1\n2@\u0017uk2\nL2(@):\nObserve that\n\u0000\nLu;P\u0003\nhu\u0001\nL2=\u0000\nLu;f\u00002Mhu\u0001\nL2=o(1)\u00002\u0000\nLu;Mhu\u0001\nL2=o(1)\u00002\u0000\nLu;hra\u0001ru+ha\u0001u\u0001\nL2:\nSinceLis a \frst order di\u000berential operator and from Corollary 3.2 that ka1\n2h\u0001ukL2=o(1), we have\nj\u0000\nLu;hra\u0001ru+ha\u0001u\u0001\nL2j6khrukL2krarukL2+ka1\n2rukL2ka1\n2h\u0001ukL2=o(1):\nTherefore,\nkh@\u0017uk2\nL2(@)+ihka1\n2@\u0017uk2\nL2(@)=O(1):\n\u0003Strictly speaking, the symbol of Lis not C1, but here we only need1\nh[Ph;0; hL] =O(h2) onL2.KELVIN-VOIGT DAMPING 5\nThe proof of Lemma 3.3 is then completed by taking real and imaginary parts. \u0003\nLet\u001f2C1\nc(R) such that \u001f(z)\u00111 forjzj61 and\u001f(z)\u00110 forjzj>2. We decompose\n(3.5) uh=vh+wh; vh=\u001f\u0000\nah\u00001\u0001\nuh; wh=\u0000\n1\u0000\u001f\u0000\nah\u00001\u0001\u0001\nuh:\nIn the rest of this note, we always assume that a2C1(\n) is a nonnegative function satisfying (1.3).\nLemma 3.4. We have\nkwhkL2+khrwhkL2=o(h1\n2);ka1\n2vhkL2+ka1\n2hrvhkL2=o(h);\nand\nkuhkH1\nh(a>ch)+kvhkH1\nh(a>ch)=o(h1\n2):\nProof. By de\fnition,Z\n\n\u0000\njwj2+jhrwj2\u0001\n6Z\na>h\u0000\njuj2+jhruj2+jraj2juj2\u0001\n:\nThe conclusion then follows from Corollary 3.2 and the fact that jraj26Ca. Similarly, for any other cuto\u000b\nto the region a>ch, we deduce that kuhkH1\nh(a>ch)=o(h1\n2):For the estimate of v, note that a1\n2hrv=\na1\n2\u001fhru+a1\n2ra\u001f0v, from Corollary 3.2, we have\nka1\n2hrvkL26ka1\n2hrukL2+k\u001f0a1\n2(a1\n2v)kL2=o(h):\nThis completes the proof of Lemma 3.4. \u0003\n4.Geometry, semi-classical measures\nHaving the a priori estimates of the previous section at hand, we can now study vh. For some subsequence\nofvh, we will associate it a semi-classical measure and then prove the invariance of the measure under the\ngeneralized geodesic \row. First recall some geometric preliminaries from [7].\n4.1.Geometry. Denote bybT\n the bundle of rank dwhose sections are the vector \felds tangent to @\n,bT\u0003\nthe dual bundle (Melrose's compressed cotangent bundle) and j:T\u0003\n!bT\u0003\n the canonical map. In any\ncoordinate system where \n = fx= (xd>0;x0)g), the bundlebT\n is generated by the \felds@\n@x0,xd@\n@xdandj\nis de\fned by\n(4.1) j(xd;x0;\u0018d;\u00180) = (xd;x0;v=xd\u0018d;\u00180):\nDenote by Car P0the semi-classical characteristic manifold of P0=\u0000h2\u0001\u00001 andZits projection\n(4.2) Car P0=\b\n(x;\u0018) = (x0;xd;\u00180;\u0018d)2T\u0003Rdj\n;p(x;\u0018) = 0\t\n; Z =j(CarP0):\nThe setZis a locally compact metric space.\nConsider, near a point x02@\n a geodesic system of coordinates for which x0= (0;0), \n =f(xd;x0)2\nR+\u0002Rd\u00001gand the operator P0has the form (near x0)\n(4.3) Ph;0=\u0000h2\u0001\u00001 =h2D2\nxd\u0000R(xd;x0;hDx0) +hQ(x;hDx);\nwithRa second order tangential operator and Qa \frst order operator.\nWe recall now the usual decomposition of T\u0003@\n (in this coordinate system). Denote by r(x0;xd;\u00180) the\nsemi-classical principal symbol of Randr0=rjxd=0. ThenT\u0003@\n is the disjoint union of E[G[H with\n(4.4) E=fr0<0g;G=fr0= 0g;H=fr0>0g:\nRemark that jgives a natural identi\fcation between Zj@MandH[G\u001aT\u0003@M. InGwe distinguish between the\ndi\u000bractive pointsG2;+=fr0= 0;r1=@xdrjxd=0>0gand the gliding pointsG\u0000=fr0= 0;r1=@xdrjxd=060g.\nWe will make the assumption (\n has no in\fnite order contact with its tangents) that for any %02T\u0003@M, there\nexistsN2Nsuch that\nHN\nr0(r1)6= 06 N. BURQ AND C-M. SUN\nThe de\fnition of the generalized bicharacteristic \row, 'sassociated to the operator P0is essentially the\nde\fnition given in [11]:\nDe\fnition 4.1. A generalized bicharacteristic curve \r(s) is a continuous curve from an interval I\u001aRtoZ\nsuch that\n(1) ifs02Iand\r(s0)2T\u0003\n then close to s0,\ris an integral curve of the Hamiltonian vector \feld Hep\n(2) Ifs02Iand\r(s0)2H[G2;+then there exists \">0 such that for 0 <js\u0000s0j<\",xd(\r(s))>0\n(3) Ifs02Iand\r(s0)2G\u0000then for any function f2C1(T\u0003Rdj\n) satisfying the symmetry condition\n(4.5) 8%02Z;8b%0;e%02j\u00001(%0)\\Car(eP);f(b%0) =f(e%0)\nthen\nd\ndsf(j(\r(s))js=s0=Hepjj\u00001(\r(s0))f(j\u00001(\r(s0)))\nIt is proved in [11] that under the assumption of no in\fnite order contact, through every point %o2bT\u0003Mnf0g\nthere exists a unique generalized bicharacteristic (which is furthermore a limit of bicharacteristics having only\nhyperbolic contacts with the boundary). This de\fnes the \row \b.\n4.2.Wigner measures. Consider functions a=ai+a@withai2C1\n0(T\u0003M), anda@2C1\n0(R2d\u00001). Such\nsymbols are quantized in the following way: take 'i2C1\n0(M) (resp'@2C1\n0(Rd)) equal to 1 near the\nx-projection of supp( ai) (resp the x-projection of supp( a@)) and de\fne\n(4.6) Op'i;'@\nh(a)(x;hDx)f=1\n(2\u0019h)dZ\nei(x\u0000y)\u0001\u0018=hai(x;\u0018)'i(y)f(y)dyd\u0018\n+1\n(2\u0019h)d\u00001Z\nei(x0\u0000y0)\u0001\u00180=ha\u000e(xd;x0;\u0018)'\u000e(xd;y0)f(xd;y0)dy0d\u00180:\nRemark that according to the symbolic semi-classical calculus, the operator Op'i;'@\nh(a) does not depend on the\nchoice of functions 'i;'@, modulo operators on L2of norms bounded by O(h1). For conciseness we shall in\nthe sequel drop the index 'i;'@.\nDenote byAhthe space of the operators which are a \fnite sum of operators obtained as above in suitable\ncoordinate systems near the boundary and for B2A, byb=\u001b(B) the semiclassical symbol of the operator A.\nFor such functions bwe can de\fne \u0014(b)2C0(Z) by\n(4.7) \u0014(b)(\u001a) =b(j\u00001(\u001a))\n(the value is independent of the choice of j\u00001(\u001a) since the operator is tangential).\nThe set\n(4.8) f\u0014(b);b=\u001b(B);B2Ahg\nis a locally dense subset of C0\nc(Z).\n4.2.1. Elliptic regularity. The sequence vhsatis\fes (with \u001fh=\u001f(a=h))\nPhvh=\u001fhPhuh\u0000h2div(r\u001fhuh)\u0000h2r\u001fhruh\u0000ihdiv(ar\u001fhuh)\u0000ihar\u001fh\u0001ruh:\nSincer\u001fh=h\u00001\u001f0(a=h)raandjraj.a1\n2, by Corollary 3.2, we have\nPhvh=oL2(h) +oH\u00001(h2):\nThus\n(h2\u0001 + 1)vh=\u0000ihdiv(ar(\u001fhu)) +oL2(h) +oH\u00001(h2):\nUsing Corollary 3.2 again and the fact that a.hon the support of \u001fh, we deduce that har(\u001fhu) =oL2(h3\n2),\nhence\n(h2\u0001 + 1)vh=o(h3\n2)H\u00001(\n)+oL2(h):\nWe deduce, by standard elliptic regularity resultsKELVIN-VOIGT DAMPING 7\nProposition 4.2. Ifaiis equal to 0near Car(P0)then\n(4.9) lim\nk!+1(Ophk(ai)vhk;vhk)L2= 0;\nwhile near the boundary (see e.g. [7, Appendice A.1] in a slightly di\u000berent context) we get\nProposition 4.3. Ifa@is equal to 0nearZ(i.e.aiis supported in the elliptic region) then\n(4.10) lim\nk!+1(a@(x0;xd;hkDx0)vhk;vhk)L2= 0:\nRemark 4.4. Note that if we regard the damping term hdiv(ar(\u001fhu)) =oH\u00001(h3\n2) as a source term, we are\nnot able to use the classical propagation theorem for h2\u0001 + 1 as a black box, as such a strategy would require\nsmaller r.h.s., namely oH\u00001(h2) +oL2(h). On the other hand, an integration by parts shows from Lemma 3.4\n\u0010\ndiv(ar(\u001fhu));\u001fhu\u0011\nL2=\u0000ka1=2rx\u001fhuk2\nL2=o(1);\nand this will ensure that in the propagation estimates such terms are invisible. The key of our analysis in the\nsequel will be to systematically uses this procedure: testing the damping term on expressions like Qh\u001fhu, doing\nthe integration by part and then balancing a1\n2to the other side. It is to perform this analysis that we need the\nconditionjraj6Ca1\n2to ensure the gain O(h) from the commutator [ a1\n2;Qh]. More precisely, we shall need the\nfollowing lemma:\nLemma 4.5. Assume that Qh;B0;h;B1;hare tangential h-pseudodi\u000berential operators of order 0andBh=\nB0;h+B1;hhDxd, then\nh\u00001\u0000\nQhMhu;Bhu\u0001\nL2=o(1);\nwhereMh=\u0000hdivar.\nProof. SinceMhu=\u0000rahru\u0000ah\u0001u, from Corollary 3.2, we have\nh\u00001\f\f\u0000\nQhrahru;Bhu\u0001\nL2\f\f6CkrarukL2kukH1\nh=o(1):\nTo estimate\u0000\nQha\u0001u;Bhu\u0001\nL2, we writeQha\u0001u=a1\n2Qha1\n2\u0001u+ [Qh;a1\n2]a1\n2\u0001u. From Corollary 3.2, a1\n2\u0001u=\noL2(h\u00001). By Corollary A.2, [ Qh;a1\n2];[Bh;a1\n2] =OL(L2)(h). Therefore,\n\u0000\nQha\u0001u;Bhu\u0001\nL2=\u0000\nQha1\n2\u0001u;Bha1\n2u\u0001\nL2+o(1):\nAgain from Corollary 3.2, we have Bha1\n2u=OL2(h), hence (Qha\u0001u;Bhu)L2=o(1). The proof of Lemma 4.5\nis complete. \u0003\n4.2.2. De\fnition of the measure. The following results gives the existence of semi-classical measures.\nProposition 4.6. Let(vhkp)be a sequence bounded in L2(\n). There exists a subsequence (kp)and a Radon\npositive measure \u0016onZsuch that\n(4.11) 8Q2Ahkplim\np!1(Qvhkp;vhkp)L2=h\u0016;\u0014(\u001b(Q))i:\nThe proof of this result relies on the G\u0017 arding inequality for tangential operators (see G. Lebeau [10] for a\nproof in the classical context and [3, 9] for the semi-classical construction). As before, we drop the indexes kp\nand denote by ( vh) the extracted sequence.\nProposition 4.7 (First properties of the measure \u0016).We have\n(4.12) \u0016(H) = 0:\nMoreover, for any tangential symbol b,\n(4.13) lim sup\nk!+1j(Oph(b)hkDxdvhk;vhk)L2j6C sup\n%2supp(b)jrj1=2jbj:8 N. BURQ AND C-M. SUN\nProof. The \frst property (4.12) follows from the fact that the trajectories near a hyperbolic point is transversal\nto the boundary. It follows from [7], with additional attention to the damping term ihdivarvh. We factorize\nPh;0=\u0000h2\u0001\u00001 as (hDxd\u0000L+\nh)(hDxd\u0000L\u0000\nh)+OH1(h1) (see [7, Lemme 6.1]) near \u001a02Hand choose L\u0006\nhwith\nprincipal symbols \u0006l(x0;xd;\u00180) =\u0006p\nr(x0;xd;\u00180). we denote by q0(x0;\u00180)2C1\nc(H) andq\u0006(y;x0;\u00180) solutions\nof\n@xdq\u0006\u0007fl;q\u0006g= 0; q\u0006jxd=0=q0:\nDenote by\nu\u0006:= (xd)Q\u0006\nh(hDxd\u0000L\u0007\nh)u;\nwhere (xd)\u00111 if 06xd6\u000f0. We have\n(hDxd\u0000L\u0006\nh)u\u0006= (xd)[hDxd\u0000L\u0006\nh;Q\u0006\nh](hDxd\u0000L\u0007\nh)u+Q\u0006\nhfh+h\ni 0(xd)Q\u0006\nh(hDxd\u0000L\u0007\nh)u\n+iQ\u0006\nhMhu+OL2(h1)\nwhereMh=hdivarandfh=oL2(h). By de\fnition of q\u0006, the \frst term of r.h.s. is O(h2), hence\n(hDxd\u0000L\u0006\nh)u\u0006=g\u0006\nh\u0000ih 0(xd)Q\u0006\nh(hDxd\u0000L\u0007\nh)u+iQ\u0006\nhMhu; g\u0006\nh=oL2(h): (4.14)\nWe have\nhd\ndxd(u\u0006;u\u0006)L2(@)=\u00002 Im\u0000\ng\u0006\nh\u0000ih 0(xd)Q\u0006\nh(hDxd\u0000L\u0007\nh)u+iQ\u0006\nhMhu;u\u0006\u0001\nL2(@)+i\u0000\n(L\u0006\nh\u0000L\u0006;\u0003\nh)u\u0006;u\u0006\u0001\nL2(@):\nFory06\u000f0, we have\nku\u0006(y0)k2\nL2(@)6ku\u0006(0)k2\nL2(xd=0)+Ch\u00001kg\u0006\nhkL2(xd6y0)ku\u0006kL2(xd6y0)+Cku\u0006k2\nL2(xd6y0)\n+Ch\u00001\f\f(Q\u0006\nhMhu;u\u0006)L2(xd6y0)\f\f\nThe second line of r.h.s. is o(1), due to Lemma 4.5, and the \frst line of r.h.s. can be bounded by\nku\u0006(0)k2\nL2(xd=0)+Cku\u0006k2\nL2(xd6y0)+o(1):\nIntegrating both sides over y06\u000f0, lettingh!0 and then \u000f0!0, we deduce that h\u00161y=0;q0i= 0. This proves\n(4.12).\nFor (4.13), it su\u000eces to prove the inequality for uinstead ofv. By Cauchy-Schwarz,\n\f\f\u0000\nOph(b)h@xdu;u\u0001\nL2\f\f6\f\f\u0000\nOph(b)\u0003Oph(b)h@xdu;h@xdu\u0001\nL2\f\f1\n2kukL2:\nDoing integration by part, we have\n\u0000\nOph(b)\u0003Oph(b)h@xdu;h@xdu\u0001\nL2=\u0000\u0000\nOph(b)\u0003Oph(b)h2@2\nxdu;u\u0001\nL2+O(h):\nReplacingh2@2\nxduby equation h2@2\nxdu=\u0000Rhu\u0000iMhu+OL2(h);we deduce that\n\f\f\u0000\nOph(b)\u0003Oph(b)h2@2\nxdu;u\u0001\nL2\f\f6\f\f\u0000\nOph(jbj2)Rhu;u\u0001\nL2\f\f+O(h) +\f\f\u0000\nOph(b)\u0003Oph(b)Mhu;u\u0001\nL2\f\f:\nFrom Lemma 4.5, the third term of r.h.s. is o(h). Passingh!0, we complete the proof of Proposition 4.7. \u0003\n4.3.Invariance of the measure. The key to prove the invariance of the measure will be to apply the propa-\ngation theorem in [7, Th\u0013 eor\u0012 eme 1].\nTheorem. The two following properties are equivalent\n(1)The measure \u0016is invariant along the generalised \row.\n(2)The measure \u0016satis\fes _\u0016= 0and\u0016(G+\n2) = 0 in the sense thath\u0016;fp;qgi= 0holds for any even symbol\nq2C1\nc(Car(P0)), i.e.q(x0;xd= 0;\u00180;\u0018d) =q(x0;xd= 0;\u00180;\u0000\u0018d).KELVIN-VOIGT DAMPING 9\nRemark 4.8. Technically, Theorem 4.3 is proved in [7] for time dependent measures, i.e. measures depending\nin addition on two additional variables ( t;\u001c)2T\u0003R, andpis replaced by p\u0000\u001c2. However, it is easy to apply\nthe results from [7] by considering the measure\n(4.15) e\u0016=\u0016x;\u0018\ndt\n\u000e\u001c=1;\nwhich is supported in Car( \u0000\u0001 +@2\nt) and satis\fes _e\u0016= 0 in the sense that h\u0016;fp\u0000\u001c2;qgi= 0 holds for any even\nsymbolq2C1\nc(Car(P0\u0000\u001c2)), i.e.q(x0;xd= 0;t;\u00180;\u0018d;\u001c) =q(x0;xd= 0;t;\u00180;\u0000\u0018d;\u001c). Remark that though we\nshall not use it, the measure e\u0016is, in the sense of [7, Section 2], the microlocal defect measure on the sequence\nvn(t;x) =hneith\u00001\nnun(x) (the pre-factor hncomes from the H1normalisation of the sequence vnin [7]). Now,\nthe generalised bicharacteristic \row for p0\u0000\u001c2, \tsis given in terms of the generalised bicharacteristic \row for\np0, sby\n\ts(t;x;\u001c = 1;\u0018) = (t\u00002s;\u001c= 1; s(x;\u0018));\nThe set of di\u000bractive points eG2;+in the time dependent frame-work is given by\neG2;+=G2;+\u0002R\u0002f\u001c=\u00061g\nand consequently,\n\u0016(G2;+) = 0,e\u0016(eG2;+) = 0;\nand in view of the particular form (4.15), the invariance of e\u0016by \tsis equivalent to the invariance of \u0016by s.\nLet us now brie\ry explain the procedure we are going to follow.\n\u000fFirst from Proposition 4.6 and the elllipticity (Proposition 4.9, Proposition 4.10), the measure \u0016is\nde\fned on Z=j(Car(P0)) by testing on symbols of the form q=qi+q@;qi2C1\nc(T\u0003\n) andq@\ntangential (which is dense in C0(Z)).\n\u000fUsing the fact \u0016(H) = 0 (Proposition 4.7), the measure \u0016can be extended to test on functions of\nCar(P0) which admits a representation (thanks to Malgrange's theorem)\nq(x0;xd;\u00180;\u0018d) =q0(x0;xd;\u00180) +\u0018dq1(x0;xd;\u00180);on\u00182\nd=r(x0;xd;\u00180):\nThen, we will show in Proposition 4.9 that for tangential h-pseudodi\u000berential operators B0;h;B1;h, the\nquadratic form\n((B0;hk+B1;hk1\nihk@xd))vhk;vhk)\nconverges toh\u0016;b0+b1\u0018d1\u001a=2Hi, by a suitable limit procedure for symbols in Ah. Consequently, for any\nq2C1\nc(Car(P0)), we can make sense of the expression\nh\u0016;fp;qgi\n\u0016-a.e., by viewing fp;qg= 2\u0018d@xdq1\u001a=2H\u0000fr;qg:We remark that to calculate fp;qg, it is enough to\nchoose one representation q=q0+q1\u0018don Car(P0), sincefp;pg= 0 andp= 0 on supp( \u0016).\n\u000fFinally, to prove that the measure \u0016is invariant along the Melrose-Sj ostrand \row, we apply Theorem 4.3,\nfor which we need to verify the following conditions:\n(a)\u0016(G2;+) = 0\n(b) _\u0016= 0, in the sense that h\u0016;fp;qgi= 0 holds for any even symbol q2C1\nc(Car(P0)), i.e.q(x0;xd=\n0;\u00180;\u0018d) =q(x0;xd= 0;\u00180;\u0000\u0018d).\nThe veri\fcation of (a),(b) in our context is based on the propagation formula: Proposition 5.1 and Proposi-\ntion 6.1. Especially, starting from Proposition 6.1, by choosing suitable test symbols of the form q0+q1\u0018d, we\nare able to verify the conditions (a) and (b).\nProposition 4.9. IfB0;h;B1;hare two tangential h-pseudodi\u000berential operators of with principal symbols b0;b1\nof order 0, then we have\nlim\nk!1((B0;hk+B1;hk1\nihk@xd)vhk;vhk)L2=h\u0016;b0+b1\u0018d1\u001a=2Hi:10 N. BURQ AND C-M. SUN\nProof. SinceB0;handB1;hare all tangential, by the de\fnition of the measure, the \frst term ( B0;hkvhk;vhk)L2\nconverges toh\u0016;b0i. It remains to prove the convergence of the second term ( B1;hk1\nihk@xdvhk;vhk)L2. For this,\nwe pick\u000f>0;\u000e> 0 and de\fne\nB1;hk;\u000f=\u0000\n1\u0000 \u0000xd\n\u000f\u0001\u0001\nB1;hk\u0000\n1\u0000 \u0000xd\n2\u000f\u0001\u0001\n; B\u000f\n1;hk=B1;hk\u0000B1;hk;\u000f;\nB\u000f;\u000e\n1;hk= Ophk\u0000\n \u0000r\n\u000e\u0001\u0001\nB\u000f\n1;hk; B\u000f\n1;hk;\u000e=B\u000f\n1;hk\u0000B\u000f;\u000e\n1;hk;\nwhere is a cuto\u000b function which is 1 near 0. Now by the de\fnition of \u0016and the dominating convergence,\nlim\n\u000f!0lim\nk!1(B1;hk;\u000f1\nihk@xdvhk;vhk)L2=h\u0016;b1\u0018d1xd>0i=h\u0016;b1\u0018d1\u001a=2Hi;\nsince\u0016(E) =\u0016(H) = 0. Now from Proposition 4.7, the contribution of\nlim\n\u000f!0lim sup\nk!1j(B\u000f;\u000e\n1;hkhk@xdvhk;vhk)L2j6C\u000e1\n2;\nwhich converges to 0 if we let \u000e!0. Finally, by Cauchy-Schwarz,\nj(B\u000f\n1;hk;\u000ehk@xdvhk;vhk)L2j6khk@xdvhkkL2k(B\u000f;\u000e\n1;hk)\u0003vhkkL2:\nNotice that\nlim\nk!1k(B\u000f;\u000e\n1;hk)\u0003vhkk2\nL2= lim\nk!1(B\u000f;\u000e\n1;hk(B\u000f;\u000e\n1;hk)\u0003vhk;vhk)L2\n=h\u0016;\u0000\n1\u0000 \u0000r\n\u000e\u0001\u0001\u0002\n \u0000xd\n\u000f\u0001\u0000\n1\u0000 \u0000xd\n2\u000f\u0001\u0001\nb1+\u0000\n1\u0000 \u0000xd\n\u000f\u0001\n \u0000xd\n2\u000f\u0001\u0001\nb1\u0003\ni;\ntaking the double limit lim sup\u000e!0lim sup\u000f!0, we obtain that\nlim sup\n\u000e!0lim sup\n\u000f!0lim\nk!1k(B\u000f;\u000e\n1;hk)\u0003vhkk2\nL26h\u0016;b2\n11xd=01r6=0i= 0;\nsince\u00161E[H= 0. This completes the proof of Proposition 4.9. \u0003\n5.Interior propagation estimate\nProposition 5.1 (Interior propagation) .LetQh=e\u001fQhe\u001fbe ah-pseudodi\u000berential operator of order 0, where\ne\u001f2C1\nc(\n), then we have\n1\nih\u0000\n[h2\u0001 + 1;Qh]vh;vh\u0001\nL2=o(1):\nProof. Denote byPh=Ph;0+iMhwithMh=\u0000hdivarandPh;0=\u0000h2\u0001\u00001, we have\n1\nih\u0000\n[Ph;0;Qh]v;v\u0001\nL2=1\nih\u0000\nQhv;Ph;0v\u0001\nL2\u00001\nih\u0000\nPh;0v;Q\u0003\nhv\u0001\nL2\n=1\nih\u0000\nQhv;\u001fPh;0u\u0001\nL2\u00001\nih\u0000\n\u001fPh;0u;Q\u0003\nhv\u0001\nL2+R1\nwithR1=1\nih\u0000\nQhv;[Ph;0;\u001f]u\u0001\nL2\u00001\nih\u0000\n[Ph;0;\u001f]u;Q\u0003\nhv\u0001\nL2:\nBy using the equation Ph;0u=f\u0000iMhu, we have\n1\nih\u0000\n[Ph;0;Qh]v;v\u0001\nL2=R1+R2+o(1); (5.1)\nwhere\nR2=1\nh\u0000\nQhv;\u001fMhu\u0001\nL2+1\nh(\u001fMhu;Q\u0003\nhv)L2:\nNote that\n\u0000[Ph;0;\u001f] =hr\u0000\nra\u001f0(a=h)\u0001\n+ 2ra\u001f0(a=h)hr;\nsincehr(\u001f(a=h)) =ra\u001f0(a=h).\n\u000fClaim 1:R1=o(1)KELVIN-VOIGT DAMPING 11\nIt su\u000eces to show that ih\u00001\u0000\nBhv;[Ph;0;\u001f]u\u0001\nL2=o(1) for any compact supported h-pseudoBhof degree 0. By\nintegration by part,\n\u00001\nih\u0000\nBhv;[Ph;0;\u001f]u\u0001\nL2=\u00001\nih\u0000\nBhv;hdiv (\u001f0(a\nh)rau) +\u001f0(a\nh)rahru\u0001\nL2\nand we simply apply Corollary 3.2, to get for each term o(1).\n\u000fClaim 2:R2=o(1)\nIt su\u000eces to prove that ( Qhv;\u001fdiv (aru))L2=o(1). We write\n\u0000\nQhv;\u001fdivaru\u0001\nL2=\u0000\u0000\n(r\u001f)Qhv;aru\u0001\nL2\u0000\u0000\n\u001f[r;Qh]v;aru\u0001\nL2\u0000\u0000\n\u001fQhrv;aru\u0001\nL2:\nSinceja1\n2r\u001fj=h\u00001ja1\n2\u001f0raj6C, from Corollary 3.2, the \frst term of r.h.s. can be bounded by\nkQhvkL2ka1\n2rukL2=o(1):\nThe second term of r.h.s. can be bounded by o(h). Observe that r(a1\n2) =1\n2a\u00001\n2rais bounded, thus from\nCorollary A.2,\n[a1\n2;Qh] =OL(L2)(h):\nTherefore,\f\f\u0000\n\u001fQhrv;aru\u0001\nL2\f\f6\f\f\u0000\n\u001fQha1\n2rv;a1\n2ru\u0001\nL2\f\f+\f\f\u0000\n\u001f[a1\n2;Qh]rv;a1\n2ru\u0001\nL2\f\f:\nThe second term is bounded by ChkrvkL2ka1\n2rukL2=o(1), and the \frst term can be bounded by o(1), due\nto Lemma 3.4. This completes the proof of Proposition 5.1. \u0003\n6.Propagation near the boundary\nRecall that vh=\u001f(a=h)uh. Consider the operator\nBh=B0;h+B1;hh\ni@xd\nwhereBj;h=e\u001f1Oph(bj)e\u001f1,j= 0;1 are two tangential operators and e\u001f1has compact support near a point\nz02@\n. Note that in the local coordinate system,\nPh;0=\u0000h2\u0001\u00001 =\u00001p\njgjh@xdp\njgjh@xd\u0000Rh;\nwhereRhis a self-adjoint tangential operator of order 2. The operator involving the damping can be written as\nMh=\u0000hp\njgj@xdp\njgja@xd\u0000hp\njgj@x0\nkp\njgjagjk@x0\nj\nProposition 6.1 (Boundary propagation) .\n1\nih\u0000\n[Ph;0;Bh]v;v\u0001\nL2=\u0000\nB1;hjxd=0(h@xdv)jxd=0;(h@xdv)jxd=0\u0001\nL2(@)+o(1):\nProof. We give the proof in the case B0;h= 0. TheB0;hterms are handled by slightly simpler versions of the\nsame computations. By developing the commutator, we have\n1\nih\u0000\n[Ph;0;Bh]v;v\u0001\nL2=1\nih\u0000\nBhv;Ph;0v\u0001\nL2\u00001\nih\u0000\nBhPh;0v;v\u0001\nL2+\u0000\nB1;hjxd=0(h@xdv)jy=0;h@xdvjxd=0\u0001\nL2(@);\nwhere the boundary term (the third) comes from the integration by part of the term\n1\nih\u00001p\njgjh@xdp\njgjh@xdv;v\u0001\nL2;\nsinceRhis self-adjoint tangential operator. It su\u000eces to show that\nIh:=1\nih\u0000\nB1;hh@xdv;Ph;0v\u0001\nL2\u00001\nih\u0000\nB1;hh@xdPh;0v;v\u0001\nL2=o(1): (6.1)12 N. BURQ AND C-M. SUN\nSincev=\u001fuandPh;0u=Phu\u0000iMhu=fh\u0000iMhu, we have\nPh;0v=\u001fPh;0u+ [Ph;0;\u001f]u=\u001ffh\u0000i\u001fMhu+ [Ph;0;\u001f]u:\nTherefore,\nIh=o(1) + Ih;1+ Ih;2;\nwhere\nIh;1=1\nih\u0000\nB1;hh@xdv;[Ph;0;\u001f]u\u0001\nL2\u00001\nih\u0000\nB1;hh@xd[Ph;0;\u001f]u;v\u0001\nL2\nand\nIh;2=1\nh\u0000\nB1;hh@xdv;\u001fMhu\u0001\nL2\u00001\nh\u0000\nB1;hh@xd\u001fMhu;v\u0001\nL2\n\u000fClaim 1:Ih;1=o(1).\nIndeed, from integration by part, the second term\nih\u00001(B1;hh@xd[Ph;0;\u001f]u;v)L2=ih\u00001([Ph;0;\u001f]u;h@xdAhv)L2\nfor some tangential operator Ah, hence it has the same structure as the \frst term. It su\u000eces to show that\nh\u00001\u0000\nB1;hh@xdv;[Ph;0;\u001f]u\u0001\nL2=o(1):\nSince\n\u0000[Ph;0;\u001f]u=hr\u0001(ra\u001f0(a=h))u+ 2ra\u001f0(a=h)\u0001hru;\ndoing integration by part, we obtain that\n\u0000h\u00001\u0000\nB1;hh@xdv;[Ph;0;\u001f]u\u0001\nL2=\u0000\u0000\nr(uB1;hh@xdv);ra\u001f0\u0001\nL2+ 2h\u00001\u0000\nB1;hh@xdv;ra\u001f0hru\u0001\nL2\n=\u0000\u0000\nrB1;hh@xdv;ra\u001f0u\u0001\nL2+h\u00001\u0000\nB1;hh@xdv;ra\u001f0hru\u0001\nL2:\nNote thatv=\u001fu, if one of the derivatives h@xd,hrfall on\u001f(a=h) we can bound them from Corollary 3.2 by\no(h). If all the derivatives fall on uin anyone of the two terms, by Lemma 3.1 and Corollary 3.2, these terms\ncan be bounded by\nkhr@xdukL2kraukL2+h\u00001kh@xdukL2krahrukL2=o(1):\n\u000fClaim 2: Ih;2=o(1).\nIt su\u000eces to prove that\nh\u00001\u0000\nB1;hh@xd(\u001fu);\u001fMhu\u0001\nL2=o(1):\nNote that\u0000Mhu=rahru+ah\u0001u=oL2(1) andh@xd(\u001fu) =@xda\u001f0u+\u001fh@xdu. We have\nh\u00001j\u0000\nB1;h@xda\u001f0u;\u001fMhu\u0001\nL2j6h\u00001kraukL2k\u001fMhukL2=o(1);\nsincekraukL2=o(h) from Corollary 3.2. It remains to show that\nh\u00001\u0000\nB1;h\u001fh@xdu;\u001f(ra\u0001hru+ah\u0001u)\u0001\nL2=o(1):\nSincekrahrukL2=o(h), we haveh\u00001\u0000\nB1;h\u001fh@xdu;\u001f(ra\u0001hru)\u0001\nL2=o(1). Finally, we show that\nh\u00001\u0000\nB1;h\u001fh@xdu;\u001fah \u0001u\u0001\nL2=o(1):\nRecall that from jr(a1\n2)j6Cand Corollary A.2,\n[B1;h;a1\n2] =OL(L2)(h);\nwe have\nh\u00001j\u0000\nB1;h\u001fh@xdu;\u001fah \u0001u\u0001\nL2j6h\u00001j\u0000\nB1;ha1\n2\u001fh@xdu;\u001fa1\n2h\u0001u\u0001\nL2j+h\u00001j\u0000\n[B1;h;a1\n2]\u001fh@xdu;\u001fa1\n2h\u0001u\u0001\nL2j\n6Ch\u00001ka1\n2hrukL2ka1\n2h\u0001ukL2+CkhrukL2ka1\n2h\u0001ukL2=o(1):\nThis completes the proof of Proposition 6.1. \u0003KELVIN-VOIGT DAMPING 13\nTo show that the semi-classical measure \u0016of (vhk) is invariant along the Melrose-Sj ostrand \row (to complete\nthe proof of Proposition 2.3), we need to verify the condition (2) in Theorem 4.3. We will make use of the\npropagation formula, i.e. Proposition 6.1. Formally, for Bh=B0;h+B1;h1\nih@xd, the principal symbol of\ni\nh[Ph;0;Bh] is given by\nf\u00112\u0000r;b0+b1\u0018dg=a0+a1\u0018d+a2\u00182\nd;\nwhere\na0=b1@xdr\u0000fr;b0g0; a 1= 2@xdb0\u0000fr;b1g0; a 2= 2@xdb1; (6.2)\nandf\u0001;\u0001g0is the Poisson bracket for ( x0;\u00180) variables. On the other hand, by calculating the commutator, we\n\fnd\ni\nh[Ph;0;Bh] =A0+A1hDxd+A2h2D2\nxd+hOph(S0\n@+S0\n@\u0018d); (6.3)\nwhereA0;A1;A2are tangential operators with symbols a0;a1;a2, with respectively. We will prove the following\npropagation formula:\nCorollary 6.2. Assume that Bh=Bh;0+Bh;1hDxd, whereBh;0;Bh;1are tangential operators of order 0 with\nsymbolsb0;b1, with respectively. Assume that b=b0+b1\u0018d. De\fne the formal Poisson bracket\nfp;bg= (a0+a2r) +a1\u0018d1\u001a=2H;\nwherea0;a1;a2are given by (6.2) . Then the defect measure \u0016satis\fes the equation\nh\u0016;fp;bgi=\u0000h\u0017@;b1i;\nwhere\u0017@is the semiclassical measure of (h@xdvhjxd=0):Moreover, if bis an even symbol (i.e. b(x0;xd=\n0;\u00180;\u0018d) =b(x0;xd= 0;\u00180;\u0000\u0018d)), then we have\nh\u0016;fp;bgi= 0:\nIn particular, by combining Proposition 5.1, we have _\u0016= 0.\nProof. From Proposition 6.1 and the decomposition (6.3), we have\n\u0000\n(A0+A1hkDxd+A2h2\nkD2\nxd)vhk;vhk\u0001\nL2=\u0000h\u0017@;b1i+o(1): (6.4)\nFrom Lemma 3.4, we can also replace the function vhkon the l.h.s. by uhk. Using the equation of uhk:\n(h2D2\nxd\u0000Rhk)uhk=iMhkuhk\u0000fhk+OL2(hk);\nwe deduce that\n(A2h2\nkD2\nxduhk;uhk) = (A2Rhkuhk;uhk)L2+o(1);\nthanks to Lemma 4.5. Therefore, from Proposition 4.9,\nlim\nkh!1((A0+A1hkDxd+A2Rhk)uhk;uhk)L2=h\u0016;a0+a1\u0018d1\u001a=2H+a2ri=h\u0016;fp;bgi:\nNow ifb=b0+b1\u0018dis an even symbol, we must have b1jxd=0= 0, therefore,h\u0016;fp;bgi=\u0000h\u0017@;b1i= 0. The\nproof of Lemma 6.2 is complete. \u0003\nCorollary 6.3. We have\u0016(G2;+) = 0 .\nProof. We will make use of the formula h\u0016;fp;bgi=\u0000h\u0017@;b1iby choosing b=b1;\u000f\u0011with\nb1;\u000f(x0;xd;\u00180) = \u0000xd\n\u000f1\n2\u0001\n \u0000r(xd;x0;\u00180)\n\u000f\u0001\n\u0014(xd;x0;\u00180);\nwhere 2C1\nc(R) equals to 1 near the origin and \u0014(y;x0;\u00180)>0 near a point \u001a02G2;+. Sincefp;b\u000fg=\n(a0+a2r) +a1\u0018d1\u001a=2H, anda0;a1;a2are given by the relation (6.2). In particular for our choice, by direct\ncalculation we have\na0=b1;\u000f@xdr; a 1=\u0000fr;\u0014g0 \u0000xd\n\u000f1\n2\u0001\n \u0000r\n\u000f\u0001\n;14 N. BURQ AND C-M. SUN\nand\na2= 2@xdb1;\u000f= 2\u000f\u00001\n2 0\u0000xd\n\u000f1\n2\u0001\n \u0000r\n\u000f\u0001\n\u0014+ 2@xdr\n\u000f \u0000xd\n\u000f1\n2\u0001\n 0\u0000r\n\u000f\u0001\n\u0014+ 2 \u0000xd\n\u000f1\n2\u0001\n \u0000r\n\u000f\u0001\n@xd\u0014:\nNote thata2is uniformly bounded in \u000fand for any \fxed ( y;x0;\u00180),ra2!0 as\u000f!0. Thus by dominating\nconvergence, we have\nlim\n\u000f!0h\u0016;fp;b\u000fgi=h\u0016;\u0014jxd=0@xdr1r=0i>0\nsince@xdr>0 onG2;+. However,\u0000h\u0017@;b\u000fi60, we must have \u00161G2;+= 0. This completes the proof of Lemma\n6.3. \u0003\nFrom Lemma 6.2 and Lemma 6.3, we have veri\fed that _ \u0016= 0 and\u0016(G2;+) = 0, thus from Theorem 4.3,\nthe semi-classical \u0016is invariant along the Melrose-Sj ostrand \row. Thanks to the geometric control condition\nand the fact that a1\n2vhk=oL2(1), we deduce that \u0016= 0. This contradicts to the assumption that kvhkkL2=\nkuhkkL2= 1 +o(1), ask!1 . The proof of Proposition 2.3 is now complete.\nAppendix A.Some commutator estimates\nLemma A.1. Assume that b(x;y;\u0018 )2L1(R3d\nx;y;\u0018)such that\nj@\u000b\n\u0018b(x;y;\u0018 )j.\u000bh\u0018i\u0000(j\u000bj+1)\nfor all multi-index \u000b2Nd;j\u000bj6d+ 1. Then the operator Thassociated with the Schwartz kernel\nKh(x;y) :=1\n(2\u0019h)dZ\nRdb(x;y;\u0018 )ei(x\u0000y)\u0001\u0018\nhd\u0018\nis bounded on L2(Rd), uniformly in 0<h61.\nProof. Using the Littlewood-Paley decomposition, we can decompose the operator Th=P\nj>0Th;jwhere each\nTh;jhas the Schwartz kernel\nKh;j(x;y) =1\n(2\u0019h)dZ\nRdbj(x;y;\u0018 )ei(x\u0000y)\u0001\u0018\nhd\u0018;\nwithbj(x;y;\u0018 ) =b(x;y;\u0018 ) j(\u0018) and j(\u0018) = (2\u0000j\u0018) for some  2C1\nc(1\n26j\u0018j62), ifj>1 and 0(\u0018) is\nsupported onj\u0018j61. Note that\n(x\u0000y)\u000bKh;j(x;y) =i\u0000j\u000bjhj\u000bj\n(2\u0019h)dZ\nRdD\u000b\n\u0018bj(x;y;\u0018 )\u0001ei(x\u0000y)\u0001\u0018\nhd\u0018;\nwe have\njKh;j(x;y)j.\u000bhj\u000bj\u0000d\njx\u0000yjj\u000bj\u00012\u0000j(j\u000bj+1\u0000d):\nWe have another trivial bound jKh;j(x;y)j.\u000b2jdh\u0000d:Therefore, for \fxed x2Rd, by choosingj\u000bj=d+ 1, we\nhaveZ\nRdjKh;j(x;y)jdy.Z\nRdmin\b\n2\u0000j2\u0000jh\njx\u0000yjd+1;2jdh\u0000d\t\ndy\n6Z\njzj62\u0000j\nd+1\u00012\u0000jh2jdh\u0000ddz+Z\njzj>2\u0000j\nd+1\u00012\u0000jh2\u0000j2\u0000jh\njzjd+1dz.2\u0000jd\nd+1:\nSimilarly, for \fxed y2Rd,Z\nRdjKh;j(x;y)jdx.2\u0000jd\nd+1:\nBy Schur's test, we have kTh;jkL(L2).2\u0000jd\nd+1. Using the triangle inequality, we obtain that This bounded on\nL2(Rd), uniformly in 0 <h61. The proof of Lemma A.1 is now complete. \u0003KELVIN-VOIGT DAMPING 15\nCorollary A.2. Assume that \u00142W1;1(Rd)andb2S0(R2d)is a symbol of order zero, then we have\nk[Oph(b);\u0014]kL(L2)=O(h):\nProof. The kernel of [Oph(b);\u0014] is given by\nK(x;y) =1\n(2\u0019h)dZ\nRdb(x;\u0018)(\u0014(y)\u0000\u0014(x))ei(x\u0000y)\u0001\u0018\nhd\u0018:\nSince\u00142W1;1, there exists \t2L1(Rd;Rd) such that\n\u0014(y)\u0000\u0014(x) = (y\u0000x)\u0001\t(x;y):\nThus\nK(x;y) =\u0000dX\nj=1h\ni(2\u0019h)dZ\nRd@\u0018jb(x;\u0018)\tj(x;y))ei(x\u0000y)\u0018\nhd\u0018\nApplying Lemma A.1 to each @\u0018jb(x;\u0018)\tj(x;y), the proof of Corollary A.2 is complete. \u0003\nReferences\n[1] Charles J. K. Batty and Thomas Duyckaerts. Non-uniform stability for bounded semi-groups on Banach spaces. J. Evol. Equ. ,\n8(4):765{780, 2008.\n[2] A. Borichev and Y. Tomilov. Optimal polynomial decay of functions and operator semigroups. Math. Ann. , 347(2):455{478,\n2010.\n[3] N. Burq. Mesures semi-classiques et mesures de d\u0013 efaut. S\u0013 eminaire Bourbaki , Mars 1997.\n[4] N. Burq. D\u0013 ecroissance de l'\u0013 energie locale de l'\u0013 equation des ondes pour le probl\u0012 eme ext\u0013 erieur et absence de r\u0013 esonance au\nvoisinage du r\u0013 eel. Acta Mathematica , 180:1{29, 1998.\n[5] N. Burq and H. Christianson. Imperfect geometric control and overdamping for the damped wave equation. Comm. Math.\nPhys. , 336(1):101{130, 2015.\n[6] N. Burq and P. G\u0013 erard. Stabilization of wave equations on the torus with rough dampings. To appear Pure and Applied\nAnalysis. preprint arxiv https://arxiv.org/abs/1801.00983 , 2020.\n[7] N. Burq and G. Lebeau. Mesures de d\u0013 efaut de compacit\u0013 e, application au syst\u0012 eme de Lam\u0013 e. Ann. Sci. \u0013Ecole Norm. Sup. (4) ,\n34(6):817{870, 2001.\n[8] Nicolas Burq. Decays for Kelvin-Voigt damped wave equations I: the black box perturbative method. SIAM J. Control Optim. ,\n58(4):1893{1905, 2020.\n[9] P. G\u0013 erard and E. Leichtnam. Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Mathematical Journal ,\n71:559{607, 1993.\n[10] G. Lebeau. Equation des ondes amorties. In A. Boutet de Monvel and V. Marchenko, editors, Algebraic and Geometric Methods\nin Mathematical Physics , pages 73{109. Kluwer Academic, The Netherlands, 1996.\n[11] R.B. Melrose and J. Sj ostrand. Singularities of boundary value problems II. Communications in Pure Applied Mathematics ,\n35:129{168, 1982.\n[12] Louis Tebou. A constructive method for the stabilization of the wave equation with localized Kelvin-Voigt damping. C. R.\nMath. Acad. Sci. Paris , 350(11-12):603{608, 2012.\nUniversit \u0013e Paris-Saclay, Laboratoire de math \u0013ematiques d'Orsay, UMR 8628 du CNRS, B ^atiment 307, 91405 Orsay\nCedex, France and Institut Universitaire de France\nEmail address :nicolas.burq@math.u-psud.fr\nUniversit \u0013e de Cergy-Pontoise, Laboratoire de Math \u0013ematiques AGM, UMR 8088 du CNRS, 2 av. Adolphe Chauvin\n95302 Cergy-Pontoise Cedex, France\nEmail address :chenmin.sun@u-cergy.fr"    },    {        "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": "1512.05408v2.Parity_time_symmetry_breaking_in_magnetic_systems.pdf",        "content": "Parity-time symmetry breaking in magnetic systems\nAlexey Galda1and Valerii M. Vinokur1\n1Materials Science Division, Argonne National Laboratory,\n9700 South Cass Avenue, Argonne, Illinois 60439, USA\n(Dated: July 1, 2016)\nThe understanding of out-of-equilibrium physics, especially dynamic instabilities and dynamic phase transi-\ntions, is one of the major challenges of contemporary science, spanning the broadest wealth of research areas that\nrange from quantum optics to living organisms. Focusing on nonequilibrium dynamics of an open dissipative\nspin system, we introduce a non-Hermitian Hamiltonian approach, in which non-Hermiticity reflects dissipation\nand deviation from equilibrium. The imaginary part of the proposed spin Hamiltonian describes the e \u000bects of\nGilbert damping and applied Slonczewski spin-transfer torque. In the classical limit, our approach reproduces\nLandau-Lifshitz-Gilbert-Slonczewski dynamics of a large macrospin. We reveal the spin-transfer torque-driven\nparity-time symmetry-breaking phase transition corresponding to a transition from precessional to exponen-\ntially damped spin dynamics. Micromagnetic simulations for nanoscale ferromagnetic disks demonstrate the\npredicted e \u000bect. Our findings can pave the way to a general quantitative description of out-of-equilibrium phase\ntransitions driven by spontaneous parity-time symmetry breaking.\nINTRODUCTION\nA seminal idea of parity-time ( PT)-symmetric quantum\nmechanics [1,2], that has stated that the condition of Hermitic-\nity in standard quantum mechanics required for physical ob-\nservables and energy spectrum to be real can be replaced by\na less restrictive requirement of invariance under combined\nparity and time-reversal symmetry, triggered an explosive de-\nvelopment of a new branch of science. The interpretation of\nPTsymmetry as “balanced loss and gain” [3] connected PT\nsymmetry breaking to transitions between stationary and non-\nstationary dynamics and established its importance to under-\nstanding of the applied field-driven instabilities. Experiments\non a diverse variety of strongly correlated systems and phe-\nnomena including optics and photonics [4–10], superconduc-\ntivity [11–13], Bose-Einstein condensates [14], nuclear mag-\nnetic resonance quantum systems [15], and coupled electronic\nand mechanical oscillators [16–18] revealed PT symmetry-\nbreaking transitions driven by applied fields. These observa-\ntions stimulated theoretical focus on far-from-equilibrium in-\nstabilities of many-body systems [12,13,19] that are yet not\nthoroughly understood.\nHere we demonstrate that the non-Hermitian extension\nof classical Hamiltonian formalism provides quantitative de-\nscription of dissipative dynamics and dynamic phase transi-\ntions in out-of-equilibrium systems. Focusing on the case of\nspin systems, we consider the zero-temperature spin dynamics\nunder the action of basic nonconservative forces: phenomeno-\nlogical Gilbert damping [20] and Slonczewski spin-transfer\ntorque [21] (STT). The latter serves as the most versatile way\nof directly manipulating magnetic textures by external cur-\nrents. We propose a general complex spin Hamiltonian, in\nwhich Slonczewski STT emerges from an imaginary magnetic\nfield. ThePT-symmetric version of the Hamiltonian is shown\nto exhibit a phase transition associated with inability of the\nsystem to sustain the balance between loss and gain above a\ncertain threshold of external nonconservative field.\nIn the classical limit of a large spin, our formalism repro-\nduces the standard Landau-Lifshitz-Gilbert-Slonczewski [20–22] (LLGS) equation of spin dynamics and predicts the PT\nsymmetry-breaking phase transition between stationary (con-\nservative) and dissipative (nonconservative) spin dynamics. In\nthis Letter we focus on a single spin, yet our theory can be ex-\ntended to coupled spin systems in higher dimensions. More-\nover, as spin physics maps onto a wealth of strongly corre-\nlated systems and phenomena ranging from superconductiv-\nity to cold-atom and two-level systems, our results provide\nquantitative perspectives on the nature of phase transitions as-\nsociated withPTsymmetry breaking in a broad class of far-\nfrom-equilibrium systems.\nWe introduce the non-Hermitian Hamiltonian for a single\nspin operator ˆS:\nˆH=E\u0000ˆS\u0001+ij\u0001ˆS\n1\u0000i\u000b; (1)\nwhere E\u0000ˆS\u0001denotes the standard Hermitian spin Hamil-\ntonian determined by the applied magnetic field Hand\nmagnetic anisotropy constants kiin the x;y;zdirections:\nE\u0000ˆS\u0001=P\nikiˆS2\ni+\rH\u0001ˆS. A schematic system setup is shown\nin Fig. 1. The phenomenological constant \u000b>0 in Eq. (1)\ndescribes damping; the imaginary field ijis responsible for\nthe applied Slonczewski STT, with jS=ep(~=2e)\u0011Jbeing\nthe spin-angular momentum deposited per second in the di-\nrection epwith spin polarization \u0011=(J\"\u0000J#)=(J\"+J#) of the\nincident current J; and\r=g\u0016B=~is the absolute value of the\ngyromagnetic ratio; g'2,\u0016Bis the Bohr magneton, ~is the\nPlanck’s constant, and eis the elementary charge. We conjec-\nture that Eq. (1) serves as a fundamental generalization of the\nHamiltonian description of both quantum and classical spin\nsystems, which constitutes one of our core results. This form\nof the Hamiltonian proves extremely useful for the general un-\nderstanding of STT-driven dissipative spin dynamics. In this\nwork we focus primarily on the classical limit of spin dynam-\nics, while the semiclassical limit of finite spin will be consid-\nered elsewhere.\nSpin dynamics in the classical limit is conveniently\nobtained by studying expectation value of the Hamilto-\nnian (1) with respect to SU(2) spin-coherent states [23,24]:arXiv:1512.05408v2  [cond-mat.other]  30 Jun 20162\nFIG. 1. Schematic representation of the system setup. Ferromagnetic\ncylinder (blue) is placed in magnetic field Happlied along the xaxis,\nand STT-inducing electric current Jis polarized in the direction ep\nalong theyaxis. Spin-polarized current passes through a nonmag-\nnetic metallic spacer and induces torque (Slonczewski STT, shown\nby the small red arrow) on the total spin S.\njzi=ezˆS+jS;\u0000Si, where ˆS\u0006\u0011ˆSx\u0006iˆSy, and z2Cis the stan-\ndard stereographic projection of the spin direction on a unit\nsphere, z=(sx+isy)=(1\u0000sz), with si\u0011Si=S. Note that such\nparametrization of the phase space for a classical single-spin\nsystem (i.e., in the limit S!1 ) guarantees the invariance of\nthe traditional equation of motion [24] under generalization to\nnon-Hermitian Hamiltonians (see Appendix A):\n˙z=i(1+¯zz)2\n2S@H\n@¯z; (2)\nwhere zand ¯zform a complex conjugate pair of stereographic\nprojection coordinates, and\nH(z;¯z)=hzjˆHjzi\nhzjzi(3)\nis the expectation value of the Hamiltonian (1) in spin-\ncoherent states (for a detailed review see, e.g., Ref. [25]). In\nthis formulation, the eigenstates of ˆHcorrespond to the fixed\npoints ziof the equation of motion for H, while the eigen-\nvalues (i.e., energy values) are equal to Hevaluated at the\ncorresponding fixed points, Ei=H(zi;¯zi).\nAssuming a constant magnitude of the total spin, ˙S=0,\nEq. (2) reduces to the following equation of spin dynamics in\nthe classical limit:\n˙S=rS(ReH)\u0002S+1\nS\u0002rS(ImH)\u0002S\u0003\u0002S: (4)\nHere we refer to the real and imaginary parts of the Hamil-\ntonian functionHwritten in the spin Srepresentation. For\nthe non-Hermitian Hamiltonian (1), Eq. (4) reproduces theLLGS equation describing dissipative STT-driven dynamics\nof a macrospin:\n\u0010\n1+\u000b2\u0011˙S=\rHe\u000b\u0002S+\u000b\r\nS[\rHe\u000b\u0002S]\u0002S+1\nSS\u0002[S\u0002j]\n+\u000bS\u0002j; (5)\n\rHe\u000b=rSE(S): (6)\nThe first two terms in Eq. (5) describe the standard Landau-\nLifshitz torque and dissipation, while the last two are respon-\nsible for the dissipative (antidamping) and conservative (e \u000bec-\ntive field) Slonczewski STT contributions, correspondingly,\nboth of which appear naturally from the imaginary magnetic\nfield term in the Hamiltonian (1).\nPT-SYMMETRIC HAMILTONIAN\nSlonczewski STT turns the total spin-angular momentum,\nS, in the direction of spin-current polarization, ep, without\nchanging its magnitude. On the S-sphere this can be repre-\nsented by a vector field converging in the direction of epand\noriginating from the antipodal point. It is the imaginary mag-\nnetic field ijthat produces exactly the same e \u000bect on spin dy-\nnamics, according to Eq. (2). The action of STT is invari-\nant under the simultaneous operations of time reversal and\nreflection with respect to the direction ep, which is the un-\nderlying reason behind the inherent PTsymmetry of certain\nSTT-driven magnetic systems, including the one considered\nbelow.\nBefore turning to the PT-symmetric form of Hamilto-\nnian (1), we note that PT-symmetric systems play an im-\nportant role in the studies of nonequilibrium phenomena\nand provide a unique nonperturbative tool for examining the\nphase transition between stationary and nonstationary out-\nof-equilibrium dynamics. We show that despite being non-\nHermitian, such systems can exhibit both of the above types\nof behavior, depending on the magnitude of the external non-\nconservative force. In the parametric regime of unbroken\nPTsymmetry, systems exhibit physical properties seemingly\nequivalent [26] to those of Hermitian systems: real energy\nspectrum, existence of integrals of motion (see Appendix C),\nand, notably, the validity of the quantum Jarzynski equal-\nity [27]. However, in the regime of brokenPT symme-\ntry, one observes complex energy spectrum and nonconser-\nvative dynamics. Therefore, the true transition between sta-\ntionary and nonstationary dynamics can be identified as the\nPTsymmetry-breaking phase transition.\nSpin systems are generally subject to various non-\nlinear magnetic fields including ones originating from\nshape, exchange, and magnetocrystalline and magnetoelastic\nanisotropies. Restricting ourselves for simplicity to a second-\norder anisotropy term, we arrive at the following Hamiltonian\nfor a nonlinear magnetic system with uniaxial anisotropy and\napplied Slonczewski STT:\nˆHPT=\rH0\u0010\nkzˆS2\nz+hxˆSx+i\fˆSy\u0011\n; (7)3\nFIG. 2. Real (a) and imaginary (b) parts of energy spectrum of the Hamiltonian (7) as a function of the STT parameter \fforhx=1 and\nD=20. Blue and red lines correspond to the eigenvalues E1;2andE3\u00006, respectively. The first PTsymmetry-breaking transition occurs at\nj\fj=\f1\u00194:5.\nwhere the applied magnetic field hxis measured in units of\nsome characteristic magnetic field H0, and\fis a dimension-\nless STT parameter determining the relative to Samount of\nangular momentum transfered in time \u001c\u0011(\rH0)\u00001(charac-\nteristic timescale of the dynamics, used as a unit of dimen-\nsionless time in what follows). The Hamiltonian (7) modeling\nthe dynamics of the free magnetic layer in a typical nanopil-\nlar device with fixed polarizer layer (see Fig. 1) is PTsym-\nmetric: It is invariant under simultaneous action of parity and\ntime-reversal operators ( y!\u0000y,t!\u0000t,i!\u0000i). Because\nthe Hamiltonian ˆHPTcommutes with an antilinear operator\nPT, its eigenvalues are guaranteed to appear in complex con-\njugate pairs. Notice that PT-symmetric Hamiltonian (7) does\nnot contain damping, which is assumed to be negligibly small,\nas is the case in many experimental systems.\nCLASSICAL SPIN SYSTEM\nIn order to best illustrate the mechanism of PT symme-\ntry breaking, we focus on the classical limit, S!1 and\nkzS!D=2, where Dis the dimensionless uniaxial anisotropy\nconstant. Formula (2) then yields the following equation of\nmotion for the Hamiltonian (7):\n˙z(t)=\u0000i(hx+\f)\n2 \nz2\u0000hx\u0000\f\nhx+\f!\n\u0000i D z1\u0000jzj2\n1+jzj2; (8)\nwith up to six fixed points zk,k=1;:::; 6.\nShown in Fig. 2 are the real and imaginary parts of the en-\nergy spectrum E1\u00006as functions of the STT amplitude \f. It\nreveals that in a system with strong anisotropy, D\u001d1,PT\nsymmetry breaking occurs in three separate transitions, with\nthe first one atj\fj=\f1=jhxjq\u00021+p\n1+(2D=jhxj)2\u0003=2,\nwhich corresponds to the smallest amplitude of STT at which\nIm(E),0. Therefore,PTsymmetry is not broken in the en-\ntire phase space of initial spin directions simultaneously, at\nvariance to the linear spin system with D=0 (see AppendixB). Instead, the regions of broken and unbroken PTsymme-\ntry may coexist in the phase diagram of a nonlinear spin sys-\ntem.\nIn what follows we consider a system described by\nthe Hamiltonian (7) with hx=1 and D=20. For all\nj\fj<\f 1\u00194:5,PT symmetry is unbroken and the character\nof spin (magnetization) dynamics is oscillatory in the entire\nphase diagram, i.e., for all possible initial conditions z. At\nj\fj=\f1the phase transition (first of the three, see Fig. 2) oc-\ncurs sharply in a wide region around the easy plane, jzj=1,\ni.e. near the equator of the unit S-sphere, shown in gray in\nFigs. 3(a) and 3(b) in Cartesian and stereographic projection\ncoordinates, respectively. It this region the nature of spin dy-\nnamics becomes fundamentally di \u000berent—all spin trajectories\nfollow the lines connecting the fixed points z1andz2, where\nz1;2=\u0000\u0000Dhx\u0006ip\n\f4\u0000\f2h2x\u0000D2h2x\u0001=(hx+\f)\f, and no closed\ntrajectories are possible; see Fig. 3(b).\nAsj\fjis increased further, the region of broken PTsymme-\ntry expands until it eventually closes around the fixed point z5\nat\f2\u00199:3 (second bifurcation in Fig. 2) and, eventually, the\nlast region of unbroken PT symmetry near z3disappears at\n\f3\u001910:8. The second and third phase transitions are less rel-\nevant experimentally as they occur in the vicinity of the least\nfavorable spin directions (parallel and antiparallel to the hard\naxisz) and at considerably higher applied currents.\nThe predicted transition from precessional dynamics (un-\nbrokenPTsymmetry) to exponentially fast saturation in the\ndirection z1(hx;\f) for any initial spin position around the easy\nplane (brokenPT symmetry) occurs in the setup with mu-\ntually perpendicular applied magnetic field and Slonczewski\nSTT. Such a transition in nanoscale magnetic structures can\nbe used for STT- or magnetic-field-controlled magnetization\nswitching in spin valves and a variety of other experimental\nsystems. This e \u000bect can further be used for direct measure-\nments of the amplitude of the applied STT, which, unlike the\napplied current, can be hard to quantify experimentally.4\nFIG. 3. (a, b) Spin dynamics described by Eq. (8) with hx=1,\f=4:7, and D=20.PTsymmetry is broken in the shaded region around\nthe easy planejzj=1 (dashed line), encompassing two fixed points, z1;2(blue dots), appearing as source and sink nodes. The green line\ndepicts a typical nonoscillatory spin trajectory in the region of broken PTsymmetry. Red dots represent the fixed points z3\u00006. (c) Results of\nmicromagnetic simulations for \f\u0003as a function of stereographic projection of the initial spin direction z. In the blue region, 4 :6.\f\u0003.4:8, the\nPTsymmetry is broken at all j\fj<\f\u0003, and the spin takes under 0 :5 ns to saturate in the direction of z1, which is in full agreement with the\nanalytical result.\nNUMERICAL SIMULATIONS OF PTSYMMETRY\nBREAKING\nHere we present the results of numerical simulations con-\nfirming thePT symmetry-breaking phase transition in the\nclassical single-spin system (7) by modeling magnetization\ndynamics of a ferromagnetic disk 100 nm in diameter and\nd=5 nm thick, which is consistent with the anisotropy con-\nstant D=20 in Eq. (8). We used the following typical\npermalloy material parameters: damping constant \u000b=0:01,\nexchange constant Aex=13\u000210\u000012J/m and saturation mag-\nnetization Msat=800\u0002103A/m . The simulations were car-\nried out using the open-source GPU-accelerated micromag-\nnetic simulation program MuMax3 [28] based on the LLGS\nequation (5) discretized in space. We used a cubic discretiza-\ntion cell of 5 nm in size, which is smaller than the exchange\nlength in permalloy, lex=(2Aex=\u00160M2\nsat)1=2\u00195:7 nm.\nThe permalloy disk was simulated in an external magnetic\nfield applied along the xaxis, H0=400 Oe, which corre-\nsponds to the characteristic time \u001c\u00190:14 ns. The STT was\nproduced by applying electric current perpendicular to the\ndisk in the zdirection with spin polarization \u0011=0:7 along\nep=ˆy(see Fig. 1) and current density \fmeasured in dimen-\nsionless units of 2 eH0Msatd=\u0011~\u00190:7\u0002108A/cm2. While\nsuch current density is comparable to typical switching cur-\nrent densities in STT-RAM devices [29,30], its magnitude can\nbe optimized for various practical applications by changing\nH0and adjusting the size, shape, and material of the ferro-\nmagnetic element.\nFor all possible initial spin directions z, we calculated the\ncritical amplitude of the applied STT, \f\u0003, for which the char-\nacter of spin dynamics changes from oscillatory (at j\fj< \f\u0003)\nto exponential saturation. Shown in Fig. 3(c) is the color map\nof\f\u0003as a function of zin complex stereographic coordinates.\nThe region shown in the shades of blue corresponds to the\ninitial conditions z, for which the minimum values of \fthat\nwould guarantee saturation of spin dynamics in the directionofz1in under 0:5 ns are between 4.6 and 4.8. This is in full\nagreement with the region of broken PTsymmetry at \f=4:7\ncalculated analytically, i.e., the shaded area in Fig. 3(b) [the\noutline is repeated in Fig. 3(c) for comparison]. Outside of\nthis region, a considerably larger magnitude of the applied\nSTT is required to break PTsymmetry.\nThe agreement between theoretical results and micromag-\nnetic simulations is remarkable considering the non-zero\nGilbert damping parameter ( \u000b=0:01) and nonlinear e \u000bects\n(demagnetizing field, finite size and boundary e \u000bects, etc.)\ninherently present in the micromagnetic simulations but not\nincluded in the model Hamiltonian (7).\nCONCLUSION\nThe presented non-Hermitian Hamiltonian formulation of\ndissipative nonequilibrium spin dynamics generalizes the pre-\nvious result [31], where the classical Landau-Lifshitz equa-\ntion was derived from a non-Hermitian Hamilton operator,\nto open STT-driven spin systems. The introduction of Slon-\nczewski STT in the imaginary part of the Hamiltonian re-\nvealed the possibility of STT-driven PTsymmetry-breaking\nphase transition. Micromagnetic simulations confirm the\nPTsymmetry-breaking phenomenon in realistic mesoscopic\nmagnetic systems and its robustness against weak dissipation,\nindicating high potential for impacting spin-based informa-\ntion technology. The way STT enters the complex Hamilto-\nnian (1), i.e. as imaginary magnetic field, provides a unique\ntool for studying Lee-Yang zeros [32] in ferromagnetic Ising\nand Heisenberg models and, more generally, dynamics and\nthermodynamics in the complex plane of physical parame-\nters. We envision further realizations of the PT symmetry-\nbreaking phase transitions in diverse many-body condensed-\nmatter systems and the expansion of practical implementa-\ntions of thePT symmetry beyond the present realm of op-\ntics [33] and acoustics [34].5\nFIG. 4. Real (a) and imaginary (b) parts of energy spectrum of the linear Hamiltonian ˆH0PTas functions of \fforhx=1.PTsymmetry-\nbreaking transition occurs at j\fj=1.\nACKNOWLEDGEMENTS\nWe thank Alex Kamenev for critical reading of the\nmanuscript and valuable suggestions. This work was sup-\nported by the U.S. Department of Energy, O \u000ece of Science,\nBasic Energy Sciences, Materials Sciences and Engineering\nDivision.\nAPPENDIX A. GENERALIZED EQUATION OF MOTION\nFOR NON-HERMITIAN SPIN HAMILTONIANS IN THE\nCLASSICAL LIMIT\nThe remarkable simplicity of the equation of motion (2)\nfor an arbitrary non-Hermitian spin Hamiltonian function\nHstems from the choice of parametrization of the phase\nspace, i.e., the complex stereographic projection coordinates\nfz;¯zg. The extension of classical equations of motion to non-\nHermitian Hamiltonians in terms of canonical coordinates\nfq;pghas the following generalized form [35]:\n \nq\np!\n= \n\u00001rq;p(ReH)\u0000G\u00001rq;p(ImH); (A1)\nwhere \nandGare the symplectic structure and metric of\nthe underlying classical phase space, respectively, which must\nsatisfy the compatibility condition [36] written in the matrix\nform as\n\n\u00001=\u0010\n\u0000\u00001\n \u0000\u00001\u0011T: (A2)\nIn the stereographic projection coordinates, one obtains the\nfollowing symplectic structure and metric:\n\n =2\n\u00001+jzj2\u00012 \n0i\n\u0000i0!\n; G=2\n\u00001+jzj2\u00012 \n0 1\n1 0!\n:(A3)\nIt is the form of these matrices that leads to Eq. (2), where the\nreal and imaginary parts (as written in the Srepresentation)\nof the Hamiltonian combine naturally into a single complexfunctionH. Therefore, when written in stereographic projec-\ntion coordinates, the generalized classical equation of motion\nfor non-Hermitian Hamiltonians coincides with that for tradi-\ntional Hermitian Hamiltonians.\nAPPENDIX B.PTSYMMETRY BREAKING IN LINEAR\nSPIN SYSTEM\nIn the absence of magnetic anisotropy fields, the Hamilto-\nnian (1) from the main text becomes linear:\nˆH0= \rH+ij\n1\u0000i\u000b!\n\u0001ˆS; (B1)\nwith e \u000bects of applied magnetic field, damping and Slon-\nczewski STT contributions all incorporated in the complex\nmagnetic field (in parentheses). The PT-symmetric version\nof this Hamiltonian has mutually perpendicular real and imag-\ninary parts of the complex magnetic field:\nˆH0PT=\rH0\u0010\nhxˆSx+i\fˆSy\u0011\n: (B2)\nThe quantum spin-1\n2version of this Hamiltonian describes\na two-level quantum system with balanced loss and gain and\nis known [37,38] to exhibit PTsymmetry-breaking transition\nathx=\u0006\f. Whenjhxj>j\fj, the Hamiltonian ˆH0PThas real\neigenvalues, \u00151;2=\u0006p\nh2x\u0000\f2, while in the parametric region\njhxj<j\fjeigenvalues are imaginary, see Fig. 4.\nThe generalized Hamilton’s equation of motion (2) for this\nHamiltonian takes the form\n˙z(t)=f(t)=\u0000i(hx+\f)\n2 \nz2\u0000hx\u0000\f\nhx+\f!\n: (B3)\nIt has two fixed points: z1;2=\u0006p\n(hx\u0000\f)=(hx+\f), which\ncorrespond to stereographic coordinates of the spin equi-\nlibria directions on a unit Bloch sphere. The character\nof spin dynamics around the fixed points is fully deter-\nmined by the eigenvalues of the complex Jacobian matrix6\nFIG. 5. Spin trajectories for classical linear Hamiltonian H0PTin the regime of unbroken, (a) and (c), (at \f=0:8) and broken, (b) and (d), (at\n\f=1:2)PTsymmetry for hx=1.\nJC=@(Ref;Imf)=@(Rez;Imz)in their vicinity. The solu-\ntion of Eq. (B3) takes the form of a M ¨obius transformation:\nz(t)=z(0)+ihx+\fp\nh2x\u0000\f2tan p\nh2x\u0000\f2\n2t!\nihx\u0000\fp\nh2x\u0000\f2tan p\nh2x\u0000\f2\n2t!\nz(0)+1: (B4)\nThe parametric region jhxj>j\fjdefines the regime of un-\nbrokenPT symmetry with real Hamiltonian eigenvalues,\nE1;2=\u0006p\nh2x\u0000\f2. In the classical approximation, the spin\nperforms persistent oscillations along circular orbits about the\nfixed points z1;2situated on the real axis, see Figs. 5(a) and\n5(c). The eigenvalues of JCatz1;2are purely imaginary, iden-\ntifying the fixed points are centers , according to the stan-\ndard classification [39]. Closed trajectories represent PT-\nsymmetric dynamics with balanced loss and gain: the spin\nsystem gains and loses equal amounts of energy from the non-\nconservative term i\fSyon they <0 andy >0 segments of\ntrajectories, respectively.As the driving parameter j\fjis increased, z1;2move towards\neach other until they eventually collide at j\fj=jhxj, which\nmarks the point of PT symmetry breaking. In the regime\nofbrokenPT symmetry,jhxj<j\fj, the energy eigenvalues\nare imaginary, E1;2=\u0006ip\n\f2\u0000h2x, and no closed trajectories\nare possible. The eigenvalues of the Jacobian JC(z1;2) in this\nregime are real and of the same sign, defining z1;2assink\nandsource nodes [39]. All trajectories follow circle arcs con-\nnecting the fixed points with coordinates in three-dimensional\nspace\u0010\n0;\u0006p\n1\u0000(hx=\f)2;\u0000hx=\f\u0011\n, which are now out of the xz\nplane, see Figs. 5(b) and 5(d).\nWe emphasize that in the linear Hamiltonian (B1) the ef-\nfects of damping can always be fully compensated by the\nappropriate choice of the applied STT. For instance, the\nPT-symmetric Hamiltonian (B2) describes a single spin\nplaced in external magnetic field H=H0(hx;\u000b\f; 0)and STT\nj=\rH0(\u0000\u000bhx;\f;0). Note that in the general case of nonlin-\near spin Hamiltonian, dissipation cannot be completely can-\nceled by STT. However, in many magnetic systems dissipative\nforces are extremely weak, which justifies the approximation\nof zero damping.7\nAPPENDIX C. INTEGRALS OF MOTION\nThere exists a remarkable similarity between spin dynam-\nics in the regime of unbroken PT symmetry and that of a\nfully Hermitian system. Indeed, it follows from Eq. (B3) that\nall spin trajectories are circular with the precession frequency\nequal top\nh2x\u0000\f2. The equivalent [26] to (B2) Hermitian\nHamiltonian reads:\nH0=\rH0q\nh2x\u0000\f2S0\nx; (B5)for which the equation of motion is obtained from Eq. (B3) by\nthe circle-preserving M ¨obius transformation\nz0=s\nhx+\f\nhx\u0000\fz: (B6)\nThe relation between time evolution of complex linear spin\nHamiltonians and M ¨obius transformations will be described\nelsewhere. The existence of this transformation leads to the\nnon-Hermitian system (B2) having an integral of motion:\nI[z;¯z]=(hx+\f)z+¯z\n1+hx+\f\nhx\u0000\fjzj2; (B7)\nwhich is nothing but the magnetic energy conserved by the\nHermitian Hamiltonian (B5):\nI\u0002z0;¯z0\u0003=q\nh2x\u0000\f2s0\nx=q\nh2x\u0000\f2z0+¯z0\n1+jz0j2(B8)\nafter the transformation (B6).\n1C. M. Bender and S. Boettcher. Real spectra in non-Hermitian\nHamiltonians having PT symmetry. Phys. Rev. Lett. 80, 5243\n(1998).\n2C. M. Bender, S. Boettcher, and P. N. Meisinger. PT-symmetric\nquantum mechanics. J. Math. Phys. 40, 2201-2229 (1999).\n3A. Ruschhaupt, F. Delgado, and J. G. Muga. Physical realization\nof PT-symmetric potential scattering in a planar slab waveguide.\nJ. Phys. A 38, L171 (2005).\n4S. Klaiman, U. G ¨unther, and N. Moiseev. Visualization of branch\npoints in PT-symmetric waveguides. Phys. Rev. Lett. 101, 080402\n(2008).\n5S. Longhi. Bloch oscillations in complex crystals with PT symme-\ntry.Phys. Rev. Lett. 103, 123601 (2009).\n6S. Longhi. Optical realization of relativistic non-Hermitian quan-\ntum mechanics. Phys. Rev. Lett. 105, 013903 (2010).\n7H. Schomerus. Quantum noise and self-sustained radiation of PT-\nsymmetric systems. Phys. Rev. Lett. 104, 233601 (2010).\n8H. Ramezani, T. Kottos, R. El-Ganainy, and\nD. N. Christodoulides. Unidirectional nonlinear PT-symmetric\noptical structures. Phys. Rev. A 82, 043803 (2010).\n9C. E. R ¨uteret al. Observation of paritytime symmetry in optics.\nNature Phys. 6, 192-195 (2010).\n10J. Sheng, M. A. Miri, D. N. Christodoulides, and M. Xiao. PT-\nsymmetric optical potentials in a coherent atomic medium. Phys.\nRev. A 88, 041803 (2013).\n11J. Rubinstein, P. Sternberg, and Q. Ma. Bifurcation diagram and\npattern formation of phase slip centers in superconducting wires\ndriven with electric currents. Phys. Rev. Lett. 99, 167003 (2007).\n12J. Rubinstein, P. Sternberg, and K. Zumbrun. The resistive state in\na superconducting wire: bifurcation from the normal state. Arch.\nRation. Mech. Anal. 195, 117 (2010).\n13N. M. Chtchelkatchev, A. A. Golubov, T. I. Baturina, and\nV . M. Vinokur. Stimulation of the fluctuation superconduc-\ntivity by PT symmetry. Phys. Rev. Lett. 109, 150405 (2012).\n14E. M. Graefe, H. J. Korsch, and A. E. Niederle. Mean-field dynam-\nics of a non-Hermitian Bose-Hubbard dimer. Phys. Rev. Lett. 101,\n150408 (2008).15C. Zheng, L. Hao, and G. L. Long. Observation of fast evolu-\ntion in paritytime-symmetric system. Phil. Trans. R. Soc. A 371,\n20120053 (2013).\n16C. M. Bender, B. K. Berntson, D. Parker, and E. Samuel. Observa-\ntion of PT phase transition in a simple mechanical system. Am. J.\nPhys. 81, 173 (2013).\n17C. M. Bender et al. Twofold transition in PT-symmetric coupled\noscillators. Phys. Rev. A. 88, 062111 (2013).\n18J. Schindler et al. . Experimental study of active LRC circuits with\nPT symmetries. Phys. Rev. A 84, 040101(R) (2011).\n19N. Poccia et al. Critical behavior at a dynamic vortex insulator-to-\nmetal transition. Science 349, 1202-1205 (2015).\n20T. L. Gilbert. A phenomenological theory of damping in ferro-\nmagnetic materials. IEEE Trans. Magn. 40, 3443 (2004).\n21J. C. Slonczewski. Current-driven excitation of magnetic multilay-\ners.J. Magn. Magn. Matter ,159, L1-L7 (1996).\n22L. D. Landau and E. M. Lifshitz. On the theory of the dispersion\nof magnetic permeability in ferromagnetic bodies. Phys. Z. Sow-\njetunion 8, 101-114 (1935).\n23E. H. Lieb. The classical limit of quantum spin systems. Commun.\nMath. Phys. 34, 327-340 (1973).\n24M. Stone, K.-S. Park, and A. Garg. The semiclassical propagator\nfor spin coherent states. J. Math. Phys. 41, 8025-8049 (2000).\n25J. M. Radcli \u000be. Some properties of coherent spin states. J. Phys. A\n4, 313 (1971).\n26A. Mostafazadeh. Exact PT-symmetry is equivalent to Hermitic-\nity.J. Math. Phys. 36, 7081 (2003).\n27S. De \u000bner and A. Saxena. Jarzynski equality in PT-symmetric\nquantum mechanics. Phys. Rev. Lett. 114, 150601 (2015).\n28A. Vansteenkiste et al. . The design and verification of MuMax3.\nAIP Adv. 4, 107133 (2014).\n29K. L. Wang, J. G. Alzate, and P. K. Amiri. Low-power non-volatile\nspintronic memory: STT-RAM and beyond. J. Phys. D: Appl.\nPhys. 46, 074003 (2013).\n30A. D. Kent, B. ¨Ozyilmaz, and E. Del Barco. Spin-transfer-induced\nprecessional magnetization reversal. Appl. Phys. Lett. 84, 3897\n(2004).8\n31R. Wieser. Comparison of quantum and classical relaxation in spin\ndynamics. Phys. Rev. Lett. 110, 147201 (2013).\n32C. N. Yang and T. D. Lee. Statistical theory of equations of state\nand phase transitions. II. Lattice gas and Ising model. Phys. Rev.\n87, 410 (1952).\n33B. Peng et al. Parity-time-symmetric whispering-gallery micro-\ncavities. Nature Phys. 10, 394 (2014).\n34R. Fleury, D. Sounas, and A. Al `u. An invisible acoustic sensor\nbased on parity-time symmetry. Nat. Commun. 6, 5905 (2015).\n35E. M. Graefe, M. H ¨oning, and H. J. Korsch. Classical limit of non-\nHermitian quantum dynamics - a generalized canonical structure.\nJ. Phys. A 43, 075306 (2010).36D. C. Brody and L. P. Hughston. Geometric quantum mechanics.\nJ. Geom. Phys. 38, 19 (2001).\n37C. M. Bender, P. N. Meisenger, and Q. Wang. Calculation of the\nhidden symmetry operator in PT-symmetric quantum mechanics.\nJ. Phys. A 36, 6791 (2003).\n38D. C. Brody. Biorthogonal quantum mechanics. J. Phys. A 47,\n035305 (2014).\n39D. V . Anosov, V . I. Arnold, and S. Kh. Aranson. Ordinary di \u000ber-\nential equations and smooth dynamical systems, Vol. 1 (Springer\nScience & Business Media, 1997)."    },    {        "title": "2005.14153v2.Spintronics_meets_nonadiabatic_molecular_dynamics__Geometric_spin_torque_and_damping_on_noncollinear_classical_magnetism_due_to_electronic_open_quantum_system.pdf",        "content": "Spintronics meets nonadiabatic molecular dynamics: Geometric spin torque and\ndamping on noncollinear classical magnetism due to electronic open quantum system\nUtkarsh Bajpai and Branislav K. Nikoli´ c\nDepartment of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA\nWe analyze a quantum-classical hybrid system of steadily precessing slow classical localized mag-\nnetic moments, forming a head-to-head domain wall, embedded into an open quantum system of\nfast nonequilibrium electrons. The electrons reside within a metallic wire connected to macroscopic\nreservoirs. The model captures the essence of dynamical noncollinear and noncoplanar magnetic\ntextures in spintronics, while making it possible to obtain the exact time-dependent nonequilib-\nrium density matrix of electronic system and split it into four contributions. The Fermi surface\ncontribution generates dissipative (or damping-like in spintronics terminology) spin torque on the\nmoments, and one of the two Fermi sea contributions generates geometric torque dominating in the\nadiabatic regime. When the coupling to the reservoirs is reduced, the geometric torque is the only\nnonzero contribution. Locally it has both nondissipative (or field-like in spintronics terminology)\nand damping-like components, but with the sum of latter being zero, which act as the counter-\nparts of geometric magnetism force and electronic friction in nonadiabatic molecular dynamics.\nSuch current-independent geometric torque is absent from widely used micromagnetics or atomistic\nspin dynamics modeling of magnetization dynamics based on the Landau-Lifshitz-Gilbert equation,\nwhere previous analysis of Fermi surface-type torque has severely underestimated its magnitude.\nOne of the most fruitful applications of geometric (or\nBerry) phase [1] concepts is encountered in quantum-\nclassical hybrid systems where separation of time scales\nmakes it possible to consider fast quantum degrees of free-\ndom interacting with the slow classical ones [2, 3]. The\namply studied example of this kind are fast electrons in-\nteracting [4, 5] with slow nuclei in molecular dynamics\n(MD) [6–9] problems of physics, chemistry and biology.\nThe parameters driving adiabatic evolution of quantum\nsubsystem, with characteristic frequency smaller that its\nlevel spacing, are nuclear coordinates elevated to the\nstatus of dynamical variables. The electronic system\nthen develops geometric phase in states evolving out of\nan instantaneous energy eigenstate, while also acquiring\nshifts in the energy levels. Conversely, nuclei experience\nforces due to back-action from electrons. The simplest\nforce is the adiabatic Born-Oppenheimer (BO) force [4, 5]\nwhich depends only on the coordinates of the nuclei, and\nit is associated with electronic adiabatic potential sur-\nfaces [6, 7]. Even small violation of BO approximation\nleads to additional forces—the first nonadiabatic correc-\ntion generates forces linear in the velocity of the nuclei,\nand being Lorentz-like they are dubbed [2, 10] “geomet-\nric magnetism.” The “magnetism” is not a not a real\nmagnetic field, but an emergent geometrical property of\nthe Hilbert space [11], and akin to the true Lorentz force,\nthe emergent geometric force is nondissipative .\nAdditional forces appear upon making the quantum\nsystem open by coupling it to a thermal bath [10, 12]\n(usually modeled as an infinite set of harmonic oscilla-\ntors [13]) or to macroscopic reservoirs of particles [14].\nIn the latter case, one can also introduce chemical po-\ntential difference between the reservoirs to drive particle\nflux (i.e., current) through the quantum system which is,\nthereby, pushed out of equilibrium [14–16, 18, 19]. In\nFIG. 1. (a) Schematic view of a two-terminal system where\na single classical LMM, precessing steadily with frequency ω\nand cone angle θ, interacts with an open quantum system of\nconduction electron spins. The electrons hop along 1D infinite\ntight-binding chain which terminates into the left and right\nmacroscopic reservoirs kept at the same chemical potential µ.\nPanel (c) depicts 7 LMMs, M1–M7forming a head-to-head\nBloch domain wall, which precess with the same frequency\nbut are noncollinear andnoncoplanar . Both (a) and (c) can\nbe mapped in the rotating frame to a time-independent four-\nterminal system in (b) with an effective bias voltage ~ω/e\nbetween the left or right pair of leads.\nboth equilibrium and nonequilibrium cases, the energy\nspectrum of the quantum system is transformed into a\ncontinuous one, and frictional forces [8–10, 14–19] linear\nin the velocity of the nuclei become possible. Also, due\nto continuous spectrum, adiabaticity criterion has to be\nreplaced by a different one [14]. Stochastic forces also ap-\npear, both in equilibrium and in nonequilibrium, where\nin the former case [10, 12] they are due to fluctuations at\nfinite temperature while in the latter case they includearXiv:2005.14153v2  [cond-mat.mes-hall]  14 Jun 20202\nadditional contribution from nonequilibrium noise [14–\n16]. Finally, specific to nonequilibrium is the emergence\nof nonconservative forces [14–16, 18, 19]. The derivation\nof all of these forces is achieved by computing nonadia-\nbatic corrections to the density matrix (DM) [10, 12, 14–\n16, 18, 19]. This yields a non-Markovian stochastic\nLangevin equation, with nonlocal-in-time kernel describ-\ning memory effects [20], as the most general [16, 19] equa-\ntion for nuclei in nonadiabatic MD.\nThe analogous problem exists in spintronics, where the\nfast quantum system is comprised of conduction electron\nspins and slow classical system is comprised of localized-\non-atoms spins and associated localized magnetic mo-\nments (LMMs) described by unit vectors Mi(t). The\ndynamics of LMMs is accounted by the Landau-Lifshitz-\nGilbert (LLG) type of equation [21]\n∂Mi\ndt=−gM×Beff\ni+λMi×∂Mi\n∂t\n+g\nµM/parenleftBig\nTi/bracketleftBig\nISα\next/bracketrightBig\n+Ti[∂Mi/∂t]/parenrightBig\n. (1)\nThis includes phenomenological Gilbert damping, whose\nparameter λcan be measured or independently calcu-\nlated [22] by using electronic Hamiltonian with spin-orbit\ncoupling and impurities. It can also include Slonczewski\nspin-transfer torque (STT) term Ti/bracketleftBig\nISα\next/bracketrightBig\ndue to exter-\nnally supplied spin current ISα\next. The STT is a phe-\nnomenon [28] in which spin angular momentum of con-\nduction electrons is transferred to local magnetization\nnot aligned with electronic spin-polarization. Finally,\nsome analyses [23–25] also consider current-independent\ntorque Ti[∂Mi/∂t] as a back-action of electrons pushed\nout of equilibrium by time-dependent Mi(t). Neverthe-\nless, such effects have been deemed negligible [23, 26] or\neasily absorbed into Eq. (1) by renormalizing gandλ[23].\nHeregis the gyromagnetic ratio; Beff\ni=−1\nµM∂H/∂Mi\nis the effective magnetic field as the sum of external field,\nfield due to interaction with other LMMs and magnetic\nanisotropy field in the classical Hamiltonian Hof LMMs;\nandµMis the magnitude of LMM [21].\nThe STT vector, T=TFL+TDL, can be decomposed\n[Fig. 1(a)] into: ( i) even under time-reversal or field-like\n(FL) torque, which affects precession of LMM around\nBeff\ni; and ( ii) odd under time-reversal or damping-like\n(DL) torque, which either enhances the Gilbert damp-\ning by pushing LMM toward Beff\nior competes with\nGilbert term as “antidamping.” For example, negative\nvalues ofTDL=TDL·eDLin Figs. 2 and 3, where\neDL= (Mi×∂Mi/∂t)|Mi×∂Mi/∂t|−1, means that TDL\nvector points away from the axis of precession which is\nantidamping action. Similarly, TFL=TFL·eFL, where\neFL= (∂Mi/∂t)|∂Mi/∂t|−1, is plotted in Figs. 2 and 3.\nThe current-driven STT Ti/bracketleftBig\nISα\next/bracketrightBig\nacts as the coun-\nterpart of nonconservative force in nonadiabatic MD.\nThe Gilbert damping plus current-independent torque\nFIG. 2. The FL and DL components [Fig. (1)] of three spin\ntorques contributions in Eq. (4) exerted by nonequilibrium\nspin density of electrons onto a single localized precessing\nmagnetic moment in the setup of Fig. 1(a) as a function of\ncoupling to the leads. Black dotted line is the sum of the three\ntorques. In panels (a) and (c) Jsd= 0.1γ, while in panels\n(b) and (d) Jsd= 20γensures perfectly adiabatic regime [32],\nJsd/~ω/greatermuch1, for the chosen precession frequency ~ω= 0.001γ.\nTi[∂Mi/∂t] appear as the counterpart of electronic\nfriction [8, 9, 14–19], but Gilbert damping requires\nagents [22] other than electrons alone considered in nona-\ndiabatic MD. Thus, the geometric torque and damping,\nas counterparts of geometric magnetism force [2] and fric-\ntion [10], are absent from standard modeling of classi-\ncal magnetization dynamics. Geometric torque has been\nadded ad hoc into the LLG equation applied to spe-\ncific problems, such as spin waves within bulk magnetic\nmaterials [29–31]. A recent study [32] of a single clas-\nsical LMM embedded into a closed (i.e., finite length\none-dimensional wire) electronic quantum system finds\nthat nonequilibrium electronic spin density always gener-\nates geometric torque, even in perfectly adiabatic regime\nwhere electron-spin/LMM interaction is orders of mag-\nnitude larger than the characteristic frequency of LMM\ndynamics. It acts as a purely FL torque causing anoma-\nlous frequency of precession that is higher than the Lar-\nmor frequency. By retracing the same steps [14, 15] in\nthe derivation of the stochastic Langevin equation for\nelectron-nuclei system connected to macroscopic reser-\nvoirs, Ref. [33] derived the stochastic LLG equation [34–\n37] for a single LMM embedded into an open electronic\nsystem out of equilibrium. The novelty in this derivation\nis damping, present even in the absence of traditional\nspin-flip relaxation mechanisms [23, 25], while the same3\nFIG. 3. Spatial profile of FL and DL components of Tgeo\ni,Tsea\niandTsurf\nispin torques on precessing LMMs depicted in Fig. 1(c)\nfor closed or open electronic quantum system and for two different values of Jsd. Insets on the top of each row mark positions\nand static configuration of LMMs within the Bloch DW, with their x-component depicted by the colorbar next to panel (a).\nconclusion about geometric torque changing only the pre-\ncession frequency of LMM has been reached (in some\nregimes, geometric phase can also affect the stochastic\ntorque [38]). However, single LMM is a rather special\ncase, which is illustrated in Fig. 1(a) and revisited in\nFig. 2, and the most intriguing situations in spintronics\ninvolve dynamics of noncollinear textures of LMMs. This\nis exemplified by current- or magnetic-field driven dy-\nnamics of domain walls (DWs) and skyrmions [25, 37, 39–\n43] where a much richer panoply of back-action effects\nfrom fast electronic system can be expected.\nIn this Letter, we analyze an exactly solvable\nmodel of seven steadily precessing LMMs, M1(t)–M7(t)\n[Fig. 1(c)], which are noncollinear and noncoplanar and\nembedded into a one-dimensional (1D) infinite wire host-\ning conduction electrons. The model can be viewed as\na segment of dynamical noncollinear magnetic texture,\nand it directly describes magnetic field-driven [43] head-\nto-head Bloch DW [44] but without allowing it to prop-\nagate [41, 43]. Its simplicity makes it exactly solvable—\nwe fins the exact time-dependent DM via the nonequi-\nlibrium Green function (NEGF) formalism [45] and an-\nalyze its contributions in different regimes of the ratio\nJsd/~ωofsdexchange interaction Jsd[23] between elec-\ntron spin and LMM and frequency of precession ω. In\nboth Figs. 1(a) and 1(c), the electronic subsystem is an\nopen quantum system and, although no bias voltage is\napplied between the macroscopic reservoirs, it is pushed\ninto the nonequilibrium state by the dynamics of LMMs.\nFor example, electronic quantum Hamiltonian becomes\ntime-dependent due to M1(t) [Fig. 1(a)] or M1(t)–\nM7(t) [Fig. 1(c)], which leads to pumping [25, 27, 46]\n[Fig. 4(b),(c)] of spin current locally within the DW re-\ngion, as well as into the leads [Fig. 4(a)]. Pumping ofcharge current will also occur if the left-right symmetry\nof the device is broken statically [27] or dynamically [47].\nThe electrons are modeled on an infinite tight-binding\n(TB) clean chain with Hamiltonian in the lab frame\nˆHlab(t) =−γ/summationdisplay\n/angbracketleftij/angbracketrightˆc†\niˆcj−Jsd/summationdisplay\niˆc†\niˆ\u001bˆci·Mi(t). (2)\nHere ˆc†\ni= (ˆc†\ni↑,ˆc†\ni↓) and ˆc†\niσ(ˆciσ) creates (annihilates) an\nelectron of spin σ=↑,↓at sitei. The nearest-neighbor\nhoppingγ= 1 eV sets the unit of energy. The active re-\ngion in Figs. 1(a) or 1(c) consists of one or seven sites,\nrespectively, while the rest of infinite TB chain is taken\ninto account as the left (L) and the right (R) semi-infinite\nleads described by the same Hamiltonian in Eq. (2), but\nwithJsd= 0. The hopping between the leads and the\nactive region is denoted by γc. The leads terminate at\ninfinity into the macroscopic particle reservoirs with iden-\ntical chemical potentials µL=µR=EFdue to assumed\nabsence of bias voltage, and EF= 0 is chosen as the\nFermi energy. In contrast to traditional analysis in spin-\ntronics [23, 25], but akin to Refs. [32, 33], Hamiltonian in\nEq. (2) does not contain any spin-orbit or impurity terms\nas generators of spin-flip relaxation.\nThe spatial profile of Bloch DW is given by\nMx\ni=−sech[(hDW−zi)/W] tanh[(ZDW−zi)],My\ni=\nsech2[(ZDW−zi)/W] andMz\ni= tanh[(ZDW−zi)/W],\nwhereZDW= 4 andW= 0.9. Instead of solving LLG\nequations [Eq. (1)] for M1(t)–M7(t), we impose a so-\nlution where LMMs precess steadily around the z-axis:\nMx\ni(t) = sinθicos(ωt+φi);My\ni(t) = sinθisin(ωt+\nφi); andMz\ni(t) = cosθi. Using a unitary transfor-\nmation into the rotating frame (RF), the Hamiltonian\nin Eq. (2) becomes time-independent [25, 27], ˆHRF=4\nˆU†(t)ˆHlab(t)ˆU(t)−i~ˆU†∂ˆU/∂t =ˆHlab(t= 0)−~ωˆσα/2,\nwith LMMs frozen at t= 0 configuration from the lab.\nThe unitary operator is ˆU(t) = exp(−iωtˆσα/2) forα-axis\nof rotation. In the RF, the original two-terminal Lan-\ndauer setup for quantum transport in Figs. 1(a) and\n1(c) is mapped, due to ~ωˆσα/2 term, onto an effective\nfour-terminal setup [27] [illustrated for single LMM in\nFig. 1(b)]. Each of its four leads is an effective half-metal\nferromagnet which accepts only one spin species, ↑or↓\nalong theα-axis, and effective dc bias voltage ~ω/eacts\nbetween L or R pair of leads.\nIn the RF, the presence of the leads and macro-\nscopic reservoirs can be taken into account exactly us-\ning steady-state NEGFs [45] which depend on time\ndifferencet−t/primeand energy Eupon Fourier trans-\nform. Using the retarded, ˆG(E), and the lesser, ˆG<(E),\nGreen functions (GFs), we find the exact nonequilib-\nrium DM of electrons in the RF, ˆ ρRF=1\n2πi\u0001\ndEˆG<(E).\nHere the two GFs are related by the Keldysh equa-\ntion, ˆG<(E) = ˆG(E)ˆΣ<(E)ˆG†(E), where ˆΣ<(E) is\nthe lesser self-energy [45] due to semi-infinite leads and\nˆG(E) = [E−ˆHRF−ˆΣ(E,~ω)]−1with ˆΣ(E,~ω) =/summationtext\np=L,R,σ=↑,↓ˆΣσ\np(E−Qσ\nα~ω) being the sum of retarded\nself-energies for each of the four leads p,σin RF. We\nuse shorthand notation Q↑\np=−1/2 andQ↓\np= +1/2.\nSince typical frequency of magnetization dynamics is\n~ω/lessmuchEF, we can expand [48] ˆ ρRFin small ~ω/EF\nand then transform it back to the lab frame, ˆ ρlab(t) =\nˆU(t)ˆρRFˆU†(t) to obtain ˆ ρlab(t) = ˆρad\nt+ ˆρgeo(t)+ ˆρsea(t)+\nˆρsurf(t) where:\nˆρad\nt=−1\nπˆU+∞\u0002\n−∞dEImˆG0f(E)ˆU†, (3a)\nˆρgeo(t) =1\nπˆU+∞\u0002\n−∞dEIm/bracketleftbigg\nˆG0/parenleftbigg\ni~ˆU†∂ˆU\n∂t/parenrightbigg\nˆG0/bracketrightbigg\nf(E)ˆU†,(3b)\nˆρsea(t) =−~ω\n2πˆU/summationdisplay\np+∞\u0002\n−∞dEIm/bracketleftbigg\nˆG0/parenleftbigg∂ˆΣ↑\np\n∂E−∂ˆΣ↓\np\n∂E/parenrightbigg\nˆG0/bracketrightbigg\n×f(E)ˆU†, (3c)\nˆρsurf(t) =~ω\n4πˆU/summationdisplay\np+∞\u0002\n−∞dEˆG0(ˆΓ↑\np−ˆΓ↓\np)ˆG†\n0∂f\n∂EˆU†.(3d)\nWe confirm by numerically exact calculations [39] that\nthus obtained ˆ ρlab(t) is identical to ~G<(t,t)/icomputed\nin the lab frame. Here ˆG0(E) = [E−ˆHRF−ˆΣ(E,0)]−1\nisˆG(E) with ~ω= 0; ˆΓσ\np(E) =i[ˆΣσ\np(E)−ˆΣσ\np(E)†] is\nthe level broadening matrix due the leads; and fσ\np(E) =\nf(E−[EF+Qσ\nα~ω]) is the the Fermi function of macro-\nscopic reservoir p,σin the RF.\nThe total nonequilibrium spin density, /angbracketleftˆsi/angbracketright(t) =\nTr[ˆρlab(t)|i/angbracketright/angbracketlefti|⊗ˆ\u001b] =/angbracketleftˆsi/angbracketrightad\nt+/angbracketleftˆsi/angbracketrightgeo(t) +/angbracketleftˆsi/angbracketrightsea(t) +\n/angbracketleftˆsi/angbracketrightsurf(t), has the corresponding four contributions fromDM contributions in Eq. (3). Here /angbracketleftˆsi/angbracketrightad\ntis the equilib-\nrium expectation value at an instantaneous time twhich\ndefines ‘adiabatic spin density’ [23, 25, 30–32]. It is com-\nputed using ˆ ρad\ntas the grand canonical equilibrium DM\nexpressed via the frozen (adiabatic) retarded GF [14, 15,\n33], ˆGt(E) = [E−ˆHt−ˆΣ]−1, for instantaneous configu-\nration of Mi(t) while assuming ∂Mi/∂t= 0 [subscript\ntsignifies parametric dependence on time through slow\nvariation of Mi(t)]. The other three contributions—from\nˆρgeo(t) and ˆρsea(t) governed by the Fermi sea and ˆ ρsurf(t)\ngoverned by the Fermi surface electronic states—contain\nfirst nonadiabatic correction [14, 15, 33] proportional to\nvelocity∂Mi/∂t, as well as higher order terms due to\nˆρlab(t) being exact. These three contributions define STT\nout of equilibrium [23, 39, 48]\nTi=Jsd/angbracketleftˆsi/angbracketright(t)×Mi(t) =Tgeo\ni+Tsea\ni+Tsurf\ni.(4)\nEach term Tgeo\ni,Tsea\ni,Tsurf\nican be additionally sepa-\nrated into its own DL and FL components [Fig. 1(a)], as\nplotted in Figs. 2 and 3. Note that Tsea\niis insignificant\nin both Figs. 2 and 3, so we focus on Tgeo\niandTsurf\ni.\nTo gain transparent physical interpretation of Tgeo\niand\nTsurf\ni, we first consider the simplest case [32, 33]—a single\nM1(t) in setup of Fig. 1(a). The STT contributions as a\nfunction of the coupling γcto the leads (i.e., reservoirs)\nare shown in Fig. 2. We use two different values for Jsd,\nwhere large ratio of Jsd= 20 eV and ~ω= 0.001 eV is\nperfect adiabatic limit [30–32]. Nevertheless, even in this\nlimit and for γc→0 we find Tgeo\n1/negationslash= 0 in Fig. 2(b) as the\nonly nonzero and purely FL torque. This is also found\nin closed system of Ref. [32] where Tgeo\n1was expressed\nin terms of the spin Berry curvature. As the quantum\nsystem becomes open for γc>0,Tgeo\n1is slightly reduced\nwhile Tsurf\n1emerges with small FL [Fig. 2(b)] and large\nDL [Fig. 2(d)] components. The DL torque Tsurf,DL\n1\npoints toward the z-axis and, therefore, enhances the\nGilbert damping. In the wide-band approximation [49],\nthe self-energy ˆΣ(E) =−iΓˆI2is energy-independent for\nEwithin the bandwidth of the lead, which allows us to\nobtain analytical expression (at zero temperature)\nTgeo\n1(t) =~ω\n2π/bracketleftbigg\nπ−2 tan−1/parenleftbiggΓ\nJsd/parenrightbigg/bracketrightbigg\nsinθeφ(t).(5)\nHere eφ(t) =−sinωtex+ cosωtey. Thus, in per-\nfect adiabatic limit, Jsd/~ω→∞ , or in closed system,\nΓ→0,Tgeo\n1is independent of microscopic parameters\nas expected from its geometric nature [29]. The always\npresent Tgeo\ni/negationslash= 0 means that electron spin is never along\n‘adiabatic direction’ /angbracketleftˆsi/angbracketrightad\nt.\nSwitching to DW [Fig. 1(c)] embedded into a closed\nquantum system ( γc= 0) shows in Fig. 3(a)–(d) that\nonlyTgeo\ni/negationslash= 0, which also acquires DL component lo-\ncally with damping or antidamping action depending on\nthe position of LMM. Upon opening the quantum sys-\ntem (γc=γ), Fig. 3(e)–(h) shows emergence of ad-\nditional Tsurf\ni/negationslash= 0 which, however, becomes negligible5\nFIG. 4. (a) The z-component of total DL torques which act\non DW in Fig. 1(c) as a function of Jsdforγc=γ. Cir-\ncles show that sum of spin currents pumped into the leads\nmatches/parenleftbig/summationtext\niTsurf,DL\ni/parenrightbig\nz≡ISz\nL+ISz\nR. Panel (b) and (c),\nwhich correspond to Fig. 3(g), show spatial profile of lo-\ncal spin currents ISz\ni→jpumped between sites iandjfor\nJsd= 0.1γ, with their sum being identically zeroin panel (c).\nDashed black line in panels (a) and (b) is pumped local spin\ncurrent by SMF [24, 26], ISz\nSMF(x) =gµB~G0\n4e2[∂M(x,t)/∂t×\n∂M(x,t)/∂x]z, whereG0=G↑+G↓is the total conductivity.\n[Fig. 3(f),(h)] in the perfectly adiabatic limit Jsd/~ω/greatermuch1.\nAt first sight, Tgeo,DL\ni/negationslash= 0 violates Berry and Robbins\noriginal analysis [2] according to which an isolated quan-\ntum system, with discrete energy spectrum, cannot exert\nfriction onto the classical system. This apparent contra-\ndiction is resolved in Fig. 4(a) where we show that total/summationtext\niTgeo,DL\ni≡0 is always zero. Conversely, Fig. 4(a) con-\nfirms that total/parenleftBig/summationtextTsurf,DL\ni/parenrightBig\nz≡ISz\nL+ISz\nRis identical\nto net spin current pumped into the leads via which the\nconduction electrons carry away excess angular momen-\ntum of precessing LMMs [46]. Such identity underlies\nphysical picture where spin current generated by time-\ndependent magnetization becomes DL torque [24, 46].\nNote that pumped spin current ISz\ni→jdue to ˆρgeoor ˆρsea\nin Fig. 4(c) can be nonzero locally, but they sum to zero.\nThe nonuniform pumped spin current due to spatially\nand time varying magnetization has prompted propos-\nals [24, 26] to amend the LLG equation by adding the\ncorresponding DL torque M×D·∂M/∂twith 3×3 damp-\ning tensorDwhose spatial dependence is given by the so-\ncalled spin-motive force (SMF) formula. However, SMF\ncorrection was estimated to be small [26] in the absence\nof spin-orbit coupling in the band structure. 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Sloan3\nJanuary 31, 2019\nAbstract\nA key quantity that occurs in the error analysis of several numeric al methods\nfor eigenvalue problems is the distance between the eigenvalue of int erest and the\nnext nearest eigenvalue. When we are interested in the smallest or f undamental\neigenvalue, we call this the spectral orfundamental gap . In a recent manuscript\n[Gilbert et al., https://arxiv.org/abs/1808.02639 ], the current authors, to-\ngether with Frances Kuo, studied an elliptic eigenvalue problem with ho mogeneous\nDirichlet boundary conditions, and with coefficients that depend on a n infinite num-\nber of uniformly distributed stochastic parameters. In this settin g, the eigenvalues,\nand in turn the eigenvalue gap, also depend on the stochastic param eters. Hence, for\na robust error analysis one needs to be able to bound the gap over a ll possible real-\nisations of the parameters, and because the gap depends on infinit ely-many random\nparameters, this is not trivial. This short note presents, in a simplifie d setting, an im-\nportant result that was shown in the paper above. Namely, that, u nder certain decay\nassumptions on the coefficient, the spectral gap of such a random elliptic eigenvalue\nproblem can be bounded away from 0, uniformly over the entire infinit e-dimensional\nparameter space.\n1 Introduction\nEigenvalue problems are useful for modelling many phenomena from a pplications in\nengineering and physics, e.g, structural mechanics, acoustic sca ttering, elastic mem-\nbranes, criticality of neutron transport/diffusion and band gap ca lculations for pho-\ntonic crystal fibres.\nIn this work, we consider the following eigenvalue problem\n−∇·/parenleftbig\na(x,y)∇u(x,y)/parenrightbig\n=λ(y)u(x,y), forx∈D,(1.1)\nu(x,y) = 0, forx∈∂D,\nwhere the derivatives are taken with respect to x, andy= (y1,y2,...) is a ran-\ndom, infinite-dimensional vector with independent uniformly distribu ted components\nyj∼U([−1\n2,1\n2]). We assume that the physical domainD⊂Rd, ford= 1,2,3, is\nbounded with Lipschitz boundary, and denote the stochastic/parameter domain by\nU:= [−1\n2,1\n2]N.\n1Institute for Applied Mathematics and Interdisciplinary C enter for Scientific Computing, Universit¨ at\nHeidelberg, 69120 Heidelberg, Germany\na.gilbert@uni-heidelberg.de ,r.scheichl@uni-heidelberg.de\n2Department of Mathematical Sciences, University of Bath, B ath BA2 7AY UK\ni.g.graham@bath.ac.uk\n3School of Mathematics and Statistics, University of New Sou th Wales, Sydney NSW 2052, Australia\ni.sloan@unsw.edu.au\n1In many uncertainty quantification (UQ) applications, the coefficien ta(x,y) is\ngiven by a Karhunen–Lo` eve expansion of a random field. Taking this as motivation,\nwe assume that the coefficient is an affine map of y, and satisfies\n0< a(x,y) =a0(x)+∞/summationdisplay\nj=1yjaj(x)<∞,for allx∈D,y∈U.\nFurtherassumptionsonthecoefficientwillbegivenexplicitlyinAssump tionA1below.\nIf we ignore the ydependence, then (1.1) is a self-adjoint eigenvalue problem,\nwhich has been studied extensively in the literature, see, e.g., [7, 9]. I n particular, it\nis well known that (1.1) has countably many eigenvalues\n0< λ1< λ2≤λ3≤ ···,\nand that the smallest eigenvalue is simple. However, in our setting the eigenvalues\ndepend on y:λk=λk(y), and since the parameter domain is infinite-dimensional,\ncare must be taken when transferring classical results to our set ting. In particular,\nalthough it is well known that in the unparametrised setting the spec tral gap, λ2−λ1,\nis some fixed positive number, in our setting the spectral gap, λ2(y)−λ1(y), is a\nfunction defined on an infinite-dimensional domain, which could be arb itrarily close\nto 0.\nOur main result (see Theorem 3 for a full statement) is as follows. As suming that\nthe terms ajin the coefficient decay sufficiently fast (in a suitable norm), then the re\nexists aδ >0, independent of y, such that the spectral gap of the eigenvalue problem\n(1.1) satisfies\nλ2(y)−λ1(y)≥δ,for ally∈U .\nAs an example of the important role that the spectral gap plays in er ror analysis,\nconsider the random elliptic eigenvalue problem from [8]. There, an algo rithm using\ndimension truncation, Quasi–Monte Carlo (QMC) quadrature and fin ite element (FE)\nmethods was used to approximate the expectation with respect to the stochastic pa-\nrameters of the smallest eigenvalue. Throughout the error analys is the reciprocal of\nthe spectral gap occurred in: 1) the bounds on the derivatives of the eigenvalues with\nrespect to y(required for the QMC and dimension truncation error analysis); 2) the\nconstants for the FE error; and 3) the convergence rate for th e eigensolver (by Arnoldi\niteration). In short, the entire error analysis in [8] fails unless the g ap can be bounded\nfrom below uniformly in y.\nIn the remainder of this section we frame (1.1) as a variational eigen value problem,\nintroduce the function space setting and summarise some known pr operties of the\neigenvalues. Then, in Section 2 we prove that the spectral gap is un iformly bounded.\nFinally, in Section 3we perform a numericalexperiment fora specific e xample of (1.1),\nand present results on the size of the gap over different realisation s of the parameter\ngenerated by a QMC pointset.\n1.1 Variational eigenvalue problems\nIt is often useful to study the eigenvalue problem (1.1) in its equivale nt variational\nform. In this section we introduce the variational eigenvalue proble m, then present\nsome well known properties and tools that are required for our ana lysis.\nFirst, we clarify the assumptions on the coefficient aand our setting.\nAssumption A1.\n1. The coefficient is of the form\na(x,y) =a0(x)+∞/summationdisplay\nj=1yjaj(x), (1.2)\nwithaj∈L∞(D), for allj≥0.\n22. There exists 0< amin< amax<∞such that amin≤a(x,y)≤amax, for all\nx∈D,y∈U.\n3. For some p∈(0,1),\n∞/summationdisplay\nj=1/ba∇dblaj/ba∇dblp\nL∞<∞.\nThe last condition (Assumption 1.3) is the same as is required for the Q MC and\ndimension truncation analysis for corresponding source problems ( see [11]). Note that\nin that paper they also allow p= 1, but with an extra condition on the size of the\nsum.\nAlso, the assumption from the Introduction that each yjis uniformly distributed\nis not a restriction, the important point is that each yjbelongs to a bounded interval.\nLetV=H1\n0(D) equipped with the norm /ba∇dblv/ba∇dblV:=/ba∇dbl∇v/ba∇dblL2, and let V∗denote\nthe dual of V. We identify L2(D) with its dual, and denote the inner product on\nL2(D) by/an}b∇acketle{t·,·/an}b∇acket∇i}ht, which can be continuously extended to a duality pairing on V×V∗,\nalso denoted /an}b∇acketle{t·,·/an}b∇acket∇i}ht. Note that we have the following chain of compact embeddings\nV⊂⊂L2(D)⊂⊂V∗. The parameter domain Uis equipped with the topology and\nmetric of ℓ∞.\nNext, define the (parametric) symmetric bilinear form A:U×V×V→Rby\nA(y;w,v):=/integraldisplay\nDa(x,y)∇w(x)·∇v(x) dx,\nwhich is also an inner product on V.\nIn this setting, for each y∈U, the variational eigenvalue problem equivalent to\n(1.1) is: Find 0 /ne}ationslash=u(y)∈Vandλ(y)∈Rsuch that\nA(y;u(y),v) =λ(y)/an}b∇acketle{tu(y),v/an}b∇acket∇i}ht,for allv∈V , (1.3)\n/ba∇dblu(y)/ba∇dblL2= 1.\nItfollowsfromAssumptionA1thatthebilinearform A(y)iscoerciveandbounded,\nuniformly in y:\nA(y;v,v)≥amin/ba∇dblv/ba∇dbl2\nV, for allv∈V ,and (1.4)\nA(y;w,v)≤amax/ba∇dblw/ba∇dblV/ba∇dblv/ba∇dblV, for allw,v∈V . (1.5)\nAsaconsequence,foreach ywehaveaself-adjointandcoerciveeigenvalueproblem.\nTherefore, it is well known (see, e.g., [3, 9]) that (1.3) has a countab le sequence of\npositive, real eigenvalues, which (counting multiplicities) we write as\n0< λ1(y)≤λ2(y)≤ ···,\nand the corresponding eigenvectors are denoted by u1(y),u2(y),...∈V.\nThemin-max principle [3, (8.36)]\nλk(y) = min\nVk⊂V\ndim(Vk)=kmax\nv∈VkA(y;v,v)\n/an}b∇acketle{tv,v/an}b∇acket∇i}ht,\nallows us to bound each eigenvalue above and below independently of y. Indeed, by\n(1.4) and (1.5) we have\naminmin\nVk⊂V\ndim(Vk)=kmax\nv∈Vk/an}b∇acketle{t���v,∇v/an}b∇acket∇i}ht\n/an}b∇acketle{tv,v/an}b∇acket∇i}ht≤λk(y)≤amaxmin\nVk⊂V\ndim(Vk)=kmax\nv∈Vk/an}b∇acketle{t∇v,∇v/an}b∇acket∇i}ht\n/an}b∇acketle{tv,v/an}b∇acket∇i}ht.\nNow, using the min-max properties of the kth eigenvalue of the Dirichlet Laplacian\nonD, which we denote by χk, the bounds above simplify to\naminχk≤λk(y)≤amaxχk. (1.6)\n3To consider our problem in the framework of Kato [10] for perturb ations of linear\noperators, we introduce, for each y∈U, the solution operator T(y) :V∗→V, which\nforf∈V∗is defined by\nA(y;T(y)f,v) =/an}b∇acketle{tf,v/an}b∇acket∇i}htfor allv∈V . (1.7)\nClearly,µ= 1/λis an eigenvalue of T(y) if and only if λis an eigenvalue of (1.3),\nand their eigenspaces coincide. Alternatively, we can consider the o peratorT(y) :\nL2(D)→L2(D). In this case, T(y) is self-adjointwith respect to the L2inner product\ndue to the symmetry of A(y); it is compact because V⊂⊂L2(D); and finally, it is\nbounded due to the Lax-Milgramtheorem, which states that for ea chf∈L2(D) there\nis a unique T(y)f∈Vsatisfying (1.7), with\n/ba∇dblT(y)f/ba∇dblV≤1\namin/ba∇dblf/ba∇dblV∗≤1\namin√χ1/ba∇dblf/ba∇dblL2. (1.8)\nIn the last inequality, we have used the Poincar´ e inequality:\n/ba∇dblv/ba∇dblL2≤χ−1/2\n1/ba∇dblv/ba∇dblV,forv∈V , (1.9)\nand a standard duality argument. Note that we have expressed th e Poincar´ e constant\nin terms of the smallest eigenvalue of the Dirichlet Laplacian on D, using again the\nmin-max principle.\n2 Bounding the spectral gap\nThe Krein-Rutman theorem guarantees that for every ythe fundamental eigenvalue\nλ1(y) is simple, see, e.g., [9, Theorems 1.2.5 and 1.2.6]. However, it does not provide\nany quantitative statements about the size of the spectral gap, λ2(y)−λ1(y), for dif-\nferent parameter values y. As discussed in the Introduction, when studying numerical\nmethods for eigenvalue problems in a UQ setting (see, e.g., [8]) severa l areas of the\nerror analysis require uniform positivity of this gap over all y∈U. Here, we prove\nthe required uniform positivity under the conditions of Assumption A 1, in particular\nA1.3.\nAn explicit bound on the spectral gap can be obtained in slightly differe nt settings\nor by assuming tighter restrictions on the coefficients. For example , for Schr¨ odinger\noperators ( −∆+V) onDwith a weakly convex potential Vand Dirichlet boundary\nconditions, [2] gives an explicit lower bound on the fundamental gap. Alternatively,\nusing the upper and lower bounds on the eigenvalues (1.6), we can de termine restric-\ntions on aminandamaxsuch that the gap is bounded away from 0. Explicitly, if the\ncoefficient ais such that aminandamaxsatisfy\namin\namax>χ1\nχ2, (2.1)\nthen, by (1.6), λ2(y)−λ1(y)≥aminχ2−amaxχ1>0. However, the condition (2.1)\nmay prove to be too restrictive.\nThe general idea of our proof is to use the continuity of the eigenva lues to show\nthat a non-zero minimum of the gap exists. A complication that arises in this strategy\nis that the parameter domain Uis not compact, so we cannot immediately conclude\nthe existence of such a minumum; we know that Ucannot be compact in the topology\nofℓ∞because it is the unit ball of ℓ∞, and the unit ball of an infinite-dimensional\nBanachspace isnot compact. Our solutionis basedon the fact that althoughthere are\ninfinitely-many parameters, because of the decay of the terms in t he coefficient (see\nAssumption A1.3), the contribution of a parameter yjdecreases as jincreases. Specif-\nically, we reparametrise (1.3) as an equivalent eigenvalue problem who se parameters\ndo belong to a compact set.\nThe first step is the following elementary lemma, which shows that sub sets ofℓ∞\nthat are majorised by an ℓqsequence (for some 1 < q <∞) are compact.\n4Lemma 1. Letα∈ℓqfor some 1< q <∞. The set U(α)⊂ℓ∞given by\nU(α):=/braceleftbigg\nw∈ℓ∞:|wj| ≤1\n2|αj|/bracerightbigg\nis a compact subset of ℓ∞.\nProof.Sinceℓ∞is a normed (and hence a metric) space, U(α) is compact if and\nonly if it is sequentially compact. To show sequential compactness of U(α), take\nany sequence {y(n)}n≥1⊂U(α). Clearly, by definition of U(α), eachy(n)∈ℓqand\nmoreover,\n/ba∇dbly(n)/ba∇dblℓq≤1\n2/ba∇dblα/ba∇dblℓq<∞for alln∈N.\nSoy(n)is a bounded sequence in ℓq. Sinceq <∞,ℓqis a reflexive Banach space, and\nso by [4, Theorem 3.18] {y(n)}n≥1has a subsequence that converges weakly to a limit\ninℓq. We denote this limit by y∗, and, with a slight abuse of notation, we denote the\nconvergent subsequence again by {y(n)}n≥1.\nWe now prove that y∗∈U(α) and that the weak convergence is in fact strong, i.e.\nwe show y(n)→y∗inℓ∞, asn→ ∞. For any j∈N, consider the linear functional\nfj:ℓq→Rgiven by fj(w) =wj, wherewjdenotes the jth element of the sequence\nw= (wj)j≥1∈ℓq. Clearly, fj∈(ℓq)∗(the dual space)and usingthe weakconvergence\nestablished above, it follows that\ny(n)\nj=fj(y(n))→fj(y∗) =y���\njasn→ ∞,for each fixed j.\nThat is, we have componentwise convergence. Furthermore, sinc e|y(n)\nj| ≤1\n2|αj|it\nfollows that |y∗\nj| ≤1\n2|αj|for each j, and hence y∗∈U(α).\nNow, for any J∈Nwe can write\n/ba∇dbly(n)−y∗/ba∇dblq\nℓq=J/summationdisplay\nj=1|y(n)\nj−y∗\nj|q+∞/summationdisplay\nj=J+1|y(n)\nj−y∗\nj|q\n≤Jmax\nj=1,2,...,J|y(n)\nj−y∗\nj|q+∞/summationdisplay\nj=J+1|αj|q. (2.2)\nLetε >0. Since α∈ℓq, we can choose J∈Nsuch that\n∞/summationdisplay\nj=J+1|αj|q≤εq\n2,\nand since y(n)converges componentwise we can choose K∈Nsuch that\n|y(n)\nj−y∗\nj| ≤(2J)−1/qεfor allj= 1,2,...,Jandn≥K.\nThus, by (2.2) we have /ba∇dbly(n)−y∗/ba∇dblq\nℓq≤εqfor alln≥K, and hence /ba∇dbly(n)−y∗/ba∇dblℓq→0\nasn→ ∞. Because /ba∇dblw/ba∇dblℓ∞≤ /ba∇dblw/ba∇dblℓqwhenw∈ℓqand 1< q <∞, this also implies\nthaty(n)→y∗inℓ∞, completing the proof.\nA key property following from the perturbation theory of Kato [10] is that the\neigenvalues λk(y) are continuous in y, which for completeness is shown below in\nProposition 2. First, recall that T(y) is the solution operator as defined in (1.7), and\nlet Σ(T(y)) denote the spectrum of T(y).\nProposition 2. Let Assumption A1 hold. Then the eigenvalues λ1,λ2,...are Lips-\nchitz continuous in y.\n5Proof.We prove the result by establishing the continuity of the eigenvalues µk(y) of\nT(y). Lety,y′∈Uand consider the operators T(y),T(y′) :L2(D)→L2(D) as\ndefined in (1.7). Since T(y),T(y′) are bounded and self-adjoint with respect to /an}b∇acketle{t·,·/an}b∇acket∇i}ht,\nit follows from [10, V, §4.3 and Theorem 4.10] that we have the following notion of\ncontinuity of µ(·) in terms of T(·)\nsup\nµ∈Σ(T(y))dist(µ,Σ(T(y′)))≤ /ba∇dblT(y)−T(y′)/ba∇dblL2→L2. (2.3)\nForaneigenvalue µk(y)∈Σ(T(y)), (2.3)impliesthatthereexistsa µk′(y′)∈Σ(T(y′))\nsuch that\n|µk(y)−µk′(y′)| ≤ /ba∇dblT(y)−T(y′)/ba∇dblL2→L2. (2.4)\nNote that this means there exists an eigenvalue of T(y′) close to µk(y), but does\nnot imply that the kth eigenvalue of T(y′) is close to µk(y), that is, in (2.4) k\nis not necessarily equal to k′. However, consider any µk(y) and let mdenote its\nmultiplicity. Since m <∞, we can assume without loss of generality that the\ncollection µk(y) =µk+1(y) =···=µk+m−1(y) is afinite system of eigenvalues\nin the sense of Kato. It then follows from the discussion in [10, IV, §3.5] that\nthe eigenvalues in this system depend continuously on the operator with multiplic-\nity preserved. This preservation of multiplicity is key to our argumen t, since it\nstates that for T(y′) sufficiently close to T(y) there are mconsecutive eigenvalues\nµk′(y′),µk′+1(y′),...,µ k′+m−1(y′)∈Σ(T(y′)), no longer necessarily equal, that are\nclose toµk(y).\nA simple argument then shows that each µkis continuous in the following sense\n|µk(y)−µk(y′)| ≤ /ba∇dblT(y)−T(y′)/ba∇dblL2→L2. (2.5)\nTo see this, consider, for k= 1,2,..., the graphs of µkonU. Note that the separate\ngraphs can touch (and in principle can even coincide over some subse t ofU), but\nby definition cannot cross (since at every point in Uthe successive eigenvalues are\nnonincreasing); and by the preservation of multiplicity a graph cann ot terminate and\na finite set of graphs cannot change multiplicity at an interior point. T hus by (2.23)\nthe ordered eigenvalues µkmust be continuous for each k≥1 and satisfy (2.5).\nIt then follows from the relationship µk(y) = 1/λk(y) along with the upper bound\nin (1.6) that we have a similar result for the eigenvalues λkof (1.3):\n|λk(y)−λk(y′)| ≤(amaxχk)2/ba∇dblT(y)−T(y′)/ba∇dblL2→L2. (2.6)\nAll that remains is to bound the right hand side of (2.6) by CLip/ba∇dbly−y′/ba∇dblℓ∞, with\nCLip>0 independent of yandy′. To this end, note that since the right hand side of\n(1.7) is independent of ywe have\nA(y;T(y)f,v) =A(y′;T(y′)f,v) for all f∈L2(D),v∈V .\nRearranging and then expanding this gives\nA(y;(T(y)−T(y′))f,v)) =A(y′;T(y′)f,v)−A(y;T(y′)f,v)\n=/integraldisplay\nD[a(x,y′)−a(x,y)]∇[T(y′)f](x)·∇v(x) dx.\nLettingv= (T(y)−T(y′))f∈V, the left hand side can be bounded from below using\nthe coercivity (1.4) of A(y), and the right hand side can be bounded from above using\nthe Cauchy-Schwarz inequality to give\namin/ba∇dbl(T(y)−T(y′))f/ba∇dbl2\nV≤ /ba∇dbla(y)−a(y′)/ba∇dblL∞/ba∇dblT(y′)f/ba∇dblV/ba∇dbl(T(y)−T(y′))f/ba∇dblV.\nDividing by amin/ba∇dbl(T(y)−T(y′))f/ba∇dblVand using the upper bound in (1.8) we have\n/ba∇dbl(T(y)−T(y′))f/ba∇dblV≤1\na2\nmin√χ1/ba∇dblf/ba∇dblL2/ba∇dbla(y)−a(y′)/ba∇dblL∞.\n6Then, applying the Poincar´ e inequality (1.9) to the left hand side and taking the\nsupremum over f∈L2(D) with/ba∇dblf/ba∇dblL2≤1, in the operator norm we have\n/ba∇dblT(y)−T(y′)/ba∇dblL2→L2≤1\na2\nminχ1/ba∇dbla(y)���a(y′)/ba∇dblL∞.\nUsingthis inequalityasanupperbound for(2.6)weseethatthe eigen valuesinherit\nthe continuity of the coefficient, and so\n|λk(y)−λk(y′)| ≤a2\nmaxχ2\nk\na2\nminχ1/ba∇dbla(y)−a(y′)/ba∇dblL∞. (2.7)\nwhere the constant is clearly independent of yandy′.\nFinally, to establish Lipschitz continuity with respect to y, we recall Assump-\ntions A1.1 and A1.3, expand the coefficients in (2.7) above and use the triangle in-\nequality to give\n|λk(y)−λk(y′)| ≤a2\nmaxχ2\nk\na2\nminχ1∞/summationdisplay\nj=1|yj−y′\nj|/ba∇dblaj/ba∇dblL∞≤a2\nmaxχ2\nk\na2\nminχ1/parenleftBigg∞/summationdisplay\nj=1/ba∇dblaj/ba∇dblL∞/parenrightBigg\n/ba∇dbly−y′/ba∇dblℓ∞.\nBy Assumption 1 the sum is finite, and hence the eigenvalue λk(y) is Lipschitz in y,\nwith the constant clearly independent of y.\nNow that we have shown Lipschitz continuity of the eigenvalues and id entified\nsuitable compact subsets, we can prove the main result of this pape r: namely, that\nthe spectral gap is bounded away from 0 uniformly in y. The strategy of the proof is\nto rewrite the coefficient as\na(x,y) =a0(x) +∞/summationdisplay\nj=1/tildewideyj/tildewideaj(x),\nwith/tildewideyj=αjyjand/tildewideaj(x) =aj(x)/αj, choosing α∈ℓqto decay slowly enough\nsuch that/summationtext∞\nj=1/ba∇dbl/tildewideaj/ba∇dblL∞<∞continues to hold. Then, using the intermediate re-\nsult (2.7) from the proof of Proposition 2 we can show that the eigen values of the\n“reparametrised” problem are continuous in the new parameter /tildewidey, which now ranges\nover the compact set U(α). The required bound on the spectral gap is obtained by\nusing the equivalence of the eigenvalues of the original and reparam etrised problems.\nTheorem 3. Let Assumption A1 hold. Then there exists a δ >0, independent of y,\nsuch that\nλ2(y)−λ1(y)≥δ. (2.8)\nProof.We can assume, without loss of generality, that p >1/2, because if Assump-\ntion A1.3 holds with exponent p′≤1/2 then it also holds for all p∈(p′,1). Conse-\nquently, set ε= 1−p∈(0,1/2) and consider the sequence αdefined by\nαj=/ba∇dblaj/ba∇dblε\nL∞+1/j,for each j∈N. (2.9)\nSettingq=p/ε=p/(1−p)∈(1,∞), using Assumption A1.3 and the triangle\ninequality, it is easy to see that α∈ℓq. Moreover, the inclusion of 1 /jin (2.9) ensures\nthatαj/ne}ationslash= 0, for all j≥1. Hence, from now on, for w= (wj)∞\nj=1∈ℓ∞, we can\ndefine the sequences αw= (αjwj)∞\nj=1andw/α= (wj/αj)∞\nj=1. Then, recalling the\ndefinition of U(α) in Lemma 1, it is easy to see that /tildewidey∈U(α) if and only if /tildewidey/α∈U,\nand moreover, y∈Uif and only if αy∈U(α).\nNow for x∈Dand/tildewidey∈U(α), we define\n/tildewidea(x,/tildewidey) =a0(x)+∞/summationdisplay\nj=1/tildewideyjaj(x)\nαj,\n7from which it is easily seen that\n/tildewidea(x,/tildewidey) =a(x,/tildewidey/α). (2.10)\nThen we set\n/tildewideA(/tildewidey;w,v):=/integraldisplay\nD/tildewidea(x,/tildewidey)∇w(x).∇v(x) dxforw,v∈V ,\nand we consider the following reparametrised eigenvalue problem: Fin d/tildewideλ(/tildewidey)∈Rand\n0/ne}ationslash=/tildewideu(/tildewidey)∈Vsuch that\n/tildewideA(/tildewidey;/tildewideu(/tildewidey),v) =/tildewideλk(/tildewidey)/an}b∇acketle{t/tildewideu(/tildewidey),v/an}b∇acket∇i}htfor allv∈V ,\n/ba∇dbl/tildewideu(/tildewidey)/ba∇dblL2= 1. (2.11)\nNote that because we have equality between the original and repar ametrised coeffi-\ncients (2.10), for each y∈U, and corresponding /tildewidey=αy∈U(α), (2.10) implies that\nthere is equality between eigenvalues λk(y) of (1.3) and /tildewideλk(/tildewidey) of the reparametrised\neigenvalue problem (2.11)\nλk(y) =/tildewideλk(/tildewidey) fork∈N, (2.12)\nand their eigenspaces coincide.\nMoreover, for an eigenvalue /tildewideλk(/tildewidey) of (2.11), using (2.12) in the inequality (2.7) we\nhave\n|/tildewideλk(/tildewidey)−/tildewideλk(/tildewidey′)| ≤a2\nmaxχ2\nk\na2\nminχ1/ba∇dbla(/tildewidey/α)−a(/tildewidey/α)/ba∇dblL∞,\nwhich after expanding the coefficient and using the triangle inequality becomes\n|/tildewideλk(/tildewidey)−/tildewideλk(/tildewidey′)| ≤/parenleftBigg\na2\nmaxχ2\nk\na2\nminχ1∞/summationdisplay\nj=11\nαj/ba∇dblaj/ba∇dblL∞\n/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright\n/tildewideCLip/parenrightBigg\n/ba∇dbl/tildewidey−/tildewidey′/ba∇dblℓ∞,\nwhere/tildewideCLipis clearly independent of /tildewideyand/tildewidey′. Now by (2.9) together with Assump-\ntion A1, we have\n∞/summationdisplay\nj=1/ba∇dblaj/ba∇dblL∞\nαj≤∞/summationdisplay\nj=1/ba∇dblaj/ba∇dbl1−ε\nL∞=∞/summationdisplay\nj=1/ba∇dblaj/ba∇dblp\nL∞<∞.\nThus,/tildewideCLip<∞and hence the reparametrised eigenvalues are continuous on U(α).\nIt immediately follows that the spectral gap /tildewideλ2(/tildewidey)−/tildewideλ1(/tildewidey) is also continuous on\nU(α), whichbyLemma1isacompactsubset of ℓ∞. Therefore, thenon-zerominimum\nis attained giving that the spectral gap /tildewideλ2(/tildewidey)−/tildewideλ1(/tildewidey) is uniformly positive. Finally,\nbecause there is equality between the original and reparametrised eigenvalues (2.12)\nthe result holds for the original problem over all y∈U.\nThe paper [8] proves a similar result to Theorem 3 in a more general se tting, the\nsame strategy is used there also.\n3 Numerical results\nFor our numerical results, we consider the 1-dimensional domain D= (0,1), and let\nthe basis functions that define the coefficient in (1.2) be given by\naj(x) =\n\na0 forj= 0,\nc0\nj2sin(jπx),forj= 1,2,...,s,\n0 for j > s,(3.1)\n8wherea0>0 andc0∈Rwill determine the values of aminandamax. Note that the\nstochastic dimension is here fixed at s= 100. Since multiplying the coefficient by any\nconstant factor will simply rescale the eigenvalues by that same fac tor, without loss\nof generality we can henceforth set a0= 1 and vary c0.\nFor a coefficient (1.2) given by the basis functions (3.1), we can obta in a formula\nfor the bounds amin,amaxas follows\namin= 1−c0\n2s/summationdisplay\nj=11\nj2= 1−0.81c0,\namax= 1+c0\n2s/summationdisplay\nj=11\nj2= 1+0.81c0.\nWe remark that these bounds are not sharp.\nWe also consider the so-called log-normal coefficient:\na(x,y) =a∗+exp/parenleftBiggs/summationdisplay\nj=1Φ−1(yj+1\n2)aj(x)/parenrightBigg\n, (3.2)\nwherea∗≥0,ajare as in (3.1), and Φ−1is the inverse of the normal cumulative\ndistribution function. In this way each Φ−1(yj+1\n2)∼ N(0,1). Since Φ−1maps [0,1]\ntoR, the coefficient (3.2) is unbounded, and although it is positive ( amin=a∗), for\na∗= 0 it could be arbitrarily close to 0. So in this case Assumption 1.2 does n ot hold,\nour compactness argument fails and we have no theoretical predic tion.\nTo approximate the eigenvalue problem in space we use piecewise linear finite\nelements on a uniform mesh, with a meshwidth of h= 1/64. Numerical tests for\ndifferent meshwidths ( h= 1/8 toh= 1/128) produced qualitatively the same results.\nThe purpose of this section is to provide supporting evidence that f or the problem\nabove the gap remains bounded. We do this in a brute force manner b y studying the\nminimum of the gap over a large number of parameter realisations. Th e parameter\nrealisations are generated by a QMC point set; specifically, a base-2 embedded lattice\nrule(see[5])with up to220points andasinglerandomshift. Togeneratethe pointswe\nuse the generating vector exod2base2m20CKNfrom [12]. We choose QMC points\nas the test set because they can be shown to be well-distributed in h igh dimensions,\nsee e.g. [5, 6]. The goal is not to find the minimum, but to provide eviden ce that\nas more and more of the parameter domain is searched (which corre sponds to more\nrealisations),the minimum ofthe gap overallrealisationsapproache saconstant value.\nThe results are given in Figures 1–5, where in each figure we plot the m inimum\nof the gap against the number of realisations N(blue circles) and an estimate (see\nthe next paragraph) of the distance between the estimated minimu m and the true\nminimum (black triangles), along with a least-squares fit to αN−β(dashed red line)\nof this estimate. Each data point corresponds to a doubling of the n umber of QMC\npoints:N= 1,2,4,...,220, and the axes are in loglog scale.\nLettingδNdenote the approximate minimum over the first Nrealisations, we\nestimate the distance to the true minimum by\nδN−min\ny∈U/parenleftbig\nλ2(y)−λ1(y)/parenrightbig\n≈δN−δN∗,\nwhereδN∗corresponds to the most accurate estimate of the minimum, with N∗= 220.\nThe purpose of including such an estimate of the distance to the tru e minimum is to\ndemonstrate that not only does the minimum of the gap appear to pla teau, but that\nthe differences also decay like a power of N.\nFirst we considerthe affine coefficient in (1.2). The three different ch oices ofc0are:\nin Figure 1 c0= 1, which gives amin= 0.18 andamax= 1.82; in Figure 2 c0= 1.223,\nwhich gives amin= 2×10−4andamax= 2; and in Figure 3 c0= 0.5, which gives\namin= 0.59 andamax= 1.41. Then, in Figure 4 we plot the log-normal coefficient\n(3.2) with amin=a∗= 0 and c0= 1, and in Figure 5 we plot the lognormal coefficient\nwithamin=a∗= 0.18 andc0= 1.\n9Foreachdifferentchoiceoftheaffinecoefficient(1.2)(Figures1,2, 3),theminimum\nvalue of the gap appears to plateau and approach a nonzero minimum . The minimum\nof the gap seems fairly insensitive to changes in c0. However, in Figures 4 and 5 for\nthe log-normal coefficient (which, we recall is not covered by the th eory of the current\nwork) the results are inconclusive. It appears that the gap tends to 0 in the case of\na true lognormal coefficient ( a∗= 0, Figure 4), and that it is bounded away from 0\nfor the “regularised” lognormal coefficient with a∗= 0.18 in Figure 5. However, in\nFigure 4 the smallest computed value of the gap is close to 1, and it is po ssible that\nit will plateau for a denser set of QMC points. Also, in Figure 5 the plate au is not\nas clearly developed in as in Figures 1–3. It remains an open question if the gap can\nbe bounded in the case of the lognormal coefficient (3.2), and wheth era∗needs to be\nstrictly positive.\n100102104106\nNumber of realisations (N)10-210-1100101102\nFigure 1: Estimate of the minimum of the spectral gap and esti mate of the distance to\nthe true minimum: affine coefficient (1.2) with a0=c0= 1,amin= 0.18,amax= 1.82.\n100102104106\nNumber of realisations (N)10-1100101102\nFigure 2: Estimate of the minimum of the spectral gap and esti mate of the distance to the\ntrue minimum: affine coefficient (1.2) with a0= 1,c0= 1.223,amin= 2×10−4,amax= 2.\n10100102104106\nNumber of realisations (N)10-210-1100101102\nFigure 3: Estimate of the minimum of the spectral gap and esti mate of the distance to\nthe true minimum: affine coefficient (1.2) with a0= 1,c0= 0.5,amin= 0.59,amax= 1.41.\n100102104106\nNumber of realisations (N)10-1100101102103\nFigure 4: Estimate of the minimum of the spectral gap and esti mate of the distance to\nthe true minimum: log-normal coefficient (3.2) with amin=a∗= 0 and c0= 1.\n100102104106\nNumber of realisations (N)10-1100101102\nFigure 5: Estimate of the minimum of the spectral gap and esti mate of the distance to\nthe true minimum: log-normal coefficient (3.2) with amin=a∗= 0.18 andc0= 1.\n114 Conclusion\nThe spectral gap is an important quantity that occurs throughou t several areas of the\nnumerical analysis of eigenvalue problems, and in this work we proved that, under\ncertain conditions on the coefficient, the spectral gap of a random elliptic eigenvalue\nproblem is uniformly bounded from below. In all of our numerical expe riments the\nresults strongly suggest that the minimum of the gap approaches a nonzero constant\nvalue. The only exception is the lognormal coefficient, which is not cov ered by our\ntheory.\nReferences\n[1] R. Andreev and Ch. Schwab. Sparse tensor approximation of pa rametric eigen-\nvalue problems. In I. G. Graham et al, editor, Numerical Analysis of Multiscale\nProblems, Lecture Notes in Computational Science and Engin eering, 203–241.\nSpringer-Verlag, Berlin Heidelberg, Germany, 2012.\n[2] B. Andrews and J. Clutterbuck. Proof of the fundamental gap conjecture. J.\nAmer. Math. Soc. 24, 899–916, 2011.\n[3] I. Babuˇ ska and J. Osborn. Eigenvalue problems. In P. G. Ciarlet and J. L. Lions,\neditors,Handbook of Numerical Analysis, Volume 2: Finite Element Me thods\n(Part 1), 641–787. Elsevier Science, Amsterdam, The Netherlands, 1991.\n[4] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differenti al Equations .\nUniversitext, Springer, New York, 2011.\n[5] R. Cools, F. Y. Kuo and D. Nuyens, Constructing embedded lattic e rules for\nmultivariate integration. SIAM J. Sci. Comp. ,28, 2162–2188, 2006.\n[6] J. Dick, F. Y. Kuo, and I. H. Sloan. High-dimensional integration: The quasi-\nMonte Carlo way. Acta Numerica ,22, 133–288, 2013.\n[7] N. Dunford and J. T. Schwartz, Linear Operators, Part II: Spectral Theory, Self\nAdjoint Operators in Hilbert Space . Interscience Publishers, New York London,\n1963.\n[8] A. D. Gilbert, I. G. Graham, F. Y. Kuo, R. Scheichl and I. H. Sloan , Analysis of\nquasi-Monte Carlo Methods for elliptic eigenvalue problems with stoch astic coef-\nficients.Submitted, arXiv: https://arxiv.org/abs/1808.02639 , 2018.\n[9] A. Henrot. Extremum Problems for Eigenvalues of Elliptic Operators . Birkh¨ auser\nVerlag, Basel, Switzerland, 2006.\n[10] T. Kato. Perturbation Theory for Linear Operators . Springer-Verlag, Berlin\nHeidelberg, Germany, 1984.\n[11] F. Y. Kuo, Ch. Schwab, and I. H. Sloan. Quasi-Monte Carlo finite element meth-\nods for a class of elliptic partial differential equations with random co efficients.\nSIAM J. Numer. Anal. ,50, 3351–3374, 2012.\n[12] D.Nuyens, https://people.cs.kuleuven.be/ ˜dirk.nuyens/qmc-generators/ ,\naccessed November 2, 2018.\n12"    }]
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